[QUE/TH-03001] TH-PROBLEM

Consider \(N\) molecules of a gas obeying van der Waals
equation of state given by
\[\left(P+ \frac{a N^2}{V^2}\right)\big(V-Nb\big)
= Nk_B T\]
where \(a\) is a measure of the attractive forces between
the molecules and \(b\) is another constant proportional to
the size of a molecule. The other symbols have their
usual meanings. Show that during an isothermal expansion (Temperature is kept
constant) from volume \(V_1\) to
volume \(V_2\) quasi-statically and reversibly, the work done is
\[ W =-Nk_B T \log\left(\frac{V_2-Nb}{V_1-Nb}\right) + a^2
\Big(\frac{1}{V_1}-\frac{1}{V_2}\Big)\]