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. 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 and, therefore, lower ground-state vibrational energies, 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 targeted replacement of metabolically vulnerable C–H bonds with "stronger" C–D bonds (e.g., deutetrabenazine).
Background 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 broken as the C–CN bond is formed. 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. 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; 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).
Classification
Primary kinetic isotope effects A primary kinetic isotope effect (PKIE) may be found when a bond to the isotopically labeled atom is being formed or broken. 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.) 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.
Secondary kinetic isotope effects A secondary kinetic isotope effect (SKIE) is observed when no bond to the isotopically labeled atom in the reactant is broken or formed. 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.
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.
Theory
General theory 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. 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. 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, allowing both primary and secondary isotope effects to be easily measured and interpreted. In contrast, primary KIE are small and secondary KIE are very small for heavier elements, necessitating special experimental techniques to measure them precisely and complicating their interpretation. In the context of isotope effects, hydrogen often means the light isotope, protium (1H), specifically. In the rest of this article, reference to hydrogen (H) and deuterium (D) in the subscripts of equations and in parallel constructions or direct comparisons in the text should be interpreted to refer to 1H and 2H. The theory of KIEs was first formulated by Jacob Bigeleisen in 1949. 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–1H bond. In the equation, subscript H or D refer to the species with 1H or 2H (or, more generally, the light and heavy isotopes), respectively; quantities with or without the double-dagger, ‡, refer to transition state or reactant ground state, respectively. In its full form, the Bigeleisen equation gives the KIE as
k H k D = ( σ H σ D ‡ σ D σ H ‡ ) ( M H ‡ M D M D ‡ M H ) 3 2 ( I x H ‡ I y H ‡ I z H ‡ I x D ‡ I y D ‡ I z D ‡ I x D I y D I z D I x H I y H I z H ) 1 2 ( ∏ i = 1 3 N ‡ − 7 1 − e − u i D ‡ 1 − e − u i H ‡ ∏ i = 1 3 N − 6 1 − e − u i D 1 − e − u i H ) e − 1 2 [ ∑ i = 1 3 N ‡ − 7 ( u i H ‡ − u i D ‡ ) − ∑ i = 1 3 N − 6 ( u i H − u i D ) ] {\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 (for X = H or D, with or without ‡)
u i X ( ‡ ) := h ν i X ( ‡ ) k B T = h c N A ν ~ i X ( ‡ ) R T {\displaystyle u_{i\mathrm {X} }^{(\ddagger )}:={\frac {h\nu _{i\mathrm {X} }^{(\ddagger )}}{k_{\mathrm {B} }T}}={\frac {hcN_{\mathrm {A} }{\tilde {\nu }}_{i\mathrm {X} }^{(\ddagger )}}{RT}}} . Here, h = Planck constant; kB = Boltzmann constant; ν ~ i X {\displaystyle {\tilde {\nu }}_{i\mathrm {X} }} = frequency of vibration, expressed in wavenumber; c = speed of light; NA = Avogadro constant; and R = universal gas constant. The σX are the symmetry numbers for the reactants and transition states. The MX are the molecular masses of the corresponding species, and the IqX (for q = x, y, or z) terms are the moments of inertia about the three principal axes. The reduced frequencies uiX are dimensionless quantities directly proportional to the corresponding vibrational frequencies, νiX, and the vibrational zero-point energies (ZPE) (see below). The integers N and N‡ are the number of atoms in the reactants and the transition states, respectively. The complicated expression given above can be interpreted as the product of four separate factors:
k H k D = S × M M I × E X C × Z P E {\displaystyle {\frac {k_{{\ce {H}}}}{k_{{\ce {D}}}}}=\mathbf {S} \times \mathbf {MMI} \times \mathbf {EXC} \times \mathbf {ZPE} } , the meanings of which are described in the next section. (Strictly speaking, an additional κ H / κ D {\displaystyle \kappa _{\mathrm {H} }/\kappa _{\mathrm {D} }} term resulting from an isotopic difference in transmission coefficients should also be included.)
Simplifying approximations 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, resulting in a greatly simplified form of the Bigeleisen equation. The first factor S (containing the 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. 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) is a function of the ratios 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). For hydrogen/deuterium KIEs, the observed values are then typically dominated by the last factor, ZPE, an exponential function of vibrational zero-point energy (ZPE) differences:
k H k D ≈ Z P E = exp { − 1 2 [ ∑ i = 1 3 N ‡ − 7 ( u i H ‡ − u i D ‡ ) − ∑ i = 1 3 N − 6 ( u i H − u i D ) ] } = exp [ ∑ i ( r e a c t . ) 1 2 Δ u i − ∑ i ( T S ) 1 2 Δ u i ‡ ] {\displaystyle {\begin{aligned}{\frac {k_{{\ce {H}}}}{k_{{\ce {D}}}}}\approx \mathbf {ZPE} &=\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\}\\&=\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 (with or without ‡)
Δ u i ( ‡ ) := u i H ( ‡ ) − u i D ( ‡ ) {\displaystyle \Delta u_{i}^{(\ddagger )}:=u_{i\mathrm {H} }^{(\ddagger )}-u_{i\mathrm {D} }^{(\ddagger )}}
to be the isotopic differences in reduced frequencies. 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 either 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 Δ u i {\displaystyle \Delta u_{i}} and Δ u i ‡ {\displaystyle \Delta u_{i}^{\ddagger }} that are nearly equal, and cancellations occur when the sums in the exponent are calculated. As a result, 2H KIEs often depend largely on a handful of key vibrational modes that do not cancel, making qualitative predictions and analyses of kH/kD possible. 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. 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}} , and so the isotopic difference in ZPE is given by ( Δ Z P E ) i = 1 2 k B T Δ u i {\displaystyle (\Delta \mathrm {ZPE} )_{i}={\frac {1}{2}}k_{\mathrm {B} }T\Delta u_{i}} . Thus, we can readily interpret the difference between reactant and transition state of the sums of the 1 2 Δ u i {\displaystyle {\frac {1}{2}}\Delta u_{i}} over vibrational modes i that appears in the simplified Bigeleisen equation above as accounting for the collective changes in the isotopic differences in ZPE on progressing from the reactant to the transition state. More precisely, the ZPE term takes the sum of the isotopic ZPE differences over the vibrational modes of the reactants and the sum over the vibrational modes of the transition state, exponentiating the difference between the two (after scaling by 1/kBT):
Z P E = exp [ 1 k B T ( ∑ i ( r e a c t . ) ( Δ Z P E ) i − ∑ i ( T S ) (