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A system is said to be in thermodynamic equilibrium iff it is in thermal, mechanical and chemical equilibrium. That is it has everywhere the same $(T,P,μ)$. Does it make sense to say that two systems are in any type of equilibrium (thermodynamic or thermal)? Or we should consider them as one system? I don't understand if equilibrium is a property of the system we are interested in or a property of the system + surroundings.

What we mean when we say that a system is in an equilibrium state or in thermodynamic equilibrium? Do we mean with respect to surroundings?

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    $\begingroup$ I suggest to read about the "0-th law of thermodynamics" that defines equilibrium, and 'detailed balance' - the current practical best definition of equilibrium. $\endgroup$
    – Alexander
    Feb 3, 2021 at 2:43

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A system is in equilibrium if its properties do not change with time. A system isolated from its surroundings can itself be in equilibrium but not in equlilbrium with respect to its surroundings. An isolated system is one with no energy transfer (through heat or work) and no mass transfer to/from its surroundings. The isolated system can be in equilibrium, regardless of the conditions of its surroundings.

Consider a closed container of constant volume with pressure P and temperature T with surroundings at pressure P1 < P and temperature T. There is no work on the system and no mass transfer and no heat transfer (since the surroundings and system are at same temperature). The system is in equilibrium but is not in equilibrium with respect to its surroundings, since the system is at a different pressure.

Consider a closed perfectly insulated container of constant volume with pressure P and temperature T with surroundings at pressure P1 < P and temperature T1 > T. There is no work on the system and no mass transfer and no heat transfer (since the system container is perfectly insulated). The system is in equilibrium but is not in equilibrium with respect to its surroundings, since the system is at a different pressure and temperature.

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  • $\begingroup$ Should the term equilibrium with a constraint make sense? $\endgroup$
    – Anton
    Jul 9, 2021 at 11:30
  • $\begingroup$ I think this terminology may cause confusion. "Isolated system" is the accepted terminology. $\endgroup$
    – John Darby
    Jul 10, 2021 at 13:11
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In statistical physics one speaks about one system only one dealing with the microcanonical distribution. The canonical and the grand canonical distributions are explicitly derived for the case where the system of interest interacts with another system or where the system of interest is a part of a bigger system. This other system often referred to as a thermostat. The equality of temperatures, pressures and chemical potentials between the two systems are referred to as thermal equilibrium, mechanical equilibrium, and diffusive equilibrium.

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  • $\begingroup$ Sorry, I have not go into in statistical physics. I have covered some basic courses of classical thermodynamics. $\endgroup$ Feb 2, 2021 at 23:18
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In brief, equilibrium is a state or condition of two or more thermodynamic systems under consideration which tells about the way they interact with one another w.r.t. temperature, pressure and chemical potential (concentration).

Yes, it make sense to say two or more thermodynamic systems under consideration are in a particular equilibrium i.e. thermal, mechanical, chemical or thermodynamic equilibrium.

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  • $\begingroup$ So when we say that the system under consideration is in a equilibrium state we mean that its state variables aren't change irrespective of the fact that it may not be in thermal equilibrium with its surroundings? $\endgroup$ Feb 2, 2021 at 23:16
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It is possible to have 2 systems in equilibrium. For example, a block of ice floating in a glass of liquid water inside a thermally isolated container.

We can say that they are 2 systems because one phase is different from the other. But the ice needs energy to melt and there is no source. And for the water to freeze, some energy must be released, and there is no sink.

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