J. Phys. Chem. 1991, 95, 4918-4921 TABLE 1% c Vatvcr (in k u l / d )
w of A J ~ ,
GW,, Val-
'fH0
compound acetylene ProPYne 1-butyne 2-butyne I-pentyne
198
from heats of hydrogenation the overall agreement is exd* ~ x ~ m h some l AAfHO298
expt group (expt - group) 54.51 54.4 0.1 1 -0.43 44.17 44.6 39.46 39.9 -0.44 34.81 34.8 0.01 34.50 29.88 32.87
34.9 30.1 32.9
-0.40 -0.22 0.00
2-pentyne 3-methyl-1-butyne 3,3-dimethyl-1-butyne 25.42 25.2 0.22 trans-3-penten-1 -yne 61.61 61.1 0.54 cis-3-penten- 1-yne 60.98 60.5 0.51 1,S-hcxadiyne 100.14 99.8 0.34 phenylacetylene 73.3 73.7 -0.4 1,7-octadiync 90.14 89.8 0.34 1 -0cten-3-yne 43.72 44.2 -0.52 rruns-3-decen-1-ync 36.16 36.3 -0.15 cis-3-decen-l-yne 35.44 35.7 -0.27 5,7-dodccadiyneg 56.98 57.0 (-0.02) 3.9-dodecadiyne 60.79 60.8 -0.01 3,3,6,6-tetramethyl-1,7-octadiyne 61.86 62.0 -0.14 3,3-dimethyl-1 ,4-pentadiyneo 91.23 91.2 (0.03) I-phenyl-I-propync 64.1 63.9 0.2 I-phenyl-]-butyne 59.7 59.2 0.5 1 -phenyl-l-hexyne 49.5 49.2 0.3 diphenylacetylene 93.8 93.0 0.8 diphenylbutadiyne 142.5 144.6 -2.1 #Compound was used to derive unique group contribution. being 0.54 kcal/mol for trans-3-penten-1-yne among the smaller molecules. For the larger molecules we note a deviation of +0.8 kcal/mol for diphenylacetylene and -2.1 kcal/mol for diphenylbutadiyne. We could reduce this last by making C,-C, more stable but the accuracy of the data does not warrant it and it would increase the deviation of 5,7-dodecadiyne. Both of these deviations are well within the uncertainties connected with the original data. It is thus reasonable to conclude that as a class of compounds the group additivity estimates are probably more reliable than any individual piece of data. This is based on the consideration that the groups average the uncertainties over all the compounds in a given class. Since some of the experimental observations come from combustion data and
cellent. Some Predictions. Using the group contributions in Table 111, it is possible to derive heats of formations for new compounds. Some interesting examples are as follows: comwund butadiyne (diacetylene)
butenync (vinylacetylene) 3,s-octadiyne 1,3-hcxadiync-5-ene
A,Holdd / kcal mol-' 106.0 68.7 77.0 120.3
The value for butadiyne is significantly lower than most previous estimates of 110-1 13 kcal/mol. However, it is in good agreement with that of Stein and Fahr.14 Perhaps the most unusual feature in the thermochemistry of acetylenes is the stability of the cis-yne-ene configuration. For the two examples shown in Table IV,the experimental data show quite clearly that the cis form (Z) is more stable than the trans form (E) by about 0.6 kcal/mol. In butene-2 the cis is less stable than the trans by 0.9 kcal/mol. This has been attributed to a repulsion of the H atoms on the cis-CH, groups in cis-butene-2. Acetylene has no H atoms so a cis-CHS as in cis-pentenyne-3 would see no repelling H atoms but possibly an attractive electron T cloud. In ref 15, the authors were led to make their experiments because of what seemed like an obvious discrepancy in the heats of hydrogenation of phenylacetylene compared to methylacetylene. Ahydff was reported to be -71 kcal for the former while it was -69 kcal for propyne. This suggested to the authors that the phenyl group was less stabilizing than the methyl group, a paradoxical result. However, it is not possible to interpret heats of hydrogenation so simply. Thus Ah,,,H of vinyl acetylene to butyne-1 is -29.3 kcal, only 0.7 kcal less than Ah@ of butene-1 to butane, suggesting the triple bond is 0.7 kcal more stabilizing than the ethyl group, and much less stabilizing than a double bond since Ah& of butadiene-l,3 to butene-1 is -26.3 kcal. There are many changes that occur in a molecule when a double or triple bond is hydrogenated and it is an oversimplificationto single out a single interaction as the one source of the observation. Acknowledgment. This work has been supported by a grant from the National Science Foundation (CHE-8714647). Registry No. Acetylene, 74-86-2.
On the Concept of Stoichiometry of Reactlon Mechanisms Raiil E.ValdbP6rez School of Computer Science, Carnegie Mellon Uniuersity, Pittsburgh, Pennsylvania I5213 (Received: May 21, 1990)
The usual method to determine the stoichiometry implied by a reaction mechanism has an ambiguity arising from the need to specify the intermediates,leading in some cases to several possible stoichiometries. A formalizationof mechanistic stoichiometry as an optimization problem removes the reliance on the concept of an intermediate,and clarifies the relation between stoichiometry of a mechanism and ideal yield of a target product. The proposed formalization is in terms of linear programming.
1. Introduction
The concept of stoichiometry is used in chemistry in different contexts. h e use refem to an abstract, hian& transformation ofa set of spacia into another set of species characterized well by Smith and Missen.' Another use refers
( 1 ) Smith, W.
R.; Missen. R. W. Chrm. Eng. Edue. 1979, 13, 26-32.
0022-3654/91/2095-491 8$02.50/0
to the stoichiometry of a reaction mechanism, and is concerned roughly with a balanced transformation of starting materials into final products that is implied by the mechanism. This note examines the concept of stoichiometry only in this sense of a transformation imp1ied by a mechanism* The stoichiometric coefficients of starting materials are often construed as the ideal proportions of starting materials, in the sense of not providing more initial concentration than needed to obtain, 0 1991 American Chemical Society
On the Concept of Stoichiometry of Reaction Mechanisms in the idealized case, a certain yield of a target product. The purpose of this note is to formalize the concept of reaction mechanism stoichiometry by relating the concept explicitly to ideal yield. The current, semiformal notion of mechanism stoichiometry, as described by Coria,' has an ambiguity due to an uncertain, somewhat arbitrary decision of which mechanism species to regard as intermediates. This ambiguity makes the concept of stoichiometry less valuable for the discussion of ideal mechanism yields. We shall illustrate by example how our formalization in terms of linear programming removes this ambiguity, while gaining a clearer understandingof the reldtion between stoichiometry and ideal yield.
2. Concept of Reaction Extent The formal development of stoichiometry relies on the wellknown concept of reaction extent, or hereafter simply extent (e& see Ark3)). The extent of a reaction is a number associated with each mechanistic step that represents the net flow in moles from left-side species to right-side species. A step A + B C of extent 1.5 means that uia this step 1.5 moles of A,B are transformed to c. 3. Stoichiometry and Reaction Extent One procedure to determine stoichiometry is sketched by Corio? This procedure is often followed implicitly when one is finding stoichiometry by hand in simple cases. First, one arranges the mechanistic steps in order from starting materials to target product. Next, one decides which species are intermediates and which are final. products, i.e. not intermediates or starting materials. Then, one selects values for the extents such that the net formation of intermediates across the steps is zero. (The extent contributes negatively to the net formation of a step’s left-side species, and positively to the net formation of the right-side species.) The species of negative net formation are the mechanism’s starting materials, and those of positive net forma”&ionare the final products. The stoichiometry is derived from these nonzero net formations, adopting their magnitudes as stoichiometric coefficients. This procedure is illustrated on the following schematic mechanism I, where the Et above the arrows are the extents. The A.
+B
II 4
X
+ 2Y
I2
B+X-2Z
The Journal of Physical Chemistry, Vol. 95, No. 12, 1991 4919 are intermediates. We derive equations as before to force their net formations to zero. x: €1 - 5 2 = 0 Y: €1 - €3 = 0 z: 212 - €3 = 0 The only solution to these equations has the extents identically zero, meaning that the intermediates cannot all have zero net formation, unless no reactions occur at all. That is, there exists no ratio of starting materials that yields T with no by-product. Therefore, one of these intermediates will appear as a stoichiometric product, in which case the net formation of the remaining two can be forced to zero. A plausible candidate for status as stoichiometricproduct is species Z, which from inspection of the last two steps appears to be formed in excess. If we proceed this T Z. way, the derived stoichiometry would be A 2B There are two lessons from mechanism 11. First, the concept of an intermediateas a species formed and later consumed, which is on the path to a target product, cannot be identified with a set of mechanism species that disappear from the stoichiometry. We saw above that not all the intermediates could be eliminated. Second, the stoichiometry of a mechanism need not be unique, since it depends in some quite ordinary cases on an arbitrary decision of which intermediate shall be regarded as a stoichiometric product. This raises the question of how best to define stoichiometry in a manner compatible with results on nonproblematic cases such as mechanism I, while gaining on problematic mechanisms such as I1 a correspondence to ideal yield.
-
+
+
4. Stoichiometry via Minimization of Starting Materials Among the possible mechanism stoichiometries, we shall single out that which affords the best yield of a target product. Our formalization of stoichiometry shall take the form of a linear minimization problem. In this section, we formulate the equations, inequalities, and minimization criterion for mechanism 11. Then the stoichiometry determined by our procedure is contrasted with the stoichiometry seen above. The intuitive idea is to select, as before, values of the reaction extents, but without any preconceived decision to force the net formation of certain species to zero; our method has no need for the concept of intermediate. In addition, minimal values are selected for the initial concentrations of starting materials, such that there exist reaction extents resulting in an arbitrary yield of 1 mole for the target product. To illustrate, we write the following equation derived from mechanism 11:
Y+Zb-T intermediatesare X, Y, Z, and the target product is T. The three equations that enforce zero net formation of the intermediates are
x: €1 - €2 = 0
Y: 25, - 5 3 = 0
z: 252 - €3 = 0
S3
which possess the solutions 5, = t2= a, = 2a. Summing over the steps the species formations determined by the extents, and equating u to 1 for convenience, we obtain the stoichiometry A + 2B 2T. The above semiformal procedure has some drawbacks, which we illustrate by the following mechanism 11, differing from I only
-
A+B-X+Y B+X-22 Y+Z-T in the coefficient of Y in the first step; the mechanism is quite ordinary; e.g., it exhibits no feedback! The species X, Y, Z again (2) Corio. P. L. Theory of reaction mechanisms. Topics in Current Chemistry: Relationships and Mechanisms in the Periodic Table; Springer-Verlag: Secaucus, NJ, 1989; Vol. 150. (3) Ark, R. Elementary Chemical Reacfor Analysis; Prentice Hall: Englewood Cliffs, NJ, 1969.
This states that a transition from concentrations at time zero to those at time t is induced by the reaction extents. The matrix premultiplying the extents is derived from the stoichiometriesof the individual mechanistic steps. All species except the starting materials have zero initial concentrations. The next equation T(f) = 1
(2)
arbitrarily fixes the yield of T at 1 mol/L. All the other variables receive no fixed value, but are constrained to be nonnegative: A(f),
WO),
W t ) , X(r), y ( t ) , z(t),ti, Ez,
43
20
(3)
The objective is to find the minimum amounts of starting materials that yield 1 mole of T via mechanism 11. This objective is formalized by the following statement: Minimize the quantifyA(0) (4) It is important to realize that mechanism I1 is consistent. One set of consistent instances has A, B, X, Y, 2 all sharing the Same molecular formula, and T double that formula. Whether such instances are empirically plausible is another matter.
4920 The Journal of Physical Chemistry, Vol. 95, No. 12, 1991
+ B(0) subject to the constraints of eqs 1 and 2 and inequalities 3.
We discuss how to solve this minimization problem shortly; its solution is A(0) = 1.0, B(0) = 1.5, X ( t ) = 0.5, A(t) = B(t) = Y ( t ) = Z(r) = 0
-
Multiplying all values by 2, we obtain a stoichiometry of 2A
-
+ 3B
2T
+X
which improves on the yield of T implied by the previous stoichiometry A + 2B T + Z, which was derived by arbitrarily forcing the net formations of intermediates X,Y to zero. Following the same procedure, we calculate for the unproblematic mechanism I a stoichiometry of A + 2B 2T, the same as obtained before with the usual procedure.
-
5. The General Case In section 4, we formulated a specific minimization problem to obtain a stoichiometry for mechanism 11. The general case is
treated here. Given a mechanism, we derive from it a matrix Rrxsof r rows, one for each mechanistic step, and s columns, one for each species. The entries of a row are the stoichiometric coefficients of an individustl step, with negative coefficients for the left-side species, positive coefficients for the right-side species, and zero coefficients for any species not appearing in the step. If a species appears both on the left and right sides, then its row entry is simply the signed sum of the two coefficients. This equation says that the species concentrationsat time t equal the concentrations at time 0 plus the contributions to the concentrations from the reaction extents, represented by the vector E, having r rows, one for each step. This equation is the general case of eq 1 above. The other constraints needed are eq 2 and a generalization of the inequalities 3: (1) The target product has a concentration = 1 mol/L (or any convenient value) at time t. (2) All variables (Le., species concentrations, extents) are nonnegative. The generalized minimization problem is that of minimizing at t = 0 the sum of the concentrations of all starting materials. There is one subtlety in the procedure of this section. If any of the mechanistic steps is reversible, one should include the reversed step explicitly in the mechanism. Otherwise, since extents are constrained to be nonnegative, a net reaction in the reverse direction would be precluded. 6. Solving the Minimization Problem
The above minimization problem is an instance of linear programming. A linear programming (LP) problem consists partly of a set of constraints, which can be linear equations or linear inequalities. By convention, all variables appearing in the constraints assume only nonnegative values. The remaining part of an LP problem is the minimization criterion, required to be a linear expression (e.g., A(0) B(0) in the example above). An LP problem is solved by reporting one of the following: (1) that the constraints are inconsistent; (2) that the quantity to be minimized is unbounded; or (3) values for all variables, as well as the value of the expression to be minimized. Papadimitriou and Steiglitz5 have given a detailed discussion of linear programming. These reports are interpreted in our case as follows: ( I ) Inconsistent constraim mean that the mechanism is inconsistent, in one of two ways. If the mechanism is schematic, then inconsistency means that no assignment of molecular formulas to the variables can result in balanced steps. If the mechanism is a specific instance, then inconsistency means it is unbalanced. (2)
+
(5) Papadimitriou, C.; Steiglitr, K. Combinaforfol Opffmiraffon: Algorifhms and Complexify; Prentice-Hall: Englewood Cliffs, NJ, 1982.
Valdis-Ptrez The quantity that we minimize cannot be unbounded, because concentrations cannot be negative. Hence, the lower bound on the summed quantity of starting materials is zero. (3) If the minimization succeeds, then a stoichiometry maximizing the yield of target product exists. We have chosen a minimization criterion rather arbitrarily. In general, the criterion can be any suitable, weighted sum of the starting materials, Le. a A BB yC ..., of which our choice of unit weights is a special case. The weights might reflect relative costs, for example. In general, the resulting stoichiometry depends on the weights. For example, a zero weight for starting material B of mechanism I1 would result in the inferior stoichiometry of A + 2B T + Z). Computer programs to solve LP problems are readily available; the most common are based on the Simplex algorithm. Press et a1.6 list-a program for one implementation of Simplex. We have used this Simplex implementation as a foundation for another program that finds stoichiometry from the symbolic inputs of mechanism, starting materials, and target product. 6.1. Comparison with Usual Procedure. The remaining comments of this section shall assume good familiarity with the Simplex algorithm. Below, we let #sp be the total number of mechanistic species, #st the number of mechanistic steps, and #sm the number of starting materials. We have already remarked that the usual method to find stoichiometry consists of selecting certain intermediates XI,and forcing their net formations Xi(?) to zero. The Simplex-based procedure is a generalization of the same idea, as will be seen shortly. The linear-programmingformulation builds a constraints matrix of rank #sp, since there is a constraint for each species (as in eq 1). The number of variables nvars in the constraints matrix equals (#sp - 1) + #st + #sm, corresponding to the species concentrations at time t (except for the known target concentration of 1 mol/L), the reaction extents, and the initial concentrations of starting materials. The Simplex a l g d h m in effect optimizes the objective function over all of these cases: find a subset of the nvars variables of size nvars - rank, set equal to zero those variables in the subset, and then infer the values of the nonsubset variables (some of these may turn out to be zero also). Hence, any solution to the linear program has at least
+ +
-
nvars - rank = ((#sp - 1)
+ #st + #sm)
- #sp =
#st
+ #sm-
1 (4)
zeroed variables. If we assume that for the calculated stoichiometry the final concentrationsof the #sm starting materials are zero, the initial concentrationsof the #sm starting materials are positive, and the reaction extents are positive (all mechanistic steps have reacted to some degree), then we obtain that at least #st - 1 (non-starting-material)species have final values of zero, i.e. are stoichiometric intemediates. We can illustrate this result with a mechanism consisting of steps Xk Xk+l,k = 1, ..., N. The stoichiometry is X I XN+,. the final concentration of XI is zero, and the extents are identically 1. The number of stoichiometric intermediates is seen to be N - 1, one less than the number of steps, consistent with the formula
-
#st
-
-
1.
7. Empirical Interpretation
The stoichiometry found by the LP formulation is an idealized input/output relation between starting materials and target product. The reaction extents are freely chosen to optimize this relation, although physically a choice of extents might correspond to “freezing” a reaction step while its reactants are present. In addition, empirical reaction yields depend on energetic and kinetic considerations. Hence, it may not be feasible actually to achieve (6) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes: The Arf of Scfentffic Cotnpufing Cambridge University Press: Port Chester, NY, 1986.
J. Phys. Chem. 1991, 95,4921-4922 the ideal yield determined by the stoichiometry. Another point is that a certain species might not appear at all in the stoichiometry, although the reaction could not proceed without it. An example is a species that is consumed by an initial step but is regenerated completely by a later step. Hence, the stoichiometric reactants may be only a subset of the necessary starting materials. These empirical issues are quite separate from the concept of mechanism stoichiometry,which it is our purpose here to clarify. 8. Hand Calculation of Stoichiometry Unfortunately, the cursory analysis of this section will indicate that a hand simulation of the proposed method is intractable even for relatively simple mechanisms. Section 6.1 explained that the Simplex algorithm works by combinatoriallychoosing subsets of size nvars - rank from the nvars variables, setting the subset variables to zero, and inferring the values for the rest of the variables. Now, since nvars = (#sp - 1) + #st + #sm we can reduce the combinatorial problem by assuming that the #st extent variables cannot be assigned to zero (all steps undergo net positive reaction). Also by use of eq 4, the combinatorial problem becomes choose #st + #sm - 1 from (#sp - 1) + #sm If we further assume that the initial concentrations of starting maltrials are positive, then the combinatorial problem becomes choose #st #sm - 1 from (#sp - 1)
+
As a final assumption, the final concentrations of starting materials are set to zero, so the problem becomes choose #st - 1 from #sp - #sm - 1 This means that under the above assumptions, one still must
4921
examine all ways to choose #st - 1 variables from among the species that are not starting materials and are not the target product. In a simple, typical case of 10 total species, 2 starting materials, and 5 steps, one must attempt choose(4,7) = 35 ways to find the ideal stoichiometry. Even if one could use some mechanism-dependent heuristics to reduce the combinatorial problem further, the number of cases likely remains too great for convenient hand calculation. We conclude from this analysis that a computer program should be used in nontrivial situations where an optimal-yield stoichiometry is desired. 9. Conclusion
We have clarified the concept of mechanistic stoichiometry, and its relation to ideal yield, by formalizing the derivation of stoichiometry. A derived stoichiometry corresponds to the ideal yield of a target product, in the sense of minimizing the summed quantity of starting materials. The formalization is in terms of linear algebraic equations and inequalities, using the concept of reaction extent, but without using the concept of reaction intermediate. Finding the stoichiometry involves solving a simple, linear-programming problem. The usual method of finding stoichiometry, as described by Corio,zcontains an ambiguity due to its reliance on the concept of reaction intermediate. Our example above has illustrated that one cannot correctly identify the mechanistic intermediates with the set of species not appearing stoichiometrically. Hence, an arbitrary decision is usually made regarding which species to omit from the stoichiometry. A wrong decision, from the viewpoint of ideal yield, will lead to an inferior stoichiometry. Acknowledgment. The author is supported by an Air Force Laboratory Graduate Fellowship, administered by the Southeastern Center for Electrical Engineering Education (SCEEE). Professor Herbert A. Simon commented valuably on a draft of this paper.
COMMENTS
-+
Comment on “Efiect of Rotatlonai Ouantum States ( J = 0, 1) on the Tunneling Reaction H, H H H, In Parahydrogen Soild at 4.2 K”
+
Sir: Miyazaki et al. recently reported measurements of the rate coefficients for H p H 2 at 4.2 K.’ (At this temperature H, is dominantly in j = 0.) By combining those measurements with previous ones, they reported that the ratio of the rate coefficients, &j.,/kj*o, was approximately 1 / 3 . (The interpretation of the experiment in terms of bimolecular rate processes has been discussed’ and also assumed in previous theoretical calculations of nonrotational state selected rate coeffi~ients.2~)In this Comment I propose a simple explanation of this result. The rate coefficient for the initial rotational state j , averaged over its projection quantum number j , , is given by
+
(1) Miyazaki, T.; Hiraku, T.; Fueki, K.; Tsuchihashi. Y. J . fhys. Chem. 1991, 95, 26.
(2) (a) Takayanagi. T.; Masaki, N.; Nakamura, K.; Okamoto, M.; Sato,
S.;Schatz. 0. C. J . Chem. fhys. 1987,86,6133. (b) Takayanagi, T.; Na-
kamura, K.; Sato, S. J. Chem. fhys. 1989,90,1641. (c) Takayanagi. T.; Sato, S. J. Chem. Phys. 1990, 92, 2862. (3) Hancock, G. C.; Mead, C. A.; Truhlar, D. G.;Varandas, A. J. C. J . Chem. Phys. 1989, 91, 3492.
where kjjJT) is the rate coefficient for the fully rotationally state-selected quantum state. The projection quantum number j , can be defined with respect to a space-fixed, helicity4 or body-fixed5v6z axis. A body-fixed z axis which has been used in reactive scattering is the Jacobi vector R, which for H + H i the position vector of H with respect to the center of mass of Ht?: The projection quantum number referred to this z axis is usually denoted fl to distinguish it from the spacsfixed projection quantum number, which is usually denoted m,. In the discussion below, which is based on the adiabatic-bend theory of reactive scattering, I will make use of the projection quantum number Q in eq 1, and so
The adiabatic treatment of the H3bending motion, together with accurate semiclassical7or reduced dimensionality quantum* treatments of tunneling, has been shown to give accurate thermal, Le., not state-selected, rate coefficients, at least at room tem(4) (5) (6) (7) 4755. (8)
Miller, W. H. J . Chem. fhys. 1969, 50, 407. Pack, R. T. J . Chem. Phys. 1974.60.633. Schatz, G. C.; Kuppermann. A. J . Chem. fhys. 1976,65. 4642. Garrett, B. C.; Truhlar. D. G. froc. Natl. Acad. Sci. U S A . 1919.76, Bowman, J. M.; Ju, G.-Z.; Lee, K.-T. J . fhys. Chem. 1982,86,2232.
0022-3654191/2095-4921$02.50/0 0 1991 American Chemical Society
