Module 2 : Normal metals
Lecture 1 : Electrical conductivity and heat capacity followed by problem solving
 
For ideal metals, the thermal conductivity due to electrons is given by $\frac{1}{3}$$v^{2}$$\tau$$c_{v}$ where $v^{2}$ is the mean square electronic speed, $\tau$ is the relaxation time, and $c_{v}$ is the specific heat capacity of the electrons. The ratio of the thermal conductivity $\kappa$ to the electrical conductivity times temperature ($\sigma$T) of an electron gas is a universal constant (Lorentz number) and this is called Wiedmann-Franz law. The value of the Lorentz number is of the order of $10^{-8}$ $\frac{\mathrm{Watt\, ohm}}{\mathrm{K}^{2}}$ in typical metals.
Heat capacity
The temperature dependence of the specific heat capacity of an electron gas is given by $\frac{c_{v}}{k_{B}}=\gamma T=\frac{\pi^{3}}{3}g(\epsilon_{F})k_{B}T$ where $k_{B}$ is the Boltzmann constant and $g(\epsilon_{F})$ is the density-of-states at the Fermi level. The value of $\gamma$ in typical metals is of the order of 1 $\frac{\mathrm{mJ}}{\mathrm{mole\,}\mathrm{K}^{2}}$. In the Debye model, the phonon contribution to the heat capacity per atom is given by $\frac{c_{v}}{k_{B}}=9(\frac{T}{\Theta_{D}})^{3}\ensuremath{\int\limits _{0}^{\Theta_{D/T}}}\frac{x^{4}e^{x}dx}{(e^{x}-1)^{2}}$ where $\Theta_{D}$ is the Debye temperature.