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 (kL) and the heavy (kH) isotopically substituted reactants (isotopologues): KIE = kL/kH.
This change in reaction rate is a quantum effect that occurs mainly because heavier isotopologues have lower vibrational frequencies than their lighter counterparts. In most cases, this implies a greater energy input needed for heavier isotopologues to reach the transition state (or, in rare cases, dissociation limit), and therefore, a slower reaction rate. The study of KIEs can help elucidate reaction mechanisms, and is occasionally exploited in drug development to improve unfavorable pharmacokinetics by protecting metabolically vulnerable C-H bonds.
KIE is considered one of the most essential and sensitive tools for studying reaction mechanisms, the knowledge of which allows improvement of the desirable qualities of said reactions. For example, KIEs can be used to reveal whether a nucleophilic substitution reaction follows a unimolecular (SN1) or bimolecular (SN2) pathway.
In the reaction of methyl bromide and cyanide (shown in the introduction), the observed methyl carbon KIE is 1.082, a small effect which indicates an SN2 mechanism in which the C-Br bond is formed as the C-CN bond is broken. For SN1 reactions in which the leaving group leaves first to form a trivalent carbon transition state, the KIE is close to the maximum observed value for a secondary KIE (SKIE, see below) of 1.22.[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. Thus, replacing normal hydrogen (1H) with its isotope deuterium (D or 2H), doubles the mass; whereas in replacing carbon-12 with carbon-13, the mass increases by only 8%. The rate of a reaction involving a C–1H bond is typically 6–10x faster than with a C–2H bond, whereas a 12C reaction is only 4% faster than the corresponding 13C reaction;[4]: 445 even though, in both cases, the isotope is one atomic mass unit (amu) (dalton) heavier.
Isotopic substitution can modify the reaction rate 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, have lower zero-point energy (ZPE). With a lower ZPE, 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
A primary kinetic isotope effect (PKIE) may be found when a bond to the isotopically labeled atom is being formed or broken.[3][4]: 427 Depending on the way a KIE is probed (parallel measurement of rates vs. intermolecular competition vs. intramolecular competition), the observation of a PKIE 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 PKIE must reflect bond cleavage/formation to the isotope at the rate-limiting step is often repeated in textbooks and the primary literature: see the section on experiments below.)[5]
For the aforementioned nucleophilic substitution reactions, PKIEs have been investigated for both the leaving groups, the nucleophiles, and the α-carbon at which the substitution occurs. Interpretation of the leaving group KIEs was difficult at first due to significant contributions from temperature independent factors. KIEs at the α-carbon can be used to develop some understanding into the symmetry of the transition state in SN2 reactions, though this KIE is less sensitive than what would be ideal, also due to contribution from non-vibrational factors.[1]
A secondary kinetic isotope effect (SKIE) is observed when no bond to the isotopically labeled atom in the reactant is broken or formed.[3][4]: 427 SKIEs tend to be much smaller than PKIEs; however, secondary deuterium isotope effects can be as large as 1.4 per 2H atom, and techniques have been developed to measure heavy-element isotope effects to very high precision, so SKIEs are still very useful for elucidating reaction mechanisms.
For the aforementioned nucleophilic substitution reactions, secondary hydrogen KIEs at the α-carbon provide a direct means to distinguish between SN1 and SN2 reactions. It has been found that SN1 reactions typically lead to large SKIEs, approaching to their theoretical maximum at about 1.22, while SN2 reactions typically yield SKIEs that are very close to or less than 1. KIEs greater than 1 are called normal kinetic isotope effects, while KIEs less than 1 are called inverse kinetic isotope effects (IKIE). In general, smaller force constants in the transition state are expected to yield a normal KIE, and larger force constants in the transition state are expected to yield an IKIE when stretching vibrational contributions dominate the KIE.[1]
The magnitudes of such SKIEs at the α-carbon atom are largely determined by the Cα-H(2H) vibrations. For an SN1 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 Cα-H(2H) bond order, an IKIE would be expected if only the stretching vibrations were important. The observed large normal KIEs 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 SN2 reactions, bending vibrations still play an important role for the KIE, but stretching vibrational contributions are of more comparable magnitude, and the resulting KIE may be normal or inverse depending on the specific contributions of the respective vibrations.[1][6][7]
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 KIE 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 tunneling may also make a large contribution to an observed KIE 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 KIE. The accurate prediction of the numerical value of a 2H KIE using density functional theory calculations is now fairly 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 2H 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 2H and 1H and the attendant differences in vibrational frequency, the isotope effect is larger than for any other pair of isotopes except 1H and 3H,[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 often means 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 taken to mean 1H and 2H.[a]
The theory of KIEs was first formulated by Jacob Bigeleisen in 1949.[11][4]: 427 Bigeleisen's general formula for 2H KIEs (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 kH and kD. 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–2H bond compared to a C–H bond. In the equation, subscript H or D refer to the species with 1H or 2H, respectively; quantities with or without the double-dagger, ‡, refer to transition state or reactant ground state, respectively.[7][12] (Strictly speaking, a κ H / κ D {\displaystyle \kappa _{\mathrm {H} }/\kappa _{\mathrm {D} }} term resulting from an isotopic difference in transmission coefficients should also be included.[13])
where we define
Here, h = Planck constant; kB = Boltzmann constant; ν ~ i {\displaystyle {\tilde {\nu }}_{i}} = frequency of vibration, expressed in wavenumber; c = speed of light; NA = Avogadro constant; and R = universal gas constant. The σX (X = H or D) are the symmetry numbers for the reactants and transition states. The MX are the molecular masses of the corresponding species, and the IqX (q = x, y, or z) terms are the moments of inertia about the three principal axes. The uiX are directly proportional to the corresponding vibrational frequencies, νi, and the vibrational zero-point energy (ZPE) (see below). The integers N and N‡ are the number of atoms in the reactants and the transition states, respectively.[7] The complicated expression given above can be represented as the product of four separate factors:[7]
For the special case of 2H isotope effects, we will argue that the first three terms can be treated as equal to or well approximated by unity. The first factor S (containing σX) is the ratio of the symmetry numbers for the various species. This will be a rational number (a ratio of integers) that depends on the number of molecular and bond rotations leading to the permutation of identical atoms or groups in the reactants and the transition state.[12] For systems of low symmetry, all σX (reactant and transition state) will be unity; thus S can often be neglected. The MMI factor (containing the MX and IqX) refers to the ratio of the molecular masses and the moments of inertia. Since hydrogen and deuterium tend to be much lighter than most reactants and transition states, there is little difference in the molecular masses and moments of inertia between H and D containing molecules, so the MMI factor is usually also approximated as unity. The EXC factor (containing the product of vibrational partition functions) corrects for the KIE caused by the reactions of vibrationally excited molecules. The fraction of molecules with enough energy to have excited state A–H/D bond vibrations is generally small for reactions at or near room temperature (bonds to hydrogen usually vibrate at 1000 cm−1 or higher, so exp(-ui) = exp(-hνi/kBT) < 0.01 at 298 K, resulting in negligible contributions from the 1–exp(-ui) factors). Hence, for hydrogen/deuterium KIEs, the observed values are typically dominated by the last factor, ZPE (an exponential function of vibrational ZPE differences), consisting of contributions from the ZPE differences for each of the vibrational modes of the reactants and transition state, which can be represented as follows:[7]
The sums in the exponent of the second expression can be interpreted as running over all vibrational modes of the reactant ground state and the transition state. Or, one may interpret them as running over those modes unique to the reactant or the transition state or whose vibrational frequencies change substantially upon advancing along the reaction coordinate. The remaining pairs of reactant and transition state vibrational modes have very similar Δ u i {\displaystyle \Delta u_{i}} and Δ u i ‡ {\displaystyle \Delta u_{i}^{\ddagger }} , and cancellations occur when the sums in the exponent are calculated. Thus, in practice, 2H KIEs are often largely dependent on a handful of key vibrational modes because of this cancellation, making qualitative analyses of kH/kD possible.[12]
As mentioned, especially for 1H/2H substitution, most KIEs arise from the difference in ZPE between the reactants and the transition state of the isotopologues; this difference can be understood qualitatively as follows: in the Born–Oppenheimer approximation, the potential energy surface is the same for both isotopic species. However, a quantum treatment of the energy introduces discrete vibrational levels onto this curve, and the lowest possible energy state of a molecule corresponds to the lowest vibrational energy level, which is slightly higher in energy than the minimum of the potential energy curve. This difference, known as the ZPE, is a manifestation of the uncertainty principle that necessitates an uncertainty in the C-H or C-D bond length. Since the heavier (in this case the deuterated) species behaves more "classically", its vibrational energy levels are closer to the classical potential energy curve, and it has a lower ZPE. The ZPE differences between the two isotopic species, at least in most cases, diminish in the transition state, since the bond force constant decreases during bond breaking. Hence, the lower ZPE of the deuterated species translates into a larger activation energy for its reaction, as shown in the following figure, leading to a normal KIE.[14] This effect should, in principle, be taken into account all 3N−6 vibrational modes for the starting material and 3N‡−7 vibrational modes at the transition state (one mode, the one corresponding to the reaction coordinate, is missing at the transition state, since a bond breaks and there is no restorative force against the motion). The harmonic oscillator is a good approximation for a vibrating bond, at least for low-energy vibrational states. Quantum mechanics gives the vibrational ZPE as ϵ i ( 0 ) = 1 2 h ν i {\displaystyle \epsilon _{i}^{(0)}={\frac {1}{2}}h\nu _{i}} . Thus, we can readily interpret the factor of 1/2 and the sums of u i = h ν i / k B T {\displaystyle u_{i}=h\nu _{i}/k_{\mathrm {B} }T} terms over ground state and transition state vibrational modes in the exponent of the simplified formula above. For a harmonic oscillator, vibrational frequency is inversely proportional to the square root of the reduced mass of the vibrating system:
where kf is the force constant. Moreover, the reduced mass is approximated by the mass of the light atom of the system, X = H or D. Because mD ≈ 2mH,
In the case of homolytic C–H/D bond dissociation, the transition state term disappears; and neglecting other vibrational modes, kH/kD = exp(1/2Δui). Thus, a larger isotope effect is observed for a stiffer ("stronger") C–H/D bond. For most reactions of interest, a hydrogen atom is transferred between two atoms, with a transition-state [A···H···B]‡ and vibrational modes at the transition state need to be accounted for. Nevertheless, it is still generally true that cleavage of a bond with a higher vibrational frequency will give a larger isotope effect.
To calculate the maximum possible value for a non-tunneling 2H KIE, we consider the case where the ZPE difference between the stretching vibrations of a C-1H bond (3000 cm−1) and C-2H bond (2200 cm−1) disappears in the transition state (an energy difference of [3000 – 2200 cm−1]/2 = 400 cm−1 ≈ 1.15 kcal/mol), without any compensation from a ZPE difference at the transition state (e.g., from the symmetric A···H···B stretch, which is unique to the transition state). The simplified formula above, predicts a maximum for kH/kD as 6.9. If the complete disappearance of two bending vibrations is also included, kH/kD values as large as 15-20 can be predicted. Bending frequencies are very unlikely to vanish in the transition state, however, and there are only a few cases in which kH/kD values exceed 7-8 near room temperature. Furthermore, it is often found that tunneling is a major factor when they do exceed such values. A value of kH/kD ~ 10 is thought to be maximal for a semi-classical PKIE (no tunneling) for reactions at ≈298 K. (The formula for kH/kD has a temperature dependence, so larger isotope effects are possible at lower temperatures.)[15] Depending on the nature of the transition state of H-transfer (symmetric vs. "early" or "late" and linear vs. bent); the extent to which a primary 2H isotope effect approaches this maximum, varies. A model developed by Westheimer predicted that symmetrical (thermoneutral, by Hammond's postulate), linear transition states have the largest isotope effects, while transition states that are "early" or "late" (for exothermic or endothermic reactions, respectively), or nonlinear (e.g. cyclic) exhibit smaller effects. These predictions have since received extensive experimental support.[16]
For secondary 2H isotope effects, Streitwieser proposed that weakening (or strengthening, in the case of an inverse isotope effect) of bending modes from the reactant ground state to the transition state are largely responsible for observed isotope effects. These changes are attributed to a change in steric environment when the carbon bound to the H/D undergoes rehybridization from sp3 to sp2 or vice versa (an α SKIE), or bond weakening due to hyperconjugation in cases where a carbocation is being generated one carbon atom away (a β SKIE). These isotope effects have a theoretical maximum of kH/kD = 20.5 ≈ 1.4. For a SKIE at the α position, rehybridization from sp3 to sp2 produces a normal isotope effect, while rehybridization from sp2 to sp3 results in an inverse isotope effect with a theoretical minimum of kH/kD = 2-0.5 ≈ 0.7. In practice, kH/kD ~ 1.1-1.2 and kH/kD ~ 0.8-0.9 are typical for α SKIEs, while kH/kD ~ 1.15-1.3 are typical for β SKIE. For reactants containing several isotopically substituted β-hydrogens, the observed isotope effect is often the result of several H/D's at the β position acting in concert. In these cases, the effect of each isotopically labeled atom is multiplicative, and cases where kH/kD > 2 are not uncommon.[17]
The following simple expressions relating 2H and 3H KIEs, which are also known as the Swain equation (or the Swain-Schaad-Stivers equations), can be derived from the general expression given above using some simplifications:[8][18]
i.e.,
In deriving these expressions, the reasonable approximation that reduced mass roughly equals the mass of the 1H, 2H, or 3H, was used. Also, the vibrational motion was assumed to be approximated by a harmonic oscillator, so that u i X ∝ μ X − 1 / 2 ≅ m X − 1 / 2 {\displaystyle u_{i\mathrm {X} }\propto \mu _{\mathrm {X} }^{-1/2}\cong m_{\mathrm {X} }^{-1/2}} ; X = 1,2,3H. The subscript "s" refers to these "semi-classical" KIEs, which disregard quantum tunneling. Tunneling contributions must be treated separately as a correction factor.
For isotope effects involving elements other than hydrogen, many of these simplifications are not valid, and the magnitude of the isotope effect may depend strongly on some or all of the neglected factors. Thus, KIEs for elements other than hydrogen are often much harder to rationalize or interpret. In many cases and especially for hydrogen-transfer reactions, contributions to KIEs from tunneling are significant (see below).
In some cases, a further rate enhancement is seen for the lighter isotope, possibly due to quantum tunneling. This is typically only observed for reactions involving bonds to hydrogen. Tunneling occurs when a molecule penetrates through a potential energy barrier rather than over it.[19][20] Though not allowed by classical mechanics, particles can pass through classically forbidden regions of space in quantum mechanics based on wave–particle duality.[21]
Tunneling can be analyzed using Bell's modification of the Arrhenius equation, which includes the addition of a tunneling factor, Q:
where A is the Arrhenius parameter, E is the barrier height and
where α = E R T {\displaystyle \alpha ={\frac {E}{RT}}} and β = 2 a π 2 ( 2 m E ) 1 / 2 h {\displaystyle \beta ={\frac {2a\pi ^{2}(2mE)^{1/2}}{h}}}
Examination of the β term shows exponential dependence on the particle's mass. As a result, tunneling is much more likely for a lighter particle such as hydrogen. Simply doubling the mass of a tunneling proton by replacing it with a deuteron drastically reduces the rate of such reactions. As a result, very large KIEs are observed that can not be accounted for by differences in ZPEs.
Also, the β term depends linearly with barrier width, 2a. As with mass, tunneling is greatest for small barrier widths. Optimal tunneling distances of protons between donor and acceptor atom is 40 pm.[23]
Tunneling is a quantum effect tied to the laws of wave mechanics, not kinetics. Therefore, tunneling tends to become more important at low temperatures, where even the smallest kinetic energy barriers may not be overcome but can be tunneled through.[19]
Peter S. Zuev et al. reported rate constants for the ring expansion of 1-methylcyclobutylfluorocarbene to be 4.0 × 10−6/s in nitrogen and 4.0 × 10−5/s in argon at 8 kelvin. They calculated that at 8 kelvin, the reaction would proceed via a single quantum state of the reactant so that the reported rate constant is temperature independent and the tunneling contribution to the rate was 152 orders of magnitude greater than the contribution of passage over the transition state energy barrier.[24]
So even though conventional chemical reactions tend to slow down dramatically as the temperature is lowered, tunneling reactions rarely change at all. Particles that tunnel through an activation barrier are a direct result of the fact that the wave function of an intermediate species, reactant or product is not confined to the energy well of a particular trough along the energy surface of a reaction but can "leak out" into the next energy minimum. In light of this, tunneling should be temperature independent.[19][3]
For the hydrogen abstraction from gaseous n-alkanes and cycloalkanes by hydrogen atoms over the temperature range 363–463 K, the H/D KIE data were characterized by small preexponential factor ratios AH/AD ranging from 0.43 to 0.54 and large activation energy differences from 9.0 to 9.7 kJ/mol. Basing their arguments on transition state theory, the small A factor ratios associated with the large activation energy differences (usually about 4.5 kJ/mol for C–H(D) bonds) provided strong evidence for tunneling. For the purpose of this discussion, it is important is that the A factor ratio for the various paraffins they used was roughly constant throughout the temperature range.[25]
To determine if tunneling is involved in KIE of a reaction with H or D, a few criteria are considered:
Also for reactions where isotopes include H, D and T, a criterion of tunneling is the Swain-Schaad relations which compare the rate constants (k) of the reactions where H, D or T are exchanged:
In organic reactions, this proton tunneling effect has been observed in such reactions as the deprotonation and iodination of nitropropane with hindered pyridine base[26] with a reported KIE of 25 at 25°C:
and in a 1,5-sigmatropic hydrogen shift,[27] though it is observed that it is hard to extrapolate experimental values obtained at high temperature to lower temperatures:[28][29]
It has long been speculated that high efficiency of enzyme catalysis in proton or hydride ion transfer reactions could be due partly to the quantum mechanical tunneling effect. Environment at the active site of an enzyme positions the donor and acceptor atom close to the optimal tunneling distance, where the amino acid side chains can "force" the donor and acceptor atom closer together by electrostatic and noncovalent interactions. It is also possible that the enzyme and its unusual hydrophobic environment inside a reaction site provides tunneling-promoting vibration.[30] Studies on ketosteroid isomerase have provided experimental evidence that the enzyme actually enhances the coupled motion/hydrogen tunneling by comparing primary and secondary KIEs of the reaction under enzyme-catalyzed and non-enzyme-catalyzed conditions.[31]
Many examples exist for proton tunneling in enzyme-catalyzed reactions that were discovered by KIE. A well-studied example is methylamine dehydrogenase, where large primary KIEs of 5–55 have been observed for the proton transfer step.[32]
Another example of tunneling contribution to proton transfer in enzymatic reactions is the reaction carried out by alcohol dehydrogenase. Competitive KIEs for the hydrogen transfer step at 25°C resulted in 3.6 and 10.2 for primary and secondary KIEs, respectively.[33]
Isotopic effect expressed with the equations given above only refer to reactions that can be described with first-order kinetics. In all instances in which this is not possible, transient KIEs should be taken into account using the GEBIK and GEBIF equations.[34]
Simmons and Hartwig refer to the following three cases as the main types of KIE experiments involving C-H bond functionalization:[5]
In this experiment, the rate constants for the normal substrate and its isotopically labeled analogue are determined independently, and the KIE is obtained as a ratio of the two. The accuracy of the measured KIE is severely limited by the accuracy with which each of these rate constants can be measured. Furthermore, reproducing the exact conditions in the two parallel reactions can be very challenging. Nevertheless, a measurement of a large kinetic isotope effect through direct comparison of rate constants is indicative that C-H bond cleavage occurs at the rate-determining step. (A smaller value could indicate an isotope effect due to a pre-equilibrium, so that the C-H bond cleavage occurs somewhere before the rate-determining step.)
This type of experiment, uses the same substrates as used in Experiment A, but they are allowed in to react in the same container, instead of two separate containers. The KIE in this experiment is determined by the relative amount of products formed from C-H versus C-D functionalization (or it can be inferred from the relative amounts of unreacted starting materials). One must quench the reaction before it goes to completion to observe the KIE (see the Evaluation section below). Generally, the reaction is halted at low conversion (~5 to 10% conversion) or a large excess (> 5 equiv.) of the isotopic mixture is used. This experiment type ensures that both C-H and C-D bond functionalizations occur under exactly the same conditions, and the ratio of products from C-H and C-D bond functionalizations can be measured with much greater precision than the rate constants in Experiment A. Moreover, only a single measurement of product concentrations from a single sample is required. However, an observed kinetic isotope effect from this experiment is more difficult to interpret, since it may either mean that C-H bond cleavage occurs during the rate-determining step or at a product-determining step ensuing the rate-determining step. The absence of a KIE, at least according to Simmons and Hartwig, is nonetheless indicative of the C-H bond cleavage not occurring during the rate-determining step.
This type of experiment is analogous to Experiment B, except this time there is an intramolecular competition for the C-H or C-D bond functionalization. In most cases, the substrate possesses a directing group (DG) between the C-H and C-D bonds. Calculation of the KIE from this experiment and its interpretation follow the same considerations as that of Experiment B. However, the results of Experiments B and C will differ if the irreversible binding of the isotope-containing substrate takes place in Experiment B prior to the cleavage of the C-H or C-D bond. In such a scenario, an isotope effect may be observed in Experiment C (where choice of the isotope can take place even after substrate binding) but not in Experiment B (since the choice of whether C-H or C-D bond cleaves is already made as soon as the substrate binds irreversibly). In contrast to Experiment B, the reaction need not be halted at low consumption of isotopic starting material to obtain an accurate kH/kD, since the ratio of H and D in the starting material is 1:1, regardless of the extent of conversion.
One non-C-H activation example of different isotope effects being observed in the case of intermolecular (Experiment B) and intramolecular (Experiment C) competition is the photolysis of diphenyldiazomethane in the presence of t-butylamine. To explain this result, the formation of diphenylcarbene, followed by irreversible nucleophilic attack by t-butylamine was proposed. Because there is little isotopic difference in the rate of nucleophilic attack, the intermolecular experiment resulted in a KIE close to 1. In the intramolecular case, however, the product ratio is determined by the proton transfer that occurs after the nucleophilic attack, a process which has a substantial KIE of 2.6.[35]
Thus, Experiments A, B, and C will give results of differing levels of precision and require different experimental setup and ways of analyzing data. As a result, the feasibility of each type of experiment depends on the kinetic and stoichiometric profile of the reaction, as well as the physical characteristics of the reaction mixture (e.g. homogeneous vs. heterogeneous). Moreover, as noted in the paragraph above, the experiments provide KIE data for different steps of a multi-step reaction, depending on the relative locations of the rate-limiting step, product-determining steps, and/or C-H/D cleavage step.
The hypothetical examples below illustrate common scenarios. Consider the following reaction coordinate diagram. For a reaction with this profile, all three experiments (A, B, and C) will yield a significant primary KIE:
On the other hand, if a reaction follows the following energy profile, in which the C-H or C-D bond cleavage is irreversible but occurs after the rate-determining step (RDS), no significant KIE will be observed with Experiment A, since the overall rate is not affected by the isotopic substitution. Nevertheless, the irreversible C-H bond cleavage step will give a primary KIE with the other two experiments, since the second step would still affect the product distribution. Therefore, with Experiments B and C, it is possible to observe the KIE even if C-H or C-D bond cleavage occurs not in the rate-determining step, but in the product-determining step.
A large part of the KIE arises from vibrational ZPE differences between the reactant ground state and the transition state that vary between the reactant and its isotopically substituted analog. While one can carry out involved calculations of KIEs using computational chemistry, much of the work done is of simpler order that involves the investigation of whether particular isotopic substitutions produce a detectable KIE or not. Vibrational changes from isotopic substitution at atoms away from the site where the reaction occurs tend to cancel between the reactant and the transition state. Therefore, the presence of a KIE indicates that the isotopically labeled atom is at or very near the reaction site.
The absence of an isotope effect is more difficult to interpret: It may mean that the isotopically labeled atom is away from the reaction site, but it may also mean there are certain compensating effects that lead to the lack of an observable KIE. For example, the differences between the reactant and the transition state ZPEs may be identical between the normal reactant and its isotopically labeled version. Alternatively, it may mean that the isotopic substitution is at the reaction site, but vibrational changes associated with bonds to this atom occur after the rate-determining step. Such a case is illustrated in the following example, in which ABCD represents the atomic skeleton of a molecule.
Assuming steady state conditions for the intermediate ABC, the overall rate of reaction is the following:
If the first step is rate-determining, this equation reduces to:
Or if the second step is rate-determining, the equation reduces to:
In most cases, isotopic substitution at A, especially if it is a heavy atom, will not alter k1 or k2, but it will most probably alter k3. Hence, if the first step is rate-determining, there will not be an observable kinetic isotope effect in the overall reaction with isotopic labeling of A, but there will be one if the second step is rate-determining. For intermediate cases where both steps have comparable rates, the magnitude of the kinetic isotope effect will depend on the ratio of k3 and k2.
Isotopic substitution of D will alter k1 and k2 while not affecting k3. The KIE will always be observable with this substitution since k1 appears in the simplified rate expression regardless of which step is rate-determining, but it will be less pronounced if the second step is rate-determining due to some cancellation between the isotope effects on k1 and k2. This outcome is related to the fact that equilibrium isotope effects are usually smaller than KIEs.
Isotopic substitution of B will clearly alter k3, but it may also alter k1 to a lesser extent if the B-C bond vibrations are affected in the transition state of the first step. There may thus be a small isotope effect even if the first step is rate-determining.
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.[8] One can also, in case of 13C 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.[37] 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:[8] Assume that two isotopic molecules, A1 and A2, undergo irreversible competition reactions:
The KIE for this scenario is found to be:
Where F1 and F2 refer to the fraction of conversions for the isotopic species A1 and A2, respectively.
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:
Since f([B],[C],...) does not depend on the isotopic composition of A, it can be solved for in both rate expressions with A1 and A2, and the two can be equated to derive the following relations:
Where [A1]0 and [A2]0 are the initial concentrations of A1 and A2, respectively. This leads to the following KIE expression:
Which can also be expressed in terms of fraction amounts of conversion of the two reactions, F1 and F2, where 1-Fn=[An]/[An]0 for n = 1 or 2, as follows:
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 [A2]0/[A1]0=R0), in the substrate after some conversion ([A2]/[A1]=R), or in the product ([P2]/[P1]=RP). 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 [A2]0 / ([A1]0+[A2]0) ≈ [A2]0/[A1]0 = R0 in the initial substrate, [A2] / ([A1]+[A2]) ≈ [A2]/[A1] = R in the substrate after some conversion, and [R2] / ([R1]+[R2]) ≈ [R2]/[R1] = RP, 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.[38]
When the substrate composition is followed, the following KIE expression in terms of R0 and R can be derived:
Taking the ratio of R and R0 using the previously derived expression for F2, one gets:
Isotopic enrichment of the starting material can be calculated from the dependence of R/R0 on F1 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.
When the products are followed, the KIE can be calculated