Expression for Thermal Conductivity

EXPRESSION FOR THERMAL CONDUCTIVITY

Let us consider a uniform rod AB with the temperatures T₁ (Hot) at end 'A' and T₂ (cold) at end B. Heat flows from hot end 'A' to the cold end 'B'. Let us consider a cross sectional area 'C' which is at a distance equal to the mean free path (λ) of the electron between the ends A and B of the rod as shown in Fig. 1.5.

The amount of heat (Q) conducted by the rod from the end A to B of length 2 λ is given by

(or)….(1)

where K→ Coefficient of thermal conductivity

A→ Area of cross section of the rod

T₁-T₂ → Temperature difference between the ends A and B

t → Time for conduction

2 λ → Length of rod (A to B)

From equation (1) we can write

Coefficient of thermal conductivity per unit area per unit time

….(2)

Let us assume that there is equal probability for the electrons to move in all the six directions as shown in Fig. 1.6. Since each electron travels with thermal velocity v, if 'n' is the free electron density, then on an average 1/6 nv electrons will travel in any one direction.

The number of electrons crossing unit area per unit time at

 ….(3)

According to kinetic theory of gas, [Since free electrons are assumed to be gas molecules which are freely moving]

The average kinetic energy of an electron at hot end 'A' of temperature (T₁)

The average kinetic energy of an electron at cold end 'B' of temperature (T₂)

where K_B→ Boltzmann constant = 1.380 × 10^-23 J/K

…..(4)

Similarly, the heat energy transferred per unit area per unit time from end B to A across

…..(5)

The net heat energy transferred from end A to B per unit area per unit time across 'C' can be got by subtracting equation (5) from equation (4)

…..(6)

Substituting equation (6) in equation (2) we have

(or) …..(7)

We know for metals.

Relaxation time (τ) = Collision time (τ_c)

 (i.e.,) τ = τ _c= λ/v

τ v =λ ….(8)

Substituting equation (8) in equation (7) we have

…..(9)

Equation (9) is the classical expression for thermal conductivity.