Discussion
Match each quantity in Column M with the appropriate relation from Column N. Here, \(\psi\) is Helmholtz function, \(P\) is pressure, \(v\) is specific volume, \(T\) is temperature, \(h\) is specific enthalpy, and \(s\) is specific entropy.
| Column M | Column N | |
|---|---|---|
| (M1) \(v\) | (N1) \(-\left.\frac{\partial \psi}{\partial v}\right|_T\) | (N5) \(\left.\frac{\partial h}{\partial P}\right|_s\) |
| (M2) \(P\) | (N2) \(\left.\frac{\partial \psi}{\partial v}\right|_T\) | (N6) \(-\left.\frac{\partial h}{\partial s}\right|_P\) |
| (M3) \(s\) | (N3) \(\left.\frac{\partial \psi}{\partial T}\right|_v\) | (N7) \(-\left.\frac{\partial h}{\partial P}\right|_s\) |
| (M4) \(T\) | (N4) \(-\left.\frac{\partial \psi}{\partial T}\right|_v\) | (N8) \(\left.\frac{\partial h}{\partial s}\right|_P\) |
Answer: C
The correct matching is M1-N5, M2-N1, M3-N4, M4-N8.
The Helmholtz function satisfies \(d\psi = -s\,dT - P\,dv\), so
\[ \begin{aligned} P &= -\left.\frac{\partial \psi}{\partial v}\right|_T \\ s &= -\left.\frac{\partial \psi}{\partial T}\right|_v \end{aligned} \]
giving M2-N1 and M3-N4. Enthalpy satisfies \(dh = T\,ds + v\,dP\), so
\[ \begin{aligned} T &= \left.\frac{\partial h}{\partial s}\right|_P \\ v &= \left.\frac{\partial h}{\partial P}\right|_s \end{aligned} \]
giving M4-N8 and M1-N5. The first option gives T the wrong sign (N7), the second gives the wrong signs throughout, and the last swaps the relations for v and T.
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