Question

Consider a linear collection of \( N \) independent spin 1/2 particles, each at fixed location. The entropy of the system is:

• A. \( 0 \)
• B. \( \frac{N k_B}{2} \)
• C. \( \frac{N k_B}{2} \ln 2 \)
• D. \( N k_B \ln 2 \)

Hint:

For an ideal spin system, entropy depends on the number of accessible spin states.

Answer 1

The Correct Option is D

For \( N \) independent spin-1/2 particles, each spin can be in two possible states. The number of accessible microstates is:
\[ W = 2^N \] The entropy is given by Boltzmann’s relation:
\[ S = k_B \ln W = k_B \ln(2^N) = N k_B \ln 2 \]