# Example 4-2: Distribution coefficients in an ideal mixture (propane + n-pentane)

Describe the impact of pressure on the phase compositions of an ideal mixture. As an example, we use the propane + pentane mixture at 343.15 K. Following data are available (Table 1). The A, B and C parameters are those used for the Antoine equation for calculating the vapour pressure expressed in non-dimensional units ( and ):

Table 1: characteristic properties of propane (C3) and n-pentane (C5)
C3 C5
Tc(K) 369.8 469.6
Pc(MPa) 4.24866 3.37662
Zc 0.281 0.262
Acentric factor 0.152 0.251
A 5.7544 6.0008
B -5.4826 -5.5603
C -0.0472 -0.0734

## Analysis:

Phase equilibrium for a binary mixture is calculated from the distribution coefficients, as follows:

(1)

yielding:

(2)

and therefore

(3)

If equilibrium is considered at high pressure, fugacity coefficients and Poynting corrections must be taken into account:

(4)

Assuming an ideal mixture (i.e. the equilibrium coefficient is independent of composition), the activity coefficient equals one:

(5)

In expression (4), the fugacity coefficients are approximated with that of the virial equation:

(6)

and

(7)

The Poynting correction is written as:

(8)

Introducing (8), (7) and (6) into (4), one finds:

(9)

which may be written as:

(10)

or:

(11)

where:

(12)

and

(13)

The distribution coefficients thus depend only on pure component properties (liquid molar volume , vapour pressure and second virial coefficient ). Equation (11) shows that the product of equilibrium constant and pressure should be considered. Raoult's law suggests that this product is only a function of temperature. In practice, a linear relationship with pressure is required above 0.5 MPa.

## Solution:

### See complete results in file (xls):

Some help on nomenclature and tips to use this file can be found here.

In order to use the above equations, values for vapour pressure, liquid molar volume and second virial coefficient are needed. Unless specific parameters are provided, we use the corresponding states methods:

for vapour pressure, the Antoine parameters are provided

for liquid molar volume, the Rackett equation is used:

(14)

for second virial coefficient, the method of Tsonopoulos is used:

(15) with:

with:

(16) with:

and:

(17) with:

Phase equilibrium can now be computed from (2) and (3) using either the basic Raoult equation or the more complete equation (11). The results are as follows:

P(MPa)= 0.5
Ki Raoult 5.3062 0.5780
x 0.0893 0.9107
y 0.4736 0.5264
Ki (11) 3.9161 0.6209
x 0.1151 0.8849
y 0.4506 0.5494

We can notice that the complete equation yields smaller distribution coefficients than the Raoult approximation if the value is larger than one, and larger distribution coefficients if it is smaller than one. This results in smaller differences in volatilities, and as a result phase compositions that are less different.

The distribution coefficients as a function of pressure for propane and n-pentane at 343.15 K are shown in figure 1. We can see that their respective slopes are such that the equilibrium coefficients converge at higher pressures. The practical meaning of this observation is that the volatility of the two components becomes closer when pressure is increased, and therefore more equilibrium stages will be needed to reach the same separation level.