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Consider a box with two compartments separated by a semipermeable membrane. The first compartment is initially at pressure $P_0$ and contains the solvent ; the second compartment is initially at pressure $P_1$ and contains the solvent and a solute.

In deriving the van't Hoff my textbook says that we consider the system at equilibrium , yet it says that the two pressures are different at equilibrium. Isn't equality of pressures of subsystems a condition for equilibrium?

Qmechanic
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lohey
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  • No, equality of partial pressures applies for a semipermeable membrane. Only one component can sample both volumes and come to equilibrium between them, the other (that can't pass the membrane) cannot sample the other side. – Jon Custer May 24 '23 at 17:04
  • @Jon Custer I don't quite understand , if we wait some time the concentrations hence the pressures should equalize , no? Hence the system is not at equilibrium to begin with – lohey May 24 '23 at 17:41
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    Is the membrane movable? Regardless of the presence of osmosis, a fixed barrier can certainly sustain a pressure difference. In addition, matter moves to the region of lowest chemical potential, not necessarily to the region of lowest pressure. – Chemomechanics May 24 '23 at 18:11
  • See, for example, https://physics.stackexchange.com/q/565575/ – Jon Custer May 24 '23 at 18:19
  • @Chemomechanics the membrane is fixed, – lohey May 24 '23 at 18:20
  • @Chemomechanics so you are saying that the initial pressures/concentration remain fixed for all time ? Shouldn't the system tend to evolve to a state where the concentrations/pressures are equal ? – lohey May 24 '23 at 18:25
  • Matter moves to equalize chemical potential gradients, not necessarily concentration or pressure gradients. – Chemomechanics May 24 '23 at 18:31
  • @Chemomechanics ok yes I agree, but the definition of equilibrium requires all of the parts of a system to have equal pressure ? Am I missing something here ? – lohey May 24 '23 at 18:38
  • There can be any number of "transport complexes" in which certain interactions are coupled so that in equilibrium some individual potentials do not have to be equalized, but their joint effort is. For example, a fluid with surface tension, where the internal potential is not the same as external potential if surface tension is included. Other example is a vertical column gas in gravitational field, where the gravito-chemical potential must be equalized but the chemical potential itself may have a nontrivial gradient in equilibrium. – hyportnex May 24 '23 at 19:23
  • @lohey If the boundary is fixed, the system can't progress any closer to mechanical equilibrium between two chambers at different pressures. – Chemomechanics May 24 '23 at 19:28

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In thermodynamics we can have partial equilibrium depending on the presence or absence of constraints. Suppose we divide a box into two parts: the equilibrium conditions between the two parts depend on the the properties of the wall:

  • If the wall is impermeable, fixed, and conducting, we only have thermal equilibrium. The pressure and chemical potentials in each part are not allowed to equilibrate.

  • If the wall is impermeable, movable and insulating, then we only have mechanical equilibrium. i.e., pressure is equalized but temperature and chemical potential are not.

  • In the osmotic experiment the wall is fixed, conducting and semipermeable. This leads to thermal and chemical equilibrium between two parts that are at different pressures.

Themis
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