One of the most important principles of thermodynamics is the equipartition theorem:

A system in thermodynamic equilibrium will have an internal thermal energy of \(\frac{1}{2}k_BT\) in each degree of freedom.

But there’s a subtlety here: what exactly are degrees of freedom? There are (at least) two slightly different kinds:

• A mechanical degree of freedom is any way in which a system can freely change its spatial configuration
• A thermodynamic degree of freedom is any way in which a system can freely increase its stored energy

The degrees of freedom the equipartition theorem mentions are the thermodynamic variety. It’s important to know this because the equipartition theorem predicts the heat capacity for many substances in the high-temperature limit, and if if you count the wrong kind of degrees of freedom, you’ll get the wrong answer.

Diatomic molecules

One simple example of this is a diatomic molecule. If you want to figure out how many mechanical degrees of freedom this molecule has, you just count up all the various distances that you need to completely specify the molecule’s spatial configuration. They break down like this:

• Three positions \(x\), \(y\), \(z\) to specify the center of …