AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Andhra Pradesh BIEAP AP Inter 2nd Year Chemistry Study Material 9th Lesson Biomolecules Textbook Questions and Answers.

AP Inter 2nd Year Chemistry Study Material 9th Lesson Biomolecules

Very Short Answer Questions

Question 1.
Define Carbohydrates.
Answer:
The compounds which are primarily produced by plants and form a very large group of naturally occuring organic compounds are called Carbohydrates.
Eg: Glucose, Fructose, Starch.
-H> Carbohydrates are the polyhydroxy aldehydes (or) ketones.

Question 2.
Name the different types of carbohydrates on the basis of their hydrolysis. Give one example for each.
Answer:
On the basis of the hydrolysis, carbohydrates are classified as

  1. Monosaccharides, Eg : Glucose, fructose AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 1 No saccharides
  2. Oligosaccharides, Eg : Sucrose, maltose AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 1 two monosaccharides
  3. Polysaccharides, Eg : Starch, cellulose AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 1 large number of monosaccharides.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 3.
Why are sugars classified as reducing and non-reducing sugars ?
Answer:
Carbohydrates that reduce Fehling’s reagent, Tollen’s reagent are called reducing sugars.
Eg: Glucose.
Carbohydrates that doesnot reduce Fehling’s reagent, Tollen’s reagent are called non reducing sugars.
Eg : Sucrose.

Question 4.
What do you understand from the names
(a) aldo pentose and
(b) ketoheptose ?
Answer:
a) Aldo pentose : If a monosaccharide contains 5 carbon atoms with aldehyde group then it is known as aldo pentose.
b) Ketoheptose : If a monosaccharide contain seven carbons with a ketone group then it is called ketoheptose.

Question 5.
Write two methods of preparation of glucose.
Answer:
Methods of preparations of glucose :
1) From Sucrose.: Sucrose when boiled with di/.HC/ in alcoholic solution then glucose, fructose are obtained in equal portions.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 2
2) From Starch : Starch undergo hydrolysis, with dil. H2S04 at 393K under pressure to form glucose.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 3

Question 6.
Glucose reacts with bromine water to give gluconic acid. What information do you get from this reaction about the structure of glucose ?
Answer:
Glucose reacts with bromine water to give gluconic acid. This reaction indicates that the carbonyl group present ip,glucose is an aldehyde group.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 4

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 7.
Glucose and gluconic acid on oxidation with nitric acid give saccharic acid. What Information do you get from this reaction about the structure of glucose ?
Answer:
Glucose and gluconic acid on oxidation with nitric acid give saccharic acid. This indicates the presence of primary alcoholic (-OH) group in glucose.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 5

Question 8.
Glucose reacts with acetic anhydride to form penta acetate. What do you understand about the structure of glucose from this reaction ?
Answer:
Glucose reacts with acetic anhydride to form peta acetate. It confirms the presence of five- OH groups in glucose which are attached to different carbon atoms.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 6

Question 9.
Give any two reasons to understand that glucose molecule has no open chain structure.
Answer:
Open chain structure of glucose could not explain

  • why glucose doesnot respond to Schiffs test,
  • why it does not react with NaHSO3 and NH3,
  • why there is mutarotation.

Question 10.
D – glucose means dextro rotatory glucose. Is it true ? Why ?
Answer:
In glucose the carbon atom attached to the lowest CH2-OH group is found to contain H atom on left side. So it is correlated to glyceraldehyde molecule and gives D-symbol. D-Glucose means not dextro rotatory glucose. D-symbol has nothing to do with optical activity of compound.

Question 11.
What are anomers ?
Answer:
Anomers : The two isomeric structures of a compound which differ in configuration at C-l only are called Anomers.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 12.
Write the ring structure of D-glucose. What are their names?
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 7

Question 13.
Write the ring structure and open chain structure of fructose.
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 8

Question 14.
What do you understand by invert sugars?
Answer:
During the hydrolysis of sucrose there is a change in the sign of rotation, from dextro (+) to laevo (-) and the product is named as invert sugar.

Question 15.
What are amino acids? Give two examples.
Answer:
The organic compounds which contain amino (-NH2) functional group and carboxyl (-COOH) functional group are called amino acids.
Eg: Glycine, Alanine etc.

Question 16.
Write the structure of alanine and aspartic acid.
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 9

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 17.
What do you mean by essential amino acids ? Give two examples for non essential amino acids? [T.S. Mar. 16]
Answer:
Essential amino acids : The amino acids which cannot be synthesized in the body and must be obtained through diet are known as essential aminoacids.
Eg : valine,. leucine etc.
Examples of non essential amino acids are Alanine, glycine, Aspartic acid.

Question 18.
What is zwitter ion ? Give an example.
Answer:
Zwitter ion : In aqueous solution of amino acids, the carboxyl group can lose a proton and amino acid can accept that proton to form a dipolar ion. This ion is called as zwitter ion.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 10

Question 19.
What are proteins ? Give an example.
Answer:
Proteins : A poly peptide with more than hundred amino acid residues, having molecular mass higher than 10,000 units is called a protein.
Eg : keratin, myosin, insulin.

Question 20.
What are fibrous proteins ? Give examples.
Answer:
When the poly peptide chains run parallel and are held together by hydrogen and disulphide bonds then fibre – like structure is formed. These are called fibrous proteins. These are insoluble in water.
Eg : keratin, myosin.

Question 21.
What are globular proteins ? Give examples.
Answer:
When the chains of polypeptides coil around to give a spherical shape then globular proteins are formed. These are usually soluble in water.
Eg : insulin and albumins.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 22.
How are proteins classified with respect to peptide bond ?
Answer:
With respect to peptide bond proteins classified into two types.

  1. Fibrous proteins
  2. Globular proteins.

Question 23.
What are the components of a nucleic acid ?
Answer:

  • Nucleic acids are long chain polymers of nucleotides i.e poly nucleotides,
  • Nucleic acids are constituted by pentose sugar, phosphoric acid, and nitrogenous hetero cyclic base (purine (or) pyrimidine).

Question 24.
Write the names of three types of RNA.
Answer:
Three types of RNA are :

  1. Messenger RNA (m – RNA)
  2. Ribosomal RNA (r – RNA)
  3. Transfer RNA (t – RNA).

Question 25.
Write the biological functions of nucleic acids.
Answer:
Biological functions of Nucleic acids :

  • DNA is the chemical basis of heredity.
  • DNA is the reserve of genetic information.
  • DNA is capable of self duplication during cell division and identical DNA strands are transfered to daughter cells.
  • RNA molecules are used in the protein synthesis in the cell
  • The message for the synthesis of a protien is present in DNA.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 26.
Name the vitamin responsible for the coagulation of blood.
Answer:
The vitamin responsible for the coagulation of blood is vitamin K.

Question 27.
What are monosaccharides ?
Answer:
A carbohydrate which donot form any saccharide unit on hydrolysis (or) which cannot be hydrolysed is called monosaccharide.
Eg : glucose, fructose.

Question 28.
What are reducing sugars ?
Answer:
Carbohydrates that reduce Fehling’s reagent, Tollen’s reagent are called reducing sugars.
Eg : glucose.

Question 29.
Write two main functions of carbohydrates in plants.
Answer:
Carbohydrates are essential for life of plants.

  • Carbohydrates are used as storage molecules as starch in plants.
  • Cell wall of plants is made up of cellulose.

Question 30.
Classify the following into monosaccharides and disaccharides,

  1. ribose
  2. 2-deoxy ribose
  3. maltose
  4. fructose.

Answer:

  1. Ribose – Monosaccharide
  2. 2 – deoxy ribose – Monosaccharide
  3. Maltose – Disacch&ride .
  4. Fructose – Monosaccharide.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 31.
What do you understand by the term glycosidic linkage ?
Answer:
The linkage between two monosacharides of disacharides is called glycosidic linkage.

Question 32.
What is glycogen ? How is it different from starch ?
Answer:
The carbohydrate which is stored in animal body is called glycogen. It is also known as animal starch. Starch is a polysaccharide which is stored in plants where as glycogen is stored in animals (liver, muscles and brain).

  • Glycogen mainly resembles amylopectin of starch in structure but glycogen is more branched than amylopectin.

Question 33.
What are the hydrolysis products of
i) sucrose and
ii) lactose ?
Answer:
i) The hydrolysis products of sucrose are glucose and fructose.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 11
ii) The hydrolysis products of lactose are galactose and glucose.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 12

Question 34.
What is the basic structural difference between starch and cellulose ?
Answer:
In cellulose a straight chain polysaccharide composed only of β – D – glucose units which are joined by glycosidic linkage between C – 1 of one glucose unit and C – 4 of the next glucose unit.

Starch consists of two components – Amylose and Amylopectin. Amylose is a long un branched chain with 200 – 1000 α – D – glucose units held by C – 1 to C – 4 glycosidic linkage. Amylopectin is a branched chain polymer of α – D – glucose units in which Chain is formed by C – 1 to C – 4 glycosidic linkage where as branching occurs by C – 1 to C – 6 glycosidic linkage.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 35.
What happens when D – glucose is treated with the following reagents :
i) HI
ii) Bromine water
iii) HNOs.
Answer:
i) D – glucose on prolonged heating with HI, glucose forms n – hexane.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 13

Question 36.
Enumerate the reactions of D – glucose which cannot be explained by its open chain structure.
Answer:
The following reactions cannot be explained by the open chain structure of D – glucose.

  • Glucose has aldehyde group, but it does not respond to the reagents like Schiff s reagent, NaHSO3, NH3 etc., .
  • The penta acetyl derivative of glucose doesnot react with hydroxyl amine.
  • α and β – methyl glucosides cannot be explained by its open chain structure.

Question 37.
What are essential and non – essential amino acids ? Give one example for each.
Answer:
Essential amino acids : The amino acids that cannot be synthesised by the body and must be supplied in the diet are called essential amino acids.
Eg : valine, leucine, phenyl alanine etc.,

Non-essential amino acids : Other amino acids synthesised by the tissues of the body are called non-essential amino acids.
Eg : Glycine, alanine etc.,

Question 38.
Define the following as related to proteins :
i) Peptide linkage
ii) Primary structure
iii) Denaturation
Answer:
i) Peptide linkage :
Amide AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 14 linkages between amino acids are known as peptide linkages. The product obtained from two amino acid molecules through peptide bond is called a dipeptide. Peptide chain may be extended to many amino acids.

Proteins are naturally occuring polypeptides that contain more than 50 amino acid units.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 15

ii) Primary structure : The primary structure of protein is defined as the linear sequence of amino acid residues making up its polypeptide chain. Proteins may be formed of one or more polypeptide chains. The amino acid residues are linked by peptide bonds. The peptide bond is formed between carboxyl group of one amino acid and amino group of adjacent amino acid.
Eg : fibroin of silk.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 16

iii) Protein Denaturation : The phenomenon of disorganization of native protein structure is known as denaturation. Denaturation results in the loss of secondary, tertiary nad quaternary structure of proteins. This involves a change in physical, chemical and biological properties of protein molecules.
Agents of denaturation:
Physical agents – Heat, violent shaking, X – rays, UV – radiation
Chemical agents – Acids, alkalies, organic solvents, urea, salts of heavy metals.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 39.
What are the common types of secondary structure of proteins ?
Answer:
Secondary structure: The secondary structure explains the shape of the polypeptide chain. It is derived from the primary structure by the formation of hydrogen bond interactions between amino acid residues.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 17
Disulfide bonds also occur either in same chain or between polypeptide chains. The secondary structure is classified into two types namely helical structure and pleated sheets, e.g.: Keratin of hair.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 18

Question 40.
What type of bonding helps in stabilizing the α – helix structure of proteins ?
Answer:
The α – helix structure of proteins is stabilized by intramolecular hydrogen bonding between AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 19 = O and AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 20 – H groups of dlifferent peptide bonds.

Question 41.
Differentiate between globular and fibrous proteins.
Answer:
Globular proteins

  • In this proteins the chains of poly peptides coil around to give a spherical shape
  • These are soluble in water
  • Eg: Insulin

Fibrous proteins

  • In this proteins the poly peptides run parallel and are held together by hydrogens and disulphite – bondy
  • These are insoluble in water
  • Eg: keratin

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 42.
How do you explain the amphoteric behaviour of amino acids ?
Answer:
Amino acids are the compounds having amino group (-NH2) i.e., a basic group and carboxyl group (-COOH) i.e., an acidic group

Zwitter ion,: In aqueous solution of amino acids, the carboxyl group can lose a proton and amino acid can accept that proton to form a dipolar ion. This is called as zwitter ion.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 21
In zwitter ion form amino acids behave both acidic as well as basic nature. Therefore amino acids are amphoteric in nature.

Question 43.
Why are vitamin A and vitamin C essential to us ? Give their important sources.
Answer:
vitamin A and vitamin C are essential to us.
Explanation :

  • Deficiency of vitamin A causes night blindness, red ness in eyes, xerophthalhia
  • Deficiency of vitamin C causes pernicious anaemia (RBC deficient in haemoglobin)

Sources :

  • Vitamin – A : Fish liver oil, carrots, butter and milk.
  • Vitamin – C : Citrous fruits, amla, green leafy vegetables.

Question 44.
What are nucleic acids ? Mention their two important functions.
Answer:

  • Nucleic acids are long chain polymers of nucleotides i.e poly nucleotides.
  • Nucleic acids are constituted by pentose sugar, phosphoric acid, and nitrogenous hetero cyclic base (purine (or) pyrimidine).

Biological functions of Nucleic acids :

  • DNA is the chemical basis of heredity.
  • DNA is the reserve of genetic information.
  • DNA is capable of self duplication during cell division and identical DNA strands are transfered to daughter cells.
  • RNA molecules are used in the protein synthesis in the,cell.
  • The message for the synthesis of a protien is present in DNA.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 45.
What is the difference between a nucleoside and a nucleotide ?
Answer:
Nucleosides: Nucleic acid bases combine with pentose sugars to form N – glycosides known as nucleosides.
Eg: Cytidine
Structure of Cytidine
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 22
Nucleotides : Nucleotide is a phosphate ester of nucleoside and consists of a purine or pyrimidine base. Nucleotide contains a base moiety, a sugar moiety (i.e.,) a nucleoside moiety and phosphate group.
Nucleotide = base (purine / pyrimidine) + sugar (ribose/deoxyribose) + phosphate
Ex: Adenosine Triphosphate (ATP)
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 23

Short Answer Questions

Question 1.
How are the carbohydrates classified on the basis of their
a) Taste
b) Hydrolysis
c) Functional groups
Answer:
a) Taste : On the basis of their taste carbohydrates are classified into
i) Sugars, ii) Non sugars.

  1. Sugars : Carbohydrates which are sweet in taste.
    Eg : Sucrose.
  2. Non sugars : Carbohydrates which are not sweet in taste.
    Eg : Cellulose.

b) Hydrolysis : On the basis of the hydrolysis, carbohydrates are classified as

  1. Monosaccharides, Eg: Glucose, fructose AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 1 No saccharides
  2. Oligosaccharides. Eg: Sucrose, maltose AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 1 two monosaccharides
  3. Polysaccharides, Eg: Starch, cellulose AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 1 large number of monosacchandes.

c) Functional groups: Carbohydrates classified into two types on the basis of functional group present.

  1. Aldoses: Carbohydrates having aldehyde group.
    Eg: glucose.
  2. Ketoses: Carbohydrates having ketone group.
    Eg: fructose.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 2.
Write a brief note on the structure of glucose.
Answer:
1. The molecular formula of glucose is found to be C6H12O6. It can be written as CHO – (CHOH)4 – CH6OH’.

2. Acylation of glucose with acetic anhydride gives glucose pentacetate. Hence glucose molecule has five -OH groups.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 24

3. Glucose reacts with hydroxylamine and hydrogen cyanide forming corresponding oxime and cyano hydrin respectively. Hence glucose molecule has carbonyl group.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 25

4. Glucose on oxidation with Tollens reagent and Fehling’s solution gives gluconic acid. Hence carbonyl group present in glucose is aldehydic group.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 26

5. Glucose on oxidation with nitric acid gives glucaric (saccharic) acid. This suggests that glucose molecule has primary alcoholic group (-CH2OH).
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 27
6. On reduction with HI, glucose gives n – hexane.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 28

7. Open chain structure of glucose:
Based on above observations, glucose is given the open chain structure.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 29
8. Objections against open chain structure of glucose :
Open chain structure of glucose could not explain

  1. Why glucose does not respond to schiffs test.
  2. Why it does not react with NaH SO3, and NH3.
  3. Why there is mutarotation.

Mutarotation :
Change in specific rotation of a compound from certain value to fixed value is known as “Mutarotation”.
The specific rotation of α – glucose is + 110° and that of β – form is + 19.7°. If each of them is separately dissolved in water and allowed to stand, their specific rotation gradually change and reach to a specific constant value + 52.5°.

9. Cyclic structure of glucose :
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 30
These two structures have difference in configuration at ‘C1‘ only and are called ANOMERS.
Cyclic structure of glucose explains all the properties of glucose.

10. Pyranose structure of glucose :
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 31

Question 3.
Write short notes on sucrose.
Answer:
Sucrose is a disaccharide. It under go hydrolysis and forms equimolar mixture of D(+) glucose and D(-) fructose.
C12 H22 O11 + H12O → C6 H12 O6 (glucose) + C6 H12 O6 (fructose).

  • In sucrose two monosaccharide units are held together by a glycosidic linkage between C – 1 of α – glucose and C – 2 of β – glucose.
  • Sucrose is a non reducing sugar.
    AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 32
    During the hydrolysis of sucrose there is a change in the sign of rotation, from dextro (+) to laevo (-) and the product is named as invert sugar.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 4.
Write the structure of maltose and lactose. What are the products of hydrolysis of maltose and lactose?
Answer:
Structure of maltose:

  • Maltose is formed from two α – glucose units (Disaccharide)
  • C – 1 of one glucose linked to C -4 of another glucose
  • It is a reducing sugar because C – 1 of 2nd glucose is solution have free aldehyde group.
    AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 33
  • maltose on hydrolysis to form two glucose units
    C12 H22 O11 (Maltose) + H2O → C6H12O6 (glucose) + C6H12O6 (glucose)

Structure of lactose:

  • Lactose is milk sugar.
  • Lactose is formed by the combination of β – glucose and β – galactose (Disaccharide)
  • The linkage between C – 1 of galactose and c – 4 of glucose.
  • It is a reducing sugar.
    AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 34
  • Lactose on hydrolysis to form glucose and galactose
    C12 H22 O11 (Lactose) + H2O → C6H12O6 (glucose) + C6H12O6 (galactose)

Question 5.
Write about polysaccharides with starch and cellulose as examples.
Answer:
Polysaccharides : The saccharides which on hydrolysis to form large number of monosaccharides are called polysaccharides.
Eg : Starch and cellulose Starch :

  • Starch is the most important dietary source for human beings.
  • Vegetables, roots, cereals are important sources of starch.
  • It is a polymer of α – glucose.
  • It is constituted by two components Amylose and amylopectin.

Amylose :

  • It constitutes 15 – 20% of starch.
  • Amylose is water soluble component.
  • Amylose is a branched chain with 200 – 1000 α – D – glucose units held by C – 1 to C -4 glycosidic linkage.

Amylopectin:

  • Amylopectin constitutes 80 – 85% of starch.
  • It is a branched chain polymer of a – glucose units in which chain is formed by C – 1 to C – 4 glycosidic linkage whereas branching occurs by C – 1 to C – 6 glycosidic linkage.

Cellulose :

  • ellulose occurs in plants and it is the most abundant organic substance.
  • It is a major constituent of cell wall of plant cells.
  • Cellulose is a straight chain polysaccharide composed only of β – D – glucose units which are joined by glycosidic linkage between C – 1 of one glucose and C – 4 of the next glucose.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 6.
Write the importance of carbohydrates.
Answer:
Importance of carbohydrates :

  • Carbohydrates are essential for life of plants.
  • Carbohydrates are used as storage molecules as starch in plants.
  • Cell wall of plants is made up of cellulose.
  • Carbohydrates are also essential for life of animals. Carbohydrates are used as storage molecules as glycogen in animals.
  • Carbohydrate source hohey is used for a long time as an instant source of energy.

Question 7.
Explain the classification of proteins as primary, secondary, tertiary and quartemary proteins with respect ot their structure.
Answer:
Protein structure’s and shapes are studied at four different levels as
i) Primary ii) Secondary iii) Tertiary iv) Quaternary structures

i) Primary structure : The primary structure of protein is defined as the linear sequence of amino acid residues making up its polypeptide chain. Proteins ‘may be formed of one or more polypeptide chains. The amino acid residues are linked by peptide bonds. The peptide bond is formed between carboxyl group of one amino acid and amino group of adjacent amino acid.
Eg : fibroin of silk.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 35
R1 – amino acid 1
R2 – amino acid 2

ii) Secondary structure : The secondary structure explains the shape of the polypeptide chain. It is derived from the primary structure by the formation of hydrogen bond interactions between amino acid residues.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 36
Disulfide bonds also occur either in same chain or between polypeptide chains. The secondary structure is classified into two types namely helical structure and pleated sheets.
Eg. : Keratin of hair
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 37

iii) Tertiary structure : Tertiary structure is exhibited by proteins having only one polypeptide chain. The three dimensional structure of a protein is referred to as tertiary structures. Fibrous and globular proteins are formed by tertiary structures. Various bonds are responsible for tertiary structure.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 38

  • Hydrophobic interactions
  • Hydrogen bonds
  • Disulphide bonds (- CH2 – S – S – CH2 -)
  • Ionic or electrostatic interactions
  • Vander Waal forces
    e.g : Myoglobin, Ribonuclease

iv) Quaternary structure : Two or more poly- peptide chains associate together to protect quaternary structure. A protein with a quaternary structure is referred to as an oligomes.
Eg. : Hemoglobin a chain
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 39

Question 8.
Explain the denaturation of proteins. [T.S. Mar. 16]
Answer:
The phenomenon of disorganization of native protein structure is known as denaturation. Denaturation results in the loss of secondary, tertiary and quaternary structure of proteins. This involves a change in physical, chemical and biological properties of protein molecules.
Agents of denaturation :
Physical Agents – Heat, violent shaking, X – rays, UV – radiation.
Chemical Agents – Acids, alkalies, organic solvents, urea, salts of heavy metals.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 9.
What are enzymes ? Give examples ?
Answer:
The group of complex proteinoid organic compounds, elaborated by living organism ‘which catalyse specific organic reactions are called enzymes.
Eg.: Lipases, Rennin, Maltase, Invertase etc. Practic ally all biological processes such as digestion, respiration etc., are carried on through the agency of enzymes.
Enzymes may be defined as biocatalysts synthesised by living cells.
The functional unit of enzyme is known as holo enzyme made up at apo enzyme (protein part) a co enzyme (non-protein part)
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 40

Question 10.
Explain the role of sucrose in its hydrolysis.
Answer:

  • Sucrose is dextrorotatory but after hydrolysis gives dextrorotatory glucose and laevorotatory fructose.
  • The laevorotation of fructose (-92.4°) is more than dextrorotation of glucose (+52.5°). This mixture is leavo rotators.
  • During the hydrolysis of sucrose there is a change in the sign of rotation, from dextro (+) to laevo (-) and the product is named as invert sugar.

Question 11.
Write notes on vitamins.
Answer:
Vitamin is defined as an “accessory food factor which is essential for growth and healthy maintenance of the body.”
Classification : Vitamins are broadly classified into two major groups.
a) the fat soluble Eg : vitamin A, D, E and K.
b) water soluble Eg : vitamin B – complex and C.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 41
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 42

Question 12.
What do you understand by “The two strands of DNA are complementary to each other” ? Explain.
Answer:
Structure of DNA : To understand the structure of DNA, we should know the charagaff s rule, which states that:

  • Base composition on DNA for a particular organism is constant throughout all the somatic cells.
  • Base composition always varies from one organism to another and is expressed by the dissymmetry ratio \(\frac{(A+T)}{(G+C)}\).
  • Organisms closely related to each other often have similar base composition and hence – have closed values for their dissymmetry ratios.
  • In a given organism the amount of adenine is always equal to the amount of thymine (A = T) and amount of guanine is equal to amount of cytosine (G = C).
  • In a given organism the total amount of purine, bases
    are always equal to total number of pyrimidine bases (i.e.,) (A + G = T + C).
    M. Wilkins found that within the crystal there is a repeat distance of 3.4 nm and there are ten subunits per him.
    AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 43

Watson – Crick model for DNA : From the above observations Watson and Crick constructed a model for DNA. This model consists of two right handed polynucleotide chains that are complimentary and coiled about the same axis to form double-helix. Some specific base pairs can be spatially accommodated and these are A – T and G – C pairs. The bases are closely associated with each other by hydrogen bonding.

The helix has diameter of about 2.0 nm and contains 10 nucleotide pairs in each turn of helix as shown above :

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 13.
What are harmones ? Give one example for each. [A.P. & T.S. Mar. 19, 18] [Mar. 14]
i) steroid hormones
ii) Poly peptide hormones and
iii) amino acid derivatives.
Answer:
Hormones : Hormone is defined as an “organic compound synthesised by the ductless glands of the body and carried by the blood stream to another part of the body for its function”.
Eg : testosterone, estrogen.

  1. Example for steroid hormones : Testosterone, Estrogen
  2. Example for poly peptide hormones : Insulin
  3. Example for Amino acid derivative : Thyroidal hormones thyroxine.

Question 14.
Give the sources of the following vitamins and name the diseases caused by their deficiency (a) A (b) D (c) E and (d) K [A.P. Mar. 16, 15; T.S. Mar. 15]
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 44

Long Answer Questions

Question 1.
Explain the classification of carbohydrates.
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 45
Saccharides that reduce Fehling’s reagent, Tollen’s reagent are called reducing sugars.
Eg : glucose.
Saccharides that does not reduce these reagents are called non-reducing sugars.
Eg : sucrose.

Question 2.
Discuss the structure of glucose on the basis of its chemical propeties.
Answer:
1. The molecular formula of glucose is found to be C6H12O6. It can be written as ‘CHO – (CHOH)4 – CH2OH’.

2. Acylation of glucose with acetic anhydride gives glucose pentacetate. Hence glucose molecule has five ‘-OH’ groups.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 46

3. Glucose reacts with hydroxylamine and hydrogen cyanide forming corresponding oxime and cyano hydrin respectively. Hence glucose molecule has carbonyl group.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 47

4. Glucose on oxidation with Tollen’s reagent and Fehling’s solution gives gluconic acid. Hence carbonyl group present in glucose is aldehydic group.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 48

5. Glucose on oxidation with nitric acid gives glucaric’(saccharic) acid. This suggests that glucose molecule has primary alcoholic group (-CH2OH)
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 49

6. On reduction with HI, glucose gives n – hexane.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 50
This reaction suggests that all six carbons in glucose are linearly linked.

7. Open chain structure of glucose:
Based on above observations, glucose is given the open chain structure.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 51

8. Objections against open chain structure of glucose :
Open chain structure of glucose could not explain

  1. Why glucose does not respond to schiff s test.
  2. Why it does not react with NaHSO3 and NH3.
  3. Why there is mutarotation.

Mutarotation :
Change in specific rotation of a compound from certain value to fixed value is known as “Mutarotation”.
The specific rotation of α – glucose is + 110° and that of β – form is + 19.7°. If each of them is separately dissolved in water and allowed to stand, their specific rotation gradually change and reach to a specific constant value + 52.5°.

9. Cyclic structure of glucose :
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 52
These two structures have difference in configuration at ‘C1‘ only and are called
ANOMERS.
Cyclic structure of glucose explains all the properties of glucose.

10. Pyranose structure of glucose :
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 53

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 2.
Write notes on
a) fructose
b) sucrose
c) maltose
d) lactose
Answer:
a) Fructose :

  • Fructose is an important ketohexose.
  • It is formed by the hydrolysis of sucrose.
    AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 54
  • It has the ketonic group at C – 2 and six carbons in straight chain as in the glucose.
  • It has furanose ring structure.
  • Open chain structures:
    AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 55
    α – D (-) – Fructo furanose structure
    β – D (-) – Fructo furanose structure
    Haworth structures :
    AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 56

b) Sucrose :
Sucrose is a disaccharide. It under go hydrolysis and forms equimolar mixture of D(+) – glucose arid D(-) fructose.
C12 H22 O11 + H2O → C6 H12 O6 (glucose ) + C6 H12 O6 (fructose)

In sucrose two monosaccharide units are held together by a glycosidic linkage between C- 1 of α – glucose and C – 2 of β – glucose.

Sucrose is a non reducing sugar.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 57
During the hydrolysis of sucrose there is a change in the sign of rotation, from dextro (+) to laevo (-) and the product is named as invert sugar.

c) Structure of maltose :
Maltose is formed from two α – glucose units (Disaccharide)
C – 1 of one glucose linked to C – 4 of another glucose
It is a reducing sugar because C – 1 of 2nd glucose is solution have free aldehyde group.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 58
maltose on hydrolysis to form two glucose units
C12 H22 O11 (Maltose) + H2O → C6 H12 O6 (Glucose ) + C6 H12 O6 (Glucose)

d) Structure of lactose :

  • Lactose is milk sugar
  • Lactose is formed by the combination of β – glucose and β – galactose (Disaccharide).
  • The linkage between C – 1 of galactose and c – 4 of glucose >
  • It is a reducing sugar
    AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 59
  • Lactose on hydrolysis to form glucose and galactose
    C12 H22 O11 (Lactose) + H2O → C6 H12 O6 (glucose ) + C6 H12 O6 (galactose)

Question 3.
Write notes on
a) starch
b) cellulose
c) importance of carbohydrates.
Answer:
a) Starch:
Polysaccharides : The saccharides which on hydrolysis to form large number of monosaccharides are called polysaccharides.
Eg : Starch and cellulose Starch :

  • Starch is the most important dietary source for human beings.
  • Vegetables, roots, cereals are important sources of starch.
  • It is a polymer of a – glucose.
  • It is constituted by two components Amylose and amylopectin

Amylose :

  • It constitutes 15 – 20% of starch -4 Anylose is water soluble component.
  • Amylose is a branched chain with 200 – 1000 α – D – glucose units held by C – 1 to C – 4 glycosidic linkage.

Amylopectin :

  • Amylopectin constitutes 80 – 85% of starch.
  • It is a branched chain polymer of a – glucose units in which chain is formed by C – 1 to C – 4 glycosidic linkage whereas branching occurs by C – 1 to C – 6 glycosidic linkage.

b) Cellulose :

  • Cellulose occurs in plants and it is the most abundant organic substance.
  • It is a major constituent of cell wall of plant cells.
  • Cellulose is a straight chain polysaccharide composed only of β – D – glucose units – which are joined by glycosidic linkage between C – 1 of one glucose and C – 4 of the next glucose.

c) Importance of carbohydrates :

  • Carbohydrates are essential for life of plants.
  • Carbohydrates are used as storage molecules as starch in plants.
  • Cell wall of plants is made up of cellulose.
  • Carbohydrates are also essential for life of animals. Carbohydrates are used as storage molecules as glycogen in animals.
  • Carbohydrate source honey is used for a long time as an instant source of energy.

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 4.
Write notes on amino acids.
Answer:
The organic compounds which contain amino (-NH2) functional group and carboxyl (-COOH) functional group are called amino acids.
Eg : Glycine, Alanine etc.

Essential amino acids : The amino acids that cannot be synthesised by the body and must be supplied in the diet are called essential amino acids.
Eg : valine, leucine, phenyl alanine etc.,

Non-essential amino acids : Other amino acids synthesised by the tissues of the body are called non-essential amino acids.
Eg : Glycine, alanine etc.,

Zwitter ion : In aqueous solution Of amino acids, the carboxyl group can lose a proton and amino acid can accept that proton to form a dipolar ion. This is called as zwitter ion.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 60

Peptide linkage :
Amide AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 61 linkages between amino acids are known as peptide linkages. The product obtained from two amino acid molecules through peptide bond is called a dipeptide. Peptide chain may be extended to many amino acids.
Proteins are naturally occufing polypeptides that contain more than 50 amino acid units.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 62

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 5.
Write notes on proteins.
Answer:
Proteins : A poly peptide with more than hundred amino acid residues, having molecular mass higher than 10,000 units is called a protein.
g : keratin, myosin, insulin.
With respect to peptide bond proteins classified into two types.

  1. Fibrous proteins
  2. Globular proteins.

When the chains of polypeptides coil around to give a spherical shape then globular proteins are formed. These are usually soluble in water.
Eg : insulin and albumins.

When the poiy peptide chains run parallel and are held together by hydrogen and disulphide bonds then fibre – like structure is formed. These are called fibrous proteins. These are insoluble in water.
Eg: keratin, myosin.

Protein Denaturation: The phenomenon of disorganization of native protein structure is known as denaturation. Denaturation results in the loss of secondary; tertiary,. nad quaternary structure of proteins. This involves a change in physical, chemical and biological properties of protein molecules.

Agents of denaturation:
Physical agents – Heat, violent shaking, X – rays, Uy – radiation
Chemical agents – Acids, alkalies, organic solvénts, urea, salts of heavy metals.
Protein strúctures and shapes are studied at four different levels as:
i) Primary
ii) Secondary
iii) Tertiary
iv) Quaternary structures

i) Primary structure: The primary structure of protein is defined as the linear sequence of amino acid residues making up its polypeptide chain. Proteins may be formed of one or more polypeptide chains. The amino acid residues are linked by peptide bonds, The peptide bond is formed between carboxyl group of one amino açid and amino group of adjacent amino acid.
Eg: fibroin of silk.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 63
R1 – amino acid 1
R2 – amino acid 2

ii) Secondary structure : The secondary structure explains the shape of the polypeptide chain. It is derived from the primary structure by the formation of hydrogen bond interactions between amino acid residues.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 36
Disulfide bonds also occur either in same chain or between polypeptide chains. The secondary structure is classified into two types namely helical structure and pleated sheets.
Eg. : Keratin of hair
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 37

iii) Tertiary structure i Tertiary structure is exhibited by proteins having only one polypeptide chain. The three dimensional structure of a protein is referred to as tertiary structures. Fibrous and globular proteins are formed by tertiary structures. Various bonds are responsible for tertiary structure.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 38

  1. Hydrophobic interactions
  2. Hydrogen bonds
  3. Disulphide bonds (- CH2 – S – S – CH2 -)
  4. Ionic or electrostatic interactions
  5. Vander Waal forces
    e.g : Myoglobin, Ribonuclease

iv) Quaternary structure : Two or more poly- peptide chains associate together to protect quaternary structure. A protein with a quaternary structure is referred to as an oligomes.
Eg. : Hemoglobin a chain
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 39

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 6.
Write notes on
(a) enzymes and
(b) vitamins.
Answer:
Enzymes :
The group of complex proteinoid organic compounds, elaborated by living organism which catalyse specific organic reactions are called enzymes. Eg.: Lipases, Rennin, Maltase, Invertase etc. Practically all biological processes such as digestion, respiration etc., are carried on through the agency of enzymes.
Enzymes may be defined as biocatalysts synthesised by living cells.
The functional unit of enzyme is known as holo enzyme made up at apo enzyme (protein part) and co enzyme (non-protein part)
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 64
b) Vitamins :
Vitamin is defined as an “accessory food factor which is essential for growth and healthy maintenance of the body.”
Classification : Vitamins are broadly classified into two major groups.
a) the fat soluble Eg : vitamin A, D, E and K.
b) water soluble Eg : vitamin B – complex and C.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 41
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 42

Question 7.
Explain the structures of DNA and RNA.
Answer:
Structure of DNA : To understand the structure of DNA, we should know the charagaff s rule, which states that :

  1. Base composition on DNA for a particular organism is constant throughout all the somatic cells.
  2. Base composition always varies from one organism to another and is expressed by the dissymmetry ratio \(\frac{(\mathrm{A}+\mathrm{T})}{(\mathrm{G}+\mathrm{C})}\)
  3. Organisms closely related to each other often have similar base composition and hence have closed values for their dissymmetry ratios.
  4. In a given organism the amount of adenine is always equal to the amount of thymine (A = T) and amount of guanine is equal to amount of cytosine (G = C).
  5.  In a given organism the total amount of purine bases are always equal to total number of pyrimidine bases (i.e.,) (A + G = T + C).

M. Wilkins found that within the crystal there is a repeat distance of 3.4 nm and there are ten subunits per turn.

Watson – Crick model for DNA : From the above observations Watson and Crick constructed a model for DNA.
This model consists of two right handed polynucleotide chains that are complimentary and coiled about the same axis to form double-helix. Some specific base pairs can be spatially accommodated and these are A – T and G – C pairs. The bases are closely associated with each other by hydrogen bonding.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 43
The helix has diameter of about 2.0 nm and contains 10 nucleotide pairs in each turn of helix as shown above :

Structure of RNA: The native RNA is single stranded rather than a double stranded helical structure characteristic of DNA. However, given the complementary base sequence with opposite polarity, the single strand of RNA may fold back on itself like a hairpin and thus acquire double stranded pattern. In the region of hairpin loops, “A” pairs with ‘U’ and ‘G’ pairs with ‘C’. The base pairing in RNA hairpins is frequently imperfect. The proportion of helical regions in various types of RNA varies over a wide range.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 65

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 8.
Write notes on the functions of different hormones in the body.
Answer:
Functions of Hormones :

  • Hormones help to maintain the balance of biological activities in the body.
  • Insulin maintains the blood glucose level with in the limit.
  • Growth hormones and sex hormones play role in growth and development.
  • Low level of thyroxine (produced from thyroid gland) causes hypothyroidism. High level of thyroxine causes hyper thyroidism. .
  • gluco corticoids control the carbohydrate metabolism, modulates the inflammatory reactions.
  • The mineralo corticoids control the level of excretion of water and salt by the kidney.
  • Adrenal cortex does not function properly then results in Addison’s disease.
  • Hormones released by gonads are responsible for development of secondary sex characters.
  • Testosterone is responsible for development of secondary sex hormone produced in male.
  • Estradiol is the main female sex hormone responsible for development of secondary female characterstics like control of menustrual cycle.
  • Progesterone is responsible for preparing the uterus for implantation of fertilised egg.

Intext Questions

Question 1.
Glucose or sucrose are soluble in water but cyclohexane and benzene (simple six membered ring compounds) are insoluble in water. Explain.
Answer:
In glucose (C6H12O6), 5 – OH groups are present and in sucrose (C12H22O11), B – OH groups are present, which are polar in nature. They are involved in intermolecular hydrogen bonding with water molecules, so these compounds are readily soluble in water.

Benzene (C6H6) and cyclohexame (C6H12) are hydrocarbons. They do not have any polar group so they cannot make hydrogen bonds with water. Therefore, they are insoluble in water.

Question 2.
What are the expected products of hydrolysis of lactose ?
Answer:
Lactose forms two molecules of monosaccharides on hydrolysis – One molecule of D-(+) – glucose and one molecule of D – (+) – galactose.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 66

Question 3.
How do you explain the formation of furanose structure for fructose while glucose forms pyranose structure with same molecular formula C6H12O6 ?
Answer:
D-glucose, an aldohexose given the characteristic reactions of aldehydic group (e.g., with HCN Tollen’s reagent, Fehling reagent etc.,) but penta acetae of D-glucose does not give these reactions. It means that either – CHO group is absent or it is not available for the chemical reactions in penta acetyl glucose. Infact, the aldehydic group is a part of hemiacetal structure which penta acetyl glucose has. So, it is not available for reaction.
AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules 67

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 4.
The melting points and solubility in water of amino acids are generally higher than that of the corresponding halo acids. Explain.
Higher the polarity of a group, more is its solubility in water.
Answer:
Amino acids are dipolar in nature (N+ H3 – CHR – COO) and have strong dipolar interatctions.
Therefore, these have high melting points. When dissolved in water, these form hydrogen bonds with water molecules, therefore, these are soluble in water, (except at isoelectric point, where they have least solubility) In case of halo acids, these are not dipolar. Only their carboxyl group is involved in hydrogen bonding, not the halogen atom. Therefore, halo acids, have lower melting points and lesser solubility than amino acids.

Question 5.
Why cannot vitamin C be stored in our body ?
Answer:
VitaminC (ascorbic acid) is water soluble. Therefore, it is regularly excreted in urine from the body and cannot be stored.

Question 6.
What products would be formed when a nucleotide from DNA containing thymine is hydrolysed ?
Give the composition of DNA molecule as on hydrolysis it gives its all constituents.
Answer:
When a nucleotide from DNA is hydrolysed, it will give deoxyribose sugar molecules, phosphoric acid, pyrimidine bases, i.e., guanine (G) and adenine (A) and purine bases, i.e., thymine (T) and cytosine (C).

AP Inter 2nd Year Chemistry Study Material Chapter 9 Biomolecules

Question 7.
When RNA is hydrolysed, there is no relationship among the quantities of different bases obtained. What does this fact suggest about the structure of RNA ?
Single stranded structure of RNA.
Answer:
DNA molecule has a double strand structure and four complementary bases paired with each other. Cytosine (C) pairs with guanine (G) while thymine (T) pairs with adenine (A). Due to such structure, DNA produces the products in definite molar ratio.

The case is not same in RNA, ie., base pairing rule is not followed. Thus, when RNA is hydrolysed, the quantities of different bases are different. This suggests that RNA has single stranded structure.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Andhra Pradesh BIEAP AP Inter 2nd Year Chemistry Study Material 8th Lesson Polymers Textbook Questions and Answers.

AP Inter 2nd Year Chemistry Study Material 8th Lesson Polymers

Very Short Answer Questions

Question 1.
Define the terms monomer and polymer.
Answer:
Monomer : The repeating structural units of polymer which are derived from simple and reactive molecules linked to one another by covalent bonds are called monomers.

Polymer : A large molecular weight complex compound which is formed by the.repeated combination of simple units (monomers) is called polymer.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 2.
What are polymers ? Give example.
Answer:
Polymer : A large molecular weight complex compound which is formed by the repeated combination of simple units (monomers) is called polymer.
E.g. : Nylon 6, 6, Buna-S,. rubber etc…..

Question 3.
What is polymerization ? Give an example of polymerization reaction.
Answer:
Polymerization: The process of formation of polymers from respective monomers is called polymerization.
(or)
A large molecular weight complex compound which is formed by the repeated combination of simple units is called polymer. This process is called polymerisation.
E.g. : Formation of polyethene from ethene and reaction of hexamethylene diamihe and adipic acid leading to the formation of Nylon 6, 6 are examples of two different types of polymerisation reactions.

Question 4.
Give one example each for synthetic and semi-synthetic polymers.
Answer:
Examples of synthetic polymers : Neoprene, Buna-S, Buna-N
Examples of Semi synthetic polymers : Cellulose rayon, cellulose nitrate.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 5.
How are the polymers classified on the basis of structure ?
Answer:
On the basis of structure, polymers are classified into three types :

  1. Linear polymers : These contains long and straight chains.
    E.g.: PVC, polythene (high density) etc.
  2. Branched chain polymers : These contains linear chains having some branches.
    E.g.: low density polythene.
  3. Cross linked polymers (or) Network polymers : These are usually formed from bi functional and tri functional monomers and contain strong covalent bonds between various linear polymer chains.
    E.g.: Bakelite, melamine etc. .

Question 6.
Give one example each for linear and branched chain polymers.
Answer:

  1. Linear polymers : These contains long and straight chains.
    E.g.: PVC, polythene (high density) etc.
  2. Branched chain polymers : These contains linear chains having some branches.
    E.g. : low density polythene.

Question 7.
What are cross linked (or network) polymers ? Give example.
Answer:
Cross linked polymers (or) Network polymers: These are usually formed from bi functional and tri functional monomers and contain strong covalent bonds between various linear polymer chains. E.g. : Bakelite, melamine etc.

Question 8.
What is addition polymer ? Give example. [T.S. Mar. 15]
Answer:
Addition Polymer: The polymer which is formed by the addition of molecules of monomers of same type (or) different type containing double bonds is called addition polymer.
E.g.: Polyethene, poly acrylonitrile.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 9.
What is condensation polymer ? Give example.
Answer:
Condensation polymer : The polymer which is formed by the condensation reaction between molecules having more than one functional group is called condensation polymer.
E.g.: Nylon 6, 6, Poly ethylene terephthalate.

Question 10.
What are homopolymers ? Give example.
Answer:
Homopolymers : The polymers which are formed by the polymerisation of a single monomeric species are known as homopolymers. E.g.: Polyethene, Poly styrene.

Question 11.
What are copolymers ? Give example.
Answer:
Copolymers : A polymer which is formed by the polymerisation of two (or) more chemically different types of monomer units is called copolymer. E.g.: Butadiene – Styrene polymer (Buna-S)

Question 12.
Is – [-CH2 – CH (C6H5)-]n – a homopolymer or a copolymer ?
Answer:
-[-CH2 – CH (C6H5-]n is polystyrene. It is a homopolymer.
It is formed by the polymerisation of single monomer styrene.
(C6H5 – CH = CH2)

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 13.
Is (NH – CHR- CO)n a homopolymer or a copolymer ?
Answer:
[NH – CHR – CO]n is a homopolymer. It is formed by the polymerisation of single monomer a-amino acid (NH2 – CHR-COOH).

Question 14.
What are the classes of the polymer based on molecular forces ?
Answer:
On the basis of molecular forces polymers are classified into four types.

  1. Elastomers: These are rubber like solids with elastic properties.
    E.g.: Buna-S, Buna-N.
  2. Fibres : Fibres are the thread forming Solids which possess high tensile strength.
    E.g. : Nylon 6, 6, polyesters
  3. Thermoplastic Polymers : These are the linear (or) slightly branched long chain
    molecules capable of softening on heating and hardening on cooling.
    E.g.: Polystyrene, polythene.
  4. Thermo setting polymers : These polymers are cross linked (or) heavily branched molecules which on heating undergo extensive cross linking in moulds and again become infusible.
    E.g. : Bakelite, urea-formaldehyde resin etc….

Question 15.
What are elastomers ? Give example. .
Answer:
Elastomers: These are rubber like solids with elastic properties. In elastomers the polymer chains are held together by the weak inter molecular forces.
E.g.: Buna – S, Buna – N etc.

Question 16.
What are fibres ? Give example.
Answer:
Fibres : Fibres are the thread forming solids which possess high tensile strength.
E.g.: Nylon 6, 6, polyesters

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 17.
What are thermoplastic polymers ? Give example.
Answer:
Thermoplastic Polymers: These are the linear (or) slightly branched long chain molecules capable of softening on heating and hardening on cooling.
E.g.: Polystyrene, polythene.

Question 18.
What are thermosetting polymers ? Give example.
Answer:
Thermo setting polymers: These polymers are cross linked (or) heavily branched molecules which on heating undergo extensive cross linking in moulds and again become infusible.
E.g.: Bakelite, urea-formaldehyde resin etc….

Question 19.
Write the name and structure of one of the common initiators used in free radical polymerization reaction.
Answer:
One of the common initiator used in free radical – polymerisation reaction is benzoyl peroxide.
Structure:
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 1

Question 20.
How can you differentiate between addition and condensation polymerization?
Answer:

Addition polymerisation

  1. Monomers used are unsaturated com­pounds.
  2. Polymer formation takes place without loss of atoms (or) groups.
  3. It is chain growth polymerisation.
  4. These polymers are called addition (or) ‘ chain (or) vinyl polymers.

Condensation polymerisation

  1. Monomers are bifunctional, tri-functional compounds.
  2. Polymer formation takes place with loss of atoms (or) groups like NH3, H20 etc.
  3. It is step growth polymerisation.
  4. These are called condensed polymers.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 21.
What is Ziegler – Natta catalyst? [T.S. Mar. 19]
Answer:
A mixture of Tri alkyl aluminium and titanium chloride is called Ziegler – Natta catalyst
E.g.: (C2H5)3 Al + TiCl4

Question 22.
How is Dacron obtained from ethylene glycol and terepthalic acid?
Answer:
Formation of dacron is an example of condensation polymerisation. It is obtained from ethylene glycol and terephthalic acid as follows.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 2

Question 23.
What are the repeating monomeric units of Nylon 6 and Nylon 6, 6?
Answer:
The repeating monomeric units of Nylon – 6 is Capro lactam
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 3
The repeating monomeric units of Nylon 6, 6 are hexamethylene diamine and Adipic acid.
H2N-(CH2)6 – NH2
Hexamethylene
HOOC – (CH2)4 – COOH
Adipic acid

Question 24.
What is the difference between Buna – N and Buna – S ?
Answer:
Buna – N : Buna – N is the copolymer which is formed by the polymerisation of 1, 3 – Butadiene and acrylonitrile.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 4

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 25.
Arrange the following polymers in increasing order of their molecular forces.

  1. Nylon 6, 6 Buna – S, Polythene
  2. Nylon 6, Neoprene, Polyvinyl chloride

Answer:

  1. Increasing order of inter molecular forces of given polymers is
    Buna – S < polythene < Nylon -6, 6
  2. Increasing order of inter molecular forces of given polymers is
    Neoprene < polyvinyl chloride < Nylon – 6.

Question 26.
Identify the monomer in the following polymeric structures
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 5
Answer:

  1. Monomers present in Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 6 are decane dioic acid HOOC -CH2-)8 COOH and hexamethylene diamine H2N – (CH2-)6 NH2
  2. Monomers present in (-NH – CO – NH – CH2 -)n are urea CO (NH2)2, formaldehyde HCHO.

Question 27.
Name the different types of molecular masses of polymers.
Answer:
The different types of important molecular masses of polymers are

  1. Number average molecular mass (\(\overline{\mathbf{M}}_{\mathrm{n}}\))
  2. Weight average molecular mass (\(\overline{\mathbf{M}}_{\mathrm{w}}\))

Question 28.
What is PDI (Poly Dispersity Index) ? [A.P. Mar. 19]
Answer:
Poly Dispersity Index (PDI) : The ratio between weight average molecular mass (\(\overline{\mathbf{M}}_{\mathrm{w}}\)) and the number average molecular mass (\(\overline{\mathbf{M}}_{\mathrm{n}}\)) of a polymer is called Poly Dispersity Index (PDI).

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 29.
What is vulcanization of rubber ? [T.S. Mar. 19, 16; A.P. Mar. 17]
Answer:
Vulcanization of rubber : The process of heating the raw rubber with sulphur (or) with sulphur compounds to improve it’s physical properties is called vulcanization of rubber.

Question 30.
What is the cross linking agent used in the manufacture of type rubber ?
Answer:
In the manufacture of tyre rubber 5% sulphur is used as cross linking agent. ‘

Question 31.
What is biodegradable polymer ? Give one example of a biodegradable polyester ?
Answer:
Biodegradable poiymers : The polymers degradable by enzymatic hydrolysis and to some extent by oxidation are called biodegradable polymers.
E.g. : Nylon – 2, Nylon – 6, PHBV, Polyglycolic acid, Polylactic acid etc.

Question 32.
What is PHBV ? How is it useful to man ? [A.P. Mar. 19, 18, 16; T.S. Mar. 18, 16, 15]
Answer:
Poly β – hydroxy butyrate – CO – β – hydroxy Valerate (PHBV) : It is a Copolymer of 3 -hydroxy butanoic acid and 3 – hydroxy pentanoic acid.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 7
Properties & Uses : The properties of PHBV vary according to the ratio of both the acids, 3-hydroxy butanoic acid provides stiffness and 3-hydroxy pentanoic acid imparts flexibility to copolymer.
It is used in medicine for making capsules.
PHBV also undergoes degradation by bacteria.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 33.
Give the structure of nylon 2 – nylon 6 ?
Answer:
Nylon 2 – Nylon 6 :
an alternating polyamide copolymer of glycine (Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 8 – COOH) and amino caproic acid (H2N – CH2)5 – COOH). It is a biodegradable polymer.
Structure of Nylon 2 – Nylon – 6
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 9

Short Answer Questions

Question 1.
Classify the following into addition and condensation polymers [A.P. Mar. 16]
i) Terylene
ii) Bakelite
iii) Polyvinyl chloride
iv) Polythene
Answer:
i) Terylene is a condensation polymer.
ii) Bakelite is a condensation polymer
iii) Polyvinyl chloride is an addition polymer
iv) Polythene is an addition polymer.

Question 2.
How do you explain the functionality of a polymer ?
Answer:
The number of bonding sites present in the monomers of the polymer is called functionality.
E.g.:

  1. Functionality of ethene, propene is one.
  2. Functionality of ethylene glycol is two.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 3.
Distinguish between the terms homo polymer and co polymer. Give one example of each.
Answer:
Homopolymers : The polymers which are formed by the polymerisation of a single monomeric species are known as homopolymers.
E.g. : Polyethene; Poly styrene.
Copolymers: A polymer which is formed by the polymerisation of two (or) more chemically different types of monomer units is called copolymer.
E.g.: Butadiene – Styrene polymer (Buna-S)

Question 4.
Define thermoplastics and thermosetting polymers with two examples of each.
Answer:
Thermoplastic Polymers : These are the linear (or) slightly branched long chain molecules capable of softening on heating and hardening on cooling. E.g. : Polystyrene, polythene.
Thermo setting polymers: These polymers are cross linked (or) heavily branched molecules which on heating undergo extensive cross linking in moulds and again become infusible.
Eg.: Bakelite, urea-formaldehyd, resin etc….

Question 5.
Explain copolymerization with an example.
Answer:
Copolymers: A polymer which is formed by the polymerisation of two (or) more chemically different types of monomer units is called copolymer.
E.g.: Butadiene-Styrene polymer(Buna-S)
The process of formation of copolymer is called copolymerisation.
E.g. : Buna – S : Buna – S is the copolymer which is formed by the polymerisation of 1,3- Butadiene and Styrene.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 10

Question 6.
Explain free radical mechanism for the polymerization of ethene.
Answer:
Polymerisation of ethene to polythene proceeds through free radical mechanism. The initiator used in this reaction is benzoyl peroxide. This mechanism involves three steps.
1) Chain initiation : The addition of phenyl free radical formed by the peroxide to the ethene double bond and forms a larger free radical.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 11
C6H5 + CH2 = CH2 → C6H5 – CH2 – CH2

2) Chain propagation : The above formed radical reacts with another molecule of ethene another bigger sized radical is formed. The repetetion of this sequence takes places.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 12
3) Chain Termination : The product radical formed reacts with another radical to form polymerised product.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 13

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 7.
Write the names and structures of the monomers used for getting the following polymers . [Mar. 14]
i) Polyvinyl chloride
ii) Teflon
iii) Bakelite
iv) Polystyrene.
Answer:
i) Polyvinyl chloride
Monomer : Vinyl chloride
Structure : CH2 = CH – Cl

ii) Teflon
Monomer : Tetrafluoro ethylene
Structure : CF2 = CF2

iii) Bakelite
Monomers : Phenol, Formal dehyde
Structure : Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 14

iv) Polystyrene
Monomer : Styrene
Structure : Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 15

Question 8.
Write the names and structures of the monomers of the following polymers.
i) Buna – S
ii) Buna – N
iii) Dacron
iv) Neoprene
Answer:
i) Buna – S
Monomers : 1,3- Butadiene, Styrene
Structure :
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 16

ii) Buna – N
Monomers : 1,3- Butadiene, Acrylonitrile
Structure : CH2 = CH – CH = CH2, CH2 = CH – CN

iii) Dacron
Monomers : Ethylene glycol, Terephthalic acid
Structure : HOCH2 – CH2OH, C HOOC Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 17 COOH

iv) Neoprene
Monomers : 2 – chloro -1,3- Butadiene
Structure :
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 18

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 9.
What is natural rubber ? How does it exhibit elastic properties ?
Answer:

  1. Natural rubber is a polymer and possesses elastic properties.
  2. It is an elastomer and it is manufactured from rubber latex. Latex is a colloidal dispersion of rubber in water.
  3. Natural rubber may be considered as a linear polymer of isoprene. It is also called as cis-1, 4-Poly isoprene.
  4. The cis poly isoprene molecule consists of various chains held together by weak vander waals interactions and has a colloid structure. Thus it stretches like a spring and exhibits elastic properties.

Question 10.
Explain the purpose of vulcanization of rubber. [T.S. Mar. 17]
Answer:

  1. Natural rubber becomes soft at high temperatures and brittle at low temperatures. It shows high water absorption capacity.
  2. Natural rubber is soluble in non-polar solvents and is non-resistant to oxidising agents.
  3. To improve these physical properties rubber can be vulcanised.
  4. The process of heating the raw rubber with sulphur (or) with sulphur compounds to improve it’s physical properties is called vulcanisation of rubber.
  5. The vulcanized rubber has improved properties like elasticity, minimum water absorbing tendency, high resistance to chemical oxidation as well as organic solvents.

Question 11.
Explain the difference between natural rubber and synthetic rubber.
Answer:
Natural rubber : The rubber which is obtained from natural sources such as plants and animals rubber called natural rubber. E.g.: Starch, cellulose, rubber etc.

Synthetic rubber : The rubber which are artificially prepared i.e. man-made are called synthetic rubbers.
These have wide applications in daily life as well as in industry.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 12.
How does the presence of double bonds in rubber molecules influence their structure and reactivity ?
Answer:

  1. Natural rubber is cis. poly isoprene. It is obtained by polymerisation of isoprene. The polymerisation takes place at 1, 4 positions. .
  2. In rubber molecule double bonds are located between C2 and C3 of the monomer
    isoprene.
  3. These double bonds (cis) do not allow the polymer chain to come closer. Therefore only weak vander waal’s forces are present.
  4. The chains are not linear, they can be stretched just like springs and exhibits elastic properties.

Question 13.
What are LDP and HDP ? How are they formed ?
Answer:
Polythenes are two types : .

  1. LDP (Low Density Polythene),
  2. HDP (High Density Polythene)

1) Low Density Polythene : LDP is formed by the polymerisation of ethene under high pressure of 1000 to 2000 atm. at a pressure of 350 to 570K in the presence of traces of dioxygen (or) a peroxide initiator.
Properties :
a) This is obtained through the free radical addition.
b) LDP is chemically inert and tough.
c) LDP is flexible and a poor conductor of electricity.
Uses:
a) It is used in the insulation of electric cables.
b) It is used in the manufacture of pipes in agriculture irrigation.

2) High Density Polythene : HDP is formed by the polymerisation of ethene in a hydro carbon solvent in presence of Ziegler Natta catalyst at a temperature of’333K to 343 K and under a pressure of 6 – 7 atm
Properties :
a) HDP consists of linear molecules and has high density due to close packing.
b) It is chemically inert and more tough, hard.
Uses:
a) It is used in manufacture of house hold articles like buckets, dustbins etc.
b) It is used in manufacture of pipes.

Question 14.
What are natural and synthetic polymers ? Give two examples of each type.
Answer:
Natural polymers : The polymers which are obtained from natural sources such as plants and animals are called natural polymers.
E.g.: Starch, cellulose, rubber etc.

Synthetic polymers: The polymers which are artificially prepared i.e. man-made are called synthetic polymers.
These have wide applications in daily life as well as in industry.
E.g.: Plastics, Nylon 6, 6, synthetic rubbers.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 15.
Write notes on different types molecular masses of polymers.
Answer:
The molecular mass of the polymer doesnot remain constant as in the case of simple chemical substances. So the molecular weight of a polymer is expressed in terms Of “average value”.
The average molecular masses of polymers expressed in different ways. The following are the important among them.

  1. Number average molecular mass (\(\overline{\mathrm{M}}_{\mathrm{n}}\)),
  2. Weight average molecular mass (\(\overline{\mathrm{M}}_{\mathrm{w}}\))

i) Number average molecular mass (\(\overline{\mathrm{M}}_{\mathrm{n}}\)) : If a polymer sample contains N1 particles of mass M1 each and N2 particles of Mass M2 each and so on.
The number average molecular weight (\(\overline{\mathrm{M}}_{\mathrm{n}}\)) of the polymer = \(\frac{\sum_{\mathrm{n}_{\mathrm{i}}=1}^{\infty} \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}{\sum_{\mathrm{n}_{\mathrm{i}}=1}^{\infty} \mathrm{N}_{\mathrm{i}}}\)
(\(\overline{\mathrm{M}}_{\mathrm{n}}\)) can be determined by end group analysis method (or) physical methods based on the number of particle present.

ii) Weight average molecular weight (\(\overline{\mathrm{M}}_{\mathrm{w}}\)) : Weight average molecular weight (\(\overline{\mathrm{M}}_{\mathrm{w}}\)) is calculated by multiplying the molecular weight of each type of particle with contribution of the species to the total weight of the sample.
Weight average molecular weight (\(\overline{\mathrm{M}}_{\mathrm{w}}\)) of the polymer = \(\frac{\sum_{n_{i}=1}^{\infty} N_{i} M_{i}^{2}}{\sum_{n_{i}=1}^{\infty} N_{i} M_{i}}\)

Long Answer Questions

Question 1.
Write an essay on i) Addition polymerization and ii) Condensation polynerization
Answer:
i) Addition Polymerization : Addition polymerization involves the self addition of unsaturated monomers without loss of any small molecules to form polymer.
The polymer so formed is called addition polymer.
This polymerisation is also called chain growth polymerisation.
In addition polymerisation, monomers should contain double bonds.
This type of polymerisation can be broadly subdivided into 2 groups.
a) ionic polymerisation (cationic & anionic)
b) free radical polymerization
Addition polymerization mechanism involves three steps. They are : i)-Chain initiation step ii) Chain propagation step and iii) Chain termination step.
E.g. : Vinyl Chloride molecules polymerises to PVC.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 19

ii) Condensation polymerization : In this polymerisation monomers combine together with the loss of simple molecules like H2O, NH3 etc. The formed polymer is called condensation polymer.
Condensation polymers are generally derived from di (or) polyfunctional monomers. This polymerization is also called step polymerization.
E.g.: i) Nylon – 6,6 is formed from hexamethylene diamine and adipic acid by condensation.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 20
ii) Polyethylene terephthalate (PET) from ethylene glycol and terephtholic acid by condensation reaction.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 21

Question 2.
Explain the classification of polymers based on their source and structure.
Answer:
On the basis of source polymers are classified into three types.

  1. Natural polymers
  2. Semi synthetic polymers
  3. Synthetic polymers

1) Natural polymers : The polymers which are obtained from natural sources such as plants and animals are called natural polymers.
E.g.: Starch, cellulose, rubber etc.

2) Semi synthetic polymers : The polymers which are synthetic derivatives of the natural polymers are called semisynthetic polymers.
E.g. : Cellulose rayon, cellulose nitrate.

3) Synthetic polymers : The polymers which are artificially prepared i.e. man-made are called synthetic polymers. .
These have wide applications in daily life as well as in’ industry.
E.g.: Plastics, Nylon 6, 6, synthetic rubbers.

Classification of polymers on the basis of structure :
i) Linear polymers : It is one in which the repeating units are similar to the links in a very long chain.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 22
E.g.: Polythene, PVC etc.

ii) Branched chain polymers : It is one in which some of the molecules are attached as side chains to the linear chains.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 23
E.g.: Low density polythene (LDPE) etc.

iii) Cross linked (or) network polymers : More branching at random points connecting many chains give rise to network (or) cross linked polymers.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 24
E.g.: Bakelite, melamine etc.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 3.
Explain the classification of polymers based on the mode of polymerization and nature of molecular forces.
Answer:
On the basis of mode of polymerisation polymers are classified into two types :
1) Addition polymers
2) Condensation polymers.
1) Addition Polymer : The polymer which is formed by the addition of molecules of monomers of same type (or) different type containing double bonds is called addition polymer.
Eg.: Polyethene, poly acrylonitrile.
2) Condensation polymer : The polymer which is formed by the condensation reaction between molecules having more than one functional group is called condensation polymer.
Eg.: Nylon 6, 6, Poly Ethylene terephthalate.

On the basis of molecular forces polymers are classified into four types.

  1. Elastomers : These are rubber like solids with elastic properties.
    Eg.: Buna-S, Buna-N.
  2. Fibres : Fibres are the thread forming solids which possess high tensile strength.
    Eg.: Nylon 6, 6, polyesters.
  3. Thermoplastic Polymers : These are the linear (or) slightly branched long chain molecules capable of softening on heating and hardening on cooling.
    Eg.: Polystyrene, polythene.
  4. Thermo setting polymers : These polymers are cross linked (or) heavily branched molecules which on heating undergo extensive cross linking in moulds and again become infusible.
    Eg.: Bakelite, urea-formaldehyde resin etc….

Question 4.
What are synthetic rubbers ? Explain the preparation and uses of the following
i) Neoprene
ii) Buna-N
iii) Buna-S.
Answer:
The polymer which is capable of getting stretched almost to double the length and which is also vulcanizable as in the case of natural rubber is called a synthetic rubber.
1) These are homo polymers of 1, 3 – butadiene derivatives (or) copolymers in which one of the monomer is 1, 3 – Butadiene (or) its derivatives and the other is any unsaturated monomer.
2) Synthetic rubber return to its original shape and size when the stretched force is removed.
Eg.: Neoprene, Buna – S, Buna – N.

i) Neoprene : Neoprene is formed by the free radical polymerisation of chloroprene (2-Chloro 1, 3 – Butadiene)
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 25
Uses:
1) It has superior resistance to vegetable and animal oils.
2) It is used in the manufacture of gaskets, conveyor belts.

ii) Buna – N : Buna – N is the copolymer which is formed by the polymerisation of 1, 3 – Butadiene and acrylonitrile.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 26
Uses:
1) It is resistant to the action of petrol, lubricating oil and organic solvents.
2) It is used in. making oil seals, tank lining.

iii) Buna – S : Buna – S is the copolymer which is formed by the polymerisation of 1, 3 – Butadiene and Styrene.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 27
Uses :
1) It is a good substitute for natural rubber.
2) It is used in manufacture of auto tyres.
3) It is used in manufacture of floor tiles.
4) It is used in manufacture of foot wear components, cable insultation

Textual Examples

Question 1.
Is [-CH2 – CH (C6H5-)] n a homopolymer or a copolymer ?
Solution:
It is a homopolymer and the monomer from which it is obtained is styrene C6H5CH = CH2.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 2.
A polymer contains 10 molecules with molecular mass 10,000 and 10 molecules with molecular mass 1,00,000. Calculate number – average molecular mass.
Solution:
\(\overline{\mathrm{M}}_{\mathrm{n}}=\frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}{\sum \mathrm{N}_{\mathrm{i}}}=\frac{10 \times 10000+10 \times 100000}{10+10}\) = 55,000

Intext Questions

Question 1.
What are polymers ?
Answer:
Polymers are high molecular mass substances consisting of large -number of repeating structural units. They are also called as macromolecules. Some examples of polymers are polythene, bakelite, rubber, nylon 6, 6 etc.

Question 2.
How are polymers classified on the basis of structure ?
Answer:
On the basis of structure, the polymers are classified as below :

  1. Linear polymers such as polythene, polyvinyl chloride, etc.
  2. Branched chain polymers such as low density polythene.
  3. Cross linked polymers such as bakelite, melamine, etc.

Question 3.
Write the names of monomers of the following polymers.
Inter 2nd Year Chemistry Study Material Chapter 8 Polymers 28
Answer:

  1. Hexamethylene diamine and adipic acid
  2. Caprolactam
  3. Tetrafluoroethene.

AP Inter 2nd Year Chemistry Study Material Chapter 8 Polymers

Question 4.
Classify the following as addition and condensation polymers : Terylene, Bakelite, Polyvinyl chloride, polythene.
Answer:
Additon polymers : Polyvinyl chloride, polythene. Condensation polymers : Terylene, Bakelite.

Question 5.
Explain the difference between Buna – N and Buna – S. .
Answer:
Buna – N is a copolymer of 1, 3 – butadiene and acrylonitrile and Buna-S a copolymer of 1, 3—butadiene and styrene.

Question 6.
Arrange the following polymers in increasing order of their intermolecular forces.
i) Nylon 6, 6 Buna — S, Polythene. ii) Nylon 6, Neoprene, Polyvinyl chloride.
Answer:
In order of increasing intermolecular forces.

  1. Buna—S, Polythene, Nylon 6, 6.
  2. Neoprene, Polyvinyl chloride, Nylon 6.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Andhra Pradesh BIEAP AP Inter 2nd Year Chemistry Study Material 7th Lesson d and f Block Elements & Coordination Compounds Textbook Questions and Answers.

AP Inter 2nd Year Chemistry Study Material 7th Lesson d and f Block Elements & Coordination Compounds

Very Short Answer Questions

Question 1.
What are transition elements ? Give examples.
Answer:
Transition elements are the elements which contains partially filled d-subshells in their ionic state (or) in their elementary state.
Eg : Mn, Co, Ag etc.

Question 2.
Which elements of 3d, 4d, and 5d series are not regarded as transition elements and why ?
Answer:
Zn (3d-series), Cd (4d-series), Hg (5d-series) are not regarded as transition elements because these contains completely filled d-subshells.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 3.
Why are d-block elements called transition elements ?
Answer:
The name transition elements given to d-block is due to their properties which are transitioned between the electropositive s-block metals and electronegative p-block elements.

Question 4.
Write the general electronic configuration of transition elements.
Answer:
General electronic configuration of transition elements is (n – 1)d1-10 ns1-2.

Question 5.
In what way is the electronic configuration of transition elements different from non . transition elements ?
Answer:

  • The general electronic configuration of transition elements is (n – 1) d1-10 ns1-2.
  • The general electronic configuration of non-transition elements is (n – 1)d1-10 ns2.

Question 6.
Write the electronic configuration of chromium (Cr) and copper (Cu).
Answer:

  • Electronic configuration of chromium (Cr) – [Ar]4s1d5.
  • Electronic configuration of copper (Cu) – [Ar]4s13d10.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 7.
Why do transition elements exhibits characteristic properties ?
Answer:
Due to the presence of partially filled d-orbitals transition elements exhibit characteristic properties such as variable oxidation states, colour property, magnetic property, complex tendency etc.

Question 8.
Scandium is a transition element. But Zinc is not. Why
Answer:
Scandium has electronic configuration [Ar] 4s23d1.
Zinc has electronic configuration [Ar] 4s23d10.
Scandium has one unpaired d-electron where as Zinc has zero unpaired d-electrons so Scandium is transition element but Zinc is not.

Question 9.
Even though silver has d10 configuration, it is regarded as transition element. Why ?
Answer:
The outer electronic configuration of silver is – 4d105s1. It is having general electronic configuration of a transition element (n – 1) d1-10 ns1-2.
So silver is a transition element.

Question 10.
Write the electronic configuration of Co2+ and Mn2+.
Answer:

  • The electronic configuration of Co+2 is [Ar] 4s0 3d7.
  • The electronic configuration of Mn+2 is [Ar] 4s0 3d5.

Question 11.
Why are Mn2+ compounds more stable than Fe2+ towards oxidation to their +3 state ?
Answer:

  • Mn+2 has electronic configuration [Ar] 4s0 3d5.
  • Fe+2 has electronic configuration [Ar] 4s1 3d6.
  • Mn+2 has half filled d-subshell which is more stable.
  • Hence Mn+2 compounds are more stable than Fe+2 toward oxidation to their +3 state.

Question 12.
Which metal in the first series of transition metals exhibits +1 oxidation state most frequently and why ?
Answer:
Copper exhibits +1 oxidation state most frequently because Cu+ has electronic configuration [Ar] 4s03d10 which has a fulfilled d-subshell which is more stable.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 13.
Why do transition elements exhibit more than one oxidation state (variable oxidation states) ?
Answer:
Transition elements exhibits more than one oxidation state.

Reason:
The energy difference between (n -1) d subshell and ns subshell is very low. So both of these subshells compelete to lose the electrons.

Question 14.
Though Sc is a transition element, it does not exhibit variable oxidation state. Why ?
Answer:
Scandium has electronic configuration [Ar] 4s23d1. It has only one unpaired d-electron. So it does not exhibits variable oxidation state. It exhibits +3 stable oxidation state.

Question 15.
Why is it difficult to obtain M3+ oxidation state in Ni, Cu and Zn ?
Answer:

  • Ni has electronic configuration [Ar] 4s23d8.
  • Ni+2 has electronic configuration [Ar] 4s03d8
  • It is difficult to remove an electron from 3d8. So, Ni+3 is difficult to obtain. (Ni has high negative enthalpy of hydration).
  • Cu has electronic configuration [Ar] 4s13d10.
  • Cu+ has electronic configuration [Ar] 3d10
  • It is difficult to remove the electrons from 3d10 (stable). So, Cu+3 is difficult to obtain.
  • Zn has electronic configuration [Ar] 4s23d10
  • Zn+2 has electronic configuration [Ar] 4s03d10
  • It is difficult to remove the electron from 3d10 (stable). So Zn+3 is difficult to obtain.

Question 16.
Why is Cr2+ reducing and Mn3+ oxidizing even though both have the same d4 electronic configuration.
Answer:
Cr+2 is reducing as its configuration changes from d4 to d3, the latter having a half filled t2g level. On the other hand the change from Mn+2 to Mn+3 results in the half filled (d5) configuration which has extra stability.

Question 17.
Although Cr, Mo and W belong to the same group (group 6) Cr (VI) is a strong oxidizing agent while Mo (VI) and W (VI) are not. Why ?
Answer:
In group 6, Mo (VI) and W (VI) are more stable than Cr (VI). Thus Cr (VI) in the form of dichromate in acidic medium is a strong oxidising agent. Where as MoO3 and WO3 are not.

Question 18.
What do you infer from the fact that M3+/M2+ standard electrode potential of Mn is comparatively higher and that of Fe is comparitively lower ?
Answer:
M3+/M2+ standard electrode potential of Mn is comparatively higher than that of Fe is comparatively lower. This is because much larger 3rd ionisation energy of Mn (d5 to d4).

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 19.
Transition elements have high melting points. Why ?
Answer:
Transition elements have high melting points because of involvement of greater number of electrons from (n – 1)d in addition to ns electrons in the interatomic metallic bonding of these metals.

Question 20.
Among the first transition series (3d series) Chromium has highest melting point. Why? Ans. Among first transition series (3d-series) chromium has highest melting point.
Reason : In chromium one unpaired electron per d-orbital is particularly favourable for strong inter atomic interactions.

Question 21.
Compared to s-block elements, the transition elements exhibit higher enthalpy of atomization. Why ?
Answer:
Because of large no. of unpaired electrons in their atoms transition elements have stronger inter atomic interaction and hence stronger bonding between atoms resulting in higher enthalpies of atomisation.

Question 22.
Among the first transition series (3d series) zinc has lowest enthalpy of atomization. Why ?
Answer:
Among the first transition series zinc has lowest enthalpy of atomisation because zinc has no unpaired electrons in their atomic state or in ionic state.

Question 23.
How do you expect the density of transition elements to vary in a given series and why ?
Answer:
In a given series the density of transition elements increases.

Eg. : From Titanium to copper there is a significant increase in the density.
This is due to decrease in metallic radius coupled with increase in atomic mass.

Question 24.
How do the atomic and ionic sizes vary among transition metals in a given series ?
Answer:
In a given series of transition elements atomic and ionic sizes progressively decreased. This is due to entering of a new electron into d-orbital each time.

Question 25.
Why do Mn, Ni and Zn exhibit more negative E values than expected ?
Answer:
E0 values of Mn, Ni and Zn are more negative than expected from general trend. This is due to the stabilities of half filled d-subshell (d5) in Mn+2 and completely filled d-subshell (d10) in Zinc. For Nickel E0 value is related to the highest negative enthalpy of hydration.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 26.
Among the first transition series (3d series) only copper has positive EM2+/M value. Why?
Answer:
Among the first transition series only copper has positive EM2+/M value. This is due to high ∆aH0 and low ∆hydH0 values.

Question 27.
Cu+2 forms halides like CuF2, CuCl2 and CuBr2 but not CuI2. Why ?
Answer:
Cu+2 forms halides like CuF2, CuCl2 and CuBr2 but not CuI2 because Cu+2oxidises I to I2
2Cu+2 + 4I → Cu2I2 + I2

Question 28.
The highest Mn fluoride is MnF4 where as the highest oxide is Mn2O7 Why ?
Answer:
The ability of oxygen to stabilize the high oxidation states exceeds that of fluoride. Thus the highest Mn fluoride is MnF4 where as highest oxide is Mn2O7.

Question 29.
In its fluoride or Oxide, in which a transition metal exhibits highest oxidation state and why ?
Answer:

  • In fluorides highest oxidation numbers are achieved in TiF4, VF5 and CrF6.
  • The +7 state of Mn is represented in MnO3F.
  • The highest oxidation number in oxides acheived from Sc2O3 to Mn2O7.
  • In Mn2O7, Mn oxidation state is +7.

Question 30.
Why Zn2+ is diamagnetic whereas Mn2+ is paramagnetic ? [T.S. Mar. 15]
Answer:

  • Zn+2 electronic configuration is [Ar] 4s03d10. It has no unpaired electrons. So it is dia magnetic.
  • Mn+2 electronic configuration is [Ar] 4s03d5. It has five unpaired electrons so it is paramagnetic.

Question 31.
Write ‘spin only’ formula to calculate the magnetic moment of transition metal ions.
Answer:
Spin only formula to calculate the magnetic moment of transition metal ions is
μ = \(\sqrt{n(n+2)}\) BM
BM = Bohr Magneton.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 32.
Calculate the ‘spin only’ magnetic moment of Fe2+(aq) ion.
Answer:
Fe+2 ion has electronic configuration [Ar] 4s03d6
It has four unpaired electrons n = 4
Spin only magnetic moment μ = \(\sqrt{n(n+2)}\) BM = \(\sqrt{4(4+2)}\) = \(\sqrt{24}\) BM = 4.9 BM

Question 33.
What is meant by ‘disproportionation’ ? Give an example of disproportionation reaction in aqueous solution.
Answer:
The reactions in which only one element undergo both oxidation as well, as reduction are called disproportion reactions.

When a particular oxidation state becomes less stable relative to other oxidation states, one lower, one higher, it is said to undergo disproportionation.
Eg: Cu+ ion is not stable in aqueous solution because it undergo disproportionation in aqueous solution.
2Cu+(aq) → Cu+2(aq) + Cu(s)

Question 34.
Aqueous Cu2+ ions are blue in colour, where as Aqueous Zn2+ ions are colourless. Why ?
Answer:

  • Electronic configuration of Cu+2 ion is [Ar] 4s03d9. If contains one unpaired electron due to presence of this unpaired electron aq. Cu+2 ions are blue in colour.
  • Electronic configuration of Zn+2 ion is [Ar] 4s03d10. It contains no unpaired electrons, due to absence of unpaired electrons aq. Zn+2 ions are colourless.

Question 35.
What are complex compounds ? Give examples.
Answer:
Complex compounds: Transition metal atoms or ions form a large number of compounds in which anions or neutral groups are bound to metal atom or ion through co-ordinate covalent bonds. Such compounds are called co-ordination compounds (or) complex compounds.
Eg.: [Fe(CN)6]4-, [Co(NH3)6]3+.

Question 36.
Why do the transition metals form a large number of complex compounds ?
Answer:
Transition elements (metals) forms a large number of complex compounds this is due to

  1. Small size of these ions
  2. High effective nuclear charge,
  3. Possessing in completely filled d-orbitals.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 37.
How do transition metals exhibit catalytic activity ?
Answer:
Catalytic properties:

  • Transition metals and their compounds form important catalysts in industry and in biological systems.
  • The catalytic activity of transition metals depends on their ability to exist in different oxidation states of oxidation (or) to form co-ordination compounds.
    Eg : 1) V2O5 is used as catalyst in manufacturing of SO3 from SO2.
    AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 1
    2) Fe is used as a catalyst in manufacturing of NH3.
    AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 2

Question 38.
Give two reactions in which transition metals or their compounds acts as catalysts.
Answer:
1) V2O5 is used as catalyst in manufacturing of SO3 from SO2.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 1
2) Fe is used as a catalyst in manufacturing of NH3.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 2

Question 39.
What is an alloy ? Give example.
Answer:
Alloy : An intimate mixture having physical properties similar to that of the metal formed by a metal with other metals or metalloids or sometimes a non metal is called as an alloy.
Eg.: Invar – 64% Fe, 35% Ni, Mn 8cc in traces
Nichrome – 60% Ni, 25% Fe, 15% Cr.

Question 40.
Why do the transition metals readily form alloys ?
Answer:
Because of similar atomic or ionic radii and similar characterstic properties of transition elements alloys are readily formed by these elements.

Question 41.
How do the ionic character and acidic nature vary among the oxides of first transition series ?
Answer:

  • As the oxidation number of a metal increases ionic character decreases in case of transition elements. Eg : Mn2O7 is a covalent green oil.
  • In CrO3 and V2O5 the acidic character is predominant.
  • V2O5 is however amphoteric though mainly acidic V2O5 reacts with alkali as well as acids to give V04-3 and VO+4.
  • CrO is basic
  • Mn2O7gives HMnO4 and CrO3 gives H2CrO4 and H2Cr2O7.

Question 42.
What is the effect of increasing pH on a solution of potassium dichromate ?
Answer:
On increasing pH of K2Cr2O7 (orange) it changes into K2CrO4 (yellow)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 3

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 43.
Name the oxometal anions of the first series of transition metals in which the metal exhibits the oxidation state equal to its group number.
Answer:
V4-3 ion exhibits ‘+5’ oxidation state which is equal to the V group number
V4-3 + 4(-2) = -3, x = +5

Question 44.
Permanganate titrations are carried out in the presence of sulphuric acid but not in presence of hydrochloric acid. Why ?
Answer:
Per manganate titrations are carried out in the presence of sulphuric acid but not in presence of hydrochloric acid because hydrochloric acid is oxidised to chlorine.

Question 45.
What is lanthanoid contraction ? [T.S. Mar. 19]
Answer:
The slow decrease of atomic and ionic radii in lanthanides with increase in atomic number is called lanthanide contraction.

Question 46.
What are the different oxidation states exhibited by the lanthanoids ?
Answer:

  • Lanthanoids exhibits +2, +3 states majorly. Sometimes +2 and +4 states exhibited in solid compounds.
  • The common oxidation state of lanthanoids is +3.

Question 47.
What is mischmetall ? Give its composition and uses. [A.P. Mar. 19, 16]
Answer:

  • Mischmetall is an alloy which consists of a lanthanoid metal (~ 95%) and iron (~ 5%) and traces of S, C, Ca and Al.
  • It is used in Mg- based alloy to produce bullets, shell and lighter flint.

Question 48.
What is actinoid contraction ?
Answer:
The gradual decrease in the size of atoms or M+3 ions across, the actinoid series is called actinoid contraction.

Question 49.
What are co-ordination compounds ? Give two examples.
Answer:
Co-ordination compounds : Transition metal atoms or ions form a large number of compounds in which anions or neutral groups are bound to metal atom or ion through coordinate covalent bonds. Such compounds are called co-ordination compounds (or) complex compounds.
Eg.: [Fe(CN)6]4-, [Co(NH3)6]+3.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 50.
What is a coordination polyhedron ?
Answer:
The spatial arrangement of the ligands which are directly bonded to the central atom or ions defines the geometry about the central atom is called co-ordination polyhedron
Eg. : Octahedral, tetrahedral etc.

Question 51.
What is a double salt ? Give example.
Answer:
The salts which contains two cations and one anion are called double salts.
These dissociates into simple ions completely when dissolved in water.
Eg : KCl.MgCl2.6H3O -Camallite.

Question 52.
What is difference between a double salt and a complex compound ?
Answer:
Double salts dissociate into simple ions completely when dissolved in water while complex compounds dissociate to give complex ion and the counter ions.

Question 53.
What is a ligand ?
Answer:
Ligand : A co-ordinating entity which is bound to the central atom by donating electron pairs is called a ligand.
Eg.: Cl, NH3, CN etc.

Question 54.
Give one example each for ionic and neutral ligands.
Answer:
Examples for ionic ligands – CN, I, Cl
Examples for Neutral ligands – NH3, H2O

Question 55.
How many moles of AgCl is precipitated when 1 mole of CoCl3 is treated with AgNO3, solution ?
Answer:
3 moles of AgCl is precipitated by the reaction of 1 mole of CoCl3 with AgNO3 solution
3 AgNO3 + CoCl3 → CO(NO3)3 +3 AgCl ↓

Question 56.
What is a chelate ligand ? Give example.
Answer:
The ligands which can form two co-ordinate covalent bonds through two donar atoms are called bidentate ligands. These bidentate ligands are also called chelate ligands.
Eg.: C2O4-2, CO3;-2 etc.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 57.
What is an ambidentate ligand ? Give example. [A.P. Mar. 16]
Answer:
A unidentate ligand containing two possible donor atoms can co-ordinate through either of donor atoms. Such ligands are called ambidentate ligands.
Eg : NO2

Question 58.
CUSO4.5H2O is blue in colour where as anhydrous CuSO4 is colourless. Why ?
Answer:
CUSO4.5H2O is blue in colour whereas anhydrous CuSO4 is colourless because in the absence of ligand, crystal field splitting does not occurs.

Question 59.
FeSO4 solution mixed with (NH4)2SO4 solution in 1:1 molar ratio gives the test for Fe2+ ion but CuS04 mixed with aqueous ammonia in 1 : 4 molar ratio does not give the test of Cu2+ ion. why ?
Answer:

  • FeSO4 solution mixed with (NH4)2SO4 solution in 1 : 1 molar ratio gives the test for Fe2+ because of formation of double salt (FeSO4(NH4)2 SO4 6H2O Mohr’s salt).
  • CuSO4 mixed with aq.ammonia in 1 : 4 molar ratio does not give the test of Cu+2 ion because of formation of complex compound [Cu(NH3)4]SO4.

Question 60.
How many geometrical isomers are possible in the following coordination entities ?

  1. [Cr(C2O4)3]3-
  2. [CO(NH3)3Cl3]

Answer:

  1. [Cr(C2O4)3]3- : Two geometrical isomers are possible i) cis-isomer, ii) Trans-isomer.
  2. [CO(NH3)3Cl3] : Two geometrical isomers are possible i) cis-isomer, ii) Trans-isomer.

Question 61.
What is the coordination entity formed when excess of aqueous KCN is added to an aqueous solution copper sulphate ? Why ?
Answer:
When excess of aq. KCN is mixed with aq.CuSO4 a complex named potassium tetra cyano cuprate (II) is formal
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 4

Question 62.
[Cr(NH3)6]3+ is paramagnetic while [Ni(CN)4]2- is diamagnetic Why ?
Answer:
[Cr(NH3)6]3+ is paramagnetic due to the presence of three unpaired electrons.
[Ni(CN)4]2- is diamagnetic due to the absence of unpaired electrons.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 63.
A solution of [Ni(H2O)6]2+ is green but a solution of [Ni(CN)4]2- is colourless. Why ?
Answer:

  • In the complex [Ni(H2O)6]2+, H2O molecules are weak ligands they do not cause pairing.
    So the complex has two unpaired electrons, d-d-transitions takes place due to absorption of red light radiation and emission of green colour occurs.
  • In the complex [Ni(CN)4]2-, CN ions are strong ligands they cause pairing. So there is no unpaired electrons so, no d-d transition. Hence it is colourless.

Question 64.
[Fe(CN)4]2- and [Fe(H2O)6]2+ are of different colours in dilute solutions. Why ?
Answer:
In the given complexes Fe has + 2 oxidation state with 3d6 outer electronic configuration. It has four unpaired electrons in presence of weak ligand H2O. But in presence of strong ligand CN the electrons are paired up. Due to the difference in the no. of unpaired electrons both complex have different colours in dilute solutions.

Question 65.
What is the oxidation state of cobalt in (i) K[Co(CO)4] and (ii) [Co(NH3)6]3+ ?
Answer:

  1. K[CO(CO)4] : 1 + x + 4(0) = 0, x = -1 .
  2. [CO(NH3)6]3+ : x + 6(0) = + 3, x = + 3

Short Answer Questions

Question 1.
Compared to 3d series the corresponding transition metals of 4d and 5d transition series show high enthalpy of atomization. Explain.
Answer:
Compared to 3d-series the corresponding transition metals of 4d and 5d transition series show high enthalpy of atomisation.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 5
Reason : This is due to the occurrence of much more frequent metal-metal bonding in compounds of heavy transition metals.

Question 2.
Compared to the changes in atomic and ionic sizes of elements of 3d and 4d series, the change in radii of elements of 4d and 5d series is virtually the same Comment.
Answer:
Compared to the change in atomic and ionic sizes of elements 3d and 4d series, the change in radii of elements 4d and 5d series is virtually the same.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 6
This is due to the intervention of the 4f orbitals which must be filled before the 5d series of elements begin. The filling of 4f before 5d orbitals results in a regular decrease in atomic radii called lanthanoid contraction. The consequence of lanthanoid contraction is that the 4d and 5d series exhibit same size of radii.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 3.
Account for the zero oxidation state of Ni and Fe in [Ni(CO)4] and [Fe(CO)5] respectively.
Answer:
In [Ni(CO)4] and [Fe(CO)5] the oxidation state of Ni and Fe is zero.
These low oxidation states found when the complex compound has ligands capable of π-acceptor character in addition to the σ-bonding.

Question 4.
Why do the transition metal ions exhibit characteristic colours in aqueous solution. Explain giving examples.
Answer:
Colour property of transition metal ions in aqueous solution is due to the presence of unpaired d-electrons. These unpaired d-electrons from a lower energy level of a metal ion in a complex is excited to a higher energy d-orbital of the same n value. The energy of excitation corresponds to the frequency of light absorbed and this frequency lies in visible region. The colour observed corresponds to the complementary colour of light obsorbed. The frequency of light absorbed is determined by the nature of the ligand. In aqueous solutions water molecules are ligands the colour of the ions observed are listed in the following table.
Colours of Some of the First Row (aquated) Transition Metal Ions
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 7

Question 5.
Explain the catalytic action of Iron(DI) in the reaction between I arid S2O82- ions.
Answer:
Transition metal ions can change their oxidation states and become more effective catalysts. Iron (III) Catalyses the reaction between iodide and per sulphate ions
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 8
Catalytic action explained as follows
2Fe+3 + 2I → 2Fe+2 + I2
2Fe+2 + S2O8-2 → 2Fe+3 + 2SO4-2

Question 6.
What are interstitial compounds ? How are they formed ? Give two examples.
Answer:
The compounds which are formed when small atoms like H, C or N are trapped inside the crystal lattices of metal are called interstitial compounds.
Eg : TiC, Fe3H, VH0.56 etc.

  • These are non stoichiometric and are neither typically ionic nor covalent.
  • They have high melting points, higher than of pure metals.
  • They are very hard, some borides approach diamond in hardness.
  • They retain metallic conductivity.
  • They are chemically inert.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 7.
Write the characteristics of interstitial compounds.
Answer:
Characteristics of interstitial compounds :

  • These are non stoichiometric and are neither typically ionic nor covalent.
  • They have high melting points, higher than of pure metals.
  • They are very hard, some borides approach diamond in hardness.
  • They retain metallic conductivity.
  • They are chemically inert.

Question 8.
Write the characteristic properties of transition elements. [A.P. Mar. 15]
Answer:
Transition elements exhibits typical characteristic properties.

  • Electronic configurations.
  • Para and ferro magnetic properties.
  • Alloy forming ability
  • Complex forming ability.
  • Interstitial compounds.
  • Variable oxidation states.
  • Formation of coloured hydrated ions.
  • Catalytic property.
  • Metallic character.

Question 9.
Write down the electronic configuration of ‘

  1. Cr3+
  2. Cu+
  3. Co2+
  4. Mn2+

Answer:

  1. Cr3+ electronic configuration is [Ar] 4s03d3
  2. Cu+ electronic configuration is [Ar] 4s03d10
  3. Co2+ electronic configuration is [Ar] 4s03d7
  4. Mn2+ electronic configuration is [Ar] 4s03d5

Question 10.
What may be the stable oxidation state of the transition element with the following d electron configurations in the ground state of their atoms : 3d3 3d5 3d8 and 3d4 ?
Answer:

  1. 3d3-stable oxidation states are +2,+3,+4 and+5 (V)
  2. 3d5-stable oxidation states are +3, +4 and +6 (Cr)
  3. 3d5-stable oxidation states are +2, +4, +6 and +7 (Mn)
  4. 3d8-stable oxidation states are +2, +3 (Co)
  5. 3d4-This configuration does not exist.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 11.
What is lanthanoid contraction ? What are the consequences of lanthanoid contraction?
Answer:
Lanthanoid contraction : The overall decrease in atomic and ionic radii from lanthanum to leutetium is observed in the lanthanoids. This phenomenon is called lanthanoid contraction. It is due to the fact that with every additional proton in the nucleus, the corresponding electron goes into a 4f-subshell. This is too diffused to screen the nucleus as effectively as the more localised inner shell. Hence, the attraction of the nucleus for the outermost electrons increases steadily with the atomic number.

Consequences of Lanthanoid Contraction : The important consequences of lanthanoid contraction are as follows :
i) Basic character of oxides and hydroxides : Due to the lanthanoid contraction, the covalent nature of La-OH bond increases and thus, the basic character of oxides and hydroxides decreases from La(OH)3 to Lu(OH)3.
ii) Similarity in the size of elements of second and third transition series : Because of
lanthanoid contraction, elements which follow the third transition series are considerably smaller than would otherwise be expected. The normal size increases from Sc → Y → La and disappears after lanthanides. Thus, pairs of elements such as Zr/Hf, Nb/Ta and Mo/w are almost identical in size.
Due to almost similar size, such pairs have very sijnilar properties which makes their separation difficult.

iii) Separation of lathanoids : Due to lanthanoid contraction, there is a difference in some properties of lanthanoid like solubility, degree of hydration and complex formation. These difference enable the separation of lanthanoids by ion exchange method.

Question 12.
How is the variability in oxidation states of transition metals different from that of the non transition metals ? Illustrate with examples.
Answer:

  • In transition elements the oxidation states vary by unity (due to incomplete filling of d- orbitals)
    Eg : Mn exhibits +2, +3, +4, +5, +6 and +7 all differing by 1.
  • In non-transition element, this variation is selective, always differing by 2.
    Eg : S exhibits 2, 4, 6 oxidation states. N exhibits 3, 5 etc.

Question 13.
Describe the preparation of potassium dichromate from iron chromite ore.
Answer:
Preparation of K2Cr2O7 from chromite ore :
i) a) Potassium dichromate is obtained by the fusion of chromite ore (FeCr2O4) with sodium (or potassium) carbonate in the presence of excess of air.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 9
b) The solution is filtered and treated with sulphuric acid.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 10
c) Now sodium dichromate is treated with potassium chloride. As a result, potassium dichromate is produced.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 11

ii) Effect of increasing pH on K2Cr2O7 solution :
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 12
On increasing pH, K2Cr2O7 changes into K2CrO4 (orange to yellow).

Question 14.
Describe the oxidising action of potassium dichromate and write the ionic equations for its.
With (i) iodide (ii) iron (II) solution (iii) H2S and (iv) Sn(II)
Answer:
Potassium dichromate is a strong oxidising agent. In acidic solution, its oxidising action may be represented as :
Cr2O72- + 14H+ + 6e → 2Cr3+ + 7H2O (E° = 1.33 V)
Ionic reactions :

  1. Reaction of K2Cr2O7 with I
    Cr2O72- + 14H+ + 6I → 2Cr3+ + 3I2 + 7H2O
  2. Reaction of K2Cr2O7 with Fe2+ (aq)
    Cr2O72- + 14H+ + 6Fe2+ → 2Cr3+ + 6Fe3+ + 7H2O
  3. Reaction of K2Cr2O7with H2S
    Cr2O72- + 8H+ + 3H2S → 2Cr+3 + 3S + 7H2O

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 15.
Describe the preparation of potassium permanganate.
Answer:
Preparation of KMnO4 Potassium permanganate is prepared by the fusion of MnO2 with an alkali metal hydroxide and an oxidising agent like KNO3. It forms dark green, K2MnO4 which disproportionates in a neutral or acidic solution to give permanganate.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 13

Question 16.
How does the acidified permanganate solution react with

  1. iron (II) ions
  2. SO2 and
  3. oxalic acid.

Write the ionic equations for the reactions.

Answer:
Reactions of KMnO4 in acidic medium
MnO4 + 8H+ + 5e → Mn2+ + 4H2O …………. (i)
1) Iron (II) ions : Ferrous is oxidised to ferric.
[Fe2+ → Fe3+ + e] × 5 ………………. (ii)
From equation (i) and (ii) 5Fe2+ + MnO4 + 8H+ → Mn2+ + 4H2O + 5Fe3+

2) SO2 : It is oxidised to SO42- by acidified KMnO4.
SO2 + 2H2O → SO42- + 4H+ + 2e ……………….(iii)
From equation (i) and (iii) .
5SO2 + 2MnO4 + 2H2O → 2Mn2+ + 4H+ + 5SO42-

3) Oxalic acid : C2O42- → 2CO2 + 2e ……………… (iv)
From equation (1) and (iv)
5C2O42- + 2MnO4 + 16H+ → 2Mn2+ + 8H2O + 10CO2

Question 17.
Predict which of the ions Cu+, Sc3+, Mn2+, Fe2+ are coloured in aqueous solution ? Give reasons.
Answer:
Only those ions will be coloured which have incomplete d-orbitals. Ions which has complete or vacant d-orbitals are colourless.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 14
As Sc3+ and Cu+ have 3d0 and 3d10 configuration in their valence shell so their aqueous solutions are colourless. All others, i.e., Ti3+, V3+, Mn2+ Fe+2 and Co2+ are coloured in aqueous medium.

Question 18.
Compare the stability of +2 oxidation state of the elements of the first transition series.
Answer:
Element (+2 state) 21Sc2+ Ti2+ 22 22V2+ 24Cr2+ 25Mn2+,
Electronic configuration 3d1 3d2 3d3 3d4 3d5
In all the elements listed, the removal of two 4s. Electrons (in Cr2+, i.e., from 4s and 1e from 3d), the 3d-orbitais get gradually occupied. Since, the number of empty d-orbitais decreases or the number unpaired electrons in 3d-orbitais increases with increase in atomic number of cations, so the stability of the cations (M2+) increases from Sc2+ to Mn2+.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 19.
Use Hund’s rule to derive the electronic configuration of Ce3+ ion and calculate its magnetic moment on the basis of ‘spin-only’ formula.
Answer:
Ce(Z = 58) = [Xe] = 4f1 5d1 6s2
Ce3+ = [Xe]4f1 (only one unpaired electron)
By ‘spin-only’ formula, Magentic moment of Ce3+(μ) = \(\sqrt{n(n+2)}\)
[∵ n = 1 (unpaired electron)] = \(\sqrt{1(1+2)}\) = \(\sqrt{3}\) = 1.73 BM

Question 20.
Write down the number of 3d electrons in each of the following ions : Ti2+, V2+ Cr3+ and Mn2+ Indicate how would you expect the five 3d orbitais to be occupied for these hydrated ions (octahedral).
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 15

Question 21.
Explain Werner’s theory of coordination compounds with suitable examples. [A.P. Mar. 18; T.S. Mar. 18, 15] [Mar. 14]
Answer:
Werner’s theory:
Postulates :
1) Every complex compound has a central metal atom (or) ion.

2) The central metal shows two types of valencies namely primary valency and secondary valency.

A) Primary valency: The primary valency is numerically equal to the oxidation state of the
metal. Species or groups bound by primary valencies undergo complete ionization. These valencies are identical with ionic bonds and are non-directional. These valencies are represented by discontinuous lines (…..)
Eg- : CoCl3 contains Co3+ and 3Cl ions. There are three Primary Valencies or three ionic bonds.

B) Secondary Valency : Each’ metal has a characteristic number of Secondary Valencies. They are directed in space around the central metal.
The number of Secondary Valencies is called Coordination numbe (C.N.) of the metal. These valencies are directional in Nature.
For example in CoCl3. 6NH3
Three Cl ions are held by primary Valencies and 6NH3 molecules are held by Secondary Valencies. In CuSO4 . 4NH3 complex SO42- ion is held by two Primary Valencies and 4NH3 molecules are held by Secondary Valencies.

3) Some negative ligands, depending upon the complex, may satisfy both primary and secondary valencies. Such ligands, in a complex, which satisfy both primary as well as secondary valencies do not ionize.

4) The primary valency of a metal is known as its outer sphere of attraction or ionizable valency while the Secondary valencies are known as the inner sphere of attraction or coordination Sphere. Groups bound by secondary valencies do not undergo ionization in the complex.

Example to Clarify Werner’s Theory
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 16
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 17

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 22.
Give the geometrical shapes of the following complex entities

  1. [Co(NH3)6]3+
  2. [Ni(CO)4]
  3. [Pt Cl4]2- and
  4.  [Fe(CN)8]4-.

Answer:

  1. Geometrical shape of [Co(NH3)6]3+ is octahedral
  2. Geometrical shape of [Ni(CO)4] is tetrahedral
  3. Geometrical shape of [Pt Cl4]2- is square planar
  4. Geometrical shape of [Fe(CN)8]4- is octahedral

Question 23.
Explain the terms
(i) Ligand
(ii) Coordination number
(iii) Coordination entity
(iv) Central metal atom/ion.
Answer:
i) Ligand : The ions or molecules bound to the central atom/ion in the coordination entity are called ligands. These may be
a) simple ions such as Cl
b) small molecules such as H2O or NH3
c) large molecules such as H2NCH2CH2NH2 or N(CH2CH2NH2)3 or even (d) macromolecules, such as proteins.
On the basis of the number of donor atoms available for coordination, the ligands can be classified as :
a) Unidentate : One donor atom, Eg.: AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 18
b) Bidentate : Two donor atoms, Eg. : H2NCH2CH2NH2
(ethane-1, 2-diamine or en), C2O42- (oxalate), etc.
c) Polydentate : More than two donor atoms, Eg. : N(CH2CH2NH2)3 EDTA, etc.

ii) Coordination number: The coordination number (CN) of metal ion in a complex can be defined as the number of ligands or donor atoms to which the metal is directly bonded.
Eg : In [PtCl6]2-, CN of Pt = 6, In [Ni(NH3/sub>)4]2+, CN of Ni 4.

iii) Coordination entity: A central metal atoms or ion bonded to a fixed number of molecules or ions (ligands) is known as coordination entity. For example [CoCl3(NH3)3. Ni(CO)4/sub>], etc.

iv) Central metal atom/ion : In a coordination entity, the atom/ion to which a fixed no. of ions/groups are bound in a definite geometrical arrangement around it is called central metal atom or ion. Eg: K4[Fe(CN)6] ‘Fe’ is central metal.

Question 24.
Explain the terms (i) Unidentate ligand (ii) bidentate ligand (iii) polydentate ligand and (iv) ambidentate ligand giving one example for each.
Answer:
i) Unidentate : The negative ion or neutral molecule having only one donor atom is called unidentate ligand
Eg : AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 19
ii) Bidentate (or didentate) : The ions or molecules having two donor atoms are called bidentate ligands.
Eg: AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 20
iii) Polydentate ligands : Ligands having more than one donor atom in the coordinating group and are capable of forming two or more coordinate bonds with same central atom simultaneously are called poly dentate ligands.
Eg : C2O4-2.
iv) Ambidentate: Ligand which can ligate through two different atoms is called ambidentate ligand. Eg: NO2, SCNions, NO2 ion can coordinate either through nitrogen or through oxygen to a central metal atom/ion. Similarly, SCN ion can coordinate through the sulphur or nitrogen atom.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 21

Question 25.
What is meant by chelate effect ? Give example.
Answer:
When a didentate or a polydentate ligand contains donor atoms positioned in such a way that when they coordinate with the central metal ion, a 5-or 6-membered ring is formed, the effect is known as chelate effect. Example
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 22

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 26.
Give the oxidation numbers of the central metal atoms in the following complex entities
(i) [Ni(CO)4]
(ii) [CO(NH3)6]3+
(iii) [Fe(CN)6]4- and
(iv) [Fe(C2O4)3]3-
Answer:
i) [Ni(CO)4] :
x + 4(0) = 0
x = 0
‘Ni’ oxidation state is zero.

ii) [CO(NH3)6]3+
x + 6(0) = + 3
x = +3
‘Co’ oxidation state is + 3.

iii) [Fe(CN)6]4-
x + 6(-1) = -4
x = + 2
‘Fe’ oxidation state is + 2.

iv) [Fe(C2O4)3]3-
x + 3(-2) = -3
x = + 3
‘Fe’ oxidation state is + 3.

Question 27.
Using IUPAC norms write the formulas for the following.

  1. Tetrahydroxozincate (II)
  2. Hexaamminecobalt (III) sulphate
  3. Potassium tetrachloropalladate (II) and
  4. Potassium tri(oxalato) chromate (III)

Answer:

  1. Tetrahydroxozincate (II) – [Zn(OH)4]-2
  2. Hexa ammine cobalt (III) sulphate – [Co(NH3)6]2 (SO4)3
  3. Potassium tetrachloropalladate – K2[PdCl4]
  4. Potassium tri(oxalato) chromate (III) – K3[Cr(C2O4)3]

Question 28.
Using IUPAC norms write the systematic names of the following. [A.P. Mar 19]

  1. [CO(NH3)6]Cl3
  2. [Pt(NH3)2Cl(NH2CH3)]Cl
  3. [Ti(H2O)6]3+ and
  4. [NiCl4]2-

Answer:

  1. Hexa ammine cobalt (III) chloride
  2. Diammine chlorido (methyl amine) platinum (II) chloride
  3. Hexa aquo titanium (III) ion
  4. Tetra chloro nickelate (III) ion

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 29.
Explain geometrical isomerism in Co-ordination compounds giving suitable examples. [A.P. Mar 19]
Answer:

  • Geometrical isomerism arises in Co-ordination complexes due to different possible geometric arrangements of the ligands.
  • This isomerism found in complexes with Co-ordination numbers 4 and 6.
  • In a square planar complex of formula [MX2L2] the two ligands may be arranged in adjacent to each other in a cis isomer (or) opposite to each other in a trans isomer.
    AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 23
  • Square planar complex of type [MAB XL] shows three isomers two-cis and one trans. (A, B, X, L are unidentate ligands is square planar complex).
  • Geometrical isomerism is not possible in tetrahedral geometry.
  • In octahedral complexes of formula [MX2L4] in which two ligands X may be oriented cis or trans to each other.
    AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 24
  • Another type of geometrical isomerism occurs in octahedral Co-ordination compounds of type [Ma3b3] if three donar atoms of the same ligands occupy adjacent positions at the corners of an octahedral face then it is facial (fac) isomer. When the positions occupied are around the meridian of the octahedran then it is meridonial (mer). isomer.
    AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 25

Question 30.
What are homoleptic and heteroleptic complexes ? Give one example for each.
Answer:
Homoleptic complexes : These are the complexes in which a metal is bound by only ore kind of ligands.
eg.: [Co(NH3)6]3+
Heteroleptic complexes: These are the complexes in which a metal is bound by more than one kind of ligends. eg.: [Co(NH3)4Cl2]+

Long Answer Questions

Question 1.
Explain giving reasons :
(i) Transition metals and many of their compounds show paramagnetic behaviour.
(ii) The enthalpies of atomisation of the transition metals are high.
(iii) The transition metals generally form coloured compounds.
(iv) Transition metals and their many compounds act as good catalysts.
Answer:
i) Transition elements have unpaired electrons. Each unpaired electron has a magnetic moment associated with its spin angular momentum and orbital angular momentum. This is the reason of paramagnetism in transition metals.

ii) The reason for the high enthalpy of atomisation is the presence of large number of unpaired electrons in their atoms. These atoms have strong interatomic interaction and hence, stronger bonding between them.

iii) Formation of coloured compounds by transition metals is due to partial adsorption of visible light. The electron absorbs the radiation of a particular frequency (of visible region) and jumps into next orbital.

iv) Catalysts, at the solid surface, involve the formation of bonds between reactants molecules and atoms of the surface of the catalyst (I row transition metals utilise 3d and 48- electrons for bonding). This has the effect of increasing the concentration of the reactants at the catalyst surface and also lowering of the activation energy.
Transition metal ions show variable oxidation states, so they are effective catalysts.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 26

Question 2.
Describe the preparation of potassium permanganate. How does the acidified permanganate solution react with
(i) iron (II) ions
(ii) SO2 and
(iii) oxalic acid ?
Write the ionic equations.
Answer:
Preparation of KMnO4 Potassium permanganate is prepared by the fusion of MnO2 with an alkali metal hydroxide and an oxidising agent like KNO3. It forms dark green, K2MnO4 which disproportionates in a neutral or acidic solution to give permanganate.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 27
Reactions of KMnO4 in acidic medium
MnO4 + 8H+ + 5e → Mn2+ + 4H2O …………. (i)
i) Iron (II) ions : Ferrous is oxidised to ferric.
[Fe2+ → Fe3+ + e] × 5 ………………. (ii)
From equation (i) and (ii) 5Fe2+ + MnO4 + 8H+ → Mn2+ + 4H2O + 5Fe3+

ii) SO2 : It is oxidised to SO42- by acidified KMnO4.
SO2 + 2H2O → SO42- + 4H+ + 2e ……………….(iii)
From equation (i) and (iii) .
5SO2 + 2MnO4 + 2H2O → 2Mn2+ + 4H+ + 5SO42-

iii) Oxalic acid : C2O42- → 2CO2 + 2e ……………… (iv)
From equation (1) and (iv)
5C2O42- + 2MnO4 + 16H+ → 2Mn2+ + 8H2O + 10CO2

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 3.
Compare the chemistiy of actinoids with that of the lanthanoids with special reference to :
(i) electronic configuration (ii) oxidation state (iii) atomic and ionic sizes and (iv) chemical reactivity.
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 28
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 29

Question 4.
How would you account for the following.
(i) Of the d4 species, Cr2+ is strongly reducing while manganese (HI) is strongly oxidising.
(ii) Cobalt (II) is stable in aqueous solution but in the presence of complexing reagents it is easily oxidised,
(iii) The d1 configuration is very unstable in ions.
Answer:
i) E° value of Cr3+/Cr2+ is negative (-0.41 V) while that of Mn3+/Mn2+ is positive (+1.57 V). This means that Cr2+ ions can lose electrons to form Cr3+ ions and act as a reducing agent while Mn3+ ions can accept electrons and can act as an oxidising agent.

ii) Cobalt (III) ion has greater tendency to form complexes than cobalt (II) ion. Therefore, Co (II) ion, being stable in aqueous solution, changes to Co (III) ion, in the presence of complexing reagents and gets oxidised.

iii) Ions of transition metals with d1 configuration tend to lose one electron to acquire d0 configuration that is quite stable. Therefore, such ions (with d1) undergo either oxidation or disproportionation hence unstable.

Question 5.
Give examples and suggest reasons for the following features of the transition medals.
(i) The lowest oxide of transition metal is basic, the highest is amphoteric/acidic.
(ii) A transition metal exhibits highest oxidation state in oxides and fluorides.
(iii) The highest oxidation state is exhibited in oxoanions of a metal.
Answer:
i) Acidic strength of oxides increases with the increase in oxidation state of the element Eg.: MnO(Mn2+) is basic whereas Mn2O7(Mn7+) is acidic in nature.

ii) Both oxygen and fluorine being highly electronegative can increase the oxidation state of a particular transition metal. In certain oxides, the element oxygen is involved in multiple bonding with the metal .and this is responsible for the higher oxidation state of the metal.

iii) This is also due to high electronegativity of oxygen eg.: chromium exhibits oxidation states of +6 in oxoanion [CrO4]2- and manganese shows oxidation state of +7 in oxoanion [MnO4]

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 6.
Compare the chemistry of the actinoids with that of lanthanoids with reference to : (i) electronic configuration (ii) oxidation states and (iii) chemical reactivity
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 30

Question 6.
Explain IUPAC nomenclature of Co-ordination compounds with suitable examples.
Answer:
IUPAC nomenclature : The formula of a compound is an abreviated description of the constitution of the compound. The fpllowing rules are prescribed by IUPAC for naming of Coordination compounds.
i) Positive ions are named first followed by negative ions,
eg.: Potassium hexacyanoferrate (II), K4[Fe(CN)6]

ii) Within the Co-ordination sphere ligands are named before the metal atom/ion. However, in formulae, metal ion is written first.
eg.: Tetraammine copper (II) sulphate [Cu(NH3)4]SO4

iii) Prefixes are used to denote the number of same ligands that the present in the Co¬ordination sphere. Complex ions are denoted in paranthesis ( ) and prefixed by bis, tris etc.
Examples:
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 31
eg. : [CO(NH2CH2CH2NH2)Cl2] Cl is named as dichloro bis (ethylenediamine) cobalt (III) chloride.

iv) Ligands are named in alphabetical order.
eg.: [PtCl2(NH3)2 diammine dichloro platinum (II)

v) Anionic ligands are denoted by a suffix ‘O’ and neutral ligands are denoted by their original names.
eg.: Cl – Chloro, CN – Cyano
Exception for the above are indicated below.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 32
vi) Oxidation state of the metal ion is indicated by Roman numerical in parenthesis, eg.: [Ag(NH3)2] [Ag(CN)2] is named as diammine silver (I) dicyanoargentate (I).

vii) If the charge of the Co-ordination entity is negative, the name of the metal ends with a suffix-ate.
eg.: [CO(SCN)4]2- – tetrahiocyanato cobaltate (II)
Some metal ions are denoted by their names from which their symbols are derived
eg.: Fe – ferrate
Pb – plumbate
Sn – stannate .
Ag-argentate
Au-aurate

viii) Prefixes cis – and trans are used to designate adjacent and opposite geometric locations of the ligands, in a complex.
eg. :
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 33
ix) Bridging ligands between two metal ions in a Co-ordination entity are denoted by prefix p(greek letter’mu).
eg.: [(NH3)4 CO(OH) (NH2)Co(NH3)4]+ is named as μ-amido-μ hydroxo bis (tetraammine) cobalt (IV) .

  1. Tetrahydroxozincate (II) – [Zn(OH4]-2
  2. Hexa ammine cobalt (III) sulphate – [Co(NH3)6]2 (SO4)3
  3. Potassium tetrachloropalladate – K2[PdCl4)
  4. Potassium tri(oxalato) chromate (III) – K3[Cr(C2O4)3]

Question 7.
Explain different types of isomerism exhibited by Co-ordination compounds, giving suitable examples.
Answer:
Isomerism in Co-ordination compounds : Isomers are compounds that have the same chemical formula but different arrangement of atoms. Two principal types of isomerism are known among Co-ordination compounds namely stereo isomerism and structural isomerism.
a) Stereoisomengni: Stereoisomerism is a form of isomerism in which two substances have the same composition and structure but differ in the relative spatial positions of the ligands. This can be sub divided into two classes namely.

  1. Geomethcal isomerism and
  2. optical isomerism

b) Structural isomerism:

  1. Linkage isomerism
  2. Co-ordination isomerism
  3. Ionisation isomerism
  4. Hydrate isomerism

a) (1) Geometrical isomerism:

  • Geometrical isomerism arises in Co-ordination complexes due to different possible geometric arrangements of the ligands.
  • This isomerism found in complexes with Co-ordination numbers 4 and 6.
  • In a square planar complex of formula [MX2L2] the two ligands may be arranged in adjacent to each other in a cis isomer (or) opposite to each other in a trans isomer.
    AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 23
  • Square planar complex of type [MAB XL] shows three isomers two-cis and one trans. (A, B, X, L are unidentate ligands is square planar complex).
  • Geometrical isomerism is not possible in tetrahedral geometry.
  • In octahedral complexes of formula [MX2L4] in which two ligands X may be oriented cis or trans to each other.
    AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 24
  • Another type of geometrical isomerism occurs in octahedral Co-ordination compounds of type [Ma3b3] if three donar atoms of the same ligands occupy adjacent positions at the corners of an octahedral face then it is facial (fac) isomer. When the positions occupied are around the meridian of the octahedran then it is meridonial (mer). isomer.
    AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 25

a(2) Optical isomerism : Optical isomerism arises when two isomers of a compound exist such that one isomer is a mirror image of the other isomer. Such isomers are called optical isomers or enantiomers. The molecules or ions that cannot be superimposed are called chiral. The two forms are called dextro (d) and laevo (1) depending upon the direction they rotate the plane of polarised light in a polarimeter (d rotates to the right, l to the left). Optical isomerism is common in octahedral complexes involving bidentate ligands.

b(1) Linkage isomerism: Linkage isomerism arises in Co-ordination compound containing ambidentate ligand. A simple example is provided by complexes containing the thiocyanate ligand-NCS, which may bind through the nitrogen to give M-NCS or through sulphur to give M-SCN.
eg. : [Mn(CO)5SCN] and [Mn(CO)5NCS]

(2) Co-ordinate isomerism : This type isomerism arises from the interchange of ligands between cationic and anionic entities of different metal ions present in a complex.
eg. : [CO(NH3)6] [Cr(CN)6] and [Co(CN)6] [Cr(NH3)6]

(3) Ionisation isomerism: This form of isomerism arises when the counter ion in a complex salt is itself a potential ligand and can displace a ligand which can then become the counter ion
eg. : [CO(NH3)5SO4] Br and [Co(NH3)5Br]SO4

(4) Hydrate isomerism : This form of isomerism is known as ‘hydrate isomerism since water is involved as a solvent. Hydrate isomers differ by whether or not a hydrate molecule is directly bonded to the metal ion or merely present as free solvent molecules in the crystal lattice.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 8.
Discuss the nature of bonding and magnetic behaviour in the following Co-ordination entities on the basis of valence bond theory.
(i) [Fe(CN6)]4-
(ii) [FeF6]3-
(iii) [Co(C2O4)3]3- and
(iv) [CoF6]3-
Answer:
i) Fe(CN6)]4- : In this complex Fe is present as Fe2+.
Fe = [Ar] 3d64s2
Outer configuration of Fe2+ = 3d64s0
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 34
CN being strong field ligand, pair up the unpaired d electrons Thus, two 3d-orbital are now available for CN ions.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 35
Since, all the electrons are paired, the complex is diamagnetic. Moreover (n – 1) d- orbitals are involved in bonding, so, it is an inner orbital or low spin complex.

ii) [FeF6]3- : In this complex, the oxidation state of Fe is + 3.
Fe3+ = 3d54s0
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 36
F- is not a strong field ligand. It is a weak field ligand, so no pairing occurs. Thus, 3d- orbitals are not available to take part in bonding.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 37
Because of the presence of five unpaired electrons, the complex is paramagnetic. Moreover, nd-orbitals are involved in bonding, so it is an outer orbital or high spin complex.

iii) [Co(C2O4)3]3- : In this complex, the oxidation state of Co is + 3.
Outer configuration of Co = 3d7 4s2
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 38
Oxalate ion being a strong field ligand pair up the 3d electrons, thus two out of the five 3d-orbitals are available for oxalate ions.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 39
Since, all the electrons are paired, this complex is diamagnetic. It is an inner orbital complex because of the involvement of (n – 1) d-orbital for bonding.

iv) [CoF6]3- : In this complex, Co is present as Co3+.
Outer configuration of Co3+ = 3d6 4s0
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 40
Because of the presence of four unpaired electrons, the complex is paramagnetic. Since, nd orbitals take part in bonding, it is an outer orbital complex or high spin complex.

Question 9.
Sketch the spliting of d orbitals in an octahedral crystal field.
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 41

Question 10.
What is spectrochemical series ? Explain the difference between a weak field ligand and a strong field ligand.
Answer:
The arrangement of ligands in order of their increasing field strengths, i.e., increasing crystal field splitting energy (CFSE) values is called spectrochemical series.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 42
The ligands with a small value of CFSE (∆0) are called weak field ligands. For such ligands ∆0 < P where P is the pairing energy. Whereas the ligands with a large value of CFSE are called strong field ligands. In case of such ligands ∆0 > P.

When ligands approach a transition metal ion, the d-orbitals split into two sets (t2g and eg), one with lower energy and the other with higher energy. The difference of energy between the two sets of orbitals is called crystal field splitting energy, ∆0 for octahedral field (and ∆t for tetrahedral field)

If ∆0 < P (pairing energy), the 4th electron enters one of the eg orbitals giving the configuration \(t_{2 g}^{3} e_{g}^{1}\) thereby forming high spin complex. Such ligands for which ∆0 < P are known as weak field ligands.

If ∆0 > P, the 4th electron pairs up in one of the t2g orbitals giving the configuration \(\mathrm{t}_{2 \mathrm{~g}}^{4} \mathrm{e}_{\mathrm{g}}^{0}\), thus forming low spin complexes. Such ligands for which ∆0 > P are called strong field ligands.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 11.
Discuss the nature of bonding in metal carbonyls.
Answer:
The metal-carbon bond in metal carbonyls have both sigma (σ) and pi (π) characters. The metal-carbon σ-bond is formed by the donation of lone pair of electrons of the carbonyl carbon to a vacant orbital of the metal. The metal- carbon re-bond is formed by the donation of a pair of electrons from a filled d-orbital of metal into the vacant
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 43

Question 12.
Explain the applications of Co-ordination compounds in different fields.
Answer:
Applications of Co-ordination compounds : Co-ordination compounds are of great importance. These compounds are widely present in the mineral, plant and animal worlds and are known to play many important functions in the area of analytical chemistry, metallurgy, biological systems, industry and medicine. These are described below.

Co-ordination compounds find use in many qualitative and quantitative chemical analysis. The familiar colour reactions given by metal ions with a number of ligands (especially chelating ligands), as a result of formation of co-ordination enitites, form the basis for their detection and estimation by classical and instrumental methods of analysis. Examples of such reagents include EDTA, DMG (dimethylglyoxime), oc-nitroso-p-naphtol, cupron etc.

Hardness of water is estimated by simple titration with Na2EDTA. The Ca2+ and Mg2+ ions form stable complexes with EDTA. The selective estimation of these ions can be done due to difference in the stability constants of calcium and magnesium complexes.

Some important extraction processes of metals, like those of silver and gold, make use of complex formation. Gold for example, combines with cyanide in the presence of oxygen and water to form the co-ordination entity [Au(CN)2] in aqueous solution. Gold can be separated in metallic from this solution by the addition of zinc.

Similarly, purification of metals can be achieved through formation and subsequent decomposition of their coordination compounds. For example, impure nickel is converted to [Ni(CO)4], which is decomposed to yield pure nickel.

Co-ordination compounds are of great importance in biological systems. The pigment responsible for photosynthesis, chlorophyll, is a co-ordination compound of magnesium. Haemoglobin, the red pigment of blood which acts as oxygen carrier is a co-ordination compound of iron. Vitamin B12, cyanocobalamine, the anti-pernicious anaemia factor, is a coordination compound of cobalt. Among the other compounds of biological importance with coordinated metal ions are the enzymes like, carboxypeptidase A and carbonic anhydrase (catalysts of biological systems).

Co-ordination compounds are used as catalysts for many industrial processes. Examples include rhodium complex, [(Ph3P)3RhCl], a Wilkinson catalyst, is used for the hydrogenation of alkenes.

Articles can be electroplated with silver and gold much more smoothly and evenly fr6m solutions of the complexes, [Ag(CN)2] and. [Au(CN)2] than from a solution of simple metal ions.

In black and white photography, the developed film is fixed by washing with hypo solution which dissolves the undecomposed AgBr to form a complex ion, [Ag(S2O3)2]3-.

There is growing interest in the use of chelate therapy in medicinal chemisty. An example is the treatment of problems caused by the presence of metals in toxic proportions in plant/animal systems. Thus, excess of copper and iro are removed by the chelating ligands D-penicillamine and desferrioxime B via the formation of coordination compounds. EDTA is used in the treatment of lead poisoning. Some co-ordination compounds of platinum effectively inhibit the growth of tumours. Examples are : cis-platin and related compounds.

Textual Examples

Question 1.
On what ground can you say that scandium (Z = 21) is a transition element but zinc (Z = 30) is not ?
Solution:
On the basis of incompletely filled 3d orbitals in case of scandium atom in its atomic state (3d1), it is regarded as a transition element. On the other hand, zinc atom has completely filled d orbitals (3d10) in its ground state as well as in its oxidised state, hence it is not regarded as a transition element.

Question 2.
Why do the transition elements exhibit higher enthapies of atomisation ?
Solution:
Because of large number of unpaired electrons in their atoms they have stronger interatomic interaction and hence stronger bonding between atoms resulting in higher enthalpies of atomisation.

Question 3.
Name at transition element which does not exhibit variable oxidation states.
Solution:
Scandium (Z = 21) does not exhibit variable oxidation states.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 4.
Why is Cr2+ reducing and Mn3+ oxidising when both have d4 configuration ?
Solution:
Cr22+ is reducing as its configuration changes from d4 to d3, the latter having a half-filled t2g level. On the other hand; the change from Mn2+ to Mn3+ results in the half-filled (d<sup5) configuration which has extra stability.

Question 5.
How would you account for the increasing oxidising power in the series VO2+ < Cr2O72- < MnO4?
Solution:
This is due to the increasing stability of the lower species to which they are reduced.

Question 6.
For the first row transition metals the E values are
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 44
Explain the non-regularity in the above values.
Solution:
The E (M2+/M) values are not regular which can be explained from the non-regular variation of ionisation enthalpies (∆iH1 + ∆iH2) and also the sublimation enthalpies which are relatively much less for manganese and vanadium.

Question 7.
Why is the E value for the Mn3+/Mn2+ couple much more positive than that for Cr3+/ Cr2+ or Fe3+/Fe2+ ? Explain.
Solution:
Much larger third ionisation energy of Mn (where the required change is d5 to d4) is mainly responsible for this. This also explains why the +3 state of Mn is of little importance.

Question 8.
Calculate the magnetic moment of a divalent ion in aqueous solution if its atomic number is 25. [A.P. Mar. 16]
Solution:
With atomic number 25, the divalent ion in aqueous solution will have d5 configuration (five unpaired electrons). The magnetic moment, μ is μ = \(\sqrt{5(5+2)}\) = 5.92 BM

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 9.
What is meant by ‘disproportionation’ of an oxidation state ? Give an example.
Solution:
When a particular oxidation state becomes less stable relative to other oxidation states, one lower, one higher, it is said to undergo disproportionation. For example, manganese (VI) becomes unstable relative to manganese (VII) and manganese (TV) in acidic solution.
3MnVIO42- + 4H+ → 2MnVIIO4 + MnIVO2 + H2O

Question 10.
Name a member of the lanthanoid series which is well known to exhibit +4 oxidation state.
Solution:
Cerium (Z = 58)

Question 11.
On the basis of the following observations made with aqueous solutions, assign secondary valencies to metals in the following compounds.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 45
Solution:
i) Secondary 4 ii) Secondary 6 iii) Secondary 6 iv) Secondary 6 v) Secondary 4

Question 12.
Write the formulas for the following Co-ordination compounds
a) Tetraammineaquachloro cobalt (III) chloride
b) Potassium tetrahydroxozincate (II)
c) Potassium trioxalatoaluminate (III)
d) Dichlorobis (ethane-1, 2-diamine) cobalt (III)
e) Tetracarbonylnickel (0)
Solution:
a) [Co(NH3)4(H2O)Cl]Cl2
b) K2[Zn(OH)4]
c) K3[Al(C2O4)3]
d) [CoCl2(en)2]+
e) [Ni(CO)4]

Question 13.
Write the IUPAC names of the following coordination compounds.
a) [Pt(NH3)2Cl(NO2)]
b) K3[Cr(C2O<sub4)3]
c) [CoCl2(en)2]Cl
d) [Co(NH3)5(CO3)]Cl
e) Hg[Co(SCN)4]
Solution:
a) Diamminechloronitrito-N-platinum (II)
b) Potassium trioxalatochromate (III)
c) Dichlorobis (ethane-1, 2-diamine) cobalt (III) chloride
d) Pentaairuninecarbonatocobalt (III) chloride
e) Mercury tetrathiocyanocobaltate (III)

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 14.
Why is geometrical isomerism not possible in tetrahedral complexes having two different types of unidentate ligands Co-ordinated with the central metal ion ?
Solution:
Tetrahedral complexes do not show geometrical isomerism because the relative positions of the unidentate ligands attached to the central metal atom are the same with respect to each other.

Question 15.
Draw structures of geometrical isomers of [Fe(NH3)2(CN)4]
Solution:
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 46

Question 16.
Out of the following two Co-ordination entities which is chiral (optically active) ?
a) cis-(CrCl2(ox)2]3-
b) trans-[CrCl2(ox)2]3-
Solution:
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 47
Out of the two, (a) cis-(CrCl2(ox)2]3- is chiral (optically active).

Question 17.
The spin only magnetic moment of [MnBr4]2- is 5.9 BM. Predict the geometry of the complexion?
Solution:
Since the Co-ordination number of Mn2+ ion in the complex ion is 4, it will be either tetrahedral (sp3 hybridisation) or square planar (dsp2 hybridisation). But the fact that the magnetic moment of the complex ion is 5.9 BM, it should be tetrahedral in shape rather than square planar because of the presence of five unpaired electrons in the d-orbitals.

Intext Questions

Question 1.
Silver atom has completely filled d-orbitals (4d10) in its ground State. How can you say that it is a transition element ?
Solution:
Silver (atomic no. = 47), in its +1 oxidation state exhibits 4d105s0 configuration. But in some compounds it also shows +2 oxidation state, i.e., 4d95s0 configuration. Here d-orbital is not completely filled. Therefore, silver is a transition element.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 2.
In the series Sc (Z = 21) to Zn (Z = 30), the enthalpy of atomisation of zinc is the lowest, i.e., 126 kJ mol-1, why ?
Solution:
In zinc (3d104s1) d-orbital is complete so the electrons of d-orbital are not involved in bonding. So, metallic bond is weaker than the other elements of the series, where the electrons of d-orbitals are involved in metallic bonds. So, enthalpy of atomisation of zinc is lowest in its transition series.

Question 3.
Which of the 3d series of the transition metals exhibits the largest number of oxidation states and why ?
Solution:
Manganese (atomic no. = 25) has electronic configuration [Ar] 3d54s2. It shows maximum number of oxidation states, i.e., from +2 to +7 (+2, +3, +4, +5, +6, +7) in its compounds.

Question 4.
The E°(M2+/M) value for copper is positive (+0.34 V). What is possibly the reason for this ?
Answer:
E° (M2+/M) value for any metal depends on three factors : •
i) ∆aH (Enthalpy of atomisation); M(s). + ∆aH → M(g)
ii) ∆iH (Enthalpy of ionisation); M(g) + ∆iH → M2+(g)
iii) ∆hydH (Hydration enthalpy); M2+(g) + (aq → M2+(aq)
Copper has high value of enthalpy of atomisation and low value of enthalpy of hydration. It means that ∆iH required is not compensated by the energy released. Therefore, E0(Cu2+/Cu) is positive.

Question 5.
How would you account for the irregular variation of ionisation enthalpies (first and second) in the first series of the transition elements ?
Solution:
In the first transition series, there is irregular variation of ionisation enthalpies because the stability of 3d configuration differs to some extent. Generally the ionisation enthalpy increases with increase in effective nuclear charge, Thus, it is lower for Cr due to absence of any change in the d-configuration while high for Zn as it represents an jonisation from the 4s level. Second ionisation energy, shows the removal of second. The configurations like d5 and d10 are exceptionally stable. So, they have high ionisation enthalpies.

Question 6.
Why is the highest oxidation state of a metal exhibited in its oxide or fluoride only ?
Solution:
Oxygen as well as fluorine both have high values of electronegativity. So, in their compounds (oxides and fluorides) they can oxidise the metal to their highest oxidation states.

Question 7.
Which is the stronger reducing agent Cr2+ or Fe2+ and why ?
Solution:
Cr2+ is a stronger reducing agent than Fe2+. The E° values are
Cr3+/Cr2+ = -0-41 V and E0Fe3+/Fe2+ = 0.77 V
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 48
Therefore, Cr2+ is stronger reducing agent (itself gets oxidised easily) than Fe2+.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 8.
Calculate the ‘spin only’ magnetic moment of M2+(aq) ion (Z = 27).
Solution:
Electronic configuration of M(Z = 27) is [Ar] 3d7 4s2
Electronic configuration of M2+ = [Ar]3d7 or AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 49
Three unpaired electrons are present in M2+ (aq) ion. i.e., n = 3. Applying ‘spin only1 formula for magnetic moment.
μ = \(\sqrt{n(n+2)}\) = \(\sqrt{3(3+2)}\) = \(\sqrt{15}\) BM = 3.87 BM
Note : Unit of magnetic moment is Bohr Magnetons (BM).

Question 9.
Explain why Cu+ ion is not stablein aqueous solutions ?
Solution:
In aqueous solution, Cu+(aq) undergoes disproportionation reaction as follows.
2Cu+(aq) → Cu2+(aq) + Cu(s)
The higher stability of Cu2+ (aq) in aqueous solution is due to higher negative enthalpy of hydration ∆hydH° than that of Cu+(aq). It compensates the second IE of Cu involved in the formation of Cu2+ ion. Hence, Cu+ ion changes to more stable Cu2+ ion in aqueous medium.

Question 10.
Actinoid contraction is greater from element to element than lanthanoid contraction ?
Solution:
The decrease in atomic (or ionic) radii in actinoid elements (actinoid contraction) is greater than lanthanoid contraction because 5f-electrons have poor shielding effect as compared to 4f- electrons. Therefore, the effect of increased nuclear charge leading to contraction in size is more in case of actinoid elements.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 11.
Write the formulae for the following Co-ordination compounds
i) Tetraamminediaquacobalt (III) chloride,
ii) Potassium tetracyanonickelate (II)
iii) Tris-(ethane-l, 2-diamine) chromium (m) chloride
iv) Amminebromidochloridonitrito-N-platinate (II)
v) Dichlorido bis-(ethane-1,2-diamine) platinum (IV) nitrate
Solution:
i)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 50
To find the value of x, we have to find the charge on the complex.
III
[CO(NH3)4 (H2O)2]x+ + 3 + 4 × 0 + 2 × 0 = x, × = +3
o, the formula of the complex is [CO(NH3)4 (H2O)2]Cl3.

ii)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 51
To find the value of x, find the charge on the complex.
[Ni(CN)4]x- (as K+ is positive)
+2 + (-1) x 4 = – x
-x = -2 or x = + 2
So, the formula of the complex is K2[Ni(CN)4].
Similarly
iii) [Cr(en)3]Cl3
iv) [Pt(NH3) BrCl(NO2)]
v) [PtCl2(en)2] (NO3)2
vi) Fe4[Fe(CN)6]3

Question 12.
Write the IUPAC names of the following Co-ordination compounds.
(i) [Co(NH3)6]Cl3
(ii) [Co(NH3)5Cl]Cl2
(iii) K3[Fe(CN)6]
(iv) K3[Fe(C2O4)3]
(v) K2[PdCl4]
(vi) [Pt(NH3)2Cl(NH2CH3)Cl.
Solution:
i)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 52
Let the oxidation state of Co be x.
x + (0)6 + (-1) × 3 = 0, x + 0 – 3 = 0, x = +3
So, the name of this complex is hexaamminecobalt (III) chloride,

ii) [CO(NH3)5Cl]Cl2
x + (0) × 5 + (-1) × 1 + (-1) × 2 = 0, x + 0 – 3 = 0, x =.+ 3
So, the name of the complex is pentaamminechloridocobalt (III) chloride
[K3[Fe(CN)6]

iii)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 53
Let the oxidation state of Fe is x.
(+ 1) 3 + x + (-1) 6 = 0, + 3 + x – 6 = 0, x = + 3
So, the name of the complex is potassium hexacyanoferrate (III) .

iv)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 54
Let the oxidation state of Fe is x.
(+ 1) 3 + x + (-2) 3 = 0 [∵ Oxalate ion (C2O42-) bears – 2 charge.]
3 + x- 6 = 0, x = +3
So, the name of the complex is potassium trioxalateferrate (III).

v)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 55
Let the oxidation state of Pd is x.
(+ 1) 2 + x + (-1) 4 = 0, 2 + x – 4 = 0, x = + 2 .
So, the name of the complex is potassium tetrachloridopalladate (II).

vi)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 56
Let the oxidation state of Pt is x.
x + (0) 2 + (-1) × 1 + 0 + (-1) × 1 = 0
x + 0- 1 + 0 -1 = 0, x – 2 = 0, x = + 2
So, the name of the complex is diamminechlorido (methylamine) platinum (II) chloride.

Question 13.
Indicate the types of isomerism exhibited by the following complexes and draw the structures for these isomers :
(i) K[Cr(H2O)2(C2O4)2]
(ii) [Co(en)3]Cl3
(iii) [Co(NH3)5(NO2)](NO3)2
(iv) [Pt(NH3)(H2O)Cl2]
Solution:
i) [Cr(H2O)2(C2O4)2] or K[Cr(H2O)2(ox)2] :
(where, ox = oxalate ion)
a) It exists as geometrical isomers : cis and trans forms.
(In cis form, the same groups occupy adjacent positions while in trans form, they are present at alternate positions.)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 57
b) The cis isomer can also exist as a pair of optical isomers (i.e., d-and 1- forms) (due to absence of plane of symmetry).

ii) [Co(en)3]Cl3 : It has two optical isomers (i.e. d-and 1-forms)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 58

iii) [Co(NH3)5(NO2)](NO3)2 : It can exist as a pair of ionization isomers as well as linkage iosmers.
Ionization isomers :
[Co(NH3)5(NO2)](NO3)2 and [Co(NH3)5(NO3)](NO2) (NO3) as they give different ions on ionization.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 59
Linkage isomers : [Co(NH3)5(NO2)](NO3)2 and [CO(NH3)5ONO](NO3)2

iv) [Pt(NH3)(H2O)Cl2] : It can exist as two geometrical iosmers. .
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 60

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 14.
Give evidence that [Co(NH3)5Cl]SO4 and [CO(NH3)5SO4]Cl are ionization isomers.
Solution:
We dissolve both the compounds in water in separate test tubes. To the both test tube
I Step or I Test: Add BaCl2 solution One compound give white ppt., indicating the presence of SO42- ions.
The other compound does not give white ppt, indicating the absence of SO42- ions.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 61

II Test: Add AgNO3 solution to both the Compounds in separate test tubes :
Only [II] compound gives white ppt, not the [I] one, due to absence of Cl as counter ion.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 62
These two tests prove that the given two compounds are a pair of ionization isomers.

Question 15.
Explain on the basis of valence bond theory that [Ni(CN)4]2- ion with square planar structure is diamagnetic and the [NiCl4] ion with tetrahedral geometry is paramagnetlc.
Solution:
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 63
(Cl being weak field ligand, cannot cause pairing.)
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 64
[NiCl4]2- ion is paramagnetic in nature as it has two unpaired electrons.

Question 16.
[NiCl4]2- is paramagnetic while [Ni(CO)4] is diamagnetic though both are tetrahedral.
Solution:
In the complex [NiCl4]2- Ni is in +2 oxidation state and has the configuration 3d84s0. The cr ion being a weak field ligand cannot pair the two unpaired electrons present in 3d-orbitals. This means that 3d-orbitals are not involved in hybridization. Thus, the complex is sp3 hybridized (tetrahedral) and.is paramagnetic in nature.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 65
In the complex [Ni(CO)4], the oxidation state of nickel is zero and electronic configuration is 3d84s2. In the presence of the ligand CO, the 4s-electrons shift to the two half-filled 3d-orbitals and make all the electrons paired. The valence 4s and 4p-orbitals are involved in hybridization. Thus, the complex is tetrahedral but diamagnetic in nature.
Outer configuration of Ni atom = 3d8 4s2.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 66

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 17.
[Fe(H2O)6]3+ is strongly paramagnetic whereas [Fe(CN)6]3- is weakly paramagnetic Explain.
Solution:
Outer configuration of 26Fe = 3ds64s2. In both the complexes Fe is present as Fes3+ ion.
Fes3+ ion = 3d54s0
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 67
In the presence of CN (a strong field ligand) the 3d-electrons pair up leaving only one unpaired electron. The hybridization is d2sp3 forming an inner orbital complex.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 68
In the presence of H2O, (a weak field ligand), 3d-electrons do not pair up. The hybridization is sp3d2 forming an outer orbital complex containing five unpaired electrons. Hence, it is strongly paramagnetic.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 69

Question 18.
Explain [Co(NH3)6]3+ is an inner orbital complex whereas [Ni(NH3)6]2+ is an outer orbital complex.
Solution:
In [Co(NH3)6]3+, CO is present as Co3+ arid has 3d0 configuration.
In the presence of NH3, the 3d-electrons pair up leaving two d-orbitals empty to the involved in d2sp3 hybridization forming inner orbital complex in the case of [Co(NH3)6]3+.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 70
Since, (n – 1) d-orbitals are not available but the nd-orbitals are used in bond formation i.e., in hybridization, the complex is called outer orbital complex.

Question 19.
Predict the number of unpaired electrons in the square planar [Pt(CN)4]2- ion.
Solution:
78Pt is present in group 10 of the Periodic Table. Its outer configuration is 5d96s1.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 71
For square plannar shape, the hybridization is dsp2. Hence, the unpaired electrons in 5d- orbital pair up to make one d-orbital empty for dsp2 hybridization. Thus, there is no unpaired electron.

AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds

Question 20.
The hexaquo manganese (II) ion contains five unpaired electrons, while the hexacyano ion contains only one unpaired electron. Explain using crystal field theory.
Solution:
Mn(II) ion has 3d5 configuration. In the presence of H2O molecules (acting as weak field ligands), the distribution of these five electrons is \(\mathrm{t}_{2 \mathrm{~g}}^3 \mathrm{e}_{\mathrm{g}}^2\) all the electrons remain unpaired to form a high spin complex.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 72
However, in the presence of CN (acting as strong field ligands), the distribution of these electrons is \(\mathrm{t}_{2 \mathrm{~g}}^5 \mathrm{e}_{\mathrm{g}}^0\), i.e., two t2g orbitals contain paired electrons while the third t2g orbital contains one unpaired electron. The complex formed is a low spin complex.
AP Inter 2nd Year Chemistry Study Material Chapter 7 d and f Block Elements & Coordination Compounds 73

Question 21.
Calculate the overall complex dissociation equilibrium constant for the Cu(NH3)42+ ions, given that β4 for this complex is 2.1 × 1013.
Solution:
Overall complex dissociation equilibrium constant
= \(\frac{1}{\beta_4}\) = \(\frac{1}{2.1 \times 10^{13}}\)
= 4.7 × 10-14

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

Andhra Pradesh BIEAP AP Inter 2nd Year Chemistry Study Material Lesson 6(d) Group-18 Elements Textbook Questions and Answers.

AP Inter 2nd Year Chemistry Study Material Lesson 6(d) Group-18 Elements

Very Short Answer Questions

Question 1.
What inspired Bartlett for carrying out reaction between Xe and PtF6 ?
Answer:

  1. At first Bartlett prepared a red compound \(\mathrm{O}_2^{+}+\mathrm{PtF}_6^{-}\)
  2. As the first Ionisation Enthalpy of molecular oxygen is identical with that the Xe.
  3. He made efforts to prepare same type of compound with Xe and was successful in preparing another red colour compound Xe+ \(\mathrm{PtF}_6^{-}\) by mixing PtF6 and Xe.

Question 2.
Which of the following does not exist ?
a) XeOF4
b) NeF2
c) XeF2
d) XeF6
Answer:
Given compounds are XeOF4, NeF2, XeF2, XeF6.
NeF2 does not exist among the above compounds.

Reason : Due to small size and high I.E. of ‘Ne’. It does not form chemical compounds.

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

Question 3.
Why do noble gases have comparatively large atomic sizes ?
Answer:
Noble gases have comparatively large atomic sizes as they have vander waals radii which is larger than both the ionic and covalent radii.

Question 4.
List out the uses of Neon. [A.P. Mar. 18]
Answer:
Uses of Ne:

  1. Ne is used in discharge tubes and fluor escent bulbs for advertisement display purpose.
  2. ‘Ne’ – bulbs are used in botanical gardens and in green houses.

Question 5.
Write any two uses of argon.
Answer:
Uses of Ar :

  1. ‘Ar’ is used to create inert atmosphere in high temperature net allurgical process.
  2. ‘Ar’ is ued in filling electric bulbs.

Question 6.
In modern diving apparatus, a mixture of He and O2 is used – Why ? [A.P. Mar. 16]
Answer:
In modem diving apparatus, a mixture of He and O2 is used because He is very low soluble in blood.

Question 7.
Helium is heavier than hydrogen. Yet helium is used (instead of H2) in filling baloons for meteorological observations – Why ?
Answer:
‘He’ is a non-inflammable and light gas. Hence it is used in filling baloons for meterological observations.

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

Question 8.
How is XeO3 prepared ?
Answer:
XeF6 on hydrolysis produce XeO3
XeF6 + 3H2O → XeO3 + 6HF

Question 9.
Give the preparation of
a) XeOF4 and
b) XeOaF2
Answer:
Partial hydrolysis of XeF6 gives oxy fluorides XeOF4 and XeO2F2
XeF6 + H2O → XeOF4 + 2HF
XeF6 + 2H2O → XeO2F2 + 4HF

Question 10.
Explain the structure of XeO3. [T.S. Mar. 16]
Answer:
Structure of XeO3

  1. Central atom is ‘Xe’
  2. ‘Xe’ under goes sp3 hybridisation in 3rd excited state.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 1
  3. ‘Xe’ forms 3σ-bonds and 3π-bonds with three oxygens.
  4. Shape of molecule is pyramidal with bond angle 103°.

Question 11.
Noble gases are inert – explain.
Answer:
Noble gases are chemically inert:

  1. Noble gases have stable electronic configuration coctet configuration except He)
  2. Noble gas have high Ionisation energy values and have large positive values of electron gain Enthalpy.

Question 12.
Write the name and formula of the first noble gas compound prepared by Bertlett.
Answer:
The first noble gas compound prepared by Bertlett is XePtF6. Name of the compound is xenon hexa fluoro platinate.

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

Question 13.
ExplaIn the shape of XeF4 on the basis of VSEPR theory
Answer:
Shape of XeF4:

  1. Central atom in XeF4 is ‘Xe’.
  2. Xe undergoes sp3d2 hybridisation in it’s 2nd excited. state.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 2
  3. Shape of the molecule is square planar with bond angle 90° and bond length 1 .95Å.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 3
  4. Xe – forms four o-bonds by the overlap of sp3d2 – 2pz(F) orbitals.

Question 14.
Give the outer electronic configuration of noble gases.
Answer:
The outer electronic configuration of noble gases is ns2np6 (except He (1s2)

Question 15.
Why do noble gases form compounds with fluorine and oxygen only ?
Answer:
Noble gases form compounds with flourine and oxygen only.
Reason : Oxygen and Fluorine are most electronegative elements.

Question 16.
How is XeOF4 prepared ? Describe its molecular shape.
Answer:
Partial hydrolysis of XeF6 gives XeOF4
AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 4
XeF6 + H2O → XeOF4 + 2HF
XeOF4 is a colourless volatile liquid it has a square pyramidal shape.

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

Question 17.
What is the major source of helium ?
Answer:
The major source of helium is natural gas.

Question 18.
Which noble gas is radioactive ? How is it formed ?
Answer:
Radon (Rn) is radio active noble gas. Radon is obtained as decay product of 86Ra226
86Ra22686Rn222 + 2He4

Question 19.
Name the following :
a) most abundant noble gas in atmosphere
b) radioactive noble gas
c) noble gas with least boiling point
d) noble gas forming large number of compounds
e) noble gas not present in atmosphere
Answer:
a) Argon is the most abundant noble gas in atmosphere.
b) Radon is the radio active noble gas.
c) Helium has lowest boiling point (4.2K)
d) Xenon forms large number of compounds.
e) Radon is not present in atmosphere.

Short Answer Questions

Question 1.
How are Xenon fluorides XeF2, XeF4 and XeF6 obtained?
Answer:
Xenon forms the binary fluorides XeF2, XeF4, XeF6 as follows. These are formed by direct combination of Xe and F2.
AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 5

Question 2.
How are XeO3 and XeOF4 prepared? [Mar. 14]
Answer:
XeF6 on hydrolysis produce XeO3
XeF6 + 3H2O → XeO3 + 6HF
Partial hydrolysis of XeF6 gives XeOF4
XeF6 + H2O → XeOF4 + 2HF

Question 3.
Give the formulae and describe the structures of a noble gas species, isoelectronic with
a) ICl4
b) IBr2
c) BrO3
Answer:
a) ICl4 is also electronic with XeF4 and it has square planar shape.
b) IBr2 is also electronic with XeF2 and it has linear shape.
c) BrO3 is also electronic with XeO4 and it has tetrahedral shape.

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

Question 4.
Explain the reaction of the following with water.
a) XeF2
b) XeF4
c) XeF6
Answer:
a) XeF2 is hydrolysed to form Xe, HF and O2
2XeF2 + 2H2O → 2Xe + 4HF + O2

b) XeF4 is hydrolysed to give XeO3
6XeF4 + 12H2O → 4Xe + 2XeO3 + 24 HF + 3O2

c) XeF6 undergo hydrolysis to form XeO3
XeF6 + 3H2O → XeO3 + 6HF
XeF6 undergo partial hydrolysis to form XeOF4 + XeO2F2
XeF6 + H2O → XeOF4 + 2HF
XeF6 + 2H2O → XeO2F2 + 4HF

Question 5.
Explain the structures of [A.P. Mar. 18] [Mar. 14]
a) XeF2 and
b) XeF4
Answer:
a) Structure of XeF2 :

  1. In XeF2 central atom is ‘Xe’.
  2. ‘Xe’ undergoes sp3d hybridisation in it’s 1st excited state
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 6
  3. Shape of molecule is linear
  4. Xe form two σ – bonds with two fluorines (sp3 – 2pz overlap)
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 7

b) Structure of XeF4 :

  1. Central atom in XeF4 is ‘Xe’.
  2. Xe undergoes sp3d2 hybridisation in it’s 2nd excited, state.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 8
  3. Shape of the molecule is square planar with bond angle 90° and bond length 1.95Å.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 9
  4. Xe – forms four a – bonds by the overlap of sp3d2 – 2pz(F) orbitals.

Question 6.
Explain the structures of
a) XeF6 and
b) XeOF4
Answer:
a) Structure of XeF6 :

  1. Central atom in XeF6 is ‘Xe’
  2. Xe under goes sp3d3 hybridisation in it’s 3rd excited state.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 10
  3. Shape of molecule is distorted octahedral.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 11

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

b) Structure of α XeOF4

  1. In XeOF4 molecule ‘Xe’ under goes sp3d2 hybridisation.
  2. Shape of the molecule is square pyramid.
  3. There is one Xe-O double bond containing
    pπ = dπ overlaping.
    Partial hydrolysis of XeF6 gives XeOF4
    XeF6 + H2O → XeOF4 + 2HF
    XeOF4 is a colourless volatile liquid it has a square pyramidal shape.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 12

Question 7.
Complete the following.
a) XeF2 + H2O →
b) XeF2 + PF5
c) XeF4 + SbF5
d) XeF6 + ASF5
e) XeF4 + O2F2
f) NaF + XeF6
Answer:
a) 2XeF2 + 2H2O → 2Xe + 4HF + O2
b) XeF2 + PF5 → [XeF]+ + A[F6]
c) XeF4 + SbF5 → [XeF3]+ [SbF6]
d) XeF6 + ASF5 → [Xe2F11]+ [ASF6]
e) XeF4 + O2F2 → XeF6 + O2
f) NaF + XeF6 → Na+ [XeF7]

Question 8.
How are XeF2 and XeF4 prepared ? Give their structures. [T.S. Mar. 18]
Answer:
Xenon forms the binary fluorides XeF2, XeF4, XeF6 as follows. These are formed by direct combination of Xe and F2.
AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 13
Structure of XeF2

  1. In XeF2 central atom is Xe’.
  2. ‘Xe’ undergoes sp3d hybridisation in it’s 1st excited state
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 14
  3. Shape of molecule is linear.
  4. Xe form two σ bonds with two fluorines.

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

b) Structure of XeF4 :

  1. Central atom in XeF4 is ‘Xe’.
  2. Xe undergoes sp3d2 hybridisation in it’s 2nd excited state.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 15
  3. Shape of the molecule is square planar with bond angle 90° and bond length 1.95Å.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 16
  4. Xe – forms four σ-bonds by the overlap of sp3d2 – 2pz (F) orbitals.

Long Answer Question

Question 1.
How are XeF2, XeF4 and XeF6 prepared ? Explain their reaction with water. Discuss their structures. [A.P. Mar. 15]
Answer:
Xenon forms the binary fluorides XeF2, XeF4, XeF6 as follows. These are formed by direct combination of Xe and F2.
AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 17
Reaction with water:

  1. XeF2 is hydrolysed to form Xe, HF and O2
    2XeF2 + 2H2O → 2Xe + 4HF + O2
  2. XeF4 is hydrolysed to give XeO3
    6XeF4 + 12H2O → 4Xe + 2XeO3 + 24 HF + 3O2
  3. XeFg undergo hydrolysis to form XeO3
    XeF6 + 3H2O → XeOF3 + 6HF
  4. XeF6 undergo partial hydrolysis to form XeOF4 and XeO2F2
    XeF6 + H2O → XeOF4 + 2HF
    XeF6 + 2H3O → XeO2F2 + 4HF

Structure of XeF2:

  1. In XeF2 central atom is ‘Xe’.
  2. ‘Xe’ undergoes sp3d hybridisation in it’s 1st excited state
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 18
  3. Shape of molecule is linear.
  4. Xe form two σ-bonds with two fluorines.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 19

b) Structure of XeF4 :

  1. Central atom in XeF4 is ‘Xe’.
  2. Xe undergoes sp3d2 hybridisation in it’s 2nd excited state.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 20
  3. Shape of the molecule is square planar with bond angle 90° and bond length 1.95Å.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 21
  4. Xe – forms four σ-bonds by the overlap of sp3d2 – 2pz (F) orbitals.

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

Structure of XeF6 :

  1. Central atom in XeF6 is ‘Xe’
  2. Xe under goes sp3d3 hybridisation in it’s 3rd excited state.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 10
  3. Shape of molecule is distorted octahedral.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements 11

Textual Examples

Question 1.
Why are the elements of group 18 known as noble gases ?
Solution:
The elements present in Group 18 have their valence shell orbitals completely filled and, therefore, react with a few elements only under certain conditions. Therefore, they are now known as noble gases.

Question 2.
Noble gases have very low boiling points. Why ?
Solution:
Noble gases being monoatomic have no interatomic forces except weak dispersion forces and therefore, they are liquefied at very low temperatures. Hence, they have low boiling points.

AP Inter 2nd Year Chemistry Study Material Chapter 6(d) Group-18 Elements

Question 3.
Does the hydrolysis of XeF6 lead to a redox reaction ?
Solution:
No, the products of hydrolysis are XeOF4 and XeO2F2 where the oxidation states of all the elements remain the same as it was in the reacting state.

Intext Questions

Question 1.
Why is helium used in diving apparatus ?
Solution:
Helium is used in diving apparatus due to its very low solubility in blood.

Question 2.
Balance the following equation :
XeF6 + H2O → XeO2F2 + HF
Solution:
XeF6 + 2H2O → XeO2F2 + 4HF

Question 3.
Why has it been difficult to study the chemistry of radon ?
Solution:
Radon is a radioactive element with very short half-life of 3.82 days. That is why, the study of chemistry of radon is a difficult task.

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Andhra Pradesh BIEAP AP Inter 2nd Year Chemistry Study Material Lesson 6(b) Group-16 Elements Textbook Questions and Answers.

AP Inter 2nd Year Chemistry Study Material Lesson 6(b) Group-16 Elements

Very Short Answer Questions

Question 1.
Why is dioxygen a gas but sulphur a solid?
Answer:
Dioxygen is a gas but sulphur a solid.
Explanation:

  • Due to small atomic size and high electronegativity oxygen forms Pπ – Pπ multiple bond and exists as O2 molecules held together by weak vander waal’s forces. Thus oxygen exists as a gas at room temperature.
  • Due to large atomic size and less electronegativity sulphur forms strong S—S single bonds and exists as S8 molecules with puckered ring structure. Hence sulphur is a solid at room temperature.

Question 2.
What happens when
a) KClO3 is heated with MnO2
b) O3 is passed through KI soultion
Answer:
a) When KClO3 is heated with Mn02 and liberates oxygen gas.
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 1
b) O3 is passed through KI solution I2 is liberated
2KI + O3 + H2O → 2KOH + I2 + 2O2

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 3.
Give two examples each for amphoteric oxides and neutral oxides.
Answer:

  • Examples of amphoterc oxides – Al2O3, SiO2, PbO.
  • Examples of neutral oxides – CO, NO and N2O.

Question 4.
Oxygen generally exhibits an oxidation state of -2 only while the other members of the group show oxidation states of +2, +4 and +6 also – explain.
Answer:

  • Oxygen exhibits -2 oxidation state due to its high electronegativity. The tendency of exhibiting -2 oxidation state decrease down the group.
  • Due to decrease of electronegativity down the group the other elements exhibit +2, +4 and +6 oxidation states also.

Question 5.
Write any two compounds, in which oxygen shows an oxidation state different from -2. Give the oxidation states of oxygen in them.
Answer:
OF2 and O2 F2 are two compounds in which oxygen shows an oxidation state different from-2.

  • In OF2 the oxidation state of oxygen is +2
  • In O2F2 the oxidation state of oxygen is +1.

Question 6.
Oxygen molecule has the formula O2 while sulphur has S8 – explain.
Answer:
Due to small atomic size and high electronegativity oxygen forms Pπ – Pπ multiple bond and exists as O2 molecule.
Due to large atomic size and less electronegativity sulphur forms strong S – S single bonds and exists S8 molecule.

Question 7.
Why is H2O a liquid while H2S is a gas ?
Answer:
H2O is liquid due to the presence of intermolecular hydrogen bonding. While H2S is gas because it is not having such type of bonding.

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 8.
H2O is neutral while H2S is acidic – explain.
Answer:
H2O is neutral while H2S is acidic.
Reason: The O-H bond dissociation Enthalpy is greater than the S – H bond dissociation Enthalpy.

Question 9.
Name the most abundant element present in earth’s crust.
Answer:
The most abundant element present in earth’s crust is oxygen (about 46.6%).

Question 10.
Which element of group-16 shows highest catenation ?
Answer:
Sulphus shows highest catenation among group – 16 elements and exists as S9 molecule with puckered ring structure.

Question 11.
Among the hydrides of chalcogens, which is most acidic and which is most stable ?
Answer:

  • Among hydrides of chalcogens, H2Te is most acidic.
  • Among hydrides of chalcogens, H2O is most stable.

Question 12.
Give the hybridization of sulphur in the following.
a) SO2
b) SO3
c) SF4
d)SF6
Answer:
a) Hybridisation of’S’ in SO2 is Sp2
b) Hybridisation of ‘S’ in SO3 is Sp2
c) Hybridisation of ‘S’ in SF4 is Sp3d
d) Hybridisation of ‘S’ in SF6 is Sp3d2

Question 13.
Write the names and formulae of any two oxyacids of sulphur. Indicate the oxidation state of sulphur in them.
Answer:

  • Peroxy mono sulphuric acid – H2SO5 ‘S’ oxidation state +6 .
  • Peroxy di sulphuric acid – H2S2O8 ‘S’ oxidation state +6

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 14.
Explain the structures of SF4 and SF6.
Answer:
Structure of SF4:

  • It SF4 ‘S’ undergoes sp3d hybndisation.
  • It has trigonal bipyramidal structure in which one of the equitorial positions is occupied by a lone pair of electrons. This geometry is also known as see – saw geometry.

Structure of SF6:

  • In SF6 ‘S’ undergoes sp3d2 hybridisation.
  • It has octahedral structure.

Question 15.
Give one example each for
a) a neutral oxide
b) a peroxide
c) a super oxide
Answer:
a) CO, N2O are neutral oxides.
b) Na2O2, BaO2 are pernxides.
c) KO2, RbO2 are super oxides.

Question 16.
What is tailing of mercury? How is it removed? [A.P. & T.S. (Mar. 15)]
Answer:
Mercury loses it’s lustreness, meniscus and consequently sticks to the walls of glass vessel when it reacts with ozone. This phenomenon is called tailing of mercury.
2Hg + O3 → Hg2O + O2
It is removed by shaking it with water which dissolves Hg2O.

Question 17.
Write the principle involved in the quantitative estimation of ozone gas.
Answer:
When ozone reacts with an excess of Kl solution buffered with a borate buffer (pH 9.2) iodine is liberated which can be titrated against a standard solution of sodium thiosulphate. In this way O3 is estimated quantitatively.

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 18.
Write the structure of ozone.
Answer:
Structure of Ozone:
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 2

  • O3 is angular molecule with bond angle 117°
  • O – O bond length is 128 pm.

Question 19.
SO2 can be used as an anti-chior. Explain.
Answer:
SO2 gas is used as anti-chlor. Anti-chior means the substance which removes the excess of on clothes. SO2 reacts with chlorine in presence of charcoal to give suiphuryl chloride
SO2(g) + Cl2(g) → SO2Cl2l

Question 20.
How is ozone detected ?
Answer:
Ozone is a pale blue gas, dark blue liquid and violet black solid and it has characterstic smell.

  • It is detected by the reaction with ‘Hg’ which is called as tailing of mercury.
    2Hg + O3 → Hg2O + O2
  • Ozone turns benzidene paper to brown colour.

Question 21.
How does ozone react with Ethylene ?
Answer:
Ethylene reacts with ozone to form Ethylene ozonoid followed by the hydrolysis to form formaldehyde.
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 3

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 22.
Out of O2 and O3, which is paramagnetic ?
Answer:

  • O2 is paramagnetic due to presence of unpaired electrons
  • O3 (gaseous) is diamagnetic due to absence of unpaired electrons.

Question 23.
Between O3 and O2, ozone is a better oxidizing agent – why ?
Answer:
Ozone is next to fluorine in the oxidising capacity. It is best oxidising agent than O2. (Fluorine is the powerful oxidising agent). Ozone liberates nascent oxygen easily.

Question 24.
Write any two uses each for O3 and H2SO4.
Answer:
Uses of O3:

  • Ozone is used in sterilisation of water.
  • Ozone is used in manufacture of artificial silk and camphor etc.
  • Ozone is used to identify unsaturation in carbon compounds.

Uses of H2SO4:

  • H2SO4 is used in the manufacture of fertilisers.
  • H2SO4 is used in petrol refining.
  • H2SO4 is used in detergent industry.

Question 25.
Which form of sulphur shows paramagnetism ?
Answer:
In vapour state sulphur partly exists as S2 molecule which has two unpaired electrons in the antibonding π (π*)orbitals like O2. Hence exhibits paramagnetism.

Question 26.
How is the presence of SO2 detected ?
Answer:
SO2 has a pungent odour SO2 presence can be detected by the following tests.

  1. SO2 changes the colour of acidified potassium dichromate solution from orange to green.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 4
  2. SO2 decolourises acidified KMnO4 solution.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 5

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 27.
Why are group – 16 elements called chalcogens ?
Answer:
Chalcogens means mineral forming (or) ore forming elements. Most of elements exist in earth crust as oxides, sulphides, selinides, telurids etc. So Group – 16 elements are called as chalcogens.

Question 28.
Among chalcogens, which has highest eletronegativity and which has highest electron gain enthalpy?
Answer:

  • Among chalcogens Oxygen has high electronegativity.
  • Among chalcogens Sulphur has high electron gain Enthalpy.

Question 29.
Which hydride of group – 16 has highest boiling point and weakest acidic character ?
Answ:

  • Among group – 16 hydrides water (H2O) has high bioling point.
  • Among group – 16 hydrides water (H2O) has weakest acidic, character.

Short Answer Questions

Question 1.
Justify the placement of O, S, Se, Te and Po in the same group of the periodic table in terms of electronic configuration, oxidation states and hydride formation.
Answer:
1) Electronic configurations :
Oxygen (O) – [He] 2s2 2p2
Sulphur (S) – [Ne] 3s2 3p2
Selenium (Se) – [Ar] 3d10 4s24p4
Tellurium (Te) – [Kr] 4d10 5s2 sp4
Polonium (Po) – [Xe] 4f14 5d10 6s2 6p4
All the above elements has general outer electronic configuration ns2np4.

2) Oxidation states :
All the gives elements (chalcogens) exhibits common oxidation state of-2
O-2, S-2, Se-2 etc.

3) Hydride formation:
All these elements (chalcogens) forms hydrides of type EH2(E = chalcogen)
Eg : H2O, H2S, H2Se, H2Te, H2Po.
The above mentioned concepts evident that the elements O, S, Se, Te and Po are present in the same group of periodic table.

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 2.
Describe the manufacture of H2SO4 by contact process.
Answer:
Manufacture of H2SO4 by contact process:
Manufacturing of H2SO4 involves three main steps.
Step – 1.
SO2 production : The required SO2 for this process is obtained by burning S(or) Iron pyrites in oxygen.
S + O2 → SO2
4FeS2 + 15O2 → 2Fe2O3 + 8SO3
Step – 2
SO3 formation: SO2 is oxidised in presence of catalyst with atmosphric air to form SO3
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 6
Step – 3
Formation of H2SO4: SO3 formed in the above step absorbed in 98% H2SO4 to get oleum (H2S2O7). This oleum is diluted to get desired concentration of H2SO4.
SO3 + H2SO4 → H2S2O7
H2S2O7 + H2O → 2H2SO4

Question 3.
How is ozone prepared ? How does it react with the following? [Mar. 14]
a) PbS
b) KI
c) Hg
d) Ag
Answer:
Preparation of Ozone:
A slow dry stream of oxygen under silent electric discharge to form ozone (about 10%). The product obtained is known as ozonised oxygen.
3O2 → 2O3 ∆ H° = 142kJ/mole

  • The formation of ozone is an endothermic reaction.
  • It is necessary to use silent electric discharge in the preparation of O3 to prevent its decomposition

a) Reaction with PbS : Black lead suiphide oxidised to white lead sulphate in presence of ozone.
PbS + 4O3 → PbSO4 + 4O2

b) Reaction with KI: Moist Kl is oxidised to Iodine in presence of ozone.
2KI + H2O + O3 -→ 2KOH + I2 + O2

c) Reaction with Hg : Mercury loses it’s lustreness, meniscus and consequently sticks to the walls of glass vessel when it reacts with ozone. This phenomenon is called tailing of mercury.
2Hg + O3 → Hg2 O + O2
It is removed by shaking it with water which dissolves Hg2O.

d) Reaction with Ag : Ag metal oxidised to Ag2O (Ag metal is tarnished):
2Ag + O3 → Ag2O + O2

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 4.
Write a short note on the allotropy of sulphur.
Answer:
The important allotropes of sulphur are
a) yellow rhombic (α. sulphur)
b) Monoclinic (β – sulphur)

  • The stable from is α-sulphur (at room temperature)

Rhombic sulphur (α – Sulphur):

  • Colour : Yellow. ,
  • Melting point: 385.8K.
  • Specific gravity : 2.06.
  • It is insouble in water and partially soluble in alcohol, benzene etc. and readily soluble in CS2.

Monoclinic sulphur (β – Sulphur):

  • Melting point: 392K
  • Specific gravity : 1.98.
  • It is soluble in CS2.
  • Rhombic sulphur transforms to monoclinic sulphur by heating above 369K. This temperature is called transition temperature.

Question 5.
How does SO2 react with the following ?
a) Na2SO3(aq)
b) Cl2
c) Fe+3 ions
d) KMnO4
Answer:
a) Sodium sulphite (aq) reacts with So2 to form sodium hydrogen sulphite.
Na2SO3 + H2O + SO2 → 2NaHSO3

b) SO2 as reacts with chlorine gas in the presence of charcoal to form sulphuryl chloride.
SO2(g) + Cl2(g) → SO2Cl2(l)

c) Fe+3 ions are reduced to Fe+2 ions by SO2.
2Fe+3 + SO2 + 2H2O → 2Fe+2 + SO-24 + 4H+

d) SO2 gas decolourises acidified potassium permanganate (VII) solution.
5SO2 + 2MnO4 + 2H2O → 5SO-24 + 4H++ 2Mn+2

Question 6.
Starting from elemental sulphur, how is H2SO4 prepared ?
Answer:
Manufacture of H2SO4 by contact process :
Manufacturing of H2SO4 involves three main steps.
Step – 1.
SO2 production : The required SO2 for this process is obtained by burning S(or) Iron pyrites in oxygen.
S + O2 → SO2
4FeS2 + 15O2 → 2Fe2O3 + 8SO3
Step – 2
SO3 formation: SO2 is oxidised in presence of catalyst with atmosphric air to form SO3
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 6
Step – 3
Formation of H2SO4: SO3 formed in the above step absorbed in 98% H2SO4 to get oleum (H2S2O7). This oleum is diluted to get desired concentration of H2SO4.
SO3 + H2SO4 → H2S2O7
H2S2O7 + H2O → 2H2SO4

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 7.
Describe the structures (shapes) of SO-24 and SO3.
Answer:
Structure of SO3:
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 7

  • In SO3 sulphur undergoes sp2 hybridisation.
  • Shape : Trigonal planait s
  • Bondangle: 120°.
  • S -0 bond length: 143 pm.

Structure of SO-24:
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 8

  • In SO-24 sulphur undergo sp3 hybridisation.
  • Shape : Tetrahedral.
  • It has several resonance structures.
  • In this two Pπ – dπ bonds are present.

Question 8.
Which oxide of sulphur can act as both oxidizing and reducing agent? Give one example each.
Answer:
Sulphur dioxide (SO2) acts as both oxidising as well as reducing agent.
SO2 as Oxidising agent:
Sodium suiphide oxidises to hypo with SO2.
2Na2S + 3SO2 → 2Na2S2O2 + S
SO2 as Reducing agent:
SO2 reduces Fe+3 ions to Fe+2 ions.
2Fe+3 + SO2 + 2H2O → 2Fe+2 + SO-24 + 4H+

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 9.
Explain the conditions favourable for the formation of SO3 from SO2 in the contact process of H2SO4.
Answer:
Le Chatlier’s principle – Application to produce SO3:
The oxidation of SO2 to SO3 in the presence of a catalyst is a reversible reaction. The thermochemical equation for the conversion is written as
2SO2(g) + O2(g) ⇌ 2SO3(g); ∆H = -196 kJ
The equation reveals the following points :

  1. 3 volumes of the reactants convert into 2 volumes of SO3. i.e., a decrease of volume accompanies the reaction.
  2. the reaction is an exothermic change.
  3. the catalyst may be present to increase the SO3 yields.

According to Le Chatlier’s principle,
i) a decrease in volume of the system is favoured at high pressures. But in practice only about 2 bar pressure is used. The reason for not using high pressures is acid resisting towers which can withstand high pressures cannot be built.

ii) exothermic changes are favoured at low temperatures. It is not always convenient in the industry to work at low temperatures. In such situations an optimum temperature is maintained. At this temperature considerable amounts of the product are obtained. In the manufactrue of H2SO4, the optimum temperature suitable for the conversion of SO2 into SO3 is experimentally found to be 720K.

iii) The rate of formation of SO3 is enhanced by the use of a catalyst. (V2 O5 (or) Pt – asbestos).
Favourable Conditions :
Temperature : 720K
Pressure : 2 bar
Catalyst: V2O5 (or) platinized asbestos.

Question 10.
Complete the following
a) KCl + H2SO4 (Conc) →
b) Sucrose AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 9
c) Cu + H2SO4 (Conc) →
d) C + H2SO4 (Cone) →
Answer:
a) 2KCl + Cone. H2SO4 → 2HCl + K2SO4
b) C12H22O11 AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 9 12C + nH2O
c) Cu + 2H2SO4(Conc) → CuS04 + SO2 + 2H2O
d) C + 2H2SO4(Conc) → CO2 + 2SO2 + 2H2O

Question 11.
Which is used for drying ammonia ?
Answer:
For drying ammonia quick line (CaO) is used.
For drying ammonia cone. H2SO4, P4O10 and anhydrous CaCl2 cannot be used because they react with ammonia and forms (NH4)2SO4, (NH4)3PO4 and CaCl2. 8NH3
H2SO4 + 2NH3 → (NH4)2 SO4
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 10
CaCl2 + 8NH3 → CaCl38NH3.

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 12.
Why cone H2SO4, P4O10 and anhydrous CaCl2 cannot be used to dry ammonia ?
(Hint: ammonia reacts with them forming (NH4)2 SO4; (NH4)3 PO4 and CaCl2, 8NH2)
Answer:
For drying ammonia quick line (CaO) is used.
For drying ammonia cone. H2SO4, P4O10 and anhydrous CaCl2 cannot be used because they react with ammonia and forms (NH4)2SO4, (NH4)3PO4 and CaCl2. 8NH3
H2SO4 + 2NH3 → (NH4)2 SO4
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 10
CaCl2 + 8NH3 → CaCl38NH3.

Long Answer Questions

Question 1.
Explain in detail the manufacture of sulphuric acid by contact process. [T.S. Mar. 18, 16]
Answer:
Manufacture of H2SO4 by contact process :
Manufacturing of H2SO4 involves three main steps.
Step – 1.
SO2 production : The required SO2 for this process is obtained by burning S(or) Iron pyrites in oxygen.
S + O2 → SO2
4FeS2 + 15O2 → 2Fe2O3 + 8SO3
Step – 2
SO3 formation: SO2 is oxidised in presence of catalyst with atmosphric air to form SO3

Le Chatlier’s principle – Application to produce SO3 :
The oxidation of SO2 to SO3 in the presence of a catalyst is a reversible reaction. The thermochemical equation for the conversion is written as
2SO2(g) + O2(g) ⇌ 2SO3(g); ∆H = -196 kJ
The equation reveals the following points : .

  1. 3 volumes of the reactants convert into 2 volumes of SO3. i.e., a decrease of volume accompanies the reaction.
  2. the reaction is an exothermic change.
  3. the catalyst may be present to increase the SO3 yields.

According to Le Chatlier’s principle, .
i) a decrease in volume of the system is favoured at high pressures. But in practice only about 2 bar pressure is used. The reason for not using high pressures is acid resisting towers which can withstand high pressures cannot be built.

ii) exothermic changes are favoured at low temperatures. It is not always convenient in the industry to work at low temperatures. In such situations an optimum temperature is maintained. At this temperature considerable amounts of the product are obtained. In the manufactrue of H2SO4, the optimum temperature suitable for the conversion of SO2 into SO., is experimentally found to be 720K.

iii) The rate of formation of SO3 is enhanced by the use of a catalyst. (V2 O5 (or) Pt – asbestos).
Favourable Conditions :
Temperature : 720K .
Pressure : 2 bar
Catalyst: V2O5 (or) platinized asbestos.
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 11
Step – 3 :
Formation of H2SO4: SO3 formed in the above step absorbed in 98% H2SO4 to get oleum (H2S2O7). This oleum is diluted to get desired concentration of H2SO4.
SO3 + H2SO4 → H2S2O7
H2S2O7 + H2O → 2H2SO4

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 2.
How is ozone prepared from oxygen ? Explain its reaction with [A.P. Mar. 19, 18, 17, 16] [Mar. 14]
a) C2H4
b) KI
c) Hg
d) PbS.
Answer:
Preparation of Ozone :
A slow dry stream of oxygen under silent electric discharge to form ozone (about 10%). The product obtained is known as ozonised oxygen.
3O2 → 2O3; ∆H° = 142kJ/mole

  • The formation of ozone is an endothermic reaction.
  • It is necessary to use silent electric discharge in the preparation of O3 to prevent its decomposition

a) Reaction with C2H3 : Ethylene reacts with ozone to form Ethylene ozonoid followed by the hydrolysis to form formaldehyde.
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 12

) Reaction with KI: moist KI is oxidised to Iodine in presence of ozone.
2KI + H2O + O3 → 2KOH + I2 + O2

c) Reaction with Hg : Mercury loses it’s lustreness, meniscus and consequently sticks to the walls of glass vessel when it reacts with ozone. This phenomenon is called tailing of mercury.
2Hg + O3 → Hg2O + O2
It is removed by shaking it with water which dissolves Hg2O.

d) Reaction with PbS : Black lead sulphide oxidised to white lead sulphate in presence of ozone.
PbS + 4O3 → PbSO4 + 4O2.

Intext Questions

Question 1.
List the important sources of sulphur.
Answer:
Occurrence or sources of sulphur in the earth’s crust, percentage of sulphur is only 0.03 to 0. 1%. In combined state, it occurs

  1. In the form of sulphates, eg., gypsum (CaSO4 . 2H2O). epsom salt (MgSO4 . 7H2O), baryte
    (BaSO4).
  2. In the form of sulphides, eg., galena (PbS), zinc blende (ZnS), Copper pyrites (CuFeS2). In volcanoes, traces of sulphur occur as H2S. In organic materials such as eggs, proteins, garlic, onion,mustard, hair and wool, sulphur is present in trace amounts.

Question 2.
Write the order of thermal stability of the hydrides of group 16 elements.
Answer:
The thermal stability of hydrides of group 16 elements is directly proportional to the bond dissociation enthalpy of H – E bond. On moving down the group, bond dissociation energy decreases because bond length increases.
Thus, the order of bond dissociation energy is
H2O > H2S > H2Se > H2Te > H2Po
This is also the order of thermal stability.

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 3.
Why is H2O a liquid and H2S a gas ?
Answer:
The difference in electronegativity values of 0(3.5) and H(2.1) is more than the difference between the electronegativity values of H(2.1) and S(2.5) i.e., O – H bond is more polar than S – H bond. That is why H-bonding is present among water molecules but absent in H2S. Thus,
strong intermolecular interactions causes water to exist as a liquid but due to weak van der waals’ force H2S exists as a gas.
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 13

Question 4.
Which of the following does not react with oxygen directly ? Zn, Ti, Pt, Fe.
Answer:
Platinum (Pt).

Question 5.
Complete the following reactions.
i) C2H4 + O2
ii) 4Al + 3O2
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 14

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 6.
Why does O3 act as a powerful oxidising agent ?
Answer:
Because ozone liberates nascent oxygen very easily.
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 15

Question 7.
How is Oa estimated quantitatively ?
Answer:
When ozone reacts with an excess of potassium iodide solution buffered with a borate buffer (pH 9.2), iodine is liberated which can be titrated against a standard solution of sodium thiosulphate. In this way O3 can be estimated quantitatively.

Question 8.
What happens when sulphur dioxide is passed through an aqueous solution of Fe (III) salt ?
Answer:
When SO2 is passed through an aqueous solution of Fe (III), i.e., ferric salt, it is reduced to Fe (II) i,e., ferrous salt. Here, SO2 acts as a reducing agent.
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 16

Question 9.
Comment on the nature of two S – O bonds formed in SO2 molecule. Are the two S – O bonds in this molecule equal ?
Answer:
The two S – O bonds in SO2 molecule are covalent in nature. These are equal with bond length = 143 pm. The resonating structures of SO2 are as follows.
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 17
SO2 is a resonance hybrid of these two canonical forms.

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 10.
How is the presence of SO2 detected ?
Answer:
SO2 has a pungent odour. Two tests to detect the presence of SO2 are as follows :

  1. SO2 decolourises acidified KMnO4 solution.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 18
  2. SO2 Changes the colour of acidified potassium dichromate solution from orange to green.
    AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 19

Question 11.
Mention three areas in which H2S04 plays an important role.
Answer:
Uses of sulphuric acid.

  1. It is used in the manufacture of pigments, paints and dyestuff intermediate.
  2. It is used in petroleum refining.
  3. It is also used in fertilizer industry.

AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements

Question 12.
Write the conditions to maximize the yield of H2SO4 by contact process.
Answer:
The key step in the manufacture of H2SO4 is catalytic oxidation of SO2 to produce SO3 in presence of V2O5.
AP Inter 2nd Year Chemistry Study Material Chapter 6(b) Group-16 Elements 20
The reaction is exothermic, reversible and the forward reaction results in the decrease in volume. Thus according to Le-Chatelier’s principle, the forward reaction should be favoured by low temperature and high pressure. But the temperature should not be very low otherwise the. rate of reaction will become very slow.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Andhra Pradesh BIEAP AP Inter 2nd Year Chemistry Study Material Lesson 6(a) Group-15 Elements Textbook Questions and Answers.

AP Inter 2nd Year Chemistry Study Material Lesson 6(a) Group-15 Elements

Very Short Answer Questions

Question 1.
Why does the reactivity of nitrogen differ from phosphorus?
Answer:
Nitrogen gas exists as diatomic molecule. Dut to the presence of triple bond between Base N – Atoms bond dissociation energy is high (941.4 KJ /mol). Hence nitrogen is inert and unreactive.
Phosphorus is a tetra atomic molecule and P-P single bond is weaker than N ≡ N. P – P bond dissociation energy is 213 KJ/mole. Hence phosphorus is more reactive than Nitrogen.

Question 2.
How is nitrogen prepared in the laboratory ?
Write the chemical equations of the reactions involved.
Answer:
Preparation of di Nitrogen :

  • Very pure nitrogen is obtained by the thermal decomposition of sodium (or) barium a-zide.
    Ba(N3)2 → Ba + 3N2
  • In the laboratory dinitrogen is prepared by treating an aqueous solution of NH4Cl with NaNO2
    NH4Cl(aq) + NaNO2(aq) → N2(g) + 2H2O(I) + NaCl(aq)
  • Nitrogen can also be obtained by the thermal decompostion of ammonium dichromate.
    (NH4)3 Cr2O7 AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 1 N2 + 4H2O + Cr2O3

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 3.
Nitrogen exists as diatomic molecule and phosphorus as P4 – Why ?
Answer:
Nitrogen exists as diatomic molecule :

  • Nitrogen has small size and high electronegativity and nitrogen atom forms Pπ – Pπ multiple bonds with it self (triple bond). So it exists as a discrete diatomic molecule in elementary state.

Phosphorus exists as tetra atomic molecule :

  • Phosphorus has large size and less electronegative and it forms P-P single bonds. So it exists as tetra atomic i.e., P4.

Question 4.
Why does nitrogen show catenation properties less than phosphorus ?
Answer:
Explanation :
The single N-N bond is weaker than the single P-P bond due to high inter electronic repulsion of the non-boriding electrons in N2 because of small bond length. Therefore the catenation property is weaker in nitrogen as compared to phosphorus.

Question 5.
Nitrogen molecule is highly stable – Why ?
Answer:
Nitrogen Molecule is more stable because in between two nitrogen atoms of N2, a triple bond is present. To break this triple bond high energy is required (941.4KJ/mole).

Question 6.
Why are the compounds of bismuth more stable in +3 oxidation state ?
Answer:
Bismuth compounds are more stable in +3 oxidation state because ‘Bi’ exhibits +3 stable oxidation state instead of +5 due to inert pair effect.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 7.
What is allotropy ? Explain the different allotropic forms of phosphorus.
Answer:
Allotropy: The existance of an element in different physical forms having similar chemical properties is called allotropy.
Allotropes of ‘P’: → White ‘P’ (or) Yellow ‘P’.

  • Red ‘P’
  • Scarlet ‘P’
  • Violet ‘P’
  • α – black ‘P’
  • β – black ‘P’.

White phosphorus :

  • It is poisonous and insoluble in water and soluble in carbon disulphide and glows in dark. It is a translucent white waxy solid. ‘
  •  It dissolves in boiling NaOH solution and gives PH3.
    P4 + 3NaOH + 3H2 → PH3 + 3NaH2PO2. (sodium hypo phosphite)
  •  It is more reactive than other solid phases.
  • Bond angle is 60° and it readily catches fire.

Red phosphorus :
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 2

  • Red ‘F possesses iron grey lustre.
  • In is odour less, non poisonous and insoluble in water as well as CS2.
  • Red F’ is much less reactive than white ‘P’.
    Black ‘P’:
  • α – Black ‘P’: It is formed when red ‘P’ is heated in a sealed tube 803K.
  • β – Black F : It is prepared by heating white P’ at 473 K under high pressure.

Question 8.
How do you account for the inert character of dinitrogen ?
Answer:
Di Nitrogen is chemically inert. .
Explanation : In nitrogen molecule there exists a triple bond between two nitrogen atoms. To break this triple bond high energy (bond dissociation energy) is required (941.4 KJ/mole).

Question 9.
Explain the difference in the structures of white and red phosphorus.
Answer:
White ‘P’ molecule has tetrahedral structure (discrete molecule). Discrete ‘P’ molecules are held by vander waal’s forces
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 3
White phosphorus
Red ‘P’ is polymeric consisting of chains of P4 tetrahedron linked together through covalent bonds.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 4

Question 10.
How is α – black phosphorus prepared from red phosphorus?
Answer:
α – Black ‘P: It is formed when red? is heated in a sealed tube 803K.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 11.
Write the difference between the properties of white phosphorus and red phosphorus.
Answer:
White ‘P’

  • It is white waxy solid (translucent).
  • It is insolable in water and solable in CS2.
  • It is more reactive.
  • It is poisonous.

Red ‘P’

  • It possesses iron grey lustre.
  • It is in soluble in CS2 as cool water.
  • It is more reactive than white ‘P’.
  •  It is non – poisonous.

Question 12.
What is inert pair effect ?
Answer:
Inert pair effect: The reluctance of ns pair of electrons to take part in bond formation is called inert pair effect.
Bi exhibits +3 oxidation state instead of +5 due to inert pair effect.

Question 13.
Explain why is NH3 basic while BiH3 is only feebly basic.
Answer:
NH3 is basic while BiH3 is only feebly basic.

Explanation :
Due to small atomic size of nitrogen, the electron density on nitrogen atom is greater than that on Bi atom. So electron releasing tendency is greater in NH3.
Hence NH3 is basic and BiH3 is feebly basic.

Question 14.
Arrange the hydrides of group – 15 elements in the increasing order of basic strength and decreasing order of reducing character.
Answer:

  1. Increasing order of basic strength of Group – 15 elements hydrides is
    BiH3 < SbH3 < ASH3 < PH3 < NH3.
  2. Decreasing order of reducing character of Group – 15 elements hydrides is
    BiH3 > SbH3 > ASH3 > PH3 > NH3.

Question 15.
PH3 is a weaker base than NH3 – Explain.
Answer:
PH3 is a weaker base than NH3.

  • In NH3 nitrogen atom undergoes SP3 hybridisation and due to small size it has high electron density than in ‘P’ of PH3.
  • Due to large size of ‘P’ atom and availability of large surface area, lonepair of electron spread in PH3. Hence PH3 is weaker base then NH3.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 16.
A hydride of group -15 elements dissolves in water to form a basic solution. This solution dissolves the AgCl precipitate. Name the hydride. Write the chemical equations involved.
Answer:
Given a hydride of group – 15 elements dissolves in water to form a basic solution;This solution dissolves the Agcl precipitate.

The given hydride is ammonia. It forms basic solution when dissolved in water due to formation of OH ions.
NH3(g) + HxOl ⇌ \(\mathrm{NH}_4^{+}(\mathrm{aq})\) + OH(aq)
This solution dissolves AgCl. Ppt due to formation of complex compound.
AgCl(s) (white Ppt) + 2NH3(aq) → [Ag(NH3)2]Cl(aq) (colourless)

Question 17.
What happens when white phosphorus is heated with cone. NaOH solution in an inert atmosphere of CO2? [A.P. Mar. 19, 15]
Answer:
When white phosphorus heated with con. NaOH solution in an inert atomosphere of CO2 forms PH3.
P4 + 3NaOH + 3H2O → PH3 + 3NaH2PO2.

Question 18.
NH3 forms hydrogen bonds but PH3 does not – Why ?
Answer:
NH3 forms hydrogen bonds but PH3 does not.
Reason : Ammonia forms hydrogen bonds because it it a polar molecule and N-H bond is highly polar. Nitrogen has more electronegativity than phosphrous. In case of PH3 P-H bond polarity decreases.

Question 19.
The HNH angle is higher than HPH, HAsH and HSbH angles – Why?
Answer:
The central atom ε (where, ε = N, P, As, Sb, Bi) in all given hydrides is Sp3 hybridized. However, its electronegativity decreases and atomic size increase on moving down the group. Therefore is a gradual decrease in the force of repulsion between the shared electron pairs around the central atom. Thus, bond angle decreases as we move down the group.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 5

Question 20.
How do calcium phosphide and heavy water react?
Answer:
Calcium phosphide reacts with heavy water to form Deutero phosphine.
Ca3P2 + 6D2O → 3 Ca (OD)2 + 2PD3

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 21.
Ammonia is a good complexing agent – Explain with an example. [Mar. 14]
Answer:
NH3 is a lewis base and it donates electron pair to form dative bond with metal ions. This results in the formation of complex compound.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 6

Question 22.
A mixture of Ca3P2 and CaC2 is used in making Holme’s signal – Explain. [A.P. Mar. 16]
Answer:
A Mixture of Ca3P2 and CaC2 is used in Holme’s signal.This Mixture containing containers are pierced and thrown in the sea, when the gas is evolved bum and serve as a signal.
The spontaneous combustion of PH3 is the technical use of Holme’s signal.

Question 23.
Which chemical compound is formed in the brown ring test of nitrate ions ?
Answer:
In the brown ring test of nitrate salts a brown ring is formed. It’s formula is [Fe(H2O)5N0]+2.

Question 24.
Give the resonating structures of NO2 and N2O5.
Answer:
Resonance structure of NO2:
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 7
Resonance structure of N2O5
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 8

Question 25.
Why does R3P = O exist but R3N = O does not (R = alkyl group) ?
Answer:
R3P = O exist but R3N = O does not.
Explanation :
Nitrogen does not form dπ – Pπ multiple bond with oxygen because of lack of d – orbitals in Nitrogen atom. But in case of R3N = O the value of nitrogen should be 5. So these compounds do not exist where as in case of ‘P’ atom d-orbitals are available. So P-atom can able to form dπ – Pπ multiple bonds hence R3P = O exist.

Question 26.
How is nitric oxide (NO) prepared ?
Answer:
The catalytic oxidation of NH3 by atmospheric oxygen gives nitric oxide.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 9

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 27.
Give one example each of normal oxide and mixed oxide of nitrogen.
Answer:

  • Nitric oxide (NO) is an example of normal oxide of Nitrogen.
  • Dinitrogen trioxide (N2O3) is an example of mixed oxide of nitrogen.

Question 28.
NO is paramagnetic in gaseous state but diamagnetic in liquid and solid states – Why ?
Answer:
In gaseous state NO2 exists as a Monomer and contains one unpaired electron but in solid state it dimerises to N2O4 so it doesnot contain unpaired electron.
Hence NO2 is parce magnetic is geseous state but diamagnetic in solid state.

Question 29.
Give an example of
a) acidic oxide of phosphorus
b) neutral oxide of nitrogen.
Answer:
a) P2O5 (or) P4O10, phosphorus pentoxide is an example of acidic oxide of phosphorus
b) Nitrous oxide (N2O) and Nitric oxide (NO) are neutral oxides of nitrogen.

Question 30.
Explain the following
a) reaction of alkali with red phosphorus.
b) reaction between PCl3 and H3PO3.
Answer:
a) Red phosphorus reacts with an alkali to form hypophosphoric acid (H4P2O6)
b) PCl3 undergoes hydrolysis to form H3PO3
PCl3 + 3H2O → H3PO3 + 3HCl

Question 31.
How does PCl3 react with
a) CH3COOH
b) C2H5OH and
c) water.
Answer:
a) PCl3 reacts with CH3COOH and form phosphorous acid, acetyl chloride.
3CH3COOH + PCl3 → 3CH3COCl + H2PO3

b) PCl3 reacts with phosphorous acid and forms phosphorus acid, Ethyl chloride
3C2H5OH + PCl3 → 2C2H5Cl3 + H3PO3

c) PCl3 reacts with water (hydrolysis) to form phosphorus acid. It undergo hydrolysis in presence of moisture.
PCl3 + 3H2O → H3PO3 + 3HCl

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 32.
PCl3 can act as an oxidizing as well as a reducing agent – Justify.
Answer:
PCl3 acts as reducing agent. It is evidented by the following reaction.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 10
PCl3 acts as oxidising agent. It is evidented by the following reaction.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 11

Question 33.
Which of the following are not known ?
PCl3, AsCl3, SbCl3, NCl5, BiCl5, PH5
Answer:
NCl5, BiCl5, PH5 are not known in the given compounds.

Question 34.
Which of the following is more covalent – SbCl5 or SbCl3 ?
Answer:
SbCl5 (Penta halide) is more covalent than SbCl3 (Tri halide). Because Sb in the higher oxidation state exert more polarising power.

Question 35.
Write the oxidation states of phosphorus in solid PCl5.
Answer:

  • In solid state PCl5 exists as an ionic solid [PCl4)+ [PCl6]
  • ‘P’ exhibits +5 oxidation state in [PCl4)+ [PCl6]

Question 36.
Illustrate how copper metal can give different products on reaction with HNO3.
Answer:
Reaction of Cu metal with dilute HNO3.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 12
Reaction of Cu Metal with cone. HNO3
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 13

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 37.
Which oxide of nitrogen has oxidation number of N same as that in nitric acid ?
Answer:

  • In HNO3, ‘N’ has oxidation state +5.
  • Among oxides of nitrogen N2O5 exhibits ‘+5’ oxidation state.

Question 38.
Write the chemical reactions that occur in the manufacture of nitric acid.
Answer:
Chemical reactions involved in the manufacturing of HNO3.

  • Oxidation of NH3
    AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 14
  • NO2 formation :
    2NO + O2 ⇌ 2NO2
  • Formation of HNO3:
    3NO2 + H2O → 2HNO3 + NO

Question 39.
Iron becomes passive in cone. HNO3 – Why ?
Answer:
Iron becomes passive in cone. HNO3 due to formation of a passive film of oxide on the surface of iron.

Question 40.
Give the uses of
a) nitric acid and
b) ammonia.
Answer:
Use of HNO3:

  • HNO3 is used in the manufacture of ammonium nitrate for fertilisers and other nitrates for use in explosives and pyrotechnics.
  • HNO3 is used in the pickling of stainless steel.
  • HNO3 is used as oxidiser in rocket fuels.

Question 41.
What are the oxidation states of phosphorus in the following ?
i) H3PO3
ii) PCl3
iii) Ca3P2
iv) Na3PO4
v) POF3
Answer:
i) H3PO3
3(1) + x + 3(-3) = 0
x = +3

ii) PCl3
x + 3(-1) = 0
x = 3

iii) Ca3P2
3(+2) + 2x = 0
x = -3

iv) Na3PO4
3(1) + x + 4(-2) = 0
x = +5

v) POF3
x + (-2) + 3(-1) = 0
x = +5

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 42.
H3PO3 is diprotic while H3PO2 is monoprotic – Why ?
Answer:
H3PO3 is diprotic :
Structure of H3PO3:
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 15

H3PO2 is monoprotic :
Structure of H3PO2 :
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 16

Question 43.
Give the disproportionation reaction of H3PO3.
Answer:
Orthophosphoric acid (H3PO3) on heating disproportionates to give orthophosphoric acid and phosphine.
4H3PO3 → 3H3PO4 + PH3.

Question 44.
H3PO2 is a good reducing agent – Explain with an example.
Answer:
In H3PO2, two H-atoms are bonded directly to P-atom which imparts reducing character to the acid.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 45.
Draw the structures of
a) Hypo phosphoric acid
b) Cyclic meta phosphoric acid.
Answer:
a) Structure of hypo phosphoric acid (H4P2O6) :
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 17
b) Structure of cyclic meta phosphoric acid (HPO3)3
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 18

Short Answer Questions

Question 1.
Discuss the general characteristics of Group – 15 elements with reference to their electronic configuration, oxidation state, atomic size, ionization enthalpy and electronegativity.
Answer:
1) Electronic Configuration : All these elements of this group have s2p3 configuration in their respective outmost orbits.
Nitrogen (N) : [He] 2s22p3; phosphorus (P) : [Ne] 3s23p3, Arsenic (As) : [Ar] 3d104s24p3
Atimony (Sb) : [Kr] 4d10 5s2 5p3 and Bismuth (Bi) = [Xe] 4f14 5d10 6s26p3

2) Oxidation Spates : As per their configurations, these elements may utilize either the three electrons iti the p-orbitals or the five electrons present in both s -and p -orbitals in the exhibition of oxidation states. Consequently, they exhibit common oxidation states +III and +V.

3) Atomic size: In Group-15 elements atomic size increase down the group. A considerable increase in covalent radius observed from N to P and from As to Bi only a small increase in atomic size observed.

4) Ionisation enthalpy : In group – 15 elements ionisation enthalpy decreases down the group due to gradual increase in atomic size.

5) Electronegativity : In group-15 elements electronegativity values decrease down the group with increase in atomic size.

Question 2.
Discuss the trends in chemical reactivity of group – 15 elements.
Answer:
Nitrogen gas exists as diatomic molecule. Dut to the presence of triple bond between Base N – Atoms bond dissociation energy is high (941.4 KJ /mol). Hence nitrogen is inert and unreactive.

Phosphorus is a tetra atomic molecule and P-P single bond is weaker than N ≡ N. P – P bond dissociation energy U 213 KJ/mole. Hence phosphorus is more reactive than Nitrogen.

Explanation :
The single N-N bond is weaker than the single P-P bond due to high inter electronic repulsion of the non-bonding electrons in N2 because of small bond length. There fore the catenation property is weaker in nitrogen as compoud to phosphorares.
R3 P = O exist but R3N = O does not.

Explanation :
Nitrogen does not form dπ – Pπ multiple bond with oxygen because of lack of d – orbitals in Nitrogen atom. But in case of R3N = 0 the value of nitrogen should be 5. So these compounds do not exist where as in case of P’ atom d-orbitals are available. So P-atom can able to form dπ – Pπ multiple bonds hence R3P = 0 exist.

  1. Reactivity towards hydrogen: Group -15 elements forms EH3 type hydrides (E = Group – 15 elements) .
    Eg : PH3, NH3, AsH3, BiH3, SbH3.

    • Among above hydrides NH3 is mild reducing agent while BiH3 is strong reducing agent.
    • Stability of hydrides decreases from NH3 to BiH3.
    • Basicity of hydrides decreases as follows.
      NH3 > PH3 > AsH3 > SbH3 > BiH3.
  2. Reactivity towards Oxygen : These forms two types of oxides E2O3 and E<sub2O5.
    Eg : P2O3 > N2O5, N2O3, P2O5.

    • Acidic character of oxides decrease down the group.
    • E3O3 of N and P are acidic, As and Sb are amphoteric while ‘Bi’ is basic.
  3. Reactivity towards halogens : These elements forms two types of halides EX3 and EX5.
    ‘N’ – does not form penta halides due to lack of the d-orbitals.
    Penta halides are more covalent than tri halides because the elements in higher oxidation state have more polarising power.
  4. Reactivity towards metals : All these elements react with metals to form their binary compounds containing – 3 oxidation state.
    Eg. : Ca3N2, Ca3P2 etc.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 3.
How does P4 react with the following ?
a) SOCl2
b) SO2Cl2
Answer:
a) When P4 reacts with SOCl2 to form phosphorus trichloride.
P4 + 8 SOCl2 → 4 PCl3 + 4 SO2 + 2 S2Cl2
b) When P4 reacts with SOCl2 to form phosphorus pentachloride.
P4 + 10 SOCl2 → 4 PCl5 + 10 SO2

Question 4.
Explain the anomalous nature of nitrogen in group -15.
Answer:
Anomalous properties of Nitrogen : Nitrogen differs from the remaining elements of this group due to its small size, high electronegativity, high ionisation enthalpy and non-availability of d-orbitals.

Nitrogen gas exists as diatomic molecule. Due to the presence of triple bond between base N – Atoms bond dissociation energy is high (941.4 KJ /mol). Hence nitrogen is inert and un-reactive.

Explanation :

  • The single N-N bond is weaker than the single P-P bond due to high inter electronic repulsion of the non-bonding electrons in N2 because of small bond length. Therefore the catenation property is weaker in nitrogen as compound to phosphorus.
  • Nitrogen does not form penta halides due to lack of d-orbitals.
    R3 P = O exist but R3N = O does not.

Explanation :
Nitrogen does not form dπ – Pπ multiple bond with oxygen because of lack of d – orbitals in Nitrogen atom. But in case of R3N = O the value of nitrogen should be 5. So these compounds do not exist where as in case of ‘P atom d-orbitals are available. So P-atom can able to form dπ – Pπ multiple bonds hence R3P = O exist.

Question 5.
Complete the following reactions.
a) Ca3P2 + HaO →
b) P4 + KOH →
c) CuSO4 + NH3
d) Mg + N2
d) (NH4)2 + Cr2O7 AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 1
f) Decomposition of nitrous acid →
Answer:
a) Ca3P2 + 6 H2O → 3 Ca(OH)2 + 2PH3
b) P4 + 3 KOH + 3 H2O → PH3 + 3KH2 PO2
c) CuSO4(aq) (blue) + 4NH3 → [Cu (NH3)4] SO4 (deep blue)
d) 3 Mg + N2 → Mg3N2
e) (NH4)2 + Cr2O7 AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 1 N2 + 4 H2O + Cr2O3
f) 3HNO2 → HNO3 + 2 NO + H2O

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 6.
How does PCl5 react with the following?
a) Water
b) C2H5OH
c) CH3COOH
d) Ag
Answer:
a) PCl5 undergo hydrolysis to form phosphoric acid.
PCl5 + H2O → POCl3 + 2HCl
POCl3 + 3H2O → H3PO4 + 3 HCl

b) PCl5 reacts with C2H5OH to form Ethyl chloride.
C2H5OH + PCl5 → C2H5Cl + POCl3 + HCl

c) PCl5 reacts with CH3COOH to form acetyl chloride.
CH3COOH + PCl5 → CH3COCl + POCl3 + HCl

d) PCl5 reacts with Ag to form PCl3 and AgCl
PCl5 + 2 Ag → 2 AgCl + PCl3

Question 7.
Complete the following.
a) NH4NO3 AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 1
b) HNO3 + P4O10
c) Pb(NO3)2 AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 19
d) Zn + dil.HNO3
e) P4 + conc.HNO3
f) HgCl2 + PH3
Answer:
a) NH4NO3 AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 1 N2O (Nitrous oxide) + 2 H2O
b) 12 HNO3 + P4O10 → 6 N2O5 + 4 H3PO4
c) 2 Pb(NO3)2 AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 1 2 PbO + 4 NO2 + O2
d) 4 Zn + 10 HNO3 (dil) → 4 Zn (NO3)2 + 5 H2O + N2O
e) P4 + 20 HNO3 → 4H3PO4 + 20 NO2 + 4 H2O
f) 3 HgCl2 + 2 PH3 → Hg3P2 + 6 HCl

Long Answer Questions

Question 1.
How is ammonia manufactured by Habes process? Explain the reactions of ammonia with [A.P. Mar. 18]
a) ZnSO4(aq)
b) CuSO4(aq)
c) AgCl(s)
Answer:
In Haber process ammonia is directly synthesised from elements (nitrogen and hydrogen). The principle involved in this is
N2 (g) + 3H2O (g) ⇌ 2NH3 + 92.4kJ
This is a reversible exothermic reaction.
According to Le Chatelier’s principle favourable conditions for the better yield of ammonia are low temperature and high pressure. But the optimum conditions are
Temperature : 720k
Pressure : 200 atmospheres
Catalyst : Finely divided iron in the presence of molybdenum (Promoter).
Procedure : A mixture of nitrogen and hydrogen in the volume ratio 1 : 3 is heated to 725 – 775K at a pressure of 200 atmospheres is passed over hot finely divided iron mixed with small amount of molybdenum as promotor. The gases coming out of the catalyst chamber consists of 10 – 20% ammonia gas are cooled and compressed, so that ammonia gas is liquified, and the uncondensed gases are sent for recirculation.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 20
b) Aq.CuSO4 reacts with ammonia to form a deep blue complex,
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 21
c) Solid AgCl reacts with ammonia to form a colourless complex.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 22

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 2.
How is nitric acid manufactured by Ostwald’s process ? How does it react with the following ? [T.S. Mar. 19]
a) Copper
b) Zn
c) S8
d) P4
Answer:
Ostwald’s process : Ammonia, mixed with air in 1 : 7 or 1 : 8, when passed over a hot platinum gauze catalyst, is oxidised to NO mostly (about 95%).
The reaction is
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 23
The liberated heat keeps the catalyst hot. The ‘NO’ is cooled and is mixed with oxygen to give the dioxide, in large empty towers (oxidation chamber). The product is then passed in to warm water, under pressure in the presence of excess of air, to give HNO3.
4NO2 + O2 + 2H2O → 4HNO3.
The acid formed is about 61% concentrated.

a) Copper reacts with dil. HNO3 and cone. HNO3 and liberates Nitric Oxide and Nitrogen dioxide respectively
3 Cu + 5 HNO3 (dil) → 3 Cu (NO3)2 + 2 NO + 4 H2O
Cu + 4 HNO3 (cone.) → Cu (NO3)2 + 2 NO2 + 2 H2O

b) Zn reacts with dil. HNO3 and Cone. HNO3 and liberates Nitrous oxide and Nitrogen dioxide respectively.
4 Zn + 10 HNO3 (dil) → 4 Zn (NO3)2 + 5 H2O + N2O
Zn + 4 HNO3 (cone.) → Zn (NO3)2 + 2 H2 + 2 NO2

c) S8 reacts with cone, nitric acid to form Sulphuric acid, NO2 gas.
S8 + 48 HNO3 → 8 H2SO4 + 48 NO2 + 16 H2O
d) P4 reacts with cone, nitric acid to form phosphoric acid and NO2 gas.
P4 + 20 HNO3 → 4 H3PO4 + 20 NO2 + 4 H2O

Textual Examples

Question 1.
Though nitrogen exhibits +5 oxidation state, it does not form peniahalide. Give reason.
Answer:
Nitrogen with n = 2, has s and p orbitals only. It does not have d orbitals to expand its covalence beyond four. That is why it does not form pentahalide.

Question 2.
PH3 has lower boiling point than NH3. Why ? [T.S. Mar. 16]
Answer:
Unlike NH3. PH3 molecules are not associated through hydrogen bonding in liquid state. That is why the boiling point of PH3 is lower than NH3.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 3.
Write the reaction of thermal decomposition of sodium azide.
Answer:
Thermal decomposition of sodium azide gives dinitrogen gas.
2NaN3 → 2Na + 3N2

Question 4.
Why does NH3 act as a Lewis base ?
Answer:
Nitrogen atom in NH3 has one lone pair of electrons with is available for donation. Therefore, it acts as a Lewis base.

Question 5.
Why does NO2 dimerise ?
Answer:
NO2 contains odd number of valence electrons. It be haves as a typical odd molecule. On dimerisation. It is converted to stable N2O4 molecule with even number of electrons.

Question 6.
In what way can it be proved that PH3 is basic in nature ?
Answer:
PH3 reacts with acids like HI to form PH4I which shows that it is basic in nature.
PH3 + HI →PH4I
Due to lone pair on phosphorus atom, PH3 is acting as a Lewis base in the above reaction.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 7.
Why does PCl3 fume in moisture ?
Answer:
PCl3 hydrolyses in the presence of moisture giving fumes of HCl.
PCl3+ 3H2O → H3PO3 + 3HCl

Question 8.
Are all the five bonds in PCl5 molecule equivalent ? Justify your answer.
Answer:
PCl5 has a trigonal bipyramidal structure and the three equatorial P-Cl bonds are equivalent. While the two axial bonds are different and longer than equatorial bonds.

Intext Questions

Question 1.
Why are pentahalides more covalent than trihalides ?
Answer:
Higher the positive oxidation state of central atom, more will be its polarising power, which in turn increases the covalent character of the bond formed between the central atom and the halogen atom. In pentahalides, the central atom is in +5 oxidation state while in trihalides, it is in +3 oxidation state. Therefore, pentahalides are more covalent than trihalides.

Question 2.
Why is BiH3 the strongest reducing agent amongst all the hydrides of Group 15 elements ?
Answer:
Among the 15 group elements, the size of Bi atom is largest and hence, the Bi-H bond length is largest or Bi-H bond dissociation energy is lowest. That’s why Bi-H bond dissociates more readily than the other hydrides of the group and hence, BiH3 is the strongest reducing agent.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 3.
Why is N2 less reactive at room temperature ?
Answer:
Dinitrogen is inert of less reactive because the bond enthalpy of N ≡ N bond is very high.

Question 4.
Mention the conditions required to maximise the yield of ammonia.
Answer:
Ammonia is produced by Haber’s process as.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 24
Yield of ammonia is favoured by high pressure according to Le-Chatelier’s principle, Other conditions, that favour the production of ammonia are as follows :

  1. Temperature – approximately 700 K
  2. Pressure – 200 atm or 200 × 105 Pa
  3. Catalyst – Iron oxide
  4. Promotor – Molybdenum of K2O and Al2O3

Question 5.
How does ammonia react with a solution of Cu2+?
Answer:
When ammonia (aqueous solution is ammonium hydroxide) reacts with a solution of Cu2+, a deep blue solution is obtained due to the formation of tetraamine copper (II) ion.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 25

Question 6.
What is the covalence of nitrogen in N2O5?
Answer:
Structural formula of N2O5
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 26
Since, N atom has 4 shared electron pairs; the valence of N is 4.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 7.
Bond angle in PH4+ is higher than that in PH3. Why ?
Answer:
In both PH4+ and PH3, phosphorus atom is sp3 hybridized. In PH4+, all the four orbitals are bonded whereas in PH3, there is a lone pair of electrons too. Due to lone pair-bonded pair repulsion in PH3, the bond angle is less than 109.5°.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 27

Question 8.
What happens when white phosphorus is heated with concentrated NaOH solution in an inert atmosphere of CO2? .
Answer:
White phosphorus dissolves in boiling NaOH in an inert atmosphere of CO2 producing phosphine (PH3) gas and sodium hypophosphite (NaH2PO2).
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 28

Question 9.
What happens when PCl5 is heated ?
Answer:
In PCl3, there are 5P – Cl bonds, out of which three are equatorial (longer) and two are axial (shorter). When PCl5 is heated strongly, two less stable axial bonds break and phosphorus trichloride (PCl3) is formed.
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 29
Structure of PCl5 showing axial and equatorial bonds.

AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements

Question 10.
Write a balanced equation for the hydrolytic reaction of PCl5 in heavy water.
Answer:
Reaction of PCl5 in heavy water (D2O)
AP Inter 2nd Year Chemistry Study Material Chapter 6(a) Group-15 Elements 30

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Andhra Pradesh BIEAP AP Inter 2nd Year Chemistry Study Material Lesson 3(a) Electro Chemistry Textbook Questions and Answers.

AP Inter 2nd Year Chemistry Study Material Lesson 3(a) Electro Chemistry

Very Short Answer Questions

Question 1.
What is a galvanic cell or a voltaic cell? Give one example. [T.S. Mar. 19]
Answer:
Galvanic cell: A device that converts chemical energy into electrical energy by the use of spontaneous redox reaction is called a Galvanic cell (or) voltaic cell.
Eg.: Daniell cell.

Question 2.
Write the Chemical reaction used in the construction of the Daniell cell together with the half-cell reactions.
Answer:
The chemical reactions used in the construction of the Daniell cell.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 1

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 3.
Name the two half-cell reactions that are taking place in the Daniell cell.
Answer:
The two half cell reactions that are takes place in the Daniell cell are oxidation and reduction.
Zn → Zn+2 + 2e (Oxidation)
Cu+2 + 2e → Cu (Reduction)

Question 4.
How is a galvanic cell represented on paper as per IUPAC convention ? Give one example.
Answer:
Representation of a Galvanic cell:
As per IUPAC convention

  • Oxidation half cell represented on the left side.
  • Reduction half cell represented on the right side.
  • Oxidation half cell and reduction half cell are connected by a salt bridge indicated by two vertical parallel lines.
    Eg.: Cu(s) | \(\mathrm{Cu}_{(\mathrm{aq})}^{+2}\) || \(\mathrm{Ag}_{(\mathrm{aq})}^{+}\) | Ag(s)

Question 5.
Write the cell reaction taking place in the cell
Cu(s) | \(\mathrm{Cu}_{(\mathrm{aq})}^{+2}\) || \(\mathrm{Ag}_{(\mathrm{aq})}^{+}\) | Ag(s)
Answer:
Given cell is Cu(s) | \(\mathrm{Cu}_{(\mathrm{aq})}^{+2}\) || \(\mathrm{Ag}_{(\mathrm{aq})}^{+}\) | Ag(s)
Cu → Cu+2 + 2e (Oxidation)
2Ag+ + 2e → 2Ag (Reduction)

Question 6.
What is standard hydrogen electrode ?
Answer:
The electrode whose potential is known as standard electrode (or) standard hydrogen electrode.

To determine the potential of a single electrode experimentally it combine with standard hydrogen electrode and the EMF of cell so constructed is measured with potentiometer.

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 7.
Give a neat sketch of standard hydrogen electrode.
Answer:
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 2
Standard Hydrogen Electrode (SHE)

Question 8.
What is Nernst equation ? Write the equation for an electrode with electrode reaction Mn+(aq) + ne ⇌ M(s).
Answer:
The electrode potential at any concentration measured with respect to standard hydrogen electrode is represented by Nernst equation.
Nernst equation is
E = E0 + \(\frac{\mathrm{RT}}{\mathrm{nF}}\) ln[Mn+] [Metal Electrodes]
Given electrode reaction is \(\mathrm{M}_{(\mathrm{aq})}^{\mathrm{n}+}\) + ne ⇌ M(s)
For the above electrode reaction Nemst equation is
E(Mn+/M) = \(\mathrm{E}_{\left(\mathrm{M}^{\mathrm{n}+} / \mathrm{M}\right)}^0+\frac{\mathrm{RT}}{\mathrm{nF}} \ln \left[\mathrm{M}^{\mathrm{n}+}\right]\)
Here E(Mn+/M) = Electrode potential
E0E(Mn+/M) = Standard Electrode potential
R = gas constant = 8.314 J/k.mole
F = Faraday = 96487 c/mole
T = temperature
[Mn+] = concentration of species Mn+

Question 9.
A negative E0 indicates that the redox couple is _______________ reducing couple than H+/H2, couple, (powerful or weak)
Answer:
A negative E0 value indicates that the redox couple is powerful reducing couple than H+/H2 couple.

Question 10.
A positive E0 indicates that the redox couple is a weaker _____________ couple than H+/H2 couple. (oxidising or reducing)
Answer:
A positive E0 value indicates that the redox couple is as weaker reducing couple than H+/H2 couple.

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 11.
Write the Nernst equation for the EMF of the cell .
Ni(s) |\(\mathrm{Ni}_{(\mathrm{aq})}^{2+}\) || \(\mathrm{Ag}_{(\mathrm{aq})}^{+}\) | Ag
Answer:
Given cell is Ni(s) |\(\mathrm{Ni}_{(\mathrm{aq})}^{2+}\) || \(\mathrm{Ag}_{(\mathrm{aq})}^{+}\) | Ag
Nernst equation for the cell is
Ecell = \(\mathrm{E}_{\text {cell }}^0+\frac{\mathrm{RT}}{\mathrm{nF}} \ln \frac{\left[\mathrm{Ag}^{+}\right]}{[\mathrm{Ni}]^{+2}}\)

Question 12.
Write the cell reaction for which Ecell = \(\mathrm{E}_{\text {cell }}^0-\frac{\mathrm{RT}}{2 \mathrm{~F}} \ln \frac{\left[\mathrm{Mg}^{2+}\right]}{\left[\mathrm{Ag}^{+}\right]^2}\)
Answer:
Given Ecell = \(\mathrm{E}_{\text {cell }}^0-\frac{\mathrm{RT}}{2 \mathrm{~F}} \ln \frac{\left[\mathrm{Mg}^{2+}\right]}{\left[\mathrm{Ag}^{+}\right]^2}\)
The cell reaction is Mg(s)/\(\mathrm{Mg}_{(\mathrm{aq})}^{2+}\) || \(\mathrm{Ag}_{(\mathrm{aq})}^{+}\) | Ag(s)
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 3

Question 13.
How is E0 cell related mathematically to the equilibrium constant Kc of the cell reaction?
Answer:
Relation between E0 cell and equilibrium constant KC of the cell reaction
\(\mathrm{E}_{\text {cell }}^0=\frac{2.303 \mathrm{RT}}{\mathrm{nF}}\) log KC
n = number of electrons involved
F = Faraday 96500 C mol-1,
T = Temperature
R = gas constant

Question 14.
How is Gibbs energy (G) related to the cell emf (E) mathematically ?
Answer:
Relation between Gibb’s energy (G) and emf (E) mathematically
∆G0 = -nFE(cell)
∆G = change in Gibb’s energy
n = number of electrons involved
F = Faraday = 96500 C mol-1

Question 15.
Define conductivity of a material. Give its SI units.
Answer:
The reciprocal of specific resistance (or) resistivity is called conductivity.
It is represented by (κ)
(Or)
The conductance of one unit cube of a conductor is also called conductivity.
SI units : ohm-1 m-1 (or) Sm-1 S = Siemen

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 16.
What is cell constant of a conductivity cell ?
Answer:
Resistance R = ρ . \(\frac{l}{\mathrm{~A}}\) = \(\frac{l}{\kappa \cdot \mathrm{A}}\)
\(\frac{l}{\mathrm{~A}}\) = cell constant
Cell constant G = \(\frac{l}{\mathrm{~A}}\) = R × κ
The cell constant of a conductivity cell is the product of resistance and specific conductance.

Question 17.
Define molar conductivity ∧m and how is it related to conductivity (κ) ?
Answer:
Molar conductivity: The conductivity of a volume of solution containing one gram molecular weight of the electrolyte placed between two parallel electrodes separated by a distance of unit length of 1 meter is called molar conductivity (∧m).
Relation between conductivity and molar conductivity:
m = \(\frac{\kappa}{\mathrm{c}}\);
∴ c = constant

Question 18.
Give the mathematical equation which gives the variation of molar conductivity with ∧m the molarity (c) of the solution ?
Answer:
The mathematical equation which gives the variation of molar conductivity ∧m with the molarity (c) of the solution is
m = \(\frac{\kappa}{1000\left(\text { lit } / \mathrm{cm}^3\right) \times \text { molarity (moles/lit) }}=\frac{\kappa \times 1000\left(\mathrm{~cm}^3 / \mathrm{lit}\right)}{\text { molarity (mole/lit) }}\)

Question 19.
State Kohlrausch’s law of independent migration of ions.
Answer:
Kohlrausch’s law of independent migration of ions: The limiting molar conductivity of an electrolytes can be represented as the sum of the individual contributions of the anion and the cation of the electrolytes.
\(\lambda_{\mathrm{m}(\mathrm{AB})}^0=\lambda_{\mathrm{A}^{+}}^0+\lambda_{\mathrm{B}^{-}}^0\)
\(\lambda_m^0\) = Limiting molar conductivity
\(\lambda_A^0\) = Limiting molar conductivity of cation
\(\lambda_B^0\) = Limiting molar conductivity of anion

Question 20.
State Faraday’s first law of electrolysis. [A.P. Mar. 18, 16] [Mar. 14]
Answer:
The amount of chemical reaction which occurs at any electrode during electrolysis is proportional to the quantity of current passing through the electrolyte.
(Or)
The mass of the substance deposited at an electrode during the electrolysis of electrolyte is directly proportional to quantity of electricity passed through it.
m ∝ Q; m ∝ c × t
m = ect; m = \(\frac{\text { Ect }}{96,500}\)
e = electrochemical equivalent
c = Current in amperes
t = time in seconds
E = Chemical equivalent

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 21.
State Faraday’s second law of electrolysis. [T.S. Mar. 18] [Mar. 14]
Answer:
The amounts of different substances liberated when the same quantity of electricity is passed through the electrolytic solution are proportional to their chemical equivalent weights.
m ∝ E

Qustion 22.
What are the products obtained at the platinum anode and the platinum cathode respectively in the electrolysis of fused or molten NaCl ?
Answer:
In the electrolysis of fused (or) molten NaCl with platinum anode and platinum cathode, sodium metal obtained at cathode and chlorine gas at anode
2 NaCl → 2 Na+ + 2Cl
2 Cl → Cl2 + 2e (anode)
2 Na+ + 2e → 2 Na (cathode)

Question 23.
Give the products obtained at the platinum electrodes (cathode and anode) when aqueous solution of K2SO4 is electrolysed.
Answer:
During the electrolysis of aq.K2SO4 using pt-anode and pt-cathode hydrogen gas liberated at cathode and oxygen gas at anode.
K2SO4 → 2K+ + \(\mathrm{SO}_4^{-2}\)
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 4

Question 24.
Write the chemical equation corresponding to the oxidation of H2O(l) at the platinum anode.
Answer:
The chemical equation corresponding to the oxidation of H2O(l) at the platinum anode is
2H2O(l) → O2(g) + 4H+(aq) + 4e

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 25.
Give the chemical equation that represents the reduction of liquid water H2O(l)at the platinum cathode.
Answer:
The chemical equation that represents the reduction of liquid water H2O(l) at the platinum cathode is H2O(l) + e → \(\frac{1}{2}\)H2(g) + OH

Question 26.
What is a primary battery ? Give one example. [A.P. Mar. 17]
Answer:
The batteries which after their use over a period of time, becomes dead and the cell reaction is completed and this cannot be reused again are called primary batteries.
Eg: Leclanche cell, dry cell.

Question 27.
Give one example for a secondary battery. Give the cell reaction.
Answer:
Lead storage battery is an example of secondary battery.
The cell reactions when the battery is in use are
Pb(s) + \(\mathrm{SO}_4^{-2} \text { (aq) }\) → PbSO4(s) + 2e (Anode)
PbO2(s) + \(\mathrm{SO}_4^{-2} \text { (aq) }\) → \(4 \mathrm{H}_{(\mathrm{aq})}^{+}\) + 2e → PbSO4(s) + 2 H2O(l) (Cathode)
Overall cell reaction is .
Pb(s) + PbO2(s) + 2H2SO4(aq) → 2PbSO4(s) + H2O(l)

Question 28.
Give the cell reaction of nickel-cadmium secondary battery.
Answer:
The cell reaction of nickel-cadmium secondary battery is
Cd(s) + 2Ni(OH)3(s) → CdO(s) + 2 Ni(OH)2(s) + H2O(l)

Question 29.
What is a fuel cell ? How is it different from a conventional galvanic cell ?
Answer:
A fuel cell is a galvanic cell in which the chemical energy of fuel-oxidant system is converted directly into electrical energy.

  • Conventional Galvanic cell converts chemical energy into electrical energy by spontaneous redox reactions.
  • Fuel cell convert energy of combustion of fuels like hydrogen, methane etc., into electrical energy. These cause less pollution.

Question 30.
Give the electrode reactions occurijig at the anode and at the cathode in H2, O2, fuel cell.
Answer:
At cathode : O2(g) + 2 H2, O2 + 4e → 4OH(aq)
At anode : 2H2(g) + 4 OH(aq) → 4H2O(l) + 4e
Overall reaction : 2H2(g) + O2(g) → 2H2O(l)

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 31.
What is metallic corrosion ? Give one example. [T.S. Mar. 17; A.P. Mar. 15]
Answer:
Metallic corrosion: The natural tendency of conversion of a metal into its mineral compound form i n interaction with the environment is known as metallic corrosion.
Eg : 1) Iron converts itself into its oxide [Rusting] (Fe2O3)
2) Silver converts itself into it’s sulphate
[tarnishing] [Ag2S]

Question 32.
Give the electro-chemical reaction that represents the corrosion or rusting of iron.
Answer:
Rusting of iron: Iron converts itself into its oxixe Fe2O3
The electrochemical reaction that represents the corrosion or rusting of iron is
Anode : 2 Fe(s) → 2 Fe2+ + 4e \(\mathrm{E}_{\left(\mathrm{Fe}^{2+} / \mathrm{Fe}\right)}^0\) = -0.44V
Cathode : O2(g) + 4H+(aq) + 4e → 2H2O(l) \(\mathrm{E}_{\mathrm{H}^{+} / \mathrm{O}_2 / \mathrm{H}_2 \mathrm{O}}^{\Theta}\) = 1.23V
Overall reaction: 2 Fe(s) + O2(g) + 4H+(aq) → 2Fe+2aq + 2 H2O(l) \(\mathrm{E}_{\text {(cell) }}^{\Theta}\) = 1.67 V

Short Answer Questions

Question 1.
What are galvanic cells ? Explain the working of a galvanic cell with a neat sketch taking Daniell cell as example. [T.S. & A.P. Mar. 18]
Answer:
Galvanic cell: A device which converts chemical energy into electrical energy by the use of spontaneous redox reaction is called Galvanic cell (or) voltaic cell.
Eg : Daniell cell.

Daniell cell: It is a special type of galvanic cell. It contains two half cells in the same vessel. The vessel is devided into two chambers. Left chamber is filled with ZnSO4 (aq) solution and Zn – rod is dipped into it. Right chamber is filled with aq. CuSO4 solution and a copper rod is dipped into it. Process diaphragm acts as Salt bridge. The two half cell’s are connected to external battery.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 5
Cell reactions :
Ion Zn/ZnSO4 half cell, oxidation, reaction occurs
Zn → Zn2+ + 2e
Ion Cu/CuSO4 half cell, reduction reaction occurs.
The net cell reaction is
Zn + Cu+2 ⇌ Zn+2 + Cu
Cell is represented as Zn / Zn+2 || Cu+2 / Cu

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 2.
Give the construction and working of a standard hydrogen electrode with a neat diagram.
Answer:
To determine the potential of a single electrode experimentally, it is combined with a standard hydrogen electrode (electrode whose potential is known) and the EMF of the cell. So constructed is measured with a potentiometer. Standard hydrogen electrode is constructed and is used as standard electrode or reference electrode. Standard Hydrogen Electrode (SHE) or Normal Hydrogen Electrode (NHE).

Pure hydrogen gas is bubbled into a solution of 1M HCl along a platinum electrode coated with platinum block. A platinum block electrode placed in the solution at atmospheric pressure.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 6
Generally the electrode is fitted into the tube. The tube will have two circular small holes. This tube is immersed in the acid solution such that one haf of the circular hole is exposed to air and another half in the solution.
The following equilibrium exists at the electrode.
\(\frac{1}{2}\) H2(g) (1atm) ⇌ \(\mathrm{H}_{(\mathrm{aq})}^{+}\) (1M) + e

Question 3.
State and explain Nernst equation with the help of a metallic electrode and a non-metallic electrode.
Answer:
The electrode potential at any concentration measured with respect to standard hydrogen electro represented by Nernst equation.
Nernstst equation is
E = E0 + \(\frac{\mathrm{RT}}{\mathrm{nF}}\) In [Mn+] [Metal Electrodes]
Given electrode reaction is
\(\mathrm{M}_{(\mathrm{aq})}^{\mathrm{n}+}\) + ne ⇌ M(s)
For the above electrode reaction Nernst equation is
E(Mn+ / M) = \(\mathrm{E}_{\left(\mathrm{M}^{\mathrm{n}+} / \mathrm{M}\right)}^0+\frac{\mathrm{RT}}{\mathrm{nF}} \ln \left[\mathrm{M}^{\mathrm{n}+}\right]\)
Here E(Mn+ / M) = Electrode potential
E0(Mn+ / M) = Standard Electrode potential
R = gas constant = 8.314 J/k.mole
F = Faraday = 96487 c/mole
T = temperature
[Mn+] = concentration of species Mn+
For non-metal electrodes :
E = E0 – \(\frac{\mathrm{RT}}{\mathrm{nF}}\) ln C ; C = concentration
Example for metal electrode:
Given cell is Ni(s) | \(\mathrm{Ni}_{(a q)}^{2+}\) || \(\) | Ag
Nernst equation for the cell is
Ecell = \(\mathrm{E}_{\text {cell }}^0+\frac{\mathrm{RT}}{\mathrm{nF}} \ln \frac{\left[\mathrm{Ag}^{+}\right]}{[\mathrm{Ni}]^{+2}}\)
Example for non-metal electrode: Pt, Cl2/Cl
E = E0 – \(\frac{\mathrm{RT}}{\mathrm{nF}}\) log C
E = E0 – \(\frac{\mathrm{RT}}{\mathrm{F}}\) log C

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 4.
Explain with a suitable example the relation between the Gibbs energy of chemical reaction (G) and the functioning of the electrochemical cell.
Answer:
Electrical work done in one second in an electrochemical cell is equal to the electrical potential multiplied by the total charge passing. If we want to obtain maximum work from a galvanic cell then the charge has to be passed reversibly. The reversible work done by a galvanic cell is equal to decrease in its Gibbs energy and therefore, if the emf is E and nF is the amount of charge passing and ∆rG = – nFE[cell]
It may be remembered that E[cell] is an intensive parameter but ∆rG is an extensive thermodynamic property and the value depends on n. Thus, if we write the reaction
Zn[s] + \(\mathrm{Cu}_{[\mathrm{aq}]}^{2+}\) → \(\mathrm{Zn}_{\text {[aq] }}^{2+}\) + Cu[s]
rG = -2FE[cell]
but when we write the reaction
2 Zn[S] + 2 \(\mathrm{Cu}_{[\mathrm{aq}]}^{2+}\) → \(\mathrm{Zn}_{[\text {aq] }}^{2+}\) + 2 Cu[s]
rG = -4FE[cell]
If the concentration of each of the reacting species is unity, then E[cell] = \(\mathrm{E}_{\text {[cell] }}^{\Theta}\) and we have
rG -nF\(\mathrm{E}_{\text {[cell] }}^{\Theta}\)
We can calculate equilibrium constant using the equation
rG = – RTln Kc

Question 5.
On what factors the electrical conductance of an aqueous solution of electrolyte depends?
Answer:
The electrical conductance of an aqueous solution of electrolyte depends on

  1. The nature of the electrolyte
  2. Size and solvation of the ions formed in the dissociation of the electrolyte.
  3. The nature and viscosity of the solvent.
  4. Concentration of the electrolyte
  5. Temperature

Question 6.
How is molar conductivity of an aqueous electrolyte solution measured experimentally ?
Answer:
Molar conductivity: The conductivity of a volume of solution containing one gram molecular weight of the electrolyte placed between two parallel electrodes separated by a distance of unit length of 1 meter is called molar conductivity (∧m)
Relation between conductivity and molar conductivity:
m = \(\frac{\kappa}{\mathrm{c}}\);
∴ c = constant .
The conductance of a solution measured in a conductivity cell.
By using the conductivity cells
Resistance (R) = \(\frac{l}{\kappa \times A}\)
l = distance between electrodes; A = Area of cross section; K = conductivity
G* = \(\frac{l}{\mathrm{~A}}\) = cell constant
-4 Cell constant is measured by measuring the resistance of the cell containing a solution whose conductivity is known.
-4 Cell constant determined is used for measuring the resistance (or) conductivity.
∴ molar conductivity ∧(m) = \(\frac{\kappa}{\mathrm{c}}\)
(m) = \(\frac{\kappa\left(\mathrm{Scm}^{-1}\right)}{1000\left(\mathrm{~L}^{-3}\right) \text { molarity (moles/lit) }}\)

Question 7.
Explain the variation of molar conductivity with the change in the concentration of the electrolyte. Give reasons.
Answer:
Molar conductivity of a solution at a given concentration is the conductance of the volume of solution containing one mole of electrolyte kept between two electrodes with area of cross section A and distance of unit length.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 7
Molar conductivity ∧m = \(\frac{\mathrm{\kappa A}}{l}\)
m = κ × V (∵ l = 1, A = V)
V = Volume containing 1 mole of electrolyte
Molar conductivity increase with decrease in concentration of electrolyte (or) increase in dilution. This is due to total volume V of solution containing one mole of electrolyte also increases.

When concentration approaches to zero, the molar conductivity is known as limiting molar conductivity and it is represented by
For strong electrolytes molar conductivity is given by \(\wedge_m^0\)
m = \(\wedge_m^0\) – Ac1/2
For weak electrolytes limiting’ molar conductivity is given by
λAB+ = λA+ + λB (Kohlrausch law)

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 8.
State and explain Kohlrausch’s law of independent migration of ions. [T.S. Mar. 16; A.P. Mar. 15]
Answer:
Kohlrausch’s law of independent migration of ions: The limiting molar conductivity of an electrolytes can be represented as the sum of the individual contributions of the anion and the cation of the electrolytes.
\(\lambda_{\mathrm{m}(\mathrm{AB})}^0=\lambda_{\mathrm{A}^{+}}^0+\lambda_{\mathrm{B}^{-}}^0\)
\(\lambda_{\mathrm{m}}^0\) = Limiting molar conductivity
\(\lambda_{\mathrm{A}^{+}}^0\) = Limiting molar conductivity of cation
\(\lambda_{\mathrm{B}^{-}}^0\) = Limiting molar conductivity of anion

Applications:

  1. Kohlrausch’s law is used in the calculation of the limiting molar conductivity of weak electrolytes.
    Eg : λCH3COOH = λCH3COONa + λHCl – λNaCl
    = λCH3COO + λNa+ + λH+
  2. This law is used in the calculation of degree of dissociation of a weak electrolyte.
  3. This law is used in the calculation of solubility of sparingly soluble salts like AgCl, BaSO4 etc.

Question 9.
What is electrolysis ? Give Faraday’s first law of electrolysis. [A.P. & T.S. Mar. 15]
Answer:
Electrolysis : The decomposition of a chemical compound in the molten state or in the solution state into its constituent elements under the influence of an applied EMF is called electrolysis.
The amount of chemical reaction which occurs at any electrode during electrolysis is proportional to the quantity of current passing through the electrolyte.
(Or)
The mass of the substance deposited at an electrode during the electrolysis of electrolyte is directly proportional to quantity of electricity passed through it.
m ∝ Q; m ∝ c × t
m = ect; m = \(\frac{\text { Ect }}{96,500}\)
e = electrochemical equivalent
t = time in seconds
c = Current in amperes
E = Chemical equivalent

Question 10.
What are the products obtained at the cathode and anode during the electrolysis of the following when platinum electrodes are used in the electrolysis
a) Molten KCl
b) Aq. CuSO4 solution
c) Aq. K2SO4 solution
Answer:
a) During the electrolysis of molten KCl using platinum electrodes. Potassium is obtained at cathode and chlorine at anode.
2 KCl → 2K+ + 2Cl
2 Cl → Cl2 + 2e (anode)
2K+ + 2e → 2K (cathode)

b) During the electrolysis of aq.CuS04 solution using platinum electrodes, O2 gas liberated at anode and Cu deposited at cathode.
2 CuSO4 → 2 Cu+2 + 2 SO-24
2 Cu+2 + 4e → 2 Cu (cathode)
2H2O – 4e → O2 + 4H+ (anode)

c) During the electrolysis of aq.k2SO4 using pt-anode and pt-cathode hydrogen gas liberated at cathode and oxygen gas at anode.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 8

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 11.
What are primary and secondary batteries ? Give one example for each.
Answer:
The batteries which after their use over a period of time’ becomes dead and the cell reaction is completed and this cannot be reused again are called primary batteries.
Eg : Leclanche cell, dry cell.

Secondary battery: A secondary battery is the battery in which after it’s use can be recharged and can be used again.
A good secondary battery undergoes large no. of discharging and charging cycles.
Lead storage battery is an example of secondary battery. ,
The cell reactions when the battery is in use are
Pb(s) + \(\mathrm{SO}_4^{-2}(\mathrm{aq})\) → PbSO4(s) + 2e (Anode)
PbO2(s) + \(\mathrm{SO}_4^{-2}(\mathrm{aq})\) → \(4 \mathrm{H}_{\text {(aq) }}^{+}\) + 2e → PbSO4(s) + 2 H2O(l) (Cathode)
Overall cell reaction is
Pb(s) + PbO2(s) + 2H2SO4(aq) → 2PbSO4(aq) + 2H2O(l)

Question 12.
What are the fuel cells ? How are they different from galvanic cells ? Give the construction of H2, O2 fuel cell? [T.S. Mar. 17]
Answer:
A fuel cell is a galvanic cell in which the chemical energy of fuel-oxidant system is converted directly into electrical energy.

  • Conventional Galvanic cell converts chemical energy into electrical energy by spontaneous redox reactions.
  • Fuel cell convert energy of combustion of fuels like hydrogen, methane etc., into electrical energy. These cause less pollution.

H2 – O2 fuel cell: In this cell, hydrogen and oxygen are bubbled through porous carbon electrodes into Cone. NaOH solution. Electrodes are embedded with suitable catalysts. The electrode reactions are:
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 9
Overall reaction : 2H2(g) + O2(g) → 2 H2O(l)
The cell functions as long as the reacting gases are in supply. The heat of combustion is directly converted into electrical energy.

Question 13.
What is metallic corrosion ? Explain it with respect to iron corrosion.
Answer:
Metallic corrosion: The natural tendency of conversion of a metal into its mineral compound form on interaction with the environment is known as metallic corrosion.
Eg.: 1) Iron converts itself into its oxide [Rusting] (Fe2O3)
2) Silver converts itself into it’s sulphate [tarnishing] [Ag2S]
Rusting of iron : Iron converts it self into its oxixe Fe2O3
The electrochemical reaction that represents the corrosion or rusting-of iron is
Anode : 2 Fe(s) → 2 Fe2+ + 4e \(\mathrm{E}_{\left(\mathrm{Fe}^{2+} / \mathrm{Fe}\right)}^0\) = -0.44 V
Cathode : O2(g) + \(4 \mathrm{H}_{\text {(aq) }}^{+}\) + 4e → 22O(l) \(\) = 1.23 V
Overall reaction : 2 Fe(s) + O2(g) + \(4 \mathrm{H}_{\text {(aq) }}^{+}\) → \(2 \mathrm{Fe}_{\mathrm{aq}}^{+2}\) + 2 H2O(l) \(\mathrm{E}_{(\text {cell) }}^{\Theta}\) = 1.67 V

Long Answer Questions

Question 1.
What are electro chemical cells ? How are they constructed ? Explain the working of the different types of galvanic cells ?
Answer:
Electrochemical cell is a device which makes use of a spontaneous redox reaction for the generation of electrical energy.
Eg : Galvanic cell, Daniell cell etc.
Galvanic cell: A device which converts chemical energy into electrical energy by the use of spontaneous redox reaction is called Galvanic cell (or) voltaic cell.
Eg : Daniell cell.

Daniell cell: It is a special type of galvanic cell. It contains two half cells in the same vessel.
The vessel is devided into two chambers. Left chamber is filled with ZnSO4 (aq) solution and Zn – rod is dipped into it, Right chamber is filled with aq. CuSO4 solution and a copper rod is dipped into it. Process diaphragm acts as Salt bridge. The two half cell’s are connected to external battery.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 10
Cell reactions :
Ion Zn/ZnSO4 half cell, oxidation, reaction occurs
Zn → Zn2+ + 2e
Ion Cu/CuSO4 half cell, reduction reaction occurs.
Cu+2 + 2e → Cu
The net cell reaction is
Zn + Cu+2 ⇌ Zn+2 + Cu
Cell is represented as Zn / Zn+2 // Cu2+ / Cu

  • The two vertical lines (||) represents a salt bridge.
  • It is U shaped tube filled with KCl and agar agar gel salt bridge maintains the electrical neutrality and flow of current.

The batteries which after their use over a period of time, becomes dead and the cell reaction is completed and this cannot be reused again are called primary batteries.
Eg : Leclanche cell, dry cell.

Secondary battery: A secondary battery is the battery in which after it’s use can be recharged and can be used again.
A good secondary battery undergoes large no. of discharging and charging cycles.
Lead storage battery is an example of secondaiy battery.
The cell reactions when the battery is in use are
Pb(s) + \(\mathrm{SO}_4^{-2}(\mathrm{aq})\) → PbSO4(s) + 2e (Anode)
PbO2(s) + \(\mathrm{SO}_4^{-2}(\mathrm{aq})\) → \(4 \mathrm{H}_{\text {(aq) }}^{+}\) + 2e → PbSO4(s) + 2 H2O(l) (Cathode)
Overall cell reaction is
Pb(s) + PbO2(s) + 2H2SO4(aq) → 2PbSO4(aq) + 2H2O(l)
In batteries also the principle involved same as in the galvanic cell.

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 2.
What is electrical conductace of a solution? How ¡s it measured experimentally?
Answer:
The reciprocal of resistance (R) is known as conductance (c)
\(\frac{l}{\mathrm{R}}=\kappa \times \frac{1}{l / \mathrm{A}}\)
c = \(\kappa \times \frac{1}{l / \mathrm{A}}\)
The conductance or the current conducting capacity of an electrolytic solution can be expressed as .

  1. Specific conductance (κ),
  2. Molar conductivity (∧m)

1. Specific conductance : The conductance of the solution enclosed between two parallel electrodes of unit area of cross section separated by unit distance is called specific conductance (κ).

2. Molar conductivity (∧m) : The conductivity of a volume of solution containing one gram molecular weight of the electrolyte placed between two parallel electrodes separated by a distance of unit length of 1 meter is called molar conductivity (∧m)
Relation between conductivity and molar conductivity :
m = \(\frac{\kappa}{\mathrm{c}}\); ∴ c = constant
Measurement of electrical conductance :

  • The resistance of a metallic wire can be measured with a Wheatstone bridge.
  •  In measuring the resistance of an electrolytic solution two problems are identified.
    1. On passing direct current (DC) through the solution changes the composition of the solution due to electrolysis.
    2. A solution cannot be connected to the measuring bridge like a metallic wire.
  • First problem solved by using AC instead of DC. .
  • Second problem solved by using specially designed vessel called conductivity cell.
    AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 11
    By using above cells,
    Resistance R = \(\frac{l}{\kappa \times \mathrm{A}}\)
    l = distance between electrodes; A = Area of cross section; κ = conductivity
    \(\frac{l}{\mathrm{~A}}\) = cell constant = G*
    ∴ G* = \(\frac{l}{\mathrm{~A}}\) = R × κ
    The cell constant (G*) is determined by measuring the resistance of the cell containing a solution whose conductivity is already known
  • Cell constant once determined, used for measuring the resistance (or) conductivity.
  • The set up for the measurement of the resistance is shown in following figure.

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 12

Question 3.
Give the applications of Kohlrausch’s law of independent migration of ions. [T.S. Mar. 16]
Answer:
Kohlrausch’s law of independent migration of ions: The limiting molar conductivity of an electrolytes can be represented as the sum of the individual contributions of the anion and the cation of the electrolytes.
\(\lambda_{\mathrm{m}(\mathrm{AB})}^0=\lambda_{\mathrm{A}^{+}}^0+\lambda_{\mathrm{B}^{-}}^0\)
\(\lambda_{\mathrm{m}}^0\) = Limiting molar conductivity
\(\lambda_{\mathrm{A}^{+}}^0\) = Limiting molar conductivity of cation
\(\lambda_{\mathrm{B}^{-}}^0\) = Limiting molar conductivity of anion

Applications:
1) Calculation of limiting molar conductance at infinite dilution (∧) of weak electrolytes :
Limiting molar conductance of weak electrolytes at infinite dilution- cannot be obtained graphically by extrapolation method. However the application of Kohlrausch’s law enables indirect evaluations in such cases.

Eg: Limiting molar conductance of acetic acid at infinite dilution can be calculated from the limiting molar conductances at infinite dilution of hydrochloric acid, sodium chloride and sodium acetate, as illustrated below:
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 13

2) Calculation of degree of dissociation of a weak electrolyte :
Degree of dissociation (α) = \(\frac{\lambda_{\mathrm{v}}}{\lambda_{\infty}}\)
Where λv = Limiting molar conductivity at a given concentration or dilution.
λ = Limiting molar conductivity of given electrolyte at infinite dilution.

3) Calculation of the solubility (s) of sparingly soluble salts like AgCl, BaSO4 etc.
S = \(\frac{\kappa \times 1000}{\wedge_{\infty}}\)
where K = specific conductance
= Limiting molar conductance at infinite dilution.

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 4.
Give the different types of batteries and explain the construction and working of each type of battery. [A.P. Mar. 19]
Answer:
The batteries which after their use over a period of time, becomes dead and the cell reaction is completed and this cannot be reused again are called primary batteries.
Eg.: Leclanche cell, dry cell.

Secondary battery:
A secondary battery is the battery in which after it’s use can be recharged and can be used again.
A good secondary battery undergoes large no. of discharging and charging cycles.
Lead storage battery is an example of secondary battery.
The cell reactions when the battery is in use are
Pb(s) + \(\mathrm{SO}_4^{-2}(\mathrm{aq})\) → PbSO4(s) + 2e (Anode)
PbO2(s) + \(\mathrm{SO}_4^{-2}(\mathrm{aq})\) → \(4 \mathrm{H}_{\text {(aq) }}^{+}\) + 2e → PbSO4(s) + 2 H2O(l) (Cathode)
Overall cell reaction is
Pb(s) + PbO2(s) + 2H2SO4(aq) → 2PbSO4(aq) + 2H2O(l)

Construction and working of primary batteries :

Dry cell:

  1. This is a modification of Leclanche cell. The liquid state electrolytes (in Leclanche cell) are replaced by paste electrolytes.
  2. A cylindrical ‘Zn’ vessel is covered with a cardboard. This is sealed with pitch. Zn vessel acts as negative electrode. A carbon rod is introduced at the centre of the Zn vessel. This carbon rod acts as positive electrode.
  3. Carbon rod is surrounded by a paste of (C + MnO2). The remaining space is filled with (NH4Cl + ZnCl2 paste. The two pastes are separated by a porous sheet.
  4. These cells are easy to handle and used in radios, watches, torch lights etc. The cell potential is approximately 1.5 V
    AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 14
  5. Electrode reactions:
    Cathode : MnO2 + \(\mathrm{NH}_4^{+}\) + eΘ → MnO (OH) + NH3
    Anode : Zn + 2MnO2 + 2H2O → Zn2+ + 2OHΘ + 2MnO (OH)
    Some of the secondary reactions are
    2 NH4Cl + 2OHΘ → 2NH3 + 2Cl + 2H2O
    Zn2+ + 2NH3 + 2Cl → [Zn (NH3)2] Cl2

Secondary batteries – construction and working : A secondary cell after its use can be recharged and can be used again. A good secondary cell undergoes a large number of discharging and charging cycles. The most important secondary cell in use is the lead storage battery (fig.(b)). This is commonly used in automobiles and invertors. It consists of a lead anode and a grid of lead packed with lead dioxide (PbO2) as cathode. A 38% solution of sulphuric acid is used as electrolyte.
The cell reactions when the battery is in use (discharging) are :
Anode : Pb(s) + \(\mathrm{SO}_4^{-2}(\mathrm{aq})\) → PbSO4(s) + 2e
Cathode: PbO2(s) + \(\mathrm{SO}_4^{-2}(\mathrm{aq})\) → \(4 \mathrm{H}_{\text {(aq) }}^{+}\) + 2e → PbSO4(s) + 2 H2O(l)
i.e., overall cell reaction consisting of cathode and anode reactions is
Pb(s) + PbO2(s) + 2H2SO4(aq) → 2PbSO4(aq) + 2H2O(l)
These reactions occur during discharge i.e., during use of the battery.
On charging the discharged battery the above reaction is reversed and PbSO4 (s) on anode and cathode is converted into Pb and PbO2, respectively.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 15

Problems

Question 1.
The standard potentials of some electrodes are as follows. Arrange the metals in an increasing order of their reducing power.
1) K+/K = -2.93 V
2) Ag+/Ag = 0.8OEV
3) Cu2+/Cu = 0.34 V
4) Mg2+/Mg = -2.37 V
5) Cr3+/Cr = – 0.74V
6) Fe2+/Fe = —0.44 V
Solution:
Given
E0 of cells K+/K = —2.93
Mg+2/Mg = -2.37 V
Ag+/Ag = 0.80 V
Cr+3/Cr = -0.74 V
Cu+2/Cu = 0.34 V
Fe+2/Fe = -0.44 V
Lower the reduction potential values indicates high reducing power and high value indicates the high oxidising power.
So, increasing order of the given cells reducing power is
Ag < Cu < Fe < Cr < Mg < K.

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 2.
Calculate the emf of the cell at 25°C
Cr | Cr3+ (0.1 M) || Fe2+ (0.01M) | Fe, given that \(\mathrm{E}_{\mathrm{Cr}^{3+} / \mathrm{Cr}}^0\) = – 0.74V and \(\mathrm{E}_{\mathrm{Fe}^{2+} / \mathrm{Fe}}^0\) = 0.44 V.
Solution:
Given cell is
Cr | \(\mathrm{Cr}_{(0.1 \mathrm{M})}^{+3}\) || \(\mathrm{Fe}_{(0.01 \mathrm{M})}^{+2}\) | Fe
E0 of Cr+3/Cr = -0.74 V
E0 of Fe+2/Fe = -0.44 V
ECr+3/Cr = E0 + \(\frac{0.059}{3}\) log10 [Cr+3]
= -0.74 + \(\frac{0.059}{3}\) log 0.1
= -0.76 V
EFe+2/Fe = -0.44 + \(\frac{0.059}{2}\) log 0.01
= -0.44 – 0.059 = – 0.499 V
EMF of cell = ERHS – ELHS
= (- 0.499) – (-0.76) = 0.26 1 V .

Question 3.
Calculate the potential of a Zn – Zn2+ electrode in which the molarity of Zn2+ is 0.001 M. Given that \(\mathrm{E}_{\mathrm{Zn}^{2+} / \mathrm{Zn}}^0\) = – 0.76 V
R = 8.314 JK-1 mol-1; F = 96500 C mol-1.
Solution:
Given electrode Zn | Zn+2(0.001m) \(\mathrm{E}_{\mathrm{Zn}^{+2} / \mathrm{Zn}}^0\) = -0.76V
Nemst equation is
E = E0 + \(\frac{2.303 \mathrm{RT}}{\mathrm{nF}}\) log [Mn+]
Given R = 8.314 J/K. mole
F = 96500 c/mole
E = E0 + \(\frac{0.059}{\mathrm{n}}\) log C
= -0.76 + \(\frac{0.059}{2}\) log 0.001
= -0.76 – \(\frac{0.059}{2}\) × 3
= – 0.76 – 0.0295 × 3
= -0.76 – 0.0885
= -0.8485 V

Question 4.
Determine ∆G0 for the button cell used in the watches. The cell reaction is
Zn(s) + Ag2O(s) + H2O(l)→ \(\mathrm{Zn}_{(a q)}^{2+}\) + 2 Ag(s) + 2 \(\mathrm{OH}_{(a q)}^{-}\)
\(\mathrm{E}_{\mathrm{Ag}^+ / \mathrm{Ag}}^0\) = + 0.80 V; \(\mathbf{E}_{\mathrm{Zn}^{2+} / \mathrm{Zn}}^0\) = – 0.76 V.
Solution:
Given E0 of Ag+/Ag = 0.80 V
E0 of Zn+2/Zn = – 0.76 V
Cell representation Zn/Zn+2|| Ag+ / Ag
EMF = ERHS – ELHS = 0.80 – (-0.76) = 1.56 V
∆G = -nFE0 = -2 × 96500 × 1.56 = -301.08 kJ/mole

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 5.
Calculate the emf of the cell consisting the following half cells
Al/Al3+(0.001 M), Ni/Ni2+ (0.50 M). Given that \(\mathrm{E}_{\mathrm{Ni}^{2+} / \mathrm{Ni}}^0\) = – 0.25 V \(\mathrm{E}_{\mathrm{Al}^{3+} / \mathrm{Al}}^0\) = – 1.66 v (log 8 × 10-6 = – 5.0969).
Solution:
Given E0 of Al+3/Al =-1.66 V
E0 of Ni+2/Ni = – 0.25 V
Cell representation Al/\(\mathrm{Al}_{(0.001 \mathrm{M})}^{+3}\) || \(\mathrm{Ni}_{(0.50 \mathrm{M})}^{+2}\) | Ni
Nemst equation is
EAl+3/Al = \(\mathrm{E}_{\mathrm{Al}^{+3} / \mathrm{A} l}^0\) + \(\frac{0.059}{n}\) log [Al+3]
= -1.66 + \(\frac{0.059}{3}\) log 0.001 = – 1.66 – 0.059 = – 1.719V
ENi+2/Ni = \(\mathrm{E}_{\mathrm{Ni}^{+2} / \mathrm{Ni}}^0+\frac{0.059}{\mathrm{n}}\) log [Ni+2]
= -0.25 + \(\frac{0.059}{2}\) log [0.5]
= -0.25 – 0.0295 × 0.3010 = -0.25 – 0.0088795 = -0.25888
EMF of cell = ERHS – ELHS
= -0.25888 – (-1.719) = 1.46012 V

Question 6.
Determine the values of Kc for the following reaction
Ni(s) + 2 \(\mathrm{Ag}_{\text {(aq) }}^{+}\) → \(\mathrm{Ni}_{\text {(aq) }}^{2+}\) + 2 Ag(s) E0 = 1.05 V.
Solution:
Given
Ni(s) + 2 \(\mathrm{Ag}_{\text {(aq) }}^{+}\) → \(\mathrm{Ni}_{\text {(aq) }}^{2+}\) + 2 Ag(s)
E0 = 1.05 V.
Nernst Equation is E = E0 – \(\frac{0.059}{\mathrm{n}}\) log Kc
at equilibrium E = 0
E0 = \(\frac{0.059}{\mathrm{n}}\) log Kc
1.05 = \(\frac{0.059}{2}\) log Kc
log Kc = \(\frac{2.1}{0.059}\) = 35.59
Kc = 3.42 × 1035

Question 7.
Calculate the potential of the half-cell containing 0.1 M
K2Cr2O7(aq), 0.2 M \(\mathrm{Cr}_{\mathrm{aq}}^{3+}\) and 1.0 × 10-4 M \(\mathrm{H}_{(\mathrm{aq})}^{+}\). The half-reaction \(\mathrm{Cr}_2 \mathrm{O}_{7(\mathrm{aq})}^{2-}+14 \mathrm{H}_{\text {(aq) }}^{+}+6 \mathrm{e}^{-} \longrightarrow 2 \mathrm{Cr}_{(\mathrm{aq})}^{3+}+7 \mathrm{H}_2 \mathrm{O}_{(l)}\) (E0 of \(\mathrm{Cr}_2 \mathrm{O}_7^{2-} / \mathrm{Cr}^{3+}\) = 1.33 V)
Solution:
Given half reaction
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 16

Question 8.
Calculate K for the reaction at 298 K
\(\mathrm{Zn}_{(\mathrm{s})}+\mathrm{Cu}_{(\mathrm{aq})}^{2+} \rightleftharpoons \mathrm{Zn}_{(\mathrm{aq})}^{2+}+\mathrm{Cu}_{(\mathrm{s})}\)
\(\mathrm{E}_{\mathrm{Zn}^{2+/ Z n}}^0\) = -0.76 V; \(\mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}}^0\) = + 0.34 V.
Solution:
Given \(\mathrm{Zn}_{(\mathrm{s})}+\mathrm{Cu}_{(\mathrm{aq})}^{2+} \rightleftharpoons \mathrm{Zn}_{(\mathrm{aq})}^{2+}+\mathrm{Cu}_{(\mathrm{s})}\)
E0 of Zn+2/Zn = -0.76 V
E0 of Cu+2/Cu = -0.34 V
E0 of cell = ERHS – ELHS = 0.34 – (- 0.76) = 1.1 V
∆G0 = – RTlnKc = – 2.303 RT log Kc
-212300 = -2.303 × 8.314 × 298 × log Kc
log Kc = \(\frac{212300}{2.303 \times 8.314 \times 298}\) = 37.207
Kc = 1.6 × 1037

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 9.
Calculate the emf of the cell at 298 K
Sn(s) | Sn2+ (0.05 M) || \(\mathrm{H}_{(\mathrm{aq})}^{+}\) (0.02 M) | H2 1 atm. Pt
Given that \(\mathrm{E}_{\mathrm{Sn}^{2+} / \mathrm{Sn}}^0\) = -0.144 V
Solution:
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 17

Question 10.
Calculate the concentration of silver ions in the cell constructed by using 0.1 M concentration of Cu2+ and Ag+ ions. Cu and Ag metals are used as electrodes. The cell potential is 0.422 V.
[\(\mathrm{E}_{\mathrm{Ag}^{2+} / \mathrm{Ag}}^0\) = 0.80 V; \(\mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}}^0\) = +0.34 V]
Solution:
Given E0 of Ag+/Ag = 0.80 V .
E0 of Cu+2/Cu = 0.34 V
E0 of cell = ERHS – ELHS = 0.80 – 0.34 = 0.46 V
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 18
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 19

Question 11.
Calculate the emf of the cell with the cell reaction
Ni(s) + 2 Ag+ (0.002M) → Ni2+ (0.160 M) + 2 Ag(s) ; \(\mathrm{E}_{\text {cell }}^0\) = 1.05 V.
Solution:
From the given cell reaction and Nernst equation
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 20

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 12.
Cu2+ + 2e ⇌ Cu; E0 = +0.34 V
Ag+ + e ⇌ Ag; E0 = + 0.80 V
For what concentration of Ag+ ions will the emf of the cell be zero at 25°C. The concentration of Cu2+ is 0.1 M. (log 3.919 = 0.593).
Solution:
Cell is Cu|Cu+2 | |Ag+|Ag
E0 of cell = ERHS – ELHS = 0.80 – 0.34 = 0.46 V
Nernst equation is E = E0 – \(\frac{0.059}{2} \log \frac{\left[\mathrm{Cu}^{+2}\right]}{\left[\mathrm{Ag}^{+}\right]^2}\)
EMF of cell = 0
E0 = \(\frac{0.059}{2} \log \frac{0.1}{\left[\mathrm{Ag}^{+}\right]^2}\)
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 21

Question 13.
The conductivity of 0.20 M solution of KCl at 298 K is 0.0248 S cm-1. Calculate molar conductance.
Solution:
Conductivity (K) = 0.0248 S cm-1 = 0.0248 ohm-1 cm-1
Molar concentration [c] = 0.20 mol L-1
= \(\frac{[0.2 \mathrm{~mol}]}{\left[1000 \mathrm{~cm}^3\right]}\) = 2.0 × 10-4 mol cm-3
Molar conductivity [∧m] = \(\frac{\kappa}{\mathrm{c}}=\frac{\left[0.0248 \mathrm{ohm}^{-1} \mathrm{~cm}^{-1}\right]}{\left[2.0 \times 10^{-4} \mathrm{~mol} \mathrm{~cm}^{-3}\right]}\)
= 124 ohm-1 mol-1 cm2 or 1245 mol-1 cm2
Molar conductivity = 124 mol-1 cm2

Question 14.
Calculate the degree of dissociation (α) of CH3COOH at 298 K.
Given that \(\wedge_{\mathrm{CH}_3 \mathrm{COOH}}^{\infty}\) = 11.75 cm2 mol-1
\(\wedge_{\mathrm{CH}_3 \mathrm{COO}^{-}}^0\) = 40.95 cm2 mol-1
\(\wedge_{\mathrm{H}^{+}}^0\) = 349.15 cm2 mol-1
Solution:
Given that \(\wedge_{\mathrm{CH}_3 \mathrm{COOH}}^{\infty}\) = 11.75 cm2 mol-1
\(\wedge_{\mathrm{CH}_3 \mathrm{COO}^{-}}^{\infty}\) = 40.95 cm2 mol-1
\(\wedge_{\mathrm{H}^{+}}^{\infty}\) = 349.15 cm2 mol-1
\(\wedge_{\mathrm{CH}_3 \mathrm{COOH}}^{\infty}\) = \(\wedge_{\mathrm{CH}_3 \mathrm{COO}^{-}}^{\infty}\) + \(\wedge_{\mathrm{H}^{+}}^{\infty}\) = 40.95 + 349.15 = 390.1
Degree of dissociation (α) = \(\frac{\lambda^\alpha}{\lambda^{\infty}}=\frac{11.75}{390: 1}\) = 0.03 = 3 × 10-2
∴ α = 3× 10-2

Textual Examples

Question 1.
Represent the cell in which the following reaction takes place Mg(s) + 2Ag+ (0.0001M) → Mg2+ (0.130M) + 2 Ag(s)
Calculate its E(cell) if \(\mathrm{E}_{(\text {cell) }}^{\Theta}\) = 3.17 V
Solution:
The cell is represented as Mg |Mg2+ (0.130M) || Ag+(0.0001M) | Ag
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 22

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 2.
Calculate the equilibrium constant of the reaction:
Cu(s) + 2 \({Ag}_{\text {(aq) }}^{+}\) → \(\mathrm{Cu}_{(\mathrm{aq})}^{2+}\) + 2 Ag(s) ; \(\mathrm{E}_{(\text {cell) }}^{\Theta}\) = 0.46 V
Solution:
\(\mathrm{E}_{(\text {cell) }}^{\Theta}\) = \(\frac{0.059V}{2}\) log Kc = 0.46 V or
l0g Kc = \(\frac{0.46 \mathrm{~V} \times 2}{0.059 \mathrm{~V}}\) = 156
Kc = 3.92 × 1015

Question 3.
The standard emf of Daniell cell is 1.1V. Calculate the standard Gibbs energy for the cell reactions:
Zn(s) + \(\mathrm{Cu}_{(\mathrm{aq})}^{2+}\) → \(\mathrm{Zn}_{(\mathrm{aq})}^{2+}\) + Cu(s)
Solution:
rGΘ = -nF\(\mathrm{E}_{(\text {cell) }}^{\Theta}\)
n in the above equation is 2, F = 96487 C mol-1 and \(\mathrm{E}_{(\text {cell) }}^{\Theta}\) = 1.1 V
Therefore ∆rGΘ = -2 × 1.1V × 96487 C mol-1
= – 21227 J mol-1 =-212.27 kJ mol-1

Question 4.
Resistance of a conductivity cell filled with 0.1 mol L-1 KCl solution is 100 Ω. If .the resistance of the same cell when filled with 0.02 mol L-1 KCl solution is 520 Ω, calculate the conductivity and molar conductivity of 0.02 mol L-1 KCl solution. The conductivity of 0.1 mol L-1 KCl solution is 1.29 S/m.
Solution:
The cell constant is given by the equation :
Cell constant = G* = conductivity × resistance
= 1.29 S/m 100 Ω = 129 m-1 = 1.29 cm-1
Conductivity of 0.02 mol L-1 KCl solution = cell constant / resistance
= \(\frac{\mathrm{G}^*}{\mathrm{R}}=\frac{129 \mathrm{~m}^{-1}}{520 \Omega}\) = 0.248 S m-1
Concentration = 0.02 mol L-1
= 1000 × 0.02 mol m-3 = 20 mol m-3
Molar conductivity = ∧m = \(\frac{\kappa}{\mathrm{c}}\)
= \(\frac{248 \times 10^{-3} \mathrm{Sm}^{-1}}{20 \mathrm{~mol} \mathrm{~m}^{-3}}\) = 124 × 10-4 Sm2 mol-1
Alternatively, κ = \(\frac{1.29 \mathrm{~cm}^{-1}}{520 \Omega}\) = 0.248 × 10-2 S cm-1
and ∧m = κ × 1000 cm3L-1 molarity-1
= \(\frac{0.248 \times 10^{-2} \mathrm{~S} \mathrm{~cm}^{-1} \times 1000 \mathrm{~cm}^3 \mathrm{~L}^{-1}}{0.02 \mathrm{~mol} \mathrm{~L}^{-1}}\)
= 124 S cm2 mol-1

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 5.
The electrical resistance of a column 0.05 mol L-1 NaOH solution of diameter 1 cm and length 50 cm is 5.55 × 103 ohm. Calculate its resistivity, conductivity and molar conductivity.
Answer:
A = πr2 = 31.4 × 0.52 cm2 = 0.785 cm2 = 0.785 × 10-4 m2
l = 50 cm = 0.5 m
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 23

Question 6.
Calculate \(\wedge_m^0\) for CaCl2 and MgSO4 from the data given in Table 3.4.
Solution:
We know from Kohlrausch law that
\(\wedge_{\mathrm{m}\left(\mathrm{CaCl}_2\right)}^0=\wedge_{\mathrm{Ca}^{2+}}^0+2 \lambda_{\mathrm{Cl}}^0\) = 119.0 S cm2 mol -1 + 2(76.3) S cm2 mol-1
= (119.0 + 152.6) Scm2 mol-1
= 271.6 S cm2 mol-1
\(\wedge_{\mathrm{m}\left(\mathrm{MgSO}_4\right)}^0=\lambda_{\mathrm{Mg}^{2+}}^0+\lambda_{\mathrm{SO}_4^{2-}}^0\) =106.0 S cm2 mol-1 + 160.0 S cm2 mol-1
= 266 S cm2 mol-1

Question 7.
\(\wedge_m^0\) for NaCl, HCl and NaAc are 126.4,425.9 and 91.0 S cm2 mol-1 respectively. Calculate ∧0 for HAc.
Solution:
\(\wedge_{\mathrm{m}(\mathrm{HAc})}^0=\lambda_{\mathrm{H}^{+}}^0+\lambda_{\mathrm{Ac}^0}^0=\lambda_{\mathrm{H}^{+}}^0+\lambda_{\mathrm{Cl}^{-}}^0+\lambda_{\mathrm{Na}^{+}}^0-\lambda_{\mathrm{Cl}^{-}}^0-\lambda_{\mathrm{Na}^{+}}^0\)
= \(\wedge_{\mathrm{m}(\mathrm{HCl})}^0+\wedge_{\mathrm{m}(\mathrm{NaAc})}^0-\wedge_{\mathrm{m}(\mathrm{NaCl})}^0\)
= (425.9 + 91.0 – 126.4) S cm2 mol-1
= 390.5 S cm2 mol-1

Question 8.
The conductivity of 0.001028 mol L-1 acetic acid is 4.95 × 10-5 S cm-1. Calculate its dissociation constant if \(\wedge_m^0\) for acetic acid is 390.5 S cm2 mol-1.
Solution:
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 24

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 9.
A solution of CuSO4 is electrolysed for 10 minutes with a current of 1.5 amperes. What is the mass of eopper deposited at the cathode ? [A.P. & T.S. Mar. 15] [Mar. 15]
Solution:
t = 600 s charge = current × time = 1.5A × 600 s = 900 C
According to the reaction :
\(\mathrm{Cu}_{(\mathrm{aq})}^{2+}\) + 2e = Cu(s)
We require 2F or 2 × 96487 C to deposit 1 mol or 63 g of Cu.
For 900 C, the mass of Cu deposited
= (63 g mol-1 × 900 C) / (2 × 96487 C mol-1) = 0.2938 g

Intext Questions

Question 1.
How would you determine the standard electrode potential of the system Mg2+/Mg ?
Solution:
E° value of Mg2+/Mg electrode is determined by setting up an electrochemical cell. For this purpose, a Mg electrode is dipped in 1 M MgSO4 solution, which acts as one half cell i.e., oxidation half-cell.

In the same way, the standard hydrogen electrode acts as the other half-cell i.e., reduction half cell. The deflection of voltmeter placed in the cell circuit is towards the Mg electrode which indicates the direction of flow of current.
The cell may be represented as : Mg | Mg2+ (1M) || H+ (M) | H2 (1 atm). Pt The reading as given by voltmeter gives \(\mathrm{E}_{\text {cell }}^0\)
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 25

Question 2.
Can you store copper sulphate solutions In a zinc pot?
Solution:
No, zinc pot cannot store copper sulphate solutions because the standard electrode potential (E0) value of zinc is less than that of copper. So, zinc is stronger reducing agent then copper.
\(\mathrm{Zn}_{(\mathrm{aq})}^{2+}\) + 2e → Zn(s); E0 = -0.76 V
\(\mathrm{Cu}_{(\mathrm{aq})}^{2+}\) + 2e → Cu(s); E0 = – 0.34 V
So, zinc will loss electrons to Cu2+ ions and redox reaction will occur as follows.
Zn(s) + \(\mathrm{Cu}_{(\mathrm{aq})}^{2+}\) → \(\mathrm{Zn}_{(\mathrm{aq})}^{2+}\) + Cu(s)

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 3.
Consult the table on standard electrode potentials and suggest three substances that can oxidise ferrous ions under suitable conditions.
Solution:
Fe2+ (ferrous) ions gets oxidised to Fe3+ (ferric) ion as follows
Fe2+ → Fe3+ + e \(\mathrm{E}_{0 \mathrm{x}}^0\) = -0.77V
Only those substances can oxidise Fe2+ ions to Fe3+ ions which can accept electrons released during oxidation or are placed above iron in electrochemical series. Three such substances are: Cl2(g), Br2(g) and F2(g).

Question 4.
Calculate the potential of hydrogen electrode placed in a solution of pH 10.
Solution:
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 26

Question 5.
The cell in which the following cell reaction occurs,
\(2 \mathrm{Fe}_{(\mathrm{aq})}^{3+}+2 \mathrm{I}_{(\mathrm{aq})}^{-} \longrightarrow 2 \mathrm{Fe}_{(\mathrm{aq})}^{2+}+\mathrm{I}_{2(\mathrm{~s})}\) has \(\mathrm{E}_{\text {cell }}^0\) = 0.236 V at 298 K. Calculate the standard Gibbs energy and the equilibrium constant of the cell reaction.
Solution:
Two half reactions for the given redox reaction may be written as :
\(2 \mathrm{Fe}_{(\mathrm{aq})}^{3+}+2 \mathrm{e}^{-} \longrightarrow 2 \mathrm{Fe}_{(\mathrm{aq})}^{2+} ; 2 \mathrm{I}^{-} \longrightarrow \mathrm{I}_2+2 \mathrm{e}^{-}\)
2 moles of electrons are involved in the reaction, so n = 2
rG0 = – nF\(\mathrm{E}_{\text {cell }}^0\) = (-2 mol) × (96500 mol-1) (0.236 V)
= -45548 CV = -45548 J
rG0 = – 45.55 kJ
log Kc = \(\frac{\Delta \mathrm{G}^0}{2.303 \mathrm{RT}}\)
= \(\frac{(-45.55 \mathrm{~kJ})}{2.303 \times\left(8.314 \times 10^{-3} \mathrm{kJK}^{-1}\right) \times(298 \mathrm{~K})}\) = 7.983
Kc = Antilog (7.983) = 9.616 × 107
Kc = 9.616 × 107

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 7.
Why does the conductivity of a solution decrease with dilution ?
Solution:
The conductivity of a solution is related with the number of ions present per unit volume of the solution. When the solution is diluted, the number of ions decreases. Hence, conductivity or specific conductance of the solution also decreases.

Question 8.
Suggest a way to determine the \(\wedge_{\mathrm{m}}^0\) value of water.
Solution:
The molar conductance of water at infinite dilution can be obtained from the knowledge of molar conductances at inifite dilution of sodium hydroxide, hydrochloric acid and sodium chloride (all strong electrolytes). This is in accordance with Kohlrausch’s law.
\(\wedge_{\mathrm{m}\left(\mathrm{H}_2 \mathrm{O}\right)}^0=\wedge_{\mathrm{m}(\mathrm{NaOH})}^0+\wedge_{\mathrm{m}(\mathrm{HCl})}^0-\wedge_{\mathrm{m}(\mathrm{NaCl})}^0\)

Question 9.
The molar conductivity of 0.025 mol L-1 methanoic acid is 46.1 S cm2 mol-1. Calculate its degree of dissociation and dissociation constant.
Given, λ0(H+) = 349.6 S cm-1 mol-1 and λ0 (HCOO) = 54.6 S cm2 mol-1
Solution:
Step I: Calculation of degree of dissociation (a) of HCOOH
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 27
= \(\frac{\left(3.249 \times 10^{-4} \mathrm{~mol} \mathrm{~L}^{-1}\right)}{(0.886)}\)
= 3.67 × 10-4 mol L-1
α = 0.114
Ka = 3.67 × 10-4 mol L-1

Question 10.
If a current of 0.5 ampere flows through a metallic wire for 2h, then how many electrons would flow through the wire ?
Solution:
Quantity of charge (Q) passed = Current (C) × Time (t)
= (0.5 A) × (2 × 60 × 60 s)
= (3600) Ampere sec = 3600 C ,
Number of electrons flowing through the wire on passing charge of one Faraday (96500 C) = 6.022 × 1023
Number of electrons flowing through the wire on passing a charge of 3600 C
= \(\frac{6.022 \times 10^{23} \times(3600 \mathrm{C})}{(96500 \mathrm{C})}\)
= 2.246 × 1022
Number of electrons = 2.246 × 1022

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 11.
Suggest a list to metals that are extracted electrolytically.
Solution:
Those metals which are highly reactive and have large negative E0 values can be exerted electrolytically. This type of metals are powerful reducing agents.
Eg: Sodium, Potassium, Calcium, Magnesium etc.

Question 12.
Consider the reaction,
\(\mathrm{Cr}_2 \mathrm{O}_7^{2-}\) + 14H+ + 6e → 2Cr3+ + 7H2O
What is the quantity of electricity in coulombs nefeded to reduce 1 mole of \(\mathrm{Cr}_2 \mathrm{O}_7^{2-}\) ?
Solution:
1 mole \(\mathrm{Cr}_2 \mathrm{O}_7^{2-}\) require 6 Faraday electricity (6 F)
= 6 × 96500 C = 5.79 × 105 C
Quantity of electricity = 5.79 × 105 Coulomb

Question 13.
Write the chemistry of recharging the lead storage battery, highlighting all the materials that are involved during recharging.
Solution:
When lead storage battery is recharged, electrical energy is supplied to the cell from the external source : i.e., the cell operates as an electrolytic cell during recharging. All the chemical reactions, which take place during the use of battery, are reversed at this time.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 28

Question 14.
Suggest two materials other than hydrogen that can be used as fuels in fuel cells.
Solution:
Methane (CH4) and methanol (CH3OH)

AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry

Question 15.
Explain how rusting of iron is envisaged as setting up of an electrochemical cell.
Solution:
In corrosion, a metal is oxidised by loss of electrons to oxygen with the formation of oxides. So, an electrochemical cell is set up.
Eg: Rusting of iron involves the following steps :
i) The water layer present on the surface of iron dissolves acidic oxides from air like CO2 and forms acid to produce H+ ions.
H2O + CO2 H2CO3 ⇌ 2H+ + \(\mathrm{CO}_3^{2-}\)

ii) In the presence of H+ ions, iron starts losing electrons at some spot to form ferrous ions. This spot behaves as anode.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 29

iii) The electrons released at anode move to another spot, where H+ ions and the dissolved oxygen gain these electrons. This spot becomes a cathode.
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 30

iv) Overall reaction, i.e., redox reaction is :
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 31
v) Ferrous ions are further oxidised by the atmospheric oxygen to ferric ions which combine with water molecules to form hydrated ferric oxide, Fe2O3. xH2O. (Rust).
AP Inter 2nd Year Chemistry Study Material Chapter 3(a) Electro Chemistry 32

Inter 2nd Year Maths 2B Differential Equations Important Questions

Students get through Maths 2B Important Questions Inter 2nd Year Maths 2B Differential Equations Important Questions which are most likely to be asked in the exam.

Intermediate 2nd Year Maths 2B Differential Equations Important Questions

Question 1.
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = ex – y + x2 e-y [Mar. 06; May 05]
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 1

Question 2.
x\(\frac{\mathrm{dy}}{\mathrm{dx}}\) – y = 2x2 sec22x [May 11]
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 2

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 3.
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y tan x = sin x. [T.S. Mar. 16]
Solution:
I.F. = \(e^{\int \tan x d x}\) = elog sec x = sec x
y.sec x = \(\int\) sin x . sec x dx = \(\int\) tan x dx
= log sec x + c

Question 4.
cos x . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y sin x = sec2x [Mar. 14]
Solution:
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) + tan x . y = sec3x
I.F. = e\(\int\)tan x dx = elog sec x = sec x
y . sec x = \(\int\) sec4x dx = \(\int\) (1 + tan2 x) sec2 x
dx = tan x + \(\frac{\tan ^{3} x}{3}\) + c

Question 5.
(x + y + 1)\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 1.
Solution:
\(\frac{\mathrm{dx}}{\mathrm{dy}}\) = x + y + 1
\(\frac{\mathrm{dx}}{\mathrm{dy}}\) = x + y + 1
I.F. = e\(\int\) -dy = e-y
x . e-y = \(\int\) e-y (y + 1)dy
= – (y + 1) . e-y + \(\int\) e-y . dy
= – (y + 1) e-y – e-y
= – (y + 2) e-y + c
x = – (y + 2) + c. e-y

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 6.
Find the order and degree of r
\(\frac{\mathrm{d}^{3} \mathrm{~y}}{\mathrm{dx}^{3}}\) – 3 (\(\frac{\mathrm{dy}}{\mathrm{dx}}\)) – ex = 4. [Mar. 14]
Solution:
The equation is a polynomial in \(\frac{\mathrm{dy}}{\mathrm{dx}}\) and \(\frac{\mathrm{d}^{3} \mathrm{~y}}{\mathrm{dx}^{3}}\).
The exponent of \(\frac{\mathrm{d}^{3} \mathrm{~y}}{\mathrm{dx}^{3}}\) is 2.
Hence the degree is 2.
\(\frac{\mathrm{d}^{3} \mathrm{~y}}{\mathrm{dx}^{3}}\) is the highest order derivative occuring in the equation.
Order of the equation is 3.

Question 7.
x\(\frac{1}{2}\)(\(\frac{\mathrm{d}^{2} \mathrm{~y}}{\mathrm{dx}^{2}}\))\(\frac{1}{3}\) + x . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y = 0 has order 2 and degree 1. Prove. [T.S. Mar. 15]
Solution:
The given equation can be written as
Inter 2nd Year Maths 2B Differential Equations Important Questions 3
∴ The order of the equation is 2 and its degree is 1.

Question 8.
Find the order and degree of \(\left(\frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{3}\right)^{\frac{6}{5}}\) = 6y [Mar. 16; May 11]
Solution:
Given equation is \(\left(\frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{3}\right)^{\frac{6}{5}}\) = 6y
i.e., \(\frac{\mathrm{d}^{2} \mathrm{~y}}{\mathrm{dx}^{2}}\) + (\(\frac{\mathrm{dy}}{\mathrm{dx}}\))3 = (6y)\(\frac{5}{6}\)
Order = 2, degree = 1

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 9.
Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{x(2 \log x+1)}{\sin y+y \cos y}\) [Mar. 08]
Solution:
Given equation can be written as
(sin y + y cos y) dy = x(2 log x + 1) dx
\(\int\) sin y dy + \(\int\) y cos y dy = \(\int\) 2x log x dx + \(\int\) x dx
\(\int\) sin y dy + y sin y – \(\int\) sin y dy = x2 log x – \(\int\) x2 . \(\frac{1}{x}\) dx + \(\int\) x dx + c
y sin y = x2 log x + c

Question 10.
(xy2 + x) dx + (yx2 + y) dy = 0. [A.P. Mar. 15, 07]
Solution:
(xy2 + x) dx + (yx2 + y) dy = 0
x(y2 + 1) dx + y (x2 + 1) dy = 0
Dividing with (1 + x2) (1 + y2)
\(\frac{x d x}{1+x^{2}}\) + \(\frac{y d x}{1+y^{2}}\) = 0
Integrating
\(\int \frac{x d x}{1+x^{2}}+\int \frac{y d y}{1+y^{2}}=0\)
\(\frac{1}{2}\)[(log (1 + x2) + log (1 + y2)] = log c
log (1 + x2) (1 + y2) = 2 log c = log c2
Solution is(1 + x2) (1 + y2) = k when k = c2.

Question 11.
sin-1 (\(\frac{\mathrm{dy}}{\mathrm{dx}}\)) = x + y [Mar. 07]
Solution:
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = sin (x + y)
x + y = t
1 + \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{\mathrm{dt}}{\mathrm{dx}}\)
\(\frac{\mathrm{dt}}{\mathrm{dx}}\) – 1 = sin t
\(\frac{\mathrm{dt}}{\mathrm{dx}}\) = 1 + sin t
\(\frac{d t}{1+\sin t}\) = dx
Integrating both sides we get
\(\int \frac{d t}{1+\sin t}=\int d x\)
\(\int \frac{1-\sin t}{\cos ^{2} t} d t=x+c\)
\(\int\) sec2 t dt = \(\int\) tan t . sec t dt = x + c
tan t – sec t = x + c
⇒ tan (x + y) – sec (x + y) = x + c

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 12.
(x2 – y2) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = xy [May 11]
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 4
= log y + c
\(\frac{-x^{2}}{2 y^{2}}\) = (log y + c)
-x2 = 2y2 (c + log y)
⇒ Solution is x2 + 2y2 (c + log y) = 0.

Question 13.
Solve : x dy = (y + x cos2 \(\frac{y}{x}\)) dx.
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 5

Question 14.
(2x + y + 1) dx + (4x + 2y – 1) dy = 0 [T.S. Mar. 15]
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 6
2v + log (v – 1) = 3x + c
2v – 3x + log (v – 1) = c
2(2x + y) – 3x + log (2x + y – 1) = c
4x + 2y – 3x + log (2x + y – 1) = c
Solution is x + 2y + log (2x + y – 1) = c

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 15.
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y tan x = cos3x [May 11]
Solution:
I.F. = e\(\int\) tan x dx = elog sec x = sec x
y . sec x = \(\int\) sec x. cos3 x dx
= \(\int\) cos2x dx
= \(\frac{1}{2}\) \(\int\) (1 + cos 2x) dx
= \(\frac{1}{2}\) (x + \(\frac{sin2x}{2}\)) + c
\(\frac{2 y}{\cos x}\) = x + sin x . cos x + c
Solution is 2y = x cos x + sin x . cos2 x + c . cos x^

Question 16.
(1 + x2) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y = etan-1 x [May 07] [A.P. Mar. 16] [T.S. Mar. 15]
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 7

Question 17.
Solve (1 + y2)dx = (tan-1y – x)dy. [A.P. Mar. 15]
Solution:
Given \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{\tan ^{-1} y-x}{1+y^{2}}\)
Inter 2nd Year Maths 2B Differential Equations Important Questions 8
Inter 2nd Year Maths 2B Differential Equations Important Questions 9

Question 18.
(x2 – y2)dx – xy dy = 0 [May 06]
Solution:
(x2 – y2)dx – xy dy = 0
(x2 – y2)dx = xy . dy
Inter 2nd Year Maths 2B Differential Equations Important Questions 10
–\(\frac{1}{4}\) [log (x2 – 2y2) – log x2] = log x + log c
–\(\frac{1}{4}\) log (x2 – 2y2) + \(\frac{1}{4}\) . 2 log x = log x + log c
–\(\frac{1}{4}\) log (x2 – 2y2) = \(\frac{1}{2}\) log x + log c
– log (x2 – 2y2) = – 2 log x – 4 log c
log (x2 – 2y2) = – 2 log x + log k where
k = \(\frac{1}{c^{4}}\) = log \(\frac{\mathrm{k}}{\mathrm{x}^{2}}\)
x2 – 2y2 = \(\frac{\mathrm{k}}{\mathrm{x}^{2}}\)
Solution is x2 (x2 – 2y2) = k

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 19.
\(\frac{d y}{d x}=\frac{3 y-7 x+7}{3 x-7 y-3}\) [T.S. Mar. 16]
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 11
Inter 2nd Year Maths 2B Differential Equations Important Questions 12
= 3ln (v – 1) – 3ln (v + 1) – 7ln (v + 1) – 7ln (v – 1)
14ln x – ln c = – 10 ln (v + 1) – 4 ln (v – 1)
ln (v + 1)5 + ln (v – 1)2 + ln x7 = ln c
(v + 1)5 . (v – 1)2 . x7 = c
(\(\frac{y}{x}\) + 1)5 (\(\frac{y}{x}\) – 1)2 . x7 = c
(y – x)2 (y + x)5 = c
[y – (x – 1)]2 (y + x – 1)5 = c
Solution is [y – x + 1]2 (y + x – 1)5 = c.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 20.
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) (x2y3 + xy) = 1 [Mar. 11]
Solution:
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = xy + x2y3
This is Bernoulli’s equation
x-2 . \(\frac{\mathrm{dx}}{\mathrm{dy}}\) – \(\frac{1}{x}\) . y = y3
Inter 2nd Year Maths 2B Differential Equations Important Questions 13
Inter 2nd Year Maths 2B Differential Equations Important Questions 14

Question 21.
Form the differential equation corresponding to y = A cos 3x + B sin 3x, where A and B are parameters. [AP Mar. 15]
Solution:
We have y = A cos 3x + B sin 3x
Differentiating w.r.to x
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = -3A sin 3x + 3B cos 3x
Differentiating again w.r.to. x
\(\frac{\mathrm{d}^{2} \mathrm{~y}}{\mathrm{dx}^{2}}\) = -9A cos 3x – 9B sin 3x
= – 9(A cos 3x + B sin 3x)
= -9y .
is \(\frac{\mathrm{d}^{2} \mathrm{~y}}{\mathrm{dx}^{2}}\) + 9y = 0.
Alternate method:
Eliminating A, B from the equation
y = A cos 3x + B sin 3x
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = – 3A sin 3x + 3B sin cos 3x
Inter 2nd Year Maths 2B Differential Equations Important Questions 15
This is the required differential equation.

Question 22.
Solve (x2 + y2) dx = 2xy dy [A.P. Mar. 16]
Solution:
Given equation can be written as
Inter 2nd Year Maths 2B Differential Equations Important Questions 16
log cx(1 – v2 = log 1
cx (1 – v2) = 1
cx (1 – \(\frac{\mathrm{y}^{2}}{\mathrm{x}^{2}}\)) = 1
c(x2 – y2) = x is the required solution.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 23.
Give the solution of x sin2 \(\frac{y}{x}\) dx = y dx – x dy which passes through the point (1, \(\frac{\pi}{4}\)). [Mar. 14]
Solution:
The given equation can be written as
Inter 2nd Year Maths 2B Differential Equations Important Questions 17
The given curve passes through (1, \(\frac{\pi}{4}\))
cot \(\frac{\pi}{4}\) = log 1 + c
1 = 0 + c ⇒ c = 1
Solution is cot \(\frac{y}{x}\) = log x + 1

Question 24.
Find the order and degree of the differential equation \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) = – p2y.
Solution:
The given equation is a polynomial equation in \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\)
Hence the degree is 1
\(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) is the highest order derivative occuring in the equation.
Its order is 2.

Question 25.
Find the order and degree of
(\(\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}\))2 – 3 (\(\frac{\mathrm{dy}}{\mathrm{dx}}\))2 – ex = 4 [Mar. 14]
Solution:
The equation is a polynomial in \(\frac{\mathrm{dy}}{\mathrm{dx}}\) and \(\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}\).
The exponent of \(\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}\) is 2.
Hence the degree is 2.
\(\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}\) is the highest õrder derivative occuring in the equation.
Order of the equation is 3.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 26.
x\(\frac{1}{2}\)(\(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\))\(\frac{1}{3}\) + x . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y = 0 has order 2 and degree 1. Prove. [T.S. Mar. 15]
Solution:
The given equation can be written as
x\(\frac{1}{2}\)(\(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\))\(\frac{1}{3}\) = -[x . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y]
Cubing both sides
x\(\frac{3}{2}\)(\(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\)) = -[x . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y]3
∴ The order of the equation is 2 and its degree is 1.

Question 27.
Find the order and degree of \(\left(\frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{3}\right)^{\frac{6}{5}}\) = 6y [A.P. Mar. 16; May 11]
Solution:
Given equation is \(\left(\frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{3}\right)^{\frac{6}{5}}\) = 6y
i.e., \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) + (\(\frac{\mathrm{dy}}{\mathrm{dx}}\))3 = (6y)\(\frac{5}{6}\)
Order = 2; degree = 1

Question 28.
Find the order of the differential equation corresponding to y = c(x – c)2, where c is an arbitrary constant.
Solution:
The given differential equation is
y = c(x – c)2
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 2c(x – c)
∴ Order of the differential equation is 1.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 29.
Find the order of the differential equation corresponding to y = Aex + Be3x + Ce5x; (A, B, C being parameters) is a solution.
Solution:
Required differential equation is obtained by eliminating A, B, C from y,
\(\frac{\mathrm{dy}}{\mathrm{dx}}\), \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\), \(\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}\)
Highest order deviation = \(\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}\)
Order of the differential equation = 3.

Question 30.
Form the differential equation corresponding to y = cx – 2c2, where c is a parameter.
Solution:
Given y = cx – 2c2 ………………. (1)
Differentiating (1) w.r.to
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = c
Substituting in (1), required differential equation is
y = x . (\(\frac{\mathrm{dy}}{\mathrm{dx}}\)) – 2(\(\frac{\mathrm{dy}}{\mathrm{dx}}\))2

Question 31.
Form the differential equation corresponding to y = A cos 3x + B sin 3x, where A and B are parameters. [A.P. Mar. 15]
Solution:
We have y = A cos 3x + B sin 3x
Differentiating w.r.to x
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = -3A sin 3x + 3B cos 3x
Differentiating again w.r.to. x
\(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) = -9A cos 3x – 9B sin 3x
= -9(A cos 3x + B sin 3x)
= -9y
is \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) + 9y = 0

Inter 2nd Year Maths 2B Differential Equations Important Questions

Alternate Method:
Eliminating A, B from the equation
y = A cos 3x + B sin 3x
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = -3A sin 3x = 3B sin cos 3x
\(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) = -9A cos 3x – 9B sin 3x
We get \(\left|\begin{array}{ccc}
y & -\cos 3 x & -\sin 3 x \\
\left(\frac{d y}{d x}\right) & 3 \sin 3 x & -3 \cos 3 x \\
\left(\frac{d^{2} y}{d x^{2}}\right) & 9 \cos 3 x & 9 \sin 3 x
\end{array}\right|\) = 0
y(27 sin2 3x + 27 cos2 3x) – (-9 sin 3x. cos 3x + 9 cos 3x. sin 3x) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + (3 cos2 3x + 3 sin2 3x) \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) = 0
= 27y + 3 .\(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) = 0 or \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) + 9y = 0
This is the required differential equation.

Question 32.
Form the differential equation corresponding to the family of circles of radius r given by (x – a)2 + (y – b)2 = r2, where a and b are parameters.
Solution:
We have (x – a)2 + (y – b)2 = r2 ………………… (1)
Differentiating (1) w.r.to x
2(x – a) + 2(y – b) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 0 ……………….. (2)
Differentiating (2) w.r.to. x
1 + (y – b) \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) + (\(\frac{\mathrm{dy}}{\mathrm{dx}}\))2 = 0 ……………… (3)
From (2) (x – a) = -(y – b) \(\frac{\mathrm{dy}}{\mathrm{dx}}\)
Substituting in (1), we get
Inter 2nd Year Maths 2B Differential Equations Important Questions 18
i.e., \(r^{2}\left(\frac{d^{2} y}{d x^{2}}\right)^{2}=\left(1+\left(\frac{d y}{d x}\right)^{2}\right)^{3}\)
Which is the required differential equation.

Question 33.
Form the differential equation corresponding to the family of circles passing through the origin and having centres on Y-axis.
Solution:
The equation of the family of all circles passing through the origin and having centres on Y—axis is
x2 + y2 + 2hy = 0 …………………. (1)
Where h is a parameter
Differentiating (1) w.r. to x
2x + 2y . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + 2h . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 0
or x + y . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + h . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 0
-(x + y . \(\frac{\mathrm{dy}}{\mathrm{dx}}\)) = h. \(\frac{\mathrm{dy}}{\mathrm{dx}}\)
h = \(\frac{-\left(x+y \cdot \frac{d y}{d x}\right)}{\frac{d y}{d x}}\)
Substituting in (1)
We get x2 + y2 – 2y \(\frac{\left(x+y \cdot \frac{d y}{d x}\right)}{\frac{d y}{d x}}\) = 0
x2 . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y2 . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) – 2xy – 2y2 . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 0
or (x2 – y2) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) – 2xy = 0
This is the required differential equation.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 34.
Express the following differential equations in the form f(x) dx + g(y) dy = 0.
i) \(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1+\mathrm{y}^{2}}{1+\mathrm{x}^{2}}\)
Solution:
⇒ \(\frac{d y}{1+y^{2}}=\frac{d x}{1+x^{2}}\)
\(\frac{d x}{1+x^{2}}-\frac{d y}{1+y^{2}}\) = 0

ii) y – x \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = a (y2 + \(\frac{\mathrm{dy}}{\mathrm{dx}}\))
Solution:
y – x . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = ay2 + a. \(\frac{\mathrm{dy}}{\mathrm{dx}}\)
y – ay2 = (x + a) . \(\frac{\mathrm{dy}}{\mathrm{dx}}\)
\(\frac{d x}{x+a}=\frac{d y}{y-a y^{2}}\)

iii) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = ex-y + x2 e-y
Solution:
Multiplying in the ey
ey . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = ex + x2
ey . dy = (ex + x2) dx
(ex + x2) dx – ey . dy = 0

iv) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + x2 = x2 e3y
Solution:
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = x2 . e3y – x2 = x2(e3y – 1)
\(\frac{d y}{e^{3 y}-1}\) = x2 dx ⇒ x2 dx – \(\frac{d y}{e^{3 y}-1}\) = 0
or x2dx + \(\frac{1}{\left(1-e^{3 y}\right)}\) . dy = 0

Question 35.
Find the general solution of
x + y\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 0.
Solution:
Given equation is x + y . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 0
x . dx + y . dy = 0
Integrating \(\frac{x^{2}}{2}\) + \(\frac{y^{2}}{2}\) = c
or x2 + y2 = 2 c = c’

Question 36.
Find the general solution of \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = ex+y.
Solution:
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = ex+y = ex . ey
\(\frac{d y}{e^{y}}\) = ex dx
\(\int\) e-y dy = \(\int\) ex dx
e – e-y = ex
or ex + e-y = c is the required solution.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 37.
Solve y2 – x \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = a(y + \(\frac{\mathrm{dy}}{\mathrm{dx}}\))
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 19
Inter 2nd Year Maths 2B Differential Equations Important Questions 20

Question 38.
Solve \(\frac{d y}{d x}=\frac{y^{2}+2 y}{x-1}\)
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 21

Question 39.
Solve \(\frac{d y}{d x}=\frac{x(2 \log x+1)}{\sin y+y \cos y}\) [Mar. 08]
Solution:
Given equation can be written as
(sin y + y cos y) dy = x(2 log x + 1) dx
\(\int\) sin y dy + \(\int\) y cos y dy = \(\int\) 2x log x dx + \(\int\) x dx
\(\int\) sin y dy + y sin y – \(\int\) sin y dy = x2 log x – \(\int\) x2 . \(\frac{1}{x}\) dx + \(\int\) x dx + c
y sin y = x2 log x + c

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 40.
Find the equation of the curve whose slope, at any point, (x, y) is \(\frac{y}{x^{2}}\) and which satisfies the condition y = 1 when x = 3.
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 22

Question 41.
Solve y(1 + x) dx + x(1 + y) dy = 0
Solution:
The given equation can be written as
\(\frac{(1+x)}{x}\) dx + \(\frac{(1+y)}{y}\) . dy = 0
\(\int\) (1 + \(\frac{1}{x}\))dx + \(\int\) (1 + \(\frac{1}{y}\)) dy = 0
x + log x + y + log y = c
x + y + log (xy) = c is the required solution.

Question 42.
Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = sin (x + y) + cos (x + y)
Solution:
Put x + y = t
Inter 2nd Year Maths 2B Differential Equations Important Questions 23
x = log (1 + tan \(\frac{t}{2}\)) + c
But t = x + y
Solution is x = log (1 + tan \(\frac{x+y}{2}\)) + c

Question 43.
Solve (x – y)2 \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = a2
Solution:
Put x – y = t
Inter 2nd Year Maths 2B Differential Equations Important Questions 24
Inter 2nd Year Maths 2B Differential Equations Important Questions 25

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 44.
Solve \(\sqrt{1+x^{2}} \sqrt{1+y^{2}}\) dx + xy dy = 0
Solution:
Given equation can be written as
Inter 2nd Year Maths 2B Differential Equations Important Questions 26
Inter 2nd Year Maths 2B Differential Equations Important Questions 27
Inter 2nd Year Maths 2B Differential Equations Important Questions 28

Question 45.
Solve \(\frac{d y}{d x}=\frac{x-2 y+1}{2 x-4 y}\)
Solution:
Put x – 2y = t
Inter 2nd Year Maths 2B Differential Equations Important Questions 29

Question 46.
Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\sqrt{y-x}\)
Solution:
Put y – x = t2
Inter 2nd Year Maths 2B Differential Equations Important Questions 30

Question 47.
Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + 1 = ex+y
Solution:
Put t = x + y
\(\frac{\mathrm{dt}}{\mathrm{dx}}\) = 1 + \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = et
\(\int \frac{d t}{e^{t}}=\int d x\)
\(\int\) e-t dt = \(\int\) dx
-e-t = x + c
x + e-t + c = 0
Solution is x + e-(x+y) + c = 0

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 48.
Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = (3x + y + 4)2
Solution:
Put t = 3x + y + 4
\(\frac{\mathrm{dt}}{\mathrm{dx}}\) = 3 + \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 3 + t2
\(\frac{\mathrm{dt}}{\mathrm{t}^{3}+3}\) = dx
\(\int \frac{d t}{t^{2}+3}=\int d x\)
\(\frac{1}{\sqrt{3}}\) tan-1 (\(\frac{t}{\sqrt{3}}\)) = x + c
Solution is \(\frac{1}{\sqrt{3}}\) tan-1 (\(\frac{3 x+y+4}{\sqrt{3}}\)) = x + c

Question 49.
Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}\) – x tan (y – x) = 1
Solution:
Put y – x = t
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) – 1 = \(\frac{\mathrm{dt}}{\mathrm{dx}}\)
\(\frac{\mathrm{dt}}{\mathrm{dx}}\) = x tan t + 1 – 1 = x tan t
\(\frac{\mathrm{dt}}{\tan \mathrm{t}}\) = x dx
\(\int\) cot dt = \(\int\) x dx
log |sin | = \(\frac{x^{2}}{2}\) + c
Solution is log |sin (y – x)| = \(\frac{x^{2}}{2}\) + c

Question 50.
Show that f(x, y) = 1 + ex/y is a homogeneous function of x and y.
Solution:
f(kx, xy) = 1 + ekx/ky = 1 + ex/y = f(x, y)
f(x, y) is a homogeneous function degree 0.

Question 51.
Show that f(x, y) = x\(\sqrt{x^{2}+y^{2}}\) – y2 is a homogeneous function of x and y.
Solution:
f(kx, ky) = kx\(\sqrt{k^{2} x^{2}+k^{2} y^{2}}\) – k2y2.
= k2 (x\(\sqrt{x^{2}+y^{2}}\) – y2) = k2 f(x, y)
f(x, y) is a homogeneous function of degree 2.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 52.
Show that f(x, y) = x – y log y + y log x is a homogeneous function of x and y.
Solution:
f(kx, ky) = kx – ky. log ky + ky log (kx)
= k(x – y log (ky) + y log kx)
= k(x – y log k – y log y + y log k + y log x)
= k(x – y log y + y log x)
= k. f(x, y)
f(x, y) is a homogeneous function of degree 1.

Question 53.
Express (1 + ex/y)dx + ex/y (1 – \(\frac{x}{y}\)) dy = 0 in the form \(\frac{\mathrm{dx}}{\mathrm{dy}}\) = F (\(\frac{x}{y}\))
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 31

Question 54.
Express (x\(\sqrt{x^{2}+y^{2}}\) – y2) dx + xy dy = 0 in the form \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = F (\(\frac{x}{y}\))
Solution:
Given equation is
Inter 2nd Year Maths 2B Differential Equations Important Questions 32

Question 55.
Express \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{y}{x+y e^{\frac{-2 x}{y}}}\) in the form \(\frac{\mathrm{dx}}{\mathrm{dy}}\) = F (\(\frac{x}{y}\))
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 33

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 56.
Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{y^{2}-2 x y}{x^{2}-x y}\)
Solution:
The given equation is a homogeneous equation.
Put y = vx
Inter 2nd Year Maths 2B Differential Equations Important Questions 34
\(\log v \sqrt{2 v-3}=-3 \log \frac{x}{c}=\log \frac{c^{3}}{x^{3}}\)
Inter 2nd Year Maths 2B Differential Equations Important Questions 35

Question 57.
Solve (x2 + y2) dx = 2xy dy [A.P. Mar. 16]
Solution:
Given equation can be written as
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{x^{2}+y^{2}}{2 x y}\)
This is a homogeneous function
Put y = vx
Inter 2nd Year Maths 2B Differential Equations Important Questions 36
-log (1 – v2) = log x + log c
= log cx
log cx + log (1 – v2) = 0
log cx(1 – v2) = log 1
cx (1 – v2) = 1
cx (1 – \(\frac{\mathrm{y}^{2}}{\mathrm{x}^{2}}\)) = 1
c(x2 – y2) = x is the required solution.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 58.
Solve xy2 dy – (x3 + y3) dx = 0.
Solution:
Given equation is xy2 dy = (x3 + y2) dx
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{x^{3}+y^{3}}{x y^{2}}\)
This is a homogeneous equation.
Put y = vx
Inter 2nd Year Maths 2B Differential Equations Important Questions 37

Question 59.
Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{x^{2}+y^{2}}{2 x^{2}}\)
Solution:
This is a homogeneous equation.
Inter 2nd Year Maths 2B Differential Equations Important Questions 38

Question 60.
Solve x sec (\(\frac{y}{x}\)) . (y dx + x dy) = y cosec (\(\frac{y}{x}\)) . (x dy – y dx)
Solution:
This given equation can be written as
Inter 2nd Year Maths 2B Differential Equations Important Questions 39
Inter 2nd Year Maths 2B Differential Equations Important Questions 40
log(\(\frac{\sin v}{v}\)) = log cx2
\(\frac{\sin v}{v}\) = cx2
\(\frac{x}{y}\) sin (\(\frac{y}{x}\)) = cx2
Solution is sin(\(\frac{y}{x}\)) = cxy.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 61.
Give the solution of x sin2 \(\frac{y}{x}\) dx = y dx – x dy which passes through the point (1, \(\frac{\pi}{4}\)). [Mar. 14]
Solution:
The given equation can be written as
Inter 2nd Year Maths 2B Differential Equations Important Questions 41
The given curve passes through (1, \(\frac{\pi}{4}\))
cot \(\frac{\pi}{4}\) = log 1 + c
1 = 0 + c ⇒ c = 1
Solution is cot \(\frac{y}{x}\) = log x + 1

Question 62.
Solve (x3 – 3xy2) dx + (3x2y – y3) dy = 0
Solution:
(x3 – 3xy2) dx = -(3x2y – y3) dy
Inter 2nd Year Maths 2B Differential Equations Important Questions 42
Inter 2nd Year Maths 2B Differential Equations Important Questions 43

Question 63.
Transform the following two differential equations Into linear form.
x log x \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y = 2 log x
Solution:
Given equation can be written as
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) + \(\frac{1}{x \log x}\) . y = \(\frac{2}{x}\)
This is of the form \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + Py = Q

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 64.
(x + 2y3) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = y
Solution:
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{x+2 y^{3}}{y}\) = \(\frac{x}{y}\) + 2y2
\(\frac{\mathrm{dx}}{\mathrm{dy}}\) – \(\frac{1}{y}\) . x = 2y2
This is of the form \(\frac{\mathrm{dx}}{\mathrm{dy}}\) + Px = Q.

Question 65.
(cos x) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y sin x = tan x
Solution:
Given equation can be written as
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y(tan x) = (tan x) (sec x)
P = tan x ⇒ \(\int\) P dx = \(\int\) tan x dx = log sec x
I.F. = e\(\int\) log sec x = sec x

Question 66.
Solve (2x – 10y3) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y = o
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 44

Question 67.
Solve (1 + x2) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + 2xy – 4x2 = 0
Solution:
Given equation can be written as
Inter 2nd Year Maths 2B Differential Equations Important Questions 45

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 68.
Solve \(\frac{1}{x}\) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y . ex = e(1 – x)ex
Solution:
Given equation can be written as
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) + (x.ex) . y = x . e(1 – x)ex
I.F. = e\(\int\).ex dx = e(x – 1) ex
y . e(x – 1) ex = \(\int\) x dx
= \(\frac{\mathrm{x}^{2}}{2}\) + c
2y. e(x – 1) ex = x2 + 2c is the required solution.

Question 69.
Solve sin2 x. \(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y = cot x
Solution:
Given equation can be written as
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) + y cosec2 x = cot x . cosec2 x
I.F. = e\(\int\) cosec2 x dx = e-cot x
y . e-cot x = \(\int\) e-cot x . cosec2 x. cot x dx ……………… (1)
Consider \(\int\) e-cot x . cosec2 x . cot x dx
Put -cot x = t ⇒ cosec2 x dx = dt
(1) becomes y . et = \(\int\) -t. et dt
= -(t – 1) et + c
y . e-cot x = -(-cot x – 1) e-cot x + 1
= (cot x + 1) e-cot x + c is the required solution.

Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 70.
Find the solution of the equation
x(x – 2) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) – 2(x – 1)y = x3(x – 2)
which satisfies the condition that y = 9 when x = 3.
Solution:
The given equation can be written as
Inter 2nd Year Maths 2B Differential Equations Important Questions 46
Inter 2nd Year Maths 2B Differential Equations Important Questions 47

Question 71.
Solve (1 + y2)dx = (tan-1y – x)dy. [A.P. Mar. 15]
Solution:
Inter 2nd Year Maths 2B Differential Equations Important Questions 48
is the solution.

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Students get through Maths 2B Important Questions Inter 2nd Year Maths 2B Definite Integrals Important Questions which are most likely to be asked in the exam.

Intermediate 2nd Year Maths 2B Definite Integrals Important Questions

Question 1.
\(\int_{2}^{3} \frac{2 x}{1+x^{2}} d x\) [T.S. Mar. 16; May 06]
Solution:
I = \(\left[\ln \left|1+x^{2}\right|\right]_{2}^{3}\)
= ln 10 – ln 5
= ln (10/5)
= ln 2

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 2.
\( \int_{0}^{\pi} \sqrt{2} \cdot \sqrt{2} \sqrt{\cos ^{2} \frac{\theta}{2} d \theta}\) [A.P. Mar. 16; Mar. 05]
Solution:
I = \( \int_{0}^{\pi} \sqrt{2} \cdot \sqrt{2} \sqrt{\cos ^{2} \frac{\theta}{2} d \theta}\)
= \(\int_{0}^{\pi} 2 \cdot \cos \theta / 2 d \theta\)
= \([4 \sin \theta / 2]_{0}^{\pi}\)
= 4 (sin \(\frac{\pi}{2}\) – sin 0)
= 4

Question 3.
\(\int_{0}^{2}|1-x| d x\) [A.P. Mar. 15; May 11]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 1

Question 4.
I = \(\int_{1}^{5} \frac{d x}{\sqrt{2 x-1}}\) [T.S. Mar. 15]
Solution:
Let 2x – 1 = t2
2 dx = 2t dt
dx = t dt
UL : t = 3
LL : t = 1
I = \(\int_{1}^{3} \frac{t d t}{t}\)
= \(\int_{1}^{3} d t\)
= \([\mathrm{t}]_{1}^{3}\) = 3 – 1
= 2

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 5.
I = \(\int_{0}^{1} \frac{x^{2}}{x^{2}+1} d x\) [Mar. 11]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 2

Question 6.
\(\int_{0}^{2 \pi}\) sin2x cos4; x dx [T.S. Mar. 15; Mar 14]
Solution:
sin2x cos4x is even function.
Inter 2nd Year Maths 2B Definite Integrals Important Questions 3

Question 7.
Evaluate \(\int_{0}^{a} \sqrt{a^{2}-x^{2}} d x\) [T.S. Mar. 16]
Solution:
Put x = a sin θ ⇒ dx = a cos θ . dθ
θ = 0 ⇒ x = 0, x = a ⇒ θ = \(\frac{\pi}{2}\)
Inter 2nd Year Maths 2B Definite Integrals Important Questions 4

Question 8.
Find \(\int_{-\pi / 2}^{\pi / 2}\) sin2 x cos4 x dx [A.P. Mar. 16]
Solution:
f(x) is even
∴ \(\int_{-\pi / 2}^{\pi / 2}\) sin2 x cos4 x dx = 2 \(\int_{0}^{\pi / 2} \sin ^{2} x \cdot \cos ^{4} x d x\)
= 2 . \(\frac{3}{6}\) . \(\frac{1}{4}\) . \(\frac{1}{4}\) . \(\frac{\pi}{4}\) = \(\frac{\pi}{16}\)

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 9.
\(\int_{0}^{\pi / 2} \cdot \frac{\sin ^{5} x}{\sin ^{5} x+\cos ^{5} x} d x\) [Mar. 14, 08]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 5

Question 10.
I = \(\int_{0}^{\pi / 4} \frac{\sin x+\cos x}{9+16 \sin 2 x} d x\) [Mar. 08]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 6
= –\(\frac{1}{40} \ln \left[\frac{1 / 4}{9 / 4}\right]\) = \(\frac{1}{40}\) . 2ln . 3 = \(\frac{1}{20}\) ln 3

Question 11.
y = x3 + 3, y = 0, x = -1, x = 2 [Mar. 05]
Solution:
Required area PABQ
Inter 2nd Year Maths 2B Definite Integrals Important Questions 7

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 12.
x = 4 – y2, x = 0. [Mar. 11]
Solution:
The given parabola x = 4 – y2 meets, the x – axis at A (4, 0) and Y – axis at P(0, 2) and Q(6, -2).
The parabola is symmetrical about X – axis
Inter 2nd Year Maths 2B Definite Integrals Important Questions 8
Required area = 2 Area of OAP
Inter 2nd Year Maths 2B Definite Integrals Important Questions 9

Question 13.
Evaluate the following definite integrals.
(i) \(\int_{0}^{\pi / 2} \sin ^{4} x \cos ^{5} x d x\)
(ii) \(\int_{0}^{\pi / 2} \sin ^{5} x \cos ^{4} x d x\)
(iii) \(\int_{0}^{\pi / 2} \sin ^{6} x \cos ^{4} x d x\)
Solution:
(i) \(\int_{0}^{\pi / 2} \sin ^{4} x \cos ^{5} x d x\)
= \(\frac{4}{9}\) . \(\frac{2}{7}\) . \(\frac{1}{5}\) = \(\frac{8}{315}\)

(ii) \(\int_{0}^{\pi / 2} \sin ^{5} x \cos ^{4} x d x\)
= \(\frac{3}{9}\) . \(\frac{1}{7}\) . \(\frac{4}{5}\) . \(\frac{2}{3}\) = \(\frac{8}{315}\)

(iii) \(\int_{0}^{\pi / 2} \sin ^{6} x \cos ^{4} x d x\)
= \(\frac{3}{10}\) . \(\frac{1}{8}\) . \(\frac{5}{6}\) . \(\frac{3}{4}\) . \(\frac{1}{2}\) . \(\frac{\pi}{2}\)
= \(\frac{3}{512}\) π

Question 14.
\(\int_{0}^{\pi / 2} \frac{d x}{4+5 \cos x}\) [A.P. Mar. 16, 15]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 10
= \(\frac{1}{3}\left[\ln \frac{4}{2}\right]=\frac{1}{3} \ln 2\)

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 15.
\(\int_{0}^{\pi} \frac{x}{1+\sin x} d x\) [May 11]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 11
Inter 2nd Year Maths 2B Definite Integrals Important Questions 12

Question 16.
\(\int_{0}^{\pi} \frac{x \sin ^{3} x}{1+\cos ^{2} x} d x\) [T.S. Mar. 15; Mar. 11]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 13
Inter 2nd Year Maths 2B Definite Integrals Important Questions 14

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 17.
\(\int_{0}^{1} \frac{\log (1+x)}{1+x^{2}} d x\) [Mar. 07, 05]
Solution:
Put x = tan θ
dx = sec2 θ dθ
x = 0 ⇒ θ = 0
x = 1 ⇒ θ = \(\frac{\pi}{4}\)
Inter 2nd Year Maths 2B Definite Integrals Important Questions 15
Inter 2nd Year Maths 2B Definite Integrals Important Questions 16

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 18.
\(\int_{0}^{\pi / 4} \log (1+\tan x) d x\) [A.P. Mar. 16]
Solution:
I = \(\int_{0}^{\pi / 4} \log \left[1+\tan \left(\frac{\pi}{4}-x\right)\right] d x\)
Inter 2nd Year Maths 2B Definite Integrals Important Questions 17

Question 19.
y = 4x – x2, y = 5 – 2x. [T.S. Mar. 16]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 18
y = 4x – x2 …………….. (i)
y = 5 – 2x ……………….. (ii)
y = -([x – 2]2) + 4
y – 4 = (x – 2)2
Solving equations (i) and (ii) we get
4x – x2 = 5 – 2x
x2 – 6x + 5 = 0
(x – 5) (x – 1) = 0
x = 1, 5
Required area = \(=\int_{1}^{5}\left(4 x-x^{2}-5+2 x\right) d x\)
= \(\int_{-1}^{5}\left(6 x-x^{2}-5\right) d x\)
= \(\left(3 x^{2}-\frac{x^{3}}{3}-5 x\right)_{1}^{5}\)
= (75 – \(\frac{125}{3}\) – 25) – (3 – \(\frac{1}{3}\) – 5)
= 50 – \(\frac{125}{3}\) + 2 + \(\frac{1}{3}\)
= \(\frac{150-125+6+1}{3}\) = \(\frac{32}{3}\) sq. units.

Question 20.
y2 = 4x, y2 = 4(4 – x) [May 11]
Solution:
Equations of the curves are y2 = 4x ………………… (1)
y2 = 4(4 – x) …………………. (2)
Eliminating y, we get
4x = 4 (4 – x)
2x = 4 ⇒ x = 2
Substituting in equation (1), y2 = 8
Inter 2nd Year Maths 2B Definite Integrals Important Questions 19
= 2[\(\frac{4}{3}\)(2\(\sqrt{2}\)) – \(\frac{4}{3}\)(-2\(\sqrt{2}\))]
= 2(\(\frac{8 \sqrt{2}}{3}\) + \(\frac{8 \sqrt{2}}{3}\))
= 2(\(\frac{16 \sqrt{2}}{3}\)) = \(\frac{32 \sqrt{2}}{3}\) sq. units

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 21.
Show that the area of the region bounded by \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (ellipse) is it ab. also deduce the area of the circle x2 + y2 = a2. [Mar. 14, May 05]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 20
The ellipse is symmetrical about X and Y axis
Area of the ellipse = 4 Area of CAB
= 4 . \(\frac{\pi}{4}\) ab
Inter 2nd Year Maths 2B Definite Integrals Important Questions 21
(from Prob. 8 in ex 10(a))
= πab
Substituting b = a, we get the circle
x2 + y2 = a2
Area of the circle = πa(a) = πa2 sq. units.

Question 22.
Evaluate \(\int_{\pi / 6}^{\pi / 3} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x\) [Mar. 14]
Solution:
Let A = \(\int_{\pi / 6}^{\pi / 3} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x\)
Put x = \(\frac{\pi}{2}\) – t, dx = – dt
Inter 2nd Year Maths 2B Definite Integrals Important Questions 22
= \(\int_{\pi / 6}^{\pi / 3} d x=(x)_{\pi / 6}^{\pi / 3}\)
= \(\frac{\pi}{3}-\frac{\pi}{6}=\frac{\pi}{6}\)
A = \(\frac{\pi}{12}\)

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 23.
Find \(\int_{0}^{\pi} \frac{x \cdot \sin x}{1+\sin x} d x\) [T.S. Mar. 16] [A.P. Mar. 15]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 23
Inter 2nd Year Maths 2B Definite Integrals Important Questions 24
= \(\left(2 \tan \frac{x}{2}\right)_{0}^{\pi / 2}\)
= 2 . tan \(\frac{\pi}{2}\) – 2 . 0
= 2 – 1 = 2
2A = \(\pi(x)_{0}^{\pi}\) – 2π = π(π) – 2 = π2 – 2π
A = \(\frac{\pi^{2}}{2}\) – π

Question 24.
Find \(\int^{\pi} x \sin ^{7} x \cos ^{6} x d x .\) [May 05] [T.S. Mar. 19]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 25
A = \(\pi \int_{0}^{\pi / 2} \sin ^{7} x \cos ^{6} x d x\)
= π . \(\frac{6}{17}\) . \(\frac{3}{11}\) . \(\frac{1}{9}\) . \(\frac{6}{7}\) . \(\frac{4}{5}\) . \(\frac{2}{3}\)
= π \(\frac{16}{3003}\)

Question 25.
Evaluate \(\int_{1}^{2} x^{5} d x\) dx
Solution:
\(\int_{1}^{2} x^{5} \cdot d x=\left[\frac{x^{6}}{6}\right]_{1}^{2}\)
= \(\frac{2^{6}}{6}-\frac{1}{6}=\frac{63}{6}=\frac{21}{2}\)

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 26.
Evaluate \(\int_{0}^{\pi} \sin x d x\)
Solution:
\(\int_{0}^{\pi} \sin x d x=[-\cos x]_{0}^{\pi}\)
= – cos π – (- cos 0)
= + 1 + 1= 2

Question 27.
Evaluate \(\int_{0}^{a} \frac{d x}{x^{2}+a^{2}}\)
Solution:
\(\int_{0}^{a} \frac{d x}{x^{2}+a^{2}}=\left[\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)\right]_{0}^{a}\)
= \(\frac{1}{a}\) [tan-1 (1) – tan-1 (0)]
= \(\frac{1}{a}\) (\(\frac{\pi}{4}\) – 0) = \(\frac{\pi}{4 a}\)

Question 28.
Evaluate \(\int_{1}^{4} x \sqrt{x^{2}-1} d x\)
Solution:
g(x) = x2 – 1
f(t) = \(\sqrt{t}\)
g'(x) = 2x
Inter 2nd Year Maths 2B Definite Integrals Important Questions 26

Question 29.
Evaluate \(\int_{0}^{2} \sqrt{4-x^{2}} d x\)
Solution:
Let g(θ) = 2 sin θ ⇒ g'(θ) = 2 cos θ
f(x) = \(\sqrt{4-x^{2}}\)
Inter 2nd Year Maths 2B Definite Integrals Important Questions 27

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 30.
Evaluate \(\int_{0}^{16} \frac{x^{1 / 4}}{1+x^{1 / 2}} d x\)
Solution:
Put t4 = x ⇒ dx = 4t3 . dt
Inter 2nd Year Maths 2B Definite Integrals Important Questions 28

Question 31.
Evaluate \(\int_{-\pi / 2}^{\pi / 2} \sin |x| d x\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 29
Inter 2nd Year Maths 2B Definite Integrals Important Questions 30

Question 32.
Show that \(\int_{0}^{\pi / 2} \sin ^{n} x d x=\int_{0}^{\pi / 2} \cos ^{n} x d x\)
Solution:
f(x) = sinnx.
Inter 2nd Year Maths 2B Definite Integrals Important Questions 31

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 33.
Evaluate \(\int_{0}^{\pi / 2} \frac{\cos ^{5 / 2} x}{\sin ^{5 / 2} x+\cos ^{5 / 2} x} d x\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 32
= \(\int_{0}^{\pi / 2} d x=(x)_{0}^{\pi / 2}=\frac{\pi}{2}\)
A = \(\int_{0}^{\pi / 2} \frac{\cos ^{\frac{5}{2}} x}{\sin ^{\frac{5}{2}} x+\cos ^{\frac{5}{2}} x} d x=\frac{\pi}{4}\)

Question 34.
Show that \(\int_{0}^{\pi / 2} \frac{x}{\sin x+\cos x} d x=\frac{\pi}{2 \sqrt{2}} \log (\sqrt{2}+1)\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 33
Inter 2nd Year Maths 2B Definite Integrals Important Questions 34
= \(\frac{\pi}{4 \sqrt{2}}\) log (\(\sqrt{2}\) + 1)2
= \(\frac{\pi}{4 \sqrt{2}}\) 2 log (\(\sqrt{2}\) + 1)
= \(\frac{\pi}{4 \sqrt{2}}\) log (\(\sqrt{2}\) + 1)

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 35.
Evaluate \(\int_{\pi / 6}^{\pi / 3} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x\) [Mar 14]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 35
Inter 2nd Year Maths 2B Definite Integrals Important Questions 36

Question 36.
Find \(\int_{-a}^{a}\left(x^{2}+\sqrt{a^{2}-x^{2}}\right) d x\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 37
= 2(\(\frac{a^{3}}{3}\) – 0) + 2(0 + \(\frac{a^{2}}{3}\) sin-1 (1) – 0 – 0)
= \(\frac{2a^{3}}{3}\) + a2 . \(\frac{\pi}{2}\)

Question 37.
Find \(\int_{0}^{\pi} \frac{x \cdot \sin x}{1+\sin x} d x\) [T.S. Mar. 16] [A.P. Mar. 15]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 38
Inter 2nd Year Maths 2B Definite Integrals Important Questions 39
= \(\left(2 \tan \frac{x}{2}\right)_{0}^{\pi / 2}\)
= 2 . tan \(\frac{\pi}{2}\) – 2 . 0
= 2 – 1 = 2
2A = \(\pi(x)_{0}^{\pi}\) – 2π = π(π) – 2 = π2 – 2π
A = \(\frac{\pi^{2}}{2}\) – π

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 38.
Evaluate \(\int_{0}^{\pi / 2} x \sin x d x\)
Solution:
\(\int_{0}^{\pi / 2} x \cdot \sin x d x=(-x \cdot \cos x)_{0}^{\pi / 2}+\int_{0}^{\pi / 2} \cos x d x\)
= (0 – 0) + \((\sin x)_{0}^{\pi / 2}\)
= sin \(\frac{\pi}{2}\) – sin 0 = 1 – 0 = 1

Question 39.
Evaluate \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{n}\left[\frac{n-i}{n+i}\right]\) by using the method of finding definite integral as the limit of a sum.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 40

Question 40.
Evaluate \(\lim _{n \rightarrow \infty} \frac{2^{k}+4^{k}+6^{k}+\ldots .+(2 n)^{k}}{n^{k+1}}\) by using the method of finding definite integral as the limit of a sum.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 41

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 41.
Evaluate \(\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \cdots\left(1+\frac{n}{n}\right)\right]^{\frac{1}{n}}\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 42

Question 42.
Let f: R → R be a continuous periodic function and T be the period of it. Then prove that for any positive integer n,
\(\int_{0}^{n T} f(x) d x=n \int_{0}^{T} f(x) d x\) ………………. (1)
Solution:
Let k be an integer arid define
g : [kT, (k + 1)T] → [0, T] as g(t) = t – kT.
Then g'(t) = 1 for all t ∈ [kT, (k + 1)T].
Hence by \(\int_{g(c)}^{g(d)} f(t) d t=\int_{c}^{d} f(g(x)) g^{\prime}(x) d x,(f \circ g)\)
g’ is integrable on [kT, (k + 1 )T] and
\(\int_{k T}^{(k+1) T} f(g(t)) g^{\prime}(t) d t=\int_{0}^{T} f(x) d x\) ………… (2)
We have f(g(t)) g'(t) = f(t – kT), 1 = f(t),
since f is periodic with T as the period.
Hence \(\int_{k T}^{(k+1) T} f(g(t)) g^{\prime}(t) d t=\int_{k T}^{(k+1) T} f(t) d t\) ………….. (3)
Thus from (2) and (3),
\(\int_{k T}^{(k+1) T} f(t) d t=\int_{0}^{T} f(t) d t\) ………………. (4)
Let us now prove eq. (1) by using the principle of mathematical induction.
For n = 1, clearly (1) is true.
Assume (1) is true for a positive integer m.
Inter 2nd Year Maths 2B Definite Integrals Important Questions 43
Hence eq. (1) is true for n = m + 1
Thus, eq. (1) ¡s true for any positive integer n, by the principle of mathematical induction.

Question 43.
Find
(i) \(\int_{0}^{\pi / 2} \sin ^{4} x d x\)
(ii) \(\int_{0}^{\pi / 2} \sin ^{7} x d x\)
(iii) \(\int_{0}^{\pi / 2} \cos ^{8} x d x\)
Solution:
(i) \(\int_{0}^{\pi / 2} \sin ^{4} x d x\)
= \(\frac{3}{4}\) . \(\frac{1}{2}\) . \(\frac{\pi}{2}\) = \(\frac{3\pi}{16}\)

(ii) \(\int_{0}^{\pi / 2} \sin ^{7} x d x\)
= \(\frac{6}{7}\) . \(\frac{4}{5}\) . \(\frac{2}{3}\) = \(\frac{16}{35}\)

(iii) \(\int_{0}^{\pi / 2} \cos ^{8} x d x\)
= \(\frac{7}{8}\) . \(\frac{5}{6}\) . \(\frac{3}{4}\) . \(\frac{1}{2}\) . \(\frac{\pi}{2}\) = \(\frac{35\pi}{256}\)

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 44.
Evaluate \(\int_{0}^{a} \sqrt{a^{2}-x^{2}} d x\) [T.S. Mar. 16]
Solution:
Put x = a sin θ ⇒ dx = a cos θ . dθ
θ = 0 ⇒ x = 0, x = a ⇒ θ = \(\frac{\pi}{2}\)
\(\int_{0}^{a} \sqrt{a^{2}-x^{2}} d x=\int_{0}^{\pi / 2}(a \cos \theta)(a \cos \theta) d \theta\)
= a2 \(\int_{0}^{\pi / 2}\) cos2 θ dθ
= a2 . \(\frac{1}{2}\) . \(\frac{\pi}{2}\) = \(\frac{\pi a^{2}}{4}\)

Question 45.
Evaluate the following definite integrals.
(i) \(\int_{0}^{\pi / 2}\) sin4 x . cos5 x dx
(ii) \(\int_{0}^{\pi / 2}\) sin5 x . cos4 x dx
(iii) \(\int_{0}^{\pi / 2}\) sin6 x . cos4 x dx
Solution:
(i) \(\int_{0}^{\pi / 2}\) sin4 x . cos5 x dx
= \(\frac{4}{9}\) . \(\frac{2}{7}\) . \(\frac{1}{5}\) = \(\frac{8}{315}\)

(ii) \(\int_{0}^{\pi / 2}\) sin5 x . cos4 x dx
= \(\frac{3}{9}\) . \(\frac{1}{7}\) . \(\frac{4}{5}\) . \(\frac{2}{3}\) = \(\frac{8}{315}\)

(iii) \(\int_{0}^{\pi / 2}\) sin6 x . cos4 x dx
= \(\frac{3}{10}\) . \(\frac{1}{8}\) . \(\frac{5}{6}\) . \(\frac{3}{4}\) . \(\frac{1}{2}\) . \(\frac{\pi}{2}\)
= \(\frac{3}{512}\) π

Question 46.
Find \(\int_{0}^{2 \pi}\) sin4 x . cos6 x dx [T.S. Mar. 19]
Solution:
f(x) = sin4 x . cos6 x dx
f(2π – x) = f(π – x) = f(x)
\(\int_{0}^{2 \pi}\) sin4 x . cos6 x dx = 2 \(\int_{0}^{2 \pi}\) sin4 x . cos6 x dx
= 4 \(\int_{0}^{2 \pi}\) sin4 x . cos6 x dx
= 4 . \(\frac{5}{10}\) . \(\frac{3}{8}\) . \(\frac{1}{6}\) . \(\frac{3}{4}\) . \(\frac{1}{2}\) . \(\frac{\pi}{2}\)
= \(\frac{3}{128}\) π

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 47.
Find \(\int_{-\pi / 2}^{\pi / 2}\) sin2 x . cos4 x dx [A.P. Mar. 16, 19]
Solution:
f(x) is even
∴ \(\int_{-\pi / 2}^{\pi / 2}\) sin2 x . cos4 x dx = 2 \(\int_{-\pi / 2}^{\pi / 2}\) sin2 x . cos4 x dx
= 2 . \(\frac{3}{6}\) . \(\frac{1}{4}\) . \(\frac{1}{4}\) . \(\frac{\pi}{4}\)
= \(\frac{\pi}{16}\)

Question 48.
Find \(\int_{0}^{\pi}\) x sin7 x . cos6 x dx [May 05]
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 44
\(\int_{0}^{\pi}\) x sin7 x . cos6 x dx = 2 \(\int_{0}^{\pi}\) sin7 x . cos6 x dx
A = π \(\int_{0}^{\pi}\) x sin7 x . cos6 x dx
= π . \(\frac{6}{17}\) . \(\frac{3}{11}\) . \(\frac{1}{9}\) . \(\frac{6}{7}\) . \(\frac{4}{5}\) . \(\frac{2}{3}\)
= π . \(\frac{16}{3003}\)

Question 49.
Find \(\int_{-a}^{a}\) a2 (a2 – x2)3/2 dx
Solution:
f(x) = x2 (a2 – x2)
f(x) is even
Inter 2nd Year Maths 2B Definite Integrals Important Questions 45

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 50.
Find \(\int_{0}^{1} x^{3 / 2} \sqrt{1-x} d x\)
Solution:
Put x = sin2 θ
dx = 2 sin θ . cos θ . dθ
x = 0 ⇒ θ = 0, x = 1 ⇒ θ = \(\frac{\pi}{2}\)
\(\int_{0}^{1} x^{3 / 2} \sqrt{1-x} d x\)
= \(\int_{0}^{\pi / 2} \sin ^{3} \theta \cdot \cos \theta .2 \sin \theta \cos \theta d \theta\)
= \(2 \int_{0}^{\pi / 2} \sin ^{4} \theta \cdot \cos ^{2} \theta d \theta\)
= 2 . \(\frac{3}{6}\) . \(\frac{1}{4}\) . \(\frac{1}{2}\) . \(\frac{\pi}{2}\) = \(\frac{\pi}{16}\)

Question 51.
Find the area under the curve f(x) = sin x in [0, 2π].
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 46
f(x) = sin x,
We know that in [0, π], sin x ≥ 0 and [π, 2π], sin x ≤ 0
Required area = \(\int_{1}^{\pi}\) sinx dx + \(\int_{\pi}^{2 \pi}\) (-sinx) dx
= \((-\cos x)_{0}^{\pi}[\cos x]_{\pi}^{2 \pi}\)
= – cos π + cos 0 + cos 2π – cos π
= -(-1) + 1 + 1-(-1) = 1 + 1 + 1 + 1
= 4.

Question 52.
Find the area under the curve f(x) = cos x in [0, 2π].
Solution:
We know that cos x ≥ 0 in (0, \(\frac{\pi}{2}\)) ∪ (\(\frac{3\pi}{2}\), π) and ≤ 0 in \(\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)\)
Inter 2nd Year Maths 2B Definite Integrals Important Questions 47
= sin \(\frac{\pi}{2}\) – sin 0 – sin \(\frac{3\pi}{2}\) + sin \(\frac{\pi}{2}\) + sin 2π – sin \(\frac{3\pi}{2}\)
= 1 – 0 – (-1) + 1 + 0 – (-1)
= 1 + 1 + 1 + 1 = 4.

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 53.
Find the area bounded by the parabola y = x2, the X-axis and the lines x = -1, x = 2.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 48

Question 54.
Find the area cut off between the line y = 0 and the parabola y = x2 – 4x + 3.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 49
Equation of the parabola is
y = x2 – 4x + 3
Equation of the line is y = 0
x2 – 4x + 3 = 0
(x – 1) (x – 3) = 0
x = 1, 3
The curve takes negative values for the values of x between 1 and 3.
Required area = \(\int_{1}^{3}\) -(x2 – 4x + 3)dx
= \(\int_{1}^{3}\) (-x2 + 4x – 3) dx
= \(\left(-\frac{x^{3}}{3}+2 x^{2}-3 x\right)_{1}^{3}\)
= (-9 + 18 – 9) – (-\(\frac{1}{3}\) + 2 – 3)
= \(\frac{1}{3}\) – 2 + 3 = \(\frac{4}{3}\)

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 55.
Find the area bounded by y = sin x and y = cos x between any two consecutive points of intersection.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 50
= \(\frac{1}{\sqrt{2}}\) + \(\frac{1}{\sqrt{2}}\) + \(\frac{1}{\sqrt{2}}\) + \(\frac{1}{\sqrt{2}}\) = 4\(\frac{1}{\sqrt{2}}\) = 2\(\sqrt{2}\)

Question 56.
Find the area of one of the curvilinear triangles bounded by y = sin x, y = cos x and X – axis.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 51
ln (0, \(\frac{\pi}{4}\)) cos x ≥ sin x and (\(\frac{\pi}{4}\), \(\frac{\pi}{2}\)), cos x ≤ sin x.
Required area = \(\int_{0}^{\pi / 4}\) sin x dx + \(\int_{\pi / 4}^{\pi / 2}\) cos x dx
= \((-\cos x)_{0}^{\pi / 4}+(\sin x)_{\pi / 4}^{\pi / 2}\)
= – cos \(\frac{\pi}{4}\) + cos 0 + sin \(\frac{\pi}{2}\) – sin \(\frac{\pi}{4}\)
= – \(\frac{1}{\sqrt{2}}\) + 1 + 1 – \(\frac{1}{\sqrt{2}}\)
= 2(1 – \(\frac{1}{\sqrt{2}}\)) = 2 – \(\sqrt{2}\)

Question 57.
Find the area of the right angled triangle with base b and altitude h, using the fundamental theorem of integral calculus.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 52
OAB is a right angled triangle and ∠B = 90° take ‘O’ as the origin and OB as positive X-axis
If OB = band AB = h, then Co-ordinates of A are (b, h)
Equation of OA is y = \(\frac{h}{b}\) x
Area of the triangle OAB = \(\int_{0}^{b} \frac{h}{b} x d x\)
= \(\frac{h}{b}\left(\frac{x^{2}}{2}\right)_{0}^{b}=\frac{h}{b} \cdot \frac{b^{2}}{2}=\frac{1}{2} b h .\)

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 58.
Find the area bounded between the curves y2 – 1 = 2x and x = 0.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 53
= \(\left(-\frac{y^{3}}{3}+y\right)_{0}^{1}\) = 1 – \(\frac{1}{3}\) = \(\frac{2}{3}\)

Question 59.
Find the area enclosed by the curves y = 3x and y = 6x – x2.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 54
y = 6x – x2
The straight line y = 3x meets the parabola
y = 6x – x2
3x = 6x = x2
x2 – 3x = 0
x(x – 3) = 0
x = 0 or 3
Required area = \(\int_{0}^{3}\left(6 x-x^{2}-3 x\right) d x\)
= \(\int_{0}^{3}\left(3 x-x^{2}\right) d x=\left(\frac{3 x^{2}}{2}-\frac{x^{3}}{3}\right)_{0}^{3}\)
= \(\frac{27}{2}\) – \(\frac{27}{3}\) = \(\frac{27}{6}\) = \(\frac{9}{2}\)

Question 60.
Find the area enclosed between y = x2 – 5x and y = 4 – 2x.
Solution:
Equations of the curves are
y = x2 – 5x ………. (1)
y = 4 – 2x …………….. (2)
x2 – 5x = 4 – 2x
x2 – 3x – 4 = 0
(x + 1)(x – 4) = 0
x = -1, 4
Inter 2nd Year Maths 2B Definite Integrals Important Questions 55
= 44 – \(\frac{64}{3}\) – \(\frac{3}{2}\) – \(\frac{1}{3}\)
= \(\frac{264-128-9-2}{6}\) = \(\frac{125}{6}\)

Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 61.
Find the area bounded between the curves y = x2, y = \(\sqrt{x}\).
Solution:
Inter 2nd Year Maths 2B Definite Integrals Important Questions 56
= \(\left(\frac{2}{3} x \sqrt{x}-\frac{x^{3}}{3}\right)_{0}^{1}\)
= \(\frac{2}{3}\) – \(\frac{1}{3}\) = \(\frac{1}{3}\)

Question 62.
Find the area bounded between the curves y2 = 4ax, x2 = 4by (a > 0, b > 0).
Solution:
Equations of the given curves are
y2 = 4ax …………………… (1)
x2 = 4by ……………………. (2)
From equation (2), y = \(\frac{x^{2}}{4 b}\)
Substituting in (1) \(\left(\frac{x^{2}}{4 b}\right)^{2}\) = 4ax
x4 = (16 b2) |4ax|
Inter 2nd Year Maths 2B Definite Integrals Important Questions 57
Inter 2nd Year Maths 2B Definite Integrals Important Questions 58

Inter 2nd Year Maths 2B Integration Important Questions

Students get through Maths 2B Important Questions Inter 2nd Year Maths 2B Integration Important Questions which are most likely to be asked in the exam.

Intermediate 2nd Year Maths 2B Integration Important Questions

Question 1.
\(\int\left(\frac{1}{1-x^{2}}+\frac{2}{1+x^{2}}\right)\) [May 11]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 1

Question 2.
\(\int\) sec2x cosec2x dx on I ⊂ R \ ({nπ : n ∈ Z} ∪ {(2n + 1) \(\frac{\pi}{2}\) : n ∈ Z}) [T.S. Mar. 16; Mar, May 07]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 2

Inter 2nd Year Maths 2B Integration Important Questions

Question 3.
\(\int \frac{1+\cos ^{2} x}{1-\cos 2 x}\) dx on I ⊂ R \ {nπ : n ∈ Z} [Mar. 13]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 3

Question 4.
\(\int \sqrt{1-\cos 2 x}\) dx on I ⊂ [2nπ, (n + 1) π], n ∈ Z [May 06]
Solution:
\(\int \sqrt{1-\cos 2 x}\) dx = \(\int \sqrt{2}\) sin x dx
= –\(\sqrt{2}\) cos x + C

Question 5.
\(\int \frac{1}{\cosh x+\sinh x}\) dx on R. [A.P. Mar. 16]
Solution:
\(\int \frac{1}{\cosh x+\sinh x}\) dx
= \(\int \frac{\cosh x-\sinh x}{\cosh ^{2} x-\sinh ^{2} x}\) dx
= \(\int\) (cosh x – sinh x) dx
= sinh x – cosh x + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 6.
\(\int \frac{1}{1+\cos x}\) dx on I ⊂ R \ {(2n + 1)π : n ∈ Z} [T.S. Mar. 15]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 4

Question 7.
\(\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}}\)dx, x ∈ R. [A.P. Mar. 15]
Solution:
\(\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}}\)dx
t = tan-1 x ⇒ dt = \(\frac{d x}{1+x^{2}}\)
\(\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}}\)dx = \(\int\) sin t dt
= – cos t + t
= -cos (tan-1 x) + C

Question 8.
\(\int \frac{\log (1+x)}{1+x}\)dx on (-1, ∞) [T.S. Mar. 15]
Solution:
\(\int \frac{\log (1+x)}{1+x}\)dx
t = 1 + x ⇒ dt = dx
Inter 2nd Year Maths 2B Integration Important Questions 5

Question 9.
\(\int \frac{x^{2}}{\sqrt{1-x^{6}}}\) dx on I = (-1, 1). [May 05]
Solution:
\(\int \frac{x^{2}}{\sqrt{1-x^{6}}}\)
t = x3 ⇒ dt = mx2 dx
\(\int \frac{x^{2}}{\sqrt{1-x^{6}}}\) = \(\frac{1}{3} \int \frac{d \mathrm{t}}{\sqrt{1-\mathrm{t}^{2}}}\)
= \(\frac{1}{3}\) sin-1 t + C
= \(\frac{1}{3}\) sin-1 (x3) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 10.
\(\int \frac{x^{8}}{1+x^{18}}\) dx on R. [A.P. Mar. 16]
Solution:
t = x9 ⇒ dt = 9x8 dx
\(\int \frac{x^{8} d x}{1+x^{18}}=\int \frac{x^{8}}{1+\left(x^{9}\right)^{2}} d x\)
= \(\frac{1}{9} \int \frac{d t}{1+t^{2}}\) = = \(\frac{1}{9}\) tan-1 t + C
= \(\frac{1}{9}\) tan-1 (x9) + C

Question 11.
\(\int \frac{1}{x \log x[\log (\log x)]}\) dx on (1, ∞) [Mar. 11]
Solution:
t = log (log x)
dt = \(\frac{1}{\log x} \cdot \frac{1}{x}\) dx
\(\int \frac{1}{x \log x[\log (\log x)]}\) dx = \(\int \frac{d t}{t}\)
= log |t| + C
= log |log(log x)| + C

Question 12.
\(\int \frac{1}{(x+3) \sqrt{x+2}}\)dx on I ⊂ (-2, ∞) [Mar. 14]
Solution:
x + 2 = t2
dx = 2t dt
\(\int \frac{d x}{(x+3) \sqrt{x+2}}=\int \frac{2 t d t}{t\left(t^{2}+1\right)}\)
= \(2 \int \frac{d t}{t^{2}+1}\)
= 2 tan-1 (t) + C
= 2 tan-1 (\(\sqrt{x+2}\)) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 13.
\(\int \frac{\cot (\log x)}{x}\) dx, x ∈ I ⊂ (0, ∞) \ {e : n ∈ Z). [Mar. 05]
Solution:
t = log x ⇒ dt = \(\frac{\mathrm{dx}}{\mathrm{x}}\) \(\int\)
\(\int \frac{\cot (\log x)}{x}\) dx = \(\int\) cot t dt = log (sin t) + C
= log |sin (log x)| + C

Question 14.
\(\int\) (tan x + log sec x)ex dx on ((2n – \(\frac{1}{2}\))π, (2n + \(\frac{1}{2}\))π) n ∈ Z. [May 07, Mar. 08]
Solution:
t = log |sec x| ⇒ dt = \(\frac{1}{\sec x}\) . sec x . tan x dx
= tan x dx
\(\int\) (tan x + log sec x)ex dx = ex . log|sec x| + C

Question 15.
\(\int \sqrt{x}\) log x dx on (0, ∞) [T.S. Mar. 16]
Solution:
\(\int \sqrt{x}\) log x dx
= (log x) \(\frac{2}{3}\) x3/2 – \(\frac{2}{3}\) \(\int\) x3/2 \(\frac{1}{x}\) dx
= \(\frac{2}{3}\) x3/2 (log x) – \(\frac{2}{3}\) \(\int\) x1/2 dx
= \(\frac{2}{3}\) x3/2 (log x) – \(\frac{2}{3}\) \(\frac{x^{3 / 2}}{3 / 2}\) + C
= \(\frac{2}{3}\) x3/2 (log x) – \(\frac{4}{9}\) x3/2 + C

Question 16.
\(\int\) ex (tan x + sec2 x)dx on I ⊂ R \ {(2n + 1)\(\frac{\pi}{2}\) : n ∈ Z} [Mar 06]
Solution:
f(x) = tanx = f'(x) ⇒ sec2 x dx
I = \(\int\) ex [f(x) + f'(x)] dx = ex. f(x) + C
= ex . tan x + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 17.
\(\int e^{x}\left(\frac{1+x \log x}{x}\right)\) dx on (0, ∞) [A.P. Mar. 15, Mar. 13]
Solution:
\(\int e^{x}\left(\frac{1+x \log x}{x}\right)\) dx = \(\int\) ex (log x + \(\frac{1}{x}\))dx
= ex . log x + C

Question 18.
\(\int \frac{\cos x}{\sin ^{2} x+4 \sin x+5}\)dx [Mar. 07]
Solution:
t = sin x ⇒ dt = cos x dx
I = \(\int \frac{d t}{t^{2}+4 t+5}=\int \frac{d t}{(t+2)^{2}+1}\)
= tan-1(t + 2) + C
= tan-1(sin x + 2) + C

Question 19.
\(\int \frac{d x}{(x+1)(x+2)}\) [Mar. 14, May 11]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 6

Question 20.
\(\int \frac{e^{x}(1+x)}{\cos ^{2}\left(x e^{x}\right)}\) dx on I ⊂ R \ {x ∈ R : cos (xex) = 0} [T.S. Mar. 17]
Solution:
t = x . ex
dt = (x . ex + ex) dx = ex (1 + x) dx
\(\int \frac{e^{x}(1+x)}{\cos ^{2}\left(x e^{x}\right)}\) dx = \(\int \frac{d t}{\cos ^{2} t}=\int \sec ^{2} t d t\)
= tan t + C
= tan (x . ex) + C

Question 21.
\(\int\) x tan-1 x dx, x∈ R [Mar. 05]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 7

Inter 2nd Year Maths 2B Integration Important Questions

Question 22.
\(\int \sqrt{1+3 x-x^{2}} d x\) [May 11]
Solution:
\(\int \sqrt{1+3 x-x^{2}} d x=\int \sqrt{1-\left(x^{2}-3 x\right)} d x\)
= \(\int \sqrt{1-\left(x^{2}-3 x+\frac{9}{4}-\frac{9}{4}\right)} d x\)
Inter 2nd Year Maths 2B Integration Important Questions 8

Question 23.
\(\int \frac{9 \cos x-\sin x}{4 \sin x+5 \cos x} d x\) [T.S. Mar. 17; Mar. 08]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 9

Question 24.
\(\int \frac{d x}{5+4 \cos 2 x}\) [Mar. 11]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 10

Question 25.
Obtain reduction formula for In = \(\int\) cotn x dx, n being a positive integer. n ≥ 2 and deduce the value of \(\int\) cot4 x dx. [T.S. Mar. 19] [A.P. Mar. 16; May 11]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 11

Inter 2nd Year Maths 2B Integration Important Questions

Question 26.
Obtain reduction formula for In = \(\int\) cosecn x dx, n being a positive integer. n ≥ 2 and deduce the value of \(\int\) cosec5 x dx. [T.S. Mar. 16]
Solution:
In = \(\int\) cosecn x dx = \(\int\) cosecn – 2x . cosec2 x dx
= cosecn – 2x (-cot x) + \(\int\) cot x . (n – 2) cosecn – 3 . (cot x) dx
= – cosecn – 2x . cot x + (n – 2) \(\int\) cosecn – 2 x . (cosec2 x -1) dx
= – cosecn – 2x . cot x + (n – 2) In – 2 – (n – 2)In
In (1 + n – 2) = – cosecn – 2 x . cot x + (n – 2) In – 2
Inter 2nd Year Maths 2B Integration Important Questions 12

Question 27.
Evaluate \(\int \frac{2 x+5}{\sqrt{x^{2}-2 x+10}}\) dx. [T.S. Mar. 15]
Solution:
We write
2x + 5 = A \(\frac{\mathrm{d}}{\mathrm{dx}}\) (x2 – 2x + 10) + B
= A(2x – 2) + B
On comparing the coefficients of the like powers of x on both sides of the above equation, we get A = 1 and B = 7.
Thus 2x + 5 = (2x – 2) + 7
Hence \(\int \frac{2 x+5}{\sqrt{x^{2}-2 x+10}}\) dx
= \(\int \frac{2 x-2}{\sqrt{x^{2}-2 x+10}} d x+7 \int \frac{d x}{\sqrt{x^{2}-2 x+10}}+C\)
= \(2 \sqrt{x^{2}-2 x+10}+7 \int \frac{d x}{\sqrt{(x-1)^{2}+3^{2}}}\) + C
= \(2 \sqrt{x^{2}-2 x+10}+7 \sinh ^{-1}\left(\frac{x-1}{3}\right)\) + C

Question 28.
Evaluate \(\int\) sin4 x dx. [Mar. 14]
Solution:
In = sinn x dx = \(-\frac{\sin ^{n-1} x \cdot \cos x}{n}+\frac{n-1}{n} \cdot I_{n-2}\)
Inter 2nd Year Maths 2B Integration Important Questions 13

Inter 2nd Year Maths 2B Integration Important Questions

Question 29.
\(\int \frac{2 \cos x+3 \sin x}{4 \cos x+5 \sin x} d x\) [A.P. Mar. 15]
Solution:
Let 2 cos x + 3 sin x = A(4 cos x + 5 sin x) + B(-4 sin x + 5 cos x)
Equating the coefficient of sin x and cos x we get
4A + 5B = 2
5A – 4B = 3
Inter 2nd Year Maths 2B Integration Important Questions 14

Question 30
\(\int \frac{1}{1+\sin x+\cos x}\) dx [T.S. Mar. 15]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 15

Question 31.
\(\int\) (6x + 5) \(\sqrt{6-2 x^{2}+x}\) dx [May 06]
Solution:
Let 6x + 5 = A(1 – 4x) + B
Equating the co-efficients of x
6 = -4 A ⇒ A = \(\frac{-3}{2}\)
Equating the constants
A + B = 5
B = 5 – A = 5 + \(\frac{3}{2}\) = \(\frac{13}{2}\)
Inter 2nd Year Maths 2B Integration Important Questions 16
Inter 2nd Year Maths 2B Integration Important Questions 17

Inter 2nd Year Maths 2B Integration Important Questions

Question 32.
\(\int \frac{d x}{4+5 \sin x}\) [Mar. 05]
Solution:
t = tan \(\frac{x}{2}\) ⇒ dt = sec2 \(\frac{x}{2}\) . \(\frac{1}{2}\) dx
Inter 2nd Year Maths 2B Integration Important Questions 18
Inter 2nd Year Maths 2B Integration Important Questions 19

Question 33.
\(\int \frac{d x}{(1+x) \sqrt{3+2 x-x^{2}}}\) [May 05]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 20
Inter 2nd Year Maths 2B Integration Important Questions 21

Question 34.
\(\int \frac{d x}{4 \cos x+3 \sin x}\) [Mar. 06]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 22
Inter 2nd Year Maths 2B Integration Important Questions 23

Inter 2nd Year Maths 2B Integration Important Questions

Question 35.
\(\int \frac{2 \sin x+3 \cos x+4}{3 \sin x+4 \cos x+5} d x\) [Mar. 14, 11] [A.P. & T.S. Mar. 16]
Solution:
Let 2 sin x + 3 cos x + 4
= A(3 sin x+4 cos x + 5) + 3(3 cos x – 4sin x) + C
Equating the co-efficients of
sin x. we get 3A – 4B = 2
cos x, we get 4A + 3B = 3
Inter 2nd Year Maths 2B Integration Important Questions 24
Inter 2nd Year Maths 2B Integration Important Questions 25
Substituting in (1)
I = \(\frac{18}{25}\) . x + \(\frac{1}{25}\) log |3 sin x + 4 cos x + 5| – \(\frac{4}{5\left(3+\tan \frac{x}{2}\right)}\) + C

Question 36.
\(\int \frac{x+3}{(x-1)\left(x^{2}+1\right)} d x\) dx [May 07]
Solution:
Let \(\frac{x+3}{(x-1)\left(x^{2}+1\right)}\) = \(\frac{A}{x-1}\) + \(\frac{B x+C}{x^{2}+1}\)
⇒ (x + 3) = A(x2 + 1) + (Bx + C)(x – 1) ………………… (1)
Put x = 1 in (1)
Then 4 = A(1 + 1) + 0 ⇒ A = 2
Put x = 0 in (1)
3 = A(1) + C(-1)
⇒ A – C = 3 ⇒ C = A – 3 = 2 – 3 = -1
Equating coefficient of x2 in (1)
0 = A + B
⇒ B = -A = -2
Inter 2nd Year Maths 2B Integration Important Questions 26

Inter 2nd Year Maths 2B Integration Important Questions

Question 37.
Find \(\int \frac{d x}{3 \cos x+4 \sin x+6}\) [Mar. 13]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 27
Inter 2nd Year Maths 2B Integration Important Questions 28

Question 38.
Find \(\int\) 2x7 dx on R.
Solution:
\(\int\) 2x7 dx = 2 \(\int\) x7 dx
= 2 . \(\frac{x^{8}}{8}\) + C
= \(\frac{x^{8}}{4}\) + C

Question 39.
Evaluate \(\int\) cot2x dx on I ⊂ R \ {nπ : n ∈ Z}.
Solution:
\(\int\) cot2x dx = \(\int\) (cosec2x – 1) dx
= \(\int\) cosec2 x dx – \(\int\) dx
= -cot x – x + C

Question 40.
Evaluate \(\int\left(\frac{x^{6}-1}{1+x^{2}}\right)\) dx for x ∈ R.
Solution:
\(\int\left(\frac{x^{6}-1}{1+x^{2}}\right)\) dx = \(\int\)[(x4 – x2 + 1) – \(\frac{2}{1+x^{2}}\)] dx
= \(\int\) x4 dx – \(\int\) x2 dx + \(\int\) dx – 2 \(\int \frac{d x}{1+x^{2}}\)
= \(\frac{x^{5}}{5}\) – \(\frac{x^{3}}{3}\) + x – 2 tan-1 x + C.

Inter 2nd Year Maths 2B Integration Important Questions

Question 41.
Find \(\int\) (1 – x) (4 – 3x) (3 + 2x) dx, x ∈ R.
Solution:
(1 – x) (4 – 3x) (3 + 2x) = 6x3 – 5x2 – 13x + 12
\(\int\)(1 – x) (4 – 3x) (3 + 2x) dx
= \(\int\) (6x3 – 5x2 – 13x + 12) dx
= 6\(\int\) x3dx – 5 \(\int\) x2 dx – 13 \(\int\) x dx + 12 \(\int\) dx
= \(\frac{6 x^{4}}{4}\) – 5 \(\frac{x^{3}}{3}\) – \(\frac{13 x^{2}}{2}\) + 12x + C
= \(\frac{3}{2}\)x4 – \(\frac{5}{3}\)x3 – \(\frac{13}{2}\)x2 + 12x + C.

Question 42.
Evaluate \(\int\left(x+\frac{1}{x}\right)^{3}\) dx, x > 0.
Solution:
(x + \(\frac{1}{x}\))3 = x3 + 3x + \(\frac{3}{x}\) + \(\frac{1}{x^{3}}\)
\(\int\) (x + + \(\frac{1}{x}\))3 dx = \(\int\) (x3 + 3x + \(\frac{3}{x}\) + \(\frac{1}{x^{3}}\)) dx
= \(\int\) x3 dx + 3 \(\int\) x dx + 3 \(\int\) \(\frac{\mathrm{dx}}{\mathrm{x}}\) + \(\int\) \(\frac{d x}{x^{3}}\)
= \(\frac{x^{4}}{4}\) + \(\frac{3 x^{2}}{2}\) + 3 log x – \(\frac{1}{2 x^{2}}\) + C

Question 43.
Find \(\int \sqrt{1+\sin 2 x}\) dx on R.
Solution:
1 + sin 2x = sin2 x + cos2 x + 2 sin x . cos x
= (sin x + cos x)2
\(\sqrt{1+\sin 2 x}\) = sin x + cos x
If 2nπ – \(\frac{\pi}{4}\) ≤ x ≤ 2nπ + \(\frac{3\pi}{4}\)
= -(sin x + cos x). otherwise
If 2nπ – \(\frac{\pi}{4}\) ≤ x ≤ 2nπ + \(\frac{3\pi}{4}\), then
\(\int \sqrt{1+\sin 2 x}\) dx = \(\int\) (sin x + cos x)dx
= -cos x + sin x + C
If 2nπ + \(\frac{3\pi}{4}\) ≤ x ≤ 2nπ + \(\frac{7\pi}{4}\)
\(\int \sqrt{1+\sin 2 x}\)
= \(\int\) -(sin x + cos x)dx
= –\(\int\) sin x dx – \(\int\) cos x dx
= cos x – sin x + c

Question 44.
Evaluate \(\int \frac{2 x^{3}-3 x+5}{2 x^{2}}\) dx for x > 0 and verify the result by differentiation.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 29
and it is the given expression and hence the result is correct.

Inter 2nd Year Maths 2B Integration Important Questions

Question 45.
Evaluate \(\int \frac{x^{5}}{1+x^{12}}\) dx on R.
Solution:
We define f : R → R by f(t) = \(\frac{1}{1+t^{2}}\)
g : R → R by g(x) = x6
Then g'(x) = 6x5
Define F : R → R by F(t) = tan-1 t
F is the primitive of f
\(\int \frac{x^{5}}{1+x^{12}}\) dx = \(\frac{1}{6}\) \(\int\) f(g(x)) g'(x) dx
= \(\frac{1}{6}\) (F(t) + C)t=g(x)
= \(\frac{1}{6}\) [tan-1 t + C]t=x6
= \(\frac{1}{6}\) tan-1 x6 + C

Question 46.
Evaluate \(\int\) cos3 x sin x dx on R.
Solution:
We define : f = R → R by f(x) = cosx
∴ f'(x) = – sin x
\(\int\) cos3 x sin x dx = \(\int\) (f(x))3 [-f'(x)] dx
= \(\frac{-[f(x)]^{4}}{4}\) + C
= \(\frac{-\cos ^{4} x}{4}\) + C

Question 47.
Find \(\int\) (1 – \(\frac{1}{x^{2}}\)) e(x + \(\frac{1}{x}\)) dx on I where I = (0, ∞)
Solution:
Let J = I = (0, ∞)
Define f : I → R by f(t) = et and g : J → R by g(x) = x + \(\frac{1}{x}\)
Then g(J) ⊂ I, g'(x) = 1 – \(\frac{1}{x^{2}}\)
\(\int\) (1 – \(\frac{1}{x^{2}}\)) e(x + \(\frac{1}{x}\)) dx = \(\int\) f (g(x)) g'(x) dx
= \(\int\) [f(t)dt]t = g(x)
= [\(\int\)et dt]t = g(x)
= [et + c]t=x+\(\frac{1}{x}\)
= e(x+\(\frac{1}{x}\)) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 48.
Evaluate \(\int \frac{1}{\sqrt{\sin ^{-1} x} \sqrt{1-x^{2}}}\) dx on I = (0, 1).
Solution:
We define f : I → R by f(x) = sin-1x
f'(x) = \(\frac{1}{\sqrt{1-x^{2}}}\)
\(\int \frac{1}{\sqrt{\sin ^{-1} x} \sqrt{1-x^{2}}}\) dx = \(\int \frac{f^{\prime}(x)}{\sqrt{f(x)}} d x\)
= 2\(\sqrt{f(x)}\) + C
= 2 \(\sqrt{\sin ^{-1} x}\) + C

Question 49.
Evaluate \(\int \frac{\sin ^{4} x}{\cos ^{6} x}\) dx, x ∈ I ⊂ R \ {\(\frac{(2 n+1) \pi}{2}\) : n ∈ z}
Solution:
\(\int \frac{\sin ^{4} \dot{x}}{\cos ^{6} x}\) dx = \(\int\) tan4 x . sec2 x dx
We define f : I → R by f(x)
f(x) = tan x, then f'(x) = sec2x
\(\int \frac{\sin ^{4} \dot{x}}{\cos ^{6} x}\) dx = \(\int\) [f(x)]4 . f'(x) dx = \(\frac{[f(x)]^{5}}{5}\) + C.
= \(\frac{1}{5}\) tan5x + C.

Question 50.
Evaluate \(\int\) sin2 x dx on R.
Solution:
\(\int\) sin2x dx = \(\int \frac{(1-\cos 2 x)}{2}\) dx
= \(\frac{1}{2}\) \(\int\) dx – \(\frac{1}{2}\) \(\int\) cos 2x dx
= \(\frac{1}{2}\) x – \(\frac{1}{4}\) sin 2x + C.
(since \(\int\) cos 2x dx = \(\frac{1}{2}\) sin 2x + C)

Question 51.
Evaluate \(\int \frac{1}{a \sin x+b \cos x}\) dx where a, b ∈ R and a2 + b2 ≠ 0 on R.
Solution:
We can find real numbers r and θ such that
a = r cos θ, b = r sin θ
Then r = \(\sqrt{a^{2}+b^{2}}\), cos θ = \(\frac{a}{r}\) and sin θ = \(\frac{b}{r}\)
a sin x + b cos x = r . cos θ sin x + r sin θ cos x
= r[cos θ sin x + sin θ cos x]
= r sin (x + θ)
\(
= [latex]\frac{1}{r}\) (cosec (x + θ) dx
= \(\frac{1}{r}\) log |tan \(\frac{1}{2}\)(x + θ)| + C
= \(\frac{1}{\sqrt{a^{2}+b^{2}}} \log \left|\tan \frac{1}{2}(\tilde{x}+\theta)\right|+c\)
For all x ∈ I where I is an interval disjoint with {nπ – θ : n ∈ z}.

Inter 2nd Year Maths 2B Integration Important Questions

Question 52.
Find \(\int \frac{x^{2}}{\sqrt{x+5}}\) dx on (-5, ∞)
Solution:
Put t = x + 5 so that t > 0 on (-5, ∞)
dx = dt and x = t – 5
Inter 2nd Year Maths 2B Integration Important Questions 30

Question 53.
Find \(\int \frac{x}{\sqrt{1-x}}\) dx, x ∈ I = (0, 1)
Solution:
We define f : I → R by f(x) = \(\frac{x}{\sqrt{1-x}}\)
Let J = (0, \(\frac{\pi}{2}\))
Define Φ : J → I by Φ(θ) = sin2 θ
Then Φ is a bijective mapping from J to I Further Φ and Φ-1 are differentiable on their respective domains.
put x = Φ(θ) = sin2θ
dx = 2 sin θ . cos θ dθ
Inter 2nd Year Maths 2B Integration Important Questions 31

Question 54.
Evaluate \(\int \frac{d x}{(x+5) \sqrt{x+4}}\) on (-4, ∞)
Solution:
Let I = (-4, ∞)
Define f on I as f(x) = \(\frac{d x}{(x+5) \sqrt{x+4}}\)
Let J = (0, ∞)
We define Φ : J → I by Φ(t) = t2 – 4
Then Φ is differentiable and is a bijection
Put x = Φ(t) = t2 – 4
then t = \(\sqrt{x+4}\) ⇒ dx = 2t dt
Thus \(\int \frac{d x}{(x+5) \sqrt{x+4}}\) = \(\int \frac{2 t d t}{\left(t^{2}+1\right) t}\)
= \(\int \frac{2 d t}{t^{2}+1}\)
= 2 tan-1 t + C
= 2 tan-1 (\(\sqrt{x+4}\)) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 55.
Evaluate \(\int \frac{d x}{\sqrt{4-9 x^{2}}}\) on I = (-\(\frac{2}{3}\), \(\frac{2}{3}\))
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 32

Question 56.
Evaluate \(\int \frac{1}{a^{2}-x^{2}}\) dx for x ∈ I = (-a, a)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 33

Question 57.
Evaluate \(\int \frac{1}{1+4 x^{2}}\) dx on R.
Solution:
\(\int \frac{1}{1+4 x^{2}}\) dx = \(\int \frac{d x}{4\left(\left(\frac{1}{2}\right)^{2}+x^{2}\right)}\)
= \(\frac{1}{4}\) (2 tan-1 2x) + C
= \(\frac{1}{2}\) tan-1 2x + C

Question 58.
Find \(\int \frac{1}{\sqrt{4-x^{2}}}\) dx on (-2, 2).
Solution:
\(\int \frac{1}{\sqrt{4-x^{2}}}\) dx = \(\int \frac{1}{\sqrt{2^{2}-x^{2}}}\) dx = sin-1(\(\frac{x}{2}\) dx) + C

Question 59.
Evaluate \(\int \sqrt{4 x^{2}+9}\) dx on R.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 34

Inter 2nd Year Maths 2B Integration Important Questions

Question 60.
Evaluate \(\int \sqrt{9 x^{2}-25} d x\) on [\(\frac{5}{3}\), ∞)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 35

Question 61.
Evaluate \(\int \sqrt{16-25 x^{2}}\) dx on (\(\frac{-4}{5}\), \(\frac{4}{5}\))
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 36

Question 62.
Evaluate \(\int\) x sin-1x dx on (-1, 1).
Solution:
Let u(x) = sin-1 x and v(x) = \(\frac{x^{2}}{2}\) so that
v'(x) = x
∴ u(x) v'(x) = x sin-1x
Even though the domain of u is (-1, 1) the function u ¡s differentiable only on (-1, 1).
From the same formula, we have
Inter 2nd Year Maths 2B Integration Important Questions 37
Inter 2nd Year Maths 2B Integration Important Questions 38

Inter 2nd Year Maths 2B Integration Important Questions

Question 63.
Evaluate \(\int\) x2 cos x dx.
Solution:
Let us take u(x) = x2, v(x) = sin x
so that v'(x) = cos x
u(x) v'(x) = x2 cosx
By using the formula for integration by parts, we have
\(\int\) x2 cos x dx = x2 sin x – \(\int\) sin x (x2)’ dx
= x2 sin x – 2 \(\int\) x sin x dx + C.
Again, by applying the formula for integration by parts to
\(\int\) x sin x dx, we get
\(\int\) x. sin x dx = -x cos x – \(\int\) (-cos x) dx
= -x cos x + sin x + C2
\(\int\) x2 cos x dx = x2 sin x – 2(sin x – x cos x) + C
= x2 sin x – 2 sin x + 2x cosx + C
= (x2 – 2) sin x + 2x cos x + C
In evaluating certain integrals by using the formula for integration by parts,, twice or more than twice, we come across the given integral with change of sign. This enables us to evaluate the given integral.

Question 64.
Evaluate \(\int\) ex sin x dx on R.
Solution:
Let A = ex sin x dx on R
A = \(\int\) ex . sin x dx = \(\int\) ex (-cos x)’ dx.
= ex (-cos x) – \(\int\) (-cos x) (ex)’ dx
= – ex cos x + \(\int\) ex cos x dx + C1 ………(1)
\(\int\) ex cos x dx = ex. sin x – \(\int\) ex . sin x dx
= ex sinx – A …………….. (2)
From (1) and (2)
A = – ex cos x + ex sin x – A + C1
2A = ex (sin x – cos x) + C1
A = \(\frac{1}{2}\) ex (sin x – cos x) + C where
C = \(\frac{C_{1}}{2}\)
i.e., \(\int\) ex sin x dx = \(\frac{1}{2}\) ex (sin x – cos x) + C.

Inter 2nd Year Maths 2B Integration Important Questions

Question 65.
Find \(\int\) eax cos (bx + c) dx on R, where a. b, c are real numbers and b ≠ 0.
Solution:
Let A = \(\int\) eax cos (bx + c)dx
Then from the formula for integration by parts
A = eax [latex]\frac{\sin (b x+c)}{b}[/latex] – \(\int\) a eax [latex]\frac{\sin (b x+c)}{b}[/latex] dx
= \(\frac{1}{b}\) eax sin(bx + c) – \(\frac{a}{b}\) \(\int\) eax . sin(bx + c) dx
Inter 2nd Year Maths 2B Integration Important Questions 39
By taking c = 0, we get
\(\int\) eax . cos bx dx = \(\frac{e^{a x}}{a^{2}+b^{2}}\) [a cos bx + b sin bx] + K

Question 66.
Evaluate \(\int \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\) dx, on (-1, 1)
Solution:
Put x = cos θ, θ ∈ (0, π) dx = -sin θ . dθ
Inter 2nd Year Maths 2B Integration Important Questions 40

Question 67.
Evaluate \(\int e^{x}\left(\frac{1-\sin x}{1-\cos x}\right) d x\) on I ⊂ R \ {2nπ : n ∈ z}.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 41
= -ex . cot \(\frac{x}{2}\) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 68.
Evaluate \(\int \tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right) d x\) on I ⊂ R \ {-1, 1}
Solution:
Let x = tan θ ⇒ dx = sec2 θ dθ
\(\frac{2 x}{1-x^{2}}=\frac{2 \tan \theta}{1-\tan ^{2} \theta}\) = tan 2θ
tan-1 \(\left(\frac{2 x}{1-x^{2}}\right)\) = tan-1 (tan 2θ ) = 2θ + nπ
Where n = 0 if |x| < 1 = -1 if x > 1
= 1 if x < -1
We have dθ = \(\frac{1}{1+x^{2}}\) dx and
1 + x2 = 1 + tan2 θ = sec2θ
∴ \(\int \tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right) d x\)
= \(\int\left(\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\right)\left(1+x^{2}\right) \frac{1}{1+x^{2}} d x\)
= \(\int\) (2θ + nπ) \(\int\) sec2θ dθ
= 2 \(\int\) θ sec2 θ dθ + nπ ) \(\int\) sec2 θ dθ + c
= 2 (θ tan θ – \(\int\) tan θ dθ) nπ tan θ + c
= 2 (θ tan θ + log |cos θ| + nπ tan θ + c
= (2θ + nπ) tan θ + 2 log cos θ + c
= (2θ + nπ) tan θ + log cos2 θ + c
= (2θ + nπ) tan θ + log sec2 θ + c
= x tan-1 \(\left(\frac{2 x}{1-x^{2}}\right)\) – log (1 + x2) + c

Question 69.
Find \(\int x^{2} \cdot \frac{\exp \left(m \sin ^{-1} x\right)}{\sqrt{1-x^{2}}}\)dx on (-1, 1) where m is a real number. (Here for y ∈ R, exp. {y} stands for ey).
Solution:
Let t = sin-1x, then
Inter 2nd Year Maths 2B Integration Important Questions 42
Inter 2nd Year Maths 2B Integration Important Questions 43
Inter 2nd Year Maths 2B Integration Important Questions 44

Inter 2nd Year Maths 2B Integration Important Questions

Question 70.
Evaluate \(\int \frac{d x}{4 x^{2}-4 x-7}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 45

Question 71.
Find \(\int \frac{d x}{5-2 x^{2}+4 x}\)
Solution:
5 – 2x2 + 4x = -2 (x2 – 2x – \(\frac{5}{2}\))
= -2 ((x – 1)2 – \(\frac{5}{2}\) – 1)
= -2 ((x – 1)2 – \(\left(\sqrt{\frac{7}{2}}\right)^{2}\))
\(\int \frac{d x}{5-2 x^{2}+4 x}\)
= \(-\frac{1}{2} \int \frac{1}{\left((x-1)^{2}-\sqrt{\frac{7}{2}}\right)^{2}} d x+C\)
Inter 2nd Year Maths 2B Integration Important Questions 46

Question 72.
Evaluate \(\int \frac{d x}{x^{2}+x+1}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 47

Question 73.
Evaluate \(\int \frac{d x}{\sqrt{x^{2}+2 x+10}}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 48

Inter 2nd Year Maths 2B Integration Important Questions

Question 74.
Evaluate \(\int \frac{d x}{\sqrt{1+x-x^{2}}}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 49

Question 75.
Evaluate \(\int \sqrt{3+8 x-3 x^{2}} d x\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 50
Inter 2nd Year Maths 2B Integration Important Questions 51
Inter 2nd Year Maths 2B Integration Important Questions 52

Question 76.
Evaluate \(\int \frac{x+1}{x^{2}+3 x+12}\) dx.
Solution:
We write x + 1 = A(2x + 3) + B
Equating the co-efficients of x; we get 1 = 2A.
A = \(\frac{1}{2}\)
Equating the constants 3A + B = 1
B = 1 – 3A = 1 – \(\frac{3}{2}\) = –\(\frac{1}{2}\)
x + 1 = \(\frac{1}{2}\) (2x + 3) – \(\frac{1}{2}\)
Inter 2nd Year Maths 2B Integration Important Questions 53

Inter 2nd Year Maths 2B Integration Important Questions

Question 77.
Evaluate \(\int(3 x-2) \sqrt{2 x^{2}-x+1} d x\)
Solution:
Let (3x – 2) = A(4x – 1) + B
Equating the co-efficients of x, we get 3 = 4A
A = \(\frac{3}{4}\)
Equating the constants -2 = -A + B
B = -2 + A = -2 + \(\frac{3}{4}\)
= \(\frac{-5}{4}\)
Inter 2nd Year Maths 2B Integration Important Questions 54
Inter 2nd Year Maths 2B Integration Important Questions 55

Question 78.
Evaluate \(\int \frac{2 x+5}{\sqrt{x^{2}-2 x+10}}\)dx. [T.S. Mar. 15]
Solution:
We write
2x + 5 = A \(\frac{\mathrm{d}}{\mathrm{dx}}\) (x2 – 2x + 10) + B
= A (2x – 2) + B
On comparing the coefficients of the like powers of x on both sides of the above equation, we get A = 1 and B = 7.
Thus 2x + 5 = (2x – 2) + 7
Hence \(\int \frac{2 x+5}{\sqrt{x^{2}-2 x+10}}\)dx
= \(\int \frac{2 x-2}{\sqrt{x^{2}-2 x+10}} d x+7 \int \frac{d x}{\sqrt{x^{2}-2 x+10}}+C\)
Inter 2nd Year Maths 2B Integration Important Questions 56

Inter 2nd Year Maths 2B Integration Important Questions

Question 79.
Evaluate \(\int \frac{d x}{(x+5) \sqrt{x+4}}\)
Solution:
Put t = \(\sqrt{x+4}\)
dt = \(\frac{1}{2 \sqrt{x+4}}\) dx
We have t2 = x + 4
x + 5 = t2 + 1
\(\int \frac{d x}{(x+5)(\sqrt{x+4})}=\int \frac{2}{t^{2}+1} d t\)
= 2 tan-1 t + C
= 2 tan-1 (\(\sqrt{x+4}\)) + C.

Question 80.
Evaluate \(\int \frac{d x}{5+4 \cos x}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 57

Question 81.
Find \(\int \frac{d x}{3 \cos x+4 \sin x+6}\) [Mar. 13]
Solution:
t = tan \(\frac{x}{2}\) ⇒ dx = \(\frac{2 d t}{1+t^{2}}\)
Inter 2nd Year Maths 2B Integration Important Questions 58
Inter 2nd Year Maths 2B Integration Important Questions 59

Question 82.
Find \(\int \frac{d x}{d+e \tan x}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 60
Inter 2nd Year Maths 2B Integration Important Questions 61

Inter 2nd Year Maths 2B Integration Important Questions

Question 83.
Evaluate \(\int \frac{\sin x}{d \cos x+e \sin x} d x\) and \(\int \frac{\cos x}{d \cos x+e \sin x} d x\).
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 62
Inter 2nd Year Maths 2B Integration Important Questions 63

Question 84.
Evaluate \(\int \frac{\cos x+3 \sin x+7}{\cos x+\sin x+1} d x\)
Solution:
Let cos x + 3 sin x + 7 = A(cos x + sin x + 1)’ + B(cos x + sin x + 1) + C
Comparing the coefficients
A + B = 1, A – B = 3, B + C = 7
A = -1, B = 2, C = 5
Inter 2nd Year Maths 2B Integration Important Questions 64
Inter 2nd Year Maths 2B Integration Important Questions 65

Inter 2nd Year Maths 2B Integration Important Questions

Question 85.
Find \(\int \frac{x^{3}-2 x+3}{x^{2}+x-2}\) dx.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 66
Inter 2nd Year Maths 2B Integration Important Questions 67

Question 86.
Find \(\int \frac{d x}{x^{2}-81}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 68

Inter 2nd Year Maths 2B Integration Important Questions

Question 87.
Find \(\int \frac{2 x^{2}-5 x+1}{x^{2}\left(x^{2}-1\right)}\) dx.
Solution:
Let \(\frac{2 x^{2}-5 x+1}{x^{2}\left(x^{2}-1\right)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x-1}+\frac{D}{x+1}\)
2x2 – 5x + 1 = Ax(x2 – 1) + B(x2 – 1) + Cx2 (x + 1) + Dx2 (x – 1)
x = 1 ⇒ 2 – 5 + 1 = C (1 + 1) ⇒ 2C = -2
⇒ C = -1
x = -1 ⇒ 2 + 5 + 1 = D (-1 -1)
⇒ 8 = -2B ⇒ D = -4
x = 0 ⇒ 1 = B(-1) ⇒ B = -1
Equating the coefficients of x3
0 = A + C + D ⇒ A = -C – D = 1 + 4 = 5
Inter 2nd Year Maths 2B Integration Important Questions 69

Question 88.
Find \(\int \frac{3 x-5}{x\left(x^{2}+2 x+4\right)}\) dx.
Solution:
\(\frac{3 x-5}{x\left(x^{2}+2 x+4\right)}=\frac{A}{x}+\frac{B x+C}{x^{2}+2 x+4}\)
3x – 5 = A(x2 + 2x + 4) + (Bx + C) . x
x = 0 ⇒ -5 = 4 A ⇒ A = –\(\frac{5}{4}\)
Equating the coefficients of x2
A + B = 0 ⇒ B = -A = \(\frac{5}{4}\)
Equating the coefficient of x
3 = 2 A + C
C = 3 – 2 A = 3 + 2 . \(\frac{5}{4}\) = \(\frac{11}{2}\)
Inter 2nd Year Maths 2B Integration Important Questions 70
Inter 2nd Year Maths 2B Integration Important Questions 71

Inter 2nd Year Maths 2B Integration Important Questions

Question 89.
Find \(\int \frac{2 x+1}{x\left(x^{2}+4\right)^{2}} d x\)
Solution:
Let \(\frac{2 x+1}{x\left(x^{2}+4\right)^{2}}=\frac{A}{x}+\frac{B x+C}{x^{2}+4}+\frac{D x+E}{\left(x^{2}+4\right)^{2}}\)
2x + 1 = A (x2 + 4)2 + (Bx + C) + x (x2 + 4) + (Dx + E)x
Equating the coefficients of like power of x, we obtain
A + B = 0, C = 0, 8A + 4B + D = 0,
4C + E = 2, A = \(\frac{1}{16}\)
Solving these equation, we obtain
Inter 2nd Year Maths 2B Integration Important Questions 72
Inter 2nd Year Maths 2B Integration Important Questions 73

Question 90.
Evaluate \(\int\) x3 . e5x dx.
Solution:
We take a = 5, use the reduction formula
Inter 2nd Year Maths 2B Integration Important Questions 74
Inter 2nd Year Maths 2B Integration Important Questions 75

Question 91.
Evaluate \(\int\) sin4 x dx. [Mar. 14]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 76
Inter 2nd Year Maths 2B Integration Important Questions 77

Inter 2nd Year Maths 2B Integration Important Questions

Question 92.
Evaluate \(\int\) tan6 x dx.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 78

Question 93.
\(\int\) sec5 x dx.
Solution:
Reduction formula is
Inter 2nd Year Maths 2B Integration Important Questions 79

Inter 2nd Year Maths 2B Hyperbola Important Questions

Students get through Maths 2B Important Questions Inter 2nd Year Maths 2B Hyperbola Important Questions which are most likely to be asked in the exam.

Intermediate 2nd Year Maths 2B Hyperbola Important Questions

Question 1.
Find the equations of the hyperbola whose foci are (±5, 0) the transverse axis is of length 8. [T.S. Mar. 16; May 11]
Solution:
Foci are S(±5,0) ∴ ae = 5
Length of transverse axis = 2a = 8
a = 4
e = \(\frac{5}{4}\)
b2 = a2(e2 – 1) = 16(\(\frac{25}{16}\) – 1) = 9
Equation of the hyperbola is \(\frac{x^{2}}{16}\) – \(\frac{y^{2}}{9}\) = 1
9x2 – 16y2 = 144.

Inter 2nd Year Maths 2B Hyperbola Important Questions

Question 2.
If the eccentricity of a hyperbola is \(\frac{5}{4}\), then find the eccentricity of its conjugate-hyperbola. [AP Mar. 16] [TS Mar. 15, 13]
Solution:
If e and e1, are the eccentricity of a hyper bola and its conjugate hyperbola, then
\(\frac{1}{\mathrm{e}^{2}}\) + \(\frac{1}{\mathrm{e}_{1}^{2}}\) = 1
Given e = \(\frac{5}{4}\) = \(\frac{16}{25}\) + \(\frac{1}{\mathrm{e}_{1}^{2}}\) = 1
\(\frac{1}{\mathrm{e}_{1}^{2}}\) = 1 – \(\frac{16}{25}\) = \(\frac{9}{25}\) e12 = \(\frac{25}{9}\)
e1 = \(\frac{5}{3}\)

Question 3.
Find the centre, foci, eccentricity equation of the directrices, length of the latus rectum of the x2 – 4y2 = 4 hyperbola. [A.P. Mar. 16; May 11]
Solution:
Equation of the hyperbola is \(\frac{x^{2}}{4}\)– \(\frac{y^{2}}{1}\) = 1
a2 = 4, b2 = 1
Centre is c (0, 0)
a2e2 = a2 + b2 = 4 + 1 = 5
ae = \(\sqrt{5}\)
Foci are (±ae, 0) = (±\(\sqrt{5}\), 0)
Eccentricity = \(\frac{\mathrm{ae}}{\mathrm{a}}\) = \(\frac{\sqrt{5}}{2}\)
Equations of directrices are x = ± \(\frac{\mathrm{ae}}{\mathrm{a}}\)
= ± 2 . \(\frac{2}{\sqrt{5}}\)
⇒ \(\sqrt{5}\) x = ± 4
⇒ \(\sqrt{5}\) x ± 4 = 0
Length of the latus rectum = \(\frac{2 b^{2}}{a}\) = \(\frac{2.1}{2}\) = 1

Inter 2nd Year Maths 2B Hyperbola Important Questions

Question 4.
Find the equations of the tangents to the hyperbola 3x2 – 4y2 = 12 which are
i) Parallel and
ii) Perpendicular to the line y = x – 7. [AP Mar. 15]
Solution:
i) Equation of the hyperbola is 3x2 – 4y2 = 12
\(\frac{x^{2}}{4}\) – \(\frac{y^{2}}{3}\) = 1
a2 = 4, b2 = 3.
The tangent is parallel to y = x – 7
m = slope of the tangent = 1
Equation of the parallel tangents are
y = mx ± \(\sqrt{a^{2} m^{2}-b^{2}}\)
y = x ± \(\sqrt{4-3}\)
y = x ± 1

ii) The tangent is perpendicular to y – x = 7
m – slope of the tangent = (-1)
Equation of the perpendicular tangents are
y = (-1) x ± \(\sqrt{4(-1)^{2}-3}\)
= -x ± 1
x + y = ± 1.

Question 5.
If 3x – 4y – k = 0 is a tangent to x2 – 4y2 = 5, find value of k. [T.S. Mar. 17]
Solution:
Equation of the hyperbola x2 – 4y2 = 5
\(\frac{x^{2}}{5}\) – \(\frac{y^{2}}{\left(\frac{5}{4}\right)}\) = 1
a2 = 5, b2 = \(\frac{5}{4}\)
Equation of the given line is 3x — 4y + k = 0
4y = 3x + k
y = \(\frac{3}{4}\) x + \(\frac{k}{4}\)
m = \(\frac{3}{4}\), c = \(\frac{k}{4}\),
Condition for tangency is c2 = a2m2 – b2
\(\frac{\mathrm{k}^{2}}{16}\) = 5 . \(\frac{9}{16}\) – \(\frac{5}{4}\)
k2 = 45 – 20 = 25
k = ± 5.

Inter 2nd Year Maths 2B Hyperbola Important Questions

Question 6.
Find the equations of the tangents to the hyperbola x2 – 4y2 = 4 Which are
i) Parallel
ii) Perpendicular to the line x + 2y = 0. [T.S. Mar. 15, Mar. 14, 11; May 06]
Solution:
Equation of the hyperbola is
x2 – 4y2 = 4
\(\frac{x^{2}}{4}\) – \(\frac{y^{2}}{1}\) = 1
a2 = 4, b2 = 1
i) The tangent is parallel to x + 2y = 0
m = –\(\frac{1}{2}\)
c2 = a2m2 – b2 = 4 . \(\frac{1}{4}\) = 1 – 1 = 0
c = 0
Equation of the parallel tangent is
y = mx + c
= –\(\frac{1}{2}\) x
2y = -x
x + 2y = 0.

ii) The tangent is perpendicular to x + 2y = 0
Slope of the tangent = m = \(\frac{-1}{\left(-\frac{1}{2}\right)}\) = 2
c2 = a2m2 – b2 = 4 . 4 – 1 = 153
c = ±\(\sqrt{15}\)
Equation of the perpendicular tangent is
y = 2x ± \(\sqrt{15}\)

Question 7.
Prove that the point of intersection of two perpendicular tangents to the hyperbola \(\frac{x^{2}}{a^{2}}\) – \(\frac{y^{2}}{b^{2}}\) = 1 lies on the circle x2 + y2 = a2 – b2. [T.S. Mar. 16]
Solution:
Let P (x1, y1) be the point of intersection of two perpendicular tangents to the hyperbola
\(\frac{x^{2}}{a^{2}}\) – \(\frac{x^{2}}{b^{2}}\) = 1
Equation of the tangent can be taken as
y = mx ± \(\sqrt{a^{2} m^{2}-b^{2}}\)
This tangent passes through P (x1, y1)
y1 = mx1 ± \(\sqrt{a^{2} m^{2}-b^{2}}\)
(y1 – mx1)2 = a2m2 – b2
y12 + m2x12 – 2mx1y1 = a2m2 – b2
m2x12 – a2m2 – 2mx1y1 + y12 + b2 = 0
m2(x12 – a2) – 2mx1y1 + (y12 + b2) = 0
This is a quadratic in m giving the values say m1, m1 which are the slopes of the tangents
passing through P
The tangents are perpendicular
⇒ m1m2 = – 1
\(\frac{y_{1}^{2}+b^{2}}{x_{1}^{2}-a^{2}}\) = – 1 ⇒ y12 + b2
x12 + y12 = a2 – b2
focus of P (x1, y1) is x2 + y2 = a2 – b2.
This circle is called director circle of the hyperbola.

Inter 2nd Year Maths 2B Hyperbola Important Questions

Question 8.
If e, e1 are the eccentricities of a hyperbola and its conjugate hyperbola prove that \(\frac{1}{e^{2}}\) + \(\frac{1}{\mathrm{e}_{1}^{2}}\) = 1 [Mar. 11]
Solution:
Equation of the hyperbola is \(\frac{x^{2}}{a^{2}}\) – \(\frac{x^{2}}{b^{2}}\) = 1
∴ b2 = a2(e2 – 1) ⇒ e2 – 1 = \(\frac{b^{2}}{a^{2}}\)
e2 = 1 + \(\frac{b^{2}}{a^{2}}\) = \(\frac{a^{2}+b^{2}}{a^{2}}\)
∴ \(\frac{1}{\mathrm{e}^{2}}\) = \(\frac{a^{2}}{a^{2}+b^{2}}\) …………… (1)
Equation of the conjugate hyperbola is
Inter 2nd Year Maths 2B Hyperbola Important Questions 1

Question 9.
Find the centre eccentricity, foci, directrices and length of the latus rectum of the following hyperbolas.
i) 4x2 – 9y2 – 8x – 32 = 0
ii) 4 (y + 3)2 – 9(x – 2)2 = 1.
Solution:
i) 4x2 – 9y2 – 8x – 32 = 0
4(x2 – 2x) – 9y2 = 32
4(x2 – 2x + 1) – 9y2 = 36
\(\frac{(x-1)^{2}}{9}\) – \(\frac{(y)^{2}}{4}\) = 1
Centre of the hyperbola is C (1, 0)
a2 = 9, b2 = 4 ⇒ a = 3, b = 2
e = \(\sqrt{\frac{a^{2}+b^{2}}{a^{2}}}=\sqrt{\frac{9+4}{9}}=\frac{\sqrt{13}}{3}\)
Foci are (1±3. \(\frac{\sqrt{13}}{3}\), 0) = (1±\(\sqrt{13}\), 0)
Equations of differences are x = 1 ± \(\frac{3.3}{\sqrt{13}}\)
⇒ x = 1 ± \(\frac{9}{\sqrt{13}}\)
Length of the latus rectum = \(\frac{2 b^{2}}{a}\)
= \(\frac{2.4}{3}\) = \(\frac{8}{3}\)

ii) The equation of the hyperbola is
4 (y + 3)2 – 9 (x – 2)2 = 1
\(\frac{y-(-3)^{2}}{1 / 4}\) = \(\frac{(x-2)^{2}}{1 / 9}\) = 1
Centre is C (2, -3)
Semi transverse axis = b = \(\frac{1}{2}\)
Semi conjugate axis = a = \(\frac{1}{3}\)
Inter 2nd Year Maths 2B Hyperbola Important Questions 2

Question 10.
If e, e1 are the eccentricities of a hyperbola and its conjugate hyperbola prove that \(\frac{1}{e^{2}}\) + \(\frac{1}{e_{1}^{2}}\) = 1. [Mar. 11]
Solution:
Inter 2nd Year Maths 2B Hyperbola Important Questions 3
Equation of the conjugate hyperbola is
Inter 2nd Year Maths 2B Hyperbola Important Questions 4

Inter 2nd Year Maths 2B Hyperbola Important Questions

Question 11.
i) If the line lx + my = 0 is a tangent to the hyperbola \(\frac{x^{2}}{a^{2}}\) – \(\frac{y^{2}}{b^{2}}\) = 1, then show that a2l2 – b2m2 = n2.

ii) If the lx + my = 1 is a normal to the hyperbola \(\frac{x^{2}}{a^{2}}\) – \(\frac{y^{2}}{b^{2}}\) = 1, then show that \(\frac{a^{2}}{l^{2}}\) – \(\frac{b^{2}}{m^{2}}\) = (a2 + b2)2.
Solution:
i) Equation of the given tangent ¡s
lx + my + n = 0 ……………. (1)
Equation of the tangent P(θ) is
\(\frac{x}{a}\) sec θ – \(\frac{y}{b}\) tan θ – 1 = 0 …………….. (2)
Comparing (1) and (2)
\(\frac{\sec \theta}{a l}\) = \(\frac{\tan \theta}{-\mathrm{bm}}\) = \(\frac{-1}{n}\)
sec θ = –\(\frac{\mathrm{a} l}{\mathrm{n}}\), tan θ = \(\frac{\mathrm{bm}}{\mathrm{n}}\)
sec2 θ – tan2θ = 1
= \(\frac{a^{2} l^{2}}{n^{2}}\) – \(\frac{b^{2} m^{2}}{n^{2}}\) = 1 ⇒ a2l2 – b2m2 = n2.

ii) Equation of the given line is lx + my = 1 ……………. (1)
Equation of the normal at P(θ) is
\(\frac{a x}{\sec \theta}\) + \(\frac{b y}{\tan \theta}\) = a2 + b2 ……….. (2)
Comparing (1) and (2)
Inter 2nd Year Maths 2B Hyperbola Important Questions 5

Question 12.
Find the equations of the tangents to the hyperbola 3x2 – 4y2 = 12 which are
i) Parallel and ii) Perpendicular to the line y = x – 7. [A.P. Mar. 15]
Solution:
i) Equation of the hyperbola is 3x2 – 4y2 = 12
\(\frac{x^{2}}{4}\) – \(\frac{y^{2}}{3}\) = 1
a2 = 4, b2 = 3
The tangent is parallel to y = x – 7
m = slope of the tangent = 1
Equation of the parallel tangents are
y = mx ± \(\sqrt{a^{2} m^{2}-b^{2}}\)
y = x ± \(\sqrt{4-3}\)
y = x ± 1

ii) Th tangent is perpendicular to y – x = 7
m – slope of the tangent = (-1)
Equation of the perpendicular tangents are
y = (-1) x ± \(\sqrt{4(-1)^{2}-3}\)
= -x ± 1
x + y = ±1.

Inter 2nd Year Maths 2B Hyperbola Important Questions

Question 13.
Prove that the point of intersection of two perpendicular tangents to the hyperbola \(\frac{x^{2}}{a^{2}}\) – \(\frac{y^{2}}{b^{2}}\) = 1 lies on the circle x2 + y2 = a2 – b2. [T.S. Mar. 16]
Solution:
Let P (x1, y1) be the point of intersection of two perpendicular tangents to the hyperbola
\(\frac{x^{2}}{a^{2}}\) – \(\frac{y^{2}}{b^{2}}\) = 1
Equation of the tangent can be taken as
y = mx ± \(\sqrt{a^{2} m^{2}-b^{2}}\)
This tangent passes through P (x1, y1)
y1 = mx1 ± \(\sqrt{a^{2} m^{2}-b^{2}}\)
(y1 – mx1)2 = a2m2 – b2
y12 + m2x12 – 2mx1y1 = a2m2 – b2
m2x12 – a2m2 – 2mx1y1 + y12 + b2 = 0
m2 (x12 – a2) – 2mx1y1 + (y12 + b2) = 0
This is a quadratic in m giving the values say m1, m2 which are the slopes of the tangents passing through P
The tangents are perpendicular
⇒ m1m2 = – 1
\(\frac{y_{1}^{2}+b^{2}}{x_{1}^{2}-a^{2}}\) = -1 ⇒ y12 + b2 = -x12 + a2
x12 + y12 = a2 – b2
focus of P (x1, y1) is x2 + y2 = a2 – b2.
This circle is called director circle of the hyperbola.

Question 14.
A circle cuts the rectangular hyperbola xy = 1 in the points (xr, yr), r = 1, 2, 3, 4. Prove that x1x2x3x4 = y1y2y3y4 = 1.
Solution:
Let the circle be x2 + y2 = a2.
Since (t, \(\frac{1}{t}\)) (t ≠ 0) lies on xy = 1, the points of intersection of the circle and the hyperbola are given by
t2 + \(\frac{1}{t^{2}}\) = a2
⇒ t4 – a2t2 + 1 = 0
⇒ t4 + 0 . t3 – a2t2 + 0 . t + 1 = 0.
If t1, t2, t3 and t4 are the roots of the above biquadratic, then t1t2t3t4 = 1.
If (xr, yr) = (tr; \(\frac{1}{t_{r}}\)), r = 1, 2, 3, 4
then x1x2x3x4 = t1t2t3t4 = 1,
and y1y2y3y4 = \(\frac{1}{t_{1} t_{2} t_{3} t_{4}}\) = 1

Inter 2nd Year Maths 2B Hyperbola Important Questions

Question 15.
If four points be taken on a rectangular hyperbola such that the chords joining any two points is perpendicular to the chord joining the other two, and if α, β, γ and δ be the inclinations to either asymptote of the straight lines joining these points to the centre, prove that
tan α tan β tan γ tan δ = 1.
Solution:
Let the equation of the rectangular hyperbola be x2 – y2 = a2. By rotating the X-axis and the Y-axis about the origin through an angle \(\frac{\pi}{4}\) in the clockwise sense, the equation x2 – y2 = a2 can be transformed to the form xy = c2.
Let (ctr, \(\frac{c}{t_{r}}\)), r = 1, 2, 3, 4 (tr ≠ 0) be four point on the curve. Let the chord joining
A = (ct1, \(\frac{c}{t_{1}}\)), B = (ct2, \(\frac{c}{t_{2}}\)) be perpendicular to the chord joining C = (ct3, \(\frac{c}{t_{3}}\)) and D = (ct4, \(\frac{c}{t_{4}}\)).
The slope of \(\stackrel{\leftrightarrow}{A B}\) is \(\frac{\frac{c}{t_{1}}-\frac{c}{t_{2}}}{c t_{1}-c t_{2}}=\frac{-1}{t_{1} t_{2}}\)
[No chord of the hyperbola can be vertical]
Similarly slope of \(\stackrel{\leftrightarrow}{C D}\) is –\(\frac{1}{t_{3} t_{4}}\), Since \(\stackrel{\leftrightarrow}{A B}\) ⊥ \(\stackrel{\leftrightarrow}{C D}\).
\(\left(-\frac{1}{t_{1} t_{2}}\right)\left(-\frac{1}{t_{3} t_{4}}\right)\) = -1 ⇒ t1t2t3t4 = -1 ………………… (1)
We know the coordinate axes are the asymptotes of the curve, If \(\stackrel{\leftrightarrow}{\mathrm{OA}}\), \(\stackrel{\leftrightarrow}{\mathrm{OB}}\), \(\stackrel{\leftrightarrow}{\mathrm{OC}}\) and \(\stackrel{\leftrightarrow}{\mathrm{OD}}\) make angles α, β, γ and δ with the positive direction of the X-axis, then tan α, tan β, tan γ and tan δ are their respective slopes. [O, the origin is the centre, None of \(\stackrel{\leftrightarrow}{\mathrm{OA}}\), \(\stackrel{\leftrightarrow}{\mathrm{OB}}\), \(\stackrel{\leftrightarrow}{\mathrm{OC}}\) and \(\stackrel{\leftrightarrow}{\mathrm{OD}}\) is vertical]
Inter 2nd Year Maths 2B Hyperbola Important Questions 6
If \(\stackrel{\leftrightarrow}{\mathrm{OA}}\), \(\stackrel{\leftrightarrow}{\mathrm{OB}}\), \(\stackrel{\leftrightarrow}{\mathrm{OC}}\) and \(\stackrel{\leftrightarrow}{\mathrm{OD}}\) make angles α, β, γ and δ with the other asymptote the Y-axis then cot α, cot β, cot γ and cot δ are their respective inclinations so that
cot α cot β cot γ cot δ = tan α tan β tan γ tan δ = 1.

Inter 2nd Year Maths 2B Ellipse Important Questions

Students get through Maths 2B Important Questions Inter 2nd Year Maths 2B Ellipse Important Questions which are most likely to be asked in the exam.

Intermediate 2nd Year Maths 2B Ellipse Important Questions

Question 1.
Find the value of k if 4x + y + k = 0 is a tangent to the ellipse x2 + 3y2 = 3. [AP Mar. 16, 15]
Solution:
Equation of the ellipse is
x2 + 3y2 = 3
\(\frac{x^{2}}{3}\) + \(\frac{y^{2}}{1}\) = 1
a2 =, 3, b2 = 1
Equation of the line is 4x + y + k = 0
y = -4x – k .
m = -4c = -k.
Condition for tangency is c2 = a2m2 + b2
(-k)2 = 3 (4)2 + 1 ,
k2 = 48 + 1 = 49
k = ±7.

Inter 2nd Year Maths 2B Ellipse Important Questions

Question 2.
Find the equation of tangents to the ellipse 2x2 + y2 = 8 which are
i) Parallel to x – 2y – 4 = 0 [May. 05, Mar. 06] [T.S. Mar. 17]
Solution:
Slope will be : \(\frac{1}{2}\)
Equation of tangent y = mx ± \(\sqrt{a^{2} m^{2}+b^{2}}\)
y = \(\frac{1}{2}\)x ± \(\sqrt{a^{2}\left(\frac{1}{2}\right)^{2}+b^{2}}\)
\(\frac{x^{2}}{4}\) + \(\frac{y^{2}}{8}\) = 1
y = \(\frac{1}{2}\)x ± \(\sqrt{4 \times \frac{1}{4}+8}\)
y = \(\frac{1}{2}\)x ± 3
2y – x ± 6 = 0 required equation of tangents.
x – 2y ± 6 = 0.

Question 3.
Find the equation of the ellipse in the standard form whose distance between foci is 2 and the length of latus rectum is \(\frac{1}{2}\). [T.S. Mar. 15]
Solution:
Latus rectum = \(\frac{15}{2}\)
distance between foci = 2
\(\frac{2 b^{2}}{a}\) = \(\frac{15}{2}\) ; 2ae = 2
ae = 1
⇒ b2 = a2 – a2 e2
⇒ b2 = a2 – 1
⇒ \(\frac{15}{4}\)a = a2 – 1 .
⇒ 4a2 – 15a – 4 = 0
a = 4 or a = –\(\frac{1}{4}\)
b2 = a2 – 1
= 16 – 1
Equation of the ellipse is \(\frac{x^{2}}{16}\) + \(\frac{y^{2}}{15}\) = 1

Inter 2nd Year Maths 2B Ellipse Important Questions

Question 4.
Find the equation of the ellipse in the standard form such that distance between foci is 8 and distance between directrices is 32. [Mar. 06, May. 07]
Solution:
Distance between foci = 8.
Distance between directrices = 32
2ae = 8
ae = 4
\(\frac{2 a}{e}\) = 32
\(\frac{a}{e}\) = 16
(ae) (\(\frac{a}{e}\)) = 64
a2 = 64
b2 = a2 – a2 e2
= 64 – 16 = 48
Equation of the ellipse is
∴ \(\frac{x^{2}}{64}\) + \(\frac{y^{2}}{48}\) = 1

Question 5.
Find the condition for the line x cos α + y sin α = p to be a tangent to the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1. [Mar. 14]
Solution:
Equation of the ellipse is
Inter 2nd Year Maths 2B Ellipse Important Questions 1

Question 6.
Find the equation of the ellipse with focus at (1, -1) e = \(\frac{2}{3}\) and directrix as x + y + 2 = 0. [Mar. 05] [T.S. Mar. 19]
Solution:
P(x1, y1) is any point on the ellipse. Equation of the directrix is
x + y + 2 = 0
Draw PM perpendicular to ZM, Join SP
By Definition of ellipse SP = e. PM
SP2 = e2 . PM2
(x1 – 1)2 + (y1 + 1)2 (\(\frac{2}{3}\))2[latex]\frac{x_{1}+y_{1}+2}{\sqrt{1+1}}[/latex]2
(x1 – 1)2 + (y1+ 1)2 = \(\frac{4}{9} \frac{\left(x_{1}+y_{1}+2\right)^{2}}{2}\)
9[(x1 – 1)2 + (y1 + 1)2] = 2 (x1 + y1 + 2]2
9[x12 – 2x1 + 1 + y12 + 2y1 + 1] = 2[x12 + y12 + 4 + 2x1y1 + 4x1 + 4y1]
9x12 + 9y12 – 18x1 + 18y1 + 18 = 2x12 + 2y12 + 4x1y1 + 8x1 + 8y1 + 8
7x12 – 4x1y1 + 7y12 – 26x1 + 10y1 + 10 = 0
focus of P (x1, y1) is 7x2 – 4xy + 7y2 – 26x + 10y + 10 = 0
This is the equation of the required Ellipse.
Inter 2nd Year Maths 2B Ellipse Important Questions 2

Inter 2nd Year Maths 2B Ellipse Important Questions

Question 7.
L Find tle length of major axis, minor axis, latus rectum, eccentricity, co-ordinates of centre, foci and the equations of directrices of the following ellipse. [TS Mar. 16; Mar. 14]
i) 9x2 + 16y2 = 144
ii) 4x2 + y2 – 8x + 2y + 1 = 0
iii) x2 + 2y2 – 4x + 12y + 14 = 0 [Mar. 11, May 07]
Solution:
Given equation is 9x2 + 16y2 = 144
\(\frac{x^{2}}{16}\) + \(\frac{y^{2}}{9}\) = 1
∴ a = 4, b = 3
Length of major axis = 2a = 2 . 4 = 8
Length of minor axis = 2b = 2 . 3 = 6
Length of latus rectum = \(\frac{2 b^{2}}{a}\) = \(\frac{2.9}{4}\) = \(\frac{9}{2}\)
Eccentricity = \(\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}=\sqrt{\frac{16-9}{16}}=\frac{\sqrt{7}}{4}\)
Centre is C (0, 0)
Foci are (± ae, 0) = ( ± \(\sqrt{7}\), 0)
Equations of the directrices are x = ± \(\frac{a}{e}\)
x = ± 4 . \(\frac{4}{\sqrt{7}}\) = ± \(\frac{16}{\sqrt{7}}\)
\(\sqrt{7}\)x = ± 16

(ii) Given equation is 4x2 + y2 – 8x + 2y + 1 = 0
4(x2 – 2x) + (y2 + 2y) = – 1
4(x – 1)2 + (y + 1)2 = 4 + 1 – 1 = 4
\(\frac{(x-1)^{2}}{1}\) + \(\frac{(y+1)^{2}}{4}\) = 1
Hence a < b ⇒ y – axis is major axis
a = 1, b = 2
Length of major axis = 2b = 4
Length of minor axis = 2a = 2
Length of lattis rectum = \(\frac{2 a^{2}}{b}\) = \(\frac{2}{2}\) = 1
Eccentricity = \(\sqrt{\frac{b^{2}-a^{2}}{b^{2}}}=\sqrt{\frac{4-1}{4}}=\frac{\sqrt{3}}{2}\)
Centre is C (-1, 1)
be = 2 . \(\frac{\sqrt{3}}{2}\) = \(\sqrt{3}\)
Foci are (-1, 1 ± \(\sqrt{3}\))
Equations of the directrices are y + 1 = ± \(\frac{b}{e}\)
= ± \(\frac{4}{\sqrt{3}}\)
\(\sqrt{3}\) y + \(\sqrt{3}\) = ± 4
\(\sqrt{3}\) y + \(\sqrt{3}\) ± 4 = 0

iii) Given equation is x2 + 2y2 – 4x + 12y + 14 = 0
x2 – 4x + 2 (y2 + 6y) = 14
⇒ (x2 – 4x + 4) + 2(y2 + 6y + 9) = 4 + 18 – 14
⇒ (x – 2)2 + 2(y + 3)2 = 8
⇒ \(\frac{(x-2)^{2}}{8}\) + \(\frac{(y+3)^{2}}{4}\) = 1
⇒ \(\frac{(x-2)^{2}}{(2 \sqrt{2})^{2}}\) + \(\frac{(y+3)^{2}}{2^{2}}\) = 1
a = 2\(\sqrt{2}\), b = 2, h = 2, k = -3
Length of major axis = 2a = 2(2 \(\sqrt{2}\)) = 4 \(\sqrt{2}\)
Length of minor axis = 2b = 2(2) = 4
Length of latus rectum
= \(\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}\) = \(\frac{2(4)}{2 \sqrt{2}}\) = 2\(\sqrt{2}\)
Eccentricity = \(\sqrt{1-\frac{b^{2}}{a^{2}}}=\sqrt{1-\frac{4}{8}}=\frac{1}{\sqrt{2}}\)
Centre = (h, k) = (2,-3)
Foci = (h ± ae, k) = (2 ± 2, -3)
= (4, -3), (0, -3)
Equations of the directrices are x – h = ± \(\frac{a}{e}\)
x – 2 = \(\frac{2 \sqrt{2}}{\left(\frac{1}{\sqrt{2}}\right)}\)
x – 2 = ± 4
i.e., x = 6, x = -2.

Inter 2nd Year Maths 2B Ellipse Important Questions

Question 8.
Find the equations of tangent and normal to the ellipse 2x2 + 3y2 = 11 at the point whose ordinate is 1. [T.S. Mar. 16]
Solution:
Equation of the ellipse is 2x2 + 3y2 = 11
Given y = 1
2x2 + 3 = 11
⇒ 2x2 = 8
x2 = 4
x = ±2
Points on the ellipse are P (2, 1) and Q(-2, 1)
Case i) P (2, 1)
Equation of the tangent is 2x . 2 + 3y . 1 = 11
4x + 3y = 11
The normal is perpendicular to the tangent Equation of the normal at P can be taken as
3x – 4y = k
The normal passes through p (2, 1)
6 – 4 = k ⇒ k = 2
Equation of the normal at P is 3x – 4y = 2.
Case ii) Q (-2, 1)
Equation of the tangent at Q is
2x(-2) + 3y . 1 = 11
-4x + 3y =1 1
4x – 3y + 11 = 0
Equation of the normal can be taken as
3x + 4y = k
The normal passes through Q (-2, 1)
-6 + 4 = k ⇒ k = -2
Equation of the normal at Q is 3x + 4y = -2
or 3x + 4y + 2 = 0.

Question 9.
Find the eccentricity, co-ordinates of foci. Length of latus rectum and equations of directrices of the following ellipses.
i) 9x2 + 16y2 – 36x + 32y – 92 = 0,
ii) 3x2 + y2 – 6x – 2y – 5 = 0 [T.S. Mar. 15]
Solution:
i) Given ellipse is
9x2 + 16y2 – 36x + 32y – 92 = 0
9(x2 – 4x + 4) + 16 (y2 + 2y + 1)
= 92 + 36 + 16
9(x – 2)2 + 16(y + 1)2 = 144
comparing with \(\frac{(x-2)^{2}}{16}\) + \(\frac{(y+1)^{2}}{9}\) = 1,
we get
a2 = 16, b2 = 9 ⇒ a = 4, b = 3.
Inter 2nd Year Maths 2B Ellipse Important Questions 3
Equations of the directrices are x = h ± \(\frac{\mathrm{a}}{\mathrm{e}}\)
x = 2 ± \(\frac{4 \times 4}{\sqrt{7}}\)
\(\sqrt{7x}\) = 2\(\sqrt{7}\) ± 16

Inter 2nd Year Maths 2B Ellipse Important Questions

ii) 3x2 + y2 – 6x – 2y – 5 = 0
Solution:
3(x2 – 2x) + (y2 – 2y) = 5
⇒ 3(x2 – 2x + 1) + (y2 – 2y + 1) = 9
⇒ 3(x – 1)2 + (y – 1)2 = 9
comparing with ⇒ \(\frac{(x-1)^{2}}{3}\) + \(\frac{(y-1)^{2}}{9}\) = 1,
we get
a < b ⇒ Y – axis is the major axis
a2 = 3, b2 = 9
a = \(\sqrt{3}\), b = 3, h = 1, k = 1
Length of major axis = 2b = 2(3) = 6
Length of minor axis = 2a = 2\(\sqrt{3}\)
Inter 2nd Year Maths 2B Ellipse Important Questions 4

Question 10.
Find the equation of the tangent and normal to the ellipse 9x2 + 16y2 = 144 at the end of the latus rectum in the first quadrant. [A.P. Mar. 15, Mar. 07]
Solution:
Given ellipse is 9x2 + 16y2 = 144
Inter 2nd Year Maths 2B Ellipse Important Questions 5
Equation of the normal at P is
Inter 2nd Year Maths 2B Ellipse Important Questions 6

Question 11.
Show that the points of intersection of the perpendicular tangents to an ellipse lie on a circle. [A.P. Mar. 16]
Solution:
Let the equation of the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1
(a > b). Any tangent to it in the slope-intercept form is y = mx ± \(\sqrt{a^{2} m^{2}+b^{2}}\) …………………. (1)
Let the perpendicular tangents intersect at P(x1, y1).
∴ P lies on (1) for some real m, i.e.,
y1 = mx1 ± \(\sqrt{a^{2} m^{2}+b^{2}}\)
∴ (y1 – mx1)2 = a2m2 + b2.
or .
(x12 – a2) m2 – 2x1y1m + (y12 – b2) = 0 being a
quadratic equation in ‘m’ has two roots say m1 and m2 then m1, m2 are the slopes of tangents from P to the ellipse
∴ m1m2 = \(\left(\frac{y_{1}^{2}-b^{2}}{x_{1}^{2}-a^{2}}\right)\)
∴ -1 = \(\left(\frac{y_{1}^{2}-b^{2}}{x_{1}^{2}-a^{2}}\right)\)
[∵ The tangents are perpendicular to each
other so that m1m2 = -1]
i.e., x12 + y12 = a2 + b2,
If, however, one of the perpendicular tangents is vertical, then such pair of perpendicular tangents intersect at one of the points (± a, ± b) and any of these points satisfies x2 + y2 = a2 + b2.
∴ The point of intersection of perpendicular tangents to the ellipse S = O lies on the circle x2 + y2 = a2 + b2.

Question 12.
Find the eccentricity, co-ordinates of foci, Length of latus reEtum and equations of directrices of the following ellipses.
i) 9x2 + 16y2 – 36x + 32y – 92 = 0
ii) 3x2 + y2 – 6x – 2y – 5 = 0 [T.S. Mar. 15]
Solution:
Given ellipse is :
9x2 + 16y2 – 36x + 32y – 92 = 0
9(x2 -4x + 4) + 16 (y2 + 2y + 1)
= 92 + 36 + 16
9 (x – 2)2 + 16 (y + 1)22 = 144
comparing with \(\frac{(x-2)^{2}}{16}\) + \(\frac{(y+1)^{2}}{9}\) = 1,
we get
a2 = 16, b2 = 9 ⇒ a = 4, b = 3.
e = \(\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}=\sqrt{\frac{16-9}{16}}=\frac{\sqrt{7}}{4}\)
Foci (h ± ae, k) = (2 ± 4 . \(\frac{\sqrt{7}}{4}\), -1)
= (2 ± \(\sqrt{7}\), -1)
Length of the latus rectum
= \(\frac{2 \cdot b^{2}}{a}=\frac{2.9}{4}=\frac{9}{2}\)
Equations of the directrices are x = h ± \(\frac{a}{e}\)
x = 2 ± \(\frac{4 \times 4}{\sqrt{7}}\)
\(\sqrt{7x}\) = 2\(\sqrt{7}\) ± 16

Inter 2nd Year Maths 2B Ellipse Important Questions

ii) 3x2 + y2 – 6x – 2y – 5 = 0
Solution:
3(x2 – 2x) + (y2 – 2y) = 5
⇒ 3(x2 – 2x + 1) + (y2 – 2y + 1) = 9
⇒ 3(x – 1)2 + (y – 1)2 = 9 .
comparing with ⇒ \(\frac{(x-1)^{2}}{3}\) + \(\frac{(y-1)^{2}}{9}\) = 1
we get
a < b ⇒ Y – axis is the major axis
a2 = 3, b2 = 9 .
a = \(\sqrt{3}\), b = 3, h = 1, k = 1
Length of major axis = 2b = 2(3) = 6
Length of minor axis = 2a = 2\(\sqrt{3}\)
Length of latus rectum = \(\frac{2 a^{2}}{b}=\frac{2.3}{3}\) = 2
Eccentricity = \(\sqrt{1-\frac{a^{2}}{b^{2}}}=\sqrt{1-\frac{3}{9}}=\sqrt{\frac{2}{3}}\)
Centre = (h, k) = (1, 1)
Focus = (h, k ± be) = (1, 1 ± 3 \(\left.\sqrt{\frac{2}{3}}\right)\))
= (1, 1 ± \(\sqrt{6}\))
Equation of directrices are y – k = ± \(\frac{b}{e}\)
y – 1 = ± \(\frac{3 \sqrt{3}}{\sqrt{2}}\)
y = 1 ± \(\frac{3 \sqrt{3}}{\sqrt{2}}\)

Question 13.
Find the equation of the ellipse referred to its major and minor axes as the co-ordinate axes x, y respectively with latus rectum of length 4 and the distance between foci 4\(\sqrt{2}\).
Solution:
Let the equation of the ellipse is \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1
(a > b)
Length of the latus rectum = \(\frac{2 b^{2}}{a}\) = 4
⇒ b2 = 2a.
Foci are S (ae, 0), S’ (-ae, 0)
Distance between the foci = 2ae = 4\(\sqrt{2}\)
ae = 2\(\sqrt{2}\)
b2 = a2 (1 – e2) = a2 – (ae)2
2a = a2 – 8 ⇒ a2 – 2a – 8 = 0
(a – 4) (a + 2) = 0
a = 4 or – 2
a > 0 ⇒ a = 4
b2 = 2a = 2 . 4 = 8
Equation of the ellipse is \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1
\(\frac{x^{2}}{16}\) + \(\frac{y^{2}}{8}\) = 1
x2 + 2y2 = 16.

Question 14.
If the length of the latus rectum is equal to half of its minor axis of an ellipse in the standard form, then find the eccentricity of the ellipse.
Solution:
Let \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a> b) be the ellipse in its standard form.
Length of latus rectum = \(\frac{1}{2}\) (minor axis)
2 \(\frac{b^{2}}{a}\) = \(\frac{1}{2}\) (2b)
2 \(\frac{b^{2}}{a}\) = b
a = 2b
a2 =4 b2, ⇒ a2 = 4a2 (1 – e2)
∴ 1 – e2 = \(\frac{1}{4}\) ⇒ e2 = \(\frac{3}{4}\)
⇒ e = \(\frac{\sqrt{3}}{2}\).

Inter 2nd Year Maths 2B Ellipse Important Questions

Question 15.
If θ1, θ2 are the eccentric angles of the extremeties of a focal chord (other that the vertices) of the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a > b) and e Its eccentricity. Then show that
i) e cos \(\frac{\left(\theta_{1}+\theta_{2}\right)}{2}\) = cos \(\frac{\left(\theta_{1}-\theta_{2}\right)}{2}\)
ii) \(\frac{e+1}{e-1}\) = cot \(\left(\frac{\theta_{1}}{2}\right)\) . cot \(\left(\frac{\theta_{2}}{2}\right)\)
Solution:
Equation of the ellipse is \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1,
(a > b)
Inter 2nd Year Maths 2B Ellipse Important Questions 7
sin θ1 . cos θ2 – e sin θ1 = cos θ1 sin θ2 – e . sin θ2
sin θ1 . cos θ2 – cos θ1 sin θ2 = e sin θ1 – e sin θ2
sin (θ1 – θ2) = e (sin θ1 – sin θ2)
2 sin \(\frac{\left(\theta_{1}-\theta_{2}\right)}{2}\) . cos \(\frac{\left(\theta_{1}-\theta_{2}\right)}{2}\)
= e [2 cos \(\frac{\theta_{1}+\theta_{2}}{2}\) . sin \(\frac{\theta_{1}-\theta_{2}}{2}\)]
Inter 2nd Year Maths 2B Ellipse Important Questions 8

Question 16.
C is the centre1 AA’ and BB’ are major and minor axis of the ellipse.
\(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1. If PN is the ordinate of a point P on the ellipse then show that
\(\frac{(\mathrm{PN})^{2}}{\left(\mathrm{~A}^{\prime} \mathrm{N}\right)(\mathrm{AN})}\) = \(\frac{(\mathrm{BC})^{2}}{(\mathrm{CA})^{2}}\)
Solution:
Inter 2nd Year Maths 2B Ellipse Important Questions 9
Equation of the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1
P(a cos θ, b sin θ) any point on the ellipse.
PN = b sin θ; AN = a – a cos θ,
AN = a + a cos θ; BC = b, CA = a
(A’N). (AN) = (a + a cos θ) (a – a cos θ)
= a2 – a2cos2θ
= a2 (1 – cos θ)
= a2 sin2θ
\(\frac{(\mathrm{PN})^{2}}{\left(\mathrm{~A}^{\prime} N\right)(\mathrm{AN})}=\frac{\mathrm{b}^{2} \sin ^{2} \theta}{\mathrm{a}^{2} \sin ^{2} \theta}=\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}\)
\(\frac{B C^{2}}{(C A)^{2}}=\frac{b^{2}}{a^{2}} \Rightarrow \frac{P^{2}}{\left(A^{1} N\right)(A N)}=\frac{(B C)^{2}}{(C A)^{2}}\)

Inter 2nd Year Maths 2B Ellipse Important Questions

Question 17.
S and Tare the foci of an ellipse and B is one end of the minor axis. If STB is an equilateral triangle, then find the eccentricity of the ellipse.
Solution:
Inter 2nd Year Maths 2B Ellipse Important Questions 10
Let \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a > b) be an ellipse whose foci are S and T, B is an end of the minor axis such that STB is equilateral, then SB = ST = TB. We have S(ae, 0).
T = (-ae, 0) and B(0, b)
Consider SB = ST ⇒ (SB)2 = (ST)2
⇒ (ae)2 + b2 = 4a2e2
∴ a2e2 + a2 (1 – e)2 = 4a2e2
[∵ b2 = a2 (1 – e2)]
e2 = \(\frac{1}{4}\)
∴ Eccentricity of the ellipse is \(\frac{1}{2}\).

Question 18.
Show that among the points on the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a > b), (-a, 0) is the farthest point and (a, 0) is the nearest point from the focus (ae, 0).
Solution:
Let P = (x, y) be any point on the ellipse so that – a ≤ x ≤ a and S = (ae, 0) be the focus.
Since (x, y) is on the ellipse,
y2 = \(\frac{b^{2}}{a^{2}}\) (a2 – x2)
= (1 – e2)(a2 – x2) ………….. (1) [∵ b2 = a2(1 – e2)]
Then we know that
sp2 = (x – ae)2 + y2
= (x – ae)2 + (1 – e2)(a2 – x2)
= -2xae + a2 + e2x2
= [a – ex]2
∴ SP = [a – ex]
we have – a ≤ x ≤ a
⇒ -ae ≤ xe ≤ ae
⇒ -ae – a ≤ xe – a ≤ ae – a …………………. (2)
∴ ex – a < 0
∴ SP = a – ex …………………… (3)
From (2) and (3)
ae + a ≥ SP ≥ a – ae
⇒ a – ae ≤ SP ≤ ae + a
∴ Max SP = ae + a when P = (-a, 0)
and Min SP = a – ae when P = (a, 0)
Hence the nearest point is (a, 0) and the farthest one is (-a, 0).

Inter 2nd Year Maths 2B Ellipse Important Questions

Question 19.
The orbit of the Earth is an ellipse with eccentricity \(\frac{1}{60}\) with the Sun at one of its foci, the major axis being approximately 186 × 106 miles in length. Find the shortest and longest distance of the Earth from the Sun.
Solution:
We take the orbit of the Earth to be
\(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a > b).
Since the major axis is 186 × 106 miles,
2a = 186 × 106 miles
∴ a = 93 × 106 miles
If e be the eccentricity of the orbit, e = \(\frac{1}{60}\)
We know, the longest and shortest distances of the Earth from the Sun are respectively
a + ae and a – ae (problem 7)
Here, the longest distance
= 93 × 106 × (1 + \(\frac{1}{60}\))
= 9455 × 104 miles.
and the shortest distance
= 93 × 106 × (1 – \(\frac{1}{60}\)) miles
= 9145 × 104 miles.

Question 20.
Find the equation of the tangent and normal to the ellipse 9x2 + 16y2 = 144 at the end of the latus rectum in the first quadrant. A.P. [Mar. 15, Mar. 07]
Solution:
Given ellipse is 9x2 + 16y2 = 144
Inter 2nd Year Maths 2B Ellipse Important Questions 11
Equation of the normal at P is
Inter 2nd Year Maths 2B Ellipse Important Questions 12

Question 21.
If a tangent to the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a > b) meets its major.axis and minor axis atM and N respectively, then prove that \(\frac{a^{2}}{(C M)^{2}}\) + \(\frac{b^{2}}{(C N)^{2}}\) = 1. Where C is the centre of the ellipse.
Solution:
Let P(θ) (a cos θ, b sin θ) is any point on the ellipse then Equation of the tangent at P (θ) is
Inter 2nd Year Maths 2B Ellipse Important Questions 13
a2 . \(\frac{\cos ^{2} \theta}{a^{2}}\) + b2 . \(\frac{\sin ^{2} \theta}{b^{2}}\)
= cos2 θ + sin2 θ = 1.

Inter 2nd Year Maths 2B Ellipse Important Questions

Question 22.
Find the condition for the line
i) lx + my + n = 0 to be a tangent to the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1
ii) lx+ my n = 0 to be a normal to the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1.
Solution:
i) Equation of the ellipse is \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1
Equation .of the tangent at P(θ) is
\(\frac{x}{a}\) cos θ + \(\frac{y}{b}\) sin θ = 1 ……………………… (1)
Equation of the given line is
lx + my = -n …………………. (2)
(1), (2) represent the same line. Comparing the co-efficients
Inter 2nd Year Maths 2B Ellipse Important Questions 14
Comparing (1) and (2)
Inter 2nd Year Maths 2B Ellipse Important Questions 15
Inter 2nd Year Maths 2B Ellipse Important Questions 16

Question 23.
If the normal at one end of a latus rectum of the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 passes through one end of the minor axis, then show that e4 + e2 = 1 [e is the eccentricity of the ellipse]
Solution:
Let L be the one end of the latus rectum of
\(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1. Then the coordinates of
L = (ae, \(\frac{b^{2}}{a}\))
Hence equation of the normal at L is
Inter 2nd Year Maths 2B Ellipse Important Questions 17
is a line passes through the one end
B’ = (0, -b)
or minor axis of \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 as shown in figure.
\(\frac{\mathrm{a}(0)}{\mathrm{e}}\) – a(-b) = a2 – b2
ab = a22 – a2 (1 – e2)
ab = a2e2 ⇒ e2 = \(\frac{b}{a}\) ⇒ e4 = \(\frac{b^{2}}{a^{2}}\)
= \(\frac{a^{2}\left(1-e^{2}\right)}{a^{2}}\) = 1 – e2 ⇒ e4 + e2 = 1.

Question 24.
If PN is the ordinate of a point P on the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 and the tangent at P meets the X – axis at T then show that (CN) (CT) = a2 where C is the centre of the ellipse.
Solution:
Let P(θ) = (acosθ, bsinθ) be a point on the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1. Then the equation of the tangent at P(θ) is \(\frac{x \cos \theta}{a}\) + \(\frac{y \sin \theta}{b}\) = 1 or \(\frac{x}{\frac{a}{\cos \theta}}+\frac{y}{\frac{b}{\sin \theta}}=1\) meets the X – axis at T
Inter 2nd Year Maths 2B Ellipse Important Questions 18
x – intercept (CT) = \(\frac{a}{\cos \theta}\) and the ordinate of P is PN = bsinθ
then its absicca CN = a cos θ. (see Fig.)
∴ (CN) . (CT) =, (a cos θ) (\(\frac{a}{\cos \theta}\)) = a2.

Inter 2nd Year Maths 2B Ellipse Important Questions

Question 25.
Show that the points of intersection of the perpendicular tangets to an ellipse lie on a circle. [A.P. Mar. 16]
Solution:
Let the equation of the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a > b). Any tangent to it in the slope-intercept
form is y = mx ± \(\sqrt{a^{2} m^{2}+b^{2}}\) …………………. (1)
Let the perpendicular tangents intersed at
P(x1, y1).
∴ p lies on (1) for some real m, i.e.,
y1 = mx1 ± \(\sqrt{a^{2} m^{2}+b^{2}}\)
∴ (y1 – mx1)2 = a2m2 + b2.
or
(x12 – a2) m2 – 2x1y1m + (y12 – b2) = 0 being a quadratic equation in ‘m’, has two roots say m1 and m2 then m1, m2 are the slopes of tangents from P to the ellipse
∴ m1m2 = \(\left(\frac{y_{1}^{2}-b^{2}}{x_{1}^{2}-a^{2}}\right)\)
∴ – 1 = \(\left(\frac{y_{1}^{2}-b^{2}}{x_{1}^{2}-a^{2}}\right)\)
[∵ The tangents are perpendicular to each other so that m1 m2 = -1]
i.e., \(x_{1}^{2}\) + \(y_{1}^{2}\) = a2 + b2.
If, however, one of the perpendicular tangents is vertical, then such pair of perpendicular tangents intersect at one of the points (a, ± b) and any of these points satisfies x2 + y2 = a2 + b2.
∴ The point of intersection of perpendicular tangents to the ellipse S = 0 lies on the circle x2 + y2 = a2+ b2.