Question

One mole of an ideal gas at temperature $T_1$ expands according to the law $\frac{P}{V^2}=a$ (constant). The work done by the gas till temperature of gas becomes $T_2$ is

• A. $\frac{1}{2}R(T_2-T_1)$
• B. $\frac{1}{3}R(T_2-T_1)$
• C. $\frac{1}{4}R(T_2-T_1)$
• D. $\frac{1}{5}R(T_2-T_1)$

Answer 1

The correct option is B $\frac{1}{3}R(T_2-T_1)$

Work done $W=\underset{v_1}{\overset{v_2}{\int }}PdV=\underset{v_1}{\overset{v_2}{\int }}a{V}^{2}dV$
$W=a{\left[\frac{V^3}{3}\right]}_{V_1}^{V_2}=\frac{a}{3}(V_2^3-V_1^3)$
As $P=a{V}^2$
$W=\frac{1}{3}[P_2{V}_2-P_1{V}_1]=\frac{1}{3}\left[nR\Delta T\right]=\frac{1}{3}R(T_2-T_1)$