Example 8.06 Ideal and Real Solubility of Naphthalene in the Mixture
Ethanol - Isooctane

problem:

Calculate the ideal solubility of naphthalene (3) in a mixture of ethanol (1) – 2,2,4-trimethylpentane (2) at 298.15 K as a function of the ethanol concentration. Compare the results with the results obtained taking into account the real mixture behavior using modified UNIFAC. (Tm,naphthalene = 353.35 K, Dhm,naphthalene = 19110 J/mol).

general

constants

and

definitions:

input data

melting point data for naphthalene:

equilibrium temperature:

Solution

To calculate the solubility of a solute (3) in a solvent mixture (1) + (2) in case of a eutectic mixture, the solute is added to the mixture until the solubility limit is reached, i.e. the activity in the liquid is equal to the activity of the pure solid component at the respective temperature.

Starting with 1 mol solvent mixture of composition xsolvent, the final concentration of the mixture is:

with x1 - mole fraction of component 1 in the

solvent mixture

n - moles of solute added

The activity of the pure crystalline solute can be described as

The activity of the solvent in the liquid depends on composition and temperature. In case of ideal mixture behavior in the liquid, the activity coefficient is simply given by:

and the activity can be calculated as:

The solubility can then be found by the root function:

starting value:

The following plot shows the solubility of naphthalene as function of solvent composition on a solute-free basis:

As expected, the solubility does not change with solvent composition as in an ideal mixture, the solvent activity only depends on the solvent mole fraction.

In order to calculate the real liquid mixture behavior using modified UNIFAC, several variables have to be set and the respective activity coefficient function must be defined.

There are 3 different subgroups and 2 main groups involved in the calculation. These are shown together with their ID (group number) and (in case of subgrous) the R and Q values in the following vectors. In order to identify the main group that a subgroup belongs to, the index vector s2m is defined. s2mi yields the main group address in the main group vectors, that subgroup i belongs to. The matrix contains the frequency of subgroup i in component j. The matrices , and contain the coefficients for the temperature dependent interaction parameters.

The function to calculate the activity coefficients using modified UNIFAC is in the area below.

The solute activity in the real mixture can be calculated as:

The solubility can then be found by the root function:

starting value:

The following plot shows the solubility of naphthalene as function of solvent composition on a solute-free basis:

The textbook solution reports the following equilibrium composition for an equimolar mixture of the two solvents:

The function above returns the following solubility:

This results in a nearly identical mixture composition of

At this composition the following activity coefficients are calculated by modified UNIFAC:

Textbook: g3 = 2.9998

As it can be seen, there is a slight discrepancy in the activity coefficients.