1. ${\text{λ}}_B>{\text{λ}}_A={\text{λ}}_C>{\text{λ}}_D$. Increasing the volume makes the gas less dense, so $\text{λ}$ increases. Increasing the radius makes the targets larger, so $\text{λ}$ decreases. The mean free path doesn’t depend on the atomic mass.

2. c. Each $v^2$ increases by a factor of 16 but, after averaging, $v_{\text{rms}}$ takes the square root.

3. e. Temperature is proportional to the average energy. The energy of a gas molecule is kinetic, proportional to $v^2$. The average energy, and thus T, increases by $4^2$.

4. b. The bead can slide along the wire (one degree of translational motion) and rotate around the wire (one degree of rotational motion).

5. a. Temperature measures the average translational kinetic energy per molecule, not the thermal energy of the entire system.

6. b. In this case, kinetic energy is transformed into potential energy; there is no entropy change. In the other cases, energy is transformed into thermal energy, meaning entropy increases.

7. b. The multiplicity increases by the factor $c(2V{)}^N/c{V}^N=2^N$.

8. c. Box B, with nearly equal numbers of molecules on both sides, is like a coin toss with nearly equal numbers of heads and tails. This is a very likely outcome with a high multiplicity. Box A is a very unlikely situation with a low multiplicity. Thus $S_{\text{A}}<S_{\text{B}}$ .

9. a. The entropy change of an ideal gas is the sum of two terms. In an expansion, with $V_{\text{f}}>V_{\text{i}}$, the volume term is positive. For $\text{Δ}S$ to be zero, the temperature term must be negative. Because the temperature ratio appears in a logarithm, this term is negative if $T_{\text{f}}<T_{\text{i}}$.

1.

1. 300 nm

2. 600 nm

3. $5.5\times {10}^{24}$

5.

1. $3.30\times {10}^{12}\text{m}{}^{-3}$

2. $1.71\times {10}^6\text{m}$

7. 600 nm

9.

1. $\vec{0}î+\vec{0}ĵ$

2. 57 m/s

3. 60 m/s

11. $6.5\times {10}^{25}\text{s}{}^{-1}$

13. Neon

15. 283 m/s

17. 2.5 mK

19. 2.12

21. $-24\text{°}\text{C}$

23.

1. Yes

2. Yes, $\sqrt{2}$

3. No

25. 2.5 mK

27. 580 m/s

29.

1. $0.080\text{°}\text{C}$

2. $0.048\text{°}\text{C}$

3. $0.040\text{°}\text{C}$

31.

1. $3.80\times {10}^5\text{J}$

2. $2.25\times {10}^{-9}\text{m}$

3. 0 J

33.

1. 62 J

2. 100 J

3. 150 J

35. 5000 J

37.

1. 1

2. 220

3. 924

39.

1. 0.3 J

2. 3 J

3. 30 J

41. $8.0\times {10}^2\text{J}\text{/}\text{K}$

43. $1.2\times {10}^3\text{K}$

45. 61

47. 510

49.

1. 310 m

2. 2900 m

51.

1. $v_{\text{rms}}=\sqrt{\frac{3p}{(N\text{/}V)m}}=\sqrt{\frac{3pV}{M}}$

2. $v_{\text{rms}}={\displaystyle \frac{v_{\text{rms}}{v}_{\text{piston}}}{2L}}$

3. $75{\text{m/s}}^2$

53. 1.0043

55. $1.9\times {10}^4\text{Pa}$

57. $29\text{J}\text{/}\mathrm{mol}\text{K}$

59. 482 K

61.

1. 2.5 kJ

2. $5.4\times {10}^{12}\text{rad}\text{/}\text{s}$

63. 22 kJ

65.

1. Increases by 2

2. Decreases by ${\displaystyle \frac{1}{8}}$

3. Increases by 4

4. No change

67. 2R

69.

1. $C_{\text{V}}={\displaystyle \frac{(3n_1+5n_2)}{2(n_1+n_2)}}R$

71. $\text{Δ}{S}_{\text{mix}}=-nR(f_{\text{A}}\text{ln}(f_{\text{A}})+f_{\text{B}}\text{ln}(f_{\text{B}}))$

73.

1. 436 K, 850 J from oxygen to helium

75.

1. $-20\text{J}\text{/}\text{K}$

2. $26\text{ J}/\text{K}$