X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F2_04s.tex;h=b03977cab07536a0db1e5f909f0e5ae4e24e524f;hp=7391ad03ebcc7da137b4e0309d53929564a3d67b;hb=183bd2b78445842ee859e2f10f8ab9dc84cf2776;hpb=75000006ad7eda5e22f1ca8fb47b619eb92c3cac

diff --git a/solid_state_physics/tutorial/2_04s.tex b/solid_state_physics/tutorial/2_04s.tex
index 7391ad0..b03977c 100644
--- a/solid_state_physics/tutorial/2_04s.tex
+++ b/solid_state_physics/tutorial/2_04s.tex
@@ -215,25 +215,71 @@
              {\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}