Metals — Question 9

Question 9

Electrical resistance estimates electronic heat conduction

Assume the Wiedemann–Franz relation $\kappa_e\rho=L_0T$, with $L_0=2.44\times10^{-8}\,\mathrm{W\,\Omega/K^2}$. A metal at 300 K has $\rho=2.00\times10^{-8}\,\Omega\,\mathrm m$.

1. Find its electronic thermal conductivity.

2. A rod has area $2.00\,\mathrm{mm^2}$, length .100 m, and a small temperature difference 5.00 K near 300 K. Find heat flow and the product of electrical resistance and thermal conductance.

Question 9 - Solution

Electrical resistance estimates electronic heat conduction

(a) Method and calculation

$$\boxed{\kappa_e=L_0T/\rho=366\,\mathrm{W/(m\,K)}}.$$$$\kappa_e\propto T/\rho\quad\text{within the supplied transport model}.$$

(b) Interpretation and further calculation

$$\boxed{\dot Q=\kappa_e A\Delta T/L=.0366\,\mathrm W}.$$$$R=\rho L/A,\quad G_{\mathrm{th}}=\kappa_e A/L\Rightarrow\boxed{RG_{\mathrm{th}}=L_0T=7.32\times10^{-6}\,\mathrm{W\,\Omega/K}}.$$

Physical check

The geometry cancels in the product. Lattice heat transport and departures from the assumed Lorenz number can change the total measured thermal conductivity.