Show that the first-order Joule-Thomson coefficient of a gas is given by the formula

\[\left(\frac{\partial T}{\partial P}\right)_{H}=\frac{N}{C_{P}}\left(T \frac{\partial\left(a_{2} \lambda^{3}\right)}{\partial T}-a_{2} \lambda^{3}\right)\]

where \(a_{2}(T)\) is the second virial coefficient of the gas and \(H\) its enthalpy; see equation (10.2.1). Derive an explicit expression for the Joule-Thomson coefficient in the case of a gas with interparticle interaction \[u(r)=\left\{\begin{array}{cl}+\infty & \text { for } 0and discuss the temperature dependence of this coefficient.