I am very confident that Feynman is talking about the zeroth law of thermodynamics here.

Feynman is discussing the classical Brownian motion experiment of pollen grains swimming in water in terms of kinetic theory. However, kinetic theory is only valid in ideal gases, while in liquids there are many additional hydrodynamic effects. Feynman argues this doesn't really matter, since if the grain is in equilibrium with water, it might as well be in equilibrium with "a gas of pellets". This argument is based on the zeroth law of thermodynamics.

The zeroth law of thermodynamics, introduced by Boltzmann in his book on "Theory of heat", is indeed very complicated to proof from mechanical theory. A thorough mathematical proof has been published in 2012. I am not an expert enough to judge whether this proof is comprehensive or if it will be developed further. Also, I am not aware of the proof by Ludwig Boltzmann that Feynman mentions. A proof derived from quantum mechanics, on the other hand, is relatively straightforward and well known since the 1980s (see Gorini et al. 1984).

I don't think Feynman refers to the equipartition theorem or the H-theorem here. Feynman has already touched on the equipartition theorem in Chapter 39-4. In chapter 41-1, he goes into further detail and gives additional examples. He also mentions that "if a thing is once in equilibrium it stays in equilibrium". This observation is closely related with entropy and the second law of thermodynamics, but it is also the basic definition of equilibrium: The properties do not change over time. If they would, it wouldn't be an equilibrium. Feynman uses this for this ad hoc derivation of the zeroth law, but it is not the assumption he tacitly made earlier.

The relationship for physicists is:
$$PV=NkT,$$
where $N$ is is the number of gas molecules/atoms, and $k$ is Boltzmann's constant. In chemistry they normally convert the $Nk$ to $nR$, where $n$ is measured in moles instead of being a simple count, and $R$ is the universal gas constant.

So, the proportionality from pressure to mass is incidental and depends on what the gass is made of, since, for a particular pure type of gas, you can convert from mass to number and back.

The reason that pressure is proportional to the number of atoms is because pressure is the average force per unit area exerted on the walls of the container. That force is cause by individual collisions of gas molecules with the walls, and the force is directly proportional to the collision rate. Naturally, if you double the number of molecules, all other things being equal, you'll double the collision rate, doubling the pressure. Thus pressure is proportional to particle number.

The interesting question is that when you have a collision the impulse, and thus the force, depends on both the mass and velocity of the colliding particle. So why is it proportional to number and not mass? Well, the reason is because temperature in gasses is a measure of the kinetic energy of the molecules. So, if you have two different gases at a given temperature, volume, and number of molecules, the one with more massive molecules will be moving slower in such a way that they have the same pressure.

## Best Answer

The law can be derived from the kinetic theory of gases. Several assumptions are made about the molecules, and Newton's laws are then applied. For $N$ molecules, each of mass $m$, moving in a container of volume $V$ with a root mean square speed of $c_{rms}$, the pressure, $p$, exerted on the walls by gas molecules colliding with them is given by $$pV=\tfrac 13 Nmc_{rms}^2.$$ Sir James Jeans (in

The Kinetic Theory of Gases) has a simple argument involving molecules exchanging energy with a wall (modelled as spheres on springs!) to show that for gases at the same temperature, $mc_{rms}^2$ is the same. In other words, gas temperature is determined by $mc_{rms}^2$. So for a gas at constant temperature, $c_{rms}$ is constant, and if we keep $N$ constant, too, we deduce that $pV$ is constant.