+ \left.\frac{\partial S}{\partial T}\right|_p+
+ \left.\frac{\partial S}{\partial p}\right|_T
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \]
+ and the Maxwell relation
+ \[
+ \left.\frac{\partial S}{\partial p}\right|_T=
+ -\left.\frac{\partial V}{\partial T}\right|_p
+ \]
+ and the equality
+ \[
+ dV=\left.\frac{\partial V}{\partial T}\right|_p dT+
+ \left.\frac{\partial V}{\partial p}\right|_T dp
+ \stackrel{\left.\frac{\partial}{\partial T}\right|_V}{\Rightarrow}
+ 0=\left.\frac{\partial V}{\partial T}\right|_p+
+ \left.\frac{\partial V}{\partial p}\right|_T
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \Rightarrow
+ \left.\frac{\partial p}{\partial T}\right|_V=
+ -\frac{\left.\frac{\partial V}{\partial T}\right|_p}
+ {\left.\frac{\partial V}{\partial p}\right|_T}
+ \]
+ we obtain:
+ \[
+ C_p-C_V=T\left(
+ -\left.\frac{\partial S}{\partial p}\right|_T
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \right)=T\left(
+ \left.\frac{\partial V}{\partial T}\right|_p
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \right)=T\left(
+ \frac{\left.\left.\frac{\partial V}{\partial T}\right|_p\right.^2}
+ {-\left.\frac{\partial V}{\partial p}\right|_T}
+ \right)=T\left(\frac{V^2\alpha_V^2}{V\frac{1}{B}}\right)=
+ TVB\alpha_V^2