Vapor - Liquid Equilibrium (VLE)

Most fundamental equation for VLE is $$\mu_i^L = \mu_i^V$$

It states that for every component chemical potential must be equal in both liquid and vapor phase at equilibrium. However, in my thermo textbook, this equation is derived for isolated systems while usually when dealing with VLE, systems are closed since heat is usually exchanged with surroundings like in distillation for example. Does this equation hold also for closed systems and if so how can it be derived?

• The question rather is, if the system is in equilibrium or not. Boiling liquid definitely is not. More closely the system approaches equilibrium, more closely the relation of respective ch.p. approaches equality. Aug 28 '21 at 19:34
• That isn't the question. Question is what is written in the description. Aug 29 '21 at 9:00
• In fact, it is. Is the closed system in equilibrium or not? If not the equation can be true just approximately, depending on nature and extend of the deviation. Aug 29 '21 at 9:04
• And if it is in equilibrium, it is functionally equivalent to isolated system, as energy exchange is zero. Aug 29 '21 at 9:11
• We are dealing with closed system at equilibrium. Why should energy exchanged be zero at VLE in closed system? We can have water vapor in equilibrium with liquid and add heat (latent heat)to the system which will make liquid evaporate, but this won't change the fact that chemical potential off the water in two phases is the same. Aug 29 '21 at 10:27