Entropy and the criterion for equilibrium
The Second Law says that the entropy of an isolated system can only increase, or, equivalently, that the state of equilibrium in an isolated system is the one in which entropy takes its maximum possible value for the (constant) total internal energy content of the system (we will ignore forms of energy other than U). By definition, an isolated system does not interact with anything else, so we see that the condition dU = 0 (i.e constant internal energy) must be true of an isolated system. It must also be true that dV = 0, because otherwise the system would exchange work with its environment. Moreover, an isolated system must also be closed to the exchange of chemical components. If we call the mol number (= number of mols) of the ith component ni, then for an isolated system it must be dni = 0, for all i. In the mathematical notation of thermodynamics, the maximum entropy statement of the Second Law is often written as follows:
dSU,V , ni = 0 (4.71)
d2SU,V, ni ≤ 0. (4.72)
The first equation says that the first derivative vanishes at an extremum, whereas the second one says that, if the second derivative is negative at the extremum, then the extremum is a maximum.
It is important, however, to understand what these equations are actually saying. In particular, if U, V and ni are all constant, exactly what variable are we differentiating entropy relative to, so as to find an extremum for the function? Which begs the question: what (else) is entropy a function of? Or, in physical terms, why is entropy changing in the first place?
The way to think about this is by imagining that, initially, there are restrictions, or constraints, that prevent the system from changing towards equilibrium. For instance, a partition separating two different gases that can mix by diffusion, or two different electrolyte solutions that will precipitate a solid when they mix, or a perfect thermal insula- tor separating two bodies at different temperatures. When we remove the restriction the system changes towards equilibrium, and as it does so entropy changes as a function of some physical quantity that drives the displacement towards equilibrium. This could be, for example, exchange of gas molecules between the two sides of the container in Fig. 4.7, or exchange of ions between the electrolyte solutions, or exchange of internal energy between two bodies at different temperatures. We will analyze the latter exam- ple in formal mathematical language so as to clarify the meaning of equations (4.71) and (4.72).
At constant U, V and ni, the entropy of the isolated system varies as a function of the amount of internal energy exchanged between the two bodies. Let the internal energy, entropy and temperature of body j be Uj, Sj and Tj, respectively, and the corresponding properties of the isolated system composed of the two bodies be E, S and T. We will assume that there is no exchange of matter between the bodies, and that they are incompressible. We then have:
E1 + E2 = E
and, because the system is isolated (E constant):
dE1 = -dE2