Question

Up to what temperature has one to heat classical electronic gas to make the mean energy of its electrons equal to that of free electrons in copper at $T = 0$ ? Only one free electron is supposed to correspond to each copper atom.

Answer 1

Hint: To determine this temperature, we will equate the two equations for mean kinetic energy and get the formula that will be utilised to calculate the temperature value. Then we get the answer by placing the needed values in the final phrase.

Complete step by step answer:
We know that the mean kinetic energy of the electrons present in a Fermi gas is
$\dfrac{3}{{5{E_F}}}$ where ${E_F}$ is Fermi energy.
 As The relationship between average KE and temperature is equal to $KE = \dfrac{3}{{2KT}}$
Where $K$ is Boltzmann constant and $T$ is the temperature
So $\dfrac{3}{{5{E_F}}}$ must be equal to the $\dfrac{3}{{2KT}}$.
On comparing the above two equations, we get the relation between energy and temperature.
$ \Rightarrow \dfrac{3}{{5{E_F}}} = \dfrac{3}{{2KT}}$
Dividing both sides by three, we get,
$ \Rightarrow \dfrac{1}{{5{E_F}}} = \dfrac{1}{{2KT}}$
Shifting the terms to find the value of $T$ using the method of transposition, we get,
$ \Rightarrow T = \dfrac{{2{E_F}}}{{5K}}$
For copper we know,
${E_F} = 7.01eV$
Now, on calculating the temperature upto which the classical electronic gas must be heated to make the mean energy of its electrons equal to that of free electrons in copper at \[T = 0\] is
$ \Rightarrow T = \dfrac{{2 \times 7.01}}{{5 \times 1.38 \times {{10}^{ - 23}}}}$
$ \Rightarrow T = 3.25 \times {10^4}K$.

Note:
Fermi energy is a concept which is used in quantum mechanics. It basically refers to the energy difference which is present between the highest and lowest occupied particle states in any of the quantum systems involving non-interacting fermions. This is done at absolute zero temperature.