{\left.\frac{\partial p}{\partial V}\right|_T}=0
\]
\item $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$
- and $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$
+ and $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$\\
\[
- C_p-C_V=\left.\frac{\partial E}{\partial T}\right|_p-
- \left.\frac{\partial E}{\partial T}\right|_V=
- \frac{\partial E}{\partial S}
+ C_p=\left.\frac{\partial H}{\partial T}\right|_p=
+ \left.\frac{\partial H}{\partial S}\right|_p
+ \left.\frac{\partial S}{\partial T}\right|_p=
+ T\left.\frac{\partial S}{\partial T}\right|_p
+ \]
+ \[
+ C_V=\left.\frac{\partial E}{\partial T}\right|_V=
+ \left.\frac{\partial E}{\partial S}\right|_V
+ \left.\frac{\partial S}{\partial T}\right|_V=
+ T\left.\frac{\partial S}{\partial T}\right|_V
+ \]
+ \[
+ \Rightarrow C_p-C_V=
+ T\left.\frac{\partial S}{\partial T}\right|_p-
+ T\left.\frac{\partial S}{\partial T}\right|_V=
+ T\left(
\left.\frac{\partial S}{\partial T}\right|_p-
- \frac{\partial E}{\partial S}
+ \left.\frac{\partial S}{\partial T}\right|_V
+ \right)
+ \]
+ Using the equality
+ \[
+ dS=\left.\frac{\partial S}{\partial T}\right|_p dT
+ +\left.\frac{\partial S}{\partial p}\right|_T dp
+ \Rightarrow
\left.\frac{\partial S}{\partial T}\right|_V=
- T\left.\frac{\partial S}{\partial T}\right|_p-
- T\left.\frac{\partial S}{\partial T}\right|_V
+ \left.\frac{\partial S}{\partial T}\right|_p+
+ \left.\frac{\partial S}{\partial p}\right|_T
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \]
+ 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:
+ \[
+ C_p-C_V=T\left(
+ -\left.\frac{\partial S}{\partial p}\right|_T
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \right)=T\left(
+ \left.\frac{\partial V}{\partial T}\right|_p
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \right)=T\left(
+ \frac{\left.\left.\frac{\partial V}{\partial T}\right|_p\right.^2}
+ {-\left.\frac{\partial V}{\partial p}\right|_T}
+ \right)=T\left(\frac{V^2\alpha_V^2}{V\frac{1}{B}}\right)=
+ TVB\alpha_V^2
\]
- Find an expression for $C_p-C_V$ in terms of temperature $T$,
- volume $V$, the coefficient of thermal expansion $\alpha_V$ and
- the inverse bulk modulus (isothermal compressibility)
- $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\
- $C_p=\left.\frac{\partial E}{\partial T}\right|_p$ is the heat capacity
- for constant pressure and
- $C_V=\left.\frac{\partial E}{\partial T}\right|_V$ is the heat capacity
- for constant volume.
+ For a rigorously harmonic potential $C_p=C_V$.
\end{enumerate}
\end{document}