A Statistical View of Entropy — Question 2

Question 2

Entropy additivity requires independent choices of accessible states

System A has 12 equally likely accessible states and system B has 30. Initially any A state can coexist with any B state. A later imposed constraint allows only 60 equally likely joint states. Use $S=k_B\ln\Omega$ in each ensemble.

1. Find initial joint multiplicity and prove entropy additivity for the unconstrained product ensemble.

2. Find the entropy change associated with restriction to 60 joint states. Identify why multiplying 12 by 30 is no longer justified after the constraint.

Question 2 - Solution

Entropy additivity requires independent choices of accessible states

(a) Method and calculation

$$\Omega_{AB}=\Omega_A\Omega_B=12(30)=360.$$$$S_{AB}=k_B\ln(12\cdot30)=k_B\ln12+k_B\ln30=S_A+S_B.$$

(b) Interpretation and further calculation

$$\Delta S_{AB}=k_B\ln(60/360)=-k_B\ln6\approx-1.79176k_B.$$$$\Omega_{AB,\mathrm{constrained}}=60\ne360.$$The constraint removes joint combinations. It need not preserve the marginal state probabilities, and any physical preparation must also account for the surroundings.

Physical check

A decrease in the selected system entropy alone does not establish a violation of the second law.