The entropy of a certain macropartition of a combined system is $102 \mathrm{k}_{\mathrm{B}}$. The entropy of another macropartition is $204 \mathrm{k}_{\mathrm{B}}$. How much more likely is the system to be in the second macropartition than the first? (Select the closest response.)
A) 2 times
B) $\mathrm{e}^{2}=7.4$ times
C) $10^{2}=100$ times
D) $\mathrm{e}^{102}=2 \times 10^{44}$ times
E) $e^{102 k_{B}} \approx e^{0}$; so the two macropartitions have basically the same probability

Question

Quizplus · Accepted Answer

The probability of a macropartition is proportional to $e^{S}$, where $S$ is the entropy. So the ratio of probabilities is $e^{204 k_{B}}/e^{102 k_{B}} = e^{102 k_{B}} \approx 2 \times 10^{44}$.

The Entropy of a Certain Macropartition of a Combined System 102kB102 \mathrm{k}_{\mathrm{B}}102kB​

The Entropy of a Certain Macropartition of a Combined System $102 \mathrm{k}_{\mathrm{B}}$