Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 314 / 2 / c

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- Solution c

Solution

For a perfect gas with constant specific heat capacity

c_{v}

, the first law of thermodynamics gives

d s = c_{v} d lo g (\frac{p}{ρ ^{γ}}) .

Using the pressure-density Hugoniot relation for a perfect gas, the entropy production in a perfect-gas shock is therefore

\frac{[ s ]}{c _{v}} = lo g P - γ lo g [\frac{( γ + 1 ) P + γ - 1}{( γ - 1 ) P + γ + 1}] .

This is positive for

P > 1

, as required by the Second law of thermodynamics for the physical compressive normal shock wave.

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