Isotopic exchange rate equations

This section, authored by Charles Y. Huang, originally appeared in Methods in Enzymology, volume 63, as the first part of Chapter [4], pp. 54-75 ("Derivation of Initial Velocity and Isotope Exchange Rate Equations"). All equations and references in this entry are numbered sequentially as they originally appeared. 

ENZYME RATE EQUATIONS (3. Derivation of Isotope Exchange Rate Equations )... 

The following derivations demonstrate that an isotope exchange rate equation obtained by the steady-state treatment can be converted to one of exchange at equilibrium ... 

Huang, C. Y. (1979). Derivations of initial velocity and isotope exchange rate equations. Methods in Enzymology, 63, 55-85. 

Derivation of rate equations is an integral part of the effective usage of kinetics as a tool. Novel mechanisms must be described by new equations, and famihar ones often need to be modified to account for minor deviations from the expected pattern. The mathematical manipulations involved in deriving initial velocity or isotope exchange-rate laws are in general quite straightforward, but can be tedious. It is the purpose of this entry, therefore, to present the currently available methods with emphasis on the more convenient ones. 

Simplified surface-area based rate model. A simple isotope exchange rate model derived by Northrop and Clayton (1966) was modified by Cole et al. (1983) to account for the surface area of the solid in experimental mineral-fluid systems. As a first approximation, this model assumes that the rate-limiting step involves the addition and removal of atoms (O, H, C) from the surface of the solid. The overall rate of reaction, R, can be expressed in the following pseudo-first order equation with the inclusion of a factor. As, representing the total surface area (m ) of the mineral... 

Table 2 summarizes the isotope exchange rate expressions for selected common sequential mechanisms, in addition to the Ordered Bi Bi mechanism outlined above. Since there is no such mechanism as Rapid Equilibrium Random Bi Bi without a dead-end complex, at least a dead-end complex EBQ must form (mechanism 4). If both dead-end complexes, EBQ and EAP, are formed, an extra KrP/KspB term is added in the denominator of rate equation for mechanism 4 K p represents the dissociation constant of P from the EAP complex. 

Equations (6)-(8) predict that the proportions of Fe(II)aq, FefTII), and FefOHfjfs) will change over time. However, if the rate of Fe isotope exchange is rapid between, for example, Fe(II)aq and FefTII), Fe isotope equilibrium may still be maintained between aqueous Fe species, and this may be evaluated through comparison of the residence time of FefTII) relative to the time required to attain isotopic equilibrium between Fe(II)aq and Feflll), . The residence time (r) of Fe(III)jqmay be defined as ... 

Exchange of trace components The equations for adsorption (diffusion) can be equally applied in the case of isotopic exchange (exchange of isotopes) with minor changes. The same equations can be also be used in the case of the exchange of trace components of different valences (Helfferich, 1962). This is the case where the uptake or release of an ion takes place in the presence of a large amount of another ion in both the solid and liquid phase. In such systems, the amounts removed ate so small that the concentrations in both phases are practically constant, and thus in turn the individual diffusion coefficients also remain unaffected. Moreover, the rate-controlling step is the diffusion of the trace ion. 

Since equations (45a) and (45b) are mathematically identical, it follows that the solvent isotope-effect equation (50) applies to this case too. The implications of equation (50) are, however, different for the two mechanisms. The parameter i in equation (50) for the A-SE2 case can be obtained by an independent measurement. It corresponds to the fractionation factor of the proton (in the transition state) which becomes incorporated in the product. Provided that this hydrogen nucleus does not undergo ready isotope exchange after formation of the product, 1 can therefore be measured. It follows that the ratio of rates in H20 and D20 can be used to evaluate 2 since, according to equation (47b),... 

Experimental evidence for equation (55) was first reported by Kresge (1964) for proton (deuteron) addition to 1,3,5-trimethoxybenzene in H20-D20 the rate constants are obtainable from an analysis of kinetic measurements of hydrogen isotope exchange reactions. 

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