4. The Wiedemann Franz Law (1853 states that the ratio of the thermal conduc- ti

Question

4. The Wiedemann Franz Law (1853 states that the ratio of the thermal conduc- tivity, K, to the electrical conductivity, , of metals is directly proportional to the absolute temperature with a proportionality constant independent of the metal. Let us see how one can derive this from elementary considerations of diffusion. Go back to our elementary derivation of the relation between the particle current 3p and the particle density in one dimension which, with the definition D 2/T gave 3p =-D(0p/or). Now consider a similar situation, but in which the particle density is uniform whereas the particles carry energy which varies in position, E(r). Show that the energy current, jE which is the energy which crosses from left to right per unit time per unit area is in the one-dimensional model dE dT dr dT dr where is the number of particles per unit volume and cu is the specific heat dE/dT. Thus pcD. In class I showed that, for a system with one carrier, the electrical conductivity was -pe2D/kT so that K C which is the Wiedemann Franz law. If, in addition, one assums for c the classical value of an ideal gas, 3k/2 one obtains for the ratio (3/2)(k/e)T. This coefficient turns out to be about 1/2 the value measured experimentally, so the basic idea is correct that both with the value of cy i processes are simply diffusive, but there is something wrong

Explanation / Answer

Here we assume that free electrons behave like a classical ideal gas which is not true indeed. Electrons have intermolecular interactions and have definite mass. So value of molar heat capacity of ideal gas is not valid for Electrons under consideration.

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