Effect of Concentration on Reaction Rate and Equilibrium ARTHUR A. FROST
Northwestern University, Evanston, Illinois
TEXTBOOK CONFUSION reactants, products, catalysts, etc., and fi, q, Y , etc., take on values such as 1, 2, or 3 but may sometimes T WOULD be very confusing to a student to pare discnssions of the L~~ of M~~~ ti^^ in be fractional or negative. Now according to the usual statement of the Law of Mass Action the rate expresdifferent texts of general or physical ,.hemistry, ~h~~ is no general agreement as to what the L~~ of sion is of this simple form but restricted so that A , B, Mass Action really is. In a survey of fifteen textbooks C9 etc., are reactants only and 9, ¶, +', etc.9 are the i t was found that, according to seven, the L~~ of stoichiometric coefficients of the reactants in the Mass Action is a relation between velocity of reaction chemical equation. The Law of Mass Action, if and concentration, whereas in five the law con. expressed in this form, is certainly not valid, since very sidered to be the equilibrium constant expression for few reactions have rate expressions corresponding to a reaction. one author resolves the difficulty in their stoichiometric equations. Of course, the reason terminology by stating a iqass ti^^ L~~ of ti^^ for this is that most reactions are complex, i. e., pro~~t~and also a M~~ ti^^ L~~ of chemical ~ ~ ~ i l iceed b - in steps, the rate being determined by the slowest rium. Other authors present the usual discussion of step (or combination of steps if there are steps comrate and equilibrium without using the term "mass peting with each other). By restricting the law to action" or the expression :'Law of Mass Action." s~mplereactions, the law would hold. It is correct A decision as to the validity of a particular statement to Say that for simple homogeneous reactions, or for back to a single step in a complex reaction, the rate is proof the law could be obtained by a portional to the concentration of each reactant raised ~ ~ l andd waage,l b ~ who ~ first ~ stated the law, series of papers appearing from 1864 to 1879 they to a power equal to the number of molecules of that reported measurements of reaction rate and equilibrium reactant taking Part in the Process. Although when restricted as above the Law of Mass and interpreted the effect of change in concentiation in terms of their theoretical ideas. After generalizing Action is correct, it can hardly be called a law of from experimental facts and theory that the rate of a nature. A law of nature is supposed to be a generalireaction was proportional to the concentrations of the zation from experiment. The .+hove statement of the readants, each concentration raised to a power equal law is certainly not a generalization from experiment, to the coefficientof the corresponding substance in the since to verify the law one must know whether or not chemical equation, they derived equilibrium expres- a given reaction is simple or complex, and the only sions by equating the rates of forward and reverse general experimental method for determining this is reactions, their 1879 article2 the,, apply the to work the "law" backwards after testing the deexpression - L of~M~~~ ~ &tiod' to the statementof pendence of rate on concentration. Nevertheless, the the effect of concentration on rate of 6eaction and use statement is important since i t is a result of the applib ~thei relation ~ ~ ~ , cation , of kinetic theory or of statistical mechanics. the term "condition of ~ ~ ~ i l i for between concentrations at equilibrium. on this basis Also i t is the basis for the derivation of complicated we could decide which texts are correct. However, rate expressions when the mechanism of a complex there appears to be a more fundamental difficulty than reaction is known. It would be more appropriate to speak of the Theorem, or Principle, of Mass Action just one of terminology. rather than the Law. Act1121 mechanisms are "disIS THE LAW OF MASS ACTION VALID? covered by assuming that the "theorem" of mass studies of the rate of a reaction the action applies to the steps of a reaction. Hypothetical rate may be determined as a function of concentration, steps are then set UP in such fashion that they agree other conditions assumed constant. Usually the with data. the term "Law Mass function, or rate expression, will take the simple form Action" to the equilibrium relationship between conof a product of powers of concentrations centrations are justified to the extent that the general rate = k[Al.[BlQ[Cl'.. . equilibrium relationship is a t least a law of nature and holds for both simple and complex reactions. Furtherwhere k is the rate constant, [ A ] , [El, etc., are conmore, Guldberg and Waage carried out more expericentrations of substances A , 6 etc., which may be ments relating to equilibrium than to rate, They used G ~ D B E UAND O WAAGE. Collected papers may be found the theoretical rate expressions primarily to derive the in O s r w ~ m ' s"Klassiker der Exakten Wissenschaften,"Wilhelm condition of equilibrium, Engelm-, Leipzig, 1899, No. 104, 182 pp. The conclusion from the argument of this section GULDBERG AND WAAGE,I. pmkt. Chem., 19, 71 (1879). 272
I
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is that the expression "Law of Mass Action" is both false and misleading and should be deleted from the literature. KINETIC DERIVATION OF THE CONDITION OF EQUILIBRIUM
The only completely satisfactory derivation of the general equilibrium relationship is that given by thermodynamics or statistical mechanics. However, the kinetic picture of equilibrium bas such great pedagogic value that i t should certainly be presented to students if it can be done without being misleading. Most texts follow the lead of Guldberg and Waage in giving a simple kinetic derivation of the condition of equilibrium. As ordinarily presented, the derivation is limited to simple reactions although this limitation is hardly ever mentioned. There is great need for making a clearer distinction between simple and complex reactions so that the student will not be misled to believe that rate expressions for all reactions can be predicted from their stoichiometric equations. The author has not been able to find in the literature a general kinetic derivation of the equilibrium condition which holds for complex reactions. The following are derivations which apply, first, to a particular example of a complex reactioq3 then, to any complex homogeneous reaction of perfect gases or of solutes in liquid solution at low concentration. AN EXAMPLE OF A COMPLEX REACTION
Consider the reaction AB = A + B
which takes place by the mechanism
According to the principle of microscopic reversibility, or detailed balancing, a t equilibrium each step will be in equilibrium and the rate of the forward reaction will be equal to the rate of the reverse reaction. For the first step, according to kinegc theory or the "theorem" of mass action forward rste = k t [ A B ] [ X I reverse rate = kll[AXIIB]
which is the equilibrium constant expression for the first step. For the second step, similarly
Now turning back to the chemical equations for the reaction steps, i t is clear that the sum of the reaction
Such an example is given in
HEDEBRAND, "Principles of chemistry," 4th ed., The Mamillan Co., New York City, 1940, p. 159.
equations gives the overall or stoichiometric equation, X and A X canceling out. In like manner, if the two equilibrium expressions are multiplied together the concentrations of X and A X cancel out and the result is
which is the equilibrium expression for the overall reaction. A GENERAL DERIVATION OF THE CONDITION FOR RQUILIBRIUM
The derivation for the general case of any complex homogeneous reaction of perfect gases or ideal solutions is similar to the above, the idea being that when reaction steps are added the corresponding equilibrium expressions are multiplied together. This procedure insures that concentrations of catalysts and intermediates cancel out from the final equilibrium expression for the reaction as a whole. To simplify the algebra a different notation will be used. A reaction step will be expressed by placing all molecules on the right side of the equation, reacting molecules being then given a negative coefficient. For example, the first step in the reaction discussed above would be written meaning that A X and B are formed from AB and X Let the steps in the general reaction be +P.A +%.A +,.A
PaB + PaC + . . . +++ rsB qbB + !kc + . . . + r.C + . . . .
.
where A , B,..C, etc., are molecular species involved in the reaction and the p's, q's, r's, etc., represent numerical coefficients showing the number of molecules of each kind which are involved in a particular step, a negative value of the coefficient indicating that the corresponding molecular species, is a reactant in that step. All molecular species a;e indicated in each expression for a reaction step. A zero coefficient would mean that the particular species either does not take part in that reaction step or else that if it does take part it remains unchanged. (The latter provides for the case of a third body in a collision for the removal of energy or for the reverse piocess.) .At equilibrium each step is in equilibrium and the forward rate equals the reverse rate. If kl and kl' are rate constants for the forward and reverse of the first step, k,[A]-D*[Bl-us
. ..
=
k I r [ C ][Dim ~*
. ..
where the molecular species are placed on one side or the other depending upon the signs of the coefficients, i. e., whether they are reactants in the forward step or reverse step. In the above, p. and p, were supposed to be negative, meaning that A and B were reactants in the forward step. Upon rearranging the equation so that all concentrations are on the right and the constants on the left, we have
which is the equilibrium expression for the first reaction 'step. I t should be noticed that this expression has the same albebraic form regardless of which substances are reactants or products. Similarly for the other steps K , = [ A 1 ~ [Bps[C]s[D]*a s . .., etc.
and intermediates are zero. (If these were not zero, it would mean the catalyst or intermediate molecules would appear in the stoichiometric equation.) Now combine the equilibrium expressions for each step by multiplying together after raising each to the corresponding power
These expressions all hold simultaneously so they may but be combined algebraically. Suppose that the chemical n,p. + n,q. + . . . = X.. etc. equations for the reaction steps are added together in such a way that the stoichiometric equation is ob- Let K replace the product K 7 K P . . . tained. This may require multiplying some of the steps by a simple number. Let the stoichiometric then equation be where
the coe5cients n,, n,, n,, : . . being small whole numbers, chosen in such a way that the x's for the catalysts
which is the desired equilibrium expression for the stoichiometric equation of the complex reaction as a whole. A derivation such as this would be out of place in elementary texts but it would be very desirable that when the ordinary derivation is given it be explicitly stated that it applies only to simple reactions.
