I can use PV = nRT without trouble but it feels handed down. Our notes say it comes out of kinetic theory by considering molecules hitting a wall, then skip to the result.
What are the actual steps? And where does temperature enter, given that a single molecule does not have a temperature?
Accepted Answer
Box of side L, one molecule, speed component vx towards one wall.
Each bounce reverses vx, so the momentum handed to the wall is 2mvx. The molecule returns after a round trip of 2L/vx. Average force from that one molecule:
F = 2mvx / (2L/vx) = m vx² / L
Add up N molecules and use the mean of vx²:
F = N m ⟨vx²⟩ / L
Pressure is force over area L², so P = N m ⟨vx²⟩ / L³ = N m ⟨vx²⟩ / V.
No direction is special, so ⟨vx²⟩ = ⟨v²⟩/3, which gives
PV = ⅓ N m ⟨v²⟩ = ⅔ N (½ m ⟨v²⟩)
The bracket is the average kinetic energy per molecule. Put ½m⟨v²⟩ = ³⁄₂ kT and you have PV = NkT, which is PV = nRT with R = N_A k.
Your second question is the more important one, and the answer is that temperature does not come out of the derivation at all.
½m⟨v²⟩ = ³⁄₂ kT is a definition of what temperature means for an ideal gas, not a result. Everything before that line is mechanics. That line is where thermodynamics is attached.
You are right that a single molecule has no temperature. Temperature is a property of the distribution of speeds, so it needs enough molecules for a distribution to mean anything.
Worth noting what got assumed on the way: no molecular volume, no forces between molecules, perfectly elastic collisions. That list is what "ideal" means, and why real gases drift off it once you squeeze or cool them.
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