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Does pressure in the ideal gas law and other gas laws (Boyle’s, Charles’s, and Gay-Lussac’s) mean the external or internal pressure?

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Either. A difference between external and internal pressure means that the volume is changing (indeed, the change is accelerating), and this brings you out of the range of validity of thermodynamics. Thermodynamics as it stands (barring specialist topics) is a priori valid only for a system in equilibrium; that certainly excludes the system’s volume changing. We circumvent this restriction somewhat by introducing quasistatic processes which are so slow the system is essentially in equilibrium at each particular moment, but this also precludes a significant difference in pressure.

How slow, then, is “too slow”? The answer—similar for all statements of this sort throughout physics—is that you can’t really tell without having some theory of non-equilibrium systems so that you can compute or estimate the true answer from it and see how big a difference it makes. We do, after all, neglect some states of non-equilibrium: for example, every atom that is not iron is in fact out of equilibrium, as one can extract energy from it by either fusion or fission. But for this to happen spontaneously in Earth conditions is astronomically, nay, cosmologically unlikely; so we might as well ignore these processes. That is not to they are always irrelevant to the thermodynamic problem at hand: the whole science of how stars, galaxies, clusters, etc. are formed is built around the interplay of different equilibration times.

Of course, the people who were building thermodynamics and the theory of ideal gases did not think of it this way: they just made measurements under more-or-less controlled conditions and made inferences from the results. Thermodynamics is phenomenological this way: it is a collection of empirical statements on how the macroscopic world works under certain conditions, not an explanation of why it must work this way. It is again only by recourse to a deeper theory that one can understand why the conditions are there.

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