Behaviour of Perfect Gas and Kinetic Theory Chapter-Wise Test 6

Correct answer Carries: 4.

Wrong Answer Carries: -1.

The rms speed of oxygen molecules at a certain temperature is 500 m/s. What is the rms speed of hydrogen molecules at the same temperature? (Molecular mass of \(O_2 = 32 \, \text{u}\), \(H_2 = 2 \, \text{u}\))

\(v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}\), so \(v_{\text{rms}} \propto \frac{1}{\sqrt{m}}\).

Ratio: \(\frac{v_{\text{rms,H}_2}}{v_{\text{rms,O}_2}} = \sqrt{\frac{m_{O_2}}{m_{H_2}}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4\).

\(v_{\text{rms,H}_2} = 4 \times 500 = 2000 \, \text{m/s} \).

125 m/s
500 m/s
1000 m/s
2000 m/s
4

What is the collision frequency of a gas molecule with a mean free path of \(1.0 \times 10^{-7} \, \text{m}\) and average speed of 400 m/s?

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

\(\frac{400}{1.0 \times 10^{-7}} = 4.0 \times 10^9 \, \text{s}^{-1}\).

\(2.0 \times 10^9 \, \text{s}^{-1}\)
\(3.0 \times 10^9 \, \text{s}^{-1}\)
\(4.0 \times 10^9 \, \text{s}^{-1}\)
\(5.0 \times 10^9 \, \text{s}^{-1}\)
3

A solid has a molar specific heat capacity of \(24.9 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the energy per degree of freedom per atom? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(C = f \times \frac{R}{2}\), \(24.9 = f \times \frac{8.31}{2}\), \(f = \frac{24.9 \times 2}{8.31} \approx 6\).

Energy per degree of freedom = \(\frac{1}{2} k_B T\), per mole = \(\frac{1}{2} R T\).

For 1 degree of freedom, energy per mole = \(\frac{8.31}{2} = 4.155 \, \text{J mol}^{-1} \text{K}^{-1} \).

2.08 J mol⁻¹ K⁻¹
4.16 J mol⁻¹ K⁻¹
6.24 J mol⁻¹ K⁻¹
8.31 J mol⁻¹ K⁻¹
2

A gas mixture contains 2 g of hydrogen and 32 g of oxygen. What is the ratio of their partial pressures?

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

\(\mu_{\text{H}_2} = \frac{2}{2} = 1 \, \text{mol}\), \(\mu_{\text{O}_2} = \frac{32}{32} = 1 \, \text{mol}\).

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

1:8
1:4
1:1
4:1
3

A gas mixture contains 4 g of helium and 32 g of oxygen. What is the total pressure if the volume is 22.4 litres at 273 K? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(\mu_{\text{He}} = \frac{4}{4} = 1 \, \text{mol}\), \(\mu_{\text{O}_2} = \frac{32}{32} = 1 \, \text{mol}\).

Total moles = \(1 + 1 = 2 \, \text{mol}\), \(V = 22.4 \times 10^{-3} \, \text{m}^3\).

\(P = \frac{\mu R T}{V} = \frac{2 \times 8.31 \times 273}{22.4 \times 10^{-3}} = 2.02 \times 10^5 \, \text{Pa} \approx 2 \, \text{atm} \).

1 atm
2 atm
3 atm
4 atm
2

A gas mixture has equal masses of hydrogen and argon at 300 K. What is the ratio of their rms speeds? (Molecular mass: \(H_2 = 2 \, \text{u}\), Ar = 39.9 u)

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

\(\frac{v_{\text{H}_2}}{v_{\text{Ar}}} = \sqrt{\frac{39.9}{2}} \approx \sqrt{19.95} \approx 4.47\).

1:4.47
2:1
4.47:1
8:1
3

A gas at 2 atm and 400 K has a volume of 8 litres. If the temperature decreases to 200 K at constant pressure, what is the new volume?

Charles’ law: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\).

\(V_1 = 8 \, \text{litres}\), \(T_1 = 400 \, \text{K}\), \(T_2 = 200 \, \text{K}\).

\(V_2 = V_1 \times \frac{T_2}{T_1} = 8 \times \frac{200}{400} = 4 \, \text{litres}\).

4 litres
6 litres
8 litres
12 litres
1

The number density of a gas is \(2.5 \times 10^{25} \, \text{m}^{-3}\). If the molecular diameter is \(2 \times 10^{-10} \, \text{m}\), what is the mean free path?

\(l = \frac{1}{\sqrt{2} n \pi d^2}\).

\(n = 2.5 \times 10^{25}\), \(d = 2 \times 10^{-10}\), \(\pi \approx 3.14\).

\(l = \frac{1}{1.414 \times 2.5 \times 10^{25} \times 3.14 \times (2 \times 10^{-10})^2} = \frac{1}{4.44 \times 10^{17}} \approx 2.25 \times 10^{-7} \, \text{m} \).

\(1.5 \times 10^{-7} \, \text{m}\)
\(2.25 \times 10^{-7} \, \text{m}\)
\(3.0 \times 10^{-7} \, \text{m}\)
\(4.0 \times 10^{-7} \, \text{m}\)
2

A gas has a \(C_v\) of \(20.4 \, \text{J mol}^{-1} \text{K}^{-1}\). What is its \(C_p\)? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(C_p = C_v + R\).

\(C_p = 20.4 + 8.31 = 28.71 \, \text{J mol}^{-1} \text{K}^{-1} \approx 28.7 \, \text{J mol}^{-1} \text{K}^{-1}\).

24.9 J mol⁻¹ K⁻¹
28.7 J mol⁻¹ K⁻¹
30.5 J mol⁻¹ K⁻¹
33.2 J mol⁻¹ K⁻¹
2

What is the total internal energy of 0.5 moles of a monatomic gas at 600 K? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Monatomic gas: \(U = \frac{3}{2} \mu R T\).

\(U = \frac{3}{2} \times 0.5 \times 8.31 \times 600 = 3741.75 \, \text{J} \approx 3.74 \, \text{kJ}\).

2.49 kJ
3.74 kJ
4.99 kJ
6.23 kJ
2

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