# The volume of a van der Waals gas changes from V1 to V2 at constant temperature. The work done in the process is

Q: The volume of a van der Waal gas changes from V1 to V2 at constant temperature. The work done in the process is

(a) $\displaystyle n R T \; ln(\frac{V_2 – n b}{V_1 – n b}) + a n^2 \frac{V_1 – V_2}{V_1 V_2}$

(b) $\displaystyle n R T \; ln(\frac{V_2 – n a}{V_1 – n a}) + b n^2 \frac{V_1 – V_2}{V_1 V_2}$

(c) $\displaystyle n R T \; ln(\frac{V_2 – n a}{V_1 – n a}) + n^2 \frac{V_1 – V_2}{V_1 V_2}$

(d) $\displaystyle n R T \; ln(\frac{V_2 – n b}{V_1 – n b}) + a n^2 \frac{V_1 V_2}{V_1 – V_2}$

Ans: (a)
Sol: For van der Waal gas ,

$\displaystyle ( p + \frac{a n^2}{V^2}) (V – n b) = n R T$

$\displaystyle p = \frac{n R T}{V – n b} – \frac{a n^2}{V^2}$

Work done $\displaystyle = \int_{V_1}^{V_2} p dV$

$\displaystyle = \int_{V_1}^{V_2} (\frac{n R T}{V – n b} – \frac{a n^2}{V^2} ) dV$

$\displaystyle = n R T \; ln(\frac{V_2 – n b}{V_1 – n b}) + a n^2 \frac{V_1 – V_2}{V_1 V_2}$