Wiedemann - Franz Law and Lorentz Number

WIEDEMANN - FRANZ LAW AND LORENTZ NUMBER

Law: 

The ratio between the thermal conductivity and electrical conductivity of a metal is directly proportional to the absolute temperature of the metal.

where L is a constant called as Lorentz number whose value is

2.44 × 10^-8 W Ω K^-2 (Quantum mechanical value) at temperature T = 293 K.

Proof:

(i) By Classical theory

We know electrical conductivity (from classical theory) 

Thermal conductivity (from classical theory)

……(1)

We know kinetic energy of an electron 

Substituting this in equation (1) we can write

(or)

(or)

where

Substituting the value of Boltzmann constant K_B = 1.38 × 10^-23 JK^-1

charge of electron e = 1.6021 x 10^-19 Joules, we get

L=1.12 × 10^{- 8} W Ω Κ^-2

It is found that the classical value of Lorentz number, is only one half of the experimental value (i.e.,) 2.44 × 10^-8 W Ω K^-2.

This Discrepancy in the experimental and theoretical value of 'L' is the failure of classical theory. This discrepancy can be rectified by quantum theory.

(ii) By Quantum theory

In quantum theory the mass of the electron (m) is replaced by the effective mass m*.

The electrical conductivity 

Rearranging the expression for thermal conductivity and substituting the electronic specific heat, the thermal conductivity can be written as

(or)

(or)

where

Substituting the values for Boltzmann constant (K_B) and the charge of the electron e we get

Lorentz number L=2.44 × 10^-8 W Ω K^-2

Thus quantum theory verifies Wiedeman-franz law and has good agreement with the experimental value of Lorentz number.