Constraints in Equilibrium Processes

Here we discuss qualitatively some aspects of equilibrium. We think of simple coupled systems with k components that can be described with the variables S, V, Hi. Hk. In equilibrium, the intensive variables will be adjusted dependent on the constraints of the system. For example, if there is no constraint on the volume of a single system, i.e., dV 0, the pressure will approach zero, in order to approach an extremum of the energy, df/ 0. The other extreme is a fixed volume, d V=0, where the pressure can take any value for df/ - 0. The same holds for the other variables in the set needed to describe the system in the sense of thermodynamics. 

If two subsystems ( ) and ( ) are coupled so that the total volume is constant, then the constraint equation in differential form is dV - - dV = 0, which will demand equal pressure in both subsystems. 

In the same manner, if two subsystems can exchange matter of kind (I), then dn[ + dn (=0, and the chemical potentials become equal, /rj We have such a situation in an osmotic cell. In such a cell, usually two components are present, and only the exchange of one component is allowed. The semipermeable wall is in a fixed position. For this reason, the pressures in both compartments are different. If we would allow the semipermeable wall to move, then this wall would move in the direction of decreasing pressure. Actually, one of the subsystems would disappear finally. 

If there is no wall between the two systems, then the pressures will be equal in equilibrium. We can try to treat the equilibrium in a distillation apparatus, analogous to isopiestics. Actually, there is no thermal equilibrium as the temperatures are different. However, the pressures must be equal because the system would try to expand through the cooler into the other system otherwise. In order that there is no flow of the distilling matter, the chemical potential in the high-temperature region must be equal to the chemical potential in the low-temperature region, even when the temperatures are artificially kept different. 

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