Maxwell relations are found by taking mixed derivatives of a thermodynamic potential. Does this mean that they do not hold at a first-order phase transition, where the thermodynamic potential is discontinuous?
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You are right, if, at a phase transition, second derivatives of a thermodynamic potential do not exist at that point Maxwell's relations are not valid anymore.
However, the sets of points of non-analyticity are confined on hyper-surfaces, in the thermodynamic state space, which partition it into regions of analyticity (pure phases or regions of coexistence). Thus, at each point of such hyper-surface, left- and right-limits exist (finite or infinite) and, for applications this is the only relevant thing.
answered Mar 12, 2019 at 10:48
GiorgioP-DoomsdayClockIsAt-90GiorgioP-DoomsdayClockIsAt-90
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$\begingroup$ I am missing a point here: you agree that Maxwell relations are not valid anymore at non-analyticity hyper-surfaces. This means that, e.g. $\partial \mu/\partial T \neq \partial S/\partial N$. Then I do not understand what you mean by "at each point of such hyper-surface, left- and right-limits exist (finite or infinite) and, for applications this is the only relevant thing". $\endgroup$– Alepr85Commented Mar 12, 2019 at 11:35
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$\begingroup$ I understand that I can measure, in my example above, $\partial \mu/\partial T$ and $\partial S/\partial N$, left and right of the transition line. But the two will not coincide. So I cannot deduce the value of $\partial S/\partial N$ (which, e.g., is difficult to measure) from that of $\partial \mu/\partial T$ (which, e.g., is easier). So, in my opinion, from the practical point of view it is a problem. $\endgroup$– Alepr85Commented Mar 12, 2019 at 11:35
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1$\begingroup$ @Alepr85 Certainly they do not coincide. The point is that you can measure, say $\frac{\partial{\mu}}{\partial{T}}$ to obtain $\frac{\partial{S}}{\partial{N}}$ on one side of a transition line (or surface) and the same on the other side. Of course, you cannot claim in general that the two limits are equal. But that's life with phase transition. $\endgroup$ Commented Mar 12, 2019 at 14:17