Question

Consider a system of 3 fermions which can occupy any of the 4 available energy states with equal probability. The entropy of the system is:

• A. \( k_B \ln 2 \)
• B. \( 2 k_B \ln 4 \)
• C. \( 2 k_B \ln 2 \)
• D. \( 3 k_B \ln 4 \)

Hint:

For a system of fermions, entropy depends on the available microstates considering the Pauli exclusion principle.

Answer 1

The Correct Option is C

For fermions, the number of ways to distribute 3 indistinguishable fermions in 4 states obeying the Pauli exclusion principle is given by the combinatorial formula:
\[ W = \binom{4}{3} = 4 \] The entropy is calculated using:
\[ S = k_B \ln W \] Substituting \( W = 4 \):
\[ S = k_B \ln 4 = 2 k_B \ln 2 \]