In thermodynamics, we often ask questions about the energy of the system. Here we will discuss the energy that we've already introduced as well as alternate formulations of the energy of a system.
Suppose that we seek the energy of a system U in terms of its usual variables, σ, V, and N. Unfortunately, we can't write a closed solution for U in terms of those three variables. But not all is lost. We can utilize the mathematical tool known as the differential. Then we get:
So far, this may not look helpful. But if you glance back at our previous definitions of temperature, pressure, and chemical potential, we can rewrite the above:
The result is known as the Thermodynamic Identity, and is the most basic equation in our study of thermodynamics. Notice that there is great parallel structure to the equation. All of the extensive variables appear as differentials, while the intensive variables appear alone. Note that U is still a function of just the three extensive variables, since we can think of the other three "variables" as derivable from the three extensive.
We can use another mathematical tool here to make the Thermodynamic Identity even more useful. The Legendre Transform allows us to make a variable change in our definition of U. After all, suppose we don't want the energy as a function of the three variables above, σ, V, and N.
We will utilize the Legendre Transform minimally, and not delve into the underlying mathematics. The basic idea is that you can define a new function that is related to the original by an added product of two correlated terms. Let us make this explicit by using it.