Equilibria, isotope effects

It should in principle be possible to calculate the isotope effect for an equilibrium since the reactant and product are real compounds. The problem is rather more difficult for reactions in solution but has been solved successfully for a number of reactions in the gas phase. A useful example is the ionisation of acids in water (Eqn. 

For all measurable dissociation constants XH will be a weaker acid than HjO and the proton will be bound more tightly hence the dissociation will involve a decrease in zero point energy. This decrease will be smaller for the deuterated system hence deuteration should lower the tendency to dissociate (K K ). By the same argument ATh/X should increase with decreasing acid strength. These predictions are borne out by experiment (Fig. 3) and the slope of the line is of the order of magnitude to be expected in terms of zero-point energies. 

If the vibration modes of reactant and product are known the equilibrium-isotope 

The lower acidity of deutero acids (in D2O) compared with proto acids (in H2O) provides us with a useful tool to diagnose the pre-equilibrium mechanism for acid-catalysed reactions (Eqn. 18). 

The isotope effect for this reaction is predicted by Eqn. 19 and since the 2 step does not involve proton transfer the ratio approximates to unity and the 

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The deuterium isotope effects on chemical shift consists of intrinsic isotope effect (direct perturbation of the shielding of X atom) and equilibrium isotope effect (perturbation of the equilibrium caused by the isotopic substitution). The values of deuterium isotope effects are to some extent independent of chemical shifts and allow determination of the mole fraction of the forms in equilibrium. 

These structures were then used to generate the force fields and calculate the secondary /3-deuterium-d6 equilibrium isotope effects (EIEs) for the formation of the isopropyl carbocation (Table 30). Because the transition states for formation of the carbocation will be close to the structure of the carbocation, these KIEs should be excellent approximations of the maximum secondary /3-deuterium KIEs expected for the limiting SN1 solvolytic reaction. 

In one of these studies, Kurz and Frieden (1980) observed the first unexpectedly large secondary a-deuterium KIE. They found that the secondary a-deuterium KIE for the nonenzymatic hydride ion reduction of 4-cyano-2,6-dinitrobenzenesulfonate by NADH (reaction (44)) was 1.156 0.018 and 1.1454 0.0093 using direct and competitive kinetic methods, respectively. The corresponding equilibrium isotope effects (EIEs) were found to be 1.013 0.020 and 1.0347 0.0087, respectively. Thus, the secondary deuterium KIE was much larger than the EIE. The magnitude of a secondary a-deuterium KIE is normally attributed to the rehybridization of the a-carbon that takes place when the reactant is transformed into the... 

As written Equation 4.150 applies to the case of a single isotopic substitution in reactant A with light and heavy isotopic masses mi and m2, respectively. Equation 4.150 shows that the first quantum correction (see Section 4.8.2) to the classical rate isotope effect depends on the difference of the diagonal Cartesian force constants at the position of isotopic substitution between the reagent A and the transition state. While Equations 4.149 and 4.150 are valid quantitatively only at high temperature, we believe, as in the case of equilibrium isotope effects, that the claim that isotope effects reflect force constant changes at the position of isotopic substitution is a qualitatively correct statement even at lower temperatures. 

Abstract In this chapter we discuss practical techniques and instrumentation used in experimental measurements of kinetic and equilibrium isotope effects. After describing methods to determine IE s on rate constants, brief treatments of mass spectrometry and isotope ratio mass spectrometry, NMR measurements of isotope effects, the use of radio-isotopes, techniques to determine vapor pressure and other equilibrium IE s, and IE s in small angle neutron scattering are presented. 

Equation 11.36 recognizes that Hk3/Hk4 corresponds to the equilibrium isotope effect, hK3/4 for the step containing rate constants k3 and k4. The rate ratio k4/k.5 is the commitment for catalysis for the reaction that proceeds from products to substrates, and therefore is called the reverse commitment to catalysis, Cr. Also cf = k3/k2 is the forward commitment to catalysis. Since we have assumed that these steps are the only isotope sensitive ones, HK3/4 corresponds to the overall equilibrium isotope effect, HK. 

If the isotope sensitive step is reversible the equations get more complicated and cannot be solved explicitly for the intrinsic isotope effects (unless Cf = 0, or the equilibrium isotope effect is unity). The last two equations in Equation 11.48 demonstrate that a normal deuterium kinetic isotope effect diminishes the apparent commitment if both isotopes are present. Thus 13(V/K) is smaller than 13(V/K)d when both isotope effects are related to the same step. 

Rameback H, Berglund M, Kessel R, Wellum R (2002) Modeling isotope fractionation in thermal ionization mass spectrometry filaments having diffusion controlled emission. Int J Mass Spectrom 216 203-208 Roe JE, Anhar AD, Barling J (2003) Nonhiological fractionation of Fe isotopes evidence of an equilibrium isotope effect. Chem Geol 195 69-85... 

If a single step of the reaction is much slower than all the others (i.e., it is the rate-limiting step), then the isotopic shift induced by the overall reaction is equal to the kinetic isotope effect occurring at the rate limiting step plus the sum of any equilibrium isotope effects occurring between the preceding species in the reaction chain. 

Roe JE, Anbar AD, Barling J (2003) Nonbiological fractionation of Fe isotopes evidence of an equilibrium isotope effect Chem Geol 195 69-85... 

Mo isotope fractionation in the oceans results from an equilibrium isotope effect between dissolved and Mn oxide-associated Mo (Barling et al. 2001 Siebert et al. 2003 Arnold et al. 2004 Barling and Anbar 2004), then 5 MOg,v - 8 MOox 1000 x ln(asw M Ox)- Additionally, it seems reasonable that 5 MOem approximates 8 MOs , at least in settings similar to the Black Sea and Cariaco Basin. Using these relationships and assuming / x + /e = 1 (i.e., neglecting suboxic sediments), we obtain ... 

T-secondary isotope effect can be determined. As recounted in the last item of Chart 3, such effects are expected to be measures of transition-state structure. If the transition state closely resembled reactants, then no change in the force field at the isotopic center would occur as the reactant state is converted to the transition state and the -secondary kinetic isotope effect should be 1.00. If the transition state closely resembled products, then the transition-state force field at the isotopic center would be very similar to that in the product state, and the a-secondary kinetic isotope effect should be equal to the equilibrium isotope effect, shown by Cook, Blanchard, and Cleland to be 1.13. Between these limits, the kinetic isotope effect should change monotonically from 1.00 to 1.13. 

Kurz and Frieden in 1977 and 1980 determined -secondary kinetic isotope effects for the unusual desulfonation reaction shown in Table 1, both in free solution and with enzyme catalysis by glutamate dehydrogenase. The isotope effects (H/D) were in the range of 1.14-1.20. At the time, the correct equilibrium isotope effect had not been reported and their measurements yielded an erroneous value... 

If secondary isotope effects arise strictly from changes in force constants at the position of substitution, with none of the vibrations of the isotopic atom being coupled into the reaction coordinate, then a secondary isotope effect will vary from 1.00 when the transition state exactly resembles the reactant state (thus no change in force constants when reactant state is converted to transition state) to the value of the equilibrium isotope effect when the transition state exactly resembles the product state (so that conversion of reactant state to transition state produces the same change in force constants as conversion of reactant state to product state). For example in the hydride-transfer reaction shown under point 1 above, the equilibrium secondary isotope effect on conversion of NADH to NAD is 1.13. The kinetic secondary isotope effect is expected to vary from 1.00 (reactant-like transition state), through (1.13)° when the stmctural changes from reactant state to transition state are 50% advanced toward the product state, to 1.13 (product-like transition state). That this naive expectation... 

With label in the benzyl alcohol, the secondary KIE (H/T) is 1.34-1.38 while the equilibrium isotope effect is 1.33-1.34. Substituent effects, by contrast, indicate a transition state that resembles the reactant state, not the product state. 

In 1983, Huskey and Schowen tested the coupled-motion hypothesis and showed it to be inadequate in its purest form to account for the results. If, however, tunneling along the reaction coordinate were included along with coupled motion, then not only was the exaltation of the secondary isotope effects explained but also several other unusual feamres of the data as well. Fig. 4 shows the model used and the results. The calculated equilibrium isotope effect for the NCMH model (the models employed are defined in Fig. 4) was 1.069 (this value fails to agree with the measured value of 1.13 because of the general simplicity of the model and particularly defects in the force field). If the coupled-motion hypothesis were correct, then sufficient coupling, as measured by the secondary/primary reaction-coordinate amplimde ratio should generate secondary isotope effects that... 

Note that if only one step, say the step, has a kinetic isotope isotope effect and there are no equilibrium isotope effects, then ... 

Ah/At = 7.4 and A /Ax = 1.8 and isotopic activation energy differences that are within the experimental error of zero. The values of the two A-ratios correspond to a Swain-Schaad exponent of 3.4, not much different from the semiclassical expectation of 3.3. The a-secondary isotope effects are 1.19 (H/T), 1.13 (H/D), and 1.05 (D/T), which are exactly at the limiting semiclassical value of the equilibrium isotope effect. The secondary isotope effects generate a Swain-Schaad exponent of 3.5, again close to the semiclassical expectation. At the same time that the isotope effects are temperature-independent, the kinetic parameter shows... 

The observation of exalted secondary isotope effects, i.e., those that are substantially beyond the semiclassical limits of unity and the equilibrium isotope effect. These observations require coupling between the motion at the primary center and motion at the secondary center in the transition-state reaction coordinate, and in addition that tunneling is occurring along the reaction coordinate. 

The value of the fractionation factor for any site will be determined by the shape of the potential well. If it is assumed that the potential well for the hydrogen-bonded proton in (2) is broader, with a lower force constant, than that for the proton in the monocarboxylic acid (Fig. 8), the value of the fractionation factor will be lower for the hydrogen-bonded proton than for the proton in the monocarboxylic acid. It follows that the equilibrium isotope effect on (2) will be less than unity. As a consequence, the isotope-exchange equilibrium will lie towards the left, and the heavier isotope (deuterium in this case) will fractionate into the monocarboxylic acid, where the bond has the larger force constant. 

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