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Kinetic Theory of Gases · with answers

Practice set

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Learning path · Formula sheet · Practice set

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1. Why does an ideal gas have no potential energy?

Its molecules exert no forces on each other except during collisions (which take negligible time), so there is no interaction energy: the internal energy is all kinetic.

2. Which sample is closer to ideal: (a) a gas at 500 K and 1 atm, or (b) the same gas at 100 K and 50 atm?

(a). High temperature and low pressure make attractions and molecular size negligible.

3. Which postulate of kinetic theory makes the internal energy of an ideal gas depend only on temperature? (1) No intermolecular forces 2) Elastic collisions 3) Negligible molecular size 4) Random motion)

Option 1): No intermolecular forces.

Part 2 · Gas Laws

4. A gas at 1 atm and 27 °C occupies 2 L. It is heated to 127 °C and compressed to 1 L. New pressure?

$P_{2} = \frac{P _{1} V _{1} T _{2}}{V _{2} T _{1}} = \frac{1 \times 2 \times 400}{1 \times 300}$ ≈ 2.67 atm.

5. Volume of 1 mol of an ideal gas at 27 °C and 1 atm?

$V = \frac{R T}{P} = \frac{8.314 \times 300}{101325}$ ≈ 2.46 × 10⁻² m³ = 24.6 L.

6. Density of O₂ at 2 atm and 300 K?

$ρ = \frac{P M}{R T} = \frac{2 \times 101325 \times 0.032}{8.314 \times 300}$ ≈ 2.60 kg/m³.

7. 2 g of H₂ and 28 g of N₂ fill a 10 L vessel at 300 K. Total pressure, partial pressures and mean molar mass?

1 mol each: $P = \frac{2 \times 8.314 \times 300}{0.01}$ ≈ 4.99 × 10⁵ Pa; each partial pressure ≈ 2.49 × 10⁵ Pa; $M_{mix} = \frac{30}{2} = 15$ g/mol.

8. An ideal gas at 27 °C is heated at constant pressure until its volume doubles. Find the final temperature. (1) 327 °C 2) 54 °C 3) 600 °C 4) 327 K)

Option 1): 327 °C.

Part 3 · Real Gases, Change of Phase and van der Waals Equation

9. What are the SI units of the van der Waals constants a and b?

a: $Pa m^{6} /mo l^{2}$ (so $\frac{a n ^{2}}{V ^{2}}$ is a pressure); b: m³/mol (so nb is a volume).

10. The critical temperature of CO₂ is 304 K. Can CO₂ at 320 K be liquefied by compressing it?

No. Above T_c no pressure can liquefy a gas; it must first be cooled below 304 K.

11. In the van der Waals equation (P + an²/V²)(V − nb) = nRT, what does the constant a account for? (1) Attraction between molecules 2) Finite size of molecules 3) Elastic collisions 4) Gravity on molecules)

Option 1): Attraction between molecules.

Part 4 · Pressure of Air and Barometric Relation

12. A tube open at the top holds 15 cm of mercury trapping 30 cm of air (P_atm = 75 cm Hg). It is turned slowly upside down, open end down, with no mercury spilling. New air length?

Before: 75 + 15 = 90 cm Hg. After: 75 − 15 = 60 cm Hg. 90 × 30 = 60 × L → L = 45 cm.

13. At what height does the pressure of an isothermal atmosphere at 300 K fall to half (M = 0.029 kg/mol)?

$h = \frac{R T}{M g} ln 2$ = 8.314 × 300 × 0.693 / (0.029 × 9.8) ≈ 6.1 km.

14. A barometer with some air above the mercury reads 74 cm when the true pressure is 76 cm Hg, and 71 cm at 72 cm Hg. Find the length of tube above the mercury surface.

Air: 2 cm Hg over (L − 74), then 1 cm Hg over (L − 71). 2(L − 74) = L − 71 → L = 77 cm.

15. In an isothermal atmosphere, at what height does the pressure fall to 1/e of its ground value? (1)

\frac{R T}{M g}

\frac{M g}{R T}

\frac{R T}{M}

\frac{2 R T}{M g}

)

Option 1): $\frac{R T}{M g}$ .

Part 5 · Distribution of Molecular Speeds

16. Find v_rms of O₂ at 27 °C.

$\frac{3 \times 8.314 \times 300}{0.032}$ ≈ 484 m/s.

17. At what temperature does H₂ have the same v_rms as O₂ at 47 °C?

Same T/M: $T_{H_{2}} = 320 \times \frac{2}{32} = 20 K$ .

18. Arrange v_mp, v_mean and v_rms for any gas and give their ratio.

v_mp < v_mean < v_rms, in the ratio $2 : 8/ π : 3$ ≈ 1 : 1.13 : 1.22.

19. The temperature of N₂ is raised from 27 °C until its v_rms doubles. Final temperature?

$v_{rms} \propto T$ , so T × 4 = 1200 K (927 °C).

20. O₂ has rms speed v at temperature T. At what temperature does H₂ have the same rms speed? (1)

\frac{T}{16}

16 T

\frac{T}{4}

T

)

Option 1): $\frac{T}{16}$ .

Part 6 · Pressure, Kinetic Energy and Temperature

21. A gas at 1.0 × 10⁵ Pa has density 1.2 kg/m³. Find v_rms.

$\frac{3 P}{ρ} = 2.5 \times 1 0^{5}$ = 500 m/s.

22. Average translational KE of a gas molecule at 27 °C?

$\frac{3}{2} k T = 1.5 \times 1.38 \times 1 0^{- 23} \times 300$ ≈ 6.21 × 10⁻²¹ J.

23. Translational KE per unit volume of a gas at 1 atm?

$E = \frac{3}{2} P = 1.5 \times 101325$ ≈ 1.52 × 10⁵ J/m³.

24. At constant volume, the rms speed of the molecules of a gas doubles. What happens to the pressure? (1)

4 P

2 P

2 P

\frac{P}{4}

)

Option 1): $4 P$ .

Part 7 · Degrees of Freedom and Equipartition of Energy

25. Degrees of freedom of rigid CO₂ and rigid NH₃?

CO₂ is linear: 3 + 2 = 5. NH₃ is non-linear: 3 + 3 = 6.

26.

γ = C_{P} / C_{V}

for a gas with f = 6?

$γ = 1 + \frac{2}{f} = \frac{4}{3}$ .

27. Energy of 1 mol of N₂ (rigid) at 300 K?

$\frac{5}{2} R T = 2.5 \times 8.314 \times 300$ ≈ 6.24 × 10³ J.

28. What is γ = C_P/C_V for a gas of rigid diatomic molecules? (1)

\frac{7}{5}

\frac{5}{3}

\frac{9}{7}

\frac{4}{3}

)

Option 1): $\frac{7}{5}$ .

Part 8 · Internal Energy of a Gas

29. Internal energy of 2 mol of O₂ at 27 °C?

$\frac{5}{2} \times 2 \times 8.314 \times 300$ ≈ 1.25 × 10⁴ J.

30. f_eq for 2 mol of He and 1 mol of N₂?

$\frac{2 \times 3 + 1 \times 5}{3} = \frac{11}{3}$ ≈ 3.67.

31. 2 mol of He at 400 K mix with 1 mol of O₂ at 250 K in an insulated rigid vessel. Final temperature?

$\frac{3 \times 2 \times 400 + 5 \times 1 \times 250}{6 + 5} = \frac{3650}{11}$ ≈ 332 K.

32. 1 mol of He at 600 K mixes with 1 mol of N₂ at 200 K in an insulated rigid vessel. Find the final temperature. (1) 350 K 2) 400 K 3) 450 K 4) 366.7 K)

Option 1): 350 K.

Part 9 · Mean Free Path and Diffusion

33. Mean free path of a gas at 300 K and 1 atm, with molecular diameter 3 × 10⁻¹⁰ m?

$λ = \frac{k T}{2 π d ^{2} P} = \frac{1.38 \times 1 0 ^{- 23} \times 300}{1.414 \times π \times 9 \times 1 0 ^{- 20} \times 101325}$ ≈ 1.0 × 10⁻⁷ m.

34. The temperature of a gas doubles (a) at constant pressure, (b) in a sealed rigid vessel. What happens to λ?

(a) $λ \propto \frac{T}{P}$ , so it doubles. (b) n₀ is fixed, so $λ = \frac{1}{2 π d ^{2} n _{0}}$ is unchanged.

35. At constant temperature, the pressure of a gas is doubled. What happens to its mean free path? (1) Halves 2) Doubles 3) Unchanged 4) Quarters)

Option 1): Halves.