# Vapor-liquid equilibrium

The thermodynamics of phase equilibrium establish relationships between the different properties of the system that allow to describe, in a quantitative way, the state of the equilibrium between homogeneus phases that can exchange matter freely.

The final equilibrium compositions of each phase depend on several variables: temperature, pressure and starting compositions.

The figure below shows an ebulliometer. It is made of a vessel (D), where we introduce the dissolution that we are going to bring to the boiling point, and a refrigerator (A), in which the vapor will be condensed. We then take samples at (B) and (F) that allow to evaluate the composition of the two phases in equilibrium.

A) reflux condenser, B) dropper, C) thermometer socket, D) bulb, E) heating compartment with heating coil, F) draining valve.

Next, we will analyze the isothermic vapor-liquid equilibrium. We will use the following notation:

• xi: molar fraction of the component i in the liquid phase.
• yi: molar fraction of the component i in the vapor phase.
• T: boiling temperature.
• P: vapor pressure of the dissolution.
• Pi0: vapor pressure of the pure i component.
• Bii: second coefficient of the virial of component i.
• Bij: second coefficient of the virial of the mix.

We'll suppose that the vapor mix can be treated as a mix of real gases whose behaviour (as well as the behaviour of each alone) can be described in an adequate way by the virial equation of state. In this case, the activity coefficients of each of the components in the vapor phase are given by:

while for the liquid phase we have:

$\text{ln}{\gamma }_{1}^{L}=\text{ln}\frac{{y}_{1}P}{{x}_{1}{P}_{1}^{0}}+\frac{\left(P-{P}_{1}^{0}\right)\left({B}_{11}-{\nu }_{1}^{0L}\right)+{\delta }_{12}{y}_{2}^{2}P}{RT}$

$\text{ln}{\gamma }_{2}^{L}=\text{ln}\frac{{y}_{2}P}{{x}_{2}{P}_{2}^{0}}+\frac{\left(P-{P}_{2}^{0}\right)\left({B}_{22}-{\nu }_{2}^{0L}\right)+{\delta }_{12}{y}_{1}^{2}P}{RT}$

where:

${\delta }_{12}=2{B}_{12}-{B}_{11}-{B}_{22}$