Kinetic Theory of Gases

Practice Questions:

Question 11: easy

Given below are two statements:

Assertion (A): The average translational kinetic energy per molecule of gas for various gases at the same temperature is the same.

Reason (R): At a given temperature, all molecules of a gas move with nearly the same speed.

1. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

2. Both Assertion and Reason are true but Reason is not correct explanation of Assertion.

3. Assertion is true but Reason is false.

4. Assertion and Reason are false.

View Answer

Average translational kinetic energy per molecule is \(\frac{3}{2}kT\), which depends only on temperature and is same for all gases. However, at a given temperature, gas molecules have a distribution of speeds, not the same speed. Assertion is true but Reason is false.

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Question 12: easy

Consider the following statements:

A. Work and heat are path functions in thermodynamics.

B. The internal energy of a gaseous system is state function.

C. For gaseous system, \(C_P\) is greater than \(C_V\).

D. Work done by gas at constant volume is zero.

Based on above information pick the correct option.

1. Only statement (A) is correct

2. Only statements (A), (B) and (C) are correct

3. Only statements (B), (C) and (A) are correct

4. All statements (A), (B), (C) and (D) are correct

View Answer

Work and heat depend on the path, whereas internal energy depends only on the initial and final states. For any gas, \(C_P > C_V\) due to expansion work. Since volume is constant, \(dV = 0\), so work done \(W = 0\). Hence, all statements are correct.

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Question 13: easy

Consider the following statements.

Statement A: Velocity of sound in gaseous medium depends on molar mass of gas.

Statement B: Mechanical wave require a material medium for their propagation.

Statement C: Speed of sound is less in humid air.

Which of the statement(s) is/are correct?

1. Only statements A and B

2. Only statements B and C

3. Only statements A and C

4. All statements A, B and C

View Answer

Velocity of sound is given by \(v = \sqrt{\frac{\gamma RT}{M}}\), so it depends on molar mass \(M\). Mechanical waves require a material medium. Humid air has lower density than dry air, which increases the speed of sound, making Statement C incorrect.

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Question 14: easy

Match Column – I and Column – II and choose the correct match from the given choices.

Column-I
(A) Root mean square speed of gas molecules
(B) Pressure exerted by ideal gas
(C) Average kinetic energy of a molecule
(D) Total internal energy of 1 mole of a diatomic gas

Column-II
(P) \(\frac{1}{3} n m \bar{v}^2\)
(Q) \(\sqrt{\frac{3RT}{M}}\)
(R) \(\frac{5}{2} RT\)
(S) \(\frac{3}{2} k_B T\)

1. (A) - (R), (B) - (Q), (C) - (P), (D) - (S)

2. (A) - (R), (B) - (P), (C) - (S), (D) - (Q)

3. (A) - (Q), (B) - (R), (C) - (S), (D) - (P)

4. (A) - (Q), (B) - (P), (C) - (S), (D) - (R)

View Answer

By kinetic theory: \(v_{\text{rms}} = \sqrt{\frac{3RT}{M}}\) -> (A)-(Q); Pressure \(P = \frac{1}{3} nm\bar{v}^2\) -> (B)-(P); Average KE \(= \frac{3}{2}k_BT\) -> (C)-(S); and Internal energy for diatomic gas \(= \frac{5}{2}RT\) -> (D)-(R).

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Question 15: easy

The ratio of total K.E of a molecule of Argon and Oxygen gas at 27°C is equal to:

1. 3 : 5

2. 3 : 2

3. 2 : 3

4. 5 : 7

View Answer

Total K.E. per molecule is \(\frac{f}{2}k_B T\). For monoatomic Argon, \(f=3\), and for diatomic Oxygen, \(f=5\). At equal temperature, the ratio of K.E. is \(3:5\).

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Question 16: easy

The internal energy of an ideal monoatomic gas increases by the same amount as work done on the gas, then:

1. The process should be adiabatic

2. The process should be isothermal

3. The process should be isochoric

4. The process should be isobaric

View Answer

By the first law, \(dQ = dU + dW\). If work is done on the gas, \(dW_{\text{by}} = -dW_{\text{on}}\). Given \(dU = dW_{\text{on}}\), so \(dQ = 0\). This represents an adiabatic process.

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Question 17: easy

A vessel contains gas A at pressure \(P\), volume \(V\) and temperature \(T\). Another vessel contains gas B at pressure \(2P\), volume \(\frac{V}{2}\) and temperature \(2T\). Ratio of number of molecules of B to A will be

1. \(\frac{1}{2}\)

2. 2

3. 4

4. \(\frac{1}{4}\)

View Answer

From the ideal gas law, \(PV = N k_B T\) which gives \(N = \frac{PV}{k_B T}\). Taking the ratio: \(\frac{N_B}{N_A} = \frac{P_B V_B / T_B}{P_A V_A / T_A} = \frac{2P times (V/2) / 2T}{PV / T} = \frac{1}{2}\).

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Question 18: easy

At 10°C the value of the density of a fixed mass of an ideal gas divided by its pressure is X. At 110°C this ratio is

1. \[\frac{10}{110} X\]

2. \[\frac{283}{383} X\]

3. \[X\]

4. \[\frac{383}{283} X\]

View Answer

From ideal gas law, \(P = \frac{\rho RT}{M} \Rightarrow \frac{\rho}{P} = \frac{M}{RT}\). Hence, \(\frac{\rho}{P} \propto \frac{1}{T}\). Thus, the ratio becomes \[X \times \frac{273 + 10}{273 + 110} = X \frac{283}{383}\].

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Question 19: easy

A polyatomic molecule has 3 translational, 3 rotational degrees of freedom. The molar specific heat ratio \((\gamma = \frac{C_p}{C_v})\) for this gas is

1. \(\frac{7}{5}\)

2. \(\frac{7}{6}\)

3. \(\frac{4}{3}\)

4. \(\frac{3}{2}\)

View Answer

Total degrees of freedom \(f = 3 + 3 = 6\). Thus, \(C_v = \frac{f}{2} R = 3R\), and \(C_p = C_v + R = 4R\). Therefore, \(\gamma = \frac{C_p}{C_v} = \frac{4}{3}\).

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Question 20: easy

The ratio of \(v_{\text{rms}} : v_{\text{mp}} : v_{\text{avg}}\) is (symbols have their usual meaning)

1. \(\sqrt{3} : \sqrt{2} : \sqrt{\frac{\pi}{8}}\)

2. \(\sqrt{3} : \sqrt{2} : \sqrt{\frac{8}{3}}\)

3. \(\sqrt{3} : \sqrt{\frac{8}{\pi}} : \sqrt{2}\)

4. \(\sqrt{3} : \sqrt{2} : \sqrt{\frac{8}{\pi}}\)

View Answer

\[v_{\text{rms}} = \sqrt{\frac{3RT}{M}}\], \[v_{\text{mp}} = \sqrt{\frac{2RT}{M}}\], and \[v_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}}\]. Therefore, \[v_{\text{rms}} : v_{\text{mp}} : v_{\text{avg}} = \sqrt{3} : \sqrt{2} : \sqrt{\frac{8}{\pi}}\].

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