Discussion
Consider a real gas, which obeys the following equation-of-state:
\[ v = \frac{RT}{P} + C_{1} - \frac{C_{2}}{Pv}, \]
where v is the mass specific volume, P is the pressure, T is the temperature and R is the gas constant. \(C_{1}\) and \(C_{2}\) are constants. If s is the mass specific entropy, the quantity \(\left(\frac{\partial s}{\partial v}\right)_{T}\) for the gas is given by
Answer: C
The quantity is \(\frac{R}{v-C_{1}}\).
By the Maxwell relation \((\partial s/\partial v)_{T} = (\partial P/\partial T)_{v}\). Solve the equation of state for P:
\[ \begin{aligned} v - C_{1} &= \frac{1}{P}\left(RT - \frac{C_{2}}{v}\right) \\ P &= \frac{RT - C_{2}/v}{v-C_{1}} \end{aligned} \]
\[ (\partial P/\partial T)_{v} = \frac{R}{v-C_{1}} \]
R/v is the ideal-gas result (C1 = C2 = 0), which this gas does not obey. The other two forms come from differentiating without first making P explicit.
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