- C_p-C_V=\left.\frac{\partial E}{\partial T}\right|_p-
- \left.\frac{\partial E}{\partial T}\right|_V=
- \frac{\partial E}{\partial S}
- \left.\frac{\partial S}{\partial T}\right|_p-
- \frac{\partial E}{\partial S}
- \left.\frac{\partial S}{\partial T}\right|_V=
+ C_p=\left.\frac{\partial H}{\partial T}\right|_p=
+ \left.\frac{\partial H}{\partial S}\right|_p
+ \left.\frac{\partial S}{\partial T}\right|_p=
+ T\left.\frac{\partial S}{\partial T}\right|_p
+ \]
+ \[
+ C_V=\left.\frac{\partial E}{\partial T}\right|_V=
+ \left.\frac{\partial E}{\partial S}\right|_V
+ \left.\frac{\partial S}{\partial T}\right|_V=
+ T\left.\frac{\partial S}{\partial T}\right|_V
+ \]
+ \[
+ \Rightarrow C_p-C_V=