Question

Show that, according to the free-electron model of electrical conduction in metals and classical physics, the resistivity of metals should be proportional to$\sqrt{\mathbf{T}}$where Tis the temperature in kelvins.

Short answer

The relation of resistivity to temperature of the metals $\rho \alpha \sqrt{T}$is hence, proved.

Step-by-step solution

Identification of given data

Free electron model of the electrical conduction in metals and classical physics is given.

Signification of resistivity-temperature dependence

By assuming that the conduction electronsin a metal are free to move like the molecules of a gas, it is possible to derive an expression for the resistivity of a metal. The resistivity is directly dependent on the mass of the electron, inversely proportional to the square of charge, number of elections, and meantime of collision of an electron with the atoms of the metal.

As mentioned in the problem itself, we need to use the concept of the free electron model and the classical theory of physics. In addition, we need to use the formula for resistivity and average velocity to find the relation between resistivity and temperature.

Formulae:

The resistivity of the metal due to the conductivity of electrons,$\rho =\frac{m}{n{e}^2\tau }$ …(i)

Here, is the mass of the electron,n is the number of electrons, is the charge on electrons,tmeantime of collision of an electron with the atoms of the metal.

The average speed of an electron,$V_{avg}=\sqrt{\frac{8RT}{\mathrm{\pi M}}}$ …(ii)

Here,$V_{avg}$ is the average velocity of the electron, R is the gas constant, Mis the molar mass,T is the absolute temperature.

Determining the resistivity-temperature relation

According to the free electron model, the conduction electrons in the metal are free to move like the molecules of a gas in a closed container.

So, according to the classical physics, these electrons should have a Maxwell’s speed distribution like the molecules in a gas. Thus, the average electron speed is directly proportional to the square root of the temperature considering equation (ii) as follows:

$V_{avg}\alpha \sqrt{T}$ …(iii)

The resistivity relationto the with collision time of the electron considering equation (i) as follows:

$\rho \alpha {\tau }^{-1}$ …(iv)

We know that, the average velocity relation with time as,

$V_{avg}=\frac{d}{\mathrm{t}}$

In other words,

$V_{avg}\alpha t^{-1}$ …(v)

From equation (iii), (iv) and (v) we can conclude that,

$\rho \alpha \sqrt{T}$

Hence, the resistivity-temperature relation is thus, proved.