 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/9d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

From

E = ρ V

P = ρ /3

, and

d E = T, d S - P, d V

T, d S = V, d ρ + \frac{4}{3} ρ, d V .

Using

ρ = 4 σ T^{4} / c

gives

d S = \frac{16 σ}{3 c} (T^{3} d V + 3 T^{2} V d T) = d (\frac{16 σ}{3 c} V T^{3}) .

Therefore

S = \frac{16 σ}{3 c} V T^{3}, s = \frac{S}{V} = \frac{16 σ}{3 c} T^{3},

up to an irrelevant additive entropy constant.

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