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BIRK
JAMES P. Arizona State University Tempe. AZ 85281
computer series, 137 Calculating Equilibrium Concentrations Competing, Coupled, and Catalytic Chemical Reactions E. Weltin University of Vermont Burlington, VT 05405
The equilibrium position of one reaction is influenced by the other reaction. In other words. the eouilibrium concentrations of the species that participate in one reaction deuend on the initial concentrations ofall chemicals involved in the two reactions. Coupled Reactions
The concentration of any chemical species in a reaction mixture can not be negative. This simple fact limits the possible extent of reaction to a n interval that, for a process of given stoichiometry, is determined by the initial concentrations a, box...
Two reactions are coupled when a product of one process is a reactant in the other ~rocess?When the two urocesses are coupled, a n unfavorable reaction (eq 3) can i e driven by a favorable process to form a desired product B. Coupling plays an important role in biological systems. pP+qQ
... e
F l
A + b B ... (3)
of the reagents A, B, ...
This is well-known to every chemistry student who has suffered through problems of limiting reagents. This f a d and the form of the chemical equilibrium expression also guarantee the following, as we have shown in a previous paper:' A process described by a single chemical equation reaches equilibrium at an extent of reaction that is uniquely determined by the characteristic equilibrium constant and the initial concentrationsof all reagents.
A numerical method to calculate the equilibrium position has been discussed.' It is based on the property that, inside the "chemically allowed" interval, i t is easy to test if a selected extent of reaction lies above or below the equilibri~m.~ In this paper we extend the analysis to certain cases of two equilibria affecting each other in a common reaction mixture. For given equilibrium constants we verify the following two fads. The concentrations at equilibrium are again uniquely determined by the initial concentrations. They may be calculated by a numerical method easily implemented on a personal computer. Reactions with One Common Reagent Competing Reactions
Two reactions are competing if a common reagent A reacts with different reactants to form different products. Without loss in generality we can write the chemical equations in the following normalized form
in which the stoichiometric coefficient for A is 1 and the coefficients for the remaining species are not necessarily integers. Ki and Kz are the associated equilibrium constants.
Equation 3 can be rewritten in the form of eq 1by exchanging the left-hand side and the right-hand side and by setting K, equal to the inverse ofK;. Equation 4 is already identical to eq 2. Thus, coupled and competing reactions can be treated by the same formalism. There is a practical difference: Although the initial concentration of the common reagent (a.) h i s a significant magnitude in competing processes, it is usually 0 or very small in wupledreactions. Calculating Equilibrium Concentrations
For known equilibrium constants (e.g., those derived from tabulated free energies of formation) and a given set of initial concentrations of all chemical species, the calculation of equilibrium concentrations by analytic methods is, i n general, a formidable task. A simple numerical method, however, is outlined below to calculate the unique equilibrium position for such systems. The initialconcentrations are designated by a., b,, ....Let the parameter z characterize the progress of combined reactions, while a and p indicate the separate extents of the reactions shown in eqs 1 and 2, respectively. It is convenient to set z equal to the concentration of A. Then the following equations can be generated. [Al=a,-a-p=r [Bl = (b, - ba) [Cl = (C, - 4) [PI =@,+pa)
At equilibrium the following three equations must be satisfied simultaneously.
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Homogeneous Catalysis
To find a solution we define the auxiliary function flz)= z +a@) + B(z)
(8)
In homogeneous catalysis (e.g., an enzyme reaction) the catalyst A in the first step forms a complex Q and is regenerated in the consecutive second step. Kf' A+bB = Q + p P
where ak) and P(z) are solutions of eqs 5 and 6, respectively, for a given value of the parameter z. Equation 5 is equivalent to a pseudoequilibrium problem.
K h. 0therwise continue in the lower half interval. The process is guaranteed to converge, and the individual equilibrium concentrations are evaluated by the relationships of eqs 13-18. Michaelis-Menten Enzyme Reaction In the Michaelis-Menten model, which is the simplest model for an enzyme reaction, the enzyme E binds the substrate S, forming an enzymesubstrate complex Q. This subsequently undergoes dissociation to the product P, thus regenerating free enzyme E.
in which KIand Kz are characteristic equilibrium constants. K1=- 11&1 [SI [El Kz= [PI-[El [Ql
The concentrations of the four chemical species are given below. [El=e.-a+p [SI=s.-a
where e,, q,, s., andp. are the initial concentrations, and a and p denote the extents of the reactions of eqs 21 and 22, respectively. Thus, [S1+[P1=so+p,-a+p
In terms of the parameter h = a - p, we can write the following equations. [E]=e,-h
and a vertical asymptote at h=e.
Each branch has exactly one intersection with the straight line which appears in eq 24 and has a slope of -1. The solution above the vertical asymptote e. must be rejected on chemical grounds because it would require the enzyme to have a negative concentration. With the solution below eo, the equilibrium concentrations of enzyme, enzyme-substrate complex, substrate, and product are evaluated using eq 23. Computational Considerations With the chemical constraints taken into account, the type of problems considered in this Daver have an eauilibriim state that is uniquely determined by thecharacteristic equilibrium constants and the initial concentrations of all species. The numerical computations are guaranteed to converge. They require nothing more complicated mathematically than repeated evaluations of terms of the type (b, - balb
or the calculation of one root of a quadratic equation. They are straightforward to program in any one of the standard programming languages. Except for the last case, two levels of nested iterations are required. For convergence, a good strategy is to cany out a fixed number of bisection steps. A simple check of the results can be done by comparing equilibrium constants calculated from the final equilibrium concentrations to the constants entered as input. In order to .get . mod - agreement, one finds that it is necessary to carry the calcuiation of concentrations far beyond what is usually considered chemical precision. This is especially true if the constants are eithkr very large or very small. We found, however, that most cases are handled adequately if each bisection algorithm is canied 20 steps. Since this leads to (2x 400)inner steps, computation times that reach several seconds must be expected on a PC. However, these computational speeds are well within an acceptable range for typical chemical problems. Conclusions None of the textbooks with which I am familiar even attempt to solve the type of problem discussed in this paper. Since more and more chemists have easy access to computers, the methods presented here allow the quantitative exploration of the following two areas.
The two solutions of this equation are the mots of the quadratic equation
Only the lower of the two roots is chemically significant as can be seen from the following discussion. The righehand side of eq 24 is a hyperbola with a horizontal asymptote at
How the relative magnitudes of equilibrium constants affect product concentrationsin competing or consecutive reactions How these concentrations can be manipulated by changing initial concentrations Enzyme reactions in biological systems are rarely allowed to reach equilibrium. Nevertheless, the calculation of quantities such as free enzyme or enzymesubstrate complexes are of interest because they establish limiting values to these quantities. Many students have a difficult time understanding the thermodynamic proof that a given chemical system has only one equilibrium position. However, the same students readily accept the mass-action law and the obvious fact that concentrations or partial pressures can not be negative. These are the only ingredients in the present alternate proof of a unique equilibrium. Volume 69 Number 5 May 1992
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Among other things, this shows that a molecular computer or memorv device can not be based on a closed chemical system of
