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  4. Kinetic Theory Chapter-Wise Test 2

Kinetic Theory Chapter-Wise Test 2

Correct answer Carries: 4.

Wrong Answer Carries: -1.

What is the temperature at which the rms speed of helium atoms is 1000 m/s? (Atomic mass of He = 4 u, \(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

\(v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}\), \(m = \frac{4 \times 10^{-3}}{6.02 \times 10^{23}} = 6.64 \times 10^{-27} \, \text{kg}\).

\(1000^2 = \frac{3 \times 1.38 \times 10^{-23} \times T}{6.64 \times 10^{-27}}\), \(T = \frac{10^6 \times 6.64 \times 10^{-27}}{4.14 \times 10^{-23}} \approx 1604 \, \text{K} \).

1200 K
1604 K
2000 K
2400 K
2

The mean free path of a gas molecule is \(6 \times 10^{-7} \, \text{m}\) at 0.5 atm. What will it be at 1 atm if temperature remains constant?

\(l \propto \frac{1}{n}\), \(n \propto P\). If \(P\) doubles, \(n\) doubles, \(l\) halves.

New \(l = \frac{6 \times 10^{-7}}{2} = 3 \times 10^{-7} \, \text{m}\).

\(1.5 \times 10^{-7} \, \text{m}\)
\(3.0 \times 10^{-7} \, \text{m}\)
\(6.0 \times 10^{-7} \, \text{m}\)
\(12.0 \times 10^{-7} \, \text{m}\)
2

The rms speed of a gas molecule is 400 m/s at 200 K. What is the rms speed at 800 K?

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

\(\frac{v_2}{400} = \sqrt{\frac{800}{200}} = \sqrt{4} = 2\).

\(v_2 = 400 \times 2 = 800 \, \text{m/s}\).

400 m/s
566 m/s
800 m/s
1132 m/s
3

The rms speed of a gas is 400 m/s at 100 K. At what temperature will the rms speed be 800 m/s?

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

\(\frac{800}{400} = \sqrt{\frac{T_2}{100}}\), \(2 = \sqrt{\frac{T_2}{100}}\).

Square both sides: \(4 = \frac{T_2}{100}\), \(T_2 = 400 \, \text{K}\).

200 K
300 K
400 K
500 K
3

How much heat is required to raise the temperature of 1 mole of argon gas by 20 K at constant volume? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

For monatomic gas, \(C_v = \frac{3}{2} R\).

Heat \(Q = \mu C_v \Delta T = 1 \times \frac{3}{2} \times 8.31 \times 20 = 249.3 \, \text{J} \approx 249 \, \text{J} \).

166 J
249 J
332 J
415 J
2

A gas mixture has equal numbers of hydrogen and oxygen molecules at 300 K. What is the ratio of their rms speeds? (Molecular mass: \(H_2 = 2 \, \text{u}\), \(O_2 = 32 \, \text{u}\))

\(v_{\text{rms}} \propto \frac{1}{\sqrt{m}}\), \(\frac{v_{\text{H}_2}}{v_{\text{O}_2}} = \sqrt{\frac{m_{\text{O}_2}}{m_{\text{H}_2}}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4\).

1:4
2:1
4:1
8:1
3

The rms speed of helium molecules is 1370 m/s at 300 K. What is the rms speed of oxygen molecules at the same temperature? (Atomic mass: He = 4 u, \(O_2 = 32 \, \text{u}\))

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

\(\frac{v_{\text{O}_2}}{1370} = \sqrt{\frac{4}{32}} = \sqrt{0.125} \approx 0.3535\).

\(v_{\text{O}_2} = 1370 \times 0.3535 \approx 484 \, \text{m/s}\).

430 m/s
484 m/s
600 m/s
800 m/s
2

The mean free path of a gas is \(7 \times 10^{-7} \, \text{m}\) with a number density of \(1.5 \times 10^{25} \, \text{m}^{-3}\). What is the molecular diameter?

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

\(d^2 = \frac{1}{1.414 \times 1.5 \times 10^{25} \times 3.14 \times 7 \times 10^{-7}} = \frac{1}{4.66 \times 10^{-19}} \approx 2.14 \times 10^{-20}\).

\(d = \sqrt{2.14 \times 10^{-20}} \approx 1.46 \times 10^{-10} \, \text{m}\).

\(1.0 \times 10^{-10} \, \text{m}\)
\(1.46 \times 10^{-10} \, \text{m}\)
\(2.0 \times 10^{-10} \, \text{m}\)
\(2.5 \times 10^{-10} \, \text{m}\)
2

What is the pressure exerted by 0.1 mole of an ideal gas in a 2-litre container at 127°C? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(PV = \mu R T\), \(P = \frac{\mu R T}{V}\).

\(T = 127 + 273 = 400 \, \text{K}\), \(V = 2 \times 10^{-3} \, \text{m}^3\).

\(P = \frac{0.1 \times 8.31 \times 400}{2 \times 10^{-3}} = 1.663 \times 10^5 \, \text{Pa} \approx 1.66 \, \text{atm} (1 \, \text{atm} \approx 10^5 \, \text{Pa})\).

1.25 atm
1.66 atm
2.00 atm
2.50 atm
2

A gas has a \(C_p\) of \(33.24 \, \text{J mol}^{-1} \text{K}^{-1}\). What is its \(C_v\)? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(C_p - C_v = R\), \(C_v = C_p - R\).

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

20.8 J mol⁻¹ K⁻¹
24.93 J mol⁻¹ K⁻¹
27.1 J mol⁻¹ K⁻¹
29.1 J mol⁻¹ K⁻¹
2

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