= Solution
In a <metal>, mobile <Electrons> transport heat as well as electric charge. A <temperature> gradient creates an imbalance between carriers arriving from warmer and colder regions. The resulting <heat current> obeys <Fourier's law> $\mathbf j_Q=-\kappa\nabla T$ for an isotropic material, where <thermal conductivity> $\kappa$ has units $\mathrm{W\,m^{-1}\,K^{-1}}$. For a bar of area $A$ carrying steady heat flow $P$, $P=-\kappa A\,dT/dx$ when parasitic losses are negligible.
A kinetic estimate for the electronic contribution is
$$
\kappa_e\simeq\frac13 C_ev_F\ell_\kappa=\frac13C_ev_F^2\tau_\kappa,
$$
where $C_e$ is the <electronic heat capacity> per volume, $v_F$ the <Fermi velocity>, and $\ell_\kappa=v_F\tau_\kappa$ the heat-transport mean free path. <Fermi-Dirac statistics> and the <Pauli exclusion principle> restrict thermal rearrangements to states near the <Fermi surface>, rather than allowing every <Electron> to contribute a classical heat capacity. A degenerate parabolic band gives
$$
C_e=\frac{\pi^2nk_B^2T}{2E_F},\qquad
\boxed{\kappa_e=\frac{\pi^2nk_B^2T\tau_\kappa}{3m}.}
$$
Using the <Drude model> form $\sigma=ne^2\tau_\sigma/m$ for electrical transport, the relation between <thermal and electrical relaxation times in metals> becomes
$$
\frac{\kappa_e}{\sigma T}=\frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2\frac{\tau_\kappa}{\tau_\sigma}.
$$
For scattering that relaxes heat and charge currents similarly, $\tau_\kappa\simeq\tau_\sigma$ and the <Wiedemann-Franz law> follows:
$$
\boxed{\frac{\kappa_e}{\sigma T}\simeq L_0,\qquad
L_0=\frac{\pi^2k_B^2}{3e^2}\simeq2.44\times10^{-8}\,\mathrm{W\,\Omega\,K^{-2}}.}
$$
Thus good electrical conductors are usually good thermal conductors. The correct quantum <Lorenz number> differs from a naive classical electron-gas estimate.
Scattering from impurities, boundaries and <phonons> limits the heat-transport path. In an impurity-dominated low-temperature regime with approximately constant relaxation time, $\kappa_e$ is approximately proportional to $T$. Where phonon-limited <electrical resistivity> is proportional to $T$ and the Wiedemann-Franz approximation is applicable, $\kappa_e$ can be roughly <temperature> independent. Inelastic processes can relax heat and charge currents differently, so the Lorenz ratio is not universal at all temperatures. Lattice <phonons> add a separate positive thermal contribution; the measured total is $\kappa=\kappa_e+\kappa_{\rm ph}$, and the electronic Wiedemann-Franz expression should not be applied blindly to that total.
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