Wiedemann Franz Law and its derivation

Assume that a homogeneous isotropic material is subjected to a temperature gradient dT/dx. The flow of heat will result in the direction opposite to the temperature gradient through the conducting medium.

The heat flux Q (heat flow per unit time per unit area) will be proportional to the temperature gradient i.e Q∞dT/dx

Or                        Q= -K dT/dx

Where K is the proportionality constant and is known as coefficient of thermal conductivity.  if Q is expressed in W/m2 and dT/dx in K/m the units of K will be W/mK

As discussed ,the transformation of heat in solids is due to phonons and free electrons . Thus, the coefficient of thermal conductivity K can be written as

K=Kphonon +Kelectron

In order to derive the expression for K, let us consider the heat flow from high temperature to low temperature in a metal slab having temperature gradient dT/dx.

Let cv be the heat capacity, then the heat transfer per unit area per second will be

Q=mnv/3 cv λdT/dx                                   (1)

Where v is the velocity of electrons.

λ Is mean free path of collisions

Also ,heat flux  Q=KdT/dx                                                   (2)

By comparing equations(1) and (2) ,we get

KdT/dx=mnv/3 CvdT/dx

Or               K=mnv/ Cvλ                                                        (3)

The energy of free electron is given by

M CvT=3/2 KBT

Or                  Cv=3/2m Kb (4)

Where KB is Boltzmann Constant

By putting equation (4) in (3) ,we get

Thermal conductivity K=mnv/3(3/2 KB/m)λ

Or                               K=KB(nvλ/2)                                                      (5)

Specific heat at constant volume for an ideal gas is

Cv=3/2 n KB

KB=2/3 n Cv (6)

By putting equation (6) in (5) ,we get

K=1/3 Cvλv                                                               (7)

Expression (7) represents that the thermal conductivity of solid depends upon specific heat (CV) ,mean free path of collisions (λ) and velocity of electrons (v)

Now consider the electrical conductivity σ

σ =ne2r/m                                                      (8)

And relaxation time (collision time)

r=λ/vd (9)

by putting equation (9) in (8) ,we get

σ =ne2λ/mvd (10)

Also                       ½ mv2d=3/2 KBT

Or                             m=3KBT/v2d                                               (11)

By putting equation (11) in (10) ,we get

σ =ne2λvd/3KBT                                                                               (12)

Therefore, the ratio of thermal conductivity K to electrical conductivity σ is

K/ σ =KBnvλ/2*3KBT/ne2λvd[by dividing equation (5) by (12)]

=3/2 K2B/e2.T, if we assume v=vd

Or                              K/ σ T=5.838*10-9 o cal K-sec

K/ σ T=2.44*10-8oW/K2=L

Which indicates that the ratio K/ σ is same for all metals and is a function of temperature only. This empirical law is known as Weidemann –Franz Lorenz law .Thus, we can say that best electrical conductor will be a best thermal conductor.

The L is known as the Lorenz number.

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