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% header
\begin{center}
 {\LARGE {\bf Materials Physics II}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 SS 2008\\
 \vspace{8pt}
 {\Large\bf Tutorial 4}
\end{center}

\vspace{8pt}

\section{Legendre transformation and Maxwell relations}

\begin{enumerate}
 \item Consider the total differential
       \[
       df= \sum_{i=1}^{n} u_i dx_i
       \]
       with the state function $f=f(x_1,\ldots,x_n)$ and its partial derivatives
       $u_i=\frac{\partial f}{\partial x_i}$.
       Rewrite the total differential of the function $g$ defined as
       \[
       g=f-\sum_{i=r+1}^{n} u_i x_i
       \]
       in such a way that $g$ is immediately identified to be a function of
       the variables $x_1,\ldots,x_r$ and $u_{r+1},\ldots,u_n$,
       where $u_i$ is called the conjugate variable of $x_i$.
       The transformation is called Legendre transformation.
 \item By taking the derivatives of transformed thermodynamic potentials
       with respect to the variables they depend on,
       relations between intensive and extensive variables can be gained.

       Start with the internal energy $E=E(S,V)$.
       Write down the total differential using the equalities
       $T=\left.\frac{\partial E}{\partial S}\right|_V$ and
       $-p=\left.\frac{\partial E}{\partial V}\right|_S$.
       Find more relations by doing the transformation to the potentials
       \begin{itemize}
        \item $H=E+pV$ (Enthalpy)
        \item $F=E-TS$ (Helmholtz free energy)
        \item $G=H-TS=E+pV-TS$ (Gibbs free energy)
       \end{itemize}
       and taking the appropriate derivatives.
 \item For a thermodynamic potential $\Phi(X,Y)$ the following identity
       expressing the permutability of derivatives holds:
       \[
       \frac{\partial^2 \Phi}{\partial X \partial Y} =
       \frac{\partial^2 \Phi}{\partial Y \partial X}
       \]
       Derive the Maxwell relations by taking the mixed derivatives of the
       potentials in (b) with respect to the variables they depend on.
       Exchange the sequence of derivation and use the identities gained in (b).
\end{enumerate}

\section{Thermal expansion of solids}

It is well known that solids change their length $L$ and volume $V$ respectively
if there is a change in temperature $T$ or in pressure $p$ of the system.

\begin{enumerate}
 \item The coefficient of thermal expansion of a solid is given by
       $\alpha_L=\frac{1}{L}\left.\frac{\partial L}{\partial T}\right|_p$.
       Show that the coefficient of thermal expansion of the volume
       $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$
       equals $3\alpha_L$ for isotropic materials.
 \item 
 \item
\end{enumerate}

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