nearly finished tut 4 sol

[lectures/latex.git] / solid_state_physics / tutorial / 2_04s.tex

diff --git a/solid_state_physics/tutorial/2_04s.tex b/solid_state_physics/tutorial/2_04s.tex
index 7391ad0..c115ade 100644 (file)
--- a/solid_state_physics/tutorial/2_04s.tex
+++ b/solid_state_physics/tutorial/2_04s.tex
@@ -224,16 +224,51 @@
        \frac{\partial E}{\partial S}
        \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
+       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
+       \[
+       dS=\left.\frac{\partial S}{\partial T}\right|_p dT
+       +\left.\frac{\partial S}{\partial p}\right|_T dp
+       \Rightarrow
+       \left.\frac{\partial S}{\partial T}\right|_V=
+       \left.\frac{\partial S}{\partial T}\right|_p+
+       \left.\frac{\partial S}{\partial p}\right|_T
+       \left.\frac{\partial p}{\partial T}\right|_V,
+       \]
+       the Maxwell relation
+       \[
+       \left.\frac{\partial S}{\partial p}\right|_T=
+       -\left.\frac{\partial V}{\partial T}\right|_p
+       \]
+       and (for a process with constant volume)
+       \[
+       0=dV=\left.\frac{\partial V}{\partial T}\right|_p dT+
+       \left.\frac{\partial V}{\partial p}\right|_T dp
+       \Rightarrow
+       \left.\frac{\partial p}{\partial T}\right|_V=
+       -\frac{\left.\frac{\partial V}{\partial T}\right|_p}
+       {\left.\frac{\partial V}{\partial p}\right|_T}
+       \]
+       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}