Behaviour of Perfect Gas and Kinetic Theory Chapter-Wise Test 16

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A gas has a \(C_v\) of \(24.93 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the ratio of specific heats (\(\gamma\))? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(C_p = C_v + R = 24.93 + 8.31 = 33.24 \, \text{J mol}^{-1} \text{K}^{-1}\).

\(\gamma = \frac{C_p}{C_v} = \frac{33.24}{24.93} \approx 1.33\).

1.25
1.33
1.40
1.67
2

What is the average speed of a molecule if its mean free path is \(4 \times 10^{-7} \, \text{m}\) and time between collisions is \(8 \times 10^{-10} \, \text{s}\)?

Mean free path \(l = \tau\).

\( = \frac{l}{\tau} = \frac{4 \times 10^{-7}}{8 \times 10^{-10}} = 500 \, \text{m/s} \).

250 m/s
500 m/s
750 m/s
1000 m/s
2

At what temperature is the rms speed of neon molecules 620 m/s? (Atomic mass of Ne = 20.2 u, \(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

\(v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}\), \(m = \frac{20.2 \times 10^{-3}}{6.02 \times 10^{23}} = 3.36 \times 10^{-26} \, \text{kg}\).

\(620^2 = \frac{3 \times 1.38 \times 10^{-23} \times T}{3.36 \times 10^{-26}}\), \(T = \frac{3.84 \times 10^5 \times 3.36 \times 10^{-26}}{4.14 \times 10^{-23}} \approx 312 \, \text{K}\).

250 K
312 K
400 K
500 K
2

What is the average translational kinetic energy of an argon atom at 900 K? (\(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

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

\(\frac{3}{2} \times 1.38 \times 10^{-23} \times 900 = 1.863 \times 10^{-20} \, \text{J}\).

\(1.449 \times 10^{-20} \, \text{J}\)
\(1.656 \times 10^{-20} \, \text{J}\)
\(1.863 \times 10^{-20} \, \text{J}\)
\(2.07 \times 10^{-20} \, \text{J}\)
3

A solid has a molar specific heat capacity of \(24.4 \, \text{J mol}^{-1} \text{K}^{-1}\). How many degrees of freedom per atom does it have? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(C = f \times \frac{R}{2}\), \(24.4 = f \times \frac{8.31}{2}\).

\(f = \frac{24.4 \times 2}{8.31} \approx 5.87 \approx 6\).

3
4
5
6
4

A gas at 2 atm and 400 K has a density of \(0.8 \, \text{kg m}^{-3}\). What is its molecular mass? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\), 1 atm = \(1.01 \times 10^5 \, \text{Pa}\))

\(P = \frac{\rho R T}{M}\), \(M = \frac{\rho R T}{P}\).

\(P = 2 \times 1.01 \times 10^5 = 2.02 \times 10^5 \, \text{Pa}\).

\(M = \frac{0.8 \times 8.31 \times 400}{2.02 \times 10^5} = 0.01318 \, \text{kg/mol} \approx 13.2 \, \text{g/mol}\).

10 g/mol
13 g/mol
16 g/mol
20 g/mol
2

A gas mixture has equal numbers of helium and nitrogen molecules at 400 K. What is the ratio of their rms speeds? (Atomic mass: He = 4 u, \(N_2 = 28 \, \text{u}\))

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

\(\frac{v_{\text{He}}}{v_{\text{N}_2}} = \sqrt{\frac{28}{4}} = \sqrt{7} \approx 2.65\).

1:2.65
1:1
2.65:1
7:1
3

The ratio of specific heats (\(\gamma\)) for a triatomic gas with no vibrational modes is:

Triatomic gas: 3 translational + 3 rotational = 6 degrees of freedom.

\(C_v = 3R\), \(C_p = C_v + R = 4R\).

\(\gamma = \frac{C_p}{C_v} = \frac{4R}{3R} = \frac{4}{3} = 1.33\).

1.67
1.40
1.33
1.25
3

What is the collision frequency of a gas molecule with a mean free path of \(2.4 \times 10^{-7} \, \text{m}\) and average speed of 480 m/s?

Collision frequency = \(\frac{}{l}\).

\(\frac{480}{2.4 \times 10^{-7}} = 2.0 \times 10^9 \, \text{s}^{-1}\).

\(1.5 \times 10^9 \, \text{s}^{-1}\)
\(2.0 \times 10^9 \, \text{s}^{-1}\)
\(2.5 \times 10^9 \, \text{s}^{-1}\)
\(3.0 \times 10^9 \, \text{s}^{-1}\)
2

What is the total internal energy of 0.5 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 (3 translational + 2 rotational).

\(U = \frac{5}{2} \mu R T = \frac{5}{2} \times 0.5 \times 8.31 \times 200 = 2077.5 \, \text{J} \approx 2.08 \, \text{kJ}\).

1.66 kJ
2.08 kJ
2.49 kJ
3.32 kJ
2

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