J . Chem. SOC., Faraday Trans. 1, 1982, 78, 1579-1590 Application of the Marcus Relation to Concerted Proton Transfers BY W. JOHN ALBERY Department of Chemistry, Imperial College of Science and Technology, South Kensington, London SW7 2AY Received 7th July, 198 1 A model, based on the Marcus theory, is presented for the ket-no1 tautomerism of acetone. With one adjustable kinetic parameter, the model explains the rate constants and isotope effects for the six different catalytic routes involving H,O+, OH-, H,O, CH,COOH, CH,COO- and the third-order term with both CH,COOH and CH,COO-. The range of pK, over which the third-order term is observed is also explained. General conclusions are derived and discussed concerning the catalytic advantage of the concerted mechanism described by the third-order term.In the previous paper1 we presented data for the solvent isotope effect on the kinetics of the enolisation of acetone. The reaction may be written: (Me),CO + (HA + B) or (A + HB) 1 dc 1 4* Ah (1) [B-- --H _ _ _ -CH,C(Me)O _ _ _ _ H - _ _ _ CH,: C(Me)OH + (HA + B) or (A + HB). In acetic acid/acetate buffers we have to consider the nine different transition states listed in table I , which can give rise to the six different catalytic rate constants discussed in the previous paper.l From the isotope effects and the Brsnsted slopes we deduced structures for the six different transition states. In this paper we apply the Marcus theory to the transfer from carbon and show that with only one adjustable parameter we can obtain a reasonable explanation for the free energies of the six different transition states, the isotope effects and the Brsnsted slopes.In addition we show that the three missing transition states in table 1 have higher free energy than those that are in fact observed. THE MODEL In eqn (1) one of the protons is transferred from carbon while the other is transferred from an oxygen base to the carbonyl oxygen. The fractionation factors for the carbon proton are all small1 (0.12-0.21) and show that this proton is in flight. By contrast the proton being transferred to oxygen is always at least partially bonded. Hence we assume that the main activation process is concerned with transferring the carbon proton and that negligible activation energy is involved in transferring the proton to oxygen.We further assume that the proton transfer from carbon is governed by the Marcus theory,,? but the thermodynamics for this transfer depends on the degree of proton transfer to the carbonyl oxygen. If there is no transfer then the reaction (Me),CO+B- -+ CH,:C(Me)O-+HB (1’) 15791580 MARCUS THEORY APPLIED T O PROTON TRANSFERS TABLE 1 .-NINE TRANSITION STATES FOR EQN (1) rate $ HA B constant CPKCU A P P I 2 (3 4 ( 5 6 (7 8 9 H,O+ H,O+ HAC H,O+ HAc HAc H2O H2O H2O 1 1 0 17.4)" 0 1 - 1" 1 6.4' 1 1 1 .O)' 1 2 - - - - - Free energy involved in forming caged complex from reactants. Free energy involved in forming HA and B from A and HB when B is a stronger base than A; ApK = pKLB - p&+. ' This transition state is of higher free energy than the one following it in the table and so is not observed.I 1 I 1 A H ~ = C - C - H B A HO-C=C HB HA, A I BASE I I I I I AH -0-C-C HB 0- A H 0.C-C-H B I FIG. 1 .-Location of the transition states for the base-catalysed, concerted and acid-catalysed routes for the enolisation of acetone. is very uphill. On the other hand if this proton is transferred, the reaction (Me),CO+H + B- -+ CH,: C(Me)OH + HB may well be downhill. Fig. 1 is helpful in showing different possible transition states, where the degree of proton transfer from carbon is given by p and the degree of proton transfer to oxygen by a. We assume that for intermediate values of a the thermodynamics for the Marcus transfer from carbon is given by a linear free energy relationship between the two extremes of a = 0 and a = 1.This type of model has been discussed for proton transfers to diazo corn pound^.^ Finally we assume that theW. J . ALBERY 1581 intrinsic kinetic barrier for the exchange at carbon does not depend on a. The size of this barrier is then the only adjustable kinetic parameter in the model. Before calculating the kinetic barriers, we have first however to discuss the thermodynamics which fix the free energy differences between the four corners of fig. 1 . Each of these corners represent the caged reactants, intermediates or products. These caged encounter complexes are formed from free reactants. In those cases where B is a stronger base than A then free energy, corresponding to ApK, must be expended in transferring a proton from B to A; values of ApK are given in table 1.Next we assume that the association of a catalyst with the substrate has an association (2) constant, K,, given by Kc/dm3 mol-l = 0.1. Values of CpK, are given in table 1 ; the water-catalysed reaction has CpK, = 0 since the water is always present, and the third-order term has CpK, = 2 since two molecules of catalyst are involved. Each catalytic route k, is then described by a first-order rate (3) constant given by where n is the order of the observed rate constant. With all the rate constants reduced to first-order rate constants, to describe the inter-conversion of the species depicted in fig. 1 it is convenient to express the acid-base equilibria in dimensionless equilibrium constants. We do this by considering the equilibrium k;, = (Mobs/K:-l HA + H,O $ H,O+ +A- Ka.Introduction of the concentration of H20, (55.5 mol drn-,), gives where pKHA is the more usual scale, in which KHA has the units of mol drn-,. enol content of acetone is ca. lo-’ and that for the equilibrium As regards the values for acetone, Guthrie and c o ~ o r k e r s ~ - ~ have shown that the (Me),CO+H,O~CH,:C(Me)O-+H,O+ Kkz0 the value of (PK’)~=~ is 21. There is more controversy concerning the protonation of acetone which fixes the fourth corner: (Me),CO+H + H 2 0 Me2C0 + H,O+ KiH+. E~tirnates8-l~ of pKs,+ range from - 2.2 to - 7.2. The isotopic data presented in the previous paper show that the H,O+-catalysed transition state has a < 0.5. However, values of pKsH+ of -2, implying values of pK&,+ of about zero, would mean that the transfer would be uphill with D > 0.5.For this reason we consider that pKs,+ must be at the lower end of the range and we take pK&,+ = -6. This means that in eqn (5) (Me),CO+H+H,O + H,C:C(Me)OH+H,O+ KkZ1 ( 5 ) pKk=, = 1. THE MARCUS EXPRESSION The Marcus expression is normally written in terms of free energies, but in dealing with proton transfers, where pK values are more familiar quantities, it is more instructive to write the expression using pk’ where pk’ = -log (k’h/kT) = AGJ/2.3 RT1582 MARCUS THEORY APPLIED TO PROTON TRANSFERS and AGJ refers to the first-order rate constant. In using the values of K, in eqn (2) and table (1) we have already dealt with the problem of wR. For any value of a in fig. 1 the Marcus expression gives pkh = ipk; ++ApK; +(A~K;)~/8pk;.(6) In this expression pkh refers to the rate for removing the carbon proton at the particular value of a. To this contribution must be added, first that concerned with reaching that value of a at p = 0 from the reactant corner and secondly that arising from the creation of the catalysts HA and B for those cases where B is a stronger base than A: Returning to eqn (6), pk; is the term corresponding to the degenerate reaction for Me,CO + H2C: C(Me)OH s H2C: C(Me)OH + Me,CO. acetone : There is no similar term for the catalyst, bemuse we assume that the barriers for the degenerate catalyst reactions are much smaller than that for acetone. The term ApK; describes the thermodynamics for removing the carbon proton at a, and we assume that it is given by a linear free energy relation: pk’ = pk’, + a(pK;IA - pK&+) + ApK.(7) APK; = apK:,=l+(l-a)pK,:,,-pK;,B (8) where for acetone and pK;=, = 21. pK;=, = 1 THREE TYPES OF TRANSITION STATE From the isotope effects for the carbon proton, where & < 0.21, that proton is always in flight in the transition state and this implies that p has a fractional value between 0 and 1. There are then three types of transition state depending on the value a = 0 ‘basic’ of a : a = 1 ‘acidic’. 0 < a < 1 ‘concerted’ Fig. 2-4 show typical pk surfaces for the three different types of transition state. From eqn (6)-(8) we can write down expressions for the rate constant, for p and in the case of the concerted case for a. The values of a and p for the concerted case are given by differentiation to find the transition state of lowest free energy.The results are: basic a=O acidic a = lI 0 1 U 0 II U cd *-’ E d .r. v) .-1584 MARCUS THEORY APPLIED TO PROTON TRANSFERS C- OH c =o HAc t C =OH FIG. 4.-Typical pk' surface for the acid-catalysed route; the free energy required to form H,O+, Ac- from HAc is also shown, Note that this figure is viewed from the opposite side to those in fig. 2 and 3. The parameter pKA describes the following equilibrium : CH,C(Me)CO + CH,=C(Me)OH + CH,C(Me)CO+H + CH,=C(Me)CO-. This equilibrium is independent of the catalysts. It is a property of the substrate and is concerned with the free energies of the two unstable corners of the square (NW and SE) compared to the two stable corners (SW and NE).LOCATION OF THE TRANSITION STATES We start by considering a particular base B. Then the values of /?a=o for the base-catalysed reaction by B and of Pa = for the acid-catalysed reaction by HB are given by eqn (10) and (12), respectively. Between these two extremes there will be a range of concerted transition states whose locations lie on the line given by eqn (1 5). The locus of this line depends on the substrate parameters (pk,, pKI, = and pKA) and on pKhB but not on pKhA. Hence there is a line for each base, B. The concerted transition states are only found for a limited range of acids HA with pK' values which lie between pKLA, I and pK;IA, I I where The separation between pK&A, I and pK;IA, I I does not depend on the base B. For the concerted transition states from eqn (14) the value of /? depends on the substrate parameters pKLH and pKA and on PKLA but not on pK;IB.Hence each concerted transition state is located by pKLB on a line given by eqn (15) and its value of /? is given by pKhA and eqn (14). This pattern is illustrated in fig. 5.W. J . ALBERY I 1 A HO+=C-C-H B A HO-L=b=HB 1585 AH - O - b = c HB I 1 I 1 AH O = C - C - H f3 FIG. 5.-Pattern underlying the location of the transition states. Each acid, e.g. HA, is associated with a vertical line given by eqn (14). Each base, e.g. B, or B,, is associated with a slanting line given by eqn (1 5). Possible transition states lie at the intersections of the ‘base lines’ with either the ‘acid lines’ or the edges of the diagram. THE ACETONE SYSTEM We now apply the equations introduced above to the acetone system.In the model we have assumed that there is only one kinetic parameter associated with the substrate, pk;. This assumption is almost certainly an oversimplification, but, as shown in fig. 6, a reasonable fit between the observed and calculated pk’ values for the six acetone transition states can be found using pki = 19. We can also calculate pk’ values for the other three transition states in table 1. In each case the pk’ value of these three transition states is significantly higher than the observed transition state which corresponds to the same term in the rate equation. We can then calculate the locations of the six transition states using eqn (lo), (1 2), (14) and (15). These are plotted in fig. 7.The value of p = 0.45 for the acetic acid transition state is in excellent agreement with the slope of the Brsnsted plot14 for catalysis by carboxylic acids. Rather less good agreement is found for the acetate transition state with = 0.70 as compared with a Brsnsted slope of 0.88.14 The locations of the H,O+OH- and H,O transition states agree well with the locations deduced from the solvent isotope effects in the previous paper.l Assuming that the fully developed enolate ion has fractionation of 0.50 we obtain the results in table 2. For the case of OH- we have assumed that one of the oxygen lone pairs is receiving the flying proton from carbon, leaving two solvent protons hydrogen bonded to the remaining lone pairs and one solvent derived proton, LO-. Returning to fig.7 the location of the concerted HAc/Ac- transition state also agrees with the previous arguments.l? l5 In fig. 8 we show the location of the concerted transition states that arise from different conjugate acid-base pairs HA, A, together with the pK values of the catalysing acid. The calculated values of the rate constants hardly alter as the catalyst changes. This agrees with the very low slope of the Brsnsted plot found by Hegarty1586 MARCUS THEORY APPLIED TO PROTON TRANSFERS 25- 20- FIG. 6.-Comparison of observed and calculated values of pk' for pkl, = 19. The points shown by crosses are calculated values for three alternative transition states which are not observed. HB HB FIG. 7.-Location of the transition states. The brackets indicate the alternative transition states which are not observed.W.J. ALBERY 1587 TABLE 2.-sOLVENT ISOTOPE EFFECTS OH- 0.55 0.54 0.56 H2O 0.78 0.33 0.30 A H;=~-+.-H B A HO-b=C HB AH 0-C=C HE 0- A H O=b-b-H B FIG. 8.-Location of the concerted transition states for different conjugate acid-base pairs HA, A. The numbers indicate the pK of the acid and the solid line the range of pK for which the third-order term has in fact been observed.16 Because of the competition from other routes it is difficult to observe the third-order term near the edges of the diagram. and Jencks.16 Furthermore the range of catalyst pK over which Hegarty and Jencks observed the third-order term agrees almost exactly with that found from the model. Further support for the model arises from the variation of & !he fractionation factor for the carbon proton.The results in table 3 show that & is lowest for the symmetrical transition states and is closer to unity as increases. This pattern is that predicted by Westheimer for primary kinetic isotope effects.16 The only remaining isotope effect that is curious is the factor, do, for the oxygen site for catalysis by HAc. Table 4 collects together the data on q50. The local environment of the oxygen proton will depend on the relative acidities of the protonating acid HA and the changing basicity of the carbonyl oxygen, pKb,, where we can write the linear free energy relation as In this equation pKA is defined in eqn (16) and is equal to 20 for acetone. The equation describes how the basicity of the carbonyl oxygen varies from a pK of - 6 for p = 0 to a pK of + 14 for the enol ( p = 1).Values of pKbH and the difference in pK around the oxygen proton are also given in table 4. It is interesting that for every concerted transition state from eqn (14) and (19) we find the simple result that ApK' = 0. In the transition state the base strengths on either side of the oxygen proton are in balance. This agrees with our previous conclusions.4 For the HAc/Ac- case the fractionation1588 MARCUS THEORY APPLIED TO PROTON TRANSFERS TABLE 3.-vARIATION OF 4~ WITH p catalyst D 4c HAc 0.45 0.12 H30+ 0.52 0.14 OH- 0.55 0.1 1 HAc/ Ac- 0.62 0.17 Ac- 0.70 0.2 1 TABLE 4.-VALUES FOR $0 AND P K ~ H catalyst 40 D PKOH P K H A ApK H30+ 1 .o 0.55 4.2 0.0 4.2 HAc 0.71 0.43 2.6 0.0 2.6 HAc/ Ac- 0.51 0.62 6.4 6.4 0.0 factor is similar to that found by Kreevoy et al.” for symmetrical hydrogen bonds.By contrast for the H,O+ transition state the proton is firmly on the carbonyl oxygen with a fractionation factor close to unity. For the HAc transition state the proton is still on the carbonyl oxygen but the difference in base strength is less, and this probably explains why the fractionation factor is reduced to 0.7. To summarise using one kinetic parameter, pk,, the model can explain: ( I ) the size of the rate constants, (2) the slopes of the Brarnsted plots, (3) the range of catalyst pK over which the third-order term is found, (4) the solvent isotope effects and ( 5 ) the carbon hydrogen isotope effects. THE CONCERTED MECHANISM Having established the model, we now enquire under what conditions will the third-order term be observed for other systems? The best chance of observing the third-order term is to choose the conjugate acid-base pair with the PKHA which corresponds to a = +.This PKHA is given by where pKTD describes the equilibrium between the product (the enol) and the reactant (21) (the ketone): PKTU = PK; = 1-pKiH- The optimal PKHA is the mean of pKHA, I and pKHA, 11, and from eqn (18) the concerted transition states exist for a range of PKHA of (~K,)~/dpk,. The route through the concerted transition state is in competition with the HA acid-catalysed route and the A- base route. For this particular acid-base pair (a = 8) the rate constants for these two routes are equal and whereW. J. ALBERY 1589 / 1 \ I I I 1 - 4 - 2 0 2 PH - PK,, FIG.9.-Typical variation with pH of the log of the three contributions to the rate from HA, A- and the third-order term. The acid is the optimal acid for observing the third-order term with kHA = kA-, and we have assumed that the sum of the buffer components is constant: [HA]+[A-] = c. The value ofy is given by eqn (24). Thus the parameter Y gives the largest catalytic advantage that the concerted route has for any acid-base pair. In fig. 9 we sketch the variation with pH of the three contributions to the observed rate for a solution containing a constant concentration of buffer species, c. The separation, y , between the maximum in the third-order term and the sum of the other two terms is given by y = Y+logc-1. (24) At best c/mol dm-3 cannot be much greater than 1 ; hence for the third-order term to be significant we require Y = 1.It can be seen from eqn (23) that the concerted route (large values of Y ) is favoured first for systems with large values of pK,. This is not surprising, since from eqn (16) pKA measures the height of the two ‘unstable’ corners (NW and SE) with respect to the two ‘stable’ corners (SW and NE). The larger the value of pKA the more the system will avoid the unstable corners. Secondly, for the same value of pKA the concerted route is favoured the smaller is the activation barrier given by pk,. However, pKA and pk, will probably be roughly related. Indeed it can be shown that = (pKA)2/32(pkL) = kPk:VA -8HA)2* (25) This equation suggests that systems with large barriers are more likely to have significant third-order terms.Now the rate constant for the A--catalysed reaction [eqn pka = 2Qipk;. (9)] is given by Hence finally we can obtain an estimate for the vital parameter Y in terms of the Brarnsted slopes and the rate constant for the A--catalysed rate, where HA is the This relation therefore allows Y to be calculated from accessible kinetic data.1590 MARCUS THEORY APPLIED TO PROTON TRANSFERS For instance, the ketone/enol transformation of dihydroxyacetone phosphate to glyceraldelyde phosphate catalysed by the enzyme triose phosphate isomerase has been shown to be extremely efficient.ls?l9 One suggested reason for this2*Y2l is that there is concerted catalysis with an enzyme base removing the carbon proton and an enzyme acid protonating the carbonyl group.For such reactions, where the acid and base are both on the same catalyst, the maximum catalytic advantage of the concerted mechanism is given simply by Y. Substitution of typical values in eqn (26) shows that this advantage is likely to be only one or two orders of magnitude. I am grateful to Professors J. P. 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