The preface noted that “the book's conceptual core consists of four linked elements: entropy and the Second Law of Thermodynamics, the canonical probability distribution, the partition function, and the chemical potential.” By now, three of those items are familiar. The present chapter introduces the last item and, for illustration, works out a typical application. The chemical potential plays a significant role in most of the succeeding chapters.

Discovering the chemical potential

The density of the Earth's atmosphere decreases with height. The concentration gradient—a greater concentration lower down—tends to make molecules diffuse upward. Gravity, however, pulls on the molecules, tending to make them diffuse downward. The two effects are in balance, canceling each other, at least on an average over short times or small volumes. Succinctly stated, the atmosphere is in equilibrium with respect to diffusion.

In general, how does thermal physics describe such a diffusive equilibrium? In this section, we calculate how gas in thermal equilibrium is distributed in height. Certain derivatives emerge and play a decisive role. The underlying purpose of the section is to discover those derivatives and the method that employs them. We will find a quantity that measures the tendency of particles to diffuse.

Figure 7.1 sets the scene. Two volumes, vertically thin in comparison with their horizontal extent, are separated in height by a distance H. A narrow tube connects the upper volume Vu to the lower volume Vl.