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34 {\LARGE {\bf Materials Physics II}\\}
39 {\Large\bf Tutorial 4}
44 \section{Legendre transformation and Maxwell relations}
47 \item Consider the total differential
49 df= \sum_{i=1}^{n} u_i dx_i
51 with the state function $f=f(x_1,\ldots,x_n)$ and its partial derivatives
52 $u_i=\frac{\partial f}{\partial x_i}$.
53 Rewrite the total differential of the function $g$ defined as
55 g=f-\sum_{i=r+1}^{n} u_i x_i
57 in such a way that $g$ is immediately identified to be a function of
58 the variables $x_1,\ldots,x_r$ and $u_{r+1},\ldots,u_n$,
59 where $u_i$ is called the conjugate variable of $x_i$.
60 The transformation is called Legendre transformation.
61 \item By taking the derivatives of transformed thermodynamic potentials
62 with respect to the variables they depend on,
63 relations between intensive and extensive variables can be gained.
65 Start with the internal energy $E=E(S,V)$.
66 Write down the total differential using the equalities
67 $T=\left.\frac{\partial E}{\partial S}\right|_V$ and
68 $-p=\left.\frac{\partial E}{\partial V}\right|_S$.
69 Find more relations by doing the transformation to the potentials
71 \item $H=E+pV$ (Enthalpy)
72 \item $F=E-TS$ (Helmholtz free energy)
73 \item $G=H-TS=E+pV-TS$ (Gibbs free energy)
75 and taking the appropriate derivatives.
76 \item For a thermodynamic potential $\Phi(X,Y)$ the following identity
77 expressing the permutability of derivatives holds:
79 \frac{\partial^2 \Phi}{\partial X \partial Y} =
80 \frac{\partial^2 \Phi}{\partial Y \partial X}
82 Derive the Maxwell relations by taking the mixed derivatives of the
83 potentials in (b) with respect to the variables they depend on.
84 Exchange the sequence of derivation and use the identities gained in (b).
87 \section{Thermal expansion of solids}
89 It is well known that solids change their length $L$ and volume $V$ respectively
90 if there is a change in temperature $T$ or in pressure $p$ of the system.
93 \item The coefficient of thermal expansion of a solid is given by
94 $\alpha_L=\frac{1}{L}\left.\frac{\partial L}{\partial T}\right|_p$.
95 Show that the coefficient of thermal expansion of the volume
96 $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$
97 equals $3\alpha_L$ for isotropic materials.