Behaviour of Perfect Gas and Kinetic Theory Chapter-Wise Test 11

Correct answer Carries: 4.

Wrong Answer Carries: -1.

The rms speed of nitrogen molecules is 516 m/s at 300 K. What is the rms speed of oxygen molecules at the same temperature? (Molecular mass: \(N_2 = 28 \, \text{u}\), \(O_2 = 32 \, \text{u}\))

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

\(\frac{v_{\text{O}_2}}{516} = \sqrt{\frac{28}{32}} = \sqrt{0.875} \approx 0.935\).

\(v_{\text{O}_2} = 516 \times 0.935 \approx 482 \, \text{m/s}\).

450 m/s
482 m/s
516 m/s
550 m/s
2

The rms speed of helium molecules is 1370 m/s at 300 K. What is the rms speed of argon molecules at the same temperature? (Atomic mass: He = 4 u, Ar = 39.9 u)

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

\(\frac{v_{\text{Ar}}}{1370} = \sqrt{\frac{4}{39.9}} \approx \sqrt{0.1} \approx 0.316\).

\(v_{\text{Ar}} = 1370 \times 0.316 \approx 433 \, \text{m/s}\).

433 m/s
600 m/s
800 m/s
1000 m/s
1

A solid has a molar specific heat capacity of \(24.5 \, \text{J mol}^{-1} \text{K}^{-1}\). How many degrees of freedom per atom does it have? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(C = f \times \frac{R}{2}\), \(24.5 = f \times \frac{8.31}{2}\).

\(f = \frac{24.5 \times 2}{8.31} \approx 5.9 \approx 6\).

4
5
6
7
3

A gas has a density of \(0.8 \, \text{kg m}^{-3}\) at 1 atm and 300 K. What is its molecular mass? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\), 1 atm = \(1.01 \times 10^5 \, \text{Pa}\))

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

\(M = \frac{0.8 \times 8.31 \times 300}{1.01 \times 10^5} = 0.01975 \, \text{kg/mol} = 19.75 \, \text{g/mol} \approx 20 \, \text{g/mol} \).

16 g/mol
20 g/mol
28 g/mol
32 g/mol
2

The mean free path of a gas is \(1.8 \times 10^{-7} \, \text{m}\) with a number density of \(3.0 \times 10^{25} \, \text{m}^{-3}\). What is the molecular diameter?

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

\(d^2 = \frac{1}{1.414 \times 3.0 \times 10^{25} \times 3.14 \times 1.8 \times 10^{-7}} = \frac{1}{2.4 \times 10^{-19}} \approx 4.17 \times 10^{-20}\).

\(d = \sqrt{4.17 \times 10^{-20}} \approx 2.04 \times 10^{-10} \, \text{m}\).

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

A gas at 3 atm and 600 K has a volume of 15 litres. If the pressure decreases to 1.5 atm at constant temperature, what is the new volume?

Boyle’s law: \(P_1 V_1 = P_2 V_2\).

\(P_1 = 3 \, \text{atm}\), \(V_1 = 15 \, \text{litres}\), \(P_2 = 1.5 \, \text{atm}\).

\(V_2 = \frac{P_1 V_1}{P_2} = \frac{3 \times 15}{1.5} = 30 \, \text{litres}\).

15 litres
20 litres
25 litres
30 litres
4

What is the average translational kinetic energy of a nitrogen molecule at 450 K? (\(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

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

\(\frac{3}{2} \times 1.38 \times 10^{-23} \times 450 = 9.315 \times 10^{-21} \, \text{J}\).

\(6.21 \times 10^{-21} \, \text{J}\)
\(7.59 \times 10^{-21} \, \text{J}\)
\(9.31 \times 10^{-21} \, \text{J}\)
\(12.42 \times 10^{-21} \, \text{J}\)
3

What is the volume of 0.6 moles of an ideal gas at 3 atm and 527°C? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

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

\(T = 527 + 273 = 800 \, \text{K}\), \(P = 3 \times 1.01 \times 10^5 = 3.03 \times 10^5 \, \text{Pa}\).

\(V = \frac{0.6 \times 8.31 \times 800}{3.03 \times 10^5} = 1.317 \times 10^{-2} \, \text{m}^3 \approx 13.17 \, \text{litres}\).

10.0 litres
13.2 litres
15.0 litres
18.0 litres
2

At what temperature is the rms speed of hydrogen molecules 2000 m/s? (Molecular mass of \(H_2 = 2 \, \text{u}\), \(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

\(v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}\), \(m = \frac{2 \times 10^{-3}}{6.02 \times 10^{23}} = 3.32 \times 10^{-27} \, \text{kg}\).

\(2000^2 = \frac{3 \times 1.38 \times 10^{-23} \times T}{3.32 \times 10^{-27}}\), \(T = \frac{4 \times 10^6 \times 3.32 \times 10^{-27}}{4.14 \times 10^{-23}} \approx 321 \, \text{K}\).

250 K
321 K
400 K
500 K
2

What is the pressure of 0.4 moles of an ideal gas in a 4-litre container at 327°C? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

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

\(T = 327 + 273 = 600 \, \text{K}\), \(V = 4 \times 10^{-3} \, \text{m}^3\).

\(P = \frac{0.4 \times 8.31 \times 600}{4 \times 10^{-3}} = 4.986 \times 10^5 \, \text{Pa} \approx 5.0 \, \text{atm} (1 \, \text{atm} \approx 10^5 \, \text{Pa})\).

3.5 atm
4.0 atm
5.0 atm
6.0 atm
3

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