Behaviour of Perfect Gas and Kinetic Theory Chapter-Wise Test 13

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A solid has a molar specific heat capacity of \(25 \, \text{J mol}^{-1} \text{K}^{-1}\) at room temperature. How many degrees of freedom does each atom have?

For a solid, \(C = f \times \frac{R}{2} \times N_A = f \times \frac{R}{2} \times 1 = f \times \frac{R}{2}\).

Given \(C = 25 \, \text{J mol}^{-1} \text{K}^{-1}\), \(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\).

\(25 = f \times \frac{8.31}{2}\), \(f = \frac{25 \times 2}{8.31} \approx 6\).

3
4
5
6
4

The rms speed of argon molecules is 430 m/s at 300 K. What is the rms speed of nitrogen molecules at the same temperature? (Atomic mass: Ar = 39.9 u, \(N_2 = 28 \, \text{u}\))

\(v_{\text{rms}} \propto \frac{1}{\sqrt{m}}\), \(\frac{v_{\text{N}_2}}{v_{\text{Ar}}} = \sqrt{\frac{m_{\text{Ar}}}{m_{\text{N}_2}}}\).

\(\frac{v_{\text{N}_2}}{430} = \sqrt{\frac{39.9}{28}} \approx \sqrt{1.425} \approx 1.193\).

\(v_{\text{N}_2} = 430 \times 1.193 \approx 513 \, \text{m/s}\).

430 m/s
513 m/s
600 m/s
700 m/s
2

A gas mixture contains 12 g of helium and 28 g of nitrogen. What is the ratio of their partial pressures?

\(P = \frac{\mu RT}{V}\), \(\frac{P_{\text{He}}}{P_{\text{N}_2}} = \frac{\mu_{\text{He}}}{\mu_{\text{N}_2}}\).

\(\mu_{\text{He}} = \frac{12}{4} = 3 \, \text{mol}\), \(\mu_{\text{N}_2} = \frac{28}{28} = 1 \, \text{mol}\).

Ratio = \(\frac{3}{1} = 3:1\).

1:3
1:1
3:1
7:1
3

What is the molar specific heat at constant volume (\(C_v\)) for a diatomic gas with no vibrational modes? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

For a diatomic gas (rigid rotator), 5 degrees of freedom (3 translational + 2 rotational).

\(C_v = \frac{5}{2} R = \frac{5}{2} \times 8.31 = 20.775 \, \text{J mol}^{-1} \text{K}^{-1} \approx 20.8 \, \text{J mol}^{-1} \text{K}^{-1} \).

12.5 J mol⁻¹ K⁻¹
20.8 J mol⁻¹ K⁻¹
29.1 J mol⁻¹ K⁻¹
33.2 J mol⁻¹ K⁻¹
2

The rms speed of a gas is 450 m/s at 225 K. At what temperature will the rms speed be 900 m/s?

\(v_{\text{rms}} \propto \sqrt{T}\), \(\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}}\).

\(\frac{900}{450} = \sqrt{\frac{T_2}{225}}\), \(2 = \sqrt{\frac{T_2}{225}}\).

Square both sides: \(4 = \frac{T_2}{225}\), \(T_2 = 900 \, \text{K}\).

450 K
675 K
900 K
1125 K
3

How much heat is required to raise the temperature of 0.25 moles of a diatomic gas by 20 K at constant volume, with no vibrational modes? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Diatomic gas: 5 degrees of freedom, \(C_v = \frac{5}{2} R\).

\(Q = \mu C_v \Delta T = 0.25 \times \frac{5}{2} \times 8.31 \times 20 = 207.75 \, \text{J} \approx 207.8 \, \text{J}\).

166.2 J
207.8 J
249.3 J
290.9 J
2

A gas has a \(C_p\) of \(29.8 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the ratio of specific heats (\(\gamma\))? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(C_v = C_p - R = 29.8 - 8.31 = 21.49 \, \text{J mol}^{-1} \text{K}^{-1}\).

\(\gamma = \frac{C_p}{C_v} = \frac{29.8}{21.49} \approx 1.39 \approx 1.40\).

1.33
1.40
1.67
1.80
2

A gas mixture has equal numbers of oxygen and argon molecules at 300 K. What is the ratio of their rms speeds? (Molecular mass: \(O_2 = 32 \, \text{u}\), Ar = 39.9 u)

\(v_{\text{rms}} \propto \frac{1}{\sqrt{m}}\), \(\frac{v_{\text{O}_2}}{v_{\text{Ar}}} = \sqrt{\frac{m_{\text{Ar}}}{m_{\text{O}_2}}}\).

\(\frac{v_{\text{O}_2}}{v_{\text{Ar}}} = \sqrt{\frac{39.9}{32}} \approx \sqrt{1.247} \approx 1.117\).

1:1.12
1:1
1.12:1
1.25:1
3

What is the average translational kinetic energy of a helium atom at 1000 K? (\(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Average translational KE = \(\frac{3}{2} k_B T\).

\(\frac{3}{2} \times 1.38 \times 10^{-23} \times 1000 = 2.07 \times 10^{-20} \, \text{J}\).

\(1.38 \times 10^{-20} \, \text{J}\)
\(2.07 \times 10^{-20} \, \text{J}\)
\(2.76 \times 10^{-20} \, \text{J}\)
\(4.14 \times 10^{-20} \, \text{J}\)
2

What is the total internal energy of 0.7 moles of a diatomic gas at 200 K with no vibrational modes? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Diatomic gas: 5 degrees of freedom, \(U = \frac{5}{2} \mu R T\).

\(U = \frac{5}{2} \times 0.7 \times 8.31 \times 200 = 2908.5 \, \text{J} \approx 2.91 \, \text{kJ}\).

2.33 kJ
2.91 kJ
3.49 kJ
4.07 kJ
2

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