In Section 3.3, we illustrated the thermodynamic relations that govern the conditions of equilibrium distribution of a species between two or more immiscible phases under thermodynamic equilibrium. In Section 4.1, we focus on the value of the separation factor or other separation indices for two or more species present in a variety of two-phase separation systems under thermodynamic equilibrium in a closed vessel. The closed vessels of Figure 1.1.2 are appropriate for such equilibrium separation calculations. There is no bulk or diffusive flow into or out of the system in the closed vessel. The processes achieving such separations are called equilibrium separation processes. Separations based on such phenomena in an open vessel with bulk flow in and out are studied in Chapters 6, 7 and 8. No chemical reactions are considered here; however, partitioning between a bulk fluid phase and an individual molecule/macromolecule or collection of molecules for noncovalent solute binding has been touched upon here. The effects of chemical reactions are treated in Chapter 5. Partitioning of one species between two phases is an important aspect ever present in this section.

The criteria for thermodynamic equilibrium in a single-phase system in a closed vessel subjected to an external force field were also developed in Section 3.3. Based on these criteria, we develop in Section 4.2 estimates of the separation achieved in a single phase in the closed vessel. These estimates are also developed in a closed vessel when an additional property gradient, e.g. density gradient, pH gradient, etc., exists across the vessel length. Focusing is the term often used to characterize the latter separation techniques. In this section, we cover in addition the extent of separation achieved when a temperature gradient is imposed on a single-phase system in a closed vessel not subjected to any external force field.