Kinetic isotope effect

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, KIEs 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 KIE 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. Thus, replacing normal hydrogen (¹H) with its isotope deuterium (D or ²H) means a 100% increase in 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 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

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 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 tunnelling 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 KIE. The accurate prediction of the numerical value of a ²H KIE 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 ²H and ¹H and the attendant differences in vibrational frequencies, the magnitude of the isotope effect is larger than 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 is often 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.[lower-alpha 1]

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 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][12] (Strictly speaking, a ${\displaystyle \kappa _{\mathrm {H} }/\kappa _{\mathrm {D} }}$ term resulting from an isotopic difference in transmission coefficients should also be included.[13])

${\displaystyle {\frac {k_{{\ce {H}}}}{k_{{\ce {D}}}}}=\left({\frac {\sigma _{{\ce {H}}}\sigma _{{\ce {D}}}^{\ddagger }}{\sigma _{{\ce {D}}}\sigma _{{\ce {H}}}^{\ddagger }}}\right)\left({\frac {M_{{\ce {H}}}^{\ddagger }M_{{\ce {D}}}}{M_{{\ce {D}}}^{\ddagger }M_{{\ce {H}}}}}\right)^{\frac {3}{2}}\left({\frac {I_{x{\ce {H}}}^{\ddagger }I_{y{\ce {H}}}^{\ddagger }I_{z{\ce {H}}}^{\ddagger }}{I_{x{\ce {D}}}^{\ddagger }I_{y{\ce {D}}}^{\ddagger }I_{z{\ce {D}}}^{\ddagger }}}{\frac {I_{x{\ce {D}}}I_{y{\ce {D}}}I_{z{\ce {D}}}}{I_{x{\ce {H}}}I_{y{\ce {H}}}I_{z{\ce {H}}}}}\right)^{\frac {1}{2}}\left({\frac {\prod \limits _{i=1}^{3N^{\ddagger }-7}{\frac {1-e^{-u_{i{\ce {D}}}^{\ddagger }}}{1-e^{-u_{i{\ce {H}}}^{\ddagger }}}}}{\prod \limits _{i=1}^{3N-6}{\frac {1-e^{-u_{i{\ce {D}}}}}{1-e^{-u_{i{\ce {H}}}}}}}}\right)e^{-{\frac {1}{2}}\left[\sum \limits _{i=1}^{3N^{\ddagger }-7}(u_{i{\ce {H}}}^{\ddagger }-u_{i{\ce {D}}}^{\ddagger })-\sum \limits _{i=1}^{3N-6}(u_{i{\ce {H}}}-u_{i{\ce {D}}})\right]}}$,

where we define

${\displaystyle u_{i}:={\frac {h\nu _{i}}{k_{\mathrm {B} }T}}={\frac {hcN_{\mathrm {A} }{\tilde {\nu }}_{i}}{RT}}}$ and ${\displaystyle u_{i}^{\ddagger }:={\frac {h\nu _{i}^{\ddagger }}{k_{\mathrm {B} }T}}={\frac {hcN_{\mathrm {A} }{\tilde {\nu }}_{i}^{\ddagger }}{RT}}}$.

Here, h is the Planck constant, k_B is the Boltzmann constant, ${\displaystyle {\tilde {\nu }}_{i}}$ is the frequency of the vibration, expressed in wavenumbers, c is the speed of light, N_A is the Avogadro constant, and R is the universal gas constant. The σ_X (X = H or D) are the symmetry numbers for the reactants and transition states. The M_X are the molecular masses of the corresponding species, and the I_qX (q = x, y, or z) terms are the moments of inertia about the three principal axes. The u_iX are directly proportional to the corresponding vibrational frequencies, ν_i, 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]

${\displaystyle {\frac {k_{{\ce {H}}}}{k_{{\ce {D}}}}}=\mathbf {S} \times \mathbf {MMI} \times \mathbf {EXC} \times \mathbf {ZPE} }$.

For the special case of deuterium 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 the σ_X) 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 σ_X (reactant and transition state) will be unity; thus S can often be neglected. The MMI factor (containing the M_X and I_qX) 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(-u_i) = exp(-hν_i/k_BT) < 0.01 at 298 K, resulting in negligible contributions from the 1–exp(-u_i) 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]

${\displaystyle {\begin{aligned}{\frac {k_{{\ce {H}}}}{k_{{\ce {D}}}}}&\cong \exp \left\{-{\frac {1}{2}}\left[\sum \limits _{i=1}^{3N^{\ddagger }-7}(u_{i{\ce {H}}}^{\ddagger }-u_{i{\ce {D}}}^{\ddagger })-\sum \limits _{i=1}^{3N-6}(u_{i{\ce {H}}}-u_{i{\ce {D}}})\right]\right\}\\&\cong \exp \left[\sum _{i}^{\mathrm {(react.)} }{\frac {1}{2}}\Delta u_{i}-\sum _{i}^{\mathrm {(TS)} }{\frac {1}{2}}\Delta u_{i}^{\ddagger }\right]\end{aligned}}}$,

where we define

${\displaystyle \Delta u_{i}:=u_{i\mathrm {H} }-u_{i\mathrm {D} }}$ and ${\displaystyle \Delta u_{i}^{\ddagger }:=u_{i\mathrm {H} }^{\ddagger }-u_{i\mathrm {D} }^{\ddagger }}$.

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. Alternatively, 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 ${\displaystyle \Delta u_{i}}$and ${\displaystyle \Delta u_{i}^{\ddagger }}$, and cancellations occur when the sums in the exponent are calculated. Thus, in practice, deuterium KIEs are often largely dependent on a handful of key vibrational modes because of this cancellation, making qualitative analyses of k_H/k_D possible.[12]

As mentioned, especially for hydrogen/deuterium substitution, most KIEs arise from the difference in ZPE between the reactants and the transition state of the isotopologues in question, and this difference can be understood qualitatively with the following description: within the Born–Oppenheimer approximation, the potential energy surface is the same for both isotopic species. However, a quantum-mechanical 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 Heisenberg 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 ${\displaystyle \epsilon _{i}^{(0)}={\frac {1}{2}}h\nu _{i}}$. Thus, we can readily interpret the factor of ½ and the sums of ${\displaystyle u_{i}=h\nu _{i}/k_{\mathrm {B} }T}$ terms over ground state and transition state vibrational modes in the exponent of the simplified formula above. For a harmonic oscillator, the vibrational frequency is inversely proportional to the square root of the reduced mass of the vibrating system:

${\displaystyle \nu _{\mathrm {X} }={\frac {1}{2\pi }}{\sqrt {\frac {k_{\mathrm {f} }}{\mu _{\mathrm {X} }}}}\cong {\frac {1}{2\pi }}{\sqrt {\frac {k_{\mathrm {f} }}{m_{\mathrm {X} }}}}}$,

where k_f 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 m_D is approximately 2m_H,

${\displaystyle \Delta u_{i}\cong \left(1-{\frac {1}{\sqrt {2}}}\right){\frac {h\nu _{i\mathrm {H} }}{k_{\mathrm {B} }T}}}$.

In the case of a homolytic C–H/D bond dissociation, the transition state term disappears, and neglecting other vibrational modes, k_H/k_D = exp(½Δu_i). 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 deuterium KIE, we consider the case in which the ZPE difference between the stretching vibrations of a typical carbon-hydrogen bond (3000 cm⁻¹) and carbon-deuterium bond (2200 cm⁻¹) disappears in the transition state (an energy difference of (1/2)(3000 – 2200 cm⁻¹) = 400 cm⁻¹, or about 1.15 kcal/mol), without any compensation from a zero-point energy difference at the transition state (e.g., from the symmetric A···H···B stretch, which is unique to the transition state). The simplified formula given above predicts a maximum for k_H/k_D as 6.9. If the complete disappearance of two bending vibrations is also included, k_H/k_D 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 k_H/k_D 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 k_H/k_D ~ 10 is thought to be maximal for a semi-classical primary kinetic isotope effect (no tunneling) for reactions taking place around 298 K. (The formula for k_H/k_D 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 (symmetrical vs. "early" or "late" and linear vs. bent), the extent to which a primary deuterium 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 deuterium 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 secondary kinetic isotope effect), or bond weakening due to hyperconjugation in cases where a carbocation is being generated one carbon atom away (a β secondary KIE). These isotope effects have a theoretical maximum of k_H/k_D = 2^0.5 ≈ 1.4. For a secondary KIE 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 k_H/k_D = 2^-0.5 ≈ 0.7. In practice, k_H/k_D ~ 1.1-1.2 and k_H/k_D ~ 0.8-0.9 are typical for α secondary kinetic isotope effects, while k_H/k_D ~ 1.15-1.3 are typical for β secondary kinetic isotope effect. For reactants containing several isotopically substituted β-hydrogen atoms, 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 k_H/k_D > 2 are not uncommon.[17]

The following simple expressions relating deuterium and tritium kinetic isotope effects, 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]

${\displaystyle \left({\frac {\ln \left({\frac {k_{{\ce {H}}}}{k_{{\ce {T}}}}}\right)}{\ln \left({\frac {k_{{\ce {H}}}}{k_{{\ce {D}}}}}\right)}}\right)_{s}\cong {\frac {1-{\sqrt {m_{{\ce {H}}}/m_{{\ce {T}}}}}}{1-{\sqrt {m_{{\ce {H}}}/m_{{\ce {D}}}}}}}={\frac {1-{\sqrt {1/3}}}{1-{\sqrt {1/2}}}}\cong 1.44}$;

i.e.,

${\displaystyle \left({\frac {k_{{\ce {H}}}}{k_{{\ce {T}}}}}\right)_{s}=\left({\frac {k_{{\ce {H}}}}{k_{{\ce {D}}}}}\right)_{s}^{1.44}}$.

In deriving these expressions, the reasonable approximation that reduced masses are roughly equal to the masses of the hydrogen, deuterium, or tritium, was used. Also, the vibrational motion was assumed to be approximated by a harmonic oscillator, so that ${\displaystyle u_{i\mathrm {X} }\propto \mu _{\mathrm {X} }^{-1/2}\cong m_{\mathrm {X} }^{-1/2}}$ (X = H, D, or T). The subscript "s" refers to these "semi-classical" kinetic isotope effects, 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 more difficult to rationalize or interpret. In many cases and especially for hydrogen-transfer reactions, contributions to KIEs from tunneling are significant (see below).

Simmons and Hartwig refer to the following three cases as the main types of kinetic isotope effect experiments involving C-H bond functionalization:[5]

A) KIE determined from absolute rates of two parallel reactions

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.)

B) KIE determined from an intermolecular competition

In this type of experiment, the same substrates that are used in Experiment A are employed, but they are allowed in to react in the same container, instead of two separate containers. The kinetic isotope effect from 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). It is necessary to quench the reaction before it goes to completion to observe the kinetic isotope effect (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 kinetic isotope effect, at least according to Simmons and Hartwig, is nonetheless indicative of the C-H bond cleavage not occurring during the rate-determining step.

C) KIE determined from an intramolecular competition

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 kinetic isotope effect 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 does not need to be halted at low consumption of isotopic starting material to obtain an accurate k_H/k_D, 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 for which there is 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 will depend 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 kinetic isotope effect 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 kinetic isotope effect:

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 kinetic isotope effect 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 kinetic isotope effect 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 kinetic isotope effect even if C-H or C-D bond cleavage occurs not in the rate-determining step, but in the product-determining step.

• Crossover experiment (chemistry)
• Equilibrium constant#Effect of isotopic substitution
• Isotope effect on lipid peroxidation
• Kinetic isotope effects of RuBisCO (ribulose-1,5-bisphosphate carboxylase oxygenase)
• Magnetic isotope effect
• Reaction mechanism
• Transient kinetic isotope fractionation
• Urey–Bigeleisen–Mayer equation