Position Of Fermi Energy Level

In the free electron model at absolute zero, the energy corresponding to the filling of electrons is given as , where 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 . The total energy corresponding to filling the first electrons is given by: where is Avogadro's number.
Step 2: Matching the Energy Expression. The problem provides that the energy corresponding to filling electrons is . By comparing the two expressions for the energy: canceling out from both sides:
Step 3: Solving for . Taking the logarithm of both sides: Since is Avogadro's number, we approximate . This gives: Thus, the value of is .