Behaviour of Perfect Gas and Kinetic Theory Chapter-Wise Test 10

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A gas mixture has equal masses of neon and nitrogen at 300 K. What is the ratio of their rms speeds? (Atomic mass: Ne = 20.2 u, \(N_2 = 28 \, \text{u}\))

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

\(\frac{v_{\text{Ne}}}{v_{\text{N}_2}} = \sqrt{\frac{28}{20.2}} \approx \sqrt{1.386} \approx 1.18\).

1:1.18
1:1
1.18:1
1.4:1
3

A gas occupies 33.6 litres at STP. How many molecules are present? (\(N_A = 6.02 \times 10^{23} \, \text{mol}^{-1}\), molar volume at STP = 22.4 litres)

Number of moles (\(\mu\)) = \(\frac{\text{Volume}}{\text{Molar volume}} = \frac{33.6}{22.4} = 1.5 \, \text{mol}\).

Number of molecules = \(\mu \times N_A = 1.5 \times 6.02 \times 10^{23} = 9.03 \times 10^{23}\).

\(6.02 \times 10^{23}\)
\(9.03 \times 10^{23}\)
\(12.04 \times 10^{23}\)
\(15.05 \times 10^{23}\)
2

A gas at 1 atm and 273 K has a density of \(1.43 \, \text{kg m}^{-3}\). What is its molecular mass? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\), 1 atm = \(1.01 \times 10^5 \, \text{Pa}\))

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

\(M = \frac{1.43 \times 8.31 \times 273}{1.01 \times 10^5} = 0.0321 \, \text{kg/mol} \approx 32.1 \, \text{g/mol}\).

28 g/mol
32 g/mol
36 g/mol
40 g/mol
2

What is the collision frequency of a gas molecule if its mean free path is \(2 \times 10^{-7} \, \text{m}\) and average speed is 500 m/s?

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

\(\frac{500}{2 \times 10^{-7}} = 2.5 \times 10^9 \, \text{s}^{-1}\).

\(1.0 \times 10^9 \, \text{s}^{-1}\)
\(2.5 \times 10^9 \, \text{s}^{-1}\)
\(5.0 \times 10^9 \, \text{s}^{-1}\)
\(7.5 \times 10^9 \, \text{s}^{-1}\)
2

How many degrees of freedom does a diatomic molecule have if its vibrational mode is active?

Diatomic molecule: 3 translational + 2 rotational + 1 vibrational (2 modes: KE and PE).

Total degrees of freedom = \(3 + 2 + 2 = 7\).

5
6
7
8
3

A vessel contains 1 mole of a diatomic gas at 300 K. What is its total internal energy if vibrational modes are not excited? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

For diatomic gas (rigid rotator): \(U = \frac{5}{2} \mu R T\).

\(U = \frac{5}{2} \times 1 \times 8.31 \times 300 = 6232.5 \, \text{J} \approx 6.23 \, \text{kJ} \).

4.15 kJ
5.00 kJ
6.23 kJ
8.31 kJ
3

What is the average translational kinetic energy of a neon atom at 800 K? (\(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

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

\(\frac{3}{2} \times 1.38 \times 10^{-23} \times 800 = 1.656 \times 10^{-20} \, \text{J}\).

\(1.242 \times 10^{-20} \, \text{J}\)
\(1.656 \times 10^{-20} \, \text{J}\)
\(2.07 \times 10^{-20} \, \text{J}\)
\(2.484 \times 10^{-20} \, \text{J}\)
2

What is the time between collisions for a gas molecule with a mean free path of \(4.5 \times 10^{-7} \, \text{m}\) and average speed of 450 m/s?

\(\tau = \frac{l}{} = \frac{4.5 \times 10^{-7}}{450} = 1.0 \times 10^{-9} \, \text{s}\).

\(5.0 \times 10^{-10} \, \text{s}\)
\(7.5 \times 10^{-10} \, \text{s}\)
\(1.0 \times 10^{-9} \, \text{s}\)
\(1.5 \times 10^{-9} \, \text{s}\)
3

How much heat is required to raise the temperature of 0.5 moles of neon by 25 K at constant volume? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

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

\(Q = \mu C_v \Delta T = 0.5 \times \frac{3}{2} \times 8.31 \times 25 = 155.8125 \, \text{J} \approx 155.8 \, \text{J}\).

103.9 J
155.8 J
207.8 J
259.7 J
2

A gas has a \(C_v\) of \(21.0 \, \text{J mol}^{-1} \text{K}^{-1}\). What is its \(C_p\)? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

\(C_p = C_v + R\).

\(C_p = 21.0 + 8.31 = 29.31 \, \text{J mol}^{-1} \text{K}^{-1} \approx 29.3 \, \text{J mol}^{-1} \text{K}^{-1}\).

25.4 J mol⁻¹ K⁻¹
27.1 J mol⁻¹ K⁻¹
29.3 J mol⁻¹ K⁻¹
33.2 J mol⁻¹ K⁻¹
3

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