 Solution

The Wiedemann-Franz law relates the electronic part of thermal conductivity to electrical conductivity:For degenerate electrons with elastic scattering and a transport function smooth over $k_{B}T$, a Sommerfeld expansion makes the same Fermi-level transport factor appear in charge conduction and in heat conduction. Their ratio cancels that factor and leaves the displayed Lorenz number. Heat conductivity is defined at zero electric current, so thermoelectric subtraction is part of the precise transport expression.

The law does not relate total solid heat conductivity to charge conductivity when phonons contribute substantially. Strong inelastic scattering, sharp energy dependence, nondegenerate carriers or nonstandard electronic regimes can change the ratio. Low-temperature residual elastic-scattering transport is a common setting where the standard value is recovered. The unit W·ohm/K² follows because ${\kappa }_e/(\sigma T)$ contains one extra inverse temperature beyond thermal conductivity itself.