The Kinetic Theory of Gases Problems

This section provides 100 problems to test your understanding of the kinetic theory of gases, including the ideal gas model, molecular motion, pressure, temperature, Maxwell-Boltzmann distribution, degrees of freedom, and specific heats. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering the kinetic theory, a key topic for JEE/NEET success.

Numerical Problems

1. Calculate the number of molecules in 0.5 moles of gas ($N_A = 6.022 \times 10^{23} , \text{mol}^{-1}$).

   • (a) $3.010 \times 10^{23}$
   • (b) $3.011 \times 10^{23}$
   • (c) $3.012 \times 10^{23}$
   • (d) $3.013 \times 10^{23}$

2. Find the volume of 1.2 moles of gas at $P = 2 , \text{atm}$ ($1 , \text{atm} = 1.013 \times 10^5 , \text{Pa}$) and $T = 350 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

   • (a) $0.0348 , \text{m}^3$
   • (b) $0.0349 , \text{m}^3$
   • (c) $0.0350 , \text{m}^3$
   • (d) $0.0351 , \text{m}^3$

3. Calculate the mean free path of air molecules at STP ($n = 2.7 \times 10^{25} , \text{m}^{-3}$, $d = 3 \times 10^{-10} , \text{m}$).

   • (a) $6.5 \times 10^{-8} , \text{m}$
   • (b) $6.6 \times 10^{-8} , \text{m}$
   • (c) $6.7 \times 10^{-8} , \text{m}$
   • (d) $6.8 \times 10^{-8} , \text{m}$

4. A gas with $N = 10^{22}$ molecules in $V = 2 \times 10^{-4} , \text{m}^3$ is given. Calculate the number density.

   • (a) $5.0 \times 10^{25} , \text{m}^{-3}$
   • (b) $5.1 \times 10^{25} , \text{m}^{-3}$
   • (c) $5.2 \times 10^{25} , \text{m}^{-3}$
   • (d) $5.3 \times 10^{25} , \text{m}^{-3}$

5. Calculate $v_{\text{rms}}$ for nitrogen ($M = 0.028 , \text{kg/mol}$) at $T = 400 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

   • (a) $595 , \text{m/s}$
   • (b) $596 , \text{m/s}$
   • (c) $597 , \text{m/s}$
   • (d) $598 , \text{m/s}$

6. Find the average kinetic energy per molecule at $T = 300 , \text{K}$ ($k = 1.38 \times 10^{-23} , \text{J/K}$).

   • (a) $6.20 \times 10^{-21} , \text{J}$
   • (b) $6.21 \times 10^{-21} , \text{J}$
   • (c) $6.22 \times 10^{-21} , \text{J}$
   • (d) $6.23 \times 10^{-21} , \text{J}$

7. Calculate the pressure of a gas with $N = 10^{25}$ molecules, $v_{\text{rms}} = 600 , \text{m/s}$, $m = 3.32 \times 10^{-26} , \text{kg}$ in $V = 0.02 , \text{m}^3$.

   • (a) $4.97 \times 10^5 , \text{Pa}$
   • (b) $4.98 \times 10^5 , \text{Pa}$
   • (c) $4.99 \times 10^5 , \text{Pa}$
   • (d) $5.00 \times 10^5 , \text{Pa}$

8. Find the total kinetic energy of 1.5 moles of a gas at $T = 350 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

   • (a) $4361 , \text{J}$
   • (b) $4362 , \text{J}$
   • (c) $4363 , \text{J}$
   • (d) $4364 , \text{J}$

9. Calculate $v_{\text{mp}}$ for oxygen ($M = 0.032 , \text{kg/mol}$) at $T = 500 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

   • (a) $442 , \text{m/s}$
   • (b) $443 , \text{m/s}$
   • (c) $444 , \text{m/s}$
   • (d) $445 , \text{m/s}$

10. Find $v_{\text{avg}}$ for helium ($M = 0.004 , \text{kg/mol}$) at $T = 300 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $1252 , \text{m/s}$
    • (b) $1253 , \text{m/s}$
    • (c) $1254 , \text{m/s}$
    • (d) $1255 , \text{m/s}$

11. Compare $v_{\text{rms}}$ of O₂ ($M = 0.032 , \text{kg/mol}$) and H₂ ($M = 0.002 , \text{kg/mol}$) at $T = 400 , \text{K}$. What is the ratio $v_{\text{rms, H₂}} / v_{\text{rms, O₂}}$?

    • (a) $3$
    • (b) $4$
    • (c) $5$
    • (d) $6$

12. Calculate $C_V$ for a monatomic ideal gas ($R = 8.314 , \text{J/mol·K}$).

    • (a) $12.47 , \text{J/mol·K}$
    • (b) $12.48 , \text{J/mol·K}$
    • (c) $12.49 , \text{J/mol·K}$
    • (d) $12.50 , \text{J/mol·K}$

13. Calculate $\gamma$ for a diatomic gas at 300 K.

    • (a) $1.3$
    • (b) $1.4$
    • (c) $1.5$
    • (d) $1.6$

14. Find the internal energy $U$ for 3 moles of a monatomic gas at $T = 500 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $18660 , \text{J}$
    • (b) $18661 , \text{J}$
    • (c) $18662 , \text{J}$
    • (d) $18663 , \text{J}$

15. Calculate $C_P$ for a polyatomic gas with $f = 6$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $33.25 , \text{J/mol·K}$
    • (b) $33.26 , \text{J/mol·K}$
    • (c) $33.27 , \text{J/mol·K}$
    • (d) $33.28 , \text{J/mol·K}$

16. Find the number of molecules in 0.2 moles of gas ($N_A = 6.022 \times 10^{23} , \text{mol}^{-1}$).

    • (a) $1.203 \times 10^{23}$
    • (b) $1.204 \times 10^{23}$
    • (c) $1.205 \times 10^{23}$
    • (d) $1.206 \times 10^{23}$

17. Find the volume of 0.5 moles of gas at $P = 1 , \text{atm}$ ($1 , \text{atm} = 1.013 \times 10^5 , \text{Pa}$) and $T = 273 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $0.0112 , \text{m}^3$
    • (b) $0.0113 , \text{m}^3$
    • (c) $0.0114 , \text{m}^3$
    • (d) $0.0115 , \text{m}^3$

18. Calculate $v_{\text{rms}}$ for argon ($M = 0.040 , \text{kg/mol}$) at $T = 300 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $430 , \text{m/s}$
    • (b) $431 , \text{m/s}$
    • (c) $432 , \text{m/s}$
    • (d) $433 , \text{m/s}$

19. Find the average kinetic energy per molecule at $T = 500 , \text{K}$ ($k = 1.38 \times 10^{-23} , \text{J/K}$).

    • (a) $1.035 \times 10^{-20} , \text{J}$
    • (b) $1.036 \times 10^{-20} , \text{J}$
    • (c) $1.037 \times 10^{-20} , \text{J}$
    • (d) $1.038 \times 10^{-20} , \text{J}$

20. Calculate the pressure of a gas with $N = 10^{23}$ molecules, $v_{\text{rms}} = 400 , \text{m/s}$, $m = 5 \times 10^{-26} , \text{kg}$ in $V = 0.005 , \text{m}^3$.

    • (a) $1.33 \times 10^5 , \text{Pa}$
    • (b) $1.34 \times 10^5 , \text{Pa}$
    • (c) $1.35 \times 10^5 , \text{Pa}$
    • (d) $1.36 \times 10^5 , \text{Pa}$

21. Find the total kinetic energy of 2 moles of a gas at $T = 400 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $4987 , \text{J}$
    • (b) $4988 , \text{J}$
    • (c) $4989 , \text{J}$
    • (d) $4990 , \text{J}$

22. Calculate $v_{\text{mp}}$ for nitrogen ($M = 0.028 , \text{kg/mol}$) at $T = 600 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $483 , \text{m/s}$
    • (b) $484 , \text{m/s}$
    • (c) $485 , \text{m/s}$
    • (d) $486 , \text{m/s}$

23. Find $v_{\text{avg}}$ for argon ($M = 0.040 , \text{kg/mol}$) at $T = 500 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $541 , \text{m/s}$
    • (b) $542 , \text{m/s}$
    • (c) $543 , \text{m/s}$
    • (d) $544 , \text{m/s}$

24. Calculate $C_V$ for a diatomic ideal gas ($R = 8.314 , \text{J/mol·K}$).

    • (a) $20.78 , \text{J/mol·K}$
    • (b) $20.79 , \text{J/mol·K}$
    • (c) $20.80 , \text{J/mol·K}$
    • (d) $20.81 , \text{J/mol·K}$

25. Calculate $\gamma$ for a polyatomic gas at 300 K with $f = 6$.

    • (a) $1.32$
    • (b) $1.33$
    • (c) $1.34$
    • (d) $1.35$

26. Find the internal energy $U$ for 1 mole of a diatomic gas at $T = 300 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $3117 , \text{J}$
    • (b) $3118 , \text{J}$
    • (c) $3119 , \text{J}$
    • (d) $3120 , \text{J}$

27. Calculate $C_P$ for a monatomic gas ($R = 8.314 , \text{J/mol·K}$).

    • (a) $20.78 , \text{J/mol·K}$
    • (b) $20.79 , \text{J/mol·K}$
    • (c) $20.80 , \text{J/mol·K}$
    • (d) $20.81 , \text{J/mol·K}$

28. A gas with $N = 10^{24}$ molecules in $V = 0.01 , \text{m}^3$ is given. Calculate the number density.

    • (a) $1.0 \times 10^{26} , \text{m}^{-3}$
    • (b) $1.1 \times 10^{26} , \text{m}^{-3}$
    • (c) $1.2 \times 10^{26} , \text{m}^{-3}$
    • (d) $1.3 \times 10^{26} , \text{m}^{-3}$

29. Calculate $v_{\text{rms}}$ for hydrogen ($M = 0.002 , \text{kg/mol}$) at $T = 300 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $1932 , \text{m/s}$
    • (b) $1933 , \text{m/s}$
    • (c) $1934 , \text{m/s}$
    • (d) $1935 , \text{m/s}$

30. Find the average kinetic energy per molecule at $T = 600 , \text{K}$ ($k = 1.38 \times 10^{-23} , \text{J/K}$).

    • (a) $1.242 \times 10^{-20} , \text{J}$
    • (b) $1.243 \times 10^{-20} , \text{J}$
    • (c) $1.244 \times 10^{-20} , \text{J}$
    • (d) $1.245 \times 10^{-20} , \text{J}$

31. Find the total kinetic energy of 0.5 moles of a gas at $T = 273 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $1699 , \text{J}$
    • (b) $1700 , \text{J}$
    • (c) $1701 , \text{J}$
    • (d) $1702 , \text{J}$

32. Calculate $v_{\text{mp}}$ for argon ($M = 0.040 , \text{kg/mol}$) at $T = 400 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $351 , \text{m/s}$
    • (b) $352 , \text{m/s}$
    • (c) $353 , \text{m/s}$
    • (d) $354 , \text{m/s}$

33. In a rocket engine, exhaust gas (H₂O, $M = 0.018 , \text{kg/mol}$) at $T = 3000 , \text{K}$ is analyzed. Calculate $v_{\text{rms}}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $2035 , \text{m/s}$
    • (b) $2036 , \text{m/s}$
    • (c) $2037 , \text{m/s}$
    • (d) $2038 , \text{m/s}$

34. Calculate $C_V$ for a polyatomic gas with $f = 6$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $24.94 , \text{J/mol·K}$
    • (b) $24.95 , \text{J/mol·K}$
    • (c) $24.96 , \text{J/mol·K}$
    • (d) $24.97 , \text{J/mol·K}$

35. Find the internal energy $U$ for 2 moles of a polyatomic gas ($f = 6$) at $T = 400 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

    • (a) $9976 , \text{J}$
    • (b) $9977 , \text{J}$
    • (c) $9978 , \text{J}$
    • (d) $9979 , \text{J}$

Conceptual Problems

36. What does the ideal gas model assume about intermolecular forces?

• (a) Strong attractive forces
• (b) No forces except during collisions
• (c) Repulsive forces only
• (d) Constant forces

37. What does the kinetic theory relate temperature to?

• (a) Volume of the gas
• (b) Average kinetic energy of molecules
• (c) Pressure of the gas
• (d) Number of molecules

38. What does the Maxwell-Boltzmann distribution describe?

• (a) Pressure of a gas
• (b) Distribution of molecular speeds
• (c) Internal energy of a gas
• (d) Volume of a gas

39. What happens to $v_{\text{rms}}$ when temperature increases?

• (a) Decreases
• (b) Increases
• (c) Remains the same
• (d) Becomes zero

40. What is the unit of the Boltzmann constant in SI units?

• (a) $\text{J/K}$
• (b) $\text{J/mol·K}$
• (c) $\text{m/s}$
• (d) $\text{Pa}$

41. What does the mean free path represent?

• (a) Average speed of molecules
• (b) Average distance between collisions
• (c) Average kinetic energy
• (d) Number of collisions

42. What does the equipartition theorem state?

• (a) Energy is unequally distributed
• (b) Each degree of freedom has $\frac{1}{2} k T$ energy
• (c) Energy depends on pressure
• (d) Energy is constant

43. What is the physical significance of $\frac{3}{2} k T$?

• (a) Average speed of a molecule
• (b) Average kinetic energy per molecule
• (c) Pressure of the gas
• (d) Internal energy of the gas

44. What does a monatomic gas have in terms of degrees of freedom?

• (a) 3 translational
• (b) 5 total (translational + rotational)
• (c) 6 total
• (d) 2 translational

45. What is the dimension of $v_{\text{rms}}$?

• (a) $[\text{L} \text{T}^{-1}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

46. What does a zero mean free path indicate?

• (a) No collisions
• (b) Infinite collisions
• (c) Maximum speed
• (d) No temperature

47. What is the significance of $\sqrt{\frac{3 R T}{M}}$?

• (a) Mean free path
• (b) RMS speed of gas molecules
• (c) Average kinetic energy
• (d) Pressure

48. What happens to internal energy when temperature doubles for an ideal gas?

• (a) Doubles
• (b) Halves
• (c) Quadruples
• (d) Remains the same

49. What does $\gamma = \frac{C_P}{C_V}$ represent?

• (a) Degrees of freedom
• (b) Ratio of specific heats
• (c) Internal energy
• (d) Mean free path

50. How does kinetic theory apply to rocket propulsion?

• (a) Reduces pressure
• (b) Explains high-speed exhaust gas behavior
• (c) Increases temperature
• (d) Decreases molecular speed

Derivation Problems

51. Derive the ideal gas law from the kinetic theory $P V = \frac{1}{3} N m v_{\text{rms}}^2$.

52. Derive the relation between temperature and kinetic energy $\frac{1}{2} m v_{\text{rms}}^2 = \frac{3}{2} k T$.

53. Derive the Maxwell-Boltzmann speed distribution $f(v)$.

54. Derive the most probable speed $v_{\text{mp}} = \sqrt{\frac{2 k T}{m}}$.

55. Derive the average speed $v_{\text{avg}} = \sqrt{\frac{8 k T}{\pi m}}$.

56. Derive the RMS speed $v_{\text{rms}} = \sqrt{\frac{3 k T}{m}}$.

57. Derive the internal energy for an ideal gas $U = \frac{f}{2} n R T$.

58. Derive the mean free path $\lambda = \frac{1}{\sqrt{2} \pi d^2 n}$.

59. Derive the relation $C_P - C_V = R$ (Mayer’s relation).

60. Derive the ratio of specific heats $\gamma = 1 + \frac{2}{f}$.

61. Derive Avogadro’s hypothesis using the ideal gas law.

62. Derive the total kinetic energy of a gas $KE_{\text{total}} = \frac{3}{2} n R T$.

63. Derive the number density formula $n = \frac{N}{V}$.

64. Derive the energy distribution from the Maxwell-Boltzmann speed distribution.

65. Derive the equipartition theorem for energy per degree of freedom.

NEET-style Conceptual Problems

66. What is the unit of pressure in SI units?

• (a) $\text{Pa}$
• (b) $\text{J}$
• (c) $\text{m/s}$
• (d) $\text{K}$

67. What does a zero average kinetic energy per molecule indicate?

• (a) $T = 0 , \text{K}$
• (b) Maximum speed
• (c) No collisions
• (d) High pressure

68. Which speed is the smallest in the Maxwell-Boltzmann distribution?

• (a) $v_{\text{rms}}$
• (b) $v_{\text{avg}}$
• (c) $v_{\text{mp}}$
• (d) All are equal

69. What happens to $v_{\text{rms}}$ if molar mass doubles?

• (a) Increases by $\sqrt{2}$
• (b) Decreases by $\sqrt{2}$
• (c) Remains the same
• (d) Doubles

70. What is the dimension of the gas constant $R$?

• (a) $[\text{M} \text{L}^2 \text{T}^{-2} \text{K}^{-1} \text{mol}^{-1}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

71. What does the number of degrees of freedom depend on?

• (a) Temperature
• (b) Molecular structure
• (c) Pressure
• (d) Volume

72. What is the role of molecular collisions in kinetic theory?

• (a) Increase temperature
• (b) Cause pressure via wall impacts
• (c) Decrease speed
• (d) Reduce energy

73. What happens to $\gamma$ as degrees of freedom increase?

• (a) Increases
• (b) Decreases
• (c) Remains the same
• (d) Becomes zero

74. Why does $v_{\text{rms}}$ depend on temperature?

• (a) $T$ is proportional to average kinetic energy
• (b) $T$ affects pressure
• (c) $T$ affects volume
• (d) $T$ affects collisions

75. What is the unit of molar heat capacity?

• (a) $\text{J/mol·K}$
• (b) $\text{J/kg·K}$
• (c) $\text{m/s}$
• (d) $\text{Pa}$

76. What does a constant $P V = N k T$ indicate?

• (a) Ideal gas behavior
• (b) Thermal expansion
• (c) Heat transfer
• (d) Work done

77. Which type of motion do gas molecules exhibit in kinetic theory?

• (a) Circular
• (b) Random straight-line
• (c) Oscillatory
• (d) Vibrational

78. What is the direction of molecular motion in an ideal gas?

• (a) Along pressure gradient
• (b) Random in all directions
• (c) Perpendicular to walls
• (d) Along temperature gradient

79. What does a pseudo-force do in a non-inertial frame for gas molecules?

• (a) Affects perceived motion
• (b) Affects temperature
• (c) Creates pressure
• (d) Reduces speed

80. What is the dimension of mean free path?

• (a) $[\text{L}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

81. What is the role of kinetic theory in rocket exhaust?

• (a) Increases pressure
• (b) Explains high molecular speeds for thrust
• (c) Reduces temperature
• (d) Decreases energy

82. What happens to internal energy at constant temperature for an ideal gas?

• (a) Increases
• (b) Decreases
• (c) Remains constant
• (d) Becomes zero

83. Why does pressure arise in a gas according to kinetic theory?

• (a) Due to molecular collisions with walls
• (b) Due to temperature increase
• (c) Due to volume decrease
• (d) Due to intermolecular forces

84. What is the significance of $\frac{f}{2} n R T$?

• (a) Pressure of the gas
• (b) Internal energy of an ideal gas
• (c) Mean free path
• (d) Average speed

85. What is the unit of Avogadro’s number?

• (a) $\text{mol}^{-1}$
• (b) $\text{J/K}$
• (c) $\text{m/s}$
• (d) $\text{Pa}$

86. What does a zero $v_{\text{rms}}$ indicate?

• (a) $T = 0 , \text{K}$
• (b) Maximum pressure
• (c) No collisions
• (d) Infinite speed

87. What is the physical significance of $\sqrt{\frac{2 k T}{m}}$?

• (a) Average speed
• (b) Most probable speed
• (c) RMS speed
• (d) Mean free path

88. Why does a diatomic gas have more degrees of freedom than a monatomic gas?

• (a) Due to rotational motion
• (b) Due to higher temperature
• (c) Due to vibrational motion only
• (d) Due to pressure

89. What is the dimension of kinetic energy per molecule?

• (a) $[\text{M} \text{L}^2 \text{T}^{-2}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

90. How does the Maxwell-Boltzmann distribution help in rocket design?

• (a) Increases pressure
• (b) Predicts exhaust speed distribution for thrust
• (c) Reduces temperature
• (d) Decreases energy

91. What is the role of temperature in the kinetic theory?

• (a) Determines average kinetic energy
• (b) Determines volume
• (c) Determines pressure only
• (d) Determines collisions

92. What does a high $\gamma$ value indicate?

• (a) More degrees of freedom
• (b) Fewer degrees of freedom
• (c) Higher temperature
• (d) Lower pressure

93. What is the physical significance of $4 \pi \left( \frac{m}{2 \pi k T} \right)^{3/2} v^2 e^{-\frac{m v^2}{2 k T}}$?

• (a) Pressure distribution
• (b) Maxwell-Boltzmann speed distribution
• (c) Internal energy
• (d) Mean free path

94. What is the dimension of the number density?

• (a) $[\text{L}^{-3}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

95. Why does $C_P$ exceed $C_V$ in an ideal gas?

• (a) Due to work done at constant pressure
• (b) Due to phase change
• (c) Due to thermal expansion
• (d) Due to heat transfer

NEET-style Numerical Problems

96. Find the number of molecules in 0.1 moles of gas ($N_A = 6.022 \times 10^{23} , \text{mol}^{-1}$).

• (a) $6.021 \times 10^{22}$
• (b) $6.022 \times 10^{22}$
• (c) $6.023 \times 10^{22}$
• (d) $6.024 \times 10^{22}$

97. Calculate $v_{\text{rms}}$ for helium ($M = 0.004 , \text{kg/mol}$) at $T = 400 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

• (a) $1576 , \text{m/s}$
• (b) $1577 , \text{m/s}$
• (c) $1578 , \text{m/s}$
• (d) $1579 , \text{m/s}$

98. Find the total kinetic energy of 1 mole of a gas at $T = 300 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).

• (a) $3740 , \text{J}$
• (b) $3741 , \text{J}$
• (c) $3742 , \text{J}$
• (d) $3743 , \text{J}$

99. Calculate $\gamma$ for a monatomic gas.

• (a) $1.66$
• (b) $1.67$
• (c) $1.68$
• (d) $1.69$

100. Find the internal energy $U$ for 1 mole of a monatomic gas at $T = 400 , \text{K}$ ($R = 8.314 , \text{J/mol·K}$).
     - (a) $4987 , \text{J}$
     - (b) $4988 , \text{J}$
     - (c) $4989 , \text{J}$
     - (d) $4990 , \text{J}$

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