J. Phys. Chem. 1982. 86,3321-3323
values at the two lower Cun ion concentration levels employed with the value deduced for the Cun-free system is believed to provide the unambiguous supporting evidence sought for the existence of the two Cun-complexed species deduced from this analysis.
Discussion The stability constants that have been resolved for the formation of CunA+ and CunA2in this second analysis of the Cu" ion binding data obtained with a PMA gel seem, on the basis of the above, to be more soundly based than the earlier estimates of these parameters.' There is no doubt that our earlier analysis of the data compiled for this system was distorted by disregard of the data points obtained at (Y I0.35. Apparently Cu-Cu interaction, presumed to introduce complication in this (Y range, was not an important factor. These stability constants must, however, be corrected for the neglect of the nonideality of Cun ion in the solution phase. The stoichiometric concentration of free, mobile Cu" ion was used in the computation of /3?;nA+ (pl) and &$A (f12). In 0.05 M Na2S04the CU" is competitively comptexed by in addition, electrostatic interaction of the ionized species needs to be taken into account. When this is done ycuu = 0.084; with this correction term" & = 6.5 X lo2 and 0,= 3.0 X lo2. The magnitude of (K,) is near the upper limit of the stability constant range observed for the formation of CunA+with carboxylic acid ligands. However, the K2value of 0.5 (&/& = K2) that is now resolved is a good deal smaller than is generally observed for the second stepwise constant with these ligands. This result is consistent with the fact that a second species, MEAz,has not been observed in our earlier studies&" of the binding of divalent ions other than Cu" with PMA and PAA (poly(acry1ic acid)) even though their existence in the presence of simple carboxylic (11) W. M. Anspach and J. A. Marinsky, J. Phys. Chem., 79, 433 (1975).
3321
acids closely resembling the repeating monomer unit of these polyelectrolytes is well-known. Only in the formation of the first complex (M"A+) should the complexation behavior of a polymer be expected to mimic the behavior of the repeating monomer unit after corrections for nonideality factors introduced by the polyelectrolyte have been made. These expectations are based upon the following estimate of the situation. In a polymer molecule with a high degree of polymerization, the accessibility ratio of the repeating functional unit ((HA)/ (A-) and (M"A+)/(A-)) is equal to unity even though their absolute availability, restricted to fluctuations, oscillatory and rotatory, is smaller than in the simple molecule. for this reason the pKlntHA resolved for the polyacid is exactly equal to the intrinsic pK characterizing the repeating monomer unit; the pFibA+ v a l y is for the same reason expected to be equal to the p r M ~ ~characteristic A+ of the metal complex formed with the repeating function unit. When the second complexation step is entered, however, the smaller absolute accessibility of the repeating functional unit, A-, is operative; as a consequence the formation of a second species in the polymer system is much less likely to be observed. We can now, from this reanalysis of Cun ion binding data obtained with a PMA gel, understand the fact that only the unidentate species, CulIA+,is observable when Cuu is equilibrated with the linear polyelectrolyte analogue of the gel.3 First, concentrations of A- in the polyelectrolyte undoubtedly do not reach the concentration level of A- in the gel, and, second, the absolute accessibility of A- to M"A+ in the polyelectrolyte is probably smaller than in the crowded, three-dimensional gel, to yield an apparent K2 even smaller than the 0.5 value observed with the PMA gel. We believe that with this research we have demonstrated a capability for the resolution and characterization of complex species occurring in gels with an accuracy and facility approaching that found in simple electrolyte systems.
How Should Intrinsic Barriers Be Estimated from Marcus Theory3l Wllllam H. Saunders, Jr. Department of Chemistry, University of Rochester, Rochester, New York 14627 (Received: November 13, 1981; I n Final Form: April 27, 1982)
The Marcus theory of proton transfer has been used to treat "data" generated by a computer program that determines the point of intersection of two Morse curves. When normal values of the exponential Morse constant, a, and the constant b in the Pauling relation between bond order and bond length are used, intrinsic barriers derived from the square coefficient of a quadratic least-squaresfit to the equation AE* = AEl0[l + hE/(4hE*0)]2 are as low as 10-11 % of the actual barrier height when AE = 0. Variants of this model yield somewhat better resulta, but the intrinsic barriers are usually well below the actual barrier height and quite sensitive to the precise details of the model. These results suggest that intrinsic barriers derived in this fashion from fits of the equation AG* = w, + AGlo[l + AGOR'/(~AG'~)]~ to experimental AG* vs. AGO data may be unreliable, particularly for slow proton transfers. The Marcus theory2J equation for proton transfer in solution may be written as AG* = w, + ~ ~ ' +~ AGOR'/(4AG*o)]2 1 1 (1) (1) This work wm supported by the National Science Foundation. (2) Marcus, R. A. J.Phys. Chem. 1968,72,891-9. 0022-365418212086-3321$01.25/0
where AG* is the free energy of activation and AGIOis the intrinsic barrier, the free energy of activation within the encounter complex when AGOR' = 0. This last term is defined by (3) Cohen, A. 0.; Marcus, R. A. J. Phys. Chem. 1968, 72, 4249-56.
0 1982 American Chemical Society
3322
The Journal of Physical Chemistry, Vol. 86, No.
AGoR =
W,
+ AGOR' - wP
Saunders
17, 1982
(2)
where AGOR is the observed free energy of reaction, w, is the free energy required to form the encounter complex between reactants, and wp is the free energy required to form the encounter complex between products. Equation 1can be fitted to experimental data on AG* vs. AGOR,e.g., for a series of different substrates or bases, and AG*, evaluated from the coefficient of the square term in the quadratic fit. In turn, w, can be evaluated from AGIOand the coefficients of the other terms. It is somewhat disconcerting that such fits often give w, values substantially larger than AGlo. For example, Kresge4 tabulates results for eight reactions where w, ranges from 8 to 16 kcal mol-l, and exceeds AG*,,, which ranges from 1to 10 k d mol-l, in all but one case. We have recently reported5that fits of the Marcus theory expression for k H / k Dto data showing kH/kD maxima likewise yield strikingly small AG*, values (1.4-4 kcal mol-') for reactions whose overall free energies of activation run 20-23 kcal mol-'. Such results have led to suggestion^^^^ that Marcus theory may systematicallyunderestimate intrinsic barriers. A hyperbolic free energy relationship, for example, gives a AG*, twice as large as that from Marcus theory.6 Even that formulation, however, leaves suspiciously small the AG*, values obtained from a substantial proportion of the fits to experimental data. The pattern of small intrinsic barriers and large w, values goes against a great deal of evidence showing that proton transfers to and from carbon usually have quite large free energies of activation which depend markedly on substrate structure in a way that is difficult to reconcile with the idea of an activation barrier dominated by the w, term. Thus, it is important to inquire whether there are factors that tend to make eq 1give low values for AG*, when experimental AGO vs. AG* data are fitted to it by a quadratic least-squares procedure. A possibility that should not be discounted is simple experimental error. The wider the range of AGO covered, the wider the range of rates that must be measured to get the corresponding AG* values, and the greater the likelihood that spurious curvature will be introduced. Use of a wide range of substrates also increases the likelihood of structural effects on rate (such m steric effects) that can cause deviations from fits to free energy relationships such as eq 1 by changing the intrinsic barrier. Murdoch has pointed out another source of curvature in free energy plots: the coupling of diffusion steps to proton transfer.' He demonstrates that diffusion rates can introduce sufficient curvature to cause underestimates even of substantial intrinsic barriers. In cases where the above complications are absent, can we be sure that eq 1 describes the behavior of real systems with sufficient accuracy to give reliable intrinsic barriers? The success in many instances of Marcus theory at correlating both experimental data and the results of theoretical calculations is undeniable. For example, Marcus has shown that barriers for hydrogen transfers calculated from the barriers of symmetric exchange processes agree very well with barriers derived from BEBO calculations.2 Such confirmations of the basic validity of the theory do not, however, directly test ita accuracy in deriving intrinsic barriers from the curvature of free energy plota. The model (4) Kresge, A. J. Chem. SOC.Reo. 1973,2, 475-503. (5) Miller, D. J.; Saunders, W. H., Jr. J. Org. Chem. 1981,46,4247-52. (6) Lewis, E. S.; Shen, C. C.; O'Ferrall, R. A. More J. Chem. Soc., Perkin Trans. 2 1981, 1084-8. (7) Murdoch, J. R. J. Am. Chem. SOC.1980, 102, 71-8.
described below is an attempt to provide a test, not of Marcus theory overall, but of this specific application of the theory. A simple computer program was used to calculate the barrier height at the point of intersection of two Morse curves, eq 3a and 3b. The distance between the two Vl = Vo(l)[l - exp(-a(r - r1))I2
(34
V2 = V0(2)[1- exp(-a(r2 - 4)12
(3b)
minima was adjusted by means of the Pauling relation (eq 4) so that the total bond order, nl n2, of the bonds to r - ro = - b In n (4) the proton in transit could be assigned any desired value, usually unity. In most cases the normal value of the Morse exponential parameter, a, of 1.8 A-' for a carbon-hydrogen bonde was used; the Pauling constant, b, was taken as 0.30 A, in accord with recent work in which the value giving the best fit to a variety of experimental data was determined.g It was assumed that VOcl,= Vo(2)in most cases, and different values of AE for the proton transfers were simulated by shifting one curve vertically with respect to the other. With one exception, the assumptions made in setting up this model can be shown to be identical with those of the intersecting-parabolas derivation', of eq 1. This derivation gives initially eq 5. The further assumption that hE* = hE*o[l 4- u / ( 4 h E * 0 ) ] ~ (5)
+
free energy parallels potential energy and the introduction of w, lead to eq 1. That our model is equivalent to this model can be seen if the intersecting parabolas are defined by eq 6, where n represents bond order. Marcus makes
v = v, (1 - n)2
(6)
it clear that the distance between the minima of the intersecting curves is to be defined in terms of a sum of bond orders, not an actual physical distance.2 Combination of eq 6 with eq 4 is readily seen to yield an equation of the same form as eq 3 in which a = l/b.''J2 The one respect in which the present model differs, then, is that it adopts values of a and b separately anchored in e~periment?~ with the result that a # l / b . Table I lists results of a small selection of the numerous models examined. The first column under each model lists AE*values for the model that correspond to the AE values in the column at the left of Table I. The second column under each model takes hE* at AE = 0 as AE*,and uses eq 5 to calculate other AE*values so as to give an additional comparison of each model with the Marcus theory predictions. The bottom entry for each model, in parentheses, is the ALPo value calculated from the coefficient of the square term obtained in a quadratic least-squares fit to eq 5 of the "data" in the column above. All of the models assume Vo(l)= Vo(z) = -28 kcal mol-' (simulating proton donors of pK 20) to facilitate comparisons. Model I assumes a = 1.8A-' and b = 0.3 The AE*, value derived from the coefficient of the square term of eq 5 is only 11%of the actual AE* at AE = 0. Almost identical underestimates (10-11 % of the true value) result with V, values of -10 to -100 kcal mol-', and nl + n2 = 1
-
A.e7g
(8) Wiberg, K. B. "Physical Organic Chemistry";Wiley: New York, 1964; pp 334-5. (9) Burton, F. W.; Sims,L. B.; Wilson, J. C.; Fry, A. J.Am. Chem. SOC. 1977, 99,3371-9.
(10)For a detailed description of this derivation, see: McLennan, D. J. J. Chem. Educ. 1976,53, 348-51. (11) Kurz, J. L. Chem. Phys. Lett. 1978, 57, 243-6. (12) Kurz, J. L., private communication.
The Journal of Physical Chemistry, Vol. 86, No. 17, 1982 3323
Intrinsic Barriers from Marcus Theory
TABLE I: Relation between A E and A E * for a Proton-Transfer Process Calculated from Intersecting Morse Curves” model I AE 1 2 3 -1
-2 -3 0
AE*
23.372 23.947 24.572 22.372 21.946 21.571 22.848 (2.511)b
AE*,
23.351 23.859 24.373 22.351 21.859 21.373 22.848
model I11
model I1 AE*
AE*,
AE*
25.500 26.044 26.624 24.499 24.045 23.624 24.985 (4.029)b
25.488 25.995 26.508 24.488 23.995 23.508 24.985
26.598 27.105 27.617 25.598 25.105 24.617 26.095 ( 25.991)b
AE*,
26.597 27.105 27.617 25.597 25.105 24.617 26.095
model IV
model Vc
AE*
AE+,
AE*
26.573 27.085 27.605 25.573 25.085 24.605 26.070 (15.813)b
26.572 27.080 27.592 25.572 25.080 24.592 26.070
23.534 24.055 24.627 22.534 22.055 21.626 23.064 (6.52)b
AE*,
23.567 24.075 24.588 22.567 22.075 21.588 23.064
From a computer program that calculates the point of intersection of two facing Morse curves of identical well depth V (-28 kcal in all cases) and exponential constant a , held at a horizontal distance such that n , + n, = 1 (except for model V) when n, and n, are calculated by the Pauling relation, assuming rep = 1.09 A for both. A E is the vertical displacement of one minimum from the other in kcal, and AE* is the vertical distance from the intersection point to the mimimum level of the left-hand curve. A E * , is the value calculated from eq 5, assuming that AE*o is the same as AE* at A E = 0 from the preceding column. The models (except for V ) differ only in the values of the Morse exponential constant a and the Pauling constant b. For the different models, a, b, and ( a ) ( b )are as follows: model I, 1.8, 0.3, 0.54; model 11, 2.16, 0.36, 0.7776; model 111, 2.45, 0.408 16, 1.0000; model IV, 2.45, 0.40, 0.98. Calculated from the coefficient of the square team obtained in a quadratic least-squares fit of the A E * (above column) and the corresponding A E values to eq 5. This model lets n , + n, vary linearly from 0.92 at A E = 0 to 0.98 at A E = + 3 and -3.
or
