Rate constants isotopic ratio

Table 4. Thermal Rate Constant Isotope Ratios (H -l- H2 = 1) from CS Calculations (PK2 Surface)...

It has been previously noted that the first quantum correction to the classical high temperature limit for an isotope effect on an equilibrium constant is interesting. Each vibrational frequency makes a contribution c[>(u) to RPFR and this contribution can be expanded in powers of u with the first non-vanishing term proportional to u2/24, the so called first quantum correction. Similarly, for rates one introduces the first quantum correction for the reduced partition function ratios, includes the Wigner correction for k /k2 and makes use of relations like Equation 4.103 for small x and small y, to find a value for the rate constant isotope effect (omitting the noninteresting symmetry number term)... 

Schematic illustrations of the effect of temperature and surface density (time) on the ratio of two isotopes, (a) shows that, generally, there is a fractionation of the two isotopes as time and temperature change the ratio of the two isotopes changes throughout the experiment and makes difficult an assessment of their precise ratio in the original sample, (b) illustrates the effect of gradually changing the temperature of the filament to keep the ratio of ion yields linear, which simplifies the task of estimating the ratio in the original sample. The best method is one in which the rate of evaporation is low enough that the ratio of the isotopes is virtually constant this ratio then relates exactly to the ratio in the original sample.

The origin of the isotope effect is the dependence of coq and co on the reacting particle mass. Classically, this dependence comes about only via the prefactor coq [see (2.14)], and the ratio of the rate constants of transfer of isotopes with masses mj and m2 m2 > mj) is temperature-independent and equal to... 

Broecker and Peng, p. 59 dlsscon = 7 dissolution constant in DJSS pcpcon = / carbonate precipitation constant disfac -. 01 scaling factor in dissolution rate eole/e 3/y delcorg 10 Fractionation by photosynthetic organises dcse = 2 Delta 13C Isotope ratio for sea eater, per eil... 

With respect to interpretation, the existence of alternative dating techniques has made clear the necessity for and the difficulty of this step. That is, nature seldom provides ideal dating systems with fixed injection rates, negligible losses, and constant temperature. As a result, simple dates based upon observed isotopic ratios and nuclear half-lives, for example, frequently require cautious interpretation before they can serve as accurate... 

A primary isotope effect results when the breaking of a carbon-hydrogen versus a carbon-deuterium bond is the rate-limiting step in the reaction. It is expressed simply as the ratio of rate constants, i wlky,. The full expression of k /kn measures the intrinsic primary deuterium isotope for the reaction under consideration, and its magnitude is a measure of the symmetry of the transition state, e.g., -C- H- 0-Fe+3 the more symmetrical the transition state, the larger the primary isotope effect. The theoretical maximum for a primary deuterium isotope effect at 37°C is 9. The less symmetrical the transition state, the more product-like or the more substrate-like the smaller the intrinsic isotope effect will be. 

From the last two equations expressing the rate of change of isotopic ratios, we extract a relationship between the kinetic (flux) constants k and the fraction x of Nd apportioned to each reservoir... 

The standard analysis for the kinetic isotope effect that was prevalent in the 1950s and 1960s is based upon our ability to obtain the ratio of the rate constants... 

Secondary isotope effects are small. In fact, most of the secondary deuterium KIEs that have been reported are less than 20% and many of them are only a few per cent. In spite of the small size, the same techniques that are used for other kinetic measurements are usually satisfactory for measuring these KIEs. Both competitive methods where both isotopic compounds are present in the same reaction mixture (Westaway and Ali, 1979) and absolute rate measurements, i.e. the separate determination of the rate constant for the single isotopic species (Fang and Westaway, 1991), are employed (Parkin, 1991). Most competitive methods (Melander and Saunders, 1980e) utilize isotope ratio measurements based on mass spectrometry (Shine et al., 1984) or radioactivity measurements by liquid scintillation (Ando et al., 1984 Axelsson et al., 1991). However, some special methods, which are particularly useful for the accurate determination of secondary KIEs, have been developed. These newer methods, which are based on polarimetry, nmr spectroscopy, chromatographic isotopic separation and liquid scintillation, respectively, are described in this section. The accurate measurement of small heavy-atom KIEs is discussed in a recent review by Paneth (1992). 

In the simplest case, where (+)-AH and (-)AD are isotopically pure, a = [a]H[AH]0 and a2 = [a]D[AD]0 where a is the specific rotation of the AH and AD isotopomers, respectively, and [AH]0 and [AD]0 are the concentrations of the substrates in g ml-1 at time t = 0. When the substrate is neither isotopically nor enantiomerically pure, corrections must be made in calculating fli and a2 (Bergson et al., 1977). It is important to note that the pre-exponential factors, a and a2, which contain the information about the starting conditions, can be determined with high accuracy. The extreme, ae (the maximum or minimum value of the optical rotation in the optical rotation versus time plot) and the corresponding reaction time, te, are functions of the rate constant ratio (5 = kHlkD) (65) and the difference between the rate constants (66), respectively. 

In a variation of this method, Tencer and Stein (1978), mixed the isotopic quasi-racemate to near, but not exactly, zero rotation so that at a certain time, tz, the observed optical rotation of the reaction mixture was zero. The equations for this type of kinetic experiment enable one to calculate the difference between the individual isotopic rate constants from tz and the ratio of rate constants (the KIE) from te and tz provided that the ratio of the initial rotations for the two isotopic substrates is known. Usually it is preferable to... 

In earlier sections of this chapter we learned that the calculation of isotope effects on equilibrium constants of isotope exchange reactions as well as isotope effects on rate constants using transition state theory, TST, requires the evaluation of reduced isotopic partition function ratios, RPFR s, for ordinary molecular species, and for transition states. Since the procedure for transition states is basically the same as that for normal molecular species, it is the former which will be discussed first. 

Table 6.3 Tests of variational transition state theory by comparing with exact quantum calculations isotope effects at 300 K. The numbers in the table are ratios of rate constants for the two selected reactions...

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. 

Conceptually, the simplest way to measure a kinetic isotope effect (KIE) is to use a non-competitive method, in which two separate kinetic runs are carried out, each starting with a different isotopomer of the reactant. The rate constants for both species are determined and the kinetic isotope effect (KIE) is the ratio of the two rate constants. This procedure is frequently referred to as the direct method . 

These reactions proceed through symmetrical transition states [H H H] and with rate constants kn,HH and kH,DH, respectively. The ratio of rate constants, kH,HH/kH,DH> defines a primary hydrogen kinetic isotope effect. More precisely it should be regarded as a primary deuterium kinetic isotope effect because for hydrogen there is also the possibility of a tritium isotope effect. The term primary indicates that bonds at the site of isotopic substitution the isotopic atom are being made or broken in the course of reaction. Within the limits of TST such isotope effects are typically in the range of 4 to 8 (i.e. 4 < kH,HH/kH,DH < 8). 

Using the various simplifications above, we have arrived at a model for reaction 11.9 in which only one step, the chemical conversion occurring at the active site of the enzyme characterized by the rate constant k3, exhibits the kinetic isotope effect Hk3. From Equations 11.29 and 11.30, however, it is apparent that the observed isotope effects, HV and H(V/K), are not directly equal to this kinetic isotope effect, Hk3, which is called the intrinsic kinetic isotope effect. The complexity of the reaction may cause part or all of Hk3 to be masked by an amount depending on the ratios k3/ks and k3/k2. The first ratio, k3/k3, compares the intrinsic rate to the rate of product dissociation, and is called the ratio of catalysis, r(=k3/ks). The second, k3/k2, compares the intrinsic rate to the rate of the substrate dissociation and is called forward commitment to catalysis, Cf(=k3/k2), or in short, commitment. The term partitioning factor is sometimes used in the literature for this ratio of rate constants. 

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. 

Non-unit kinetic isotope effects such as the rate-constant ratio kn/k-Q also derive from isotopic zero-point energy differences in the reactant state and in the transition state. A second manifestation of the Uncertainty Principle may also contribute to kinetic isotope eff ects, namely isotopic differences in the probability of quantum tunneling through the energy barrier between the reactant state and the product state. 

Note that the ratios of observable to microscopic rate constants (in square brackets) determine the fractional degree to which the microscopic rate constant determines the rate with protiated substrate (i.e., [(kcat/ M)H/ oiH] = 1 when ioiH = ( cat/ M)H and moiH is fully rate-limiting [(kcat/ M)H/ ioiH] =0 when ioiH ( cat/ M)H and moiH has no effect on the rate). These quantities are weighting factors for the individual isotope effects ... 

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