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In Thermodynamics And An Introduction To Thermostatistics by Herbert B. Callen, Problem 9.4-1 asks the reader to

Show that the difference in molar volumes across a coexistence curve is given by $\Delta v=-P^{-1}\Delta f$

$\Delta f$ probably refers to the change in the molar Helmholtz free energy. Now the phrase "across the coexistence" curve would make me thing that a difference of the molar volumes between two points lying on the coexistence curve $P(T)$ is to be calculated. However, this doesn't make much sense to me, as I don't even think the molar volume is clearly defined on the coexistence curve. It could take a range of values, depending on how much of each phase is present.

If Callen means the difference between molar volumes on either side of the coexistence curve, so at some point $(P,T)$ where $\Delta v$ corresponds to the difference in molar volumes of each phase at this point, then I think the solution might be as straightforward as:

$$ \mathrm{d}f=-s\mathrm{d}T-P\mathrm{d}v\\ T=\mathrm{const}, P=P(T)=\mathrm{const}\Rightarrow \mathrm{d}f=-P\mathrm{d}v\\ \Rightarrow \Delta v= -\Delta f / P $$

But it doesn't sound like this is the question to me. Am I missing something?

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You’re basically there. He does mean to consider the well defined values associated with pure phases, but the “either side” he means is arbitrarily close to one another.

Take the limits of the molar volume in the pure phases as the conditions approach a common point on the coexistence curve.

These kinds of procedural issues are better explained in some other (generally thicker) texts. Callen seems to assume that you’re had some previous exposure to the subject.

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