Let’s Derive the Ideal Gas Law from Scratch!
With nothing but a good model, a few definitions, and some math, we’ll derive a fundamental relationship of chemistry.
Anyone who has ever taken a chemistry class has seen the Ideal Gas Law:
Chemistry classes tend to teach the Ideal Gas Law as a combination of Boyle, Charles, Gay-Lussac, and Avogadro’s Laws. Although they derived these laws empirically, we’re going to take a different approach in this article. We’re going to derive the Ideal Gas Law from nothing but statistical mechanics, a few laws, and some definitions. We’re going to use the concept of entropy, so check out my article on entropy if you’re unfamiliar with entropy. If you think entropy measures disorder, check out the article because it doesn’t.
The Rules
For this derivation, I will only use the definition of an ideal gas, the Laws of Thermodynamics:
- First Law: In a closed system, the change in internal energy is equal to the heat put into the system minus the work the system does on its environment.
- Second Law: An isolated system will tend towards its most likely macrostate.
(I won’t need the Third Law and I won’t use the Zeroth Law directly, so I’ve left them out.)