- dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial}{\partial V}\right|T
- \]
- \[
- dE=TdS-pdV \Rightarrow
- \]
- Find an expression for the pressure as a function of the free energy
- $F=E-TS$.
- Rewrite this equation to express the pressure entirely in terms of
- the internal energy $E$.
- Evaluate the pressure by using the harmonic form of the internal energy.
- {\bf Hint:}
- Step 2 introduced an integral over the temperature $T'$.
- Change the integration variable $T'$ to $x=\hbar\omega_s({\bf k})/T'$.
- Use integration by parts with respect to $x$.
- \item The normal mode frequencies of a rigorously harmonic crystal
- are unaffected by a change in volume.
- What does this imply for the pressure
- (Which variables does the pressure depend on)?
- Draw conclusions for the coefficient of thermal expansion.
- \item 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.
+ dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial F}{\partial V}\right|T
+ \]
+ \[
+ \left.\frac{\partial E}{\partial T}\right|_V=
+ \left.\frac{\partial E}{\partial S}\right|_V
+ \left.\frac{\partial S}{\partial T}\right|_V=
+ T\left.\frac{\partial S}{\partial T}\right|_V
+ \Rightarrow
+ \left.\frac{\partial S}{\partial T}\right|_V=
+ \frac{1}{T}\left.\frac{\partial E}{\partial T}\right|_V
+ \]
+ \[
+ \textrm{Using } F=E-TS \textrm{ and }
+ TS=T\int_0^T\frac{\partial S}{\partial T'}dT'
+ \textrm{ (Entropy density vanishes at $T=0$)}
+ \]
+ \[
+ \Rightarrow
+ p=-\frac{\partial}{\partial V}\left(
+ E-T\int_0^T\frac{dT'}{T'}\frac{\partial E}{\partial T'}
+ \right)
+ \]
+ Harmonic approximation of the internal energy:
+ \[
+ E=E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})+
+ \sum_{{\bf k}s}
+ \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
+ \]
+ \[
+ \ldots
+ \]
+ \[
+ x=\hbar\omega_s({\bf k})/T'
+ \]
+ \[
+ \ldots
+ \]
+ \[
+ \Rightarrow
+ p=-\frac{\partial}{\partial V}\left(
+ E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
+ \right)+
+ \sum_{{\bf k}s}\left(-\frac{\partial}{\partial V}\hbar\omega_s({\bf k})
+ \right)\frac{1}{e^{\beta\hbar\omega_s({\bf k})}-1}
+ \]
+ \item The pressure depends on temperature
+ only if the normal mode frequencies depend on the volume.
+ However, the normal mode frequencies of a rigorously harmonic crystal
+ are unaffected by a change in volume.\\
+ $\Rightarrow$
+ The pressure solely depends on the volume.\\
+ $\Rightarrow$
+ The pressure required to maintain a given volume
+ does not vary with temperature.
+ \[
+ \left.\frac{\partial p}{\partial T}\right|_V=0
+ \]
+ \[
+ \left.\frac{\partial V}{\partial T}\right|_p=
+ -\frac{\left.\frac{\partial p}{\partial T}\right|_V}
+ {\left.\frac{\partial p}{\partial V}\right|_T}=0
+ \]
+ \item $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$
+ and $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$\\
+ \[
+ C_p=\left.\frac{\partial H}{\partial T}\right|_p=
+ \left.\frac{\partial H}{\partial S}\right|_p
+ \left.\frac{\partial S}{\partial T}\right|_p=
+ T\left.\frac{\partial S}{\partial T}\right|_p
+ \]
+ \[
+ C_V=\left.\frac{\partial E}{\partial T}\right|_V=
+ \left.\frac{\partial E}{\partial S}\right|_V
+ \left.\frac{\partial S}{\partial T}\right|_V=
+ T\left.\frac{\partial S}{\partial T}\right|_V
+ \]
+ \[
+ \Rightarrow C_p-C_V=
+ T\left.\frac{\partial S}{\partial T}\right|_p-
+ T\left.\frac{\partial S}{\partial T}\right|_V=
+ T\left(
+ \left.\frac{\partial S}{\partial T}\right|_p-
+ \left.\frac{\partial S}{\partial T}\right|_V
+ \right)
+ \]
+ Using the equality
+ \[
+ dS=\left.\frac{\partial S}{\partial T}\right|_p dT
+ +\left.\frac{\partial S}{\partial p}\right|_T dp
+ \Rightarrow
+ \left.\frac{\partial S}{\partial T}\right|_V=
+ \left.\frac{\partial S}{\partial T}\right|_p+
+ \left.\frac{\partial S}{\partial p}\right|_T
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \]
+ 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:
+ \[
+ C_p-C_V=T\left(
+ -\left.\frac{\partial S}{\partial p}\right|_T
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \right)=T\left(
+ \left.\frac{\partial V}{\partial T}\right|_p
+ \left.\frac{\partial p}{\partial T}\right|_V
+ \right)=T\left(
+ \frac{\left.\left.\frac{\partial V}{\partial T}\right|_p\right.^2}
+ {-\left.\frac{\partial V}{\partial p}\right|_T}
+ \right)=T\left(\frac{V^2\alpha_V^2}{V\frac{1}{B}}\right)=
+ TVB\alpha_V^2
+ \]
+ For a rigorously harmonic potential $C_p=C_V$.