Bruce K. Hamilton ond Martin J. Greenwald Deportment of Chemical Engineering Massachusetts Institute o ecnoogy Cambridge, 02139
I
I
I1
Pradicd Equilibrium Concentrations and Yields in Selective Catalyst Systems
Often catalysts are said to affect only reaction kinetics and then they are dismissed in discussions of chemical equilihrium. This treatment of catalysis can lead to confusion when equilihrium concentrations and equilihrium yields are to he calculated for a reaction system in which selective catalysts are employed. While it is true that catalysts do not affect values of equilihrium constants, they can and do affect practical equilibrium concentrations and yields obtained in selectively catalyzed chemical reaction systems. By practical equilibrium concentrations and yields, we mean those values which are approached from both ahove and below over the time span of interest to the observer. We realize that it might he said that ''true'' equilibrium concentrations and yields can not he affected by catalysts, "true" eauilihrium being" the state that a svstem reaches after an infinite period of time. I t is obvious, however, that no practicing scientist or engineer can wait forever, and so the practical definition we have just made is in order. In fact, i t should he clear that this is the definition of equilihrium that is generally used when any real system is considered. The definition is completely consistent with the second law of thermodynamics, as we will show later. We also realize that it is widely recognized (e.g., see Hougen, et al.') that in cases where many reactions are possible, restriction to a single path may he accomplished by choice of a selective catalyst which promotes only a specific reaction. For example, in the reaction of carbon monoxide with hydrogen, the number of products theoretically possible is almost unlimited; yet by choice of the proper selective catalyst essentially all products except a particular one such as methanol can he excluded. Of all compounds which might theoretically form, it is well known that i t is necessanr to have thermodvnamic information on only carbon monoxide, hydrogen, and methanol in order to calculate equilihrium concentrations and yields obtained in such a seiectively catalyzed system. What we think is perhaps not so widely recognized is the possibility of employing alternative comhinations of selective catalysts to promote conversion of given reactants to the same products in equilihrium yields which depend upon what choice of catalyst combination is made. In these alternative equilihrium conversion reactions, reactants are converted to reactant/product mixtures, all of which contain identical constituents at non-zero equilibrium concentrations, but in which the magnitudes of the equilihrium concentrations depend upon catalyst choice. In this article, the possibility of ohtaining alternative equilihrium concentrations and yields is discussed. Of particular interest is that much higher equilihrium yields of a desired product are attainable with one catalyst comhination than another. Before illustrating these points, we hriefly review criteria of chemical equilibrium in selective catalyst systems. Chemical Equilibrium in Catalyst Systems The conditions determining a state of equilihrium are a direct consequence of the second law of thermodynamics, 'Houpen, 0. A., Watson, K. M., and Ragatz, R. A,, "Chemical Process Principles, Part ll Thermodynamics," 2nd Ed., John Wiley & Sons, Inc., New York, 1959, p. 1042. 732
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which states that the entropy of a macroscopic isolated system can never decrease; i t increases in an irreversible process and remains constant in a reversible one. For a change a t constant temperature and pressure during which only pressure-volume work is done, an equivalent statement of the second law is that the Gibhs free energy of a macroscopic system can never increase. A state of equilibrium for a system subject to such changes is attained when all possible variations of the state of the system which do not alter its temperature or pressure, result in variations of its Gihbs free energy which vanish or are pmitive. The important word here is possible variations. Thermodynamic systems undergoing changes are usually assumed to he constrained in various ways. For example, an adiabatic envelooe constrains heat exchanged from an "isolated" system undergoing a state change so that, theoretically, no heat whatsoever is transferred. Likewise, in a selective catalyst system, the catalysts added constrain possible reaction paths, so that, under proper conditions, uncatalyzed reactions can he taken to he practically "forbidden" while selectively catalyzed reactions are "allowed". Although it can be argued that, strictly, there are no such things as "forbidden" uncatalyzed reactions, it can also he argued that, strictly, there are no such things as "isolated" systems. But just as the concept of isolated systems is useful to practicing scientists and engineers, so also is the concept of forbidden reactions, which is implicitly invoked in many equilihrium calculations. With this background on equilibrium kept in mind, it is now appropriate to consider a simple model reaction system in order to illustrate how selective catalysts can affect practical equilihrium concentrations and yields. Simple lsomerization Model Consider the following set of hypothetical isomerizations promoted by selective catalysts 1, 2, and 3
These reactions are assumed to occur in a liquid phase containing solutes A, B, C, and D. In the equilibrium expressions appearing ahove, K1, K2, and Ka arethermodynamic equilibrium constants, and &, 6, C, and d are equilibrium concentrations. Thermodynamic activities of solutes are assumed to be equal to their concentrations in a reacting solution divided by standard state concentrations. Standard state concentrations are all taken to he 1.0 M. The particular numerical values indicated for K1 and Kz have been chosen arbitrarily, and it is easily verified that the laws of thermodynamics require K3 = K1K2. The possihle sets of equilibrium concentrations for a batch reactor charged with a solution initially containing 1.0 M A , 1.0 M C, no B or D, and various combinations of selective catalysts 1, 2, and 3, are listed in the table. Calculation of these equilibrium concentrations is hriefly outlined in the next section. At this point, however, the final
Possible Sets of Equilibrium Concentrations for the Hypothetical lsomerization System Described in the Texta Caseb
Catalyst Charged
Equilibrium Concentrations ( M ) d
6
E
d
0.50 0.00 0.69
1.0 0.17 0.31
0.00 0.83 0.69
0.50
0.17
0.83
I I1
1
111 IV
3
0.50 1.0 0.31
VI VII
2, 3
0.50
v
a
2
stoichiometry a0=i+6=1.0M material balances c , = c + ; i = ~ . o ~ Again, the equilibrium relations a n be written in terms of equilibrium extents of reaction (& = 6/ao, .$z = q c o )
See eqns. (1)-(3).
' For all cases listed in this table a, = 1.0 M, bo = 0, ca =
1.0 M, do = 0 (a, denotes initial concentration of A).
two horizontal rows of entries in the table show that use of different combinations of selective catalysts can yield equilibrium mixtures in which equilibrium concentrations of identical constituents are unequal, and are also all nonzero. Inspection reveals that if a maximum equilibrium yield of B is desired, and if C is available at low cost, then catalyst 3 alone should be employed for equilibrium conversion of A to B. Use of any other catalyst chosen from the group 1, 2, 3, or any combination of catalysts in the gmup, results in lower equilibrium yields of desired pmduct B. Calculation ol Equilibrium Concentrations for Model Calculation of the equilibrium concentrations listed in the table is straightforward. For example, for Case 111, the reaction under consideration is as shown in eqn. (3).when the initial concentrations of A, B, C, and D in a batch reactor are 1.0 M A , 1.0 M C, and no B or D, and only catalyst 3 is charged, the following relations must be satisfied equilibrium stoichiometry material balances
b=?
where a. and co are initial concentrations of A and C (1.0 M each). In the above material balances, i t has been assumed that, a t equilibrium, the amounts of A, B, C, or D (or any intermediates involved in their interconversion) bound to catalyst are negligible compared to amounts of material found in solution. This assumption is very reasonable when initial reactant concentrations are much greater than catalytic site concentrations. Using the material balance and stoichiometry relations, the equilibrium relation given by eqn. (4) can be written in terms of the equilibrium extent of reaction ($3 = 6/ao = d/co)
Therefore, for KI = 1.0 and K2 = 5.0 -
a=Z=l/ZM F = 116 M
d = 516 M for Case IV (both catalysts 1 and 2). At equilibrium, less B is obtained in the Case N reactor than in the Case III reactor. The Gibbs Function For all the states of equilibrium involving A, B, C, and D which have been described, the Gibbs free energy of the contents of the batch reactor, subject to various constraints, is minimized. This statement is now demonstrated and explicit consistency with the second law of thermodynamics is thus proved. Continuing to assume that thermodynamic activities of solutes are equal to solute concentrations divided by solute standard state concentrations (1.0 M), the chemical potential of solutes (A,B,C,D) can be written in the form
where i = (A,B,C,D). Since the amount of solvent in the reactor does not change during the course of the reaction, if the solution is assumed to be ideal the contribution of solvent to the total Gibbs free energy of the contents of the reactor is constant and can be taken to be zero. The amounts of catalysts in the reactor are negligible and their contributions to the Gibbs free energy can be neglected. Therefore, the total specific Gibbs free energy of the solution in the reactor at any time, or the Gibbs function for the contents of the reactor, can be written in the form
where i = (A,B,C,D). For solutes A and C at their standard state concentrations of 1.0 M (in solvent solution at 25"C), let = p,.O = 0. For each reaction fj) so that pn" = pA0 -
Therefore, for Ka = 5.0
and pl,O = firo
and
- RT In
K2
= -RT In K,
-
-RT In K ?
Using these expressions for the standard state chemical potentials, the Gibbs function for the contents of the reactor becomes G = RTCa In a M-In K , In b ) + c In c d(-ln K a In d ) ] The concentrations appearing in the Gibbs function must always satisfy the relations given by eqns. (5) and (6). Written in terms of and €2, the Gibbs function is
+
for Case111 (catalyst 3 only). For Case IV, the reactions under consideration are as shown in eqns. (1) and (2). This time, the relations which must be satisfied are
RT In K ,
+
+
+
equilibrium Volume51, Number 11. November 1974
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Using eqns. (12), (13), and (14), two algebraic equations in $1and $2 are obtained
Equations (15) and (16), obtained by minimization of the unconstrained Gibbs function, are identical to eqns. (10) and (111, which were obtained through use of the equilibrium relations given by eqns. (8) and (9).The absolute minimum of the Gibbs function given by eqn. (12) therefore corresponds to the equilibrium state obtained for the Case IV (and also Cases V - VII) batch reactor. When only catalyst 3 is charged to the reactor so that 51 = 52 = 53, the Gibbsfunction given by eqn. (12) becomes G* L
Contour plot of G for the solution of reacting. A. 8. C , and D described in the text. The constraint path ( I = t2 = is indicated: the minimum 0.69 and has a value of along this path occurs at t o= I1/r[5 - "'51 G = - 1 . 4 kcal/l. The ebsolute minimum occurs at [I = 'A. [? = 51e and has a value of G = - 1 4 9 kcalll.
t3
--
Equation (12) can be used to construct a contour diagram of the Gibbs function, G = G ( h , & ) . Such a contour diagram is shown in the figure with the constraint path [I = (2 = (3 indicated. When any combination of two of the catalysts, or all three of the catalysts, is charged to the reactor, no constraints are imposed on and &. Equilibrium is therefore obtained when G as given by eqn. (12), unconstrained, is minimized. At the minimum
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+
RTladl - $,I In ( d l - &I) a&-ln K , + In (a0EJ1 + cdl - [,I In ( d l - W) c&[-ln K 2 In (cddll
=
+
+ (17)
In this case, equilibrium is obtained when G* is minimized; at the minimum
From eqns. (17) and ( l a ) , and since a,, = co and K3 = KIKZ
Equation (19), obtained by minimizing the constrained Gibhs function G*, is identical to eqn. ( I ) , which was obtained through use of the equilibrium relation given by eqn. (4). The constrained minimum of the Gibbs function therefore corresponds to the equilibrium state obtained for the Case In batch reactor.
