# When do thermal and chemical equilibrium not coincide?

What is an example for a system, which is in chemical equilibrium, but not in thermodynamical equilibrium?

And what about the other way around?

It seems to me, that as long as Parameters like temperature $T$ and pressure $p$ are changing, there cannot not be chemical equilibrium, since chemical reactions depend on these quantities temperature and pressure. Hence, if there is chemical equilibrium the parameters are not changing, enforcing thermodynamical equilibrium.

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I think generically the answer is: they're in thermal equilibrium if they have equal temperatures, in chemical equilibrium if they have equal chemical potentials, and in both thermal and chemical equilibrium if both are equal. They do not need to be at equal pressures, which is what I assume you mean by $p$. Imagine a tall column of gas in a gravitational field. Divide it (mentally) into two systems, the one at the top and the one at the bottom. The pressure at the top and bottom will differ, but they will be in chemical equilibrium.

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Chemical equilibrium is a subset of thermodynamic equilibrium. So there are no examples of systems in thermodynamic equilibrium that are not also in chemical equilibrium.

Here's an example of a hypothetical system in chemical equilibrium but not in thermodynamic equilibrium. A system containing two phases, one a condensed phase containing two equilibrating chemicals having no significant volume changes accompanying the reaction, the other an inert gas phase, under varying external pressure. The system is always in thermal equilibrium with its surroundings.

This system would not be in thermodynamic equilibrium because as the external pressure changed the gas phase would be changing its volume. It remains in chemical equilibrium because the condensed phase housing the chemical equilibrium would not respond to the pressure change, and the gas phase hosts no chemical equilibria.

This is basically a cheat because the system is, practically speaking, two separate systems that are in physical contact. Maybe someone else will be more imaginative.

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