Temperature, Heat, and the First Law of Thermodynamics Problems

This section provides 100 problems to test your understanding of thermodynamics, including temperature scales, thermal equilibrium, heat transfer mechanisms, specific heat, calorimetry, thermal expansion, and the first law of thermodynamics. 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. Convert 45°C to Kelvin.

   • (a) $317\text{K}$
   • (b) $318\text{K}$
   • (c) $319\text{K}$
   • (d) $320\text{K}$

2. Convert 77°F to Celsius.

   • (a) $24\text{°C}$
   • (b) $25\text{°C}$
   • (c) $26\text{°C}$
   • (d) $27\text{°C}$

3. An ideal gas thermometer measures pressure $P_1=1.5\text{atm}$ at 0°C and $P_2=1.8\text{atm}$ at an unknown temperature $T$. Calculate $T$ in Celsius.

   • (a) $54\text{°C}$
   • (b) $55\text{°C}$
   • (c) $56\text{°C}$
   • (d) $57\text{°C}$

4. A slab ($k=0.15\text{W/m·K}$, $A=0.4{\text{m}}^2$, $L=0.05\text{m}$) has faces at 90°C and 30°C. Calculate the heat transfer rate.

   • (a) $47\text{W}$
   • (b) $48\text{W}$
   • (c) $49\text{W}$
   • (d) $50\text{W}$

5. A body at 627°C radiates with $e=0.9$, $A=0.02{\text{m}}^2$ ($\sigma =5.67\times {10}^{-8}{\text{W/m}}^2{\text{·K}}^4$, surroundings at 27°C). Calculate the net power radiated.

   • (a) $489\text{W}$
   • (b) $490\text{W}$
   • (c) $491\text{W}$
   • (d) $492\text{W}$

6. Two slabs in series have $k_1=0.2\text{W/m·K}$, $L_1=0.04\text{m}$, $k_2=0.4\text{W/m·K}$, $L_2=0.02\text{m}$, $A=1{\text{m}}^2$, with faces at 70°C and 10°C. Calculate the heat transfer rate.

   • (a) $199\text{W}$
   • (b) $200\text{W}$
   • (c) $201\text{W}$
   • (d) $202\text{W}$

7. A rocket engine wall ($k=30\text{W/m·K}$, $A=0.15{\text{m}}^2$, $L=0.01\text{m}$) has temperatures $T_1=1100\text{K}$ and $T_2=500\text{K}$. Calculate the heat transfer rate.

   • (a) $269900\text{W}$
   • (b) $270000\text{W}$
   • (c) $270100\text{W}$
   • (d) $270200\text{W}$

8. Mix 0.25 kg of water at 90°C with 0.35 kg at 30°C ($c_{\text{water}}=4186\text{J/kg·K}$). Calculate the final temperature.

   • (a) $56\text{°C}$
   • (b) $57\text{°C}$
   • (c) $58\text{°C}$
   • (d) $59\text{°C}$

9. Melt 0.4 kg of ice at 0°C to water at 0°C ($L_f=3.34\times {10}^5\text{J/kg}$). Calculate the heat required.

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

10. A copper rod ($L_0=1.5\text{m}$, $\alpha =17\times {10}^{-6}{\text{K}}^{-1}$) is heated from 25°C to 125°C. Calculate the length increase.

    • (a) $0.0025\text{m}$
    • (b) $0.0026\text{m}$
    • (c) $0.0027\text{m}$
    • (d) $0.0028\text{m}$

11. Heat 2 kg of water from 15°C to 75°C ($c_{\text{water}}=4186\text{J/kg·K}$). Calculate the heat added.

    • (a) $501900\text{J}$
    • (b) $502000\text{J}$
    • (c) $502100\text{J}$
    • (d) $502200\text{J}$

12. 1 mole of an ideal gas expands isobarically at $P=2\text{atm}$ ($1\text{atm}=1.013\times {10}^5\text{Pa}$) from $V_1=0.015{\text{m}}^3$ to $V_2=0.025{\text{m}}^3$. Calculate the work done.

    • (a) $2024\text{J}$
    • (b) $2025\text{J}$
    • (c) $2026\text{J}$
    • (d) $2027\text{J}$

13. 3 moles of an ideal gas expand isothermally at 350 K from $V_1=0.02{\text{m}}^3$ to $V_2=0.04{\text{m}}^3$ ($R=8.314\text{J/mol·K}$). Calculate the heat added.

    • (a) $6049\text{J}$
    • (b) $6050\text{J}$
    • (c) $6051\text{J}$
    • (d) $6052\text{J}$

14. 1 mole of a monatomic ideal gas ($C_V=\frac{3}{2}R$) is heated at constant volume from 300 K to 340 K. Calculate the heat added.

    • (a) $498\text{J}$
    • (b) $499\text{J}$
    • (c) $500\text{J}$
    • (d) $501\text{J}$

15. A rocket engine gas expands adiabatically from $P_1=4\text{atm}$, $V_1=0.01{\text{m}}^3$ to $V_2=0.03{\text{m}}^3$ ($\gamma =1.4$). Calculate $P_2$.

    • (a) $0.91\text{atm}$
    • (b) $0.92\text{atm}$
    • (c) $0.93\text{atm}$
    • (d) $0.94\text{atm}$

16. Convert 50°C to Kelvin.

    • (a) $322\text{K}$
    • (b) $323\text{K}$
    • (c) $324\text{K}$
    • (d) $325\text{K}$

17. Convert 86°F to Celsius.

    • (a) $29\text{°C}$
    • (b) $30\text{°C}$
    • (c) $31\text{°C}$
    • (d) $32\text{°C}$

18. A slab ($k=0.25\text{W/m·K}$, $A=0.3{\text{m}}^2$, $L=0.02\text{m}$) has faces at 120°C and 40°C. Calculate the heat transfer rate.

    • (a) $149\text{W}$
    • (b) $150\text{W}$
    • (c) $151\text{W}$
    • (d) $152\text{W}$

19. A body at 727°C radiates with $e=0.7$, $A=0.05{\text{m}}^2$ ($\sigma =5.67\times {10}^{-8}{\text{W/m}}^2{\text{·K}}^4$, surroundings at 27°C). Calculate the net power radiated.

    • (a) $1491\text{W}$
    • (b) $1492\text{W}$
    • (c) $1493\text{W}$
    • (d) $1494\text{W}$

20. Mix 0.1 kg of water at 95°C with 0.2 kg at 25°C ($c_{\text{water}}=4186\text{J/kg·K}$). Calculate the final temperature.

    • (a) $47\text{°C}$
    • (b) $48\text{°C}$
    • (c) $49\text{°C}$
    • (d) $50\text{°C}$

21. Melt 0.6 kg of ice at 0°C to water at 0°C ($L_f=3.34\times {10}^5\text{J/kg}$). Calculate the heat required.

    • (a) $2.00\times {10}^5\text{J}$
    • (b) $2.01\times {10}^5\text{J}$
    • (c) $2.02\times {10}^5\text{J}$
    • (d) $2.03\times {10}^5\text{J}$

22. An aluminum rod ($L_0=1\text{m}$, $\alpha =24\times {10}^{-6}{\text{K}}^{-1}$) is heated from 20°C to 100°C. Calculate the length increase.

    • (a) $0.0019\text{m}$
    • (b) $0.0020\text{m}$
    • (c) $0.0021\text{m}$
    • (d) $0.0022\text{m}$

23. Heat 1.5 kg of water from 10°C to 60°C ($c_{\text{water}}=4186\text{J/kg·K}$). Calculate the heat added.

    • (a) $313900\text{J}$
    • (b) $314000\text{J}$
    • (c) $314100\text{J}$
    • (d) $314200\text{J}$

24. 2 moles of an ideal gas expand isobarically at $P=1.5\text{atm}$ ($1\text{atm}=1.013\times {10}^5\text{Pa}$) from $V_1=0.02{\text{m}}^3$ to $V_2=0.03{\text{m}}^3$. Calculate the work done.

    • (a) $1518\text{J}$
    • (b) $1519\text{J}$
    • (c) $1520\text{J}$
    • (d) $1521\text{J}$

25. 1 mole of an ideal gas expands isothermally at 400 K from $V_1=0.01{\text{m}}^3$ to $V_2=0.03{\text{m}}^3$ ($R=8.314\text{J/mol·K}$). Calculate the heat added.

    • (a) $3645\text{J}$
    • (b) $3646\text{J}$
    • (c) $3647\text{J}$
    • (d) $3648\text{J}$

26. 2 moles of a monatomic ideal gas ($C_V=\frac{3}{2}R$) are heated at constant volume from 310 K to 360 K. Calculate the heat added.

    • (a) $1246\text{J}$
    • (b) $1247\text{J}$
    • (c) $1248\text{J}$
    • (d) $1249\text{J}$

27. A gas expands adiabatically from $P_1=6\text{atm}$, $V_1=0.02{\text{m}}^3$ to $V_2=0.04{\text{m}}^3$ ($\gamma =1.4$). Calculate $P_2$.

    • (a) $2.27\text{atm}$
    • (b) $2.28\text{atm}$
    • (c) $2.29\text{atm}$
    • (d) $2.30\text{atm}$

28. A slab ($k=0.1\text{W/m·K}$, $A=0.6{\text{m}}^2$, $L=0.03\text{m}$) has faces at 150°C and 50°C. Calculate the heat transfer rate.

    • (a) $199\text{W}$
    • (b) $200\text{W}$
    • (c) $201\text{W}$
    • (d) $202\text{W}$

29. Mix 0.15 kg of water at 85°C with 0.25 kg at 35°C ($c_{\text{water}}=4186\text{J/kg·K}$). Calculate the final temperature.

    • (a) $53\text{°C}$
    • (b) $54\text{°C}$
    • (c) $55\text{°C}$
    • (d) $56\text{°C}$

30. A steel rod ($L_0=0.8\text{m}$, $\alpha =12\times {10}^{-6}{\text{K}}^{-1}$) is heated from 30°C to 130°C. Calculate the length increase.

    • (a) $0.00095\text{m}$
    • (b) $0.00096\text{m}$
    • (c) $0.00097\text{m}$
    • (d) $0.00098\text{m}$

31. Heat 0.5 kg of iron from 20°C to 100°C ($c_{\text{iron}}=450\text{J/kg·K}$). Calculate the heat added.

    • (a) $17900\text{J}$
    • (b) $18000\text{J}$
    • (c) $18100\text{J}$
    • (d) $18200\text{J}$

32. 1 mole of an ideal gas expands isobarically at $P=3\text{atm}$ ($1\text{atm}=1.013\times {10}^5\text{Pa}$) from $V_1=0.005{\text{m}}^3$ to $V_2=0.01{\text{m}}^3$. Calculate the work done.

    • (a) $1518\text{J}$
    • (b) $1519\text{J}$
    • (c) $1520\text{J}$
    • (d) $1521\text{J}$

33. A rocket component ($L_0=2\text{m}$, $\alpha =15\times {10}^{-6}{\text{K}}^{-1}$) is heated from 300 K to 900 K. Calculate the length increase.

    • (a) $0.0180\text{m}$
    • (b) $0.0181\text{m}$
    • (c) $0.0182\text{m}$
    • (d) $0.0183\text{m}$

34. 2 moles of an ideal gas expand isothermally at 500 K from $V_1=0.01{\text{m}}^3$ to $V_2=0.02{\text{m}}^3$ ($R=8.314\text{J/mol·K}$). Calculate the work done.

    • (a) $5758\text{J}$
    • (b) $5759\text{J}$
    • (c) $5760\text{J}$
    • (d) $5761\text{J}$

35. 1 mole of a diatomic ideal gas ($C_V=\frac{5}{2}R$) is heated at constant volume from 300 K to 320 K. Calculate the heat added.

    • (a) $415\text{J}$
    • (b) $416\text{J}$
    • (c) $417\text{J}$
    • (d) $418\text{J}$

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Conceptual Problems

36. What does the zeroth law of thermodynamics define?

• (a) Heat transfer
• (b) Thermal equilibrium and temperature
• (c) Work done by a system
• (d) Internal energy

37. What is the primary mechanism of heat transfer in solids?

• (a) Convection
• (b) Radiation
• (c) Conduction
• (d) Phase change

38. What does specific heat represent?

• (a) Heat required to change phase
• (b) Heat required to raise 1 kg of a substance by 1 K
• (c) Heat transfer rate
• (d) Thermal expansion

39. What happens during a phase change like melting?

• (a) Temperature increases
• (b) Temperature remains constant
• (c) Volume decreases
• (d) Internal energy decreases

40. What is the unit of thermal conductivity in SI units?

• (a) $\text{W/m·K}$
• (b) $\text{J/kg·K}$
• (c) ${\text{K}}^{-1}$
• (d) ${\text{W/m}}^2$

41. What happens to a material’s length when heated, assuming positive $\alpha$?

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

42. What does the first law of thermodynamics state?

• (a) Heat equals work
• (b) $\mathrm{\Delta }U=Q-W$
• (c) Temperature remains constant
• (d) Entropy increases

43. What is the physical significance of $P\mathrm{\Delta }V$ in thermodynamics?

• (a) Heat added
• (b) Work done at constant pressure
• (c) Internal energy change
• (d) Temperature change

44. What does an isothermal process imply?

• (a) $\mathrm{\Delta }U=0$
• (b) $Q=0$
• (c) $W=0$
• (d) $P=\text{constant}$

45. What is the dimension of specific heat?

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

46. What does a zero heat transfer rate in conduction indicate?

• (a) No temperature difference
• (b) Maximum thermal conductivity
• (c) No area for transfer
• (d) Infinite thickness

47. What is the significance of $\sigma Ae{T}^4$?

• (a) Conduction rate
• (b) Power radiated by a body
• (c) Work done
• (d) Heat capacity

48. What happens to internal energy in an adiabatic process?

• (a) Increases with heat added
• (b) Changes due to work done
• (c) Remains constant
• (d) Decreases with temperature

49. What does a constant volume process imply?

• (a) $W=0$
• (b) $Q=0$
• (c) $\mathrm{\Delta }U=0$
• (d) $P=\text{constant}$

50. How does thermal expansion affect rocket components?

• (a) Reduces length
• (b) Causes dimensional changes, impacting design
• (c) Increases thermal conductivity
• (d) Decreases heat transfer

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Derivation Problems

51. Derive the Kelvin-Celsius conversion $T(\text{K})=T(\text{°C})+273.15$.

52. Derive the Celsius-Fahrenheit conversion $T(\text{°F})=\frac{9}{5}T(\text{°C})+32$.

53. Derive the conduction equation $Q=\frac{kA\mathrm{\Delta }Tt}{L}$.

54. Derive the Stefan-Boltzmann law for net power $P_{\text{net}}=\sigma Ae(T^4-T_0^4)$.

55. Derive the heat equation $Q=mc\mathrm{\Delta }T$.

56. Derive the calorimetry equation for mixing two substances.

57. Derive the linear expansion formula $\mathrm{\Delta }L=L_0\alpha \mathrm{\Delta }T$.

58. Derive the first law of thermodynamics $\mathrm{\Delta }U=Q-W$.

59. Derive the work done in an isobaric process $W=P\mathrm{\Delta }V$.

60. Derive the work done in an isothermal process for an ideal gas $W=nRT\ln \left(\frac{V_2}{V_1}\right)$.

61. Derive the adiabatic relation $P{V}^{\gamma }=\text{constant}$.

62. Derive the thermal resistance formula $R=\frac{L}{kA}$.

63. Derive the internal energy change for an ideal gas $\mathrm{\Delta }U=n{C}_V\mathrm{\Delta }T$.

64. Derive the volume expansion formula $\mathrm{\Delta }V=V_0\beta \mathrm{\Delta }T$.

65. Derive the ideal gas thermometer relation $T\propto P$.

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NEET-style Conceptual Problems

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

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

67. What does thermal equilibrium imply?

• (a) No heat transfer between systems
• (b) Maximum heat transfer
• (c) Different temperatures
• (d) No work done

68. Which mechanism transfers heat via electromagnetic waves?

• (a) Conduction
• (b) Convection
• (c) Radiation
• (d) Phase change

69. What happens to temperature during a phase change?

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

70. What is the dimension of the coefficient of linear expansion?

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

71. What does the first law of thermodynamics conserve?

• (a) Temperature
• (b) Energy
• (c) Volume
• (d) Pressure

72. What is the role of thermal conductivity in heat transfer?

• (a) Measures resistance to heat flow
• (b) Measures ability to conduct heat
• (c) Measures heat capacity
• (d) Measures expansion

73. What happens to work done in an isochoric process?

• (a) Increases
• (b) Decreases
• (c) Is zero
• (d) Equals heat added

74. Why does thermal expansion occur in materials?

• (a) Due to increased particle vibration with temperature
• (b) Due to decreased density
• (c) Due to phase change
• (d) Due to heat loss

75. What is the unit of latent heat?

• (a) $\text{J/kg}$
• (b) $\text{J/kg·K}$
• (c) $\text{W/m·K}$
• (d) ${\text{K}}^{-1}$

76. What does a constant $PV=nRT$ indicate in a thermometer?

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

77. Which type of heat transfer occurs in fluids via bulk motion?

• (a) Conduction
• (b) Convection
• (c) Radiation
• (d) Phase change

78. What is the direction of heat flow in conduction?

• (a) From low to high temperature
• (b) From high to low temperature
• (c) Perpendicular to temperature gradient
• (d) Random

79. What does a pseudo-force do in a non-inertial frame for heat transfer?

• (a) Affects temperature gradient
• (b) Affects perceived heat flow
• (c) Creates work
• (d) Reduces internal energy

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 thermal expansion in rocket design?

• (a) Increases heat transfer
• (b) Causes structural changes, requiring design adjustments
• (c) Reduces internal energy
• (d) Increases work done

82. What happens to internal energy in an isothermal process for an ideal gas?

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

83. Why does heat transfer occur in a system?

• (a) Due to temperature difference
• (b) Due to volume change
• (c) Due to pressure change
• (d) Due to work done

84. What is the significance of $mL$ in thermodynamics?

• (a) Heat for temperature change
• (b) Heat for phase change
• (c) Work done
• (d) Internal energy

85. What is the unit of emissivity?

• (a) Dimensionless
• (b) ${\text{W/m}}^2{\text{·K}}^4$
• (c) $\text{J/kg}$
• (d) ${\text{K}}^{-1}$

86. What does a zero work done in a thermodynamic process indicate?

• (a) Isothermal process
• (b) Constant volume process
• (c) Adiabatic process
• (d) Isobaric process

87. What is the physical significance of $\frac{kA\mathrm{\Delta }T}{L}$?

• (a) Heat capacity
• (b) Heat transfer rate in conduction
• (c) Work done
• (d) Internal energy change

88. Why does an adiabatic process have no heat transfer?

• (a) $Q=0$ by definition
• (b) $\mathrm{\Delta }U=0$
• (c) $W=0$
• (d) $T=\text{constant}$

89. What is the dimension of thermal resistance?

• (a) $[\text{K}{\text{W}}^{-1}]$
• (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 first law apply to rocket engines?

• (a) Increases temperature
• (b) Governs energy conservation in expansion processes
• (c) Reduces heat transfer
• (d) Increases volume

91. What is the role of specific heat in calorimetry?

• (a) Measures phase change
• (b) Determines temperature change for given heat
• (c) Measures work done
• (d) Measures thermal expansion

92. What does a 0 K temperature theoretically indicate?

• (a) Maximum heat transfer
• (b) Zero particle motion
• (c) Maximum internal energy
• (d) No heat capacity

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

• (a) Heat capacity
• (b) Work done in an isothermal process
• (c) Internal energy change
• (d) Heat transfer rate

94. What is the dimension of molar heat capacity?

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

95. Why does heat capacity depend on the process (e.g., $C_P$ vs. $C_V$)?

• (a) Due to work done in different processes
• (b) Due to phase changes
• (c) Due to thermal expansion
• (d) Due to heat transfer

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NEET-style Numerical Problems

96. Convert 25°C to Kelvin.

• (a) $297\text{K}$
• (b) $298\text{K}$
• (c) $299\text{K}$
• (d) $300\text{K}$

97. A slab ($k=0.5\text{W/m·K}$, $A=0.2{\text{m}}^2$, $L=0.01\text{m}$) has faces at 80°C and 20°C. What is the heat transfer rate?

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

98. Mix 0.3 kg of water at 70°C with 0.2 kg at 20°C ($c_{\text{water}}=4186\text{J/kg·K}$). What is the final temperature?

• (a) $48\text{°C}$
• (b) $49\text{°C}$
• (c) $50\text{°C}$
• (d) $51\text{°C}$

99. A steel rod ($L_0=1.2\text{m}$, $\alpha =12\times {10}^{-6}{\text{K}}^{-1}$) is heated from 15°C to 115°C. What is the length increase?

• (a) $0.00143\text{m}$
• (b) $0.00144\text{m}$
• (c) $0.00145\text{m}$
• (d) $0.00146\text{m}$

100. 1 mole of an ideal gas expands isobarically at $P=1\text{atm}$ ($1\text{atm}=1.013\times {10}^5\text{Pa}$) from $V_1=0.01{\text{m}}^3$ to $V_2=0.02{\text{m}}^3$. What is the work done?
     - (a) $1012\text{J}$
     - (b) $1013\text{J}$
     - (c) $1014\text{J}$
     - (d) $1015\text{J}$

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