Suppose that we have two systems, each consisting of the same single chemical species, which come into thermal and diffusive contact (meaning that particles can move between them). Note that thermal contact alone prohibits such an exchange. Imagine what happens when you touch a radiator - there is certainly a thermal contact, as you feel the heat of the radiator. However, there isn't much of a diffusive contact, as your hand doesn't suddenly melt into the radiator and become replaced in part by metal!

Now, our chemical intuition tells us that the particles will flow from the denser system to that which is less dense. We will formalize this notion by introducing the chemical potential μ, which governs how particles will flow between two systems. For now, we can think of the chemical potential as follows:

The chemical potential can be defined in different manners as well, and we will address this shortly.

Nevertheless, we can say now that particles will flow from a system with a higher chemical potential to a system with a lower chemical potential if the two are in diffusive and thermal contact.

The final variable to define lends itself nicely to an intuitive understanding as well. We commonly think of the pressure as the force per unit area in physics. While the units work out to be the same, we define pressure in a different manner altogether here:

The equation states that pressure is related to the way that the energy changes as the volume changes. We will explore how this correlates with our intuitive notion in the problems at the end of the section.

A key distinction needs to be made here among the variables. Some of the variables we call extensive, if they obey the following property - upon the doubling of the system, they too double. We can quickly see that the volume V doubles if the system doubles, as do the number of particles N and the entropy σ. If a variable remains constant upon the doubling of the system, then we call that variable intensive. The temperature τ, the pressure p and the chemical potential μ are all intensive. We will decide upon which category to put the energy in the problems at the end of the section.