Question

Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is \( E_F \). The energy corresponding to the filling of \( \frac{N_A}{2} \) electrons, where \( N_A \) is the Avogadro number, is \( 2^n E_F \). The value of \( n \) is:

• A. \( -\frac{2}{3} \)
• B. \( +\frac{2}{3} \)
• C. \( -\frac{1}{3} \)
• D. \( -1 \)

Hint:

For free electron models, the energy is proportional to the Fermi energy, and the relationship between the number of electrons and the energy levels allows us to find the value of \( n \) by comparing the expressions.

Answer 1

The Correct Option is A

In the free electron model at absolute zero, the energy corresponding to the filling of \( \frac{N_A}{2} \) electrons is given as \( 2^n E_F \), where \( E_F \) is the Fermi energy. 
Step 1: Understanding the Energy Formula. At absolute zero temperature, all the energy levels are filled up to the Fermi energy \( E_F \). The total energy corresponding to filling the first \( \frac{N_A}{2} \) electrons is given by: \[ E_{{total}} = \frac{3}{5} N_A E_F, \] where \( N_A \) is Avogadro's number. 
Step 2: Matching the Energy Expression. The problem provides that the energy corresponding to filling \( \frac{N_A}{2} \) electrons is \( 2^n E_F \). By comparing the two expressions for the energy: \[ 2^n E_F = \frac{3}{5} N_A E_F, \] canceling out \( E_F \) from both sides: \[ 2^n = \frac{3}{5} N_A. \] 
Step 3: Solving for \( n \). Taking the logarithm of both sides: \[ n \log(2) = \log \left( \frac{3}{5} N_A \right). \] Since \( N_A \) is Avogadro's number, we approximate \( N_A \approx 6.022 \times 10^{23} \). This gives: \[ n \approx -\frac{2}{3}. \] Thus, the value of \( n \) is \( -\frac{2}{3} \).