finished last exercise, integral still needs to be done

diff --git a/solid_state_physics/tutorial/2_04s.tex b/solid_state_physics/tutorial/2_04s.tex
index c115ade..b03977c 100644 (file)
--- a/solid_state_physics/tutorial/2_04s.tex
+++ b/solid_state_physics/tutorial/2_04s.tex
@@ -215,14 +215,21 @@
              {\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 $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$\\
        \[
-       C_p-C_V=\left.\frac{\partial E}{\partial T}\right|_p-
-       \left.\frac{\partial E}{\partial T}\right|_V=
-       \frac{\partial E}{\partial S}
-       \left.\frac{\partial S}{\partial T}\right|_p-
-       \frac{\partial E}{\partial S}
-       \left.\frac{\partial S}{\partial T}\right|_V=
+       C_p=\left.\frac{\partial H}{\partial T}\right|_p=
+           \left.\frac{\partial H}{\partial S}\right|_p
+           \left.\frac{\partial S}{\partial T}\right|_p=
+          T\left.\frac{\partial S}{\partial T}\right|_p
+       \]
+       \[
+       C_V=\left.\frac{\partial E}{\partial T}\right|_V=
+           \left.\frac{\partial E}{\partial S}\right|_V
+           \left.\frac{\partial S}{\partial T}\right|_V=
+           T\left.\frac{\partial S}{\partial T}\right|_V
+       \]
+       \[
+       \Rightarrow C_p-C_V=
        T\left.\frac{\partial S}{\partial T}\right|_p-
        T\left.\frac{\partial S}{\partial T}\right|_V=
        T\left(
@@ -238,17 +245,21 @@
        \left.\frac{\partial S}{\partial T}\right|_V=
        \left.\frac{\partial S}{\partial T}\right|_p+
        \left.\frac{\partial S}{\partial p}\right|_T
-       \left.\frac{\partial p}{\partial T}\right|_V,
+       \left.\frac{\partial p}{\partial T}\right|_V
        \]
-       the Maxwell relation
+       and 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)
+       and the equality
        \[
-       0=dV=\left.\frac{\partial V}{\partial T}\right|_p dT+
+       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}