Van der Waals equation — UPSC Physics Optional IFoS 2018 Paper I

Consider N molecules of a gas obeying van der Waals' equation of state given by \left( P + \frac{a N^2}{V^2} \right) (V - Nb) = N k_B T where a is a measure of the attractive forces between the molecules and b is another constant proportional to the size of the molecules. The other symbols have their usual meanings. Show that during an isothermal expansion from volume V_1 to volume V_2 quasi-statically and reversibly, the work done is W = -N k_B T \log \left( \frac{V_2 - Nb}{V_1 - Nb} \right) + a N^2 \left( \frac{1}{V_1} - \frac{1}{V_2} \right)