Maximizing Profits in Equilibrium Processes In William M. Spicer's note entitled "A Use of Calculus in Freshman Chemistry" it was shown how to calculate maaimum precipitate formation in solubility equilibrium processes. A similar example would be t o extend the problem to include maximizing profits in industrial processes involving chemical equilibria. As an example, consider a one-step extraction process where Q males of solute, S, are distributed between two immiscible solvents, A and B. Initially all Q moles are dissolved in a given volume, VA,of solvent A. We then add volume VB of solvent B, extracting x moles of salute into the solvent B phase, leaving fQ - x ) moles in solvent A. Assume all x moles of solute can then be purified from solvent B and sold a t price P, dollars mole-'. If solvents A and B must be purchased a t a price PAand PBdollars liter-', respectively, Q costs PQ dollars mole-' t o manufacture, then what volume VB of solvent B when added to VAwill yield the maximum profit? From the equilibrium condition, SA= SB, where SArefers t o the solute in solvent A and SB to the solute in solvent B
X
Solving for x
=
KQVe ( K V B VA)
+
The profit, P, is the difference between selling price and costs
P
=
- P,V, - PqQ
PxX - P,V,
=
P
~
KQVR { + (V A~) )- ~PEVB~
- PAVA - PQQ
The condition for maximum profit, dP/dVn = 0 yields the equation .
Rearranging the equation into the general quadratic form and solving for VB yields
+ i(al/AKQ)
v, = - a V ,
where
a = P,/Px
(2)
and only the positive root is retained for obvious physical reasons. It is interesting t o note that a profit can he realized only if'kQ > alla; otherwise the expression fbr VB leads to negative values for the volume of the extracting solvent. This would indicate that the cost of the solvent would in all cases exceed the value of the extracted solute. As a simple illustration, suppose one were considering the commercial feasibility of a benzene-water separation of p-nitroaniline from the reaction mixture in which i t was prepared. A sample of the mixture that is known t o contain 0.500 moles, Q, of p-nitroaniline is dissolved in 1.00 1, VA, of water, S A ,t o which is added the unknown volume VB of benzene, SB.
If water costs $0.15/1,PA, benzene costs $5.00/1,PB,the cost of manufacturing the reaction mixture is $l.OO/mole of p-nitroaniline, PQ,and the purified p-nitroaniline can he sold for $8.00/mole, then what volume VB of benzene should be used t o yield a maximum profit for the procedure, assuming 100% recovery from the benzene phase? First, we note that k Q > OVA;therefore the choice of solvent is not cost-prohibitive. Then, substitute the above values into eqn. (2) -0.625 mole/l- X 1.00 1 v(0.625 mole/lF X 1.00 1 X 9.35 X 0.500 mole) u = s'OO = 0.6625 mole/lP V , = = 0.186 1 $8.00 mole-' 0.625 mole/lP X 9.35
+
Substitute this value into eqn. ( 1 )
P
=
@.CHI
mole-
935 X EOO mole X 0X6 I 9.35 X 0.186 1 1.00 1
(
I
+
)-
$5.00 I-'
X
0.186 1 - $0.15 I-'
X
1.00 1 -
$1.00 mole-' X 0.500 mole $0.96 The procedure can be extended to any equilibrium process where the equilibrium expression is known. Higher order equilibrium expressions give rise to polynomials of higher degree in x, which may render the problem insolvable or yield profits equations and their derivatives which are quite camplex. A computer may be needed to solve the final equation. In the classroom, the benefits of this type of problem are two-fold. First, the student can see both chemical principles and calculus applied to realistic industrial situations where profits are a must. Second, a practical use of computers in chemistry can he demonstrated. P
=
'Spieer, W. M., J. CHEM. EDUC., 50,686 (1973). 2Ewing, Galen W., "Instrumental Methods of Chemical Analysis," 3rd. Ed., McGraw-Hill Book Co., New York, 1969. p. 495. Ronald J. Rish Bueyrus High School Bucyrus. Ohio 44820
Volume 52. Number 7, July 1975 / 441
