In the Classroom
Ionic Activity Effects in Reaction Kinetics: What Happened to the Parsimony Principle? Robert de Levie Department of Chemistry, Bowdoin College, Brunswick, ME 04011;
[email protected] Equilibrium constants K are the quotients of products of the activities (rather than the concentrations) of the participating species, but they can also be considered as the ratio of forward and reverse rate constants. One must of course be careful, because kinetics involve rate-determining steps, whereas equilibrium expressions concern the overall process. Still, in view of the usual kinetic explanation of the mass action law, one might expect the chemical rate expressions to contain explicit activities of the reagents. However, kinetic expressions for reaction rates are uniformly written in terms of concentrations rather than activities. As we will see below, the explanation for this is that activity effects on the rates of chemical reactions are far weaker than one might expect on the basis of the above argument. Moreover, the usual interpretation for this effect is more complicated than necessary; that is, it does not follow the wellknown parsimony principle. Before we embark on our journey, a few general remarks are helpful. First, ionic activity effects are best understood in dilute solutions, where they are predominantly due to Coulombic interactions between ions. In fact, agreement about their quantitative description exists only at low ionic strength I = (1兾2)∑zi2ci (where ci denotes the concentration of ions i, and zi the corresponding valency), where the Debye–Hückel theory (1) shows the logarithm of the ionic activity coefficient, fi , to be proportional to zi2. Below we will therefore use this property of ionic activity coefficients. When we assign a constant value a to the distance of closest approach (a value not well-defined in electrolyte mixtures such as usually encountered in chemical reactions involving ions) we can write fi = f (zi 2 ), where f is now independent of the ions considered and where the expression for f can take various forms depending on whether one uses the Debye–Hückel limiting law, ᎑log f = A√I its full expression ᎑log f = (A√I )兾(1 + aB√I ) or one of several of its extensions, such as that of Hückel (2), ᎑log f = (A√I )兾(1 + aB√I ) − CI
Experimental Evidence We will consider the equilibrium A + B C + D and its forward and reverse reactions, A + B → C + D and A + B ← C + D, respectively. For the rate v of the forward binary reaction, A + B → products, it might be tempting to use expressions of the form v = −
d [ A] d [ B] = − = ka A aB = k f A [ A ] f B [B] dt dt
(1)
or, in view of the above, v = k [ A ][B] f A f B = k [ A ][B] f = k [ A ][B] f
(z A 2 ) f ( z B 2 ) (z A 2 + zB 2 )
(2)
but such a relation turns out not to fit the experimental results. As extensively documented by Brønsted (4), rates of reactions involving one uncharged and one charged component (i.e., with either zA or zB equal to zero) show no strong activity effects, contrary to what one would expect on the basis of eqs 1 or 2 where, for example, if A were the uncharged species, fA would be 1 and zA would be zero, but the effect of fB = f (zB2 ) would still be felt. On the other hand, Brønsted described several cases in which the ionic strength of the solution, as often determined by species not chemically involved in the reaction kinetics, strongly accelerated or decelerated the observed rates. Strong acceleration with increasing ionic strength was observed with reactions between ions of the same type, that is, anions with anions, or cations with cations, the effect being stronger the more highly charged the reacting ions. Strong deceleration was found with reactions between anions and cations. All this suggests a simple effect of the electrolyte ions on the mutual Coulombic interaction between the reagent species. In modern parlance, the mutual Coulombic attraction or repulsion is more efficiently “screened” in the presence of other ions. Examples of supporting experimental observations can be found, for example, in the original article of Brønsted (3), in an early review by La Mer (5), or in Benson’s book (6). Interpretation of Brønsted and Bjerrum
or that of Davies (3), ᎑log f = (A√I )兾(1 + √I ) − 0.3I ᎑1兾2
where, for water at room temperature, A = 0.5115 M and B = 3.291 M᎑1兾2 nm᎑1, while a is usually of the order of 0.2 to 0.6 nm, making aB of the order of 1 M᎑1兾2. However, the present discussion does not require a more specific form of f, which we will therefore leave open, so that the reader can fill in his or her favorite expression.
www.JCE.DivCHED.org
•
While Brønsted could not fit his results for A + B → C + D to eq 1, he found that he could represent them by invoking the activity coefficient of their presumed complex, AB, through v = −
Vol. 82 No. 6 June 2005
d [ A] d [ B] f f = − = k [ A ] [ B] A B f AB dt dt
•
Journal of Chemical Education
(3)
885
In the Classroom
where the charge of the complex AB was of course zAB = zA + zB. Brønsted tentatively interpreted this result in terms of a preceding equilibrium, A + B AB, forming an activated complex of valency zA + zB, followed by a subsequent unidirectional reaction, AB → products. Subsequently, Bjerrum (7) more forcefully advocated this interpretation, which postulated a thermodynamic equilibrium, A + B AB, followed by a reaction rate of the decay of AB that would then be directly proportional to the concentration of AB, v = −
d [ A] d [ B] = − = k ′ [ AB] dt dt =
f f f f k ′ [ A ][B] A B = k [ A ][B ] A B K f AB f AB
(4)
because one would obtain eqs 1 and 2 if the rate of decay of AB were proportional to its activity, [AB] fAB. This is a curious result because, in order to explain the effect of ionic strength on the reaction rates, it assumes a reaction rate proportional to concentration, coupled to an ad hoc prereaction equilibrium. Yet, this is how the concept of an activated complex, and transition state theory, got started. Interpretation of Christiansen The Debye–Hückel theory (1) appeared in 1923, a year after the Brønsted article (4) and a year before that of Bjerrum (7). The Debye–Hückel result showed the logarithm of the ionic activity coefficient to be proportional to the square of the ionic valency z, so that f A fB f = f AB
( z A 2 ) f ( zB 2 ) (z AB2 ) f
= f
(z A 2 + zB 2 − z AB2 )
= f
−2 z A z B
(5)
because zA2 + zB2 − zAB2 = zA2 + zB2 − (zA + zB)2 = ᎑2zAzB. The Brønsted–Bjerrum interpretation can indeed explain the observed experimental dependency on activity coefficients for unidirectional binary ionic reactions, but this inherently asymmetrical model of unidirectional reaction kinetics runs into conceptual difficulties when applied to equilibria, because, for the reverse reaction A + B ← C + D, one must now assume a preceding equilibrium between the products C + D and the activated complex CD, with slow kinetics for the dissociation of CD into A + B, A + B
AB
C + D
A + B
CD
C + D
This can be remedied by assuming that AB and CD are somehow different intermediates (even though they have the same stoichiometries); that is, by postulating different forward and reverse reaction pathways, in both of which the downhill process is the slow, rate-determining step. However, Christiansen (8) showed that the experimentally observed dependence on activity coefficients can be explained much more directly, without invoking any activated complexes or counterintuitive rate-determining steps, simply by considering how interionic interactions must influence ionic reaction kinetics. Because this conceptually much simpler explanation
886
Journal of Chemical Education
•
is seldom presented in contemporary textbooks, it will be briefly outlined here. For species A and B to react they must first meet, and this is explicitly recognized in the rate expression through the product cA cB, which represents the probability of an encounter of A and B. When both A and B are charged (but not when at least one of them is electrically neutral) the likelihood of their meeting depends not only on their bulk concentrations (i.e., number densities) but also on their mutual Coulombic attraction or repulsion, as described by a Boltzmann distribution, such as exp(᎑zAF ψB兾RT ), where F is the Faraday constant and ψB is the potential around ion B caused by its valency zB. In other words, when A and B have a charge of the same sign (i.e., both are anions or both are cations), they will repel each other electrostatically, thereby lowering their chances to react. Likewise, if their valencies are of opposite sign, so that one reagent is a cation and the other an anion, then their electrostatic attraction will enhance their chances of reacting. Because of the symmetry in the process, this result should be expressible as either exp(᎑zAF ψB兾RT ) when B is taken as the “central ion” in the Debye–Hückel theory, or as exp(᎑zBF ψA兾RT ) when A is assigned to play that role. One need not always solve the Debye–Hückel differential equation in all its mathematical details to extract valuable quantitative information from it (9), and we will use a somewhat similar approach here, as did Christiansen (8). In the Debye–Hückel derivation one first solves the Poisson– Boltzmann differential equation (approximately) for spherical symmetry and finds the potential ψB around B, which we will here consider the “central ion”. This potential can then be decomposed into two components: a part we will designate as ψB˝, which the ion would have at infinite dilution, that is, surrounded by solvent molecules only, and a second part, ψB´, which reflects the presence of other electrolyte ions. The contribution of ψB˝ is included in the numerical value of the rate constant at infinite dilution, and it is therefore only the term ψB´ that gives rise to the ionic activity and needs to be considered here. The remaining Debye–Hückel procedure is as follows. The lowering of the electrostatic self-energy of one mole of central ions B due to the presence of other electrolyte ions is (1兾2)zBF ψB´, which in the Debye–Hückel model is then directly identified with the activity term RT lnfB. We therefore have ψB´ = (2RT兾zBF )lnfB so that the relevant term in the Boltzmann exponent is ᎑z A F ψ B ´兾RT = ᎑(2z A 兾z B )ln f B = ᎑2zAzBln f because ln fB = zB2ln f. This result, ᎑2zAzBln f or lnf ᎑2zAzB, is fully equivalent with that of eq 5. Note also that we would have found the same answer had we started from a Boltzmann exponent ᎑zBF ψA´兾RT, as required by symmetry. No other assumptions are necessary other than that the encounter rate between the reagents is modified by their local, charge-dependent concentrations and that the activity coefficient is related to the ionic self-energy in the same general way as assumed by the Debye–Hückel theory. Incidentally, when the forward and reverse rate-determining reactions are the same as those that appear in the corresponding equilibrium expression, eq 5 is fully consistent with the thermodynamic result. This is readily illustrated, for example, for the equilibrium A + B C + D, for which eq
Vol. 82 No. 6 June 2005
•
www.JCE.DivCHED.org
In the Classroom
6 can then be rewritten as v + = k + [ A ][B] f
−2 z A z B
and v − = k − [ C][ D] f
−2 z C zD
(6)
so that K =
k+ v [C ][D ] 2 ( z A zB − z C z D ) f = + k− v − [ A ][B] =
=
[C ][D] f (zC2 + zD2 − z A 2 − zB2 ) [ A ][B] z 2 z 2 [ C ] f ( C ) [ D] f ( D )
[A] f
( zA 2 ) [ B ] f ( z B 2 )
(7) =
[C ] f C [D ] f D [ A ] f A [ B] f B
a a = C D a A aB
once we use the definition K = k+兾k− , insert the equilibrium requirement v+ = v− , and introduce conservation of charge during the reaction; that is, zA + zB = zC + zD or (zA + zB)2 = (zC + zD)2 so that 2(zAzB − zCzD) = zC2 + zD2 − zA2 − zB2. Consequently, Christiansen’s interpretation is fully compatible with thermodynamics. Note that the argument of Christiansen, as also used here in the retelling of it, does not depend on the actual mathematical solution of the problem and is valid for any activity coefficient that depends only on Coulombic ion interactions, that is, on those interactions that affect the electrostatic selfenergy of ions. The above result therefore holds also when, for example, a more complete solution for the spherical Poisson–Boltzmann potential is used (10) or when dielectric saturation is taken into account. An interesting application of eq 5 was the determination of the nature of a short-lived, highly reactive species generated in water with a van der Graaff generator as a negatively charged solvated electron rather than a “nascent” hydrogen atom (11). This conclusion could be reached because the reaction rates could be followed spectroscopically, and the valencies of the reaction partners were known, so that the dependence of the reaction rates on ionic strength according to eq 3 unambiguously provided the valency of the unknown reagent as ᎑1. Parsimony Principle Given two competing interpretations that explain the same experimental data, how do we go about selecting one over the other? In such cases it is common to use the parsimony principle (or Occam’s razor) that a model describing the observations with fewer assumptions is to be preferred. In the present case, the parsimony principle clearly favors the interpretation of Christiansen, because it does not require one or more intermediary transition states, combined with instantaneous uphill and slow downhill reactions, as does the Brønsted–Bjerrum model. It may be useful to digress here on the applicability of this “principle”. Of course, simplicity is neither a requirement nor a principle of nature. Still, the preference for the simplest possible explanation runs deep in science. After all, it was the argument Copernicus used to doubt the Ptolemaic world view of astronomical orbits following epicycles, that is, circles upon circles. By moving the center of the planetary
www.JCE.DivCHED.org
•
system from the earth to the sun, Copernicus not only simplified matters conceptually, but also (by reducing the mathematical complexity of the description) paved the way for Kepler to discover that the planetary orbits are actually ellipses rather than circles and for Newton to use that as the basis for his theory of gravitation. While there is no guarantee that the parsimony principle will always work, in the case of reaction kinetics its applicability seems warranted by comparing the natural explanation furnished by Christiansen versus the rather convoluted one of Brønsted and Bjerrum. Note that Christiansen’s explanation does not rule out that an activated complex might be formed, but merely shows it to be unnecessary for understanding the effect of ionic strength on reaction kinetics. Thus there is no need to invoke an activated complex to rationalize the effect of ionic strength on the rates of binary reactions: the experimental observations can be explained completely by the expected concentration changes resulting from bringing ions together at the Debye–Hückel distance of closest approach, a necessary prelude to their reaction. Discussion The chemical reactivity of a mole of ions i is not directly proportional to its ionic activity ai = ci f (zi 2 ) but depends critically on the valency of its reaction partner j through cicj f ᎑2zizj instead of the much stronger (and always unidirectional) dependency ai aj = cicj f (zi2+zj2 ) one might otherwise have expected. In the summary of his seminal article arguing that activities should take the place of concentrations in thermodynamic expressions, Lewis (12) stated that “a quantity named the activity … may be so defined that it serves as an ideal measure of the tendency of a given molecular species to escape from the condition in which it is.” While this may also be true for the ionic activity under equilibrium (i.e., static) conditions, it clearly does not apply to dynamic processes such as chemical reaction kinetics, regardless of whether one interprets the experimental data according to Brønsted (4) and Bjerrum (7), with a decay rate of an activated complex that is specifically proportional to its concentration rather than its activity, or according to Christiansen (8), in which case the experimental observations are understood as due to local concentration changes resulting from Coulombic interaction between the reagents that affect their chances of coming into close contact. In a footnote of the much-quoted article of Lewis and Randall (13) in which they introduced the concept of ionic strength, they commented as follows on reaction kinetics, “For any reaction which is near equilibrium conditions, at least, it is thermodynamically necessary that the speed depend upon the activity, and not upon the concentration of the reacting substances.” That statement would clearly be incorrect if the word “substances” would be interpreted to include ionic species, which in a strictly thermodynamic sense are not substances. Even so, the quoted sentence is at least confusing in the context of an article on the activity coefficient of strong electrolytes because, when ionic species are excluded, the concept of ionic strength obviously has little relevance.
Vol. 82 No. 6 June 2005
•
Journal of Chemical Education
887
In the Classroom
It is not clear why Christiansen’s interpretation has disappeared from most general physical chemistry textbooks as well as from those books more specifically concerned with reaction kinetics. It was acknowledged as an alternative explanation by Bjerrum (14), and it was mentioned in La Mer’s early review of the topic (5). I am unaware of any refutation of it in the literature. Perhaps chemists at the time did not fully appreciate the simplicity of Christiansen’s argument, or the elegance and general validity of his derivation. At any rate, whether or not the parsimony principle is still valued in chemistry, this direct application of it should no longer be ignored. Literature Cited 1. Debye, P.; Hückel, E. Physik. Z. 1923, 24, 185–206. 2. Hückel, E. Phys. Z. 1925, 26, 93–147.
888
Journal of Chemical Education
•
3. Davies, C. W. Ion Association; Butterworth: Washington DC, 1962; p 41. 4. Brønsted, J. N. Z. Physik. Chem. 1922, 102, 169–207. 5. La Mer, V. K. Chem. Revs. 1932, 10, 179–212, especially Figure 2. 6. Benson, S. W. The Foundations of Chemical Kinetics; McGrawHill: New York, 1960; Figure XV.5. 7. Bjerrum, N. Z. Physik. Chem. 1924, 108, 82–100. 8. Christiansen, J. A. Z. Physik. Chem. 1924, 113, 35–52. 9. de Levie, R. J. Chem. Educ. 1999, 76, 129–132. 10. Gronwall, T. H.; La Mer, V. K.; Sandved, K. Physik. Z. 1928, 29, 358–393. 11. Hart, E. J. Science 1964, 146, 19–25. 12. Lewis, G. N. Proc. Amer. Acad. 1907, 43, 259–293. 13. Lewis, G. N.; Randall, M. J. Am. Chem. Soc. 1921, 43, 1112– 1154. 14. Bjerrum, N. Z. Physik. Chem. 1925, 118, 251–254.
Vol. 82 No. 6 June 2005
•
www.JCE.DivCHED.org
