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If notions of thermodynamics emerge from statistical mechanics, is there a way to derive the ideal gas law?

And this derivation would be more "fundamental"?

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Yes, there are several. Check the last section of Wikipedia page: https://en.wikipedia.org/wiki/Ideal_gas_law, where it is derived from the equipartition theorem. Another way would be through the partition function and computing quantities associated with it, such as internal energy, pressure, etc, as it is done here: http://farside.ph.utexas.edu/teaching/sm1/lectures/node65.html

The derivation from statistical mechanics is more fundamental inasmuch as you start with fundamental laws, not by phenomenological ones as it's done in thermodynamics.

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