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Specific Heat Capacities of Gases — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsSpecific Heat Capacities of Gases1 Mark
MCQ
Work done by a sample of an ideal gas in a process $A$ is double the work done in another process $B.$ The temperature rises through the same amount in the two processes. If $C_A$ and $C_B$ be the molar heat capacities for the two processes:
  • A
    $C_A = C_B$
  • B
    $C_A < C_B$
  • ✓
    $C_A > C_B$
  • D
    $C_A$ and $C_B$ cannot be defined.
✓

Answer

Correct option: C.
$C_A > C_B$
According to the first law of thermodynamics, $\triangle\text{Q}=\triangle\text{U}+\triangle\text{W},$
where $\triangle\text{Q}$ is the heat supplied to the system when $\triangle\text{W}$ work is done on the system and $\triangle\text{U}$ is the change in internal energy produced.
Since the temperature rises by the same amount in both processes, change in internal energies are same,
i.e. $\triangle\text{U}_\text{A}=​​​​\triangle\text{U}_\text{B}.$
But as, $\triangle\text{W}_\text{A}=​​​​\triangle\text{W}_\text{B}$ this gives $\triangle\text{Q}_\text{A}=2\triangle\text{Q}_\text{B}.$
Now, molar heat capacity of a gas, $\text{C}=\frac{\triangle\text{Q}}{\text{n}\triangle\text{T}},$
where $\triangle\frac{\text{Q}}{\text{n}}$ is the heat supplied to a mole of gas and $\triangle\text{T}$ is the change in temperature produced.
As $\triangle\text{Q}_\text{A}=2\triangle\text{Q}_\text{B},\text{C}_\text{A}>\text{C}_\text{B}.$

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