# 5.37: Untitled Page 99

- Page ID
- 18232

## Chapter 5

densities. A clear and unambiguous definition of any and all equilibrium relations is essential to avoid errors.

The analysis of the process illustrated in Figure 5‐4 is relatively simple provided that the following conditions are valid: (1) There are negligible changes in the volumes of the organic and aqueous phases caused by the mass transfer process, (2) there is no species *A* in the aqueous phase used in the extraction process, and (3) the linear equilibrium relation given by Eq. 5‐43 is valid. If the batch process illustrated in Figure 5‐4 is repeated *N * times, the concentration of species *A* in the organic phase is given by

( *c*

)

*A*

o

( *c*

)

(5‐45)

*A* *N*

*N*

*V*

1

*V *

*eq,A*

Here we have used *V * to represent the volume of the aqueous phase,

*V* to

represent the volume of the organic phase, and ( *c*

)

*A*

to represent the

*N*

concentration of the *undesirable product* of the chemical reaction in the organic phase after *N * extractions. Equation 5‐45 indicates that repeated batch‐wise extractions can be used to reduce the concentration of species *A* in the organic phase to arbitrarily small values.

To characterize the behavior of the repeated batch extraction process, it is convenient to define an *absorption factor* according to *A * *V *

*V*

(5‐46)

*eq,A*

so that Eq. 5‐45 takes the form

( *c*

)

*A*

o

( *c*

)

(5‐47)

*A*

*N*

*N*

1 *A*

This allows us to express two important limiting cases as

*A * 0 *, *

( *c*

)

( *c *) *, *

no change occurs

(5‐48a)

*A* *N*

*A* o

*A * *, *

( *c*

)

0 *, *

maximum change occurs

(5‐48b)

*A* *N*

Here it is clear that one would like the absorption factor to be as large as possible; however, the definition given by Eq. 5‐46 indicates that the value of *A* is limited

*Two‐Phase Systems* & *Equilibrium Stages *

179

by the *process* illustrated in Figure 5‐4 and the *equilibrium coefficient* defined by Eq. 5‐43. The derivation of Eq. 5‐45 is left as an exercise for the student as is the case in which the concentration of species *A* in the original aqueous phase, ( *c*

) , is not zero.

*A* o

While the batch extraction process illustrated in Figure 5‐4 is convenient for use in an organic chemistry laboratory, a continuous process is preferred for a large‐scale commercial purification process such as we have illustrated in Figure 5‐5. There we have shown a system consisting of a *mixer* that provides a *Figure 5‐5*. Liquid‐liquid extraction

large surface area for mass transfer followed by a *settler* that separates the aqueous and organic phases. If the mixer‐settler system is efficient, the two phases will be in equilibrium as they leave the settler. In this example, we are given an equilibrium relation in the form of Henry’s law

Equilibrium relation:

*y*

*K*

*x , * at the fluid‐fluid interface

(5‐49)

*A*

*eq,A*

*A*

which is the *mole fraction* version of the equilibrium relation given earlier by Eq. 5‐43. If the flow rates of the aqueous and organic phases are *slow enough* and the mass transfer of species *A* between the two phases is *fast enough*, we can use Eq. 5‐49 to construct a process equilibrium relation as

Process equilibrium relation:

( *y *)

*K*

( *x *)

(5‐50)

*A * 3

*eq,A*

*A * 4

We refer to this as a *process equilibrium relation* because it expresses the organic phase mole fraction in Stream #3 in terms of the aqueous phase mole fraction in Stream #4. Knowing when Eq. 5‐50 is a valid approximation requires a detailed

180