From Rate Laws to Reaction Mechanisms
2323
From Rate Laws to Reaction Mechanisms Ralph G. Pearson Department of Chemistry, University of California, Santa Barbara, Californla 93 106 (Received April 26, 1977) Publication costs assisted by the Petroleum Reseamh Fund
The computer is a powerful tool for obtaining rate laws, or calculated concentration-time profiles for many components. The deduction of reaction mechanisms from such rate laws is still an inherently difficult and uncertain process. Complications due to multiple interpretations of the rate law, to nonideal behavior, and to incomplete models are discussed. Some useful applications of the computer in moderately complex systems are also given.
Introduction The computer, in its various forms, is now widely used by chemical kineticists. Uses range from the routine evaluation of rate constants from experimental data to the simulation of very complex kinetic schemes. In the latter case it is possible to obtain numerical solutions to a series of coupled rate equations. This means that rate laws can be found for the concentration variation with time of several, or many, components of a reacting system. Since rate laws are prima facie evidence for reaction mechanisms, this raises the hope that complex reaction mechanisms can be unravelled. However there are still many difficulties on the way to this objective. The purpose of this paper is to point out some of these, or to emphasize them rather, since they are well known. Actually the computer is so powerful a tool, and its applications are so successful, that kineticists are likely to become overconfident in what can now be accomplished. It is useful to remind ourselves of the nature of a reaction mechanism. It is a theoretical concept used to explain experimental results and to predict new experiments to be done. The truism remains that mechanisms can never be proved, only disproved. Rate laws, on the other hand, are experimental facts which form part of the evidence on which mechanisms are devised. Many other diverse facts, some kinetic in nature and others not, are needed to flesh out a mechanism and to make it p1ausible.l The most important evidence for the mechanism of any reaction is still the correct identification of the reactants and products, followed by the identification of intermediates which are produced en route. Next in importance would be the rate laws for the disappearance of the reactants, for the appearance of the products, and the more complex time dependence of the intermediates. Such rate laws are rich in information particularly when complex time behavior is observed. Mechanisms may now be devised which predict the time behavior, either in the form of explicit rate laws or calculated profiles of time-concentration. However the ability to solve the rate equations more accurately by the use of the computer must be matched by an equal ability to detect and measure the concentrations of all components. For reactive intermediates in very low concentration this remains a serious and often insurmountable problem. Ambiguities in t h e Rate Law The determination of a reaction mechanism usually consists of two parts which differ in the degree of sophistication involved. The first, and least sophisticated part, consists of finding all the elementary processes (unimolecular, bimolecular, etc.) which take place con-
currently or consecutively to produce the observed stoichiometric reaction. The second, and more difficult, part is to develop a detailed stereochemical picture of each elementary step as it occurs. In the first part one is finding the number, composition, and energies of all activated complexes in the reaction. In the second, one is trying to determine the geometry of each activated complex, not only in terms of the bonding between various atoms but also in terms of bond angles and distances. It is reasonable to expect rate laws to give a great deal of information about the first part above, especially when studied as a function of temperature, and other variables. It is not reasonable to expect information about the second part from the rate law directly, though a clever and experienced kineticist can often make shrewd inferences. However even in the determination of the sequence of elementary steps, rate laws must often be inadequate. A common experience is that several quite different mechanisms give the same rate law, at least in the usual case where only concentrations of reactants and products can be monitored. As an example, the oxidative addition of an alkyl halide to a transition metal complex may be considered: ML,
+ RX-.
RMXL,
(11
M is a transition metal in a low oxidation state, and L, is used to indicate several ligands attached to the metal. The rate law that has been found for reaction 1 in a number of cases is the simple second-order one rate = h[ML,][RX] (2) One interpretation of the rate law is the simplest one: a concerted addition of both the X and R groups to the metal. This raises immediate questions about stereochemistry. Will R and X add cis or trans to each other in the final product? What is the stereochemical result if R is a chiral group? In short, what is the geometry of the transition state of such a concerted process? As already stated, kinetic studies will not answer such questions, but suitable product identification studies will. Another mechanism exists which will also fit the observed rate law. This is the familiar s N 2 , or nucleophilic substitution reaction ML, t R X - RML,+ t XRML,'
+
X--. RMXL,
(3a)
(3b)
By definition, this mechanism must be a multistep one. Free halide ion will exist at some stage along the reaction coordinate. This can be verified by adding another ion to the solution to take the place of X- in reaction 3b. Stereochemical results are also important since step 3a will occur with inversion of configuration at €3. The Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
R.
2324
A free-radical mechanism is also completely consistent with rate law 2 ML,
+ R X - +XML,. + R.
R. t ML, RML,.
4
(48)
RML,.
+ RX-
RMXL, t R.
RML,. t Re -+ R,ML,
Step 4a is an initiation step, steps 4b and 4c are chaincarrying steps, and step 4d is a chain-terminating step. Applying the steady state treatment to [Re] and [9LM,] leads to rate = (k,hbk,/k,)”2[ML,][RX] (5) The free-radical mechanism can be probed by the addition of free-radical traps to inhibit the reaction, and by ESR studies to detect the free radical R. d i r e ~ t l y .In ~ fact good experimental evidence for various systems exists for all three mechanisms suggested above for the oxidativeaddition reaction2 The evidence is always extrakinetic, however. The rate law does not distinguish between them. Of course detection of Re or X- and measurement of their concentrations as a function of time would be powerful direct kinetic evidence. Until the detection and measurement problem is solved, the computer can do nothing to make the rate law more informative. Note that the kinetic ambiguity above resides in which elementary steps constitute the mechanism, not in the geometry of activated complexes or other detailed information. ‘While many other examples might be quoted, one more wi’ll suffice to show a more subtle difficulty. Consider the replacement of a solvent molecule from the coordination sphere of a metal ion, to be replaced by some ligand L:4 M(H,0),2t t L
-+
M(H,O),L’+ + H,O
(6)
The rate law is often found to be of the hyperbolic form
rate = k [ C ] [ ~ ] / ( 1+ h‘[L])
(7)
where C represents the ion M(H20)Z+,and L is usually present in considerable excess. Two mechanisms, at least, can be proposed to account for the rate law 7. One is a dissociation, of SN1, mechanisms in which a five-coordinated reactive intermediate exists: k,
M ( H 2 0 ) 6 2 +M 4 ( H 2 0 ) , 2 ++ H,O
(8)
k2
k3
M(H,O),” + L --+ M(H,O),L”
(9)
Assuming the stationary state for this intermediate gives the rate law
and hence the name. The rate law is indistinguishable from (10) by kinetic means: rate = k4K[C][L]/(1 -t K[L])
mechanism^.^
Composition of the Activated Complex It is usually considered that, at the very least, the constitution and electric charge of the activated complex may be found from the rate laws6Even here there are often difficulties in making a clear cut decision. The problem is that the rate of each elementary step is assumed to be given by the law of mass action in which either the concentrations or the activities of the reactants are taken to be the active masses. If activities are used, then the activity coefficient of the activated complex must also be added. This is an experimental unobservable, except as it appears in the rate law. For ionic reactants only an average activity coefficient can be found, involving usually a nonreactive partner. For neutral reactants activity coefficients could be determined in separate experiments. This is almost never done. The unpleasant situation then arises where a substance can affect the rate of a reaction indirectly by changing the activity coefficients. A linear relationship may well be found between rate and some power of the concentration of the substance. Nevertheless it seems unreasonable to put one or two molecules of the substance into the activated complex. The activity effect may be a long range one, as when one ion influences another, or it may reside in profound effect on solvent structure (and hence behavior) by a small amount of the added substance. The activated complex may have the same composition as before, but it is now formed in a different milieu. Such effects are particularly large for reactions of ions which interact strongly with their environment. An example is given by a study of the reversible hydrolysis of a bromo complex of r h ~ d i u m : ~ t
Rh(NH,),H203’ + Br-
( a - XI
M(H,O),,’
+
L k4
M(H,O),,LZ+
--L
K + M(H,O),,LZk
(11)
M(H,0),L2‘ t H 2 0
(12)
In the rate-determining step the outer sphere complex C,L rearranges to the products. There is an interchange between groups in the first and second coordination spheres The Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
(13)
Nevertheless the two mechanisms can be distinguished by auxiliary methods. Equilibrium 11 may be detected spectroscopically and K measured independently. Also at high ligand concentrations (10) gives a limiting rate constant equal to kl,whereas (13) gives a limiting rate constant of k4. The constant kl is a property only of the metal ion and does not depend on the nature of L, whereas It4 depends on the nature of L to some degree. Unfortunately it turns out that the dependence of k4 on L is often rather small as suggested by the Id, dissociative interchange, label. Reaction 6 is a special case in that the solvent is a reactant. The same problem of multiple interpretations of the rate law exist when one ligand is replaced by another in metal complexes. Rate laws such as (7) are often found and may be explained by as many as four distinct
Rh(NH3),BrZ+t H,O
which agrees with the experimental ( 7 ) . A second mechanism, called Id for dissociative interchange, postulates a rapid preequilibrium between C and L:
G. Pearson
&/dt = k , ( -~ X ) - I Z ~ X
X
(14)
X
(15) As indicated, the rate law found experimentally shows the reverse reaction to be first order, whereas equilibrium considerations demand that it be second order. The anomaly lies in the effect of increasing ionic strength on the activities of the two reactants, Rh(NH3)5H203+ and Br-. A reduction in the activity coefficients cancels the effect of increasing concentration. The reactions of highly charged complex ions with ions of opposite charge are particularly difficult to interpret as a function of changing concentrations. In addition to
2325
From Rate Laws to Reaction Mechanisms
0'
I
0.2
I
I
0.4
0.6
[Ll,
I
I
0.0
1.0
The real system will show a profile that includes all deviations from the idealized model. With the power of the computer available, the model can be improved. Activity coefficients can be inserted where known, or calculated from theories known to be accurate for various molecules and ions. Many more elementary steps can be included to see if they will influence the results. The test of an extended model will still be that of fitting the observed time-concentration data, usually for only one or two substances. An example of this procedure is given by a very detailed statistical analysis of mechanisms for the important ammonia synthesis reaction using heterogeneous catalysts.12 In this study a great many possible elementary steps were considered, including always the reverse processes. Ammonia synthesis N,(g) + u + N,(ads)
M
H,(g) t u + H,(ads)
Figure 1. Plot of observed first-order rate constant vs. ligand concentration for the reaction MO(CO)~ L = Mo(CO),L iCO.
+
N,(ads)
+ H,(ads) * N,H,(ads) + u
N,H,(ads) t H,(ads) + N,H,(ads) t u
long-range interactions, there are also ion pairs formed, as well as true complexes. The rate of the reaction Co(NH,),H,O3'
+ SO-:
-f
Co(NH,),SO,I
+ H,O
(16)
increases with increasing sulfate ion concentration. However the order is nonintegral and changes continuously as the concentration changes. Neutral molecules in nonpolar solvents are not free of kinetic ambiguities in the interpretation of rate laws. Figure 1 shows the effect of increasing concentrations of several reagents, L, on the rate of the reactiong Mo(CO), t L - Mo(CO),L
+ CO
(17)
There is a rate which is independent of the concentration of L, presumably a slow dissociation of the molybdenum hexacarbonyl to give MO(CO)~.This then reacts rapidly with L to give the product. There is also a modest rate contribution which is proportional to [L]. This suggests a new activated complex which contains L in addition to MO(CO)~.However in some cases the constant of proportionality (the slope of the line in Figure 1)is so small that the role of L becomes indeterminate. There are even cases where addition of L slows down reactions very similar to (17).1° If complexes are formed containing L, they still appear to react by a dissociation path and bonding participation by the incoming ligand is minimal. In effect, L has taken the place of the solvent in the second coordination shell.ll Changing the concentration of L may be considered as a change in solvent. Specifying the participation of the solvent in the activated complex has always been difficult. Incompleteness of t h e Model The previous examples serve as reminders that rate equations are based on idealized models. A model, it may be recalled, has some, but not all, of the attributes of the real thing. There can be many deviations from idealized kinetic models, including departures from ideal behavior expressed as activity coefficients. A second common error is the failure to include side reactions of minor chemical consequence but possible kinetic significance. Few chemical reactions are clean or go with 100% yields for one set of products. Unfortunately we still cannot predict with any confidence what reactions will occur when we mix chemical reagents. The computer can solve the idealized model with great exactness to give idealized concentration-time profiles.
N,H,(ads) ",(ads) N,(ads) N(ads)
+ H,(ads) * 2NH,(ads) + NH,(g)
+u
+ u + 2N(ads) + H,(ads) * ",(ads)
t u
t H,(ads) + ",(ads) t H(ads) etc. where u represents an adsorption site. Rate equations were then deduced using the Langmuir adsorption isotherms and by assuming one or another of the elementary steps to be rate determining. Purely empirical rate equations were used as well. A very large number of rate equations were tested to see how well they reproduced experimental results for ammonia production as a function of time and temperature. The results were somewhat surprising in that a number of models fit the data about equally well. Some typical results are shown in eq 18-20.
",(ads)
6.68% error
.
-
0.08% error
-
-
5.49%error The mean error is given. The C1, Cz etc. are rate constants of the Arrhenius form, and K is the known equilibrium constant. Most rate equations containing only three rate constants had errors of about 6%, but one equation, (19), was superior to all others in having an average error of only 0.08% for the calculated ammonia concentrations. This equation would be superior in applications of a chemical engineering type, to control times, yields, and so on. Nevertheless the authors concluded, quite correctly, that the equation had no mechanistic significance. This is, the particular mechanism that led to eq 19 was no more likely to be correct than other mechanisms which led to errors of 6%. In fact the mechanism associated with (19) postulates that the reaction of adsorbed NzHzwith adsorbed The Journal of Physical Chemktty, Vol. 81, No. 25, 1977
2326
R.
Hzis rate determining. It is generally considered that the chemisorption of Nz is rate determining in the ammonia synthesis.13 This leads to rate eq 18 with a large error. Adding a third constant, C3,to (18) leaves the error almost unchanged.
Some Useful Applications The discussion up to this point has stressed features which limit the usefulness of the computer in deducing mechanisms. The remarks have been meant to be cautionary and not to deny the many cases where the computer can be invaluable. As stressed by others, one area is that of very complex systems where many elementary reactions occur.14 However even systems with only a few elementary processes can often benefit from numerical solution of the rate equations. Such systems have been treated in the past by approximations such as the rate-determining step or stationary states. However such approximations have limi t a t i o n ~ .Typical ~~ cases are where several rates are of similar magnitude and where the concentrations of intermediates become appreciable. If it is possible to measure these concentrations, then a full solution is certainly needed. An example is given in a study of carbon monoxide exchange in nickel carbonyl using isotopic tracers:16 Ni(C160), t C'*O
-
Ni(C160),(C1BO),-,
(21)
In hexane solution this exchange goes by a series of steps A+B+ C+ De E
(22)
gives strong support to the assumed mechanism (24). Incidentally, the value of k,, is fo.und to be the same as the rate constant for substitution of CO by phosphines and other ligands, as of course it should be. Unexpectedly, the results of a similar study in the gas phase gives quite different results.I6 The gross rate of exchange is found to be given by eq 24 with k,, about twice as large as for hexane. The concentrations of A, B, C, D, and E are not given by (25) at all. Instead a very rapid scrambling occurs between all Ni(C0)4molecules: xNi(C160), t yNi(C180)4+ (x t y)Ni(C'60),(C180),-,
Ni(CO), t CO
w,= (4-4!r)!r! F'(1 - F)4-r Another example of intermediate complexity is provided by a study of the reduction of carbonyl compounds such as pyruvic acid and its derivatives by 1,5-dihydroflavines.17 These reactions are of biochemical importance. Scheme I shows the detailed mechanism proposed by Williams and Bruice. The reaction is monitored primarily by following the absorbance at 443 nm where the oxidized flavin (Flax) has an absorption maximum and the other components Scheme I
F H3
FH3
= k,,[Ni(CO),]
CH3
4
c
k.,
y I
-
I
0 f CH3CCOR II
H
[:lGi H3
3
O
! L
OH
t CH3CHCOR I
0
c H3
(25)
and so on. Each rate constant, k12,etc., is equal to he, times a statistical factor which changes with time. Each factor depends on the ratio of Cl6O to Cl'O in the free carbon monoxide. This can be calculated as an explicit function of time, but only if he, is known. As it happens, the set of equations (25) can be integrated to give closed solutions. However these are so complicated as to be of little use. Instead numerical integration using various assumed values of he, is far more convenient. Analytically the exchange is followed by infrared spectroscopy in the 2000-2050-~m-~ region. This is the C-0 stretch region for the nickel carbonyls. Species B, C, and D have unique bands at 2007.1, 2017.4, and 2028.8 cm-l, respectively. The bands for A and E at 2046 and 2001 cm-l cannot be used because of overlaps. By assuming a value of k,,, the expected values of B, C, and D as a function of time can be calculated and compared with the experimental results. The k,, can be varied to give the best overall fit. In principle, any one concentration could be used to determine k,,, but the ability to fit all three concentrations is much more reassuring and The Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
CH3CCOR
(24)
Isotope effects on the rate are ignored as being very small. The rate equations governing (23) are of the form d [ A I l d t = -k12[AI f k21[BI d [ B l l d t = k,z[Al + k,,[CI - kz1CBI- h23DI
0 I/
(23)
Thus the gross rate of exchange independent of isotope labeling is given by
rate
(26)
If F is the fraction of bound CO which is Cl'O at any time, then W, is the mole fraction of Ni(C0)4species containing r Cl80 groups.
where A is Ni(C160)4,B is Ni(C160)3(C180), and so on. The initial assumption is that only one CO molecule is exchanged at a time and that the mechanism of exchange is a rate-controlling dissociation: Ni(CO),%
G. Pearson
y
3
(b)
2327
From Rate Laws to Reaction Mechanisms 0.7
CH3CObOH (pH 4.90)
CH,CO$OH 0
B
(pH 3.3)
0.5 0, V
5b 0.3 a
a
0.1 0
200
400
0
400
200
600 800 Time (min)
600
SO0
1000
1203
Time (mid
Time (rnin)
1000
0
3 07
613
920
1227
1533
Time (mid
Figure 2. Examples of computer fittlngs of Scheme I1 to various concentrations as a function of time. The circles are the experimental values for [FI,,]. Reprinted with permission from R. F. Williams and T. C. Bruice, J. Am. Chem. SOC.,98, 7752 (1976).
have very small absorbance. There is an initial burst of formation of Fl,,, followed by a slow further increase in concentration until reaction is complete. This behavior is explained by a competition in which the reactants, reduced flavin, H2Fl or HF1-, and the pyruvic substrate react to form a carbinolamine CA. This then dehydrates to an imine, Im, in a reversible step. The carbinolamine by two paths can then regenerate the reduced flavin which reacts with the substrate. Kinetically Scheme I can be simplified to Scheme 11. There are eight adjustable constants in Scheme I1 and furthermore, these are functions of the pH since several of the species can protonate or deprotonate. In a buffer this complication can be ignored for any one kinetic run. The job of the computer was to find the best values of all eight constants to give a fit to the measured absorbance a t 443 nm as a function of time.I* This was done with a claimed accuracy of &5%. At the same time the computer provides the concentration of FlH2,Fl,,, CA, Im, and CT as a function of time. These results are shown in Figure 2 for several different systems. CT represents a charge transfer complex made of two molecules of half-oxidized flavin. This is similar to the more familiar quinhydrone complex. In view of the remarks made earlier on the ammonia Scheme I1 kl
4
k-1
k-2
FlH, t )C=O -.-.-, CA
FlH, Fl,, CT
+ )C=O +
k5
k-s
4 -*
H
Fl,,
k4
CA -+ CT
FlH,
Im
+ Fl,,
+ )C-OH
+ )C=O
(29) (30) (31)
synthesis, it seems absurd to claim vindication for the mechanism shown in Scheme I and to calculate eight rate constants from a single set of observables. However this conclusion must be tempered by a consideration of a great deal of other information available, both on the specific systems studied and on many related systems. For example, the concentration of dihydroflavin was independently determined in a number of spot checks, by adding formaldehyde which reacts rapidly with HzFl or HF1-. The concentration of imine, Im, at selected times, was determined by trapping experiments. These measurements provided verification of the detailed predictions of Figure 2. The carbinolamine, CA, and the charge transfer complex, CT, could not be assayed. However, scans of the total spectrum at selected times provided evidence for their existence. The chief evidence was largely by analogy, that is, the knowledge that flavins and carbonyl compounds should form carbinolamines, which will dehydrate to imines, and that a mole of reduced flavin and a mole of oxidized flavin will form a charge transfer complex. Many other observations were also available. Reference must be made to the elegant and detailed analysis by Williams and Bruicel’ to appreciate how a mechanism such as Scheme I1 can be made plausible (if not proved!), even when the data are necessarily incomplete. In particular, the computer simulated kinetics played a part without which the final conclusions could not have been drawn.
Conclusion To sum up what has been said, the computer is obviously a powerful new tool to be added to those used by chemical kineticists. Still the basic difficulty of verifying reaction mechanisms must remain because of the relationship of mechanism to theory rather than experiment. The very power and versatility of the computer can sometimes lead to its misuse. There is the temptation to allow it to substitute for the reasoning power of the human The Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
R. G. Pearson
2328 a
begin to occur a t longer times. An old-fashioned kineticist would use the equation
,
t B3
M O L E S = B,e-'Z'
kt
In
T
and relate the volumes of gas at t = 0, t = t , and t = to a and a - x. A plot of In (a - x) against time should then give a straight line, if first order, and the slope would give the rate constant. Figure 3b shows this plot. It is clear that the reaction is first order throughout and that some experimental catastrophe occurred in the middle of the experiment! Good advice to any beginner in kinetics would be to do the calculations the hard way a t first. After the results make sense, let the computer do the work from then on. The computer is not a substitute for thinking, but a tool for performing tedious and time consuming tasks.
\
i*
\*. 5
*\ IO
15 20 TIME
I
I
25
30
L f0
(33)
References and Notes
// 5
=
Acknowledgment. Grateful acknowledgment is made to the donors of the Petroleum Research Fund, admintered by the American Chemical Society, for partial support of this research.
/
0
(A)
!
15
20
25
TIME
Figure 3. (a) Computer fit of moles of gas left as a function of time using the equation shown. (b) Same data as in 3(a) plotted in conventional manner.
brain. A simple example, taken from real life in the laboratory may serve as an illustration. The reaction under study was the hydrogenation of ethylene using a rhodium complex as a homogeneous catalyst. The moles of gas remaining (a mixture of H2 and N,) were measured at various times. The proposed mechanism leads to a psuedo-first-order rate law. The following equation was used and the computer asked to find the best values of B1,B2,and B3 to fit the data: moles of gas = Ble-BZf + B3
(32)
The results are shown plotted in Figure 3a. It would appear that the first-order rate law is obeyed initially, but that the true rate law is more complex and that deviations
The Journal of Physical Chemlstry, Vffl. 8 7, No. 25, 7977
(1) For a discussion of the information which has proved useful in mechanism studies see A. A. Frost and R. G. Pearson, "Kinetics and Mechanism", 2nd ed, Wiley, New York, N.Y., 1961; R. W. Hoffman, "Auklrung von Reaktiinsmechanismen", G. Thieme Verhg, Stuttgart, 1976. (2) For a review of oxidative addition see R. G. Pearson, "Symmetry Rules for Chemical Reactions", Wiley-Interscience, New York, N.Y., 1976, Chapter 5. (3) P. J. Krusic, P. J. Fagan, and J. San Filippo, Jr., J. Am. Chem. SOC., 99, 250 (1977). (4) See F. Basolo and R. G. Pearson, "Mechanisms of Inorganic Reactions", 2nd ed, Wiley, New York, N.Y., 1967, Chapter 3; C. H. Langford and H. B. Gray, "Ligand Substition Processes", W. A. Benjamin, New York, N.Y., 1965. (5) For an example see M. Maestri, F. Bolletta, N. Serpone, L. Maggi, and V. Bolzani, Inorg. Chem., 15, 2048 (1976). (6) A useful discussion of how this and other deductions can be made is given by J. 0. Edwards, E. F. Greene, and J. Ross, J. Chem. Ed., 45, 381 (1966). (7) A. B. Lamb, J. Am. Chem. Soc., 61, 699 (1939). (8) See ref 4 for examples of ionic interaction in rate studies. (9) R. J. Angelici and J. R. Graham, J. Am. Chem. Soc., 88,3658 (1966). (10) R. J. Angelici and F. Basolo, Inorg. Chem., 2, 728 (1963). (11) J. E. Pardue and G. R. Dobson, Inorg. Chim. Acta, 20, 207 (1976). (12) G. Buzzi Ferraris, G. Donati, F. Rejna, and S.Carrl, Chem. Eng. Sci., 29, 1621 (1974). (13) M. Boudart in "Physical Chemistry, an Advanced Treatise", Vol. 7, H. Eyring, D. Henderson, and W. Jost, Ed., Academic Press, New York, N.Y., 1975, Chapter 7. (14) For example, D. Edelson, J . Chem. Ed., 52, 642 (1975). (15) R. M. Noyes, Prog. React. Kinet., 2,339 (1964); K. Frei and H. Gunthard, Helv. Chim. Acta, 50, 1294 (1967). (16) J. P. Day, F. Basolo, and R. G. Pearson, J . Am. Chem. SOC.,90, 6927 (1968). (17) R. F. Williams and T. C. Bruice, J. Am. Chem. SOC.,98, 752 (1976). (18) The kinetic simulation in ref 17 was carrled out using an analog computer. This inefficient and cumbersome method has since been replaced by a digital minicomputer package. D. Shindell and C. Magagnosc, "Proceedings of the Digital Equipment Computer Users Society", Vol. 3, No. 2, 1977 (paper given at the DECUS meeting in Las Vegas, Nev., Dec 8, 1976).
