Are thermodynamic fundamental relations for extensive properties linear?

Question

I have commonly found two definitions for extensive in various thermodynamics resources:

1. Homogeneity: $f(\lambda x) = \lambda f(x)$ From this the Euler-form can be easily found by taking the derivative with respect to $\lambda$ and a whole bunch of fun properties can be found.

2. Additivity: $f(x+y)=f(x)+f(y)$ This is the more intuitive notion of properties summing for combined systems

Suppose we have the fundamental relation $S=S(U,\vec{X})$ where $\vec{X}$ denotes the relevant extensive variables. It seems to me that taking both of these definitions for extensive together imply that fundamental relations are linear i.e. $S(a U_1 + b U_2, a \vec{X}_1 + b \vec{X}_2) = a S(U_1,\vec{X}_1) + bS(U_2,\vec{X}_2)$. But this heavily constrains fundamental relations and I've never seen them described as "linear". Is this accurate or is there something I am missing?

Answer 1

First of all, you should consider a function of more than one variable, because the only homogeneous function of degree 1 which is differentiable is a linear function ( it can be seen by chosing for each $x$ $\lambda= 1/x$ to derive $f(x) = f(1) x$). So, it is only with more than one variable that things go differently. However, every thermodynamic system must have a fundamental function which is function of not less than two variables (if reversible work and reversible heat transfer have to be different processes).

For purpose of clarity, let's analyze the case of the entropy as a function of its extensive natural variables as fundamental equation. With a trivial change, the same argument holds for internal energy as function of its extensive variables. Generalization to fundamental equations where part of the variables are extensive and part are intensive shouldn't pose particular problems: homogeneity and "additivity" (but see below about it) are confined to the dependence on extensive variables only.

Let $X$ and $Y$ be two arbitrary sets of extensive variables describing two thermodynamic states of two independent isolated systems made by the same substance, i.e. described by the same fundamental equation. Additivity just means that the entropy of the compound system made by the two isolated subsystems is $$ S_{tot}(X;Y)=S(X) + S(Y) $$ In general, $S_{tot}(X;Y)$ is not the same as $S(X+Y)$. So, we cannot write, in general, $S_{tot}(X;Y)= S(X+Y)$. It is true that the total value of the extensive variables is the sum $X+Y$. However, this does not imply, in general, that the entropy of the compound system depends only on $X+Y$. Since the way the total $X+Y$ is partitioned into the two subsystem is arbitrary, $S_{tot}$ remains a function of $X$ and $Y$ or equivalently of $X+Y$ and $X$ (or $Y$).

If one removes the constraint of isolation of the two subsystems, allowing mutual equilibrium between the two subsystems, the total system will reach in general a new state characterized by the same value of $X+Y$ and a specific value of $X$, say $X^*$. That's the value of $X$ which maximizes the entropy of the compound system with respect to the "constraint" variable $X$. In general, $X^* \neq X$.

So, one sees that in general the additivity condition is not expressed by $$ S(X+Y) = S(X) + S(Y) $$ for all $X$ and $Y$. Instead, for every total value $X+Y$, it holds only for the equilibrium value $X^*$ and $Y^*= (X+Y)-X^*$. For arbitrary $X$ and $Y$ the general relation between the equilibrium entropy of the compound system at internal equilibrium and the entropy of the original isolated systems is the super-additivity $$ S_{tot}(X+Y;X^*)=S(X+Y) \geq S(X) + S(Y) $$ expressing the principle of maximum of entropy. It is clear that super-additivity and homogeneity of degree one do not imply a linear behavior. Instead they imply the concavity of entropy.

Finally, we can notice that there is a situation where the linear behavior is actually present: it is the case of phase coexistence, corresponding to the case where the greater or equal condition becomes an equality.