Expression for Thermal Conductivity of a Metal (Derivation)

EXPRESSION FOR THERMAL CONDUCTIVITY OF A METAL (Derivation)

Consider two cross-sections A and B of a metal rod separated by a distance λ

Let A be at a high temperature (T) and B at a low temperature (T - dT).

Now, heat flows from A to B by the free electrons (Fig. 2.6)

Conduction electron per unit volume = n

Average velocity of the electrons = v

During the movement of the electrons in the metal rod, the elastic collision takes place. Hence, the electrons near A lose their kinetic energy while electrons near B gain kinetic energy.

At A, average kinetic energy of an electron = (3/2)kT .................(1)

where k - Boltzmann's constant

T- Temperature at A.

At B, average kinetic energy of the electron 

The excess of kinetic energy carried by the electron from A to B

Number of electrons crossing per unit area per unit time from A to B = (1/6) nv .................(4)

The excess of energy carried from A to B per unit area in unit time 

Similarly, the deficient of energy carried from B to A per unit area per unit time 

Hence, the net amount of energy transferred from A to B per unit area per unit time

But, from the definition of thermal conductivity, the amount of heat conducted per unit area per unit time

Note: The students are not expected to write the part of the derivation given in the box in the examination.

We know that for the metals

relaxation time (τ) = collision time (τ_c)

substituting the eqn (9) in the eqn (8), we have

The equation (10) is the expression for the thermal conductivity of a metal.

Wiedemann - Franz Law

Statement

The ratio of thermal conductivity (K) to electrical conductivity (o) is directly proportional to the absolute temperature (T). This ratio is constant for all metals at a given temperature.

where L is a proportionality constant. It is known as Lorentz number. Its value is 1.12 × 10^-8 WΩK^-2

Derivation

Wiedemann - Franz law is derived from the expressions of thermal and electrical conductivities of a metal.

We know that

Thermal conductivity of the metal

Electrical conductivity of the metal

The kinetic energy of the electron is given by

Substituting eqn (4) in eqn (3), we have

where is a constant and it is known as Lorentz number

Thus, it is proved that the ratio of thermal conductivity to electrical conductivity of the metal is directly proportional to the absolute temperature of the metal.

This law is verified experimentally and it is found to hold good at normal temperature. But, this law is not applicable at very low temperature.

Conclusion

Wiedemann - Franz law clearly shows that if a metal has high thermal conductivity, it should also have high electrical conductivity.

Among the metals, the best electrical conductors (silver, copper, aluminium) are also the best conductors of heat.

Lorentz Number

The ratio of thermal conductivity (K) to the product of electrical conductivity (σ) and absolute temperature (T) of the metal is a constant. It is known as Lorentz number and it is given by

Consider the expression 

Substituting for Boltzmann's constant k = 1.38 × 10^-23 JK^-1 and the charge of the electron e = 1.602 × 10^-19 coulomb, we get Lorentz number as

It is found that the value of Lorentz number determined using classical free electron theory is only half of the experimental value i.e., 2.44 × 10^-8 W Ω K^-2. This discrepancy in experimental and theoretical values of Lorentz number is one of the failures of the classical theory. It is rectified in quantum theory. 

ANNA UNIVERSITY SOLVED PROBLEMS

Problem 2.1

The electrical resistivity of copper at 27 °C is 1.72 × 10^-8 Ω m. Compute its thermal conductivity if the Lorentz number is 2.26 x 10^-8 W Ω K^-2. (A.U. April 2014)

Given data

Electrical resistivity p = 1.72 × 10^-8 Ω m

Temperature T = 27 °C = 27 + 273 = 300 K

Lorentz number L = 2.26 x 10^-8 W Ω K^-2

Solution

We know that Wiedemann - Franz law

Substituting the given values, we have

Problem 2.2

The thermal and electrical conductivities of copper at 20 °C are 390 W m^-1 K^-1 and 5.87 × 10⁷ Ω^-1m^-1 respectively. Calculate Lorentz number. (A.U. May 2012)

Given data

Thermal conductivity of copper K = 390 W m^-1 K^-1

Electrical conductivity of copper o copper σ = 5.87 × 10⁷ Ω^-1m^-1

Temperature T = 20 °C = (20 + 273) = 293 K

Solution

We know that Lorentz number L = K / σ T

Substituting the given values, we have

Success of Classical Free Electron Theory

i. It is used to verify Ohm's law.

ii. It is used to explain electrical and thermal conductivities of metals.

iii. It is used to derive Wiedemann-Franz law.

iv. It is used to explain the optical properties of metals.

Failures of Classical Free Electron (CFE) Theory

i. Classical theory states that all the free electrons absorb the supplied energy. But, the quantum theory states that only a few electrons absorb the supplied energy.

ii. The electrical conductivity of semiconductors insulators cannot be explained by this theory.

iii. The photo-electric effect, Compton effect and black body radiation cannot be explained on the basis of classical free electron theory.

iv. According to the classical free electron theory, the ratio K / σ T is constant at all temperatures. But, it is found that it is not constant at low temperature.

v. According to this theory, the value of specific heat capacity of a metal is 4.5R. But, the experimental value is given by 3R. (Here R is the universal gas constant.)

vi. The susceptibility of paramagnetic material is inversely proportional to temperature. But, the experimental result shows that paramagnetism of metal is independent of temperature. Moreover, ferro-magnetism cannot be explained by this theory.