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How did people come up with the formula: $$PV = nRT = k_BT \qquad$$?

I see two possible ways:

1.) We did some measurements and defined $$kB:=\frac{PV}{T},$$ remarking that $\frac{PV}{T}$ is constant.

2.) We had $k_B$ from somewhere else, derived $PV$ must equal $k_BT$. Then we made some measurements and voila it seemed to be true.

3.) Neither of the two is true. If so please explain in detail, what happened.

Remark: Beside of this question I want to understand how people came up with the equipartition theorem. Assuming that we found the following formulas independently: $$(1)\ PV = k_BT \qquad (2)\ PV = \frac{2}{3}K \qquad with\ K\ the\ kinetic\ energy\ of\ the\ gas.$$ One could argue that we get $N\frac{3}{2}k_BT = K$, so there must be $\frac{1}{2}k_BT$ kinetic energy per molecule. And assuming only translation we get the equipartition theorem. Or probably this theory of mine is again false...

I guess I kinda mix up some things a bit, but I think I made it clear what I want to understand.

Remark 2: It is important to me to understand how people came up with these ideas. It is interesting if you note that certain derivations come up with the same result and therefor confirm the idea, but what I actually want to know is, how those formulas where discovered in the first place.

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    $\begingroup$ duplicate: physics.stackexchange.com/q/215722 $\endgroup$ – user154997 Oct 8 '17 at 16:58
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    $\begingroup$ My question resembles the question you mentioned. Still there are important differences. The question you linked asks about how to derive the $\beta$. The answers given, which I wouldn't have accepted, give very general explanations. The accepted answer in fact says there were a lot of discoveries at the time and mixing them up gives us the equation in question. My question in particular asks how these ideas led to the ideal gas equation. Not so much of a duplicate imo. $\endgroup$ – Felix Crazzolara Oct 8 '17 at 17:04
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    $\begingroup$ So your question is very much historical then. Then combined gas law and Avogadro law should answer your first question. $\endgroup$ – user154997 Oct 8 '17 at 17:20
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    $\begingroup$ If it is called a 'law' then there are good odds it was first discovered empirically, and the derivations then serve to validate a candidate for a more fundamental theory rather than the other way 'round. $\endgroup$ – dmckee Oct 8 '17 at 17:26
  • $\begingroup$ @LucJ.Bourhis It seems so. Obviously it's not possible to cover everything that happened, but I'd like to see a brief summary of the most important events/laws discovered and their conclusions. $\endgroup$ – Felix Crazzolara Oct 9 '17 at 13:38

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