Isotope exchange equation

Stanko and coworkers reviewed367 their studies368 concerning radioiodine (123 125131132x) and radiobromine (77,82Br) isotopic exchange (equation 175) studied in DMSO369,370. 

In view of the kinetic Equation 51.5, the model of isotope exchange (Equation 51.1) for plug-flow reactor is reduced to the form ... 

Lee and Bethke (1996) presented an alternative technique, also based on mass balance equations, in which the reaction modeler can segregate minerals from isotopic exchange. By segregating the minerals, the model traces the effects of the isotope fractionation that would result from dissolution and precipitation reactions alone. Not unexpectedly, segregated models differ broadly in their results from reaction models that assume isotopic equilibrium. 

Fig. 16. Slow isotopic exchange at 35.0°C on Re(V) center, (a) Oxygen-17 spectra showing signal growth vs time for [Re02(CN)4]3. (b) Least-squares fit of the data to the modified exponential McKay equation (7). The total complex concentration [Re] = 0.2 m, pH = 6.6, and fi = 1.2 m (KN03). (Adapted with permission from Roodt, A. Leipoldt, J. G. Helm, L. Abou-Hamdan, A. Merbach, A. E. Inorg. Chem. 1995, 34, 560-568. Copyright 1995 American Chemical Society.)...

The H/D isotopic exchange in the uridine system (from starred nitrogen) may be explained by the mechanism (equation 8) involving dimerization of the hydroxy heterocycle70. Dimerization of hydroxy derivatives of aza-containing heterocycles plays an important role in assembling molecules, and in their tautomeric equilibria. 

Fig. 2.2 Iii(Keq) for the isotope exchange reaction, equation 2.15 plotted vs. 1,000/T (lighter...

This latter equilibrium is called an isotopic exchange equilibrium. Its equilibrium constant in terms of partition functions is from Equation 4.64... 

In addition now consider a second set of isotopomers BX01 and BXP. Then we obtain a second equation similar to Equation 4.100 for (s2/si )f for this pair of isotopomers and can write the equilibrium constant for the isotopic exchange reaction... 

At all but very high temperatures it is necessary to employ the complete equation because the vibrational frequencies for all these molecules are quite high. (Notice at room temperature u(H2) 21, and u(HI) 11). Harmonic oscillator rigid rotor calculated equilibrium constants are shown in Fig. 4.4. As expected the low temperature limiting value, while bounded, is significantly different from unity. At extremely high temperature Equation 4.95 applies and the isotope exchange constant is... 

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 ... 

In this model, Fe isotope exchange is envisioned to most likely occur between soluble Fe(III) and Fe(II) components, such as Fe(III)-LFe(in) and Fe(II)-Lpe(n) (Beard et al. 2003a), although only the Fe(II)aq component is measured. If this model is valid, a critical issue is whether isotopic equilibrium may be attained between these soluble pools of Fe, despite the changing reservoir sizes and fluxes that occur through Equations (5)-(8). As noted in the previous chapter (Chapter 1OA Beard and Johnson 2004), attainment of isotopic equilibrium will depend upon the elemental residence time in an Fe pool relative to the time required for isotopic exchange. We can define the residence time for Fe(III)-L, for example, using standard definitions, as ... 

Equation (18) illustrates that the measured 5 Fe value for Fe(ll)jq is dependent not only on t Fe(iii)L-Fe(ii)Lj but On the proportion of Fe(III)-LFe(in) in the components that are open to isotopic exchange, which additionally includes Fe(II)-LFe(n) and Fe(ll)a, we will refer to these three components as the exchangeable pool of Fe in the system. We stress that the isotopic mass balance described by Equation (18) assumes that the ligand-bound Fe(lll) component is not sampled in the aqueous phase component, but instead exists as a component that is bormd to the cells. 

Table 11.4 lists the results of calculations based on equation 11.47 for isotopic exchange reactions at T = 300 K for tetrahedrally and octahedrally coordinated heavy isotopes ... 

In light of equation 11.42, the equilibrium constant for the isotopic exchange reaction of equation 11.53 reduces to... 

Let us consider again an isotopic exchange reaction of the type shown in equation 11.53, such as, for instance, the already treated equilibrium between stannous chloride and an aqueous complex... 

The first attempts to quantify isotope exchange processes between water and rocks were made by Taylor (1974). By using a simple closed-system material balance equation these authors were able to calculate cumulative fluid/rock ratios. 

The imprint of local conditions can also be seen at other coastal and continental stations. The examples in Table 3.1 demonstrate that varying influences of different sources of vapor with different isotope characteristics, different air mass trajectories, or evaporation and isotope exchange processes below the cloud base, may often lead to much more complex relationships at the local level between 8D and 8 0 than suggested for the regional or continental scale by the global Meteoric Water Line equation. 

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. 

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 )... 

DERIVATION BY THE STEADY-STATE METHOD. Britton S first derived isotope flux equations under steady-state rather than equilibrium conditions. To illustrate his procedure, we shall again use Scheme 1, the B P exchange in Ordered Bi Bi mechanism, as an example, so that the results can be compared (Scheme la). 

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 ... 

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