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=K_phonon +K_electron

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

Q=mnv/3 c_v 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 C_vdT/dx

Or               K=mnv/ C_vλ                                                        (3)

The energy of free electron is given by

M C_vT=3/2 K_BT

Or                  C_v=3/2m K_b (4)

Where K_B is Boltzmann Constant

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

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

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

Specific heat at constant volume for an ideal gas is

C_v=3/2 n K_B

K_B=2/3 n C_v (6)

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

K=1/3 C_vλv                                                               (7)

Expression (7) represents that the thermal conductivity of solid depends upon specific heat (C_V) ,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/v_d (9)

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

σ =ne²λ/mv_d (10)

Also                       ½ mv²d=3/2 K_BT

Or                             m=3K_BT/v²d                                               (11)

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

σ =ne²λv_d/3K_BT                                                                               (12)

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

K/ σ =K_Bnvλ/2*3K_BT/ne²λv_d[by dividing equation (5) by (12)]

=3/2 K²B/e².T, if we assume v=v_d

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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