Competitive and Coupled Reactions J. N. Spencer Franklin and Marshall College, Lancaster, PA 17604 A chemical reaction alters the composition of a system. Multiple reactions introduce yet another source of variation in composition by changing the kinds and numbers of s~ecies that must be considered. In a seauence of w m ~ e t i t h e or coupled reactions, standard thermodynamic p i a m etors may be used to find the equilibrium constant, and hence the concentrations of reaitants and products, for any steady-state step. However, there are other situations that involve kinetic or mechanistic considerations for which the thermodynamic parameters may be irrelevant. The misconception that one reaction, when coupled with another, may be "driven" against its free energy comes from a misunderstanding of the use of thermodynamic parameters to describe kinetic processes. Equilibrium Calculations: The Use of Thermodynamics for Competitive Reactions Deumie and Krallafa ( I ) have made several important points concerning competitive or coupled reactions. They pointed out that for reactions such as
Kl A-B
(1)
and for reaction sequences such as
KI
Kz
A e B - C
(2)
K lis determined by the standard free energies of A and B. Thus, K lhas the same value for reaction 1as for reaction sequence 2. Further, it was shown that due to the maximization of the entropy of mixing, the same K lis obtained whether or not there are competitive reactions. However, the maximum of the mixing entropy will not be the same as when there is a competitive equilibrium. (See Appendix A,) As a consequence, the equilibrium concentrations of A and B will not be the same for reaction 1 as for reaction sequence 2, even though K lis the same for both. In reaction 1 there are fewer different molecules than in reaction sequence 2. That is, only molecules of A and B are present in reaction 1,whereas molecules ofA, B, and C are present in reaction sequence 2. Thus, the number of molecules of A and B required to maximize the entropy will be different. For example, if K l= 1 and Kz= 1, for reaction 1,the equilibrium concentration of Ais 0.50M when the initial concentration of Ais 1M. For reaction sequence 2, in which the initial wncentration of A is 1M, the concentration of Aat equilibrium is 0.33M. I t is valid to use the maximization of the entropy of mixing as the sole criterion for equilibrium concentrations only when the thermal entropy and enthalpy changes are zero. Thus, it is strictly valid only when the R s are unity for each step. (See Appendix B.) McPartland and Segel(2) have given examples of competitive or coupled reactions in which each step has different entropy and enthalpy components so that each step has a different equilibrium constant. They propose sequences analogous to the following.
Kl
A
B
Kl
A
Kz a C
where K, = 1
Kz
aB aC
where
Kl= 100
If the initial concentration of A is 1M for both reaction sequences 3 and 4, which reaction sequence will have the largest equilibrium concentration of species C? The overall reaction for both reaction sequences is
and both have a n overall equilibrium constant of 100. When the initial concentration of A is 1 M, reaction sequence 3 has a n equilibrium C concentration of 0.98,while reaction sequence 4 has C,, -- = 0.50. In reaction sequences 3 and 4, the position of maximum entro~v.and hence the concentrations of all s~ecies.would be thesame if the thermal entropy and e&lpy&anges were zero. Here the contributions from AH' and A S o give a AGO that is different for the individual steps in reaction sequences 3 and 4. The extent to which the individual reactions of the sequence proceed_is determined by standard enthalpy and entropy (i.e., AGO) considerations. Whether or not a particular reaction is thermodynamically-feasible is not determined by the standard free energy AGO. It is determined by the free e n e m AG. This distinction is frequently imored i n present ti ti^^,^^ ofthe use ofthermndynami& topred& the directionofa chemical reaction. Generally, it is pointed out that there is no connection between the thermodvnamic feasibility of a reaction and the rate at which the reaction proceeds. However, this point being made, thermodynamic arguments are often invoked to infer mechanistic implications. Kinetics and Thermodynamic Coupling: The Misuse
of Thermodvnarnics for CornDetitive Reactions
Reaction sequences such as 3 are subject to misinterpretation if the limited use of thermodynamics to describe an essentially kinetic process is not recognized. For this type of reaction, it is often said that when the free energy change for one step is positive, the second reaction will drive the first, if its free energy change is sufficiently negative. This has been called 'thermodynamic coupling". At constant temperature and pressure, the free energy for a coupled reaction sequence, such as 3 above, can be written as
dG = (VA &A) + (VB dnd + (kd n c ) When 5 is defined as the degree of advancement of the reaction
dSi)+ (dh- dSd VB + (W dS2)
dG = +A where dnA= 4
5
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Keizer (3)has shown how to expand this treatment to include reactions rates (velocities) of elementary reactions and certain elementary-complexreactions. In so doing, has provided an understanding of the dilemma just encountered. For a given reaction r
and dnc = d52
Then
aA+bB=cC+dD
the degree of advancement E,, is defined by
where A
G is~ the free energy change for
where the superscripted zero indicates the initial number of moles, and ni is the number of moles after the reaction has advanced by 5, units. The coefficient vj is negative for reactants and positive for products. At constant T and P
and AS;, is the free energy change for
d ~ = C ( pdn,) + i
dn, = vi dt,
I t is important here not to confuse with A @ . The two free energies are related by the well-known equation
and
where Q, is the ratio of the concentrations- (activities) of the products to that of the reactants. I t is AG, which determines the direction of the reaction, nbt A& , the standard free energy change. If the coupled reactions are advancing, then
where AG?is the free energy change for reaction r. Then
With the usual definition of reaction rate
d&>O
dG where V, is the rate or velocity of the reaction. Then dt becomes
and if
IAO,I> I A O ~ I where AG2 may be positive, then
may still be less than zero. But if AGB is positive, reaction sequence 2 does not proceed. Thus, for each step of the reaction sequence to proeeed AG, must be < 0. This is made clear from Euler's criterion for exact differentials, which gives
For a sequence of reactions
where A&V, refers to only one reaction in the sequence. For a flux to develop, the following must hold.
The velocity is conventionally defmed as when applied to the above equation. , both to be positive, As a con_seqnence, for del and dE2 and AG2 must both be negative. The criterion is that AG, must be of opposite sign to the degree of advancement of the reaction. Note that a common intermediate is required for the coupling of reactions. In other words, one reaction must produce a species that serves as the reactant in another reaction. However, it is not always true that the existence of a common intermediate is sufficient grounds to consider the reactions coupled. The preceding development shows thzt it is possible to obtain an overall negative dG even if AG, for certain individual steps is not negative. Yet, if each AG, is not negative, that particular step does not proceed. In part, thermodynamic coupling has been invoked to reconcile this dilemma. However, the concept that one reaction can drive another against its free energy has only led to the misuse and misunderstanding of coupled reactions.
AG~
282
Journal of Chemical Education
v,=rr--v where V+is the velocity (rate) in the forward direction, and is the velocity (rate) in the backward direction. Bothvelocities are positive. For V, to be positive, the following must hold
+>v and A& < 0
For V, to be negative, the following must hold
V>V and
AG, > 0 Now A@~=A@+RT~~Q,
If the reactions 5 and 6 were used only to describe the overall reaction
and in general (3)
Thus, the ratio of the forward velocity to the reverse velocity of the reaction depends on the ratio of the equilibrium concentrations (K,) to the concentrations when not at equilibrium (QJ. If
it mieht be concluded that reaction 5 drives reaction 6. ~husrthermodynamiccoupling would be inferred. Kinetic simificance would be hvcn where none is allowed. The act& mechanism of the reaction might be the mechanism given by reactions 7-10. Each mechanistic step is an elementary step. Each step has a negative free energy, and the reaction proceeds toward products. For the process glumse + Pi
glucose 6-phosphate
then and an equilibrium state exists. Thus,the ratio
4 -
the cellular concentrations of Pi and glucose &phosphate are approximately 10.' and lo4 M, respectively-The cellular glucose concentration is about lo4 M and AGO is +12.5 kJ mot1 (5).Then
Q,
determines the direction of the reaction. For example, if -Kr< 1
Q,
then and
v,
