${\displaystyle {\begin{matrix}\\{\ce {{CN^{-}}+{^{12}CH3-Br}->{^{12}CH3-CN}+Br^{-}}}\\{\ce {{CN^{-}}+{^{13}CH3-Br}->{^{13}CH3-CN}+Br^{-}}}\\{}\end{matrix}}\qquad {\text{KIE}}={\frac {k_{12}}{k_{13}}}=1.082\pm 0.008}$

An example of the kinetic isotope effect.
In the reaction of methyl bromide with cyanide,
the kinetic isotope effect of the carbon in the methyl group was found to be 1.082 ± 0.008.^[1][2]

In physical organic chemistry, a kinetic isotope effect (KIE) is the change in the reaction rate of a chemical reaction when one of the atoms in the reactants is replaced by one of its isotopes.^[3] Formally, it is the ratio of rate constants for the reactions involving the light (k_L) and the heavy (k_H) isotopically substituted reactants (isotopologues):

${\displaystyle {\text{KIE}}={\frac {k_{L}}{k_{H}}}}$

This change in reaction rate is a quantum mechanical effect that primarily results from heavier isotopologues having lower vibrational frequencies compared to their lighter counterparts. In most cases, this implies a greater energetic input needed for heavier isotopologues to reach the transition state (or, in rare cases, the dissociation limit), and consequently, a slower reaction rate. The study of kinetic isotope effects can help the elucidation of the reaction mechanism of certain chemical reactions and is occasionally exploited in drug development to improve unfavorable pharmacokinetics by protecting metabolically vulnerable C-H bonds.

Background

The kinetic isotope effect is considered to be one of the most essential and sensitive tools for the study of reaction mechanisms, the knowledge of which allows the improvement of the desirable qualities of the corresponding reactions. For example, kinetic isotope effects can be used to reveal whether a nucleophilic substitution reaction follows a unimolecular (S_N1) or bimolecular (S_N2) pathway.

In the reaction of methyl bromide and cyanide (shown in the introduction), the observed methyl carbon kinetic isotope effect indicates an S_N2 mechanism.^[1] Depending on the pathway, different strategies may be used to stabilize the transition state of the rate-determining step of the reaction and improve the reaction rate and selectivity, which are important for industrial applications.

Isotopic rate changes are most pronounced when the relative mass change is greatest, since the effect is related to vibrational frequencies of the affected bonds. For instance, changing a hydrogen atom (H) to its isotope deuterium (D) represents a 100% increase in mass, whereas in replacing carbon-12 with carbon-13, the mass increases by only 8 percent. The rate of a reaction involving a C–H bond is typically 6–10 times faster than the corresponding C–D bond, whereas a ¹²C reaction is only 4 percent faster than the corresponding ¹³C reaction^[4]: 445 (even though, in both cases, the isotope is one atomic mass unit heavier).

Isotopic substitution can modify the rate of reaction in a variety of ways. In many cases, the rate difference can be rationalized by noting that the mass of an atom affects the vibrational frequency of the chemical bond that it forms, even if the potential energy surface for the reaction is nearly identical. Heavier isotopes will (classically) lead to lower vibration frequencies, or, viewed quantum mechanically, will have lower zero-point energy. With a lower zero-point energy, more energy must be supplied to break the bond, resulting in a higher activation energy for bond cleavage, which in turn lowers the measured rate (see, for example, the Arrhenius equation).^{[3][4]: 427}

Classification

Primary kinetic isotope effects

A primary kinetic isotope effect may be found when a bond to the isotopically labeled atom is being formed or broken.^{[3][4]: 427} Depending on the way a kinetic isotope effect is probed (parallel measurement of rates vs. intermolecular competition vs. intramolecular competition), the observation of a primary kinetic isotope effect is indicative of breaking/forming a bond to the isotope at the rate-limiting step, or subsequent product-determining step(s). (The misconception that a primary kinetic isotope effect must reflect bond cleavage/formation to the isotope at the rate-limiting step is frequently repeated in textbooks and the primary literature: see the section on experiments below.)^[5]

For the previously mentioned nucleophilic substitution reactions, primary kinetic isotope effects have been investigated for both the leaving groups, the nucleophiles, and the α-carbon at which the substitution occurs. Interpretation of the leaving group kinetic isotope effects had been difficult at first due to significant contributions from temperature independent factors. Kinetic isotope effects at the α-carbon can be used to develop some understanding into the symmetry of the transition state in S_N2 reactions, although this kinetic isotope effect is less sensitive than what would be ideal, also due to contribution from non-vibrational factors.^[1]

Secondary kinetic isotope effects

A secondary kinetic isotope effect is observed when no bond to the isotopically labeled atom in the reactant is broken or formed.^{[3][4]: 427} Secondary kinetic isotope effects tend to be much smaller than primary kinetic isotope effects; however, secondary deuterium isotope effects can be as large as 1.4 per deuterium atom, and techniques have been developed to measure heavy-element isotope effects to very high precision, so secondary kinetic isotope effects are still very useful for elucidating reaction mechanisms.

For the aforementioned nucleophilic substitution reactions, secondary hydrogen kinetic isotope effects at the α-carbon provide a direct means to distinguish between S_N1 and S_N2 reactions. It has been found that S_N1 reactions typically lead to large secondary kinetic isotope effects, approaching to their theoretical maximum at about 1.22, while S_N2 reactions typically yield secondary kinetic isotope effects that are very close to or less than unity. Kinetic isotope effects that are greater than 1 are referred to as normal kinetic isotope effects, while kinetic isotope effects that are less than one are referred to as inverse kinetic isotope effects. In general, smaller force constants in the transition state are expected to yield a normal kinetic isotope effect, and larger force constants in the transition state are expected to yield an inverse kinetic isotope effect when stretching vibrational contributions dominate the kinetic isotope effect.^[1]

The magnitudes of such secondary isotope effects at the α-carbon atom are largely determined by the C_α-H(D) vibrations. For an S_N1 reaction, since the carbon atom is converted into an sp² hybridized carbenium ion during the transition state for the rate-determining step with an increase in C_α-H(D) bond order, an inverse kinetic isotope effect would be expected if only the stretching vibrations were important. The observed large normal kinetic isotope effects are found to be caused by significant out-of-plane bending vibrational contributions when going from the reactants to the transition state of carbenium ion formation. For S_N2 reactions, bending vibrations still play an important role for the kinetic isotope effect, but stretching vibrational contributions are of more comparable magnitude, and the resulting kinetic isotope effect may be normal or inverse depending on the specific contributions of the respective vibrations.^[1][6][7]

Theory

The theoretical treatment of isotope effects relies heavily on transition state theory, which assumes a single potential energy surface for the reaction, and a barrier between the reactants and the products on this surface, on top of which resides the transition state.^[8][9] The kinetic isotope effect arises largely from the changes to vibrational ground states produced by the isotopic perturbation along the minimum energy pathway of the potential energy surface, which may only be accounted for with quantum mechanical treatments of the system. Depending on the mass of the atom that moves along the reaction coordinate and nature (width and height) of the energy barrier, quantum mechanical tunneling may also make a large contribution to an observed kinetic isotope effect and may need to be separately considered, in addition to the "semi-classical" transition state theory model.^[8]

The deuterium kinetic isotope effect (²H KIE) is by far the most common, useful, and well-understood type of kinetic isotope effect. The accurate prediction of the numerical value of a deuterium kinetic isotope effect using density functional theory calculations is now relatively routine. Moreover, several qualitative and semi-quantitative models allow rough estimates of deuterium isotope effects to be made without calculations, often providing enough information to rationalize experimental data or even support or refute different mechanistic possibilities. Starting materials containing deuterium are often commercially available, making the synthesis of isotopically enriched starting materials relatively straightforward. Also, due to the large relative difference in the mass of deuterium and protium and the attendant differences in vibrational frequencies, the magnitude of the isotope effect is larger than any other pair of isotopes except protium and tritium,^[10] allowing both primary and secondary isotope effects to be easily measured and interpreted. In contrast, secondary effects are generally very small for heavier elements and close in magnitude to the experimental uncertainty, which complicates their interpretation and limits their utility. In the context of isotope effects, hydrogen is frequently used to refer to the light isotope, protium (¹H), specifically. In the rest of this article, reference to hydrogen and deuterium in parallel grammatical constructions or direct comparisons between them should be interpreted as referring to protium and deuterium.^[11]

The theory of kinetic isotope effects was first formulated by Jacob Bigeleisen in 1949.^{[12][4]: 427} Bigeleisen's general formula for deuterium kinetic isotope effects (which is also applicable to heavier elements) is given below. It employs transition state theory and a statistical mechanical treatment of translational, rotational, and vibrational levels for the calculation of rate constants k_H and k_D. However, this formula is "semi-classical" in that it neglects the contribution from quantum tunneling, which is often introduced as a separate correction factor. Bigeleisen's formula also does not deal with differences in non-bonded repulsive interactions caused by the slightly shorter C–D bond compared to a C–H bond. In the equation, quantities with the subscripts H or D refer to the hydrogen- or deuterium-substituted species, respectively, while quantities with or without the double-dagger, ‡, refer to the transition state or reactant ground state, respectively.^[7][13] (Strictly speaking, a ${\displaystyle \kappa _{\mathrm {H} }/\kappa _{\mathrm {D} }}$ term resulting from an isotopic difference in transmission coefficients should also be included.^[14])

${\displaystyle \frac{k_{\text{H}}}{k_{\text{D}}}=\left(\frac{{\mathrm{\sigma <!--\; \sigma \; -->}}_{\text{H}}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\text{D}}^{‡<!--\; ‡\; -->}}{{\mathrm{\sigma <!--\; \sigma \; -->}}_{\text{D}}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\text{H}}^{‡<!--\; ‡\; -->}}\right){\left(\frac{M_{\text{H}}^{‡<!--\; ‡\; -->}{M}_{\text{D}}}{M_{\text{D}}^{‡<!--\; ‡\; -->}{M}_{\text{H}}}\right)}^{\frac{3}{2}}{\left(\frac{I_{x\text{H}}^{‡<!--\; ‡\; -->}{I}_{y\text{H}}^{‡<!--\; ‡\; -->}{I}_{z\text{H}}^{‡<!--\; ‡\; -->}}{I_{x\text{D}}^{‡<!--\; ‡\; -->}{I}_{y\text{D}}^{‡<!--\; ‡\; -->}{I}_{z\text{D}}^{‡<!--\; ‡\; -->}}\frac{I_{x\text{D}}{I}_{y\text{D}}{I}_{z\text{D}}}{I_{x\text{H}}{I}_{y\text{H}}{I}_{z\text{H}}}\right)}^{\frac{1}{2}}(\frac{\underset{i=1}{\overset{3N^{‡<!--\; ‡\; -->}-<!--\; -\; -->7}{\prod <!--\; \prod \; -->}}\frac{1-<!--\; -\; -->e^{-<!--\; -\; -->u_{i\text{D}}^{‡<!--\; ‡\; -->}}}{1-<!--\; -\; -->e^{-<!--\; -\; -->u_{i\text{H}}^{‡<!--\; ‡\; -->}}}}{}}$Zdroj:https://en.wikipedia.org?pojem=Kinetic_isotope_effect
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