Thermodynamic consistency of reaction mechanisms

Thermodynamic consistency of reaction mechanisms and null cycles Guy Schmitz Citation: J. Chem. Phys. 112, 10714 (2000); doi: 10.1063/1.481715 View online: http://dx.doi.org/10.1063/1.481715 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v112/i24 Published by the American Institute of Physics.

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JOURNAL OF CHEMICAL PHYSICS

VOLUME 112, NUMBER 24

22 JUNE 2000

ARTICLES

Thermodynamic consistency of reaction mechanisms and null cycles Guy Schmitza) Universite´ Libre de Bruxelles, CP 165, Av. F. Rossevelt, 50, 1050 Bruxelles, Belgium

共Received 19 July 1999; accepted 30 March 2000兲 At equilibrium, the relationship K⫽k f /k r between the equilibrium constant and the rate constants of an elementary reaction is a consequence of the principle of detailed balancing. Out of equilibrium, it remains valid for most elementary reactions. In general, it is not valid for complex reactions out of equilibrium but its validity for the elementary steps of the mechanism may have important consequences for the interpretation of the experimental results. A related criterion of thermodynamic consistency of reaction mechanisms is the absence of null cycles defined as sets of different reactions taking place under given conditions and leading to no net stoichiometric change. © 2000 American Institute of Physics. 关S0021-9606共00兲50524-8兴

INTRODUCTION

One of the most challenging and hazardous tasks for a chemist is to propose a mechanism for a reaction. It is a hypothetical construct which is accepted as reasonable if certain conditions are fulfilled. It must explain the consumption of reactants and the formation of the observed products and it must explain the observed kinetics. Furthermore, kineticists use a set of working principles and guidelines, seldom stated explicitly in the literature, to decide whether a mechanism is reasonable or not. Many of them have been discussed by Espenson1 and by Edwards et al.2 Here we discuss the consequences of the relation between the equilibrium constant K and the rate constants in the forward, k f , and in the reverse, k r , direction, 共1兲

K⫽k f /k r .

Equation 共1兲 is generally presented as a consequence of the principle of detailed balancing:3 In a system at equilibrium the rate of every elementary reaction in one direction must be the same as the rate in the opposite direction. In this context, Eq. 共1兲 deals only with elementary reactions at equilibrium. Can we use it out of equilibrium? Let us consider an elementary reaction, reactants

products, with rate constants at equilibrium k f and k r . If the products are removed, it is expected that k f does not change and if the reactants are removed, it is expected that k r does not change. If so, Eq. 共1兲 remains valid out of equilibrium. Clearly this is true if the activated complex theory applies. Mahan3 has shown that Eq. 共1兲 holds out of equilibrium if ‘‘... the translational and internal states of the reactants have very nearly their Boltzmann equilibrium population.’’ Rice4 has discussed this under more general conditions. In this paper we assume that these conditions are

satisfied and that Eq. 共1兲 holds for the elementary reactions out of equilibrium. In general, Eq. 共1兲 does not hold for the global reaction. Nevertheless its validity for the elementary steps of the mechanisms may have important consequences for the interpretation of the experimental results.5,6 In this work we explore further these consequences for complex mechanisms.

NULL CYCLES

When we study mechanisms with many steps, it is not easy to see the consequences of Eq. 共1兲. Some become more evident using the concept of null cycles. We call null cycle a set of different reactions taking place in a given direction under given conditions and leading to no net stoichiometric change. When we specify that the reactions must be different we mean that a reaction and its reverse do not constitute a null cycle. We will show that the mechanism of a reaction cannot involve null cycles. This is not a new condition, it is a shortcut. The concept of null cycle gives a simple criterion of internal consistency of reaction mechanisms and helps to pick out the consequences of Eq. 共1兲. We will show this first with a simple example, then with three examples of real reactions. A first example

We consider two pathways for the reaction A 2B. The first one consists of reactions 共2兲–共3兲, 共2兲

X 2B.

共3兲

At equilibrium we have k 2 (A)⫽k ⫺2 (X) ⫽k ⫺3 (B) 2 giving

a兲

Electronic mail: gschmitz@ulb.ac.be

0021-9606/2000/112(24)/10714/4/$17.00

A X

10714

冋册

k 2k 3 共 B 兲2 ⫽K AB ⫽ k ⫺2 k ⫺3 共A兲

.

and k 3 (X) 共4兲

equilibrium

© 2000 American Institute of Physics


J. Chem. Phys., Vol. 112, No. 24, 22 June 2000

Thermodynamic consistency of reaction mechanisms

The second pathway consists of reactions 共5兲–共6兲, A 2Y

共5兲

Y B.

共6兲

At equilibrium we have k 5 (A)⫽k ⫺5 (Y ) 2 and k 6 (Y ) ⫽k ⫺6 (B) giving k 5 k 26 2 k ⫺5 k ⫺6

⫽K AB ⫽

冋册共 B 兲2 共A兲

.

共7兲

equilibrium

Out of equilibrium, with this combination of order one and order two reactions, the ratio of the apparent rate constants of the global forward and reverse reactions is not necessarily equal to K AB . Nevertheless, Eq. 共1兲 has a consequence: the rate constants are not independent. They must satisfy 共4兲 and 共7兲 and are related by 2 . k 2 k 3 /k ⫺2 k ⫺3 ⫽k 5 k 26 /k ⫺5 k ⫺6

In this example, we get a null cycle if the reaction A →2B follows pathway 共2兲–共3兲 and, in the same conditions, the reaction 2B→A follows pathway 共5兲–共6兲. In this situation we should have k 2 A⬎k ⫺2 X and k 3 X⬎k ⫺3 B 2 for pathway 共2兲–共3兲 and simultaneously k ⫺5 Y 2 ⬎k 5 A and k ⫺6 B ⬎k 6 Y for pathway 共5兲–共6兲. As we consider given conditions, especially given values of A and B, multiplying these in2 equalities we get k 2 k 3 k ⫺5 k ⫺6 ⬎k ⫺2 k ⫺3 k 5 k 26 . This is incompatible with the equality imposed by Eqs. 共4兲 and 共7兲 and the null cycle is ruled out.

A second example

The accepted mechanism for the oxidation of Ce共III兲 by bromate in the absence of bromide is 共8兲–共10兲. ⫹ BrO⫺ 3 ⫹BrO2H⫹H 2BrO2⫹H2O, ⫹

BrO2⫹Ce共III兲⫹H BrO2H⫹Ce共IV兲, ⫹ 2BrO2H→BrO⫺ 3 ⫹HOBr⫹H .

k9 ⫽K 9 k ⫺9

⫹ 2 k 8 共 BrO⫺ 3 兲共BrO2H兲共H 兲⬎k ⫺8 共 BrO2 兲 ,

k 9 共 BrO2兲共Ce共III兲兲共H⫹兲⬎k ⫺9 共 BrO2H兲共Ce共IV兲兲. Multiplying these inequalities and taking into account Eq. 共13兲, we get ⫹ 2 k ⫺12共 BrO⫺ 3 兲共 Ce共III兲兲共 H 兲 ⬎k 12共 BrO2 兲共 Ce共IV兲兲.

This proves that, if reaction 共12兲 exists, it takes place from right to left and cannot explain the inhibiting effect of Ce共IV兲. This was recognized later and new numerical simulations were made taking Eq. 共13兲 into account.9 If we sum the three reactions 共8兲, 共9兲, and 共12兲 taking place from left to right, the species on the left and right sides cancel out exactly and there is no stoichiometric result. This is a null cycle. We have used Eq. 共1兲 to show that if reactions 共8兲 and 共9兲 take place from left to right, then reaction 共12兲 must take place from right to left. The exclusion of null cycles leads directly to this conclusion. This is the fast qualitative approach. Then we can derive quantitative relations between the rate constants, like Eq. 共13兲. A third example

The mechanism proposed by Dagaut, Boettner, and Cathonnet10 for the combustion of ethanol involves a large number of reactions among which 共14兲–共15兲, 共16兲–共17兲, and other reaction pairs of the same form.

共8兲

C2H5OH⫹O2 sC2H5O⫹HO2,

共15兲

共9兲

C2H5OH⫹O pC2H5 O⫹OH,

共16兲

共10兲

C2H5OH⫹O sC2H5O⫹OH.

共17兲

共11兲

共12兲

If Ce共IV兲 can oxidize the BrO2 produced by 共8兲, the rate of 共9兲, and thus of the global reaction, is decreased by Ce共IV兲. Let us show that this explanation is in contradiction with the consequences of Eq. 共1兲,8 k8 ⫽K 8 k ⫺8

During the oxidation of Ce共III兲, reactions 共8兲 and 共9兲 are taking place from left to right.

共14兲

One of the kinetic features of this reaction is its inhibition by the product Ce共IV兲. The first explanation reported7 for this inhibition was reaction 共12兲. ⫹ BrO2⫹Ce共IV兲⫹H2O→BrO⫺ 3 ⫹Ce共III兲⫹2H .

共13兲

C2H5OH⫹O2 pC2H5O⫹HO2,

The global reaction 共11兲 is obtained taking 2⫻(8)⫹4⫻(9) ⫹(10). ⫹ BrO⫺ 3 ⫹4Ce共III兲⫹5H →HOBr⫹4Ce共IV兲⫹2H2O.

k8 k9 k ⫺12 ⫽ . k ⫺8 k ⫺9 k 12

10715

k 12 ⫽K 12 . k ⫺12

As the sum of the reactions 共8兲 and 共9兲 is identical to the reverse of reaction 共12兲, the equilibrium constants are related by K 8 K 9 ⫽1/K 12 . This gives relation 共13兲 between the rate constants,

sC2H5O and pC2H5O represent ethyl, 1-hydroxy, and ethyl, 2-hydroxy radicals. Subtracting reaction 共15兲 from reaction 共14兲 we get the isomerization reaction sC2H5O pC2H5O. Its equilibrium constant K iso is equal to K 14 /K 15 and Eq. 共1兲 leads to k 14 k ⫺15 ⫽K iso . k ⫺14 k 15 Subtracting reaction 共17兲 from reaction 共16兲 we get the same isomerization reaction and k 16 k ⫺17 ⫽K iso . k ⫺16 k 17 Equation 共1兲 dictates relations between the rate constants of the different steps. These relations are not obvious with mechanisms involving a very large number of steps. An efficient mean to detect them is to note the pairs of reactions leading to the same stoichiometric result. The rate constants of the different pathways must be such that there is no null cycle.


10716


Guy Schmitz

LNi⫹⫹OH⫺ L⫺⫹NiOH⫹,

A fourth example 11

Boumezioud and Tondre have studied the kinetics of formation of nickel 共II兲 complexes with 8-hydroxyquinoline in aqueous solutions. They have measured the relaxation time close to equilibrium12 and have interpreted their results in terms of the following mechanism involving the neutral 共LH兲 and the anionic 共L⫺兲 forms of the ligand, LH⫹Ni⫹⫹ LNi⫹⫹H⫹,

共18兲

L⫺⫹Ni⫹⫹ LNi⫹,

共19兲

LH⫹NiOH⫹ LNi⫹⫹H2O, ⫺

⫹

⫹

共20兲

⫺

L ⫹NiOH LNi ⫹OH .

共21兲

The following reactions are fast acid-base equilibria: LH L⫺⫹H⫹

K L ⫽ 共 L⫺兲共H⫹兲/共LH兲,

With an excess of Ni, the pseudo-first-order constant k obs can be expressed as k obs⫽k f 共 Ni⫹⫹兲 ⫹k r , where k 18共 H⫹兲 ⫹k 19K L ⫹k 20K h ⫹k 21K h K L / 共 H⫹ 兲 , 共 H⫹兲 ⫹K L

k r ⫽k ⫺18共 H⫹兲 ⫹k ⫺19⫹k ⫺20⫹k ⫺21K w / 共 H⫹兲 . K w is the ionization product of water. From the pH dependence of k f and k r the authors concluded that, under their experimental conditions, ⫹

k 18共 H 兲 Ⰷk 21K h K L / 共 H 兲

⫹

⫹

and k ⫺21K w / 共 H 兲 Ⰷk ⫺18共 H 兲 .

We have shown that these two inequalities cannot be simultaneously satisfied.13 When the system is at equilibrium, it follows from the principle of detailed balancing that all partial reactions are at equilibrium, K 18⫽

冋

k 18 共 LNi⫹兲共H⫹兲 ⫽ k ⫺18 共LH兲共Ni⫹⫹兲

册

, equilibrium

and similar relations for reactions 共19兲 to 共21兲. With the expressions of the fast acid-base equilibria, we get K 18⫽

k 19 k 20 k 18 K h K L k 21 ⫽K L ⫽K h ⫽ . k ⫺18 k ⫺19 k ⫺20 K w k ⫺21

共22兲

Since these equations do not involve concentrations but only kinetic and equilibrium constants, we can believe that they remain valid out of equilibrium. As a consequence, if for experiments at a given pH we have k 18(H⫹) ⬎K h K L k 21 /(H⫹), then for the reverse reaction at the same pH, we must have k ⫺18(H⫹)⬎K w k ⫺21 /(H⫹). In given conditions, the main reaction path must be the same in both directions. The same conclusion can be obtained using the concept of null cycle. The following four reactions taking place from left to right make a null cycle: LH⫹Ni⫹⫹ LNi⫹⫹H⫹,

L ⫹H LH, NiOH⫹ Ni⫹⫹⫹OH⫺. For these reactions to take place from left to right we must have k 18共LH兲(Ni⫹⫹)⬎k ⫺18(LNi⫹)(H⫹) and k ⫺21(LNi⫹) ⫻(OH⫺)⬎k 21(L⫺)(NiOH⫹). Multiplying these inequalities and taking the fast acid-base equilibria into account, we get k 18k ⫺21⬎k ⫺18k 21K L K h /K w . But, following Eqs. 共22兲, k 18k ⫺21 must be equal to k ⫺18k 21K L K h /K w . It cannot be larger and again the null cycle cannot exist. Using Eqs. 共22兲, we can put the expressions of k f in the form k ⫺18共 H⫹兲 ⫹k ⫺19⫹k ⫺20⫹k ⫺21K w / 共 H⫹兲 . 共 H⫹兲 ⫹K L

The expression of the ratio of the experimental rate constants simplifies to

K h ⫽共NiOH⫹兲共H⫹兲/共Ni⫹⫹兲.

⫹

共⫺21兲

⫹

k f ⫽K 18

Ni⫹⫹⫹H2O NiOH⫹⫹H⫹

kf⫽

⫺

共18兲

K 18 kf ⫽ ⫹ . k r 共 H 兲 ⫹K L This ratio is not equal to an equilibrium constant, but is determined by known equilibrium constants14 and by the pH. Even when Eq. 共1兲 is not valid for the global reaction, its validity for the elementary steps of the mechanism may dictate a relation between the experimental rate constants in the forward and backward directions. We verified this relation experimentally.13 THE GENERAL CASE

At equilibrium, there is no net stoichiometric change and all the partial reactions are at equilibrium. The existence of null cycles is prohibited by the principle of detailed balancing. Out of equilibrium we have given examples of null cycles and used Eq. 共1兲 to prove that they cannot exist. These observations can be generalized thermodynamically. The basic idea is the following: Every possible reaction is associated with a decrease of the Gibbs free energy. By definition, a null cycle leads to no net stoichiometric change, to no net decrease of free energy and thus cannot exist. This idea can be formulated more precisely. Let r i be the global rate of a reaction i. For a reversible reaction this is the rate in the forward direction minus the rate in the backward direction. Let ⌬G i be the Gibbs energy change due to reaction i at given pressure and temperature. A reaction out of equilibrium must take place in the direction such that the Gibbs energy decreases: we must have r i ⌬G i ⬍0. Consider now a set of n reactions (i⫽1 to n兲 with r i ⬎0. If the sum of these reactions gives no net stoichiometric change then the sum of the ⌬G i is null and at least one of the ⌬G i must be positive. This is inconsistent with the condition r i ⌬G i ⬍0 and a null cycle cannot exist. It is important to note that there is no contradiction between the exclusion of null cycles and the observation of Laidler15 that the favored kinetic pathway can be different for a reaction occurring in one direction and for the reverse reaction, provided that the experimental conditions are dif-



ferent. It is possible that reactions 共8兲 and 共9兲 take place from left to right in some conditions and that, in other conditions, reaction 共12兲 takes place also from left to right. CONCLUSIONS

When we propose a mechanism for a reaction we cannot forget the implications of the relationship K⫽k f /k r . At equilibrium, it is a consequence of the principle of detailed balancing. Out of equilibrium, it remains valid for most elentary reactions but is seldom valid for complex reactions. Nevertheless, written for the elementary steps of the mechanisms, it gives relations between rate constants that must be fulfilled by the proposed model. We have discussed examples showing the consequences of this relationship. A related simple criterion of thermodynamic consistency of reaction mechanisms is the absence of null cycles. It helps to pick out the qualitative consequences of Eq. 共1兲, avoiding the demonstrations using the relations between the rate constants. 1

J. H. Espenson, in Chemical Kinetics and Reaction Mechanisms, 2nd ed. 共McGraw-Hill, New York, 1995兲, p. 127.

Thermodynamic consistency of reaction mechanisms

10717

J. O. Edwards, E. F. Greene, and J. Ross, J. Chem. Educ. 45, 381 共1968兲. B. H. Mahan, J. Chem. Educ. 52, 299 共1975兲. 4 O. K. Rice, Statistical Mechanics, Thermodynamics and Kinetics 共Freeman, San Francisco, 1967兲. 5 L. Onsager, Phys. Rev. 37, 405 共1931兲. 6 J. H. Espenson, Ref. 1, p. 172. 7 R. M. Noyes, R. J. Field, and R. C. Thompson, J. Am. Chem. Soc. 93, 7315 共1971兲. 8 As the rates are functions of the concentrations, the equilibrium constants used in this paper are apparent values including the effect of the activity coefficients. 9 S. Barkin, M. Bixon, R. M. Noyes, and K. Bar-Eli, Int. J. Chem. Kinet. 9, 841 共1977兲. 10 P. Dagaut, J. C. Boettner, and M. Cathonnet, J. Chim. Phys. Phys.-Chim. Biol. 89, 867 共1992兲. 11 M. Boumezioud and C. Tondre, J. Chim. Phys. Phys.-Chim. Biol. 85, 719 共1988兲. 12 C. F. Bernasconi, Investigation of Rates and Mechanisms of Reactions, Techniques of Chemistry 共Wiley-Interscience, New York, 1986兲, Vol. VI, part II; J. H. Espenson, Ref. 1, p. 256. 13 G. Schmitz, J. Chim. Phys. Phys.-Chim. Biol. 93, 482 共1996兲. 14 Critical Evaluation of Equilibrium Constants Involving 8-Hydroxyquinoline and its Metal Chelates, IUPAC 共Pergamon, New York, 1979兲. 15 R. M. Krupka, H. Kaplan, and K. J. Laidler, Trans. Faraday Soc. 62, 2754 共1966兲; K. J. Laidler, in Chemical Kinetics, 3rd ed. 共Harper & Row, New York, 1987兲, pp. 129, 285. 2 3
