Kinetic plasticity and the determination of productratios for kinetic schemes leading to multipleproducts without rate laws — New methods basedon directed graphs

John Andraos

Abstract: This paper presents two new and fast methods of determining product ratios for kinetic schemes leading tomore than one product on which the Acree–Curtin–Hammett (ACH) principle is based. The methods involve rewritinga given kinetic scheme as a directed graph with nodes and arrows connecting the nodes and takes advantage of thedirectionality of the kinetic arrows and the enumeration of paths to the various target product nodes. The first, basedon path divergent trees, is computationally simpler but works under a specific set of conditions, whereas the second,based on an adapted version of Chou’s graphical method, works for all cases. By means of illustrated examples, bothmethods are shown to be completely verifiable with conventional more tedious treatments based on rate law determina-tions. The directed graph concept also works for kinetic schemes that involve entirely equilibrated species. In addition,the paper extends these ideas to variants of the basic ACH scheme, thereby testing the validity of the ACH principleand bringing about a deeper understanding of it. Generalization of the results yields a new parameter, called degree ofkinetic plasticity, which completely describes the dynamics of kinetic resolution between the boundary limits of ACHbehaviour (100% kinetic plasticity) and anti-ACH behaviour (100% kinetic rigidity). It is shown that this parameter is agood descriptor of all possible scenarios between and including these limits and can be determined experimentally byconducting a new kind of product study that tracks the behaviour of final product excesses as a function of initial sub-strate excesses. The resulting plot is always linear with a positive slope. The degree of kinetic plasticity is found bysimply subtracting the slope from unity. These ideas are tested on complex kinetic schemes exhibiting dynamic kineticresolution (DKR) by means of organocatalysis.

Key words: physical organic chemistry, kinetics, mechanism, directed graph, Chou digraph, Chou graphical rule,Acree-Curtin-Hammett principle, product ratio, dynamic kinetic resolution, organocatalysis. 357

Résumé : Dans ce travail on a développé deux nouvelles méthodes rapides pour déterminer les rapports des schémasconduisant à plus d’un produit sur lequel se base le principe d’Acree–Curtin–Hammett (ACH). Les méthodes impli-quent une réécriture d’un schéma cinétique donné sous la forme d’un graphique dirigé comportant des nodes et des flè-ches reliant les nodes et elles tirent avantage du caractère directionnel des flèches cinétiques et de l’énumération desvoies vers les divers nodes des produits prévus. La première, qui est basée sur des ramifications divergentes, est la plussimple d’un point de vue des calculs théoriques, mais elle ne s’applique que dans un ensemble spécifique de conditionsalors que la deuxième qui est basée sur une version adaptée de la méthode graphique de Chou s’applique à tous lescas. En faisant appel à des illustrations appropriées, on peut démontrer que les deux méthodes peuvent être entièrementvérifier à l’aide de traitements conventionnels plus fastidieux basés sur des déterminations de lois de vitesses. Leconcept du graphique dirigé fonctionne aussi pour les schémas cinétiques qui impliquent des espèces entièrement équi-librées. De plus, ce travail permet d’élargir ces idées à des variantes du schéma de base de Acree-Curtin-Hammett quipermet ainsi de vérifier la validité du principe d’ACH et de mieux la comprendre. La généralisation des résultatsconduit à un nouveau paramètre, le degré de plasticité cinétique, qui décrit complètement la dynamique de la résolutioncinétique entre les limites de la frontière entre le comportement Acree-Curtin-Hammett (100% de plasticité cinétique)et le comportement anti-Acree–Curtin–Hammett (100% de rigidité cinétique). On a montré que ce paramètre permet debien décrire tous les scénarios entre, et incluant, ces limites et qu’il peut être déterminé expérimentalement en effec-tuant un nouveau type d’étude de produit qui permet de suivre le comportement des excès de produit final en fonctiond’excès de substrats initiaux. La courbe qui en résulte est toujours linéaire avec une pente positive. On détermine ledegré de plasticité cinétique en soustrayant la pente de l’unité. On a fait appel à l’organocatalyse pour tester ces idéessur les schémas cinétiques complexes qui présentent de la résolution cinétique dynamique (RCD).

Can. J. Chem. 86: 342–357 (2008) doi:10.1139/V08-020 © 2008 NRC Canada

342

Received 23 October 2007. Accepted 16 January 2008. Published on the NRC Research Press Web site at canjchem.nrc.ca on25 March 2008.

This paper is dedicated to the memory of Keith Yates.

J. Andraos. Department of Chemistry, York University, Toronto, ON M3J 1P3, Canada (e-mail: jandraos@yorku.ca).

Mots-clés : chimie organique physique, cinétique, mécanisme, graphique dirigé, diagramme de Chou, règle graphiquede Chou, principe d’Acree–Curtin–Hammett, rapport de produits, résolution cinétique dynamique, organocatalyse.

[Traduit par la Rédaction] Andraos

Introduction

The Acree–Curtin–Hammett (ACH) principle is a funda-mental concept in mechanistic chemistry that has been usedto approximate under simplifying conditions the magnitudeof the final product ratio for a pair of products originatingfrom a pair of equilibrating starting materials as shown inScheme 1. The starting materials may be equilibrating con-formers, tautomers, or stereoisomers. Curtin and Hammettdemonstrated this principle in the 1950s for products arisingfrom equilibrating conformers (1, 2); however, it had alreadybeen described much earlier by Acree in 1907 for productsarising from equilibrating tautomers (3). Moreover, Acree’swork also derived what would be later known as theWinstein–Holness equation (4). A recent report has de-scribed the connections between these works and broughtAcree’s forgotten work to light (5). Throughout this paper,the principle is therefore referred to as the ACH principle orkinetic theorem to recognize the contributions of the threescientists. It is a concept that is well discussed in severaltextbooks of mechanistic organic chemistry (6–8), a review(9), and a pedagogical article (10).

Simply stated, the final product ratio [PX]:[PY] is equal tothe ratio of product forming rate constants k3:k4 if the mag-nitudes of the rate constants k1 and k2 involved in the equili-bration between X and Y are similar and are very largecompared with those of the product forming steps. This in-dependence of the final product ratio on the dynamics of thepreceding equilibrium and the amounts of starting materialsdefines the ACH condition and is a direct result of applyingboth kinetic conditions. When these conditions are not met,an exact analytical solution to the above kinetic scheme be-comes necessary to determine the complete form of the finalproduct ratio. Equation [1] yields the complete dependenceof the final product ratio on all four rate constants and thestarting amounts of X and Y (11).

[1][ ][ ]

( )( )

PP

X

Y

∞

∞=

+ ++ +

kk

a b k ak

a b k bk3

4

2 4

1 3

where a and b represent the initial amounts of X and Y attime zero. It is easy to verify that the right-hand side ofeq. [1] reduces to k3/k4 when (i) k1 and k2 » k3 and k4 and(ii) k1 = k2. Note that if only condition (i) holds, then the fi-nal product ratio will have a dependence on both the ratiok3/k4 and the equilibrium constant K = k1/k2, according toeq. [2].

[2][ ][ ]PP

X

Y

∞

∞=

kk K

3

4

1

In addition to applying to equilibrating conformers ortautomers, the kinetic scheme shown in Scheme 1 is applica-ble to the technique of dynamic kinetic resolution (DKR)(12–38) used in synthetic organic chemistry to optimize thesynthesis of stereoisomeric products from corresponding

enantiomeric or diastereomeric starting materials that are inequilibrium. This is arguably the most important practicalapplication of the ACH principle. In fact, it is this specialphenomenon that blends the interests of both mechanisticand synthetic organic chemists. Mechanistic chemists are at-tracted to reactions that yield multiple products via multipleintermediates but that have the potential to be perturbed toproduce one target product under the “right” set of condi-tions once their corresponding mechanism landscapes arecompletely elucidated. Synthetic chemists are of course in-terested in reactions that yield only one target product inhigh yield and high regio-, stereo-, and chemo-selectivity.Scheme 2 shows an analogous kinetic system to that inScheme 1 with appropriate stereochemical descriptors R andS for substrates and products.

If a chemical reaction produces enantiomeric or diastereo-meric products, and it is desired to carry out a subsequenttransformation on one of them, then one is faced with theprospect of first separating or resolving the product mixturefrom the first reaction, usually racemic, before carrying outthe second. Resolution of racemic mixtures by standardmethods results in an automatic loss of half the starting ma-terial, assuming that the resolution procedure goes to com-pletion. This means that if the individual stereoisomers inthe starting racemic mixture are not interconvertible then themaximum amount of desired product formed in the subse-quent step is exactly half that of the starting material, and sothe maximum yield for this second step is 50%. This situa-tion is designated as anti-ACH behaviour and corresponds tokinetic resolution. If, on the other hand, there is a possibilityto have a rapid interchange between starting materials via aracemization process to an extent that the ACH condition ap-plies, then in principle it is possible to shunt all of the start-ing material toward the product of interest in the subsequentreaction. The more dynamic the equilibration is between SRand SS, the more likely this outcome will occur. This situa-tion is designated as ACH behaviour and corresponds to dy-namic kinetic resolution.

In a recent report published in 2003, the complete dynam-ics of Scheme 2 was explored including a full kinetic analy-sis of the dependence of the initial and final product ratioson all four rate constants and starting amounts of SR and SS(11). The method used to derive expressions for these ratiosshown in eqs. [3] and [4],

[3][ ][ ]PP

R

S

0

0

3

4

=

ab

kk

[4][ ][ ]

( )( )

PP

R

S

∞

∞=

+ ++ +

kk

a b k ak

a b k bk3

4

2 4

1 3

where a and b are the initial amounts of SR and SS respec-tively, was the method of Laplace transforms (39). It is im-portant to point out that though there are no products formedinitially their ratio extrapolated to zero time is in fact a finite

© 2008 NRC Canada

Andraos 343

quantity. This observation is entirely consistent with thework of Noyori and co-workers (40, 41) who showed plotsof % final enantiomeric excess of product versus % conver-sion that had well-defined nonzero intercepts. The two ex-pressions given in eqs. [3] and [4] suggested a new kind ofexperiment that could be carried out that would quantita-tively determine the intrinsic efficiency or plasticity of dy-namic kinetic resolution that depended only on key rateconstant ratios. This parameter was shown to be able to ac-count for all experimentally observed cases between and in-cluding the limits of complete dynamic kinetic resolution(ACH conditions) and complete kinetic resolution (anti-ACH conditions) for simple kinetic schemes such as thatshown in Scheme 2.

The crux of the new experiment is to determine initial andfinal product ratios for a set of initial substrate ratios cover-ing the full range of possible optical purities of starting ma-terials beyond the simple racemic condition, which is theuniversal initial composition of starting materials. Hence fora given starting substrate ratio a/b, one records time-dependent product progress curves, and from these the initialand final product ratios are determined by extrapolation tozero and infinite times. A plot of [PR]0/[PS]0 vs. a/b yields aslope equal to r = k3/k4. A plot of final product excess withrespect to PR and pe∞ vs. initial substrate excess with respectto SR and (a–b)/(a+b), according to eq. [5],

[5] pe∞∞ ∞

∞ ∞

∞

∞

∞

∞

= −+

=−

+

[P ] [P ][P ] [P ]

[P ][P ][P ][P ]

R S

R S

R

S

R

S

1

1

=1

1 1rv ua ba b

rv urv u+ +

−+

+ −

+ +

yields a slope equal to 1/(rv + u +1) and an intercept equalto (rv – u)/(rv + u + 1) where r = k3/k4, u = k1/k3, and v =k2/k3. The rate constant ratios u and v may be determined di-rectly from the slope and intercept of such a plot usingeqs. [6] and [7],

[6] u = − −

12

11 intercept

slope

and

[7] vr

= + −

12

11 intercept

slope

An expression for the intrinsic efficiency or degree of ki-netic plasticity of dynamic kinetic resolution, εDKR, may beobtained by examining the full range of possible values forthe slope of this plot. In the ACH limit where u = v → ∞ sothat k1 = k2 and k1, k2 » k3, k4, the slope reaches a minimumvalue of zero. This result indicates that the final product ra-tio is independent of the starting amounts of SR and SS. Inthe anti-ACH limit where u = v = 0 so that k1 = k2 and k1,k2 « k3, k4, the slope reaches a maximum value of one. Thiscase indicates that the final product ratio mirrors the initialstarting material ratio. From these two boundary conditions,we set εDKR = 1 to represent the ACH case of 100% effi-ciency or complete kinetic plasticity and εDKR = 0 to repre-sent the anti-ACH case of 0% efficiency or complete kineticrigidity. Hence, we may define the intrinsic efficiency orperformance of dynamic kinetic resolution as in eq. [8] todescribe the pliability of the equilibrium toward being pulledin either direction by either of the product forming steps.

[8] εDKR 1 slope= − = −+ +

= ++ +

11

1 1rv urv u

rv u

Figure 1 illustrates graphically the two limiting cases de-scribed earlier. Equation [5] predicts that experimental datashould obey a linear relationship with a positive slope fallingbetween these two limits. The intersection of the two limit-ing lines suggests that there exists a unique combination ofinitial substrate amounts that will result in the same final

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344 Can. J. Chem. Vol. 86, 2008

Fig. 1. Relationship between final product excess and initial sub-strate excess according to eq. [8], showing the two limitingslopes corresponding to the Curtin–Hammett condition (zeroslope, u and v → 0, 100% kinetic plasticity) and anti-Curtin–Hammett condition (unit slope, u and v → ∞, 0% kinetic plastic-ity or 100% kinetic rigidity).

Scheme 1.

Scheme 2.

product ratio, and hence the same final product excess, un-der both limiting conditions. It should be noted that eq. [5]also predicts that we can never have a plot with a slope ofzero and an intercept of zero, i.e., a line falling along the x-axis. Such a condition would correspond to the nonsense sit-uation of a completely plastic kinetic system that beginswith any combination of substrates and always ends up witha racemic mixture of products. Setting the slope and inter-cept to be simultaneously zero in eq. [5] leads to a contra-diction in the allowable domains of the parameters u and v,namely, u, v → ∞ for the slope and u, v → 0 for the inter-cept. A racemic mixture of products can only arise from aracemic mixture of starting materials, which immediatelysuggests that the kinetic system is 100% rigid and hence theanti-ACH limiting line of unit slope and zero intercept is op-erative. However, ACH limiting lines with zero slopes andintercepts of +1 or –1 are allowed, i.e., lines falling alongthe top and bottom edges of the square in Fig. 1. These cor-respond to the fully dynamic situations where the equilib-rium step is fast compared with the product forming steps(u, v → ∞) and r → ∞ (or k3 ≫ k4) for the case of an inter-cept of +1, and r → 0 (or k3 ≪ k4) for the case of an inter-cept of –1. From Scheme 2, the former case essentiallycorresponds to the case when the step from SS to PS is shutdown so that the only product formed is PR, and the lattercase corresponds to the opposite situation when the stepfrom SR to PR is shut down so that the only product formedis PS. The plot shown in Fig. 1 may be used as a diagnostictest to verify if a given chemical system obeys ACH kinetic-type schemes.

Equation [8] is applicable to any kinetic scheme that lookslike Scheme 1 or 2 regardless of the isomeric relationship ofthe starting materials so long as their starting ratio is control-lable. In the case of isomeric substrates that cannot be easilyseparated and their initial ratio manipulated as describedabove, separate kinetic experiments are required to determineindividual rate constants so that the kinetic plasticity parame-ter given in eq. [8] may be determined. To date there has notyet been any experimental verification of this new idea re-ported in the literature. However, the 2003 report did survey anumber of published chemical systems where enough experi-mental detail was given so that minimum estimates of εDKRcould be made, and thus the validity of eq. [8] was confirmedin principle. In that work, several variant kinetic schemeswere examined and their properties and corresponding exactinitial and final product ratio expressions were derived. Theimportant point gleaned from that investigation was that allprior work on such kinetic systems assumed from the outsetthat the ACH condition was valid and that Winstein–Holnesskinetic behaviour (4) was applicable. These claims were thenverified by carrying out independent theoretical calculationsto determine energy barriers pertaining to the relevant transi-tion states (42). The above analysis clearly shows that theACH condition may be verified directly from wholly experi-mentally acquired data without need for further computation.Moreover, even if it turns out that this special condition maynot be met for a given chemical system it is still possible toquantify the actual extent of meeting this condition accordingto the parameter defined in eq. [8].

Having introduced the current understanding of the ACHprinciple on simple kinetic systems, we now report on fur-

ther extensions that test the validity of this principle on com-plex kinetic systems. Extended linear and cyclic kineticschemes are examined in detail. The concept of kinetic plas-ticity described above is generalized and the statement of theACH principle is revised accordingly. In doing so, we alsointroduce two new time-saving methods of determining therequired product ratio expressions for initial and final reac-tion times that serves as a check for conventional treatments,namely the Laplace transform method. As a bonus, it will bedemonstrated that they also circumvent much of the tediouscomputation that is a consequence of handling complex ki-netic systems. Limitations of the methods are also discussed.

Directed graph method

In this section we introduce easy-to-use algorithms for de-termining expressions as shown in eqs. [1] and [4], withoutthe use of rate laws. The methods are based on directedgraphs, or digraphs (43), and are applicable to competitive-type kinetic schemes leading to multiple products from oneor more substrates. The schematic method of King andAltman described in 1956 is a forerunner report describingthe use of graphs to derive rate laws for enzyme-catalyzedreactions (44). Chou has previously described powerful algo-rithms for analyzing non-steady-state and steady-state en-zyme kinetics and protein folding kinetics (45–61).

Digraphs are objects that consist of a finite set of verticesand a set of ordered pairs of distinct vertices called arcs. Ifthe ordered pair {x, y} is an arc, then the arc is said is to bedirected from x to y. The kinetic schemes shown in Schemes1 and 2 may be immediately drawn as digraphs, since thechemical entities and reaction arrows are translated as verti-ces and arcs, respectively, according to the above definitions(see Scheme 3).

The substrate vertices are designated as source nodes andthe product vertices as sink nodes. The arrows and rate con-stant descriptors are retained as before in the usual sense.For the first method based on path divergent trees, beginningwith each source node we draw a corresponding divergenttree that enumerates all paths from that node to all productnodes. In doing so, we apply the following rules. For irre-versible steps one stops when a product node is reached andfor reversible steps one stops when a previously traversednode along that branch is reached. Hence, we have forScheme 3 the two divergent trees shown in Fig. 2 in whichwe assign the initial condition that we begin with amounts aand b for substrates S1 and S2.

Next, we write expressions for the expected contributionof each product based on the attenuation of the initialamounts of each source node that lead to each product. This

© 2008 NRC Canada

Andraos 345

Scheme 3.

is done following the connectivities between nodes in thepath divergent trees. Thus, for paths leading to product P1we have eq. [9],

[9] ak

k kb

kk k

kk k

3

1 3

3

1 3

2

2 4+

+

+

+

=+

+

+

kk k

a bk

k k3

1 3

2

2 4

The first term in eq. [9] pertains to P1 originating directlyfrom S1 along the k3 branch (see Fig. 2A). Note that sincethere are two branches emanating from S1, the fraction ofstarting material a leading to P1 is k3/(k3 + k1). The secondterm pertains to P1 originating from S1, which in turn origi-nated from S2 (see Fig. 2B). The starting amount of S2, b, isfirst split along the k2 branch and then is split again alongthe k3 branch. Hence, we have the corresponding multiplica-tion of fractions.

Similarly, we have for paths leading to product P2

[10] bk

k ka

kk k

kk k

4

2 4

1

1 3

4

2 4+

+

+

+

=+

+

+

kk k

b ak

k k4

2 4

1

1 3

Note that since we have a symmetrical digraph, the pairs ofsubscripts a and b, 1 and 3, and 2 and 4, are interchangeablein both expressions. This implies that the expression ineq. [10] could have been written at once from eq. [9] usingthese variable interchanges.

The final product ratio is then given directly by dividingeq. [9] by eq. [10] thus yielding eq. [11],

[11][ ][ ]PP

1

2

∞

∞=

+

+

+

kk k

a bk

k k3

1 3

2

2 4

+

+

+

kk k

b ak

k k4

2 4

1

1 3

=

+ ++ +

kk

a b k aka b k bk

3

4

2 4

1 3

( )( )

which is identical to eqs. [1] and [4]. The initial product ra-tio is also given directly by eq. [12],

[12][P ][P ]

1 o

2 o

= akbk

3

4

which is identical to eq. [3]. Note that in eq. [12] the numer-ator refers to the shortest path from S1 to P1 and the denomi-nator refers to the shortest path from S2 to P2. Thisobservation turns out to be always true in the determinationof expressions for initial product ratios by the digraphmethod.

It is obvious how facile this method is compared with thestandard method of Laplace transforms (39) that becomesmore tedious as the number of substrate and product nodesincrease and the number of rate constant connections be-tween them increases. It may be recalled that the Laplacetransform method involves taking determinants of squarematrices whose dimension is equal to the sum of the numberof substrates, intermediates, and products involved in the ki-netic scheme. In addition, the method relies on tables of in-verse transforms that need to be worked out separately, butsuch tables are readily available (62–64). A key limitation ofthe method is that it is applicable only to unimolecular orpseudo-unimolecular kinetic systems and not biomolecularones, since the Laplace transform of a product of functionsis not equal to the product of their Laplace transform func-tions as shown in eq. [13],

[13] L f t g t L f t L g t{ ( ) • ( )} { ( )} • { ( )}≠

Figure 3 summarizes key definitions and the sequence ofsteps involved in this method. For illustration and compari-son with the directed graph method, the Supplementary Datacontains a full solution of the kinetic scheme shown inScheme 3 by the Laplace transform method.2

For verification purposes, all of the expressions for initialand final product ratios derived previously (11) were con-firmed by the new digraph method. These are also given inthe Supplementary Data.2 Reassuringly, this consistencydemonstrates that the present new method also serves as anexcellent checking device. However, this method does haveits limitations. First, it does not give analytical expressionsfor the time-dependent substrate and product concentrationprofiles. For the purposes of determining analytical expres-sions for product ratios at the beginning and at the end of re-actions, paradoxically these are not necessary. Second, like

© 2008 NRC Canada

346 Can. J. Chem. Vol. 86, 2008

Fig. 2. Path divergent trees for paths emanating from S1 (A) andS2 (B). See Scheme 3.

2 Supplementary data for this article are available on the journal Web site (canjchem.nrc.ca) or may be purchased from the Depository ofUnpublished Data, Document Delivery, CISTI, National Research Council Canada, Ottawa, ON K1A 0R6, Canada DUD 3724. For more in-formation on obtaining material, refer to cisti-icist-cnrc.gc.ca/irm/unpub_e.shtml. The Supplementary Data contains a full solution of the ki-netic scheme shown in Scheme 3 by the Laplace transform method, solutions of kinetic schemes described in ref. 11 by both directed graphmethods, a modified Chou digraph method applied to Schemes 6 to 10 in sections 6, 7, and 8.

the Laplace transform method, it is applicable to uni-molecular and pseudo-unimolecular kinetic systems. How-ever, it can still be applied to special bimolecular systems,such as those shown in Fig. 4 where multiple products arisefrom a common intermediate via unimolecular steps thoughthat intermediate itself arose from a prior bimolecular step.For these few examples, though time-dependent concentra-tion profiles for products cannot be determined analytically,their product ratios are determinable exactly by the directedgraph method.

A key observation from the path divergent trees shown inFig. 2 is that each tree is composed of two stages or tierswith only one repeat node. For Fig. 2A the repeat node is S1and for Fig. 2B it is S2. It turns out that if the constructionof path divergent trees from a digraph pertaining to a kinetic

scheme according to the above algorithm leads to trees hav-ing more than one repeat node appearing in different stages,then the above digraph method does not yield expressionsfor the product ratio that are consistent with those obtainedfrom the check Laplace transform method. Extra terms ap-pear in the numerator and the denominator for the finalproduct ratio expression, which ultimately yield an apparentdependency on the amounts of equilibrating starting materi-als when the ACH condition is applied. This would lead tothe erroneous conclusion that the ACH principle is violatedfor such cases. Hence, the third and most important limita-tion of the above digraph algorithm is that it works only forkinetic schemes having corresponding path divergent treeswith only one repeat node or ones with more than one repeatnode appearing in the same stage or tier.

An example of a kinetic scheme yielding path divergenttrees with multiple repeat nodes appearing in different stagesor tiers is shown in Scheme 4, which is a more completerepresentation of the racemization process suggested byScheme 2. Clearly, racemization between SR and SS neces-sarily takes place via a lower symmetry intermediate that isin equilibrium with both SR and SS since loss of chiralitytakes place via a bond-breaking process.

The corresponding path divergent trees are shown inFigs. 5A and 5B. Note that in Fig. 5A there is a repeat node

© 2008 NRC Canada

Andraos 347

Fig. 4. Example special bimolecular kinetic systems whose initial and final product ratios are determinable exactly by the directedgraph method based on path divergent trees.

Scheme 4.Fig. 3. Key definitions and sequence of steps involved in theLaplace transform method.

SR that appears in the second tier and another repeat node Ithat appears in the third tier. Similarly, repeat nodes in dif-ferent tiers are observed in Fig. 5B. Application of the abovedigraph algorithm leads to the incorrect expression for thefinal product ratio shown in eq. [14],

[14][P ][P ]

R

S

∞

∞=

+ + + +kk

a b k k ak k k ak k3

4

2 2 4 2 1 1( ) ( )* *1

1 1 3 1 2 2 2

*

* * *( ) ( )a b k k bk k k bk k+ + + +

The correct expression found by the Laplace transformmethod is shown in eq. [15],

[15][P ][P ]

R

S

∞

∞=

+ + ++

kk

a b k k ak k k

a b3

4

2 2 4 2 1( ) ( )( )

* *

k k bk k k1 1 3 1 2* *( )+ +

Note the extra terms appearing in the numerator and denom-inator in eq. [14]. The ACH limit applied to eq. [14] leads toeq. [16],

[16][P ][P ]

R

S ACH

∞

∞

=

++

kk

a ba b

3

4

22

which shows a dependency on the starting amounts of SRand SS, a and b. When this limit is applied to eq. [15], how-ever, the result is the expected no dependence on a and b asshown by eq. [17],

[17][P ][P ]

R

S ACH

∞

∞

= k

k3

4

A second digraph method based on a variation of the Choualgorithm is found to work for all cases and thus circum-vents the third limitation of the above path divergent treemethod, as will be shown next. Again using the same kineticschemes shown in Schemes 1 and 2, the Chou digraph isconstructed as shown Scheme 5.

The digraph is similar to that shown in Scheme 3 exceptthat the rate constants connecting the nodes are assigned anegative sign and loops are added to those nodes that havekinetic arrows emanating from them. The rate-constantweights assigned to the loop nodes correspond to the posi-tive sum of the rate constants emanating from them. Fromthis digraph, all subgraphs are enumerated according to thefollowing steps. For a given target product node, draw allpaths that lead to it from all starting substrate nodes. Allother points not traversed by this path are connected by cy-

cles that neither intersect each other nor intersect the path.Repeat the process for all other target product nodes. Thecontributing expression to each target product node is givenby eq. [18],

[18] Pxsubgraph

→ −

+ + ∏∑ s kyn c

jj

( )1 1

where for a given subgraph, sy is the starting amount of astarting node, n is the total number of nodes in the subgraph,c is the number of cycles, the product of rate-constant fac-tors corresponds to all arrow and loop contributions, and thesum is taken over the number of subgraphs leading to com-mon product node Px. Hence, the subgraphs pertaining to thedigraph in Scheme 5 are shown in Fig. 6.

© 2008 NRC Canada

348 Can. J. Chem. Vol. 86, 2008

Fig. 5. Path divergent trees for paths emanating from SR (A) and SS (B). See Scheme 4.

Scheme 5.

Fig. 6. Subgraphs from digraph shown in Scheme 5: (A) targetproduct P1; (B) target product P2.

The contributing expression for the P1 node is,

[19] [ ] ( ) ( )( )P1 ∞+ +→ − − +a k k k1 3 1 1

3 2 4

+ − − − = + ++ +b k k k a b k k( ) ( )( ) [( ) ]1 3 0 12 3 3 2 4

and that for the P2 node is,

[20] [ ] ( ) ( )( )P2 ∞+ +→ − − +b k k k1 3 1 1

4 1 3

+ − − − = + ++ +a k k k a b k k( ) ( )( ) [( ) ]1 3 0 11 4 4 1 3

Note the following transposition of variables: a ↔b, k1 ↔ k2, k3 ↔ k4. Hence, the final product ratio is foundby dividing eq. [19] by eq. [20], which leads immediately tothe previous expression given in eq. [11] found by the firstdigraph method based on path divergent trees. The applica-tion of the adapted Chou method to the kinetic schemeshown in Scheme 4 yields the digraph and subgraphs shownin Fig. 7.

The resulting contributing expressions for PR and PS aregiven by eqs. [21] and [22],

[21] [ ] [( ) ( )( )( )* *PR ∞+ +→ − − + +a k k k k k1 4 2 1

3 2 1 2 4

+ − − + − −+ + + +( ) ( )] ( ) ( )* * *1 14 1 13 1 2

4 0 12 2 3k k k b k k k

= + + +ak k k k k k k bk k k3 2 2 2 4 1 4 2 2 3( )* * *

[22] [ ] [( ) ( )( )( )* *PS ∞+ +→ − − + +b k k k k k1 4 2 1

4 1 2 1 3

+ − − + − −+ + + +( ) ( )] ( ) ( )* * *1 14 1 14 2 1

4 0 11 1 4k k k a k k k

= + + +bk k k k k k k ak k k4 1 1 1 3 2 3 1 1 4( )* * *

The ratio of these expressions is in agreement with the ex-pected result given by eq. [15].

It is clear that the present new digraph methods confer keyadvantages that can be used to tackle rather complexschemes leading to multiple products that would otherwisebe daunting to analyze by conventional means as will beshown, vide infra.

Equilibrium systems

The basic idea of keeping track of the arrow directionalityin kinetic schemes may be used to full advantage to deter-mine final equilibrium ratios of chemical species connectedonly by reversible steps by inspection, with almost no workinvolved. Figure 8 shows the results for acyclic equilibriumsystems involving two, three, and four species in a linear ar-rangement. One can observe key patterns among these ex-amples. If there are N species in an acyclic equilibrium, eachterm of the product ratio consists of N – 1 rate constants.The final equilibrium concentration expression for each spe-cies is determined by a single term equal to a product of rateconstants corresponding to the combination of reaction ar-rows that points towards that species. For example, for thecase of four species in an acyclic arrangement, the term forthe final concentration of species A consists of a singleproduct of three rate constants with descriptors 2, 4, and 6,which correspond to those that point toward A.

For cyclic equilibrium systems, the situation is more com-plicated but still can be deduced by the simple technique offollowing the reaction arrow directionality. Figure 9 showsthe results for cyclic equilibrium systems involving threeand four species. If there are N species in a cyclic equilib-rium, each equilibrium product concentration expression con-sists of a sum of N terms, each of which consists of aproduct of N – 1 rate constants. The arrow directionalityfalls between the limiting clockwise and counter-clockwise

© 2008 NRC Canada

Andraos 349

Fig. 7. Digraph and subgraphs pertaining to Scheme 4. Fig. 8. Example acyclic equilibrium systems.

senses. For example, for the case of three species in a cyclicarrangement, the term for the final concentration of speciesA consists of a sum of three terms, each of which is com-

posed of a product of two rate constants with pairwisedescriptors (2,4), (2,6), and (3,6).

Extended variants of ACH kinetic schemes

In this section we examine extended variants of Scheme 1 in acyclic and cyclic arrangements and determine their associatedinitial and final product ratio expressions by the adapted Chou directed graph method. Consider the linear and cyclic triadsshown in Schemes 6 and 7 with initial conditions [A]0 = a, [B]0 = b, and [C]0 = c.

The determination of the final product ratios as given in the Supplementary Data results in the following relationships forthe linear and cyclic triads, respectively,2

[23] [P ] :[P ] :[P ]A B C∞ ∞ ∞

= + + +k a k k k k k kA A C C{ [( )( ) ]2 4 3 + + +bk k k ck k2 4 2 4( ) }C

: { ( ) ( )( )k ak k k b k k k kB C A C1 4 1 4+ + + + + +ck k k4 1( )}A

: { ( )k ak k bk k kC 3 A1 3 1+ + + + + +c k k k k k k[( )( ) ]}A C A1 3 2

[24] [P ] :[P ] :[P ]A B C∞ ∞ ∞

= + + + + + + + + +k a k k k k k k k k b k k k kA B C C C{ [( )( ) ( )] [ ( )2 4 5 3 5 2 4 5 k k c k k k k k k3 5 5 2 3 2 4] [ ( ) ]}+ + + B

: { [ ( ) ] [( )( ) (k a k k k k k k b k k k k k k kB C A C C1 4 5 4 6 1 4 5 6+ + + + + + + + + + + +k c k k k k k k4 4 1 6 1 5)] [ ( ) ]}A

: { [ ( ) ] [ ( ) ] [(k a k k k k k k b k k k k k k c kC B A6 2 3 1 3 3 1 6 2 6+ + + + + + + + A B B+ + + + +k k k k k k k6 2 3 1 3) ( ) ( )]}

In both cases the initial product ratio based on the shortestpaths from A, B, and C to the respective target products is[PA]0:[PB]0:[PC]0 = akA:bkB:ckC. Also, in both cases theACH limit and the anti-ACH limit lead to final product ra-tios of ([PA]∞:[PB]∞:[PC]∞)ACH = kA:kB:kC and([PA]∞:[PB]∞: [PC]∞)anti-ACH = a:b:c, respectively. These re-

sults may be extended to tetrad, pentad, etc. arrangementsof chemical species in equilibrium, each producing aunique product in an irreversible product-forming step.We may conclude that the forms of final product ratio ex-pressions for both ACH anti-ACH limiting conditions areuniversal.

© 2008 NRC Canada

350 Can. J. Chem. Vol. 86, 2008

Fig. 9. Example cyclic equilibrium systems. Scheme 6.

Scheme 7.

Generalization of kinetic plasticity in DKRsystems

As demonstrated for Scheme 3, the key plot needed to de-termine the degree of kinetic plasticity is that given inFig. 1, in which the final product excess is plotted againstthe initial substrate excess. The implication is that kineticplasticity is defined in a pairwise sense since pairwise com-parisons are made between pairs of products and pairs ofsubstrates. For the trivial case of Scheme 3, there is only onepair of products and one pair of substrates. By analogy withthe previous analysis based on limiting slopes, we may gen-eralize the expression for kinetic plasticity starting from in-equality [25],

[25] slope slope slopeACH exp anti-ACH< <

Subtracting slopeACH from all terms results in

[26] 0 < slopeexp – slopeACH < slopeanti-ACH

– slopeACH

Dividing all terms by (slopeanti-ACH – slopeACH) results in

[27] 0 1<−

−<

slope slope

slope slopeexp ACH

anti-ACH ACH

Using this definition, we may revise the ACH principle sim-ply by stating that a kinetic model in which products areformed from equilibrating substrates satisfies the ACH con-dition when its degree of plasticity is 100%. For the kineticmodel in Scheme 3, slopeACH = 0 and slopeanti-ACH = 1,which implies by inequality [27] that 0 < slopeexp < 1 issatisfied. The kinetic plasticity in this case was defined asεDKR = 1 – slopeexp to make the zero value and the unit valuein inequality [27] correspond to 100% plasticity and 0%plasticity, respectively. Hence for Scheme 3, eq. [8] resulted.Maintaining this sense, the general expression for kineticplasticity becomes,

[28] εDKRexp ACH

anti-ACH ACH

slope slope

slope slopes= −

−−

= −1 1 lope exp

The determination of kinetic plasticity parameters for anykinetic scheme, such as those shown in Schemes 6 and 7,must be done on a pairwise basis such that adjacent pairs ofproducts arising from their corresponding equilibrating sub-strates are considered. For example, for Scheme 6 there arethree substrates leading to three unique products, and thereare two pairs of adjacent products to consider in the pairwiseanalysis. This will result in two kinetic plasticity indices cor-responding to the PAPB and PBPC pairs of products. Theglobal kinetic plasticity for the scheme would be describedby this set of two indices. The form of the expression for thekinetic plasticity index would be identical to eq. [8] with r =ka/kb, u = k1/ka, and v = k2/ka for the PAPB pair, and r = kb/kc,u = k3/kb, and v = k4/kb for the PBPC pair. For Scheme 7 thereare three adjacent pairs of products to consider, leading tothree kinetic plasticity parameters: PAPB, PBPC, and PAPC.Again, eq. [8] applies with r = ka/kb, u = k1/ka, and v = k2/kafor the PAPB pair; r = kb/kc, u = k3/kb, and v = k4/kb for thePBPC pair; and r = ka/kc, u = k6/ka, and v = k5/ka for the PAPC

pair. Unfortunately for both Schemes 6 and 7, it is notpossible to obtain the kinetic plasticity indices directly byconducting product study experiments by varying in apairwise fashion the initial substrate ratio as a function ofthe corresponding final product ratio, since neat expressionsof the form of eq. [5] are not obtainable. Instead, the magni-tudes of all rate constants need to be determined directly byindependent kinetic experiments to make estimates of theabove pairs of kinetic plasticity parameters using eq. [8].

Dynamic kinetic resolution byorganocatalysis

The emerging field of organocatalysis (65–86) has under-gone a rapid growth in recent years and provides several op-portunities to test ideas presented in this work. Threeexamples are presented in which the versatility of the di-rected graph methods is applied to determine initial and finalproduct ratio expressions. Corresponding expressions for thekinetic plasticity index are also determined in each case.These examples showcase the prowess of the new methodsin tackling complex schemes with apparent ease. In addition,the new suggested product study experiment and corre-sponding linear relationship according to eq. [5] serves as adiagnostic test to verify the proposed kinetic schemes forthese chemical systems and offers an opportunity to properlyquantify their kinetic plasticities directly from experiment.

The first example (87) shown in Scheme 8 involves a dou-ble feed-in reaction that first produces racemizable chiralimine intermediates from chiral aldehydes and aromaticamines, followed by catalytic production of chiral iminiumsalts that are then finally reduced to their amine analoguesvia oxidation of Hantzsch dihydropyridine esters to pyri-dines. Figure 10 shows the corresponding directed graph andSection 6 of the Supplementary Data shows the list ofsubgraphs to consider with respect to P1 as the target prod-uct by the adapted Chou method.2 The subgraphs with re-spect to P2 as the target product are found using thefollowing transposition of variables: [A]0 ↔ [B]0, k1 ↔ k8,k2 ↔ k7, k3 ↔ k6, k4 ↔ k5, k9 ↔ k10, and k11 ↔ k12. For theanalysis it was assumed that the concentrations of aromaticamine, chiral acids HX*, and Hanzsch dihydropyridine es-ters were in excess compared with the chiral aldehydes Aand B. The expressions for the initial and final product ratiosand the final product excess are given by eqs. [29], [30], and[31],

[29][P ][P ]

[A][B]

1 0

2 0

0

0

=

kk

k k

k k1

8

9 11

10 12

[30][ ][ ]

[ ] [ ][ ] [ ]

PP

A BA B

1

2

9

10

0 0 4 6

0 3 5

∞

∞=

++

k

kk k

k kα

0β

where

α = k4k6 + k10[HX*](k4 + k5)

β = k3k5 + k9[HX*](k4 + k5)

© 2008 NRC Canada

Andraos 351

[31] pe

k

kk k

k k∞ =

++

9

10

0 0 4 6

0 3 5 0

[ ] [ ][ ] [ ]A BA B

αβ

−

++

1

9

10

0 0 4 6

0 3 5 0

k

kk k

k k[ ] [ ][ ] [ ]A BA B

αβ

+ 1

= −+ + +

ru uru u

2 1 2

2 1 2 1 2

ν νν ν ν ν

1

1 [ ]( )*HX

+ −+

++ +

[ ] [ ][ ] [ ]

[ ]( )*A BA B

HX0 0

0 0

1 2

2 1 2

ν νν ν1ru u [ ]( )*HX ν ν1 2+

where r = k9/k10, u1 = k3/k10, u2 = k6/k10, v1 = k4/k10, and v2 =k5/k10. Note that eq. [31] shows a linear dependence betweenpe∞ and se0. In the ACH limit when ki = k (for i = 1 to 8)and are very large compared with kj (for j = 9 to 12), wehave,

[32] α = +k k k210 2[ ]*HX

β = +k k k29 2[ ]*HX

so that the final product ratio becomes,

[33][P ][P ]

1

2 ACH

∞

∞

=

+ +→ ∞lim

[ ] ( [ ] ) [ ][ ]

*

k

k

k

k k k k9

10

02

10 02

0

2A HX BA k k k k2

02

9 2+ +

[ ] ( [ ] )*B HX

=

+ +→ ∞

k

k

k k kk

9

10

0 02

0 10 2lim

([ ] [ ] ) [ ] ( [ ] )*A B A HX([ ] [ ] ) [ ] ( [ ] )*A B B HX0 0

20 9 2+ +

k k k

=

++

k

k9

10

[A] [B][A] [B]

0 0

0 0

= k

k9

10

which shows a dependence only on the product forming rateconstants emanating from intermediates I1 and I2.

In the anti-ACH limit when ki = k (for i = 1 to 8) and arevery small compared with kj (for j = 9 to 12), we have thesame relations as in eq. [32] so that the final product ratiobecomes,

[34][P ][P ]

1

2 anti-ACH

∞

∞

=

+ +→

lim[ ] ( [ ] ) [ ][ ]

*

k

k

k

k k k k0

9

10

02

10 02

0

2A HX BA k k k k2

02

9 2+ +

[ ] ( [ ] )*B HX

© 2008 NRC Canada

352 Can. J. Chem. Vol. 86, 2008

Scheme 8.

Fig. 10. Chou digraph pertaining to Scheme 8.

=

+ +→

k

k

k k kk

9

10 0

0 02

0 10 2lim

([ ] [ ] ) [ ] ( [ ] )*A B A HX([ ] [ ] ) [ ] ( [ ] )*A B B HX0 0

20 9 2+ +

k k k

=

k

kkk

9

10

10

9

22

[A][B]

HX0

0

[ ]*

[ ]*HX

= [A][B]

0

0

which shows a dependence only on the starting amounts ofaldehydes A and B.

The final product excess as a function of initial substrateexcess for these two limits is given by eqs. [35] and [36],

[35] ( ) limpe peACH∞ → ∞→ ∞

∞= = −+

=−

+ui

i

rr

k

kk

kν

11

1

1

9

10

9

10

[36] ( ) limpe pe seanti-ACH∞ → ∞→ ∞

∞= =ui

iν

0

where

se 00 0

0 0

= −+

[ ] [ ][ ] [ ]A BA B

Using the definition given in eq. [28], the expression forthe kinetic plasticity for this kinetic model is given by eq. [37],

[37] εDKR

d ped se

d ped se

= −

−

∞ ∞

10 0

( )( )

( )( )

exp

−

∞ ∞

ACH

anti-ACH

d ped se

d ped se

( )( )

( )( )0 0

ACH

= −

−

−

∞

1

0

1 0

0

d ped se( )( )

exp

= −

∞10

d ped se( )( )

exp

= − ++ + +

1 1 2

2 1 2 1 2

[ ]( )[ ]( )

*

*

HXHX

ν νν ν ν ν1ru u

= ++ + +

ru uru u

2 1 2

2 1 2 1 2

ν νν ν ν ν

1

1 [ ]( )*HX

The second example (88) shown in Scheme 9 is a morecomplex variant of Scheme 8. The aim is to stereoselectivelyreduce a mixture of (E/Z) α,β-unsaturated aldehydes to theirdihydroaldehyde analogs via Hanzsch dihydropyridines anda chiral organocatalyst that produces (E/Z) iminium interme-diates that are interconvertible. Figure 11 shows the corre-sponding directed graph and Section 7 of the SupplemenatryData shows the subgraphs with respect to P1 as the targetproduct.2 As in the previous case, the subgraphs with respectto P2 are found using the following transposition of vari-ables: [E]0 ↔ [Z]0, ′k1 ↔ ′k 3, k2 ↔ k4, k5 ↔ k6, k7 ↔ k9,′k8 ↔ ′k10, k11 ↔ k12, ′k13 ↔ ′k14, and ′k15 ↔ ′k16.

The expressions for the initial and final product ratios andthe final product excess are given by eqs. [38], [39], and[40],

[38][P ][P ]

[E][Z]

1 0

2 0

0

0

=

k

kk kk k

1

3

13 15

14 16

[39][P ][P ]

E Z E1

2

∞

∞= ′

′

+ + ′kk

k13

14

0 0 0 14([ ] [ ] ) [ ](

α β[ ] [ ] ) * [ ]E Z E0 0 0 13+ + ′

α βk

where

© 2008 NRC Canada

Andraos 353

Scheme 9.

α = k6β + k9 ′k8k12

α* = k5β + k7 ′k10k11

β = ′k8 ′k10 + ′k8 ′k12 + ′k10 ′k11

′k1 = k1[C], ′k 3 = k3[C], ′k8 = k8[H2O],

′k10 = k10[H2O]

′k15 = k15[H2O], ′k16 = k16[H2O],

′k13 = k13[QH], ′k14 = k14[QH]

[40] pe

E Z EE Z

∞ =

′′

+ + ′+

kk

k13

14

0 0 0 14

0

([ ] [ ] ) [ ]([ ] [

α β] ) [ ]

([ ] [ ] )

0 0 13

13

14

0 0

1α β∗ + ′

−

′′

+

E

E Z

k

kk

α βα β∗

+ ′+ + ′

+[ ]

([ ] [ ] ) [ ]E

E Z E0 14

0 0 0 13

1k

k

= −+ + ′

+ −+

′r

r k

k

r

α αα α β

β* [ ] [ ][ ] [ ]*

13

0 0

0 0

13E ZE Z α α β+ + ′

* k13

where r = ′k13/ ′k14. Note that eq. [40] shows a linear depend-ence between pe∞ and se0. In the ACH limit when ki = k (fori = 1 to 12) and very large compared with kj (for j = 13 to16), we have

[41]

α

α

β

→

→

→

4

4

3

3

3

2

k

k

k

*

so that the final product ratio becomes,

[42][ ][ ]PP

ACH

1

2

∞

∞

= ′′

+ + ′→ ∞lim

([ ] [ ] ) [ ](k

kk

k k k13

14

0 03

0 1424 3E Z E

[ ] [ ] ) [ ]E Z Z0 03

0 132

13

144 3+ + ′

′′

k k k

kk

which shows, as in the previous example, a dependence onlyon the ratio of product forming rate constants emanatingfrom intermediates I1 and I2.

In the anti-ACH limit when ki — > 0 (for i = 1 to 12), wehave the same relations as in eq. [41], so that the final prod-uct ratio becomes,

[43][ ][ ]PP

anti-ACH

1

2

∞

∞

= ′′

+ + ′→

lim([ ] [ ] ) [ ](k

kk

k k k0

13

14

0 03

0 1424 3E Z E

[ ] [ ] ) [ ][ ][ ]E Z ZEZ0 0

30 13

20

04 3+ + ′

k k k

The final product excess as a function of initial substrate ex-cess for these two limits are given by eqs. [44] and [45],

[44] (pe )ACH∞ = −+

=

′′

−

′′

+

rr

kkkk

11

1

1

13

14

13

14

[45] (pe ) seanti-ACH∞ = 0

where

se 00 0

0 0

= −+

[ ] [ ][ ] [ ]A BA B

Using the definition given in eq. [28], the expression forthe kinetic plasticity for this kinetic model is given byeq. [46],

[46] ε α αα α βDKR = +

+ + ′r

r k

*

*13

The final example (89) shown in Scheme 10 involvesstereoselective α-fluorination via an electrophilic “F+” spe-cies of an achiral aldehyde using a chiral proline analogue asan organocatalyst. The two products of interest are P1 andP2. The third difluorinated product arises from intermediateI4 which is in equilibrium with I2 and I3. All three productsarise from the same source, namely intermediate I1. Sincethe products do not arise from different equilibrating startingsubstrates, kinetic control to produce one of the productsover the others will depend on the relative magnitudes of therate constants and not on the initial amount of starting mate-rial. Figure 12 shows the corresponding directed graph andSection 8 of the Supplementary Data shows the subgraphswith respect to P1, P2, and P3 as the target products.2 Thefollowing transposition of variables applies: ′k3 ↔ ′k8, k7 ↔ k9,′k 5 ↔ ′k8, and ′k8 ↔ ′k10. The initial and final product ratios are

given by eqs. [47] and [48], respectively.

[47] [P ] :[P ] :[P ]1 0 2 0 3 0 = ′ ′ ′ ′ ′ ′ ′ + ′k k k k k k k k k3 5 4 6 11 12 3 7 4: : ( k9)

[48] [P ] :[P ] :[P ]1 2 3∞ ∞ ∞

= ′ ′ ′ + ′ ′ + ′ ′ + ′ ′k k k k k k k k k k5 3 6 9 3 8 4 8 3 11{ ( )}α

: { ( )}′ ′ ′ + ′ ′ + ′ ′ + ′ ′k k k k k k k k k k6 4 5 7 4 10 4 11 3 10α

: { ( ) ( )}′ ′ ′ + + ′ ′ +k k k k k k k k k11 3 7 6 9 4 9 5 7

where

© 2008 NRC Canada

354 Can. J. Chem. Vol. 86, 2008

Fig. 11. Chou digraph pertaining to Scheme 9.

α = ′k8 + ′k10 + ′k11

k3′ = k3[F+], ′k 4 = k4[F+], ′k11 = k11[F+]

′k 5 = k5[H2O], ′k 6 = k6[H2O], ′k8 = k8[H2O]

′k10 = k10[H2O], ′k12 = k12[H2O]

In the ACH limit when there is rapid equilibration be-tween intermediates I2, I3, and I4 so that ki = k (for i = 7 to10) is very large compared with the product forming rateconstants, then the final product ratio becomes,

[49] ([P ] :[P ] :[P ]1 2 3 ACH∞ ∞ ∞ )

= ′ ′ ′ = +k k k k k k5 6 11 5 6 11: : [ ]: [ ]: [ ]H O H O F2 2

Again, these are the product-forming rate constants ema-nating from intermediates I2, I3, and I4. On the other hand,the anti-ACH limit results in a final product ratio given byeq. [50], which predicts the absence of the difluorinatedproduct P3,

[50] ([P ] :[P ] :[P ]1 2 3 anti-ACH∞ ∞ ∞ = ′ ′) : :k k3 4 0

Since all three products arise from the same source sub-strate, it is not possible to obtain estimates of kinetic plastic-ity parameters for this scheme by varying substrate amounts.Direct estimates of the appropriate rate constants need to bemade. The two pairs of products to consider are P1P3 andP3P2 that will lead to two kinetic plasticity indices. They willbe of the form of eq. [8] with r = ′k 5/ ′k11, u = k7/ ′k 5, and v =′k8/ ′k 5 for the P1P3 pair and r = ′k11/ ′k 6, u = ′k10/ ′k11, and v =

k9 / ′k11 for the P3P2 pair.

Conclusions

The present work introduces new algorithms based on di-rected graphs to determine initial and final product ratios forsimple and complex kinetic models leading to multiple prod-ucts for unimolecular transformations and under pseudo-order conditions. These are found to circumvent the tediousand error prone computations that are characteristic of con-ventional treatments based on rate laws. A degree of kineticplasticity parameter is introduced and shown to be general in

© 2008 NRC Canada

Andraos 355

Scheme 10.

Fig. 12. Chou digraph pertaining to Scheme 10.

describing the ACH condition (100% dynamic kinetic reso-lution), the opposite anti-ACH condition (0% dynamic ki-netic resolution), and all other scenarios in between theselimits. To empirically probe the full scope of the dynamicsof a kinetic system one needs to begin from more than oneset of starting substrate conditions. Thus, it is impossible toobtain a full picture by carrying out product studies fromone set of starting substrate conditions such as a racemicmixture. For a practicing organic chemist who wishes to un-derstand their kinetic schemes more deeply without havingto invoke external assumptions, this means that they mustmove beyond the confines of racemic mixtures as the onlystarting point to obtain direct experimental measures of thedynamics of their systems. The determination of the degreeof kinetic plasticity is thus based on a key experimental plotthat relates final product excess and initial substrate excessin a pairwise fashion. Such a plot is always linear with apositive slope, and the degree of kinetic plasticity is founddirectly by subtracting the slope from unity. Furthermore, itmay be used as a diagnostic test for checking the validity ofproposed ACH-type kinetic schemes for a given chemicalsystem. A key bonus of the new methods of obtaining prod-uct ratio expressions is that the precise rate constant depend-ence of the degree of kinetic plasticity is obtained explicitly.Such dependencies may be further checked by direct deter-mination of absolute rate constants from independent kineticmeasurements. Examination of extended variants of ACH-type schemes indicates that the ACH principle holds univer-sally and may be elevated to the status of a kinetic theorem.The next challenge for organic chemists is to provide experi-mental verification of the predictions made in this work.

Acknowledgements

This work was presented at the 90th Canadian Society ofChemistry (CSC) Conference and Exhibition in Winnipeg,Manitoba, Canada (May 26–30, 2007) in honour of the 80thbirthday of Prof. Alexander Jerry Kresge and at the 35th On-tario–Quebec Physical Organic Chemistry Mini-Symposiumin Waterloo, Ontario, Canada (November 9–11, 2007) hon-ouring the late Prof. Keith Yates. Dedicated to the memoryof Keith Yates.

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