1.6 Basic equation for gas-liquid equilibrium

2024.9.04

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1.6 Basic equation for gas-liquid equilibrium

The assumption used here was that ‘at pressures as low as atmospheric pressure, gases behave as ideal gases’. However, in high-pressure gas-liquid equilibrium or in systems where compounds aggregate in the gas phase, they cannot be treated as ideal gases. Let us explain their treatment.
The basic equation for the vapour-liquid equilibrium of an N-component mixture is given by the following equation from thermodynamics

f_i^V (Fugacity of component i in the gas phase.）= f_i^L (Fugacity of component i in the liquid phase） (1-8)
i = 1, 2, …, N

Here f stands for fugacity, the superscript V for the gas phase, L for the liquid phase and i for the component i in the mixture.
Fugacity is probably a very unfamiliar term.
If I try to talk about it theoretically, I will be stumped from the start.

First, let’s make sure we understand the concept of equilibrium.
In the case of gas-liquid equilibrium, the gas phase and the liquid phase are in contact with each other. Molecules jump out of the liquid phase and into the gas phase. However, molecules also jump from the gas phase into the liquid phase. After a sufficiently long time at a certain temperature and pressure, the speed of the molecules jumping out and returning is the same, and movement apparently ceases. Such a state is called having reached equilibrium.

Fugacity describes how much a real molecule is willing to escape from one phase to another (e.g. liquid, solid or gas phase). The idea of fugacity originally originated with Willard Gibbs, who used the idea of escape tendency for thermodynamic equilibrium. It was introduced into vapour-liquid equilibrium by Gilbert Lewis. This is the same Gilbert Lewis who came up with the activity coefficient.

Since ideal gases are assumed to have no intermolecular forces, pressure arises solely from kinetic energy. The chemical potential μ_i of component i is expressed as

μ_i = μ_i⁰ + RTln P_i / P⁰ 　　(1-9)
P_i ∶Partial pressure of component i

In contrast, real gases have intermolecular forces, so it is necessary to add a correction for them.
Since it is difficult to evaluate the intermolecular forces separately, f_i is defined as the ‘effective pressure’ instead of the partial pressure of the P_i component i. This means that the chemical potential μ_i of component i can be used as it is in the ideal gas equation. Then, the chemical potential μi of component i can be directly used in the ideal gas equation.

μ_i =μ_i⁰ +RTln f_i / P⁰ 　　(1-10)

It is the same concept as the ‘effective molar concentration’ of the active volume.
In low-pressure vapour-liquid equilibrium without gas-phase association, the fugacity is equal to the partial pressure.

f_i^V= P_yi　　 (1-11)

The fugacity of component i in real solution is expressed at low pressure as follows.

f_i^L = γ_i x_i P_i^S 　　 (1-12)　

The right-hand side is a ‘real solution’ version of Raoult’s law.

Fugacity indicates ‘how much it is willing to escape’ from one phase to another (e.g. liquid, solid or gas phase). The ease of escaping from the liquid phase to the vapour phase is the activity coefficient, so the equation is valid.

Equation (1-12) says that when the fugacity of the gas phase is balanced by the fugacity of the liquid phase (when the chemical potentials are the same), the apparent movement stops and equilibrium is reached.
It should be remembered that fugacity is ‘the tendency to escape from a phase’.

If the mixed liquid is immiscible, the fugacity between the three phases must be balanced with two phases of liquid and one phase of gas. This is the basic idea in such cases. (Dealt with in heterogeneous systems).

Next section: 1.7 Handling of association systems

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