Kinetic Theory of Gases and ldeal Gas | Physics JEE previous year questions with Solutions

Q 1 :

The specific heat at constant pressure of a real gas obeying $P V^{2} = R T$ equation is: [2024]

$\frac{R}{3} + C_{v}$
$R$
$C_{v} + R$
$C_{v} + \frac{R}{2 V}$

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(4)

$d Q = d U + d W$

$n C d T = n C_{v} d T + d W,$ We need to find $d W$

we have $P V^{2} = R T, P =$ constant

differentiating, $P 2 V d V = R d T \Rightarrow P d V = \frac{R . d T}{2 V}$

Also, $d W = P d V = \frac{R . d T}{2 V}$

For one mole of gas, $C . d T = C_{v} d T + \frac{R . d T}{2 V} \Rightarrow C = C_{v} + \frac{R}{2 V}$

Q 2 :

A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature (27°C). The ratio of specific heat of gases at constant volume respectively is [2024]

$\frac{3}{2}$
$\frac{7}{5}$
$\frac{3}{5}$
$\frac{5}{3}$

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(3)

$\frac{{(C_{v})}_{m o n o}}{{(C_{v})}_{d i a}} = \frac{\frac{3}{2} R}{\frac{5}{2} R} = \frac{3}{5}$

Q 3 :

If three moles of monoatomic gas $(γ = \frac{5}{3})$ is mixed with two moles of a diatomic gas $(γ = \frac{7}{5})$ , the value of adiabatic exponent $γ$ for the mixture is [2024]

1.75
1.40
1.52
1.35

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(3)

Monoatomic gas $\to 3$ mole

Diatomic gas $\to 2$ mole

$γ_{m i x} = 1 + \frac{2}{f_{m i n}}$ ....(1)

$f_{m i x} = \frac{n_{1} f_{1} + n_{2} f_{2}}{n_{1} + n_{2}} = \frac{3 (3) + 2 (5)}{5} = \frac{19}{5}$

$γ_{m i x} = 1 + \frac{2}{19 / 5} = 1 + \frac{10}{19} = \frac{29}{19}$

$γ_{m i x} = 1.53$

Q 4 :

A gas mixture consists of 8 moles of argon and 6 moles of oxygen at temperature T. Neglecting all vibrational modes, the total internal energy of the system is [2024]

29RT
20RT
27RT
21RT

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(3)

$U = n C_{V} T$

$\Rightarrow U = n_{1} C_{V_{1}}$ $T + n_{2} C_{V_{2}} T$

$\Rightarrow 8 \times \frac{3 R}{2} \times T + 6 \times \frac{5 R}{2} \times T = 27 R T$

Q 5 :

Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is [2024]

$\frac{9}{4} R$
$\frac{7}{4} R$
$\frac{3}{2} R$
$\frac{5}{2} R$

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(1)

$C_{V} = \frac{n_{1} C_{v_{1}} + n_{2} C_{v_{2}}}{n_{1} + n_{2}}$

$= \frac{2 \times \frac{3}{2} R + 6 \times \frac{5}{2} R}{2 + 6}$

$= \frac{9}{4} R$

Q 6 :

For a diatomic gas, if $γ_{1} = (\frac{C_{p}}{C_{v}})$ for rigid molecules and $γ_{2} = (\frac{C_{p}}{C_{v}})$ for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct?

( $C_{p}$ and $C_{v}$ are specific heats of the gas at constant pressure and volume) [2025]

$γ_{2} > γ_{1}$
$γ_{2} = γ_{1}$
$2 γ_{2} > γ_{1}$
$γ_{2} < γ_{1}$

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(4)

$γ = \frac{2}{f} + 1$

without vibration : f = 5 : $γ_{1}$ = 1.4

without vibration : f = 7 : $γ_{2}$ = 1.14

$∴ γ_{2} < γ_{1}$

Q 7 :

In an adiabatic process, which of the following statements is true? [2025]

The molar heat capacity is infinite
Work done by the gas equals the increase in internal energy
The molar heat capacity is zero
The internal energy of the gas decreases as the temperature increases

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(3)

For adiabatic process, dQ = 0

$∴$ Molar heat capacity = 0

$∵$ dQ = 0 $\Rightarrow$ dU = –dW

Also, $d U = \frac{f}{2} n R d T$

$∴$ Only option (3) is correct.

Q 8 :

Match the List-I with List-II

List-I List-II

(A) Triatomic rigid gas (I) $\frac{C_{p}}{C_{v}} = \frac{5}{3}$

(B) Diatomic non-rigid gas (II) $\frac{C_{p}}{C_{v}} = \frac{7}{5}$

(C) Monoatomic gas (III) $\frac{C_{p}}{C_{v}} = \frac{4}{3}$

(D) Diatomic rigid gas (IV) $\frac{C_{p}}{C_{v}} = \frac{9}{7}$

Choose the correct answer from the options given below: [2025]

A-III, B-IV, C-I, D-II
A-III, B-II, C-IV, D-I
A-II, B-IV, C-I, D-III
A-IV, B-II, C-III, D-I

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(1)

$γ = 1 + \frac{2}{f}$ , f is degree of freedom

Triatomic rigid gas $f = 6 \Rightarrow γ = 1 + \frac{2}{6} = \frac{4}{3}$ (Triatomic)

Diatomic non-rigid gas $f = 7 \Rightarrow γ = 1 + \frac{2}{7} = \frac{9}{7}$ (Diatomic, non-rigid)

Diatomic rigid gas $f = 5 \Rightarrow γ = 1 + \frac{2}{5} = \frac{7}{5}$ (Diatomic, rigid)

Monoatomic rigid gas $f = 3 \Rightarrow γ = 1 + \frac{2}{3} = \frac{5}{3}$ (Monoatomic, rigid)

Q 9 :

The temperature of 1 mole of an ideal monoatomic gas is increased by 50°C at constant pressure. The total heat added and change in internal energy are $E_{1}$ and $E_{2}$ , respectively. If $\frac{E_{1}}{E_{2}} = \frac{x}{9}$ then the value of x is ________. [2025]

(15)

$△ T$ = 50°C = 50 K

n = 1 mole of ideal gas

$△ Q = n C_{P} △ T = E_{1}$

$△ U = n C_{V} △ T = E_{2}$

$\Rightarrow \frac{E_{1}}{E_{2}} = \frac{x}{9} = \frac{C_{P}}{C_{V}}$

For monoatomic gas $γ = \frac{C_{P}}{C_{V}} = \frac{5}{3}$

$\frac{x}{9} = \frac{5}{3} \Rightarrow x = 15$

Q 10 :

$γ_{A}$ is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. $γ_{B}$ is the specific heat ratio of polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If $\frac{γ_{A}}{γ_{B}} = (1 + \frac{1}{n})$ , then the value of n is ________. [2025]

(3)

$γ = 1 + \frac{2}{f}$

$f_{A} = 3 \Rightarrow γ_{A} = \frac{5}{3}$

$f_{B} = 3 + 3 + 2 = 8, γ_{B} = 1 + \frac{2}{8} = \frac{5}{4}$

$\frac{γ_{A}}{γ_{B}} = \frac{4}{3} = 1 + \frac{1}{3} \Rightarrow n = 3$

Q 11 :

The internal energy of air in 4 m $\times$ 4 m $\times$ 3 m sized room at 1 atmospheric pressure will be _______ $\times 10^{6} J$ . (Consider air as diatomic molecule) [2025]

(12)

$U = n C_{v} T = n \times \frac{5 R}{2} T$

$\Rightarrow U = \frac{5}{2} (n R T) = \frac{5}{2} P V$

$\Rightarrow U = \frac{5}{2} \times 10^{5} \times 4 \times 4 \times 3 = 12 \times 10^{6} J$

Q 12 :

Let $γ_{1}$ be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and $γ_{2}$ be the similar ratio of a diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio $\frac{γ_{1}}{γ_{2}}$ is [2023]

$\frac{21}{25}$
$\frac{25}{21}$
$\frac{35}{27}$
$\frac{27}{35}$

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(2)

$For monoatomic gas, γ_{1} = \frac{5}{3}$

$For diatomic gas at low temperatures, γ_{2} = \frac{7}{5}$

$∴$ $\frac{γ_{1}}{γ_{2}} = \frac{\frac{5}{3}}{\frac{7}{5}} = \frac{25}{21}$

Q 13 :

The correct relation between $γ = \frac{C_{p}}{C_{v}}$ and temperature $T$ is [2023]

$γ \propto \frac{1}{\sqrt{T}}$
$γ \propto T^{0}$
$γ \propto \frac{1}{T}$
$γ \propto T$

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(2)

$γ = \frac{C_{P}}{C_{V}}$

$At low temperature (T), γ is independent of T .$

Q 14 :

A hypothetical gas expands adiabatically such that its volume changes from 8 litres to 27 litres. If the ratio of final pressure of the gas to the initial pressure of the gas is $\frac{16}{81}$ , then the ratio $\frac{C_{P}}{C_{V}}$ will be [2023]

$\frac{4}{3}$
$\frac{3}{1}$
$\frac{1}{2}$
$\frac{3}{2}$

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(1)

$Let γ be the ratio of \frac{C_{p}}{C_{v}}$

$Then for adiabatic process$

$P V^{γ} = constant$

$\frac{P_{i}}{P_{f}} = {(\frac{V_{f}}{V_{i}})}^{γ}$

$\frac{81}{16} = {(\frac{27}{8})}^{γ}$

$γ = \frac{4}{3}$

Q 15 :

A gas mixture consists of 2 moles of oxygen and 4 moles of neon at temperature T. Neglecting all vibrational modes, the total internal energy of the system will be [2023]

8 RT
16 RT
4 RT
11 RT

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(4)

${(C_{v})}_{\max} = \frac{n_{1} C v_{1} + n_{2} C v_{2}}{n_{1} + n_{2}}$

${(C_{v})}_{\max} = \frac{2 \times \frac{5}{2} R + 4 \times \frac{3}{2} R}{2 + 4} = \frac{11 R}{6}$

$Δ U = n {(C_{v})}_{mix} Δ T$ $= 6 \times \frac{11 R}{6} \times Δ T = 11 R T$

Q 16 :

10 mole of oxygen is heated at constant volume from 30°C to 40°C. The change in the internal energy of the gas is __________ cal. (The molecular specific heat of oxygen at constant pressure, $C_{P} = 7 cal./mol ° C and R=2 cal./mol°C$ [2026]

(500)

$Δ U = n C_{V} Δ T$

$= n (C_{P} - R) Δ T$

$= 10 (7 - 2) (40 - 30)$

$Δ U = 500$

Q 17 :

Density of water at $4 ° C$ and $20 ° C$ are $1000 {kg/m}^{3}$ and $998 {kg/m}^{3}$ respectively. The increase in internal energy of $4 kg$ of water when it is heated from $4 ° C$ to $20 ° C$ is ________ J.

(Specific heat capacity of water = 4.2 J/kg, and 1 atmospheric pressure = $10^{5} Pa$ ) [2026]

315826.2
234699.2
258700.8
268799.2

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(4)

$Q = m s Δ T = 4 \times 4200 \times 16 J = 268800 J$

$W = P Δ V$

$Δ V = (\frac{m}{ρ_{f}} - \frac{m}{ρ_{i}}) = 4 [\frac{1}{998} - \frac{1}{1000}]$

$P = 10^{5} Pa$

$∴$ $W = 10^{5} \times 4 [\frac{1}{998} - \frac{1}{1000}] = \frac{8 \times 10^{5}}{10^{3} \times 998} \approx 0.8 J$

$Δ U = Q - W = 268799.2 J$

Topic Question Set

Q 1 :

The specific heat at constant pressure of a real gas obeying $P V^{2} = R T$ equation is: [2024]

Q 2 :

A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature (27°C). The ratio of specific heat of gases at constant volume respectively is [2024]

Q 3 :

If three moles of monoatomic gas $(γ = \frac{5}{3})$ is mixed with two moles of a diatomic gas $(γ = \frac{7}{5})$ , the value of adiabatic exponent $γ$ for the mixture is [2024]

Q 4 :

A gas mixture consists of 8 moles of argon and 6 moles of oxygen at temperature T. Neglecting all vibrational modes, the total internal energy of the system is [2024]

Q 5 :

Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is [2024]

Q 7 :

In an adiabatic process, which of the following statements is true? [2025]

Q 9 :

The temperature of 1 mole of an ideal monoatomic gas is increased by 50°C at constant pressure. The total heat added and change in internal energy are $E_{1}$ and $E_{2}$ , respectively. If $\frac{E_{1}}{E_{2}} = \frac{x}{9}$ then the value of x is ________. [2025]

Q 11 :

The internal energy of air in 4 m $\times$ 4 m $\times$ 3 m sized room at 1 atmospheric pressure will be _______ $\times 10^{6} J$ . (Consider air as diatomic molecule) [2025]

Q 12 :

Let $γ_{1}$ be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and $γ_{2}$ be the similar ratio of a diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio $\frac{γ_{1}}{γ_{2}}$ is [2023]

Q 13 :

The correct relation between $γ = \frac{C_{p}}{C_{v}}$ and temperature $T$ is [2023]

Q 14 :

A hypothetical gas expands adiabatically such that its volume changes from 8 litres to 27 litres. If the ratio of final pressure of the gas to the initial pressure of the gas is $\frac{16}{81}$ , then the ratio $\frac{C_{P}}{C_{V}}$ will be [2023]

Q 15 :

A gas mixture consists of 2 moles of oxygen and 4 moles of neon at temperature T. Neglecting all vibrational modes, the total internal energy of the system will be [2023]

Q 16 :

10 mole of oxygen is heated at constant volume from 30°C to 40°C. The change in the internal energy of the gas is __________ cal. (The molecular specific heat of oxygen at constant pressure, $C_{P} = 7 cal./mol ° C and R=2 cal./mol°C$ [2026]

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	List-I		List-II
(A)	Triatomic rigid gas	(I)	$\frac{C_{p}}{C_{v}} = \frac{5}{3}$
(B)	Diatomic non-rigid gas	(II)	$\frac{C_{p}}{C_{v}} = \frac{7}{5}$
(C)	Monoatomic gas	(III)	$\frac{C_{p}}{C_{v}} = \frac{4}{3}$
(D)	Diatomic rigid gas	(IV)	$\frac{C_{p}}{C_{v}} = \frac{9}{7}$

Topic Question Set

Q 1 : The specific heat at constant pressure of a real gas obeying PV2=RT equation is: [2024]

Q 2 : A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature (27°C). The ratio of specific heat of gases at constant volume respectively is [2024]

Q 3 : If three moles of monoatomic gas (γ=53) is mixed with two moles of a diatomic gas (γ=75), the value of adiabatic exponent γ for the mixture is [2024]

Q 4 : A gas mixture consists of 8 moles of argon and 6 moles of oxygen at temperature T. Neglecting all vibrational modes, the total internal energy of the system is [2024]

Q 5 : Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is [2024]

Q 6 : For a diatomic gas, if γ1=(CpCv) for rigid molecules and γ2=(CpCv) for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct? (Cp and Cv are specific heats of the gas at constant pressure and volume) [2025]

Q 7 : In an adiabatic process, which of the following statements is true? [2025]

Q 8 : Match the List-I with List-II List-I List-II (A) Triatomic rigid gas (I) CpCv=53 (B) Diatomic non-rigid gas (II) CpCv=75 (C) Monoatomic gas (III) CpCv=43 (D) Diatomic rigid gas (IV) CpCv=97 Choose the correct answer from the options given below: [2025]

Q 9 : The temperature of 1 mole of an ideal monoatomic gas is increased by 50°C at constant pressure. The total heat added and change in internal energy are E1 and E2, respectively. If E1E2=x9 then the value of x is ________. [2025]

Q 10 : γA is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. γB is the specific heat ratio of polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If γAγB=(1+1n), then the value of n is ________. [2025]

Q 11 : The internal energy of air in 4 m × 4 m × 3 m sized room at 1 atmospheric pressure will be _______ ×106 J. (Consider air as diatomic molecule) [2025]

Q 12 : Let γ1 be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and γ2 be the similar ratio of a diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio γ1γ2 is [2023]

Q 13 : The correct relation between γ=CpCv and temperature T is [2023]

Q 14 : A hypothetical gas expands adiabatically such that its volume changes from 8 litres to 27 litres. If the ratio of final pressure of the gas to the initial pressure of the gas is 1681, then the ratio CPCV will be [2023]

Q 15 : A gas mixture consists of 2 moles of oxygen and 4 moles of neon at temperature T. Neglecting all vibrational modes, the total internal energy of the system will be [2023]

Q 16 : 10 mole of oxygen is heated at constant volume from 30°C to 40°C. The change in the internal energy of the gas is __________ cal. (The molecular specific heat of oxygen at constant pressure, CP=7 cal./mol°C and R=2 cal./mol°C [2026]

Q 17 : Density of water at 4°C and 20°C are 1000 kg/m3 and 998 kg/m3 respectively. The increase in internal energy of 4 kg of water when it is heated from 4°C to 20°C is ________ J. (Specific heat capacity of water = 4.2 J/kg, and 1 atmospheric pressure = 105 Pa) [2026]

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Q 1 :

The specific heat at constant pressure of a real gas obeying $P V^{2} = R T$ equation is: [2024]

Q 2 :

A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature (27°C). The ratio of specific heat of gases at constant volume respectively is [2024]

Q 3 :

If three moles of monoatomic gas $(γ = \frac{5}{3})$ is mixed with two moles of a diatomic gas $(γ = \frac{7}{5})$ , the value of adiabatic exponent $γ$ for the mixture is [2024]

Q 4 :

A gas mixture consists of 8 moles of argon and 6 moles of oxygen at temperature T. Neglecting all vibrational modes, the total internal energy of the system is [2024]

Q 5 :

Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is [2024]

Q 7 :

In an adiabatic process, which of the following statements is true? [2025]

Q 9 :

The temperature of 1 mole of an ideal monoatomic gas is increased by 50°C at constant pressure. The total heat added and change in internal energy are $E_{1}$ and $E_{2}$ , respectively. If $\frac{E_{1}}{E_{2}} = \frac{x}{9}$ then the value of x is ________. [2025]

Q 11 :

The internal energy of air in 4 m $\times$ 4 m $\times$ 3 m sized room at 1 atmospheric pressure will be _______ $\times 10^{6} J$ . (Consider air as diatomic molecule) [2025]

Q 12 :

Let $γ_{1}$ be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and $γ_{2}$ be the similar ratio of a diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio $\frac{γ_{1}}{γ_{2}}$ is [2023]

Q 13 :

The correct relation between $γ = \frac{C_{p}}{C_{v}}$ and temperature $T$ is [2023]

Q 14 :

A hypothetical gas expands adiabatically such that its volume changes from 8 litres to 27 litres. If the ratio of final pressure of the gas to the initial pressure of the gas is $\frac{16}{81}$ , then the ratio $\frac{C_{P}}{C_{V}}$ will be [2023]

Q 15 :

A gas mixture consists of 2 moles of oxygen and 4 moles of neon at temperature T. Neglecting all vibrational modes, the total internal energy of the system will be [2023]

Q 16 :

10 mole of oxygen is heated at constant volume from 30°C to 40°C. The change in the internal energy of the gas is __________ cal. (The molecular specific heat of oxygen at constant pressure, $C_{P} = 7 cal./mol ° C and R=2 cal./mol°C$ [2026]