John 0. Edwards, Edward F. Greene, Brown University
Providence, Rhode Island and John Ross Massachusetts Institute of Technology Cambridge
1
II I
From Stoichiometry and Rate Law to Mechanism
The postulation of a suitable mechanism for a complex chemical reaction often seems to be some thing of a mystery to an undergraduate student. Even students who have had some kinetics in their course work may not understand how a chemist can write down a series of chemical steps and then suggest that this mechanism is a reasonable one. I n setting up mechanisms chemists use working rules, based on previous chemical experience, to serve as guidelines for making deductions from the observed stoichiometry and rate law. These rules are known to chemists and are frequently used, but they have not been collected in one place in the chemical literature. We wish to point out the existence of these working rules, to discuss some of them in sufficient detail so that their nature and usefulness is clear, and to refer to some places in the literature where some of these and related rules are discussed. We mention the rules in turn, we give chemical examples of each, and we try to show how these rules are consistent with chemical experience. The working rules which are presented herein, and other ones in the literature, allow a chemist t o judge the reasonableness of his mechanism. Often they tell what mechanisms are not reasonable in relation to the available data, they do not tell us what the mechanism really may be. Thus a chemist must use his knowledge of chemical systems and his imagination when he postulates a mechanism. And he will want to test his mechanisms by independent techniques, such as for example an isotope tracer experiment. Most chemical reactions take place by a sequence of steps rather than by a single step. A proposed mechanism for a chemical reaction consists of a number of elementary steps, each being defined by a stoichiometric equation and each representing the best guess of the chemical species involved in that part of the reaction. The number of reactants in each step is called the molecularity of that step. The deduction of the expected rate equations from mechanisms is almost always simplified by proposing a mechanism with either (1) a single rate determining step or (2) a sequence of steps in which the rate of change of various intermediates is assumed to be constant for some time during the reaction (the stationary state hypothesis). I n this article we discuss mostly the first of these two common cases, and unless we state otherwise the working rules apply only to that case. For a mechanism which has more than one elementary step, with one of these being rate determining, all steps other than that one may be considered to be in chemical equilibrium. We shall return to this point.
For purposes of identification and reference we give each working rule a Roman numeral. For most equations we use equal signs in order to avoid implication as to rate or equilibrium; in a few cases, we draw a single arrow to denote rate and double arrows to indicate a rapid equilibrium. The Transition State
For each step in a mechanism the molecular configuration a t the point of highest energy is termed the transition state. Rule I. The constitution and electric charge of the transition state for the rate determining step may be obtained from the empirical rate law. For the oxidation of iodide ion by arsenic acid (1) HzAs04+ 31- + 2H+ = HaAsOs+ 4HzO (1)
+
the observed rate law in the forward direction is Rate = klr[HAsOJ [I-] [H+]
(2)
If we assume a mechanism with one rate determining step for this reaction, the transition state contains an arsenic atom, an iodine atom, and at least two oxygen atoms. (Further discussion of the number of hydrogen atoms and other oxygen atoms in this transition state appears after the next working rule.) The charge on the transition state is zero since a mono-negative ion, a mono-positive ion, and a neutral molecule are being brought together. The concentrations of some chemical species are not easily varied over a range wide enough to yield a kinetic order in those species. A common example is the solvent; usually it is difficult to vary the concentration of water in aqueous solutions without changing the nature of the medium significantly. Rule II. The number of molecules contributed to the transition state by a species whose concentration cannot be experimentally varied (e.g., the solvent) i s not known. Going back to our first example we can see that we cannot say how many times the molecular equivalent of H,O appears in the transition state. Therefore we can write its constitution as AsI02:Hp,0, where x is an i n d e terminate number; the notatlon is intended to imply ignorance of the molecular structure of the transition state but not that of the solvent. Other cases where indeterminacy arises are mnltiphase reactions. For example, mercuration of benzene in water, under conditions such that the aqueous concentration of benzene is constant because some benzene is present as a second phase, is a situation where the order of the benzene concentration in the rate expression is indeterminate (2). For very small amounts of benzene where there is no Volume 45, Number 6, June 1968
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381
second phase, the order in benzene can be obtained by standard techniques. The empirical rate law does not give information about the geometry or structure of the transition state. Some such information is attainable, however, from other sources such as rate measurements with tracer isotopes, stereochemical data, and relative rates for r e lated processes.
hut if step (10) is the slow one, it is
Rapid Equilibria
A majority of known elementary steps are bimolecular, with the balance being unimolecular or termolecular. Steps of molecularity greater than three are not usually postulated, and a t the present time we have no evidence for such molecularity. For this reason, our discussion below is based on the assumption that only steps with molecularities 1, 2, or 3 need be considered. Therefore, any reaction of complicated stoichiometry, such as the oxidation of bromide ion by bromate ion in aqueous acid (5,4) BrOl- + 5Br6HC = 3Br~+ 3H*O (3)
+
is very likely to involve a multiplicity of steps. The mechanism must also involve intermediate species that are formed and used up in these steps. Under some conditions the rate law for eqn. (3) is Rate = k,[BrOs-I [Br-] [H+I2
(4)
so that the constitution of the transition state for the rate determining step is Br202.H,.0,. A preferred alternative to a tetramolecular process is a mechanism involving one or more rapid equilibria of the type H + BrOa- HBrOs (5) before the rate determining step. Here HBrOa is an intermediate. Rule I I I . When the overall order of the reaction i s greater than three, the mechanism probably has one or more equilibria and intermediates prior to the rate determining step. There is, in fact, one piece of evidence to substantiate the postulation of a rapid equilibrium in this reaction. The rate of reaction is more rapid in D20 than in HzO (4, 5), and this is explicable because D30+ is a stronger acid in D20than HaO+ is in H,O (5). If proton transfer occurred in the rate determining step, then the opposite rate effect (i.e., k , > k,) would have been observed for, according to the normal kinetic isotope effect, transfer of a lighter isotope occurs more rapidly than transfer of a heavier isotope (5,O). The oxidation of arsenious acid by tri-iodide ion (the reverse of eqn. (1)) has a rate law with concentrations in the denominator. Rule IV. Inverse orders arise from rapid equilibria prior to the rate determining step. For this oxidation of arsenious acid the observed rate law is
+
*
and a proposed mechanism is HaAsOa F? HxAsOl- + H+ 4- H,O 21- + H*OI+
+
382
/
The rate determining step is taken to be either eqn. (9) or eqn. (10) (with the arrows in parentheses for the latter case) and the remaining steps are in equilibrium by comparison. If step (9) is the slow one, the rate equation is
*
Journal of Chemical Education
(7) (8)
According to our working rules I and 11,the constitution of the transition states for these two steps are indistinguishable (they may differ only in the number of times they contain the solvent, water molecules). With the use of the equilibrium steps (7) and (8) we can easily convert eqn. (12) to the observed rate law, eqn. (6). We can do the same for eqn. (13) using equilibria (71, (8), and (9). Thus we see that inverse orders are introduced by equilibrium steps prior to the rate determining step. Concentrations appearing in the denominator of the observed rate law cancel those in the numerator for the purpose of obtaining the constitution of the transition state. This follows from the fact that in the use of equilibrium expressions conservation of atoms and charge has been assumed. According to the principle of microscopic reversibility transition probabilities for a given process for the forward and reverse direction are equal. Hence a postulated mechanism for a reaction in one direction must also serve for that reaction in the reverse direction. As a corollary, the rate determining step, and therefore the transition state for that step, must be the samc from both directions.' As an example we note that for the oxidation of iodide ion by arsenic acid, eqn. (I), and its reverse reaction, the measured rate laws, eqns. (2) and (6), do give the same constitution for the transition state. With this, we are assured that the usual relation between the observed rate coefficients in the forward, k,, and reverse lc., directions and the equilibrium constant, K, for the overall reaction holds K = k rlkr (14) Intermediates
In most chemical reactions, transient species of variable stability are formed. These chemical species, which exist for times longer than a few vibrational periods and which may react a t differentrates with different species (i.e., may be selective), are called intermediate^.^ Some intermediates are detectable, often by a spectral technique; some may be isolated (e.g., hydroperoxides from hydrocarbon antoxidation); and some remain hypothetical compounds. Intermediates are an important part of many mechanisms. However, a mechanism is a set of postulates and only that; it is by no means unique. For instance, the intermediates H201+ and HaAsOa- are written as
' There are some cases where this simple use of the principle of microscopic reversibility is not sufficient. For an interesting discussion of such cases, the reader is referred to the recent paper of Burwell and Pearson (7). % W erefer the reader to recent discussions on the evidence for intermediates and the nature of intermediates (8-10).
the active osidant and reductant, respectively, in the series of steps given by eqns. (7) to (11). Otherpossible combinations are 01- and HaAs03+,01- and H2As02+, HOI and HBAsOa,etc. Each pair of reactants should, however, be consistent with the constitution of the transition state and with the types of equilibria which are suspected to be rapid in comparison with the slow step. Other working rules concerning equilibria and intermediates are useful for reactions with a large number of reactant and product molecules. Consider the aqueous decomposition of the tetraperoxychromate (V) ion (11) 4CrOaa-
+ 4HC = 4 C r O F + 70%+ 2H20
(15)
which proceeds with the simple rate law Rate = klsr[CrOs8-I LH+l
(16)
Once the rate determining step, which involves only one CrOs3- anion and one proton, occurs, a sequence of further rapid steps ensues. I n the course of these steps three more CrOe3- anions and three more protons get involved and four chromate ions, seven oxygen mole c u l e ~ and , ~ two water molecules are formed. Clearly we must postulate a number of different intermediate species in order to write down a set of bimolecular (or termolecular) equations leading from reactants to products. The restriction to low molecularities in these steps is in keeping with the fact that molecularities greater than three for elementary steps are not presently known. As a direct consequence of this restriction, the writing down of a mechanism for this reaction involves a number of equilibria and intermediates; working rules concerning the presence and ordering of these equilibria and intermediates in a mechanistic sequence are available. Often, as in eqn. (16), the stoichiometric coefficients are larger than the orders in the same species. Rule V. If a stoichiometric coeficient on the left hand side (reactants) exceeds the species order, there are one or more intermediates after the rate determining step. Another clear example of this working rule is found in the oxidation of aniline to nitrosobenzene by peroxyacetic acid (12)
+
+ H20
CeHsNH2+ 2CHaCOsH = C6H5N0 2CH8C02H
(18)
which proceeds with the experimental rate law Rate = klai[C6H5NHd[CH&OaH]
(19)
We note that the order in peroxyacetic acid is one, while its stoichiometric coefficient is two. The probable mechanism is CHsCOIH CH,CO,H
--
+ CaH6NH2
+ CsHsNHOH
CHsC02H CH&O,H
+ CBBNHOH
(20)
+ CaH8NO + Hz0
(21)
with the first step being slower than the second and with N-phenylhydroxylamine being the intermediate formed a Occasionally a stoichiomet,rie equation is written with nonintegral coefficients. For example we could write the stoichiometry of the tetraperoxychramete decomposition as
2CrOs'-
+ 2H+ = 2Cr0.'- + '/zOz + H20
(17)
This satisfies conservation of maw and also any thermodynamic requirements. I t is not, however, appropriate for use in a mechanism since every step in a mechanism is a stoichiometric r e p resentation of molecular interactions.
in the rate determining step. In a separate rate experiment beginning with a sample of N-phenylhydroxylamine (It), the peroxyacetic acid rapidly oxidized the hydrosylamine to nitrosobenzene; this is in accord with the proposed mechanism. Occasionally fractional orders are observed in empirical rate laws. Non-integral orders in homogeneous reactions indicate that an important part of the mechanism is the splitting of a molecule. Rule V I . Whenever a rate law contains non-integral orders, there are intermediates present i n the reaction s e pence. Reactions with fractional orders do not necessarily have to proceed by radical or chain mechanisms. A simple example is a reaction of a substrate S subject to specific acid catalysis for which the source of protons is a weak acid such as acetic acid. The rate law is and the observed rate is seen to be proportional to the one-half power of the acetic acid concentration. A possible mechanism is
-
S+H+F?HS+ HS+
Products
so that the concentration of protons is proportional to the square root of the concentration of acetic acid. A more familiar type of reaction with non-integral orders is the free radical chain mechanism. For example, the oxidation of 2-propanol by peroxydisulfate (13) (CH&CHOH
-
+ SIOsZ-
+ 2HSO4-
(CHZ)d2O
(26)
has been found to proceed with the rate law Rate = kper[S20,P-][(CH&CHOH]' 1 1
--
(27)
A mechanism consistent with this law is
s>osp- 2sop-
so4-+ (CHI)ICHOH
+ S202(CH&COH + SO,--
(CH,),COH
(28)
+ (CH$)~COH (29) (CH&CO + HSOI- + SO4- (30) HSOI-
(CH,),CO
+ HS0.i
(31)
Here we have an example of a mechanism without a rate determining step; the assumption that all steps but one are rapid equilibria is not valid in this example. I n such cases, the stationary state hypothesis for intermediates is usually invoked. Two intermediate free radicals, SO4- and (CH&COH, are postulated in this mechanism, and experiments with inhibitors have given evidence for both radicals ( I S ) . Oxygen gas inhibits the chain reaction by converting the organic radical to a less reactive form. I n similar fashion, ally1 acetate inhibits the reaction, probably by reacting with the sulfate radical ion SO4-. The problem of the prediction of the existence of intermediates is usually less difficult than the problem of determining their chemical nature. There are, however, two generalizations which may help us to establish the latter. Consider the case of a reaction in which there is a single intermediate between the transition state and products. The constitution of this intermediate can be related to the constitution of the transition state from which it is derived. Volume 45, Number 6, June 1968
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383
Rule VII. The first intermediate after the transition state of an elementary step does not contain atoms that are not present in the transition state. An intermediate that is formed directly from a transition state can have either the same constitution as the transition state or a smaller number of those atoms that were present. For example, in the bromate-bromide reaction mentioned above, the first intermediate probably has the constitution Br202. I n the decomposition of Caro's acid (14) in aqueous base, the stoichiometry is and the reaction proceeds according to the rate law The first intermediate has been postulated to be HOOOS03- formed by loss of sulfate ion from the transition state of constitution HS20103- (no further solvation is assumed here). Similar conclusions concerning the first intermediate after the transition state can be drawn in the oxidation of iodide ion by arsenic acid and in the decomposition of the tetraperoxychromate ion. The other generalization concerning the constitution of intermediates is imprecise (albeit interesting and helpful), and hence must be applied with caution. Rule VIII. Thefirst guess of the structure of a n intermediate should be based n our knowledge of the structure of stable species. For example, in the reaction of the octahedral tellurate ion HsTeOe- with glycols to form the glycol-tellurate chelate complex, which also has octahedral coordination about tellurium, there is catalysis of substitution by hydroxide ion (15). The stoichiometry, relative rates with diverse glycols, and rate law suggest that TeOn2-is an intermediate. Although tetrahedral tellurates are not known, this intermediate is quite reasonable in view of the known existence and stability of sulfate ion Sod2- and selenate ion SeO? formed from elements in the same family of the periodic table as tellurium. Some other intermediates, which have been postulated in various reactions and which are chemically reasonable, include Br20zof probable structure
and HSOs- of postulated structure 1-
The Br20nstructure is related to bromate ion in that an oxide ion in BrOa- is replaced by a bromide ion. The HSOs- structure is reasonable as an intermediate since oxygen atoms are known to combine by forming two single covalent bonds and since other family members (sulfur, selenium, and tellurium) form stable compounds with homatomic chains. The intermediate formed by addition of peroxyanions to substituted olefins (16) during alkaline epoxidation has the suggested structure 384
/
Iournal of Chemical Educotion
H
R C
ROO-C
where the negative charge is dispersed on the oxygen atom as in an enolate ion. A caution which must be remembered in the application of this generalization is that a large amount of energy introduced into the reactants (by means of photochemical activation, etc.) may produce intermediates which are very unlike ground state molecules. Also reactions with higher activation energies are likely to have more reactive intermediates. Other Working Rules
The rules given here are by no means all that could he listed. Catalysts, sensitizers, and inhibitors influence rates in ways which are subject to generalization. For example, an inhibitor often acts by intercepting an intermediate in a chain reaction, a sensitizer by initiating a chain sequence which involves intermediates. The problems of mechanism for catalytic reactions in homogeneous systems have been considered by King (17). Westheimer (18)has published an interesting series of working rules for the treatment of chromate oxidation mechanisms, and these rules have considerable generality. I n the main, they deal with reactive intermediates and their probable reactions. Taube (19) has discussed the place of kinetics in the undergraduate inorganic curriculum. His article and that of IGng (17) contain discussions of some of the working rules concerning mechanisms, along with clear examples. Tauhe also has shown, in the course of his classical work on substitution reactions (20) and electron transfer reactions (21) of octahedral complexes, how the geometry of transition states may be inferred for some reactions. One type of mechanism which we do not discuss explicitly here is a reaction sequence with two or more steps of about the same rate. I n such cases, the kineticist looks for variations of experimental conditions such as pH, concentrations, and temperature which may change the rate of one step relative to the other and thus produce the simpler situation we discuss here. This problem is often found in reactions of carbonyl compounds (9),in electron transfer reactions of oxycations (22) and in the oxidation-reduction reactions of oxyanions ($3). Some of the relevant conclusions about possible mechanisms have been discussed in these references. Summary
Some of the working rules used by chemists when they go from stoichiometry and a rate law to a postulated mechanism are presented. I n addition a list of relevant references is given. The usefulness of such working rules is twofold: they help to clarify what type of mechanism is consistent with, and appropriate to, the results of experiments, and they allow the chemist to consider certain mechanisms as unreasonable.
Literature Cited ( 1 ) WILSON,N. N., AND Dm~msoN,R. G., J. Am. C h a . Sac., 59, 1358 (1937). R. M., KLAPPROTH, W.,AND WE~TREIMER, F. ( 2 ) SCHRAMM, H., J. Phvs. Coll. Chem., 55, 843 (1951). (.3.) SKRARAL. k.AND WEBER~SCH. S. R.. Manatah... 36.. 237 (1913): ' ( 4 ) ( a ) HOERING,T. C., BUTLER,R. C., AND MCDONALD, H. O., J. Am. C h a . Sac., 7 8 , 4829 (1956); ( b ) HoERlNG, T. C., ISHIMORI, F.T., AND MCDONALD, H. O., J. Am. C h a . Soc., 80, 3867 (1958); ( c ) ANBAR,M., AND G ~ M A N , S., J. Am. C h a . Sac., 8 3 , 4741 (1961). ( 5 ) ( a ) WlBERG, K., C h a . Revs., 55, 7 2 1 (1955); ( b ) SWAIN, C. G., AND BADER,R. F. W., Tetrahedmn, . 10, 182,~ 200 . (i96oj. (6) (a)WESTHEIMER, F. H., C h a . R m . , 61, 265 (1961); ( b ) BELL, R. P., "The Proton in Chemistry," Cornell Univ. Press. Ithaea. 1959. OD. 183-214. (7) BURWELL, R. L., JR.,AND PEARSON, R. G., J.P h p . Chem., 7 0 , 3 0 0 (1966). ( 8 ) BENDER,M. L., in "Technique of Organic Chemistry" (Editor: WEISSBERQER, A,) 2nd Ed. (revised), Vol. 8, Part 2. Interscience Publishers (division of John Wiley & Sons, Inc.) New York, 1963. D. 1427. JENCKB,W. P., in "Progress in Physical Organic Chemistry" (Edilors: COEEN,S. G., STRE~WIESEH, A. JR., AND T m , R. W.) Vol. 2 , Interscience Publishers (division of John Wiley & Sans, Inc.) New York, 1964, pp. 63-128. (A discussion of carbonyl group reactions and the evidence for tetrahedral intermediates.) For related problems involving tetrahedral csrbon intermediates, see also: ( a ) JENCKS, W. P., AND GILCHRIST, M., J. Am. Chem. Soc.. 86. 5616 11964): ., i,b.) KIRBY.A. J.. AND JENCK~, W. P., J. ~ m chkm. . Sac., 8 7 , 3217'(1965j; ( c ) REIMANN, J. E.. AND JENCKS, W. P., J. Am. C h a . Soc. 8 8 , 3973 (1966). , S., J. Am. C h a . Sac., 7 7 , 334 (1955); see ( 1 0 ) H M M ~ N DG. also the discussion in LANQFORD, C. H., AND GRAY,H. B., .A.
"Ligand Substitution Processes," W. A. Benjamin, Inc., New York, 1965, pp. 12-14. QUANE,D., AND EARLEY,J. E., J. Am. Chem. Soc., 87, 3823 119fi51. ~..,.. I B N L R A ~ K. , M., AND EDWARDS, J. 0.)J. Am. C h a . Sac., 84, 763 (1962). BALL,D. L., CRUTCHFIELD, M. M., AND EDWARDS, J. O., J. 079.Chem., 25, 1599 (1960). BALL,D. L., AND EDWARDS, J. O., J. Am. Chem. Sac., 7 8 , 1125 (1956). . . ELLISON,H. R., EDWARDS, J. O., AND NYBERG,L.,J. Am. C h a . Sac., 84, 1824 (1962). (a) HOUSE,H. O., AND Ro, R. S., J. Am. C h a . Soe., 80, N , E., SINQER,L., AND 2428 (1958); ( b ) Z ~ E R M A H. THYAGARAJAN, B. S., J. Am. Chem. Soc., 81, 108 (1959); ( c ) HOUSE,H. O., "Modern Synthetic Reactions," W. A. BENJAMIN, Inc., New York, 1965, pp. 116-118; and ( d ) YANG,N. C., AND FINNEGAN, R. A,, J. Am. C h a . SOC.,8 0 , 5845 (1958). KING. E. L.. in "C&t&l~sis."V01. 2. Part 2 (Editoc EMLET, P; H.) ~ e i n & l d ' ~ u b l i s h i Co., n ~ ~ e York, w 1955, pp. 3 3 7 4 5 6 . ( a ) WATANABE, W., AND WESTHEIMER, F. H., J. C h a . F . H.. Chem. Phvs.. - . 7 1.. 61 (1949): ( b ) WESTHEIMER. RWS., 45, 4 1 9 4 5 1 i i g 4 9 j . TAUBE,H., J. Chem. Educ., 36, 451 (1959). TAUBE,H., Chem. Revs., 50, 69 (1952). c.f. ( a ) TAUBE,H., in "Advances in Inorganic Chemistry and Radiochemistry" (Editors: EMELEUS,H. J., AND SHARPE,A. G.) Vol. 1, Academic Press, New York, 1959,,p. 1.; ( b ) TAUBE,H., in "Topics in Modern Inorgaulc Chemistry," Vol. 6 (Editor: MILLIQAN, W. 0.) R. A. Welch Foundation Conf., Houston, 1963, pp. 7-13. (a) HAM,A,, Inorg. Chem., 5 , 2081 (1965); ( b ) NEWTON, T. W., AND BAKER,F. B., to appear in "Advances in Chemistry Series," American Chemical Society, Washington, D. C., 1967. EDWARDS, J. O., Chem. Revs., 50, 4 5 5 4 8 2 (1952).
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