Write total differentials for $U=U(T,p),U(T,V),$ and $V=V(T,p).$ Insert the result for $V=V(T,p)$

into $U=U(T,V).$ Subtract this expression from the $U=U(T,p)$ expression to derive:

$(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{T}}{)}_p=(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_p+(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{T}}{)}_V$

Write $C_p-C_V$ in terms of their partial derivative definitions in terms of $H$ and $U.$ Substitute the

definition of $H.$ Insert the above equation to obtain Equation $5\mathrm{.}3$ in the notes:

$C_p-C_V=(p+(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T)(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_p$