[Physics] Can we derive Boyle’s law out of nothing

gasideal-gasstatistical mechanicstemperature

My textbook states Boyle's law without a proof. I saw Feynman's proof of it but found it to be too handwavy and at the same time it uses Boltzmann's equipartition theorem from statistical mechanics which is too difficult for me now.
So to state roughly what Boyle's law is, it states that at a constant temperature and mass of gas,
$$PV=k$$
Where $P$ is pressure and $V$ is the volume and $k$ is constant in this case.

Is there a proof for this that isn't based on any other gas law, perhaps based on Newtonian mechanics?

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.

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