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feat: adds full text to maxwell relation problem statement

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Deepak Mallubhotla 2022-03-09 19:28:52 -06:00
parent c7443c0f2a
commit e9cee5abf6
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GPG Key ID: BEBAEBF28083E022
1 changed files with 36 additions and 1 deletions
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tex/3.11.tex
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@ -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}
@ -125,6 +140,26 @@
\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