# Thermodynamics of VLLE

The criteria for phase equilibria in a closed, nonreacting multi-component system at constant energy and volume is that the pressure must be same on both sides and the partial molar Gibbs free energy of each species must be same in each phase (Sandler, 1999). The criterion that forms basis of phase equilibrium calculation is expressed mathematically by Equation 4.1.

*
*

For the VLLE, the vapor phase is in equilibrium with individual liquid phases and liquid phases are in equilibrium between them too. Therefore,

*
*

(liquid Lj, *L _{n}* and vapor V)

In terms of fugacity, we can write,

*
*

We explain the fugacity term by the activity coefficient model. By neglecting fugacity coefficient corrections and considering total pressure as well as vapor pressure of the species are sufficiently low,

Further, from LLE, we get,
*
*

Thus, combining Equations 4.4 and 4.5, we get the equilibrium relationship for VLLE.

*
*

where *K _{i}* is known as the distribution coefficient and is the ratio of solute concentration in two phases. When solutes are added into two partially miscible or immiscible solvents, these unequally distribute in the two liquid phases. The distribution coefficient can efficiently be calculated from excess Gibbs free energy models which will be discussed in Section 4.4.