Thermodynamics deals with large systems that consist of more particles than can be reasonably dealt with in a usual mechanics approach. We shift our focus from the variables that govern each individual particle to those that describe the system as a whole.

In the first SparkNote, we delved into the quantum basis for a statistical approach to thermodynamics, and presented the four Laws that can be viewed as postulates or quantum-verified relations and truths. We developed two variables that can be used to describe a large system, namely the entropy and the temperature.

We will pick up where we left off, defining more variables to describe a system. We will look at the pressure of a system and see how it relates to what we have already done. We will define the notion of the chemical potential. We will collect all of the variables needed to specify the state of a large system, and note the distinction between intensive and extensive variables.

Having all of the variables before us, we will look at what is known as
the thermodynamic identity, a crucial equation that we will use
throughout our entire study of thermodynamics. We will utilize a
mathematical tool known as the Legendre transform to assist in
defining three other forms of energy, namely the free energy, the
Gibbs free energy and the enthalpy in terms of the energy
*U*
and the thermodynamic variables. We will come to understand why there
are so many formulations of the energy, and realize how useful these
different forms can be for solving problems.

We will revisit the thermodynamic identity and look at what each term represents. This analysis will become especially important when we look at engines later. Finally, we will utilize some clever mathematical tricks to obtain the Maxwell Relations.