A heat pump operates between two identical bodies of specific heat \(C\) at \(T_{1}\). The operation of the pump cools down one of the bodies to \(T_{2}\). Show that for the operation of pump the minimum work input is given by:

\[W_{\min }=C\left[\frac{T_{1}^{2}}{T_{2}}+T_{2}-2 T_{1}\right]\]