Marcus plot, slope

Magnetic moment, 153, 155, 160 Magnetic quantum number, 153 Magnetization, 160 Magnetogyric ratio, 153, 160 Main reaction, 237 Marcus equation, 227, 238, 314 Marcus plot, slope of, 227, 354 Marcus theory, applicability of, 358 reactivity-selectivity principle and, 375 Mass, reduced, 189, 294 Mass action law, 11, 60, 125, 428 Mass balance relationships, 19, 21, 34, 60, 64, 67, 89, 103, 140, 147 Maximum velocity, enzyme-catalyzed, 103 Mean, harmonic, 370 Mechanism classification of. 8 definition of, 3 study of, 6, 115 Medium effects, 385, 418, 420 physical theories of, 405 Meisenheimer eomplex, 129 Menschutkin reaction, 404, 407, 422 Mesomerism, 323 Method of residuals, 73 Michaelis constant, 103 Michaelis—Menten equation, 103 Microscopic reversibility, 125... 

This discussion of sources of curvature in Br insted-type plots should suggest caution in the interpretation of observed curvature. There is a related matter, concerning particularly item 5 in this list, namely, the effect of a change in transition state structure. Br nsted-type plots are sometimes linear over quite remarkable ranges, of the order 10 pK units, and this linearity has evoked interest because it seems to be incompatible with Marcus theory, which we reviewed in Section 5.3. The Marcus equation (Eq. 5-69) for the plot of log k against log K of the same reaction series requires curvature, the slope of the plot being the coefficient a. given by Eq. (5-67). A Brjinsted plot, however, is not a Marcus plot, because it correlates rates and equilibria of different reactions. The slope p of a Br nsted plot is defined p = d log kobs/d pK, which we can expand as... 

For AG° > 4 AG (0), the Rehm-Weller equation approximates a straight line with slope 1/2.303 RT= —0.74 mol kcal-1, whereas the parabolic Marcus plot over narrow intervals, if approximated as a straight line, gives slopes < —0.74 mol kcal-1. No distinction between the two equations can be made on the basis of the data of Fig. 9. 

Electrolysis of the solution beyond the Ru(bpy)3 potential produces base-labile lesions specifically at G. Thus, the chemical reaction that leads to cycling of the metal complex is oxidation of guanine to the radical cation. Using digital simulation, the rate constant for the chemical reaction can be measured from the data shown in Fig. 18. When the potential of the mediator is varied, a Marcus plot can be constructed. If Marcus theory is obeyed, the rate constant should vary linearly with the driving force with a slope of 1/2. Such a dependence is observed for oxidation of guanine by Rulbpyla and derivatives (Fig. 19). 

Thorp and coworkers examined the oxidation of G in duplex DNA using a series of metal-polypyridyl oxidants of varying driving force [78]. A Marcus plot yields a slope of—0.8. As discussed in Section 17.3.1, a stepwise mechanism with a ratedetermining ET step produces a slope of -0.5 whereas a rate-determining PT step produces a slope of-1.0. The observations of an intermediate slope and attendant KIE of 2.1 signify that a PCET mechanism is operable. It is noted that in other... 

The one-electron R+—R potentials were calculated for the species used and, by estimating the appropriate self-exchange rate, a Marcus plot with slope 0.52 was obtained, in good agreement with the theoretical value. 

The value of k E was obtained by following the change in optical rotation (a). Plots expressing the difference between x at various times t and its value at equilibrium of ln(a, - ae) against time were made they proved to be linear with a slope of ftEE[Mn]7-. This gave an experimental value Ee = 30Lmor s, in excellent agreement with that from the Marcus correlations. 

Experimental values of AG and the pre-exponential factor were obtained from a plot of In k,. vs 1/T under the assumption that the slope is — AG /R, and the hidden assumption that AG is temperature independent (AS is zero). Comparison between the calculated and observed pre-exponential factor was used to infer significant non-adiabaticity, but one may wonder whether inclusion of a nonzero AS would alter this conclusion. From an alternative perspective, reasonable agreement was noted for the values of ke and the homogeneous self-exchange rate constant after a standard Marcus-type correction was made for the differing reaction types. 

Note that the slopes of the Br0nsted and Tafel plots, need not necesarily be constant over a large free energy range and, in fact, the Marcus—Levich theoretical treatment predicts a quadratic dependence [32c]. 

In conclusion, it can also be pointed out that in principle a large value of A is itself sufficient to account for an extended linear free energy relationship. However, as Mayr has noted this is only true if the slope of the plot is O.5.238 Moreover, if the Marcus expression offers a quantitative guide to the degree of curvature of a free energy relationship (and it is by no means clear that it does),228 it is evident that the intrinsic barriers to reactions of carbocations with familiar nucleophiles are insufficiently large to account for the lack of curvature. 

Marcus rate theory is useful to rationalize the connection between reactivity and the slope a of Bronsted plots. The derivative of Equation (19) with respect to ArG° is the slope of the Marcus curve, which corresponds to the Bronsted exponent a for a given free energy of reaction ArG°, Equation (20).74 80... 

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

When NMA+ reacts with phenyl-substituted N-phenyldihydronicotin-amides, X-PhNAH, also in anhydrous acetonitrile (Powell and Bruice, 1983b), rate and equilibrium data yield a Bronsted plot with a slope of 0.51, consistent with a centrally located transition state. The primary k.i.e. s h2/ d2, increase from 3.98 for X = />-methoxy to 4.77 for X = m-trifluoro-methyl at 50° and may indicate a trend to a more symmetrical transition state. Marcus treatment of the substituent dependence of the k.i.e. s yields an intrinsic barrier AG = 22.2 kJ mol - L. The temperature dependence of the k.i.e. for reduction by X-PhNAH with X = / -methyl gives [A ] = 7.68 kJ mol-1, but AJA = 4.3 is unusually large. A tunnelling correction of ca. 2 was estimated so that the semi-classical k.i.e. was in the range 2 to 3. 

A chemically based, mass-independent fractionation process was first observed during ozone formation through the gas-phase recombination reaction (Thiemens Heidenreich 1983) O + O2 + M - O3 + M. The product ozone possesses equally enriched heavy-oxygen isotopes I7 IS0. by approximately lOO /oo with respect to the initial oxygen, with a slope value of unity in a three-isotope oxygen plot. This discovery led to the conclusion that a symmetry-dependent reaction can produce meteoritic isotopic anomalies (Thiemens 1999, 2006). Recently, theoretical calculations of Gao Marcus (2001) established the major role of symmetry in isotopolog-specific stabilization of vibrationally excited ozone molecules that give rise to the mass-independent compositions. 

We discussed earlier the LFER for the oxygenation of Fe(II) the plot of log k versus log K given in Figure 11.8b has a slope of unity. This is in accord with the Marcus prediction (AG > 0). The data given in Figure 11.8b can be extended to the oxygenation of other transition metal ions (Wehrli, 1990) (Figure 11.20). 

Moreover, the plot of the corresponding a(E) variations in Fig. 16b shows that a depends linearly on the potential with a slope (0.249 V ), in close agreement with that predicted (0.210 V ) from the Marcus reorganization energy (29kcal/mol) determined from Eq. (107). 

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