Three moles of a gas at a temperature ' $\mathrm{T}$ ' is heated to thrice its volume by keeping the…
Three moles of a gas at a temperature ' $\mathrm{T}$ ' is heated to thrice its volume by keeping the pressure constant. If ' $\gamma$ is the ratio of specific heats, then the increase in internal energy of the gas is
$\frac{3 \mathrm{RT}}{\gamma-1}$
$\frac{6 \mathrm{RT}}{\gamma-1}$
$\frac{8 \mathrm{R}}{\gamma-1}$
$\frac{3 \mathrm{R}}{2(\gamma-1)}$
Solution
Number of moles, $\mathrm{n}=3$
Initial volume, $\mathrm{V}_{\mathrm{i}}=\mathrm{V}$
Final volume, $\mathrm{V}_{\mathrm{f}}=3 \mathrm{~V}$
Ratio of specific heat, $\gamma=\frac{C_p}{C_v}$
Change in internal energy, $\Delta \mathrm{V}=\frac{\mathrm{nfR} \Delta \mathrm{T}}{2}$
Degree of freedom;
$
\mathrm{f}=\frac{2}{\gamma-1}
$
At constant pressure,
$
\frac{\mathrm{V}_1}{\mathrm{~T}_1}=\frac{\mathrm{V}_2}{\mathrm{~T}_2} \Rightarrow \frac{\mathrm{V}}{\mathrm{T}}=\frac{3 \mathrm{~V}}{\mathrm{~T}_2} \Rightarrow \mathrm{T}_2=3 \mathrm{~T}
$
Substitute $\mathrm{T}_2$ in equation (1)
$
\begin{aligned}
\Delta \mathrm{V} & =\frac{\mathrm{nfR}}{2}\left(\mathrm{~T}_2-\mathrm{T}_1\right) \\
=\frac{3}{2} & \times \frac{2 \mathrm{R}}{\gamma-1}(3 \mathrm{~T}-\mathrm{T}) \\
& \Rightarrow \frac{3 \mathrm{R} \times 2 \mathrm{~T}}{\gamma-1} \\
\Delta \mathrm{U} & =\frac{6 \mathrm{RT}}{\gamma-1}
\end{aligned}
$