Balance equations for vapor-liquid equilibrium

To solve an isothermal vapor-liquid equilibrium (VLE) flash, we need the equilibrium condition where the chemical potential for each component is the same in both phases:

Equivalently, the fugacity of each component is the same in both phases:

For equations solved with Newton methods, we use the following material balance equation for each component:

The mole fractions in the liquid and vapor phase must sum to unity:

Therefore, the following equation gives the equilibrium between the vapor and liquid phase: