J . Phys. Chem. 1986,90, 3559-3562
3559
Marcus Theory Applied to Reactions with Double-Minimum Potential Surfaces James A. Dodd and John I. Brauman* Department of Chemistry, Stanford University, Stanford, California 94305 (Received: January 21, 1986; In Final Form: March 18, 1986)
The RRKM and Marcus formalisms have been successfully applied in the past to the interpretation of gas-phase ion-molecule reaction rates. The analysis is complicated by the necessity of incorporating ion-molecule association energies, which are often unknown. Also, precise accounting of the well depths can lead to incorrect conclusions regarding ratelequilibrium relationships. We suggest defining the reactants and products to be noninteracting, and introduce a modified Marcus equation that includes the well depth factor. With this definition, calculated activation energies are often insensitive to the well depth.
Introduction The interpretation of reaction rates plays a critical role in chemistry. There are two aspects of interpretation which require attention from the chemist. The first is the process by which the experimentally determined rate constant is related to values of the activation energy. The second is the process of relating the measured activation energy to the overall exothermicity in order to predict the behavior of other compounds with similar structure. Gas-phase ion chemistry has reached the point at which both of these aspects of reaction rate theory are important. Because the reactions are carried out at low pressure, however, the common method For relating reaction kinetics to activation energies cannot be done with the commonly used Arrenhius equation. We have used RRKM theory to analyze reaction rates, since it provides a method for dealing with a non-Boltzmann distribution of energies.l In an effort to understand the chemical significance of the rates, we have applied Marcus theory in order to derive intrinsic barriers for reactionse2 Generally speaking the two different theories for which Marcus has assumed a central role-the R R K M and Marcus formalisms-have no direct connection to each other, and the major users of each of the separate theories have worked in rather distinct areas and have made use only of the relevant theory for their own work. In fact, it is doubtful that even Marcus himself ever envisioned application of both theories in the same problem. Nevertheless, our work has made extensive use of the RRKM and Marcus theory approaches. Our reactions are bimolecular, but the “real” chemistry takes place in a unimolecular process in which one complex is converted to another (see Figure 1). Thus Marcus theory is admirably suited for the analysis, and many of the various problems associated with bimolecular reactions are obviated. RRKM theory gives us the energy difference between the top of the barrier and the separated reactants. For our applications of Marcus theory, then, we have required estimates for the depth of the wells which separate the complexes. The use of accurate well depths on a case by case basis can be misleading in applying Marcus theory, however, because the complexes may have stabilization not present either in the reatants or in the transition state. In this note we report the consequences of writing the Marcus equation in terms of the separated reactants and the transition state for ion-molecule reactions which proceed through long-lived complexes. Use of RRKM Analysis Because the mean time between ion-molecule collisions at the pressures we use ( 10-7-10-5 Torr) is much greater than the average lifetime of loosely bound ion-molecule complexes, the chemically activated3 species CR and C, are not thermalized prior to decomposition. In order to analyze the experimental efficiency for ~
~~
(1) Olmstead, W. N.; Brauman, J. I. J . Am. Chem. SOC.1977, 99, 4219. (2) Pellerite, M. J.; Brauman, J. I. J. Am. Chem. SOC.1983, 105, 2672. (3) (a) Forst, W. Theory of Unimolecular Reactions; Academic: New York, 1973. (b) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley-Interscience: New York, 1972.
0022-3654/86/2090-3559$01.50/0
a particular reaction, we use RRKM theory to calculate a branching fraction for decomposition of the initially formed complex CR,using the energy separation Edfl between the reactants and the top of the barriefl as the sole adjustable parameter (Figure 1). Thus, we generate a curve for the branching fraction as a function of Ediff,and then fix Ediffby substituting the observed branching fraction; in so doing we gain information on the energetics of the surface. The missing element is the well depth, which we must determine independently. In general, for measurably fast processes the energy of the transition state defining the top of the barrier (e.g., the SN2 transition state) lies below that of the reactants. A central barrier lower in energy than the reactants may slow a reaction because the back reaction (from the first complex to regenerate the reactants) proceeds through an orbiting transition state that typically has a large number of quantum states available to it. Also, because the orbiting transition state is higher in energy than the “tight” reaction transition state, ion-molecule reactions are predicted to display negative temperature d e p e n d e n ~ e . ~ The RRKM formalism assumes energy randomization in the reacting molecule. We have modeled the three-body collisional stabilization rates of ion-molecule complexes and found the rates to be consistent with a randomization of energy within the complex.6 More recently, the characteristics of three-body stabilization rates in ionic systems, especially temperature dependence factors, have received experimental and theoretical attention from many worker^.^
Problems with Well Depths Many ion-molecule well depths are not accurately known. Thus for some reaction groups we have made simplifying assumptions about the wells in order to determine activation energies. For instance, since the strength of ion-molecule interactions is largely determined by the nature of the uncharged molecule,* we have assumed that well depths for ion-molecule complexes involving the same neutral molecule are equaL2 In general, however, one (4) More precisely, the RRKM analysis determines the energy separation between the two transition states-the orbiting and “tight” transition states-that govern dissociation of the reactant complex CR. At room temperature the energy of the orbiting transition state is only slightly above that of the separated reactants. ( 5 ) (a) Brauman, J. I. In Kinetics of Ion-Molecule Reactions; Ausloos, P., Ed.; Plenum: New York, 1979; pp 153-164. (b) Barfknecht, A. T.; Dodd, J. A.: Salomon, K. E.; Tumas, W.; Brauman, J. I. Pure Appl. Chem. 1984, 56, 1809. (c) Caldwell, G.; Magnera, T. F.; Kebarle, P. J. Am. Chem. SOC. 1984, 106, 959. (6) (a) Jasinski, J. M.; Rosenfeld, R. N.; Golden, D. M.; Brauman, J. I. J . Am. Chem. SOC.1979, 101, 2259. (b) Olmstead, W. N.; Lev-On, y.; Golden, D. M.; Brauman, J. I. J . Am. Chem. SOC.1977, 99, 992. (7) (a) Chang, J. S.; Golden, D. M. J. Am. Chem. SOC.1981, 103, 496. (b) Bates, D. R. Chem. Phys. Lett. 1984,112. 41. fc) Herbst, E. J . Chem. Phys. 1985,82,4017. (d) Clary, D. C.; Smith, D.; Adams, N. G. Chem. Phys. Lett. 1985, 119, 320. (e) Ferguson, E. E. Chem. Phys. Lett. 1983, 101, 14. (0 Chesnavich, W. J.; Bowers, M. T. Prog. React. Kinet. 1982, 11, 137. (8) Su, T.; Bowers, M. T. In ‘Gas Phase Ion Chemistry”; Bowers, M. T., Ed.; Academic: New York, 1976; Vol. 1.
0 1986 American Chemical Society
3560
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 Y - t RX ...
X - t RY
-
f
Figure 1. Double-minimum potential energy curve for gas-phase ionmolecule reactions; shown here is an SN2 reaction. CR and Cp are loosely bound ion-molecule complexes that upon separation yield reactants and products, respectively. Ew is the well depth for CR,AE* is the activation energy for conversion of one complex to the other, AE',,, is the overall reaction energy, and AEO is the well-to-well reaction energy. Rigorously, &,ff is the energy difference between the orbiting and SN2 transition states; it is approximately equal to the energy difference between the separated reactants and the SN2 transition state.
cannot presume equal well depths for a group of reactions. In addition, for certain systems the use of accurate well depths may introduce artifacts into the analysis of the reaction energetics. For example, in our initial Marcus analysis of SN2reactions of methyl halides2 we used constant well depths of =IO kcal/mol, leading to AEo* values (intrinsic activation energies for Y- + CH,X) of 10-30 kcal/mol. In subsequent studies of the reactions of benzyl anions with methyl halides9 we used the best available value for the well depths, Le., 7.5 kcal/mol, leading to AEo* values of about 22 kcal/mol. Comparison of these AEo* values is problematic, because they reflect the different values of the well depth. As another example, consider the series of degenerate reactions C1- + ArCH2CI ArCH2Cl + C1-, which we have recently examined.I0 Since these reactions are thermoneutral, the activation energy AE* is equal to the intrinsic activation energy AEo* in each case. We studied this system in order to determine the slope of a plot of the activation energies AE* against the Hammett uo parameters. The slope p reflects the charge distribution in the transition state. The well depths for the C1-/ArCH2C1 complexes are not known. Estimates can be made of the various well depths, for instance by using the ADO formula (eq 1) for ion-dipole attraction energies.*JI For this system one cannot expect any reasonable
-
correlation with Hammett (or like) parameters, however, because of complications brought on by the anisotropy of the ion-dipole potential function. The ion is attracted most strongly when it is near the positive end of the molecular dipole. The direction of the molecular dipole, however, depends on the substituent; for a given molecule the dipole may point either toward or away from the benzylic center. Thus ion-dipole potential energy does not in general cancel between reactant (Le., the complex) and transition state. The plot of AE* vs. uo derived by using these well depths contains unacceptable scatter and has no reasonable physical interpretation (Figure 2). The well depth problem can be obviated in two ways. One can assume equal well depths, which, however, is physically incorrect. Better, one can define the reactants to be noninteracting, eg., consider t h e ion and molecule t o be a large distance apart from each other. This second option is appealing, both because we need not know precise well depths for specific reactions, and because the activation energy we seek-the energy difference &iff between the separated reactants and the top of the barrier-is the energy which results directly from the RRKM computation. Also, unlike values for 4E* (Figure I ) , Ediffvalues calculated through an RRKM analysis are independent of our choice for the well depth. (9) Dodd, J . A.: Brauman, J. I. J. Am. G e m . SOC.1984, 106, 5356. (10) Dodd, J. A. Ph.D. Thesis, Stanford University, 1985. (1 1) In eq 1 CY and C I are ~ the polarizability and dipole moment, respectively, of the molecule; q is the charge on the electron: r is the average distance between the ion and molecule: and c is the unitless "dipole locking constant",
typically 0.1S-0.20.
I
p-Me b
10.0-
m-Ye
0
" *w
6
m-CI
9.0-
0
H B.0-
0 I
I
-0.2
0. 0
0. 2
0. 4
0. 6
0. 8
-
Figure 2. Hammett plot for CI- + ArCHzCl ArCH2CI + C1- reactions; AE* (=&iff - Ew)is plotted against uo. &iff values were calculated from the rate constants usin RRKM theory; Ew values were estimated using eq 1 with r = 4.0 for all of the complexes.
x
0
1 '.
9
"
EP
w
I
-0.2
0.0
0.2
U0 Figure 3. Hammett plot for C1- + ArCH2CI tions; Edirr is plotted directly against go.
0. 4
-
0. 8
0.6
I
ArCHzCl + CI- reac-
If we draw up the Hammett plot using Ediffnumbers for the C1-/ArCH2CI system the fit is quite reasonable (Figure 3).
Modified Marcus Equation We can now explore further ramifications of defining the reactants and products as separated ion and molecule. While the above discussion points to obvious advantages of this approach, we lase in other respects. For instance, the notion that one complex is converted to the other in a single reaction step is simple and intuitive. In addition, the Marcus equation (eq 2) is easily applied to a single-step process; as mentioned in the Introduction, we have made much use of this property. According to our new definition, AEO (AEo)' AE* = AEo* + - + 2 16AEo*
however, one set of separated s p i e s is converted to the other and the wells are not explicitly considered. At this point, we may ask: Is there a Marcus-like equation that we can apply to the entire double-minimum potential surface? If so, what role would the well depth Ew play in such an equation?12 For the double-minimum potential surface, the four energy quantities relevant to a Marcus analysis are the following: Ediff, which assumes the role of "activation energy", even though it is usually negative; Mom, the overall reaction energy; Eodlff, which is the value of a t AEO,,, = 0 and thus may be thought of as the intrinsic activation energy; and Ew,the well depth, also defined ( 1 2) Murdoch has applied the Marcus equation to single minimum surfaces: Magnoli, D. E.; Murdoch, J. R. J . Am. Chem. SOC.1981, 103, 7465.
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 3561
Reactions with Double-Minimum Potential Surfaces
TABLE I: Marcus Energy Parameters for SNZReactions Y- + CHjX'
E'diff (YvW
Eodiff
Y
CH30
CI CI CI CI CI
t-BuO
HCC F
Figure 4. The effect on the barrier of an initially thermoneutral ion-
CDjS CH30
molecule reaction as the exothermicity is increased, moving from left to right. Left, thermoneutral reaction (AEorX,= 0, &iff = Eodirr); middle, moderately exothermic reaction; right, very exothermic reaction (Mom, = m o m a x , Ediff= EW).
to be negative in this analysis. As indicated in Figure 4, when the reaction exothermicity AEO,,, is equal to -lAEomaxl,the activation energy Ediffequals the well depth Ew. At this large exothermicity, AEO,,, has "pulled down" the barrier until there is no central maximum; the well depth remains constant since it is unaffected by the exothermicity. Murdoch showed some years agoi3that the Marcus equation could be derived simply by assuming a linear relation between AE* and AEo for small changes in AEO, i.e. 6AE* = a6AEo
(3)
where a is some function of AEO. Physically, eq 3 implies that a perturbation of the reaction energy results in a proportionate change in the activation energy, with 0 5 a I1. W e can write eq 3 in terms of the effect of varying the overall reaction energy AEDrX,on Ediff,viz. 6Edjff = ( Y ~ A E O , , ,
(4)
For a,Murdochi3chose a function which is linear in AE*; this function corresponds in our case to
AE", in eq 5 is the maximum allowable reaction energy within which the Marcus equation is applicable. If we substitute Murdoch's expression for a into eq 4, we have
Moan
Ediff
Br Br Br Br
t-BuO
HCC CH$O,
-42 -35
-5.6 -3.9 -2.8 -2.1 -3.0 -6.7 -5.8 -5.0 -4.8
-28 -" -29 -49 -43 -59 -17
(YX) 9.4 9.5 16.3 9.2 8.5 10.5 9.9 16.2 2.2
Equation 7 can be solved to yield, after simplification IAE'maxI = 4(Eodiff - EW)
(8)
Equation 8 is similar to the usual Marcus theory expression IhEomaxl = 4AEo*. We can use eq 8 to substitute an expression for IAEomaxlinto eq 6. If we integrate the resulting expression from AEO,,, = 0 to AEO,,,, and from Ediff= Eodiff to Ediff,we have
17.6 17.7 31.3 17.2 15.8 19.8 18.6 31.2 3.2
"All energies in kcal/mol, and defined as in Figure 1. bAssuming
Ew = -9.0 kcal/mol for (Y,Cl) reactions, and -10.0 kcal/mol for (Y,Br) reactions.
We note that eq 10 can be obtained directly from eq 2 by assuming mom = hEo,Le., that the well depths are equal. This assumption is also implicit in our derivation, in that a,eq 5, is taken to be a symmetric function about AEo,,,= 0. Thus a reaches its limiting values of 0 and 1 at the same absolute value Operationally, however, eq 10 retains much value, since of AEO,. for systems such as those we have previously examined, which have structural similarities, the well depths should also be similar. Several conclusions concerning the correct accounting for well depths can be drawn from eq 10. Wells cannot be ignored; a double-minimum surface such as shown in Figure 4 cannot be treated the same as a surface that has wells of different depth or no wells, Le., a single-minimum surface. On the other hand, Ew enters only into the quadratic term in AEO,,,; for reactions predicted by eq 10 is not of small exothermicity the value of Ediff very sensitive to the well depth. Edlffis also not very sensitive to Ew for systems in which Eodiff - Ew is large, for instance in the case of large intrinsic barriers and/or large well depths. In general, we see that by defining the reactants as we have, the calculated activation energy is sensitive to the precise values of Ewonly under certain circumstances.
Reinterpretation of Experiments With the help of eq 10 we can now reexamine some of our previously obtained data concerning gas-phase SN2 reactions. Table I lists a number of (Y,X) pairs and their Marcus energy parameters for SN2reactions Y- CH3X YCH3 X-, which we studied several years ago.2 Two columns of Table I give the intrinsic activation energies Eodiff for the cross-reactions ( Y , x ) and for the identity reactions (Y,Y). We obtain the latter quantities by assuming an additivity relation for Marcus intrinsic barriers and by using29i4Eodiff(C1,C1)= Eodlff(Br,Br)= +1.2 kcal/mol, meaning that the barriers for these identity displacements rise 1.2 kcal/mol above the energy of the separated reactants. Consistent with our previously published work,2 the well depths for Y-.CH3CI and Y--CH3Br complexes have been taken as -9 and -10 kcal/mol, respectively. The values of Eodlff in Table I support our revised version of the Marcus equation (eq 10). For instance, Eodlff(Y,Y)values obtained from different cross reactions are consistent, as were AEo* values in our original paper.* In addition, if we take the magnitude
-
+
Following Murdoch, we integrate the right-hand side of eq 6 from A E O , = 0 to -IAEomI; consequently the left-hand side integrates from Ediff = E O d i f f to Ew (as in Figure 4):
AEO* (Y,Wb 26.6 26.7 40.3 26.2 24.8 29.8 28.6 41.2 13.2
_
_
_
-
~~
~
~
+
-
~
~
(14) Recently, attempts were made to remeasure the rates for the identity
which evaluates as
Equation 10 is our result, and appears very similar in form to the Marcus equation (2). (13) Murdoch, J. R. J . Am. Chem. Soc. 1972, 94, 4410.
CH3Br + Brreactions Cl- + CH3CI CHpCl + Cl- and Br- + CH,Br in order to determine more precisely the intrinsic barriers for these two halides. Neither reaction exhibited a measurable rate. We still believe our barriers to be. approximatelycorrect,however, because the cross-reactionCl- + CHIBr CH,CI + Br- is measurably fast (ref 1 and 5c). Assuming the additivity relation holds for the intrinsic barriers, the Eodlff values for the identity reactions cannot be much larger than 1.2 kcal/mol; otherwise the cross-reaction would not occur. The E'dlff values also cannot be much smaller; otherwise the identity reactions would be measurably fast. Also, note that with Eod,fl(C1.Br)= +1.2 kcal/mol, AE',,, = -8.2 kcal/mol, and Ew = -10 kcal/mol for the reaction C1- + CH3Br CH3Cl + Br-, eq 10 predicts Ed,ll(C1,Br) = -2.5 kcal/mol. Kebarle et al. (ref 5c) determined an identical value for Ed,w in their study of the temperature dependence of the rate; while the exact agreement is surely fortuitous, it does provide further support for our values.
-
-
~
3562 The Journal of Physical Chemistry, Vol. 90, No. 16, 1986
Dodd and Brauman
+ CH,Br4 Eodiff(ArCH2,X) X = Br X = ArCH,
TABLE 11: Marcus Energy Parameters and a Values for SN2Reactions ArCH; Ar
p-MeC,H, C6H5
m-FC6H4 M-CNC~H,
Edlrf
AEO,,.
-5.25 -4.65 -3.75 -2.40
-60.4 -58.9 -53.5 -48.0
14.7 15.3 15.1 15.3
28.2 29.4 29.0 29.4
AEo*(ArCH2,X)b X = ArCH,
X = Br 26.3 26.1 26.2 26.1
42.3 43.1 42.1 42.0
cy
0.16 0.18 0.20 0.24
"All energies in kcal/mol, and defined as in Figure 1 . bAssuming Ew = -10.0 kcal/mol.
of Eodiff(Y,Y)as a measure of the nucleophilicity of Y-, we see that certain groups (CH3C02-,C1-) are very nucleophilic, others less so ( C H Q , HCC-), consistent with the conclusions reached in our earlier report2 AEo* values (A&* = E O d i f f - Ew) for the identity processes are listed in the final column of Table I as reference to our earlier work. The E'diff values are preferable measures of nucleophilicity, since they do not depend strongly on the choice for Ew. Table I1 presents revised energy data for a set of SN2reactions that we examined9 more recently, ArCH2- CH3Br ArCH2CH3 Br-. For these reactions, the quadratic term in eq 10 becomes more important because of the large exothermicity. Included in Table 11 are the intrinsic barriers Eodiff(ArCH2,Br) and Eodiff(ArCH2,ArCH2); for the latter values we have again used Eodia(Br,Br) = +1.2 kcal/mol and additivity for intrinsic barriers. A fairly accurateg estimate of 7.5 kcal/mol has been used for the well depths. The value a (defined in eq 4) is calculated for each reaction using eq l l , obtained from eq 10 by differentiation.
+
+
-
Also included in Table I1 are the values for AEo* obtained through eq 2. We have generated these numbers using unrealistically large well depths of 10 kcal/mol to remain consistent with the Y-CH3Br well depths employed in previous studies;' thus AEo*'s in Tables I and I1 are directly comparable. However, E'difflS in Tables I and 11 are directly comparable without introducing arbitrary well depths; thus we see an aspect of the appeal of our new measures &iff and Eodiff. Again, consistent with our previous work,9 the intrinsic barriers Eodiff are found to be constant through the series; this is an expected outcome for a group of similar reations. Notably, Eodiff(ArCH2,ArCH2) values are very large, indicating that the benzyl anion is on a par with the worst nucleophile in Table I, the acetylide anion. The reactions of A C H Y anions with CH3Br are fast only because they are very exothermic. Although we have
used a constant value of 7.5 kcal/mol for Ewin all of the reactions, the results are not strongly dependent on our choice. For instance, we can allow the well depths to vary by as much as 2 kcal/mol from reaction to reaction and still calculate E'diff and a values that are very similar to those in Table 11. Summary The RRKM and Marcus theories have been of great utility in the analysis of gas-phase ion-molecule reaction rates. Using them, we have been able to characterize potential surfaces for both Sh2 and proton-transfer reactions. The correct accounting for the role of potential wells in ion-molecule reactions has proven troublesome. For many reactions, forces exist in the complexes that are not present in the separated reactants or in the transition state defining the top of the barrier. On the other hand, it is not desirable to assign constant well depths arbitrarily to reaction groups for which the well depths clearly vary. Instead, one may consider ion-molecule reactions as transforming the reactant pair-the separated ion and molecule-into the product pair and not deal with the wells explicitly. Using this idea, we have introduced a modified Marcus equation that incorporates a term for the well depth Ew. The "activation energies" &ft and Eod,ff play central roles in our revised equation. Calculated activation barriers for two kinds of reactions-those of modest exothermicity, and those with large intrinsic barriers and/or deep wells-are insensitive to the precise values used for Ew. The modified equation is found to give reasonable and consistent intrinsic barriers for SN2reactions that we have studied previously, providing support for its legitimacy. Acknowledgment. We gratefully acknowledge the support of the National Science Foundation and the donors of the Petroleum Research Fund, administered by the American Chemical Society. J.A.D. thanks the N S F and Stanford University (F. R. Veatch Fellowship) for graduate fellowships. We thank C.-C. Han for many helpful discussions. This paper is dedicated to Professor R. A. Marcus on important anniversaries of his seminal work. Registry No. CH,CI, 74-87-3; CH,Br, 74-83-9
