Question

$70 \,cal $ of heat are required to raise the temperature of $2$ moles of an ideal gas at constant pressure from $30^{\circ} C$ to $35^{\circ} C$. The amount of heat required in calories to raise the temperature of the same gas through the same range $\left(30^{\circ} C -35^{\circ} C \right)$ at constant volume is :

• A. 30 cal
• B. 50 cal
• C. 370 cal
• D. 90 cal

Answer 1

Answer: (B)

The heat required to raise the temperature of gas,
at constant volume $\left(C_{V}\right)$ is
$Q_{V}=C_{V} \Delta T \ldots$ (i)
and at constant pressure is
$Q_{P}=C_{P} \Delta T \ldots$ (ii)
Also, $\gamma=\frac{C_{P}}{C_{V}} \ldots$ (iii)
From Eqs. (i), (ii) and (iii), we get
$\frac{Q_{V}}{Q_{P}}=\frac{C_{P}}{C_{V}}=\frac{1}{\gamma}$
For a diatomic gas $\gamma=\frac{7}{5}$
$\therefore Q_{V}=\frac{5}{7} \times 70=50\, cal$