Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 4 / ii

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- Solution ii

Solution

Write

R = k_{B} / (μ m_{u})

. At fixed density,

(\frac{\partial l o g P}{\partial l o g T})_{ρ} = 4 - 3 β, (\frac{\partial l o g P}{\partial l o g ρ})_{T} = β .

The specific internal energy is

u = \frac{3}{2} R T + \frac{a _{rad} T ^{4}}{ρ} .

Applying the first law of thermodynamics adiabatically gives

(\frac{d l o g T}{d l o g ρ})_{s} = \frac{8 - 6 β}{24 - 21 β} .

Combining these logarithmic derivatives yields

\nabla_{ad} (β) = \frac{8 - 6 β}{32 - 24 β - 3 β ^{2}} .

As gas pressure dominates,

β \to 1 lim \nabla_{ad} = \frac{2}{5},

the result for a monatomic perfect gas. In the radiation-pressure limit it tends to

1/4

.

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