Editing Kinetic isotope effect (section)

=== Evaluation of rate constant ratios from intermolecular competition reactions ===
In competition reactions, KIE is calculated from isotopic product or remaining reactant ratios after the reaction, but these ratios depend strongly on the extent of completion of the reaction. Most often, the isotopic substrate consists of molecules labeled in a specific position and their unlabeled, ordinary counterparts.<ref name="Saunders" /> One can also, in case of {{sup|13}}C KIEs, as well as similar cases, simply rely on the natural abundance of the isotopic carbon for the KIE experiments, eliminating the need for isotopic labeling.<ref name="Singleton_1995">{{cite journal|last1=Singleton|first1=Daniel A.|last2 = Thomas | first2 = Allen A. | name-list-style = vanc |title=High-Precision Simultaneous Determination of Multiple Small Kinetic Isotope Effects at Natural Abundance|journal=Journal of the American Chemical Society|date=September 1995|volume=117|issue=36|pages=9357–9358|doi=10.1021/ja00141a030|bibcode=1995JAChS.117.9357S }}</ref>
The two isotopic substrates will react through the same mechanism, but at different rates. The ratio between the amounts of the two species in the reactants and the products will thus change gradually over the course of the reaction, and this gradual change can be treated as follows:<ref name="Saunders" /> Assume that two isotopic molecules, A{{sub|1}} and A{{sub|2}}, undergo irreversible competition reactions:
:<math chem>\begin{align}
\ce{ {A1} + {B} + {C} + \cdots}\ &\ce{->[k_1] P1}\\
\ce{ {A2} + {B} + {C} + \cdots}\ &\ce{->[k_2] P2}
\end{align}</math>
The KIE for this scenario is found to be:

:<math>\text{KIE} = {k_1 \over k_2} = \frac{\ln (1-F_1)}{\ln (1-F_2) }</math>

Where F{{sub|1}} and F{{sub|2}} refer to the fraction of conversions for the isotopic species A{{sub|1}} and A{{sub|2}}, respectively.

{{hidden|toggle=left|1=Evaluation|2=
In this treatment, all other reactants are assumed to be non-isotopic. Assuming further that the reaction is of first order with respect to the isotopic substrate A, the following general rate expression for both these reactions can be written:

:<math chem>\text{rate} = {-d[\ce A_n]\over dt} = k_n \times [\ce A_n] \times f([\ce B],[\ce C],\cdots) \text{ where } n=1 \text{ or } 2</math>

Since f([B],[C],...) does not depend on the isotopic composition of A, it can be solved for in both rate expressions with A{{sub|1}} and A{{sub|2}}, and the two can be equated to derive the following relations:

:<math chem>{1\over k_1} \times \ce{\mathit{d}[A1]\over [A1]} = {1\over k_2} \times \ce{\mathit{d}[A2] \over [A2]}</math>

:<math chem>{1\over k_1} \times \int \limits_\ce{[A1]^0}^\ce{[A1]} {d[\ce A'_1]\over [\ce A'_1]} = {1\over k_2} \times \int \limits_\ce{[A2]^0}^\ce{[A2]}{d[\ce A'_2] \over [\ce A'_2]}</math>

Where [A{{sub|1}}]{{sup|0}} and [A{{sub|2}}]{{sup|0}} are the initial concentrations of A{{sub|1}} and A{{sub|2}}, respectively. This leads to the following KIE expression:

:<math chem>{k_1 \over k_2} = \frac\ce{\ln ([A1]/[A1]^0)}\ce{\ln ([A2]/[A2]^0) }</math>

Which can also be expressed in terms of fraction amounts of conversion of the two reactions, F{{sub|1}} and F{{sub|2}}, where 1-F{{sub|n}}=[A{{sub|n}}]/[A{{sub|n}}]{{sup|0}} for n = 1 or 2, as follows:

:<math chem>{k_1 \over k_2} = \frac{\ln (1-F_1)}{\ln (1-F_2) }</math>

[[File:F2 vs. F1 in KIE Competition Reactions.png|thumb|center|500px|Relation between the fractions of conversion for the two competing reactions with isotopic substrates. The rate constant ratio ''k''{{sub|1}}/''k''{{sub|2}} (KIE) is varied for each curve.]]

As for finding the KIEs, mixtures of substrates containing stable isotopes may be analyzed with a mass spectrometer, which yields the ratios of the isotopic molecules in the initial substrate (defined here as [A{{sub|2}}]{{sup|0}}/[A{{sub|1}}]{{sup|0}}=R{{sub|0}}), in the substrate after some conversion ([A{{sub|2}}]/[A{{sub|1}}]=R), or in the product ([P{{sub|2}}]/[P{{sub|1}}]=R{{sub|P}}). When one of the species, e.g. 2, is a radioisotope, its mixture with the other species can also be analyzed by its radioactivity, which is measured in molar activities that are proportional to [A{{sub|2}}]{{sup|0}} / ([A{{sub|1}}]{{sup|0}}+[A{{sub|2}}]{{sup|0}}) ≈ [A{{sub|2}}]{{sup|0}}/[A{{sub|1}}]{{sup|0}} = R{{sub|0}} in the initial substrate, [A{{sub|2}}] / ([A{{sub|1}}]+[A{{sub|2}}]) ≈ [A{{sub|2}}]/[A{{sub|1}}] = R in the substrate after some conversion, and [R{{sub|2}}] / ([R{{sub|1}}]+[R{{sub|2}}]) ≈  [R{{sub|2}}]/[R{{sub|1}}] = R{{sub|P}}, so that the same ratios as in the other case can be measured as long as the radioisotope is present in tracer amounts. Such ratios may also be determined using NMR spectroscopy.<ref name=Jankowski>{{cite journal | vauthors = Jankowski S | title = Application of NMR spectroscopy in isotope effects studies. | journal = Annual Reports on NMR Spectroscopy | date = January 2009 | volume = 68 | pages = 149–191 | doi = 10.1016/S0066-4103(09)06803-3 | isbn = 978-0-12-381041-0 }}</ref>

When the substrate composition is followed, the following KIE expression in terms of R{{sub|0}} and R can be derived:

:<math>\text{KIE} = \frac{k_1}{k_2} = \frac {\ln(1-F_1)}{\ln[(1-F_1)R/R_0]}</math>
}}

{{hidden|toggle=left|1=Measurement of F{{sub|1}} in terms of weights per unit volume or molarities of the reactants|2=
Taking the ratio of R and R{{sub|0}} using the previously derived expression for F{{sub|2}}, one gets:
:<math chem>{R \over R_0} = \ce{\frac {[A2]/[A1]}{[A2]^0/[A1]^0}} = \ce{\frac {[A2]/[A2]^0}{[A1]/[A1]^0}} = \frac{1-F_2}{1-F_1}=(1-F_1)^{(k_2/k_1)-1}</math>

This relation can be solved in terms of the KIE to obtain the KIE expression given above. When the uncommon isotope has very low abundance, both R{{sub|0}} and R are very small and not significantly different from each other, such that 1-''F''{{sub|1}} can be approximated with ''m''/''m''{{sub|0}} or ''c''/''c''{{sub|0}}.
}}

Isotopic enrichment of the starting material can be calculated from the dependence of ''R/R''{{sub|0}} on ''F''{{sub|1}} for various KIEs, yielding the following figure. Due to the exponential dependence, even very low KIEs lead to large changes in isotopic composition of the starting material at high conversions.

[[File:R to R0 vs. F1 in KIE Competition Reactions.png|thumb|center|500px|The isotopic enrichment of the relative amount of species 2 with respect to species 1 in the starting material as a function of conversion of species 1. The value of the KIE (''k''{{sub|1}}/''k''{{sub|2}}) is indicated at each curve.]]

When the products are followed, the KIE can be calculated using the products ratio ''R{{sub|P}}'' along with ''R''{{sub|0}} as follows:

:<math>{k_1 \over k_2} = \frac {\ln(1-F_1)} {\ln[1-(F_1R_P/R_0)]}</math>