# A formal procedure for thermodynamic relations

This is my third time taking a thermodynamics course (two in undergrad, one in grad), and I've finally become frustrated enough about something to post on here.

A lot of thermodynamic questions want you to find some quantity, given something else. Typically, the problem will be to relate volume, temperature, or pressure measurements (along with a volumetric equation of state) to other thermodynamic variables.

The math is very easy to verify, once you see it written down. The problem is that no textbook I've ever used has given a definitive procedure for going about deriving these things.

For instance, a recent problem is: Determine $C_p(P, T) - C_v(P, T)$ for such-and-such equation of state. My typical method of solving this is going to the table of Maxwell relations and plugging in and substituting things until I get what the problem is looking for. But this leads to a lot of trails to nowhere and is time-consuming. How do I know in advance that taking a partial derivative with respect to entropy is going to give me something useful in a chain of transformations?

My question is: Given an arbitrary function of two thermodynamic variables, what procedure is used for relating it to two other thermodynamic variables? (i.e. $f(\frac{\partial H}{\partial T}, S)$ -> $g(P, T)$)

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I wish I could give you +$\infty$ for this question; I've always found such thermodynamics problems to be extremely un-illuminating and frustrating for the same reasons (even after TAing for grad thermo/stat mech). I'm hopeful that someone will have something useful to say about this, but I'm somewhat pessimistic. – joshphysics Mar 11 '13 at 5:30
The blog post Good Subjective, Bad Subjective may be useful to readers considering answering this. Unless, of course, you can actually offer a algorithm. – dmckee Mar 11 '13 at 6:13
I wish I had time to write a full answer, but you might get somewhere by reading up on "Legendre transforms" if you haven't already. That's is the mathematical trick on which all this stuff is based. By itself it won't give you an algorithm, but understanding the underlying mathematical structure should help. – Nathaniel Mar 11 '13 at 8:38

There is indeed such a procedure. The starting point is the pair of equations of state which express $T$ and $S$ as functions of $p$ and $V$ and so establish the model you are using---for the ideal gas, we have $T=p V, \quad S=\frac1{\gamma-1} \ln (p V^\gamma)$ (we have omitted physical constants to simplify the expressions---mathematicians tend to do that). The general case is $T=f(p,V), \quad S=g(p,V)$ for suitable functions $f$ and $g$ of two variables. $f$ and $g$ are related by the Maxwell relations which mean that $f_1g_2-f_2g_1=1$ (subscripts denote partial derivatives with respect to $p$ and $V$). Now one can express all of the basic thermodynamical quantities in terms of $p$, $V$, $f$, $g$ and the partials of the latter two. One then verifies an identity by substituting these in both sides and checking by simple algebra if it is valid. This is a simple application of the inverse function theorem and the chain rule. For those who are not comfortable with these techniques, one can write a simple Mathematica programme which does this for you and this has been carried out in the arXiv paper "A systematic approach to thermodynamical identities" (1102.1540) which, in turn, is based on the theory developed in the arXiv paper 1102.1540. The former paper even develops a method for {\it generating} thermodynamical identities, using the technique of Groebner bases.

As a small sample, one finds that the expression $C_p-C_V$ that you mention is $\dfrac f{f_1f_2}$ and from this it follows easily that this is constant for the ideal gas (for which $f(p,V)=p V$).

One sees the advantages of this method when one considers more elaborate models, e.g., the van der Waals gas ($f(p,V)=\left(p + \frac a {V^2} \right )(V-b)$, $S=\frac 1 {\gamma-1}( \ln (p+ \frac a {V^2})+\gamma \ln (V-b))$), or the Feynman model (like the ideal gas but allowing for the fact that the adiabatic index $\gamma$ is not constant for a real gas but depends on temperature) which is analysed in the above-mentioned articles.

Added at Nick's request: $C_p$ and $C_V$ are defined as $T \frac {\partial S}{\partial T} |_V$ and $T \frac {\partial S}{\partial T} |_p$. The partial derivatives are, in the numerical code ($p\to 1$, $V \to 2$, $T \to 3$ and $S \to 4$) introduced in these articles, $(4,3,2)$ and $(4,3,1)$. This leads to the relationships $C_V= f \frac{g_1}{f_1}$ and $C_p=f\frac {g_2}{f_2}$. Taking the difference leads to the required result (since $f_1g_2-f_2g_1=1$---one of the manifestations of the Maxwell relations which can be interpreted as stating that the map from the $p,V$ plane to the $T,S$ one described by the equations of state is area preserving).

Interestingly, the difference of these two terms depends only on $f$ (i.e., temperature as a function of pressure and volume) and not on entropy, in contrast to the thermal capacities themselves. Using these formulae, we can easily compute these quantities for models which are more elborate than the ideal gas, something we have not encountered in the literature.

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I might add that I have illustrated this method in answers to the following queries on this forum if you are looking for examples: 23644, 21965, 46759, 20843, 51683, 47841, 18510. – mathuser4891 Mar 11 '13 at 13:22
I'm sorry; I've tried going over it. How exactly did you get $\frac{f}{f_1 f_2}$? – Nick Mar 18 '13 at 6:19