##### 1.1 Liquid-Vapor Equilibrium (LVE)

Separation processes are based on the phase equilibrium concept. This is a key concept within the matter treated in the TVT lecture. For this reason the following section is dedicated to the understanding of what physically "equilibrium" means.

Let's consider two phases (vapor and liquid), which are in contact with each other. What happens at the interface on a molecular scale?

Liquid molecules are continually vaporizing while gas molecules are continually condensing. When the two phases, liquid and gas, are not at equilibrium, the different chemical species present (i.e. A and B in Fig. 1) vaporise and condensate at different rate. Each phase has its own temperature, pressure and composition.

After a certain **contact time**, the two phases reach an equilibrium: temperature, pressure and composition **cease to change**.

On a microscopic scale, both liquid and gas molecules keep on evaporating and condensating continually; but each species evaporates and condensates with the same rate so that, actually, on a macroscopic scale, no further changes are observed.

From the here above observations, we can conventionally identify the equilibrium condition with three types of equilibrium conditions: thermal, mechanical and of the chemical potential.

If there is thermal equilibrium, the heat transfer stops and the temperatures of the two phases are equal:

T_{liquid}=T_{vapour}
If there is mechanical equilibrium, the forces between liquid and vapour are balanced. In case of staged separation processes, this usually implies that the pressures of the two phases are equal:

P_{liquid}=P_{vapour}
In phase equilibrium, the rate at which each species **i** is vaporizing is equal to the rate at which it is condensing. This means that there are no changes in composition. If temperature and pressure are constant, equal rates of vaporization and condensation require a minimum of free energy of the system. This implies that:

m_{i}^{liquid}=m_{i}^{vapour}
where m is the chemical potential of the species **i**. This is a function of temperature, pressure and composition of all the species present in the phase:

m_{i}^{liquid} (T,P,x) = m_{i}^{vapour}(T,P,y)
and because the chemical potential is linked with the fugacity (dm_{i}=RTdlnf_{i}), if the chemical potentials are the same also the fugacities must be the same. The above equality can be rewritten in terms of iso-fugacity condition:

liquid fugacity for component i = vapor fugacity for component i

f_{i}^{L} (x, T, P) = f_{i}^{V} (y, T, P)
The iso-fugacity condition can be written in the most general form as follows:

(1)
x_{i} g_{i}(x, T, P) f ^{L}_{i,pure}(T, P) = y_{i} f_{i}(y, T, P) P
where g_{i} is called activity coefficient and f_{i} is the fugacity coefficient for the vapor phase. They refer to a specific component i and in the most general case, they depend on temperature, pressure and **all compositions** (please note the x and y are vector compositions).

The expression (1) can be simplified for most of the practical cases by taking into account the following **assumptions**:

- The liquid fugacity of the pure component i can be expressed with its vapor pressure:

f^{L}_{i,pure}(T, P) = P^{v}_{i}

- The pressure has a negligible effect on the activity coefficient:

g_{i}(x, T, P) = g_{i}(x, T)

- The gas is ideal, which means that the fugacity coefficient for the compoment i is equal to 1:

f_{i}(y, T, P) = 1

Therefore the iso-fugacity condition can be rewritten as follows:

x_{i} g_{i}(x, T) P_{i}^{v}(T) = y_{i} P
##### Note:

Different correlations can be used to calculate the activity coefficients. One of most used are the correlations of **Margules**, which for a binary system, are:

lng_{A} = x^{2}_{A}[Λ_{AB} + 2 (Λ_{BA} - Λ_{AB}) x_{1}];

lng_{B} =
x^{2}_{B}[Λ_{BA} + 2 (Λ_{AB} - Λ_{BA}) x_{2}];

where

Λ_{AB} = ln g^{ ∞}_{A}

Λ_{BA} = ln g^{ ∞}_{B}
and g^{ ∞}_{i} = lim g_{i} for x_{i} → 0

with i = A,B