# 5.36: Untitled Page 98

- Page ID
- 18231

## Chapter 5

*B * *N*1

*X*

*c*

*c*

(5‐39)

*A*

*A*

*B*

*B * 1

When it is convenient to work in terms of this modified mole fraction, one usually needs to be able to convert from *x * to *X * and it will be left as an *A*

*A*

exercise for the student to show that this relation is given by *X*

*x*

1 *x * *, *

*A * 1 *, * 2 *,...N *

(5‐40)

*A*

*A*

*N*

In some types of analysis it is convenient to choose species *N* to be species *A*.

Under those circumstances we need to express Eq. 5‐39 as

*B * *N*

*X*

*c*

*c*

(5‐41)

*D*

*D*

*B*

*B * 1

*B * *A*

and Eq. 5‐41 takes the form

*X*

*x*

1 *x * *, *

*D * 1 *, * 2 *,...N *

(5‐42)

*D*

*D*

*A*

Similar relations can be developed for the modified mass fraction and they *A*

will be left as exercises for the student. It is important to note that the sum over *all species* of the modified mass or mole fraction is not one.

**5.5 Equilibrium Stages **

Mass transfer of a chemical species from one phase to another phase is an essential feature of the mixing and purification processes that are ubiquitous in the chemical and biological process industries. A comprehensive analysis of mass transfer requires an understanding of the prerequisite subjects of fluid mechanics, thermodynamics and heat transfer; however, there are some mass transfer processes that can be approximated as *equilibrium stages* and these processes can be analyzed using the techniques presented in this text. Most students are familiar with an equilibrium stage when it is carried out in a *batch-wise manner*, since this is a common purification technique used in organic chemistry laboratories.

If an organic reaction produces a *desirable* product that is soluble in an organic phase and an *undesirable* product that is soluble in an aqueous phase, the desirable product can be purified by liquid‐liquid extraction as illustrated in Figure 5‐4. In the first step of this process, the mixture from a reactor is placed in

*Two‐Phase Systems* & *Equilibrium Stages *

177

*Figure 5‐4*. Batch liquid‐liquid extraction

a separatory funnel. Water is added, the system is agitated, and the phases are allowed to *equilibrate and separate*. The amount of the undesirable product in the organic phase is reduced by an amount related to the volumes of the organic and aqueous phases and the equilibrium relation that determines how species *A* is distributed between the two phases. If the equilibrium relation is linear, the concentrations in the aqueous phase ( ‐phase) and organic phase ( ‐phase) can be related by

Equilibrium relation:

*c*

( *c*

)

(5‐43)

*A*

*eq,A*

*A*

This relation is based on the general concept of thermodynamic equilibrium illustrated by Eq. 5-24. In this case we have used

to represent the *equilibrium coefficient*;

*eq,A*

however, this information may be given in terms of a *distribution coefficient* defined as

*c *

distribution coefficient

*A*

(5‐44)

*dist ,A*

*cA*

When working with equilibrium coefficients or distribution coefficients, one must be very careful to note the definition since it is quite easy to invert these relations and create an enormous error. In addition, equilibrium relations are usually given in terms of mole fractions, but they are also given in terms of molar concentrations (as in Eq. 5‐43), and they are sometime given in terms of species