An ideal gas with the adiabatic exponent `gamma` goes through a process `p = p_0 - alpha V`, where `p_0` and `alpha` are positive constants, and `V` is the volume. At what volume will the gas entropy have the maximum value ?

Explanation

Let `(p_1, V_1)` be a reference point on the line
`p = p_0 - alpha V`
and let `(p, V)` be any other point.
The entropy difference
`Delta S = S (p, V) - S (p_1, V_1)`
=`C_V 1n (p)/(p_1) + C_p 1n (V)/(V_1) = C_V 1n (p_0 - alpha V)/(p_1) + C_p 1n (V)/(V_1)`
For an exetremum of `Delta S`
`(del Delta S)/(del V) = (-alpha C_V)/(p_0 - alpha V) + (C_p)/(V) = 0`
or `C_p (p_0 - alpha V) - alpha V C_V = 0`
or `gamma(p_0 - alpha V) - alpha V = 0` or `V = V_m = (gamma p_0)/(alpha(gamma + 1))`
This gives a maximum of `Delta S` because `(del^2 Delta S)/(del V^2) lt 0`
a maximum of `Delta S` is a maximum of `S(p, V))`.
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