Chemical Exchange Rate Constants

A key factor in the use of NMR for measuring dissociation constants is its sensitivity to the rate of chemical exchange . Complex formation necessarily involves the exchange of the nuclei or molecules being observed between (at least) two states - the free ligand or protein and the complex. The fact that the appearance of the NMR spectrum is sensitive not only to the position of this equilibrium but also to the rates involved has a major influence on the design of NMR experiments for measuring dissociation constants and on the analysis of such data. 

Expressions for determining rate constants from exchange contributions to observed linewidth for unequally populated systems in the fast exchange limit have been derived from the formal solutions to the Bloch equations modified for chemical exchange [3, 127-129]. These equations relate each rate constant to the site populations, chemical shift difference between sites, and spin relaxation times T and T2. For example, the forward rate A i 2 is given by [3, 127] ... 

One of the most important phenomenological aspects of electrocatalysis is the dependence of standard rate constants or exchange current densities, Iq (see Section III), for the reaction concerned on the properties and chemical identity of the electrode metal (Fig. 15) and/or the state and orientation of its surface. In fact, this is the basis of the good deffnition of electrocatalysis proposed by Busing and Kauzmann (12). 

Derivation Constants sxL Binding gal Release. There are two kinds of favorable situations. Chemical exchange between the free and the bound states affects the observed relaxation whenever the rate of chemical exchange is of the same order of magnitude as the relaxation rates. The kinetics of Na exchange were first studied by Schchorl gt al. upon the example of the dibenzo-18-crown-6 complex f 15.161. They observed virtually no chemical shift difference between the free and the complexed ion, so that the follolng simplified equations hold ... 

Transport properties (separate determination of electronic and ionic conductivity, oxygen tracer diffusion and chemical diffusion) and surface stages (rate/constant of exchange) parameters of dense ceramics can be studied by several methods, such as electronic blocking polarization methods [32-34], O tracer profile analysis by SIMS [26, 27, 29, 35], study of the isotope exchange kinetics by gas-phase analysis of the isotope composition [27,... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

Another means is available for studying the exchange kinetics of second-order reactions—we can adjust a reactant concentration. This may permit the study of reactions having very large second-order rate constants. Suppose the rate equation is V = A caCb = kobs A = t Ca, soAtcb = t For the experimental measurement let us say that we wish t to be about 10 s. We can achieve this by adjusting Cb so that the product kc 10 s for example, if A = 10 M s , we require Cb = 10 M. This method is possible, because there is no net reaction in the NMR study of chemical exchange. 

Slightly removed from this in rigor is the use of a substituent to make a pure exchange into a net chemical reaction. No isotopic label is then needed. For example, the first reliable estimate of the rate constant for the exchange of ferrocenium ions and ferrocene was made on the basis of kinetic data for processes such as... 

Fig. 5. Plot of apparent electron self exchange rate constants kf P, derived from polymer De values for films containing the indicated metals, mixed valent states, and ligands, all in acetonitrile, using Equation 2, vs. literature heterogeneous electron transfer rate constants k° for the corresponding monomers in nitrile solvents. See Ref. 6 for details. (Reproduced from Ref. 6. Copyright 1987 American Chemical Society.)...

Line shape analysis was performed for the binding of some dihydroxycholate ions to /1-cyclodextrin.205 The dihydroxycholates show different 18-CH3 signals for the complexed and free dihydroxycholate ions. To extract exchange rate constants from the NMR spectra, a complete line-shape simulation was performed. The simulation requires input of the chemical shift difference between the two sites, the line width in the absence of exchange, the residence time in each site (thg and Tg), and the relative population (fHG and fG) of each site (Equation (11)). The values were varied until the simulated and experimental spectra could be superimposed. The dissociation rate... 

The exchange of the coordinated aqua ligand of the W(IV) aqua oxo species was qualitatively studied by NMR line-broadening as a function of temperature based on Eq. (26), where the transverse relaxation time of the bound oxygen-17 nucleus is given by 1/T2b. The l/T2Qb represents the quadrupolar relaxation rate and kmi the chemical exchange rate constant... 

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