Systematic Application of the Principle of Detailed Balancing to

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A: Kinetics, Dynamics, Photochemistry, and Excited States

Systematic Application of the Principle of Detailed Balancing to Complex Homogeneous Chemical Reaction Mechanisms David M. Stanbury, and Dean Hoffman J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b03771 • Publication Date (Web): 05 Jun 2019 Downloaded from http://pubs.acs.org on June 5, 2019

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The Journal of Physical Chemistry

May 23, 2019 Systematic Application of the Principle of Detailed Balancing to Complex Homogeneous Chemical Reaction Mechanisms David M. Stanbury* and Dean Hoffman‡ Department of Chemistry and Biochemistry and Department of Mathematics and Statistics Auburn University, Auburn, AL 36849, USA Abstract: It is not uncommon for proposed complex reaction mechanisms to violate the principle of detailed balancing. Here, we draw attention to three ways in which such violations can occur: reversible reaction loops where the rate constants do not attain closure, illegal loops, and reversible steps having rate equations in the forward and reverse directions that are inconsistent with the equilibrium expressions. We present two simple methods to test whether a proposed mechanism is consistent with the first two aspects of the principle of detailed balancing. Both methods are restricted to closed homogeneous isothermal reactions having mechanisms that consist of stoichiometrically balanced reaction steps. The first method is restricted to mechanisms in which all reaction steps are reversible; values of ∆fG° are assigned to all reaction species, equilibrium constants are then computed for all steps, and all rate constants for elementary steps are constrained by the relationship Keq = kf/kr. The second method is applicable to mechanisms that can consist of a series of reversible and/or irreversible reaction steps. One first examines the subset of reversible steps to determine whether any of these steps are stoichiometrically equivalent to a combination of any of the other steps. If so, the forward and reverse rate expressions must yield equilibrium constants that are in agreement with the stoichiometric relationships. Next, the complete set of steps is examined to look for “illegal reaction loops”. Both of these procedures are performed by constructing matrices that represent the stoichiometries of the various reaction steps and then performing row reductions to identify basis sets of loops. A method based on linear programming is described that determines whether a mechanism contains any illegal loops. These methods are applied in the analysis of several published reaction mechanisms. INTRODUCTION The principle of detailed balancing is a fundamental aspect of chemical kinetics, and it can impose certain relationships between the rate constants of the steps in any reaction mechanism. Stated simply, it specifies that the equilibrium constant for an elementary reaction 1

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step is equal to the ratio of the forward and reverse rate constants: Keq = kf/kr. Despite its apparent simplicity it can be challenging to apply in complex reaction mechanisms, with the potential outcome that impossible mechanisms can be generated. These situations typically arise when there is a set of reaction steps that can be combined to form a “reaction loop”. The term “reaction loop” describes any set of reactions where one of them is stoichiometrically equivalent to a linear combination of the others. Reaction loops can be difficult to detect, but they can occur in any multi-step mechanism and they must occur in all mechanisms that have more reaction steps than species. Some reaction loops impose specific relationships between the rate constants, some others are illegal, and others are benign. In this paper we describe two systematic methods to identify such violations of the principle of detailed balancing and thereby ensure that they are not promulgated. These methods are generally applicable to closed isothermal homogeneous reaction systems; their applicability to other types of reaction systems remains to be explored. Historically, aspects of the principle of detailed balancing were discussed early on,1-3 and the first use of the term "detailed balancing" was published by Fowler and Milne in 1925.4, 5 At that time the basic concept was presented, but its implications for complex reaction mechanisms were not extensively developed. Nevertheless, by 1953 the concept had become enshrined as one of the fundamental principles of chemical kinetics, appearing in influential textbooks on chemical kinetics.6 The concept of legal and illegal reaction loops, which has special relevance to loops having irreversible steps, has been discussed at least since 1972,7 and in considerable detail by Sorensen and Stewart8 and by Bieniasz.9 A highly readable discussion of the relationships between simple three-step reaction loops and the principle of detailed balancing was published by Alberty in 2004.10 Systematic methods to apply these concepts to complex reaction mechanisms have been described, but none are optimized for application by practicing chemists to isothermal homogeneous closed chemical reaction mechanisms that include irreversible steps.8, 9, 11-15 Some software packages for modeling chemical kinetics are capable of imposing compliance with the principle of detailed balancing, but only for mechanisms where all steps are elementary and reversible (ChemKin,16 Kintecus,17 DigiSim,18 BioNetGen19). One statement of the principle of detailed balancing is that at equilibrium the forward and reverse rates of all reactions are equal. This leads directly to the relationship Keq = kf/kr for an elementary or pseudo-elementary step, as is illustrated by the following example: A+B+C

2D

(I)

K = [D]eq2/([A]eq[B]eq[C]eq) rate forward = kf[A][B][C],

(II) rate reverse = kr[D]2 2


(III)

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so at equilibrium we have

or

kf[A]eq[B]eq[C]eq = kr[D]eq2

(IV)

kf/kr = [D]eq2/([A]eq[B]eq[C]eq) = K

(V)

As the above example illustrates, the relationship between the rate constants and the equilibrium constant arises because the rate expressions for (pseudo-) elementary steps are power-law equations in terms of the reactant concentrations with the exponents being equal to the stoichiometric coefficients. However, chemists often propose mechanisms in which some of the steps are not elementary and for which the rate expressions are not directly related to the stoichiometry. With non-elementary steps the principle of detailed balancing still requires that the forward and reverse rates are equal at equilibrium, but the relationships between the rate expressions and the equilibrium constants can become more complex than for elementary steps. Suppose, for example, that the forward rate for reaction I had the form rate forward = k1[A][B]/(1 + k2[D])

(VI)

Equivalence of the forward and reverse rates at equilibrium would then define the reverse rate equation by use of eq II even though the relationship K = kf/kr is not directly applicable. Note that this aspect of detailed balancing also requires that if a reaction step is catalyzed in the forward direction then it must also be catalyzed in the reverse direction. It is essential to confirm that the forward and reverse rate expressions for reversible non-elementary steps are consistent with the equilibrium constant expression. BerkeleyMadonna is an example of a kinetic simulation software application that allows use of non-elementary reaction steps, but it does not impose compliance with this aspect of detailed balancing for such steps when they are reversible.20 To illustrate some of the basic concepts involving reaction loops it is useful to consider the simple example of an isomerization reaction loop involving three different species, A, B, and C. One might describe seven ways of interconverting these species: Case 1: all steps reversible.

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Case 1 A B C

B C A

K1 K2 K3

(VII) (VIII) (IX)

In this first case, with three reversible steps defining a reaction loop, we have reaction VII + reaction VIII = –reaction IX stoichiometrically. Since all three steps are reversible this loop constitutes a thermochemical cycle: Thus K1K2 = 1/K3.

(X)

It is important to ensure that the three equilibrium constants agree with this relationship. When the reaction steps are (pseudo)-elementary steps we have the additional constraints K1 = k1f/k1r, K2 = k2f/k2r, and K3 = k3f/k3r, which lead to the relationship (k1f/k1r)(k2f/k2r) = k3r/k3f

(XI)

This is known as Wegscheider's condition;2, 3 analogous expressions pertain to loops consisting of more than three steps and also when the steps have rate laws more complex than first order.21 One of us has recently identified a series of publications on iodate chemistry where this type of relationship is violated by many orders of magnitude.22 Case 2: one step irreversible. Illegal.

Case 2 Case 2 and other illegal loops are illegal because they can never reach equilibrium; as discussed by Alberty, illegal loops reach a steady state instead where there is a constant net flux through

4


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the system. A formal development of the concept of illegal loops has been published by Gorban and Yablonsky.23 For the present purposes a convenient device for identifying illegal loops is to consider that when a step is irreversible, as in Case 2 and in all of the following cases, the irreversibility is equivalent to having its reverse rate equal to zero, and thus the equilibrium constant is infinite. This is, of course, an approximation since reverse rate constants are never exactly zero. Case 2 thus leads to the relationship K1(infinity) = 1/K3, which is impossible; this type of description of illegality has been used previously by Bieniasz.9 Two examples of Case 2 are discussed below. Cases 3, 4, and 5: two irreversible steps. Two cases are legal; the other is illegal.

Case 3

Case 4

Case 5

In Case 3 we have two irreversible steps, both forming species C. The relationship becomes K1(infinity) = infinity, which is possible; Case 3 is legal. In Case 4, where both irreversible steps consume species C, the relationship becomes K1(1/infinity) = 1/infinity, which is also possible. In Case 5 the relationship is K1(infinity) = 1/infinity, which is impossible; Case 5 is illegal. Two examples of Case 5 are discussed below. Cases 6 and 7: all steps irreversible. One case is illegal; the other is legal.

Case 6

Case 7

The relationship for Case 6 is (infinity)(infinity) = 1/infinity, which is impossible. Case 6 is illegal. For Case 7 the relationship is (infinity)(infinity) = infinity, which is possible; Case 7 is legal. Generalizing to loops having an arbitrary number of steps, we can identify three types: 1) those consisting entirely of reversible steps, 2) those having only one irreversible step, and 3)

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those having several irreversible steps. Type 1 is legal, but the rate constants must satisfy Wegscheider's condition. Type 2 is illegal. Type 3 is illegal if all of the irreversible steps have the same sign in the linear combination that defines the loop. These concepts have been developed with formal mathematics by Gorban and Yablonsky.23 Additional complications can arise in mechanisms that contain more than one loop. These loops can be isolated or have certain reaction steps in common (linked loops). 1 4

3 6

5 8

Isolated loops

2

9

7 10

Linked loops

When isolated loops are identified it is a simple matter to use the rules described above to ensure that they meet Wegscheider's condition or are legal. Linked loops, however, can be more complex as shown in the above diagram: this example can be described by the sum of one large loop consisting entirely of the reversible steps (1,2,3,7,10,9,8,4) plus one small loop consisting of two reversible steps and two irreversible steps (2,6,9,5). Both of these loops are legal, and the first must comply with Wegscheider's condition. However, it is also possible to describe the set of linked loops with three small loops (1,5,8,4; 2,6,9,5; 3,7,10,6), two of which are illegal because they contain only one irreversible step. Thus, when dealing with mechanisms having linked loops, it is important to ensure that all possible loops are legal and comply with Wegscheider's condition when appropriate. Since the above illustrations describe mechanisms consisting of isomerization processes only, there may be some question as to whether the concepts can be extended to more complex processes such as bimolecular steps. The concern is highlighted by considering the following example, which is discussed in more detail below: I2 + CN– ICN + I– ICN + H+ + I– HCN + I2 CN– + H+ HCN

R5 R8 R9

These steps form a thermochemical cycle (reversible loop) since KR5KR8 = KR9. On first sight it might appear that the diagram for a Case 1 loop cannot be drawn because the products in R5 are not the products or reactants in either R8 or R9. This difficulty can be resolved by adding appropriate non-reacting species to both sides of each step. In this example H+ is added in R5 6


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and I2 is added in R9. These additions affect neither the net stoichiometry nor the equilibrium constants. I2 + CN– + H+ ICN + H+ + I– I2 + CN– + H+

ICN + I– + H+ HCN + I2 HCN + I2

R5' R8 R9'

With these modifications it is clear that this set of reactions can be represented by the diagram for a Case 1 loop. In this way it is possible to represent any arbitrary reaction loop by diagrams analogous to those used above, and hence the concepts of reversible loops, illegal loops, and linked loops are valid in general. It should also be mentioned that when an illegal loop is embedded within a larger irreversible mechanism it is still illegal. As one of the examples below shows, repair of an embedded illegal loop by including the requisite reverse reactions can lead to significant changes in results of the overall model. METHODS In view of the above discussion it should be clear that a mechanism consisting of several or many steps could present many possible violations of the principle of detailed balancing. It can be a significant challenge to identify these situations simply by inspection. One needs a systematic procedure to identify reaction loops and check for compliance with the principle of detailed balancing. Here we describe two such methods, one applicable to mechanisms where all the steps are reversible, and the other applicable to all mechanisms. Method 1: all reactions reversible. This method can be performed easily on paper or with a spreadsheet such as in Excel. One can simply look up the values of ∆fG° for all species in the mechanism, use these ∆fG° values to calculate ∆G° for each reaction step, calculate Keq for each reaction step from ∆G° = –RTln(Keq), and then calculate values for kr for each elementary reaction step from the relationship kr = kf/Keq. If values of ∆fG° are unknown for some of the reaction species, arbitrary values can be used and adjusted appropriately. In principle, an equivalent method is to derive a mechanism by calculating the rate constants from the Gibbs energies of the species and transition states in a Gibbs energy profile (or landscape) of the reaction. We note that this method has been reported previously in the eBNG extension of the BioNetGen modeling software for biochemical processes,19 but its exposition here is in a form that is much simpler and more suitable to homogeneous chemical reactions.

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As an example, an application of Method 1 to the reaction of iodate with NAHT (Nacetyl homocysteine thiolactone) is shown in Tables 1 and 2. In this example we have started with the 12-step mechanism proposed in the publication where this reaction was originally described,24 and then made all reaction steps reversible and elementary. ∆fG° values are available from the literature for all species except NAHT and NAHTSO (N-acetyl homocysteine thiolactone sulfoxide). A value of –100 kJ/mol was assigned arbitrarily to ∆fG° for NAHT; a value of –300 kJ/mol was then assigned to ∆fG° for NAHTSO; this later value was selected in order to make the equilibrium constants very large for the steps involving NAHT, since these steps were regarded as irreversible in the original publication. By performing the calculations in a spreadsheet it is straightforward to adjust ∆fG° for NAHTSO to achieve the desired result. The important outcome, however, is that the calculated values of kr for reactions M1 and M4 (which are independent of the value selected for ∆fG° for NAHTSO) deviate from the published values by many orders of magnitude, thus revealing a major violation of the principle of detailed balancing. Table 1. Gibbs Energies of Formation Used in the Model of the IO3–/NAHT Reaction Species I– HOI HIO2 I2 I3– a

∆fG°, kJ mol–1 -51.57 -99.2 -95 16.4 -51.4

Species

∆fG°, kJ mol–1 IO3– -128 HIO3 -132.6 NAHT -100 NAHTSO -300 + H 0

source NBS a Schmitz 2008 b Schmitz 2008 b NBS a NBS a

Reference 25. b Reference 26.

8


source NBS a NBS a guesswork guesswork NBS a

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Table 2. IO3–/NAHT Reaction Model and Calculated Reverse Rate Constants rxn

∆G° –

–

+

Keq

kf

kr calcd

kr, lit.

M1

IO3 + I + 2H = HIO2 + HOI

-14.6

3.6E+02

2.8

7.7E-03

1.44E+03

M2

HIO2 + I– + H+ = 2HOI

-51.8

1.2E+09

2.10E+08

1.7E-01

90

-69.9

1.8E+12

3.10E+12

1.7E+00

2.2

37.2

3.1E-07

8.60E+02

2.8E+09

2

-16.2

6.9E+02

6.20E+09

8.9E+06

8.50E+06

-4.6

6.4E+00

2.04E+08

3.2E+07

1.25E+09

M7

HIO3 + NAHT + H = HIO2 + NAHTSO + H -162.4

2.7E+28

8.80E-02

3.2E-30

(irreversible)

M8

IO3– + NAHT + H+ = HIO2 + NAHTSO

-167

1.8E+29

1.05E-04

6.0E-34

(irreversible)

M9

HIO2 + NAHT = HOI + NAHTSO

-204.2

5.7E+35

5.00E+01

8.7E-35

(irreversible)

M10

HOI + NAHT = NAHTSO + I– + H+

-152.4

4.8E+26

1.25E+02

2.6E-25

(irreversible)

-82.4

2.7E+14

1.16E+00

4.3E-15

(irreversible)

-66.2

3.9E+11

7.85E-01

2.0E-12

(irreversible)

M3 M4 M5 M6

–

+

HOI + I + H = I2 + H2O IO3– + HOI + H+ = 2HIO2 –

–

I2 + I = I3 IO3– + H+ = HIO3 +

+

+

–

M11

I2 + NAHT + H2O = NAHTSO + 2H + 2I

M12

I3– + NAHT + H2O = NAHTSO + 2H+ + 3I–

Method 2: mechanisms having irreversible reactions or where many of the ∆fG° values are unknown. It can become challenging to assign suitable ∆fG° values when many of them are unknown, which limits the use of Method 1. In such cases, compliance with the principle of detailed balancing can still be ensured by use of some simple matrix algebra. The basic concept is to create a matrix of the stoichiometric coefficients of the species in the mechanism, augment it with an identity matrix, and then manipulate the matrix to identify 1) reaction loops composed entirely of reversible steps, and 2) reaction loops that contain at least one irreversible step. The matrix manipulations include row reductions (or Gauss-Jordan eliminations) and linear programming that can become quite lengthy, so it is recommended to use appropriate software such as Mathematica27 or MATLAB.28 The exact procedure is described in the Supporting Information. While row reduction of the stoichiometric matrix is not a new concept,9, 12, 14 the current contribution is its extension to the identification of loops and illegal loops in homogeneous chemical reaction mechanisms by use of row reduction of the stoichiometric matrix augmented with the identity matrix. The first step is to assemble the matrix of stoichiometric coefficients. Each step in the reaction mechanism has a corresponding row in the matrix, and each column in the matrix corresponds to one of the species in the mechanism. The elements of the matrix are the stoichiometric coefficients in each of the reaction steps, with reactant coefficients being negative integers and product coefficients being positive integers. We note that some others define the stoichiometric matrix as the transpose of ours, with the steps corresponding to columns rather

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than rows; however, we consider that our definition is more intuitive and it leads more directly to final result. Note also that this method requires each reversible step to be expressed as a single step rather than individual forward and reverse steps; some others construct the stoichometric matrix with the forward and reverse steps as separate vectors.14 It is convenient first to assemble the matrix of the reversible steps, analyze it, and then add the irreversible steps and analyze the whole mechanism. Row reduction of the stoichiometric matrix reveals a basis set of loops. Loops arising from the set of reversible steps correspond to thermodynamic cycles and should be examined to ensure that the rate constants and equilibrium constants agree numerically with this constraint. Loops that include irreversible steps should be examined to determine whether they are legal. Mechanisms that contain multiple loops involving irreversible steps should be examined to determine whether the loops are linked or not. It should be determined whether any linear combinations of linked loops are illegal. This latter analysis is accomplished by 1) removing the reversible steps from the set of linked loops, 2) adding another identity submatrix, 3) performing a second row reduction, 4) looking for illegal linear combinations, and 5) searching for illegal loops by use of linear programming. Note that the number of independent loops is equal to the difference between the number of reaction steps and the rank of the matrix. This method provides an additional benefit when applied to systems of linked reversible loops. In such systems adjustments to some of the rate constants may be required in order to meet Wegscheider's condition, but if a given step contributes to more than one loop then its adjustment to repair one loop could upset another. Fortunately, our row reduction method ensures that each basis loop includes at least one step that is not shared by the other basis loops. This means that our method provides a simple alternative to some other published methods for identifying free parameters in sets of linked reversible loops.11, 15 Figures 1-6 illustrate the method applied to the NAHT/iodate reaction as implemented in Mathematica. Figure 1 shows the set of reversible steps and its corresponding submatrix.

Figure 1. The subset of reversible steps in the proposed mechanism of the reaction of NAHT with iodate, and its matrix of stoichiometric coefficients.

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Figure 2 shows the submatrix augmented with an identity matrix, with the reactions and species labels removed, and incorporated into a Mathematica operation that will perform the matrix row reduction.

Figure 2. Format of the Mathematica input for row reduction. Figure 3 shows the result of the row reduction; a vertical line has been added to help distinguish what was the stoichiometry submatrix from what was the identity submatrix. Note that row reduction transforms the matrix into “row echelon” form, where the first non-zero element of each row occurs to the right of the first non-zero element in the row above it. Reaction loops are identified as rows having all zeros in their row in the row-reduced left-hand submatrix. In Figure 3 the horizontal line has been drawn to clarify that the last row corresponds to a loop. An additional row has been added to the bottom of Figure 3 to clarify which columns correspond to which reaction steps. The matrix elements in the last row and to the right of the vertical line indicate the composition of the loop: M1 – M2 – M4 = 0, or M1 = M2 + M4. Indeed, it is simple to confirm by inspection that step M1 is the sum of steps M2 and M4. Once this dependency is identified, it is clear that K1 must equal K2K4. Since these are elementary steps, we have the relationship k1f/k1r = (k2f/k2r)(k4f/k4r). Substitution of the published rate constants into this relationship reveals that it is violated by many orders of magnitude. This is an example of a Case 1 reaction loop violation; it is legal, but the reported values of the rate constants are not consistent.

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Figure 3. Result of the row reduction operation in Fig. 2 with superimposed lines indicating a reaction loop and the corresponding coefficients in its linear combination. The next step is to analyze the whole reaction mechanism, adding the irreversible steps to the matrix of reversible steps shown above. Figure 4 shows this initial matrix for the complete NAHT/iodate reaction mechanism.

Figure 4. Complete matrix of stoichiometric coefficients for the NAHT/IO3– reaction mechanism. Figure 5 shows the matrix with the added identity matrix and the reaction and species identifiers removed.

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Figure 5. Matrix of stoichiometric coefficients plus identity matrix for the complete NAHT/IO3– reaction mechanism. Figure 6 shows the matrix in Figure 5 after row reduction, and the horizontal line shows that there are six independent reaction loops. The loops identified by row reduction constitute a basis set for all possible loops in the mechanism. The six loops in this example form a linked set, as indicated by pairs of loops having non-zero elements in the same column. Note also that none of the loops identified in Fig. 6 correspond to the reversible loop identified in Fig. 3; in fact, it exists as a linear combination of the loops in Fig. 6. The objective now is to determine whether any of the loops or their linear combinations is illegal. Recall that an illegal reaction loop is a loop where there is an irreversible step that has no opposing irreversible steps. The matrix elements in the last six columns of the lower right submatrix in Figure 6 correspond to irreversible steps, so any legal reaction loop corresponds to a row having both positive and negative elements in this submatrix. By this method it is clear that all of the identified reaction loops are legal. It is clear by inspection that there are no illegal linear combinations of loops.

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Figure 6. Result of row reduction of the matrix in Fig. 5 showing that the mechanism includes 6 independent reaction loops, all of which form one set of linked loops and all of which are legal. RESULTS and DISCUSSION One of us previously reported that the 14-step mechanism proposed for the reaction of –

IO3 with SCN– has two reversible loops, one of which fails to satisfy Wegscheider's condition by many orders of magnitude.22 These two loops were identified by inspection. Now, with matrix analysis, we find that the proposed mechanism actually includes four independent reversible loops. Three of these form a linked set. The fourth, not identified previously, is as follows: ICN + I– I2 + CN– ICN + H+ + I– HCN + I2 H+ + CN– HCN

R5 R8 R9

The sum of the first two steps equals the third, so detailed balancing requires that (kR5f/kR5r)(kR8f/kR8r) = kR9f/kR9r. The reported rate constants29 violate this relationship by seven orders of magnitude. In principle this violation can be resolved by adjusting one of the six rate constants; however, resolving the violations among the linked set of the other three loops could be more of a challenge if adjustments are made to steps that are shared between loops. Fortunately, as mentioned above, our row reduction method ensures that all basis loops have at least one step that is not shared with the others, and hence it identifies steps in the mechanism where the rate constants can be adjusted without perturbing other loops. An 8-step mechanism that has been proposed for the reaction of pentathionate with thiosulfate provides an example of an illegal loop.30 Row reduction of the stoichiometric matrix reveals that the mechanism includes one loop:

14


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S5O62– + S2O32– S6O62– + S2O32– S3O32– + H2O HSOH + SO32–

S6O62– + SO32– S5O62– + S3O32– S2O32– + HSOH S2O32– + H2O

M1 M3 M4 M6

The sum of M3, M4, and M6 is equal to the reverse of M1, and therefore the steps constitute a loop; since all the irreversible steps proceed to the right, the loop is illegal. It is an extended version of Case 5. Note that step M6 in the original publication was not balanced, and here we have balanced it by adding H2O as a product. Another application of detailed balancing is in the aqueous reaction of hydrogen peroxide with thiosulfate (H2O2 + S2O32–). Although this is an ostensibly simple reaction, careful product analyses showed that a variety of sulfur-containing products are formed, and the kinetics has a complex dependence on pH and the initial concentrations of the reactants.31 A 14-step model was proposed for the reaction mechanism with three of the steps reversible. Unfortunately, the model is incomplete because some of the steps involve H+ and others involve OH–, but the reversible autoionization of water that relates these two species was omitted. Matrix analysis of the 14-step mechanism plus the water autoionization reveals that there are no reaction loops among the four reversible steps but there are five loops involving irreversible steps. These five loops form a linked set. One of the loops is illegal as shown below: HOS2O3– + S2O32– + H+

S4O62– + H2O

R3

S2O32– + HOS2O3– S4O62– + OH– H+ + OH– H2O

R5 (R15)

The stoichiometric relationship is R3 + R5 = –R15; both R3 and R5 are irreversible and unopposed. This reaction loop is an example of Case 5 and is illegal. Although R3 and R5 might appear to be the reverse of each other, the proposed rate expressions for R3 and R5 are incompatible with the equilibrium expression for R3. Despite this formal violation of the principle of detailed balancing, the exact form of the rate law for R5 is of little significance for modeling the pH dependence of the S4O62– production because step R5 can be omitted entirely without significantly altering the results. A nine-step mechanism has been proposed for the Cu2+-catalyzed oxidation of S2O32– by S2O82–.32 Matrix analysis (see Supporting Information) of this mechanism reveals that it includes two isolated loops, one of which is illegal: Cu2+ + S2O82–

Cu3+ + SO42– + SO4•–

B6 15



Cu3+ + SO4•– Cu2+ + SO4 SO4 + SO42– 2SO4•– 2SO4•– S2O82–

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B7 B8 B9

Here reversible step B9 is equivalent to the sum of irreversible steps B6, B7, and B8, so the loop is an extended version of Case 5. This same loop also appears in a prior mechanistic proposal for this reaction.33 At this time it is not clear what should be done to repair this illegal loop. Matrix analysis of the 13-step mechanism proposed for the oxidation of formaldehyde by chlorite reveals that there are no loops within the set of 5 reversible steps. When the irreversible steps are added to the analysis it is found that there are four loops, all of which form a linked set. One of the loops is another example of an illegal loop as in Case 5 above.34 The loop consists of three reactions, two of them irreversible and unopposed as shown below: HOCl + Cl– + H+

Cl2 + H2O Cl2 + 2ClO2 2ClO2 + 2Cl– 2ClO2 + Cl– + H2O 2ClO2– + HOCl + H+

M10 M11

–

M13

Subsequent to the publication of this mechanism it was shown that reaction M13 does not occur,35 so the illegality of this reaction loop is no longer a concern. A related illegal loop appears in the mechanism proposed for the reaction of chlorite with hydroxymethanesulfinate where step M10 above is considered to proceed irreversibly to the right, step M13 above produces HClO2 and ClO2– instead of 2ClO2–, and the acid dissociation of HClO2 is included.36 An example of an illegal loop similar to Case 2 is provided by the seven-step mechanism proposed for the reaction of toluidine blue with chlorite.37 This mechanism contains a single loop. It consists of six steps (reactions R2 - R7 in the proposed mechanism) and has all steps reversible except for one (R5). Without use of the matrix method it might have been a challenge to identify this loop. A 28-step mechanism has been proposed for the reaction of periodate with thiosulfate.38 Row reduction of the stoichiometric matrix generates a basis set of 16 loops, none of which is illegal but all of which are linked. As described in the Supporting Information, four illegal loops are revealed in the second row reduction. One of these illegal loops is a Case 2 loop: S2O3OH– + IO3– S2O32– + IO4– + H+ S2O32– + IO3– + 2H+ S2O3OH– + HIO2 IO4– + HIO2 2IO3– + H+

1 5 6

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The sum of steps 5 and 6 is equal to step 1, which generates an illegal loop. An additional illegal loop is found by use of linear programming (see SI). As reaction mechanisms increase in complexity, it becomes increasingly difficult to use simple inspection to check for compliance with detailed balancing. In such circumstances, use of a systematic method to check for compliance becomes essential. For example, a report on the reaction system involving ClO2, I2, and S2O32– includes a proposed mechanism with 32 steps, 8 of them reversible.39 In this case, row reduction of the stoichiometric matrix reveals no loops among the reversible steps but a basis set of 15 loops having irreversible steps, all of which constitute a set of linked loops and all of which are legal. An illegal loop is revealed in the second row reduction, and the linear programming method reveals the existence of at least one more illegal loop. Steps M17, M18, M24, and M25 form an illegal loop because steps M17 and M25 are the only irreversible steps in the loop and they are not in opposition (0 = 2(M24) – M17 – 3(M18) – M25): IO3– + 5I– + 6H+

3I2 + 3H2O I2 + H2O HOI + I– + H+ HIO2 + I– + H+ 2HOI 2HIO2 IO3– + HOI + H+

M17 M18 M24 M25

This would have been quite difficult to confirm without a systematic method. This same illegal loop appears in the 20-step mechanisms proposed for the ClO2/I2/malonic acid reaction,40 the ClO2/I2/ethyl acetoacetate reaction,41 and the ClO2–/I2/ethyl acetoacetate reaction.42 A recent report on the reaction of ClO2– with S4O62– includes a proposed mechanism consisting of four reversible steps within the chlorine subsystem plus an additional 38 steps.43 Inspection reveals that steps M34, M23, and M22 are the stoichiometric reverse of steps M31, M33, and M21, respectively. Although step M34 is the stoichiometric reverse of step M31, the proposed rate equations are incompatible with detailed balancing because M31 is catalyzed by chloride but M34 is not. Similarly, step M23 is the stoichiometric reverse of M33 but the rate equations are incompatible with each other. Row reduction of the stoichiometric matrix excluding the reverse steps M22, M23, and M34 (and neglecting incompatibilities in the rate equations) reveals a basis set of 18 independent linked loops, three of which are illegal. The simplest of the illegal loops corresponds to Case 5 above: S4O6ClO22– ClO2 + S4O62– S4O62– + S4O6ClO22– (S4O6)2ClO24– ClO2 + (S4O6)2ClO24– 2S4O6ClO22–

M31,M34 M35 M36 17



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It is clear that step M31,M34 is equivalent to the sum of steps M35 and M36, or KM31 = KM35KM36. This would lead to a relationship between the reverse rate constants for M35 and M36 if the rate equations for M31 and M34 were modified to agree with detailed balancing. A related loop appears in the 14-step mechanism proposed earlier for the reaction of ClO2 with S4O62– where step M35 above is shown as being reversible (R2):44 ClO2 + S4O62– S4O6ClO22– S4O62– + S4O6ClO22– (S4O6)2ClO24– ClO2 + (S4O6)2ClO24– 2S4O6ClO22–

R1 R2 R8

This difference makes the loop a Case 2 type, which is also illegal. Unlike the mechanism more recently proposed for the ClO2–/S4O62– reaction, the forward and reverse rate equations proposed for step R1 are in agreement with detailed balancing. Since KR1 = KR2KR8, detailed balancing requires that kR8r = kR8fkR2fkR1r/(kR2rkR1f). Simulations of the 14-step mechanism show that the loss of ClO2 is about two times slower when kR8r is included as required by detailed balancing than when kR8r is omitted (conditions as in Figure 5 of the original publication); furthermore, the general shape of the decay curve differs, appearing essentially exponential when kR8r is omitted but having a noticeable inflection at about 100 s when kR8r is included. An example of a loop that is illegal because it contains an unnecessary process is provided by a mechanism recently proposed for the reaction of bromate with malonic acid as catalyzed by [Fe(phen)3]3+.45 This 14-step mechanism includes the following Case 2 loop: BrO3– + HOBr + H+ BrO3– + HOBr + 2H+ HBrO2 + H2BrO2+ HBrO2 + H+ H2BrO2+ 2HBrO2

R4 R4b A1

The loop is obtained by combining the steps so that R4b = R4 + A1. One conceivable repair of this loop is to make step R4b reversible; the published values for the other rate constants would require a value of 9.7  10–7 M–3 s–1 for kR4br. Simulations of the reaction under the conditions in Fig. 7 of the report on this reaction show that inclusion of this reverse step has no effect. An explanation for this is that the reverse of step R4 is very slow (kR4r = 1  10–8 M–2 s–1). Simulations show that kR4r can be set to much lower values with no effect and that it can be excluded entirely. If the reverse of R4 is removed from the mechanism instead of adding the reverse of R4b, the loop becomes a Case 3 loop, which is legal. This latter solution to the illegality is preferred because it leads to a simpler model. 18


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CONCLUSIONS In summary, there are three basic ways in which the principle of detailed balancing could be violated. First, the forward and reverse rate expressions for a step could be incompatible with the equilibrium constant expression, either numerically or in form. Second, the rate constants for the steps in a reversible loop could violate Wegscheider's condition. Third, loops containing irreversible steps could be illegal. It can be a challenge to identify all loops in a mechanism by simple inspection. Here we provide two methods to test a mechanism for compliance with these aspects of the principle of detailed balancing. Questions may arise regarding how much leeway one has in compliance. It is always essential that the rate equations for the forward and reverse of a reaction step be consistent with the form of the equilibrium expression. When using method 1 there is a significant opportunity to adjust the values of ∆fG°, since these are experimental values and may vary depending on reaction conditions. This does not, however, lift the requirement to comply with eqs V and XI and similar relationships when dealing with loops in which all reaction steps are reversible. On the other hand, reaction mechanisms are typically developed that apply under a range of conditions, and the relative importance of the various steps may vary depending on the conditions. In such situations, small deviations from eqs V and XI should be allowable. Although some illegal loops do not lead to gross errors in kinetic simulations, illegal reaction loops should be avoided. It may be a simple matter to repair illegal loops once they are identified; in some cases repair can be achieved by making certain steps reversible, making others irreversible, or removing unncessary steps. If repair is achieved by making the steps reversible it is important to ensure compliance with eqs V and XI or their equivalents (Wegscheider's condition). As the penultimate example given above shows (the 14-step ClO2/S4O62– mechanism), repair of an illegal loop by adding the requisite reverse rate constant can indeed have a significant effect on the predictions of the model. ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at XXX. Detailed description of the matrix method, row reductions, and linear programming to determine whether a mechanism contains any illegal loops. Examples illustrating the method with illegal loops identified on the first row reduction, the second row reduction, and linear programming. (PDF)

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AUTHOR INFORMATION Corresponding Author *(D.M.S.) E-mail: [email protected] ACKNOWLEDGMENTS We thank professors I. R. Epstein (Brandeis University, USA) and Q. Gao (CUMT, China) for helpful discussions. We thank professor A. Vinel (Dept. of Industrial Systems Engineering, Auburn University, USA) for his help with Mathematica. This research was supported by a grant from ACS-PRF to DMS: PRF 55078-ND3.

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Systematic Application of the Principle of Detailed Balancing to

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