Entropy and the Second Law of Thermodynamics Problems

This section provides 100 problems to test your understanding of entropy, the second law of thermodynamics, heat engines, refrigerators, and the Carnot cycle. 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 thermodynamics, a key topic for JEE/NEET success.

Numerical Problems

1. Calculate the entropy change for 1 mole of an ideal gas expanding isothermally and reversibly at 300 K from $V_1=0.02{\text{m}}^3$ to $V_2=0.04{\text{m}}^3$ ($R=8.314\text{J/mol·K}$).

   • (a) $5.76\text{J/K}$
   • (b) $5.77\text{J/K}$
   • (c) $5.78\text{J/K}$
   • (d) $5.79\text{J/K}$

2. Calculate the entropy change for 2 moles of an ideal gas ($C_V=\frac{3}{2}R$) heated at constant volume from 300 K to 450 K ($R=8.314\text{J/mol·K}$).

   • (a) $8.59\text{J/K}$
   • (b) $8.60\text{J/K}$
   • (c) $8.61\text{J/K}$
   • (d) $8.62\text{J/K}$

3. A gas undergoes free expansion (1 mole) from $V_1=0.01{\text{m}}^3$ to $V_2=0.03{\text{m}}^3$ at 400 K. Calculate $\mathrm{\Delta }{S}_{\text{total}}$ ($R=8.314\text{J/mol·K}$).

   • (a) $9.13\text{J/K}$
   • (b) $9.14\text{J/K}$
   • (c) $9.15\text{J/K}$
   • (d) $9.16\text{J/K}$

4. Heat $Q=500\text{J}$ flows from a hot reservoir at 500 K to a cold reservoir at 250 K. Calculate $\mathrm{\Delta }{S}_{\text{total}}$.

   • (a) $0.99\text{J/K}$
   • (b) $1.00\text{J/K}$
   • (c) $1.01\text{J/K}$
   • (d) $1.02\text{J/K}$

5. A heat engine absorbs $Q_H=1000\text{J}$ and rejects $Q_C=600\text{J}$. Calculate the efficiency.

   • (a) $0.39$
   • (b) $0.40$
   • (c) $0.41$
   • (d) $0.42$

6. A refrigerator removes $Q_C=800\text{J}$ with $W=200\text{J}$. Calculate the COP.

   • (a) $3.9$
   • (b) $4.0$
   • (c) $4.1$
   • (d) $4.2$

7. A heat engine performs $W=300\text{J}$ with $Q_H=900\text{J}$. Calculate $Q_C$.

   • (a) $599\text{J}$
   • (b) $600\text{J}$
   • (c) $601\text{J}$
   • (d) $602\text{J}$

8. A refrigerator transfers $Q_C=1200\text{J}$ with $COP=5$. Calculate $W$.

   • (a) $239\text{J}$
   • (b) $240\text{J}$
   • (c) $241\text{J}$
   • (d) $242\text{J}$

9. A Carnot engine operates between $T_H=700\text{K}$ and $T_C=350\text{K}$. Calculate the efficiency.

   • (a) $0.49$
   • (b) $0.50$
   • (c) $0.51$
   • (d) $0.52$

10. A Carnot refrigerator operates between $T_H=400\text{K}$ and $T_C=300\text{K}$. Calculate the COP.

    • (a) $2.9$
    • (b) $3.0$
    • (c) $3.1$
    • (d) $3.2$

11. A Carnot engine with $e=0.3$ has $T_C=280\text{K}$. Calculate $T_H$.

    • (a) $399\text{K}$
    • (b) $400\text{K}$
    • (c) $401\text{K}$
    • (d) $402\text{K}$

12. A real engine operates between $T_H=600\text{K}$ and $T_C=300\text{K}$ with $e=0.35$. Calculate the Carnot efficiency for comparison.

    • (a) $0.49$
    • (b) $0.50$
    • (c) $0.51$
    • (d) $0.52$

13. Calculate $\mathrm{\Delta }S$ for 1 mole of an ideal gas expanding isothermally and reversibly at 500 K from $V_1=0.01{\text{m}}^3$ to $V_2=0.03{\text{m}}^3$ ($R=8.314\text{J/mol·K}$).

    • (a) $9.13\text{J/K}$
    • (b) $9.14\text{J/K}$
    • (c) $9.15\text{J/K}$
    • (d) $9.16\text{J/K}$

14. Calculate $\mathrm{\Delta }S$ for 3 moles of an ideal gas ($C_V=\frac{5}{2}R$) heated at constant volume from 300 K to 600 K ($R=8.314\text{J/mol·K}$).

    • (a) $43.11\text{J/K}$
    • (b) $43.12\text{J/K}$
    • (c) $43.13\text{J/K}$
    • (d) $43.14\text{J/K}$

15. Heat $Q=800\text{J}$ flows from a hot reservoir at 800 K to a cold reservoir at 400 K. Calculate $\mathrm{\Delta }{S}_{\text{total}}$.

    • (a) $0.99\text{J/K}$
    • (b) $1.00\text{J/K}$
    • (c) $1.01\text{J/K}$
    • (d) $1.02\text{J/K}$

16. A heat engine absorbs $Q_H=1500\text{J}$ and rejects $Q_C=900\text{J}$. Calculate the efficiency.

    • (a) $0.39$
    • (b) $0.40$
    • (c) $0.41$
    • (d) $0.42$

17. A refrigerator removes $Q_C=1000\text{J}$ with $W=250\text{J}$. Calculate the COP.

    • (a) $3.9$
    • (b) $4.0$
    • (c) $4.1$
    • (d) $4.2$

18. A Carnot engine operates between $T_H=500\text{K}$ and $T_C=200\text{K}$. Calculate the efficiency.

    • (a) $0.59$
    • (b) $0.60$
    • (c) $0.61$
    • (d) $0.62$

19. A Carnot refrigerator operates between $T_H=350\text{K}$ and $T_C=250\text{K}$. Calculate the COP.

    • (a) $2.4$
    • (b) $2.5$
    • (c) $2.6$
    • (d) $2.7$

20. A gas undergoes free expansion (2 moles) from $V_1=0.02{\text{m}}^3$ to $V_2=0.04{\text{m}}^3$ at 300 K. Calculate $\mathrm{\Delta }{S}_{\text{system}}$ ($R=8.314\text{J/mol·K}$).

    • (a) $11.51\text{J/K}$
    • (b) $11.52\text{J/K}$
    • (c) $11.53\text{J/K}$
    • (d) $11.54\text{J/K}$

21. A heat engine performs $W=500\text{J}$ with $Q_H=2000\text{J}$. Calculate $Q_C$.

    • (a) $1499\text{J}$
    • (b) $1500\text{J}$
    • (c) $1501\text{J}$
    • (d) $1502\text{J}$

22. A refrigerator transfers $Q_C=1500\text{J}$ with $COP=3$. Calculate $W$.

    • (a) $499\text{J}$
    • (b) $500\text{J}$
    • (c) $501\text{J}$
    • (d) $502\text{J}$

23. A Carnot engine with $e=0.25$ has $T_C=300\text{K}$. Calculate $T_H$.

    • (a) $399\text{K}$
    • (b) $400\text{K}$
    • (c) $401\text{K}$
    • (d) $402\text{K}$

24. A real engine operates between $T_H=1000\text{K}$ and $T_C=400\text{K}$ with $e=0.5$. Calculate the Carnot efficiency for comparison.

    • (a) $0.59$
    • (b) $0.60$
    • (c) $0.61$
    • (d) $0.62$

25. Calculate $\mathrm{\Delta }S$ for 1 mole of an ideal gas expanding isothermally and reversibly at 400 K from $V_1=0.015{\text{m}}^3$ to $V_2=0.045{\text{m}}^3$ ($R=8.314\text{J/mol·K}$).

    • (a) $9.13\text{J/K}$
    • (b) $9.14\text{J/K}$
    • (c) $9.15\text{J/K}$
    • (d) $9.16\text{J/K}$

26. Calculate $\mathrm{\Delta }S$ for 1 mole of an ideal gas ($C_V=\frac{3}{2}R$) heated at constant volume from 200 K to 400 K ($R=8.314\text{J/mol·K}$).

    • (a) $8.64\text{J/K}$
    • (b) $8.65\text{J/K}$
    • (c) $8.66\text{J/K}$
    • (d) $8.67\text{J/K}$

27. Heat $Q=1000\text{J}$ flows from a hot reservoir at 600 K to a cold reservoir at 300 K. Calculate $\mathrm{\Delta }{S}_{\text{hot}}$.

    • (a) $-1.66\text{J/K}$
    • (b) $-1.67\text{J/K}$
    • (c) $-1.68\text{J/K}$
    • (d) $-1.69\text{J/K}$

28. A heat engine absorbs $Q_H=2000\text{J}$ and rejects $Q_C=1400\text{J}$. Calculate the efficiency.

    • (a) $0.29$
    • (b) $0.30$
    • (c) $0.31$
    • (d) $0.32$

29. A refrigerator removes $Q_C=500\text{J}$ with $W=100\text{J}$. Calculate the COP.

    • (a) $4.9$
    • (b) $5.0$
    • (c) $5.1$
    • (d) $5.2$

30. A Carnot engine operates between $T_H=800\text{K}$ and $T_C=200\text{K}$. Calculate the efficiency.

    • (a) $0.74$
    • (b) $0.75$
    • (c) $0.76$
    • (d) $0.77$

31. A rocket engine operates as a Carnot cycle between $T_H=2500\text{K}$ (combustion) and $T_C=500\text{K}$ (exhaust). Calculate the efficiency.

    • (a) $0.79$
    • (b) $0.80$
    • (c) $0.81$
    • (d) $0.82$

32. Calculate $\mathrm{\Delta }S$ for 2 moles of an ideal gas ($C_V=\frac{5}{2}R$) heated at constant volume from 400 K to 800 K ($R=8.314\text{J/mol·K}$).

    • (a) $28.74\text{J/K}$
    • (b) $28.75\text{J/K}$
    • (c) $28.76\text{J/K}$
    • (d) $28.77\text{J/K}$

33. A heat engine performs $W=400\text{J}$ with $Q_H=1600\text{J}$. Calculate $Q_C$.

    • (a) $1199\text{J}$
    • (b) $1200\text{J}$
    • (c) $1201\text{J}$
    • (d) $1202\text{J}$

34. A Carnot refrigerator operates between $T_H=500\text{K}$ and $T_C=400\text{K}$. Calculate the COP.

    • (a) $3.9$
    • (b) $4.0$
    • (c) $4.1$
    • (d) $4.2$

35. A Carnot engine with $e=0.2$ has $T_H=500\text{K}$. Calculate $T_C$.

    • (a) $399\text{K}$
    • (b) $400\text{K}$
    • (c) $401\text{K}$
    • (d) $402\text{K}$

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Conceptual Problems

36. What does the Kelvin-Planck statement of the second law imply?

• (a) Heat can flow from cold to hot spontaneously
• (b) No heat engine can be 100% efficient
• (c) Entropy decreases in natural processes
• (d) Work cannot be converted to heat

37. What does the Clausius statement of the second law imply?

• (a) Heat engines can be 100% efficient
• (b) Heat cannot flow from cold to hot without work
• (c) Entropy remains constant in all processes
• (d) Work cannot be done in a cycle

38. What is entropy a measure of?

• (a) Temperature
• (b) Disorder of a system
• (c) Pressure
• (d) Volume

39. What happens to total entropy in a reversible process?

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

40. What is the unit of entropy in SI units?

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

41. What does a positive $\mathrm{\Delta }{S}_{\text{total}}$ indicate?

• (a) Reversible process
• (b) Irreversible process
• (c) No change in entropy
• (d) Decrease in disorder

42. What does the efficiency of a heat engine represent?

• (a) Ratio of heat absorbed to work done
• (b) Ratio of work done to heat absorbed
• (c) Ratio of heat rejected to heat absorbed
• (d) Ratio of work done to heat rejected

43. What is the physical significance of $\frac{Q_C}{W}$ in a refrigerator?

• (a) Efficiency
• (b) Coefficient of Performance
• (c) Entropy change
• (d) Work done

44. What does the Carnot cycle represent?

• (a) Most efficient cycle possible
• (b) Least efficient cycle
• (c) Irreversible cycle
• (d) Isothermal cycle only

45. What is the dimension of entropy?

• (a) $[\text{M}{\text{L}}^2{\text{T}}^{-2}{\text{K}}^{-1}]$
• (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 $\mathrm{\Delta }{S}_{\text{total}}$ indicate?

• (a) Irreversible process
• (b) Reversible process
• (c) No heat transfer
• (d) No work done

47. What is the significance of $1-\frac{T_C}{T_H}$?

• (a) Entropy change
• (b) Carnot efficiency
• (c) Work done
• (d) Heat absorbed

48. What happens to efficiency if $T_C$ decreases in a Carnot engine?

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

49. What does a high COP in a refrigerator indicate?

• (a) Low efficiency
• (b) High efficiency in heat removal
• (c) High work input
• (d) Low heat transfer

50. How does the second law apply to rocket engines?

• (a) Increases entropy via irreversible combustion
• (b) Decreases efficiency
• (c) Increases temperature
• (d) Reduces work output

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Derivation Problems

51. Derive the equivalence of the Kelvin-Planck and Clausius statements of the second law.

52. Derive the entropy change for a reversible isothermal process $\mathrm{\Delta }S=\frac{Q_{\text{rev}}}{T}$.

53. Derive the entropy change for a constant volume process $\mathrm{\Delta }S=n{C}_V\ln \left(\frac{T_2}{T_1}\right)$.

54. Derive the entropy change in an irreversible free expansion.

55. Derive the efficiency of a heat engine $e=1-\frac{Q_C}{Q_H}$.

56. Derive the coefficient of performance for a refrigerator $COP=\frac{Q_C}{W}$.

57. Derive the Carnot efficiency $e_{\text{Carnot}}=1-\frac{T_C}{T_H}$.

58. Derive the Carnot refrigerator COP $CO{P}_{\text{Carnot}}=\frac{T_C}{T_H-T_C}$.

59. Derive the work done in the Carnot cycle $W=nR(T_H-T_C)\ln \left(\frac{V_2}{V_1}\right)$.

60. Derive the entropy change for heat transfer between two reservoirs.

61. Derive the second law in terms of entropy $\mathrm{\Delta }{S}_{\text{total}}\ge 0$.

62. Derive the relation $\frac{Q_H}{T_H}=\frac{Q_C}{T_C}$ for a Carnot cycle.

63. Derive the irreversibility of free expansion using entropy.

64. Derive the efficiency limitation of a real engine compared to a Carnot engine.

65. Derive the entropy change for an adiabatic reversible process (isentropic process).

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NEET-style Conceptual Problems

66. What is the unit of efficiency in a heat engine?

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

67. What does a negative $\mathrm{\Delta }{S}_{\text{system}}$ indicate in a process?

• (a) Increase in disorder
• (b) Decrease in disorder
• (c) No change in entropy
• (d) Irreversible process

68. Which process is inherently irreversible?

• (a) Isothermal reversible expansion
• (b) Adiabatic reversible expansion
• (c) Free expansion
• (d) Carnot cycle

69. What happens to Carnot efficiency if $T_H$ increases?

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

70. What is the dimension of COP in a refrigerator?

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

71. What does the second law of thermodynamics determine?

• (a) Amount of work done
• (b) Direction of natural processes
• (c) Temperature of a system
• (d) Pressure of a system

72. What is the role of entropy in natural processes?

• (a) Decreases total disorder
• (b) Increases total disorder
• (c) Maintains constant disorder
• (d) Reduces temperature

73. What happens to $\mathrm{\Delta }{S}_{\text{total}}$ in an irreversible process?

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

74. Why is the Carnot cycle the most efficient?

• (a) It is irreversible
• (b) It is reversible with maximum efficiency
• (c) It rejects no heat
• (d) It operates at constant temperature

75. What is the unit of work in a heat engine?

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

76. What does a constant entropy process indicate?

• (a) Irreversible process
• (b) Isentropic (reversible adiabatic) process
• (c) Isothermal process
• (d) Free expansion

77. Which type of process increases total entropy?

• (a) Reversible
• (b) Irreversible
• (c) Isentropic
• (d) Isobaric

78. What is the direction of spontaneous heat flow according to the second law?

• (a) From cold to hot
• (b) From hot to cold
• (c) No direction
• (d) Circular

79. What does a pseudo-force do in a non-inertial frame for thermodynamic processes?

• (a) Affects entropy perception
• (b) Affects heat flow
• (c) Creates work
• (d) Reduces efficiency

80. What is the dimension of heat in SI units?

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

81. What is the role of the second law in rocket engine design?

• (a) Increases efficiency
• (b) Limits efficiency due to irreversibility
• (c) Reduces entropy
• (d) Increases heat transfer

82. What happens to entropy in an isolated system over time?

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

83. Why does heat flow from a hot to a cold reservoir increase entropy?

• (a) Due to reversible process
• (b) Due to increase in total disorder
• (c) Due to decrease in temperature
• (d) Due to work done

84. What is the significance of $\frac{T_C}{T_H-T_C}$?

• (a) Carnot efficiency
• (b) Carnot refrigerator COP
• (c) Entropy change
• (d) Work done

85. What is the unit of temperature in thermodynamic calculations?

• (a) Celsius
• (b) Kelvin
• (c) Fahrenheit
• (d) Joule

86. What does a zero efficiency in a heat engine indicate?

• (a) No work done
• (b) Maximum work done
• (c) No heat absorbed
• (d) No heat rejected

87. What is the physical significance of $nR\ln \left(\frac{V_2}{V_1}\right)$?

• (a) Work done in isothermal process
• (b) Entropy change in isothermal expansion
• (c) Heat absorbed
• (d) Efficiency

88. Why is free expansion irreversible?

• (a) $\mathrm{\Delta }{S}_{\text{total}}>0$
• (b) $\mathrm{\Delta }{S}_{\text{total}}=0$
• (c) $Q>0$
• (d) $W>0$

89. What is the dimension of $Q_H$ in a heat engine?

• (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 entropy analysis help in rocket propulsion?

• (a) Increases efficiency
• (b) Quantifies irreversibility in combustion
• (c) Reduces temperature
• (d) Decreases work output

91. What is the role of $T$ in entropy calculations?

• (a) Measures disorder directly
• (b) Relates heat transfer to entropy change
• (c) Determines work done
• (d) Determines pressure

92. What does a 100% Carnot efficiency imply?

• (a) $T_C=0\text{K}$
• (b) $T_H=0\text{K}$
• (c) $Q_C=0$
• (d) $W=0$

93. What is the physical significance of $1-\frac{Q_C}{Q_H}$?

• (a) Entropy change
• (b) Efficiency of a heat engine
• (c) COP of a refrigerator
• (d) Work done

94. What is the dimension of $\mathrm{\Delta }{S}_{\text{total}}$?

• (a) $[\text{M}{\text{L}}^2{\text{T}}^{-2}{\text{K}}^{-1}]$
• (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 a real engine have lower efficiency than a Carnot engine?

• (a) Due to reversible processes
• (b) Due to irreversibilities like friction
• (c) Due to higher $T_C$
• (d) Due to lower $T_H$

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NEET-style Numerical Problems

96. Calculate $\mathrm{\Delta }S$ for 1 mole of an ideal gas expanding isothermally and reversibly at 350 K from $V_1=0.01{\text{m}}^3$ to $V_2=0.02{\text{m}}^3$ ($R=8.314\text{J/mol·K}$).

• (a) $5.76\text{J/K}$
• (b) $5.77\text{J/K}$
• (c) $5.78\text{J/K}$
• (d) $5.79\text{J/K}$

97. A heat engine absorbs $Q_H=1200\text{J}$ and rejects $Q_C=800\text{J}$. What is the efficiency?

• (a) $0.32$
• (b) $0.33$
• (c) $0.34$
• (d) $0.35$

98. A Carnot engine operates between $T_H=600\text{K}$ and $T_C=300\text{K}$. What is the efficiency?

• (a) $0.49$
• (b) $0.50$
• (c) $0.51$
• (d) $0.52$

99. A refrigerator removes $Q_C=600\text{J}$ with $W=150\text{J}$. What is the COP?

• (a) $3.9$
• (b) $4.0$
• (c) $4.1$
• (d) $4.2$

100. A Carnot refrigerator operates between $T_H=400\text{K}$ and $T_C=320\text{K}$. What is the COP?
     - (a) $3.9$
     - (b) $4.0$
     - (c) $4.1$
     - (d) $4.2$

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