The Derivation of the Equilibrium Constant Expression, K
Virtually all modern high school and first-year university chemistry textbooks teach that the general chemical equation aA + bB ⇌ c cC + d D has the equilibrium constant K
[C ] [ D]
c
d
a
b
[ A] [ B]
These textbooks also commonly include a simple derivation of this expression, which involves setting the rate equations for the forward reaction and reverse reaction equal to each other. To illustrate, one book (Brown, LeMay Jr., & Bursten, 1997) considers a forward reaction A → B, which they say has rate = k f [ A] , and its reverse reaction B → A, which they say has rate = k [ B] . r
“Eventually the reaction reaches a point at which the forward and reverse rates are the same […] compounds A and B are in equilibrium” (p. 540). And so at equilibrium, Rate of forward reaction = Rate of reverse reaction
k f [ A] k r [ B] The equation is rearranged: k f k r
[ B ] [ A]
And students are then told that this ratio is called K , the equilibrium constant. Despite its popularity, this kinetic approach to deriving the equilibrium has come under und er criticism from other educators. The first and most common criticism is given by Quílez (2004) when he writes “this approach should be avoided […] […] It is an example of obtaining a correct result (equilibrium constant) using an improper procedure that writes down two rate equations based on the stoichiometry of the reaction” (p. 81). According to this criticism, the problem is that this method
seems to require the assumption that we can write down rate equations directly from the chemical equation, which only works for reactions that occur in one-step. In other words, the critics claim that using this method with a multi-step reaction, say, aA + bB a
⇌
cC + d D, would have to go
b
something like this: assume that the forward rate equals k f [ A] [ B] and the reverse rate equals
k r [C ] [ D] , then set them equal to each other, and rearrange to get K c
d
[C ] [ D]
c
d
a
b
[ A] [ B ]
. But, they say,
a
b
this cannot be correct because the forward rate is not necessarily equal to k f [ A] [ B] for multistep reactions, and the same is true for the reverse rate. A second, less common criticism states that kinetic derivations are undermined by the fact that rate laws change as equilibrium is approached (Ashmore, 1965), as the kinetic derivation assumes a reaction has only a single, fixed rate law. Our purpose here is to evaluate these two criticisms. To this end, we will begin by consulting the history of deriving the equilibrium expression, from there go on to consider of the nature of scientific explanation and the value of demonstrations in education.
The earliest studies of how rate of reaction depended on concentration occurred in the late 19th century. Berthelot and Péan de Saint-Gilles, as well as Harcourt and Esson, made important early contributions in the 1860s. Contributions followed in the 1870s, from Guldberg and Waage, as well as from van’t Hoff (Laidler, 1985). However, the “epoch-making” work on chemical kinetics and equilibrium was made in 1884 by van’t Hoff’s book Etudes de dynamique chimique (Mellor, 1904, p. 55). In the Etudes, van’t Hoff determines the equilibrium constant K by setting the rate of the forward reaction equal to the rate of the reverse reaction and rearranging:
k 1C 1
n1
k 2 C 2
n2
k 1 k 2
C 2
n2
C 1
n1
K
In the 1899 textbook Introduction to Physical Chemistry, James Walker offers a similar derivation, and the same goes for Mellor’s 1904 textbook Chemical Statics and Dynamics. And all three – van’t Hoff, Walker, and Mellor – were aware of the fact that the rate equation could not alwa ys be directly derived from the chemical equation. Further, van’t Hoff gave the kinetic derivation approval even when he was aware of a thermodynamic proof: “experiment, thermodynamics, and kinetic theory lead, independently of one another, to the relationship” (van’t Hoff, 1886, p. 257). However, he is careful to point out that this is only true for equilibrium of gaseous systems. For equilibria of dilute aqueous systems, the constant can only be derived from thermodynamic considerations (van’t Hoff, 1886).
Although van’t Hoff and others said that a kinetic approach could be used, in certain cases at least, they did not show exactly how this worked. This would be shown later by Frost (1941) and then more rigorously by Koenig (1965). They show that, just as the coefficients of elementary chemical equations necessarily add up to the coefficients of the overall chemical equation, so too the equilibrium constants of the elementary steps, where the exponents equal the coefficients, will always multiply together to yield an overall equilibrium constant where the exponents equal the corresponding coefficients in the overall equation. In other words, if we instead write a general multi-step equation with all the terms moved to the left-hand side, so that all products have negative coefficients, x1 X 1 + x2 X 2 + x3 X 3 + … + xn X n⇌ 0
For any reaction in this notation, K
n
[ X ]
xi
and to show this with kinetic theory, we simply
i
i 1
consider the equilibrium constant for an elementary step j, call it K j . Since an elementary reaction occurs in one-step, we can safely assume that the expon ents of its forward and reverse rates match its coefficients. This means there is no problem using the kinetic approach to derive the equilibrium constant of an elementary step:
K j
n
[ X ]
x j , i
i
, where x j.i is the coefficient of xi in the jth elementary step
i 1
To find the overall equilibrium constant, K , we simply multiply together the equilibrium constants of each elementary step. If there are s elementary steps overall, then s
K K j j 1
K
n x [ X i ] j 1 i 1 s
j , i
This can be rewritten as, s
K
n
x
[ X i ] j
j , i
1
i 1
But since the elementary step coefficients must add up to coefficients of the overall reaction, we know s
x j 1
Therefore,
j ,i
xi
K
n
[ X ]
xi
i
i 1
which is the desired result.
From this, it is evident that the first criticism of the kinetic derivation of the equilibrium ex pression is misguided. The above derivation does not, as the criticism alleges, assume that the exponents of the forward and reverse rates always equal the coefficients of the chemical equation. It only assumes that the exponents of the rates of elementary reactions always equal the coefficients of the chemical equation, which is a perfectly safe assumption.
The second criticism of the derivation, however, is correct. The second criticism notes that the rate law for a reaction often changes as the reaction proceeds, and it is clear that the above derivation assumes that the rate law is constant throughout the reaction. This being said, we might wonder why the derivation still works despite this simplifying assumption. This is because “an equilibrium state does not depend on the path by which it is reached ” (Mysels, 1956, p. 179) and so the correct equilibrium constant expression will be derived even when we (wrongly) assume that the reaction occurs in such a way that the rate law remains constant throughout. This is also why we will get the correct result from a method which wrongly assumes that rate law exponents alwa ys match the coefficients of the chemical equation (i.e., the first criticism’s allegation). For such a method implicitly assumes that every reaction is an elementary reaction, and even though this is not true, such an assumption does not affect the end result because, as above, the state of equilibrium does not depend on the path used to get there.
Koenig (1965) calls his kinetic derivation “approximate” (p. 227). This is not only because, as mentioned above, the kinetic derivation requires simplifying assumptions about how rate laws change during the course of a reaction, but it also relies on collision theory, which is an approximate model of reactions. After defining an elementary reaction to be a reaction that o ccurs through a collision of particles which immediately yields the given product, he remarks that his “definition suffers from vagueness owing to the terms “collision” and immediate”” (p. 228). Presumably the problem here is that not all “collisions” occur exactly in the same way, nor do they all produce their products with the exact same speed. In contrast, thermodynamic d erivations (e.g., van’t Hoff, 1886) do not depend on modeling the mechanism of how reactions actually take place, but simply rest on evident ideas about matter, e.g., that they contain energy and that perpetual motion is impossible (Einstein, 1919). Therefore, thermodynamic derivations are “exact” (Koenig, 1965, p. 227), because they are an account of a fact (in this case, the equilibrium constant) through its causes. And, as Aristotle notes in the Posterior Analytics (I.2), this gives us certain knowledge of what the equilibrium constant is.
So it appears that the kinetic derivation falls short of proving with certainty, or even any sort of probability, the equilibrium constant. Because it requires very simplifying assumptions about how reactions actually occur, it fails to account for the condition of equilibrium through its causes. However, this does not answer the more important practical question, which was about the value of the kinetic derivation in education. Despite its flaws, the derivation helps make clear to students the meaning of the equilibrium constant expression. The whole point of the equilibrium expression is that it allows one to check if a system is in equilibrium, using only the concentrations of the
products and reactants. If one substitutes in these concentrations and it gives a number equal to the equilibrium constant at the appropriate temperature, then the system one is considering is at equilibrium. Why does this work? It works because, as the kinetic derivation shows, the equilibrium expression can be thought of as a rearrangement of the equation rate forward = ratereverse. If the equilibrium expression ‘checks out,’ one therefore knows that the forward rate equals the reverse rate, and so the system is at equilibrium. This could be presented to students with the warning that the derivation is approximate, and that a thermodynamic derivation is more rigorous. Ultimately, whether or not the kinetic derivation should be shown to students is a matter of prudence. Even if the derivation can be useful, chemistry teachers may still wisely avoid it if they think it will confuse their students, or make them think that it is acceptable to write rate equations directly from the stoichiometry of the chemical equation. But to make the best decision, teachers should be aware of what exactly are – and are not – the limitations of the kinetic approach to the equilibrium constant.
References
Ashmore, P.G. (1965). Reaction kinetics and the law of mass action. Education in Chemistry, 2, 160-166.
Brown, T. L., LeMay, H. E., Jr., & Bursten, B. E. (1997). Chemistry: the central science (7th ed.). Upper Saddle River, NJ: Prentice Hall.
Einstein, A. (1919, November 28). What is the theory of relativity? The London Times. Retrieved from http://germanhistorydocs.ghi-dc.org/pdf/eng/EDU_Einstein_ENGLISH.pdf
Frost, A. A. (1941). Effect of concentration on reaction rate and equilibrium. Journal of Chemical Education, 18 , 272-274.
Koenig, F. O. (1965). On the deduction of the equilibrium law from kinetics. Journal of Chemical Education, 42, 227-230.
Laidler, K. J. (1985). Chemical kinetics and the origins of physical chemistry. Archive for History of Exact Sciences, 32, 43-75.
Mellor, J. W. (1904). Chemical statics and dynamics. London: Longmans, Green, and Co.
Mysels, K. J. (1956). The laws of reaction rates and equilibrium. Journal of Chemical Education, 33, 178-179.
Quílez, J. (2004). A historical approach to the development of chemical equilibrium through the evolution of the affinity concept: some educational suggestions. Chemistry Education: Research and Practice, 5, 69-87.
Van’t Hoff, J. H. (1884). Etudes de dynamique chimique. Amsterdam: Frederik Muller & Co.
Van’t Hoff, J. H. (1886). L’équilibre chimique dans les systèmes gazeux ou dissous à l’état dilué. Archives Néerlandaises des Sciences Exactes et Naturelles, 20, 239-302.
Walker, J. (1899). Introduction to physical chemistry. London: Macmillan and Co.
