Question

One mole of an ideal gas at temperature $T_{1}$ expands according to the law $\left(\right.P/V\left.\right) \, $ = constant. Find the work done when the final temperature becomes $T_{2}$ .

• A. $R\left(\right.T_{2}-T_{1}\left.\right)$
• B. $\left(R/2\right)\left(\right.T_{2}-T_{1}\left.\right)$
• C. $\left(R/4\right)\left(\right.T_{2}-T_{1}\left.\right)$
• D. $PV\left(\right.T_{2}-T_{1}\left.\right)$

Answer 1

Answer: (B)

$W=\displaystyle \int _{V_{1}}^{V_{2}}PdV=\displaystyle \int _{\text{V}_{1}}^{\text{V}_{2}}KVdV$ $\left(\because \frac{\textit{p}}{\textit{V}} = \textit{K} = \text{constant}\right)$
$\therefore \textit{W}=\frac{1}{2}\textit{k}\left(\textit{V}_{2}^{2} - \textit{V}_{1}^{2}\right)$
$PV=RT$
But $P=RT$
$\therefore \textit{KV}^{2}=\textit{RT}$
or $K\left(V_{2}^{2} - V_{1}^{2}\right)=R\left(T_{2} - T_{1}\right)$
$\therefore W=\frac{R}{2}\left(T_{2} - T_{1}\right)$