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Is a system said to be in thermal equilibrium with respect to another system when no net heat exchange takes place between them?

Thus, is it true that system $A$ is in thermal equilibrium with respect to system $B$ even if the macroscopic properties of system $A$ such as volume, pressure, temperature etc. are not constant?

Note:- The striped walls are diathermic and the remaining walls are insulating. The arrow indicates the flow of heat. enter image description here

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  • $\begingroup$ Two bodies are in thermal equilibrium if they have the same temperature. $\endgroup$ Jan 4, 2020 at 8:36
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    $\begingroup$ @aditya_stack No, thermal equilibrium is established when there is no net exchange of heat between the systems. $\endgroup$
    – Sam
    Jan 4, 2020 at 8:42
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    $\begingroup$ Is same temperature not a sufficient condition for the same? $\endgroup$ Jan 4, 2020 at 13:01

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Thermodynamic equilibrium can be defined as the terminal state of a macroscopic system when all external influences have been removed, and all the properties of the system have become time-independent. The macroscopic state of a system at thermodynamic equilibrium is characterized by a few physical quantities (thermodynamic variables) whose values do not depend on the system's history.

Mention to "external influences" is important to exclude stationary systems which do not correspond to the system at equilibrium (for example, a piece of metal with different temperatures at the extreme sides; in that case, it is possible to get a stationary state but in the presence of a flux of heat).

The apparently simple concept of thermodynamic equilibrium could not be simple to check in real systems. Feynman, at the beginning of his book on Statistical Mechanics, wrote that we have thermal equilibrium.

... if all the "fast" things have happened and all the "slow" things not.

The reason is that some very slow processes on the time scale of lab measurements may be in place and eventually modify the system's state over a geological scale.

Now, in the specific case of your system, it is a composite system. Assuming that the walls without strips are not only insulating but also perfectly rigid and not permeable to diffusion of particles, subsystem B, waiting long enough, will reach thermodynamic equilibrium independently of what happens elsewhere.

Subsystem A won't be at equilibrium until fluxes are in place. However, assuming that out of the diathermal wall, there is a thermostat, it could equilibrate with the thermostat sooner or later. At this point, the global system is at thermodynamic equilibrium until some change on external or internal conditions is made.

It is meaningless to speak about equilibrium of subsystem A with respect to subsystem B if subsystem B is fully isolated. In general temperature of A and B will differ but, due to the insulating wall, this is not hampering the thermodynamic equilibrium of the composite system. If at some time, the separation wall between A and B would be transformed from insulating to diathermal, at that point, the two subsystems won't be, in general, at thermal equilibrium, and the global system will evolve towards a state where the temperature of A and B will coincide.

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  • $\begingroup$ Thank you sir for your answer. My basic question is that since the wall between A and B is insulating, no heat exchange can take place between A and B and hence, shouldn't A be in thermodynamic equilibrium with B. Note I am specifically talking with respect to B and not w.r.t the surroundings. Thanks! $\endgroup$
    – Arun
    Jan 4, 2020 at 11:37
  • $\begingroup$ @Arun B, being an insulated system, does not depend and cannot influence the equilibrium of any other part of the system. However, I feel that the real issue is that you insist in speaking about mutual equilibrium between A and B. If B is an isolated system the is no point at all to speak about an actual equilibrium of A with B. What can be said is that if suddenly the wall separating A and B would become conducting, there would be a very low probability that the system after removal of the constraint will remain at equilibrium. It will happen only if $T_A=T_B$. $\endgroup$
    – GiorgioP
    Jan 4, 2020 at 13:51
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Systems are in thermal equilibrium with one another if there is no net flow of heat when they are connected by a path that allows heat transfer. If I understand you correctly the wall between A and B is insulating so that wouldn’t permit heat transfer. If it would allow heat transfer then in order for A and B to be in thermal equilibrium they would have to be at the same temperature.

Hope this helps.

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When two or more bodies are at the same temperature, they are in thermal equilibrium. This is regardless of volume, or pressure, as long as it is constant. When the temperatures are the same, no net heat energy exchange is possible, as heat only flows from a hotter to a colder environment. In your diagram A could reach thermal equilibrium with the surroundings, but not with B unless the A/B wall were diathermal. See https://en.wikipedia.org/wiki/Laws_of_thermodynamics#Zeroth_law for more.

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The intent of stating that one system is/is not in thermodynamic equilibrium with another system requires that the two systems must exchange information with each other.

As drawn, System B is isolated. System B does not present any information about the temperature of its contents. Therefore, we cannot define it to be in thermal equilibrium with anything else.

As drawn, the walls of System B are presumed to be rigid. System B does not present any information about the pressure of its contents. Therefore, we cannot define it to be in mechanical equilibrium with anything else.

As drawn, System B is isolated. System B does not present any information about the chemical potential of its contents. Therefore, we cannot define it to be in chemical equilibrium with anything else.

Change the character of the walls around System B. Then, we have an entirely different level of discussion.

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