# How to tell if two system has same temperature?

• KFC

#### KFC

Two systems are separated 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

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