# How to tell if two system has same temperature?

## Main Question or Discussion Point

Two systems are seperated by a membrane which allow heat and partcle exchange. Both systems have same volume but different (interaction) potential. Do these two systems have same temperature?

I think they have same temperature because such system could come to equilibrium when time is longer enough, right?

ps. The whole system is isolated

## Answers and Replies

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Mapes
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The standard approach here is to write the internal energy of the systems in differential form (e.g.,

$$dU = T_1\,dS_1 +T_2\,dS_2- p_1\,dV_1 - p_2\,dV_2+ \mu_1\,dN_1+ \mu_2\,dN_2+ E_1\,dq_1+ E_2\,dq_2+\dots$$

where E is electrical potential and q is charge), solve for dS, and set this to zero to find what would happen at equilibrium.
I think they have same temperature because such system could come to equilibrium when time is longer enough, right?
Not necessarily; equilibrium doesn't mean that the properties are homogeneous. If energy is coupled to charge (which would be the case for individual particles), then a difference in electrical potential between the two systems could lead to a temperature gradient, as exemplified by the thermoelectric effect.

The standard approach here is to write the internal energy of the systems in differential form (e.g.,

$$dU = T_1\,dS_1 +T_2\,dS_2- p_1\,dV_1 - p_2\,dV_2+ \mu_1\,dN_1+ \mu_2\,dN_2+ E_1\,dq_1+ E_2\,dq_2+\dots$$

where E is electrical potential and q is charge), solve for dS, and set this to zero to find what would happen at equilibrium.

Not necessarily; equilibrium doesn't mean that the properties are homogeneous. If energy is coupled to charge (which would be the case for individual particles), then a difference in electrical potential between the two systems could lead to a temperature gradient, as exemplified by the thermoelectric effect.
I fogot to say, no interaction b/w particles