From e9cee5abf62ebe5d00ad6aba1185af5eb4ea3d73 Mon Sep 17 00:00:00 2001
From: Deepak Mallubhotla <dmallubhotla+github@gmail.com>
Date: Wed, 9 Mar 2022 19:28:52 -0600
Subject: [PATCH] feat: adds full text to maxwell relation problem statement

---
 tex/3.11.tex | 37 ++++++++++++++++++++++++++++++++++++-
 1 file changed, 36 insertions(+), 1 deletion(-)

diff --git a/tex/3.11.tex b/tex/3.11.tex
index b16e852..3d6349d 100644
--- a/tex/3.11.tex
+++ b/tex/3.11.tex
@@ -62,7 +62,22 @@
 			\left. \pdv{C_P}{P} \right|_{T} = - T \left. \pdv[2]{V}{T} \right|_{P}
 		\end{equation}
 		\item \begin{equation}
-			\left. \pdv{E}{T} \right|_{P} = - \frac{T}{C_P} \left. \pdv{S}{P} \right|_{T}
+			\left. \pdv{E}{P} \right|_{T} = - T \left. \pdv{V}{T} \right|_{P} - P \left. \pdv{V}{P} \right|_T
+		\end{equation}
+		\item \begin{equation}
+			\left. \pdv{E}{T} \right|_{P} = C_P - P \left. \pdv{V}{T} \right|_P
+		\end{equation}
+		\item \begin{equation}
+			\left. \pdv{T}{V} \right|_{S} =  - \frac{T}{C_V} \left. \pdv{S}{V} \right|_T
+		\end{equation}
+		\item \begin{equation}
+			\left. \pdv{T}{P} \right|_{S} = - \frac{T}{C_P} \left. \pdv{S}{P} \right|_{T}
+		\end{equation}
+		\item \begin{equation}
+			\left. \pdv{V}{T} \right|_{P} = - \frac{C_P}{T} \left. \pdv{T}{V} \right|_{S} \left. \pdv{V}{P} \right|_S
+		\end{equation}
+		\item \begin{equation}
+			\left. \pdv{P}{V} \right|_{S} = \left. \pdv{P}{V} \right|_{T} - \frac{T}{C_V} \left( \left. \pdv{P}{T} \right|_V \right)^2
 		\end{equation}
 	\end{enumerate}
 	
@@ -124,7 +139,27 @@
 	\begin{equation}
 		\left. \pdv{C_P}{P} \right|_{T} = - T \left. \pdv[2]{V}{T} \right|_{P}
 	\end{equation}
+	
+	By definition:
+	\begin{equation}
+		C_P = T \left. \pdv{S}{T} \right|_{P}
+	\end{equation}
+	
+	Let's start by looking at the expression and do the commutativity of derivatives thing (again subscripting for what we're holding constant).
+	\begin{align}
+		T \pdv[2]{S}{P_T}{T_P} &= T \pdv[2]{S}{T_P}{P_T} \\
+		\pdv{C_P}{P_T} &= T \pdv[2]{S}{T_P}{P_T} \\
+		\pdv{C_P}{P_T} &= - T \pdv[2]{V}{T_P}
+	\end{align}
+	where on the right hand side we've used our result from part (a).
 
+	\subsection{(c)}
+	
+	Want
+	\begin{equation}
+		\left. \pdv{E}{T} \right|_{P} = - \frac{T}{C_P} \left. \pdv{S}{P} \right|_{T}
+	\end{equation}
+	
 	\newpage
 	\listoftodos