- \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 Find an expression for the pressure as a function of the free energy
+ \item Coefficients of thermal expansion:\\
+ Consider a cube with side lengthes $L_1,L_2,L_3$.
+ Isotropic material: $\frac{1}{L_1}\frac{\partial L_1}{\partial T}=
+ \frac{1}{L_2}\frac{\partial L_2}{\partial T}=
+ \frac{1}{L_3}\frac{\partial L_3}{\partial T}=
+ \alpha_L$.
+ \begin{eqnarray}
+ \alpha_V&=&\frac{1}{V}\frac{\partial V}{\partial T}=
+ \frac{1}{L_1L_2L_3}\frac{\partial}{\partial T}(L_1L_2L_3)=
+ \frac{1}{L_1L_2L_3}\left(L_2L_3\frac{\partial L_1}{\partial T}+
+ L_1L_3\frac{\partial L_2}{\partial T}+
+ L_1L_2\frac{\partial L_3}{\partial T}\right)
+ \nonumber\\
+ &=&\frac{1}{L_1}\frac{\partial L_1}{\partial T}+
+ \frac{1}{L_2}\frac{\partial L_2}{\partial T}+
+ \frac{1}{L_3}\frac{\partial L_3}{\partial T}=3\alpha_L\nonumber
+ \end{eqnarray}
+ \item
+ Find an expression for the pressure as a function of the free energy