JEE Physics Kinetic Theory MCQs Set 12

Answer: (b) The root mean square speed is 14 m/s higher than the average speed

Explanation:
1. Average speed = 3500/5 = 700 m/s.
2. \( v_{rms}=\sqrt{(25+36+49+64+81)\times10^4/5}=\sqrt{510000}\approx714 \) m/s, which is about 14 m/s higher.

Teacher's Note:
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63. .

Answer: (c) 36.03 m/s

Explanation:
1. \( \frac12Mv_0^2=C_v\Delta T=\frac52\times8.314\times1=20.8 \) J, with M = 0.032 kg.
2. \( v_0=\sqrt{\frac{2\times20.8}{0.032}}\approx36 \) m/s.

Teacher's Note:
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Answer: (b) 2 and 4

Explanation:
1. Cp − Cv = R for every ideal gas (1 false). Monoatomic: Cv = 1.5R, Cp = 2.5R. Diatomic: Cv = 2.5R, Cp = 3.5R.
2. Sum and product are larger for the diatomic gas (2 and 4 true) while γ is 1.67 for monoatomic and 1.4 for diatomic (3 false).

Teacher's Note:
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Answer: (b) 1, 3 and 4

Explanation:
1. (1) \( \gamma=1/T \) for every ideal gas: true. (2) The average translational KE is \( \frac32KT \), not 3KT: false.
2. (3) True, as it depends only on T. (4) Mean free path is inversely proportional to the pressure: true.

Teacher's Note:
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86. .

Answer: (c) 3 and 4

Explanation:
1. \( v_p=\sqrt{2kT/m},\ \bar v=\sqrt{8kT/\pi m},\ v_{rms}=\sqrt{3kT/m} \), so \( v_p<\bar v2. \( \frac32kT=\frac34m v_p^2 \) (4 true). Molecules exist with speeds above v_rms and below v_p/√2, so 1 and 2 are false.

Teacher's Note:
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85. .

Answer: (d) 1, 2, 3 and 4

Explanation:
1. Mean velocity = (2 + 3 − 4 − 1)/4 = 0. Mean speed = (2 + 3 + 4 + 1)/4 = 2.5 m/s.
2. Mean square = (4 + 9 + 16 + 1)/4 = 7.5 and the square root is 2.74 m/s, so all four statements are true.

Teacher's Note:
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Answer: (a) 1 and 2

Explanation:
1. (1) \( PV=nRT \) and the temperature falls in an adiabatic expansion: true. (2) \( v_{rms}\propto 1/\sqrt M \): true.
2. (3) Temperatures in kelvin are 373 K and 473 K, so the KE does not double: false. (4) False because it does depend on the mass.

Teacher's Note:
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Answer: (b) 2 and 4

Explanation:
1. Same gas and same T give equal average KE and equal rms speed (1 and 3 false, 2 true).
2. P ∝ N at fixed V and T, so the number of molecules is in the ratio 1 : 2 (4 true).

Teacher's Note:
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88. .

Answer: (b) 2 and 3

Explanation:
1. Translational KE \( =\frac32PV \), independent of the gas. For N₂: \( \frac32P\times5=3000 \) J, so \( PV=2000 \) J.
2. For 10 litres at 2P, KE = 1.5 × 2P × 10 = 3 × (P × 10) = 3 × 4000 = 12000 J for any gas, so 2 and 3 are true; 1 and 4 are false.

Teacher's Note:
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Answer: (a) 1 and 3 equally (both greatest)

Explanation:
1. Mean free path \( \lambda\propto\frac VN \).
2. Values: (1) 2, (2) 1, (3) 2, (4) 1/3. The largest value 2 occurs for situations 1 and 3.

Teacher's Note:
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Answer: (a) 2 gm mole

Explanation:
1. \( C_v=\frac{1\times\frac32R+n\times\frac52R}{1+n} \) and \( \gamma=1+\frac R{C_v}=\frac{19}{13} \).
2. \( \frac{1+n}{1.5+2.5n}=\frac6{13}\Rightarrow13+13n=9+15n\Rightarrow n=2 \).

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Answer: (c) 0.17 %

Explanation:
1. \( v\propto\sqrt T \), so \( \frac{\Delta v}v=\frac12\frac{\Delta T}T=\frac1{600} \).
2. Percentage change = 0.167% ≈ 0.17%.

Teacher's Note:
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Answer: (a) 160 K

Explanation:
1. One collision with the wall occurs per 2 m of travel, so \( v=500\times2=1000 \) m/s.
2. \( v^2=\frac{3RT}M\Rightarrow T=\frac{10^6\times0.004}{3\times25/3}=160 \) K.

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Answer: (c) 0.3 g

Explanation:
1. \( n=\frac{PV}{RT}=\frac{100\times1}{(25/3)\times160}=0.075 \) mol.
2. Mass = 0.075 × 4 = 0.3 g.

Teacher's Note:
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Answer: (a) 1.1 × 10⁻⁷ m

Explanation:
1. \( \lambda=\frac{kT}{\sqrt2\pi d^2P} \) with \( d=2\times10^{-10} \) m, T = 290 K, P = 2.026 × 10⁵ Pa.
2. \( \lambda=\frac{1.38\times10^{-23}\times290}{1.414\times3.14\times4\times10^{-20}\times2.026\times10^5}\approx1.1\times10^{-7} \) m.

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Answer: (d) A is false, but R is true.

Explanation:
1. Internal energy depends only on the molecular motion relative to the container, not on the motion of the cylinder, so the assertion is false.
2. The temperature stays the same, so the reason is true.

Teacher's Note:
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Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. Higher pressure raises the boiling point of water, so food cooks faster: assertion true.
2. Aluminium is a good conductor, which is true, but it is not the reason for the faster cooking.

Teacher's Note:
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Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Because ice contracts on melting, a higher pressure lowers the melting point, so the slope of the melting curve is negative.
2. The reason explains the assertion.

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Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. A monoatomic gas has 3 translational degrees of freedom: assertion true.
2. γ is a constant for a given gas, but this does not explain the number of degrees of freedom.

Teacher's Note:
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75. .

Answer: (d) A is false, but R is true.

Explanation:
1. A gas can have any value of specific heat depending on the process, so the assertion is false.
2. The definition given is correct.

Teacher's Note:
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15. o
16. f
17.
18. a
19.
20. g
21. a
22. s
23.
24. d
25. e
26. p
27. e
28. n
29. d
30. s
31.
32. o
33. n
34.
35. t
36. h
37. e
38.
39. p
40. r
41. o
42. c
43. e
44. s
45. s
46. .

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Molecular volume and intermolecular forces are negligible at low density, so real gases follow the ideal gas equation.
2. The reason explains the assertion.

Teacher's Note:
1. L
2. o
3. w
4.
5. d
6. e
7. n
8. s
9. i
10. t
11. y
12.
13. m
14. e
15. a
16. n
17. s
18.
19. l
20. a
21. r
22. g
23. e
24.
25. s
26. e
27. p
28. a
29. r
30. a
31. t
32. i
33. o
34. n
35.
36. b
37. e
38. t
39. w
40. e
41. e
42. n
43.
44. m
45. o
46. l
47. e
48. c
49. u
50. l
51. e
52. s
53. .

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. Boyle's law gives PV = constant: true.
2. v² = 3kT/m is also true, but it is not the explanation of Boyle's law.

Teacher's Note:
1. B
2. o
3. y
4. l
5. e
6. '
7. s
8.
9. l
10. a
11. w
12.
13. f
14. o
15. l
16. l
17. o
18. w
19. s
20.
21. f
22. r
23. o
24. m
25.
26. t
27. h
28. e
29.
30. c
31. o
32. n
33. s
34. t
35. a
36. n
37. t
38.
39. t
40. o
41. t
42. a
43. l
44.
45. k
46. i
47. n
48. e
49. t
50. i
51. c
52.
53. e
54. n
55. e
56. r
57. g
58. y
59.
60. a
61. t
62.
63. a
64.
65. f
66. i
67. x
68. e
69. d
70.
71. t
72. e
73. m
74. p
75. e
76. r
77. a
78. t
79. u
80. r
81. e
82. .

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. The extra heat at constant pressure equals the work PΔV = RΔT per mole, giving Cp − Cv = R.
2. The reason explains the assertion.

Teacher's Note:
1. M
2. a
3. y
4. e
5. r
6. '
7. s
8.
9. r
10. e
11. l
12. a
13. t
14. i
15. o
16. n
17.
18. r
19. e
20. f
21. l
22. e
23. c
24. t
25. s
26.
27. t
28. h
29. e
30.
31. w
32. o
33. r
34. k
35.
36. o
37. f
38.
39. e
40. x
41. p
42. a
43. n
44. s
45. i
46. o
47. n
48. .