Editing Kinetic isotope effect (section)

=== Tunneling ===
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.<ref name="anslyn">{{cite book | vauthors = Anslyn EV, Dougherty DA
 |year=2006
 |title=Modern Physical Organic Chemistry
 | url = https://archive.org/details/modernphysicalor00ansl
 | url-access = limited
 |publisher= University Science Books
 |pages=[https://archive.org/details/modernphysicalor00ansl/page/n455 428]–437
 |isbn=978-1-891389-31-3
}}</ref><ref name="razauy">{{cite book
 | vauthors = Razauy M
 |year=2003
 |title=Quantum Theory of Tunneling
 | url = https://archive.org/details/quantumtheoryoft0000raza
 | url-access = registration
 |publisher=[[World Scientific]]
 |isbn=978-981-238-019-7
}}</ref> Though not allowed by [[classical mechanics]], particles can pass through classically forbidden regions of space in quantum mechanics based on [[wave–particle duality]].<ref name="silbey">{{cite book
 | vauthors = Silbey RJ, Alberty RA, Bawendi MG
 |year=2005
 |title=Physical Chemistry
 |pages=326–338
 |publisher=[[John Wiley & Sons]]
 |isbn=978-0-471-21504-2
}}</ref>

[[File:Energy Well Model.png|thumb|550px|right|The potential energy well of a tunneling reaction. The dash-red arrow shows the classical activated process, while the solid-red arrow shows the tunneling path.<ref name="anslyn" />]]

Tunneling can be analyzed using Bell's modification of the [[Arrhenius equation]], which includes the addition of a tunneling factor, Q:
:<math>k=QAe^{-E/RT}</math>
where A is the Arrhenius parameter, E is the barrier height and
:<math>Q = \frac{e^\alpha}{\beta-\alpha}(\beta e^{-\alpha}-\alpha e^{- \beta})</math>
where <math>\alpha=\frac{E}{RT}</math> and <math>\beta=\frac{2a \pi ^2(2mE)^{1/2}}{h}</math>

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.
[[File:Donor Acceptor Model.png|thumb|250px|center|Donor-acceptor model of a proton transfer.<ref name=Borgis>{{cite journal
 | vauthors = Borgis D, Hynes JT
 |year=1993
 |title=Dynamical theory of proton tunneling transfer rates in solution: General formulation
 |journal=[[Chemical Physics]]
 |volume=170 |issue=3 |pages=315–346
 |bibcode=1993CP....170..315B
 |doi=10.1016/0301-0104(93)85117-Q
}}</ref>]]

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&nbsp;pm.<ref name="Krishtalik_2000">{{cite journal | vauthors = Krishtalik LI | title = The mechanism of the proton transfer: an outline | journal = Biochimica et Biophysica Acta (BBA) - Bioenergetics | volume = 1458 | issue = 1 | pages = 6–27 | date = May 2000 | pmid = 10812022 | doi = 10.1016/S0005-2728(00)00057-8 | doi-access = free }}</ref>

{{hidden|toggle=left|1=Temperature dependence in tunneling|2=
[[Quantum tunneling|Tunneling]] is a quantum effect tied to the laws of wave mechanics, not [[kinetics (physics)|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.<ref name="anslyn" />

Peter S. Zuev et al. reported rate constants for the ring expansion of 1-methylcyclobutylfluorocarbene to be 4.0 × 10{{sup|−6}}/s in nitrogen and 4.0 × 10{{sup|−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.<ref name=Zuev>{{cite journal | vauthors = Zuev PS, Sheridan RS, Albu TV, Truhlar DG, Hrovat DA, Borden WT | title = Carbon tunneling from a single quantum state | journal = Science | volume = 299 | issue = 5608 | pages = 867–70 | date = February 2003 | pmid = 12574623 | doi = 10.1126/science.1079294 | bibcode = 2003Sci...299..867Z | s2cid = 20959068 }}</ref>

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.<ref name="anslyn" /><ref name="Atkins">{{cite book
 | vauthors = Atkins P, de Paula J
 |year=2006
 |edition = 8th
 |title=Atkins' Physical Chemistry
 | url = https://archive.org/details/atkinsphysicalch00atki
 | url-access = limited
 |pages=[https://archive.org/details/atkinsphysicalch00atki/page/n318 286]–288, 816–818
 |publisher=[[Oxford University Press]]
 |isbn=978-0-19-870072-2
}}</ref>

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 ''A''{{sub|H}}/''A''{{sub|D}} 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.<ref name=Fujisaki>{{cite journal
 | vauthors = Fujisaki N, Ruf A, Gaeumann T
 |year=1987
 |title=Tunnel effects in hydrogen-atom-transfer reactions as studied by the temperature dependence of the hydrogen deuterium kinetic isotope effects
 |journal=[[Journal of Physical Chemistry]]
 |volume=91 |issue=6 |pages=1602–1606
 |doi=10.1021/j100290a062
}}</ref>

The observation that tunneling is not entirely temperature independent can be explained by the fact that not all molecules of a given species occupy their vibrational ground state at varying temperatures. Adding thermal energy to a potential energy well could cause higher vibrational levels than the ground state to become populated. For a conventional kinetically driven reaction, this excitation would only have a small influence on the rate. However, for a tunneling reaction, the difference between the ZPE and the first vibrational energy level could be huge. The tunneling correction term ''Q'' is linearly dependent on barrier width and this width is significantly diminished as the number [[vibrational modes]] on the [[Morse potential]] increase. The decrease of the barrier width can have such a huge impact on the tunneling rate that even a small population of excited vibrational states would dominate this process.<ref name="anslyn" /><ref name="Atkins" />
}}

{{hidden|toggle=left|1=Criteria for KIE tunneling|2=
To determine if tunneling is involved in KIE of a reaction with H or D, a few criteria are considered:
# Δ(''E{{sub|a}}''{{sup|H}}-''E{{sub|a}}''{{sup|D}}) > Δ(''ZPE''{{sup|H}}-''ZPE''{{sup|D}}) (''E{{sub|a}}''=activation energy; ZPE=zero point energy)
# Reaction still proceeds at lower temperatures.
# The [[Arrhenius equation|Arrhenius]] pre-exponential factors ''A''{{sub|D}}/''A''{{sub|H}} is not equal to 1.
# A large negative [[entropy]] of activation.
# The geometries of the reactants and products are usually very similar.<ref name="anslyn" />
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:
:''k''{{sub|H}}/''k''{{sub|T}}=(''k''{{sub|D}}/''k''{{sub|T}})''{{sup|X}}'' and  ''k''{{sub|H}}/''k''{{sub|T}}=(''k''{{sub|H}}/''k''{{sub|D}})''{{sup|Y}}''
Experimental values of X exceeding 3.26 and Y lower than 1.44 are evidence of a certain amount of contribution from tunneling.<ref name="Krishtalik_2000" /><ref name="Laidler_1987" />{{rp|437–8}}
}}

{{hidden|toggle=left|1=Examples for tunneling in KIE|2=
In organic reactions, this proton tunneling effect has been observed in such reactions as the [[deprotonation]] and iodination of [[nitropropane]] with hindered [[pyridine]] base<ref>{{cite journal
 | vauthors = Lewis ES, Funderburk L
 |year=1967
 |title=Rates and isotope effects in the proton transfers from 2-nitropropane to pyridine bases
 |journal=[[Journal of the American Chemical Society]]
 |volume=89 |issue=10 |pages=2322–2327
 |doi=10.1021/ja00986a013
|bibcode=1967JAChS..89.2322L
 }}</ref> with a reported KIE of 25 at 25°C:

:[[File:KIE effect iodination.png|400px|KIE effect iodination]]

and in a [[Sigmatropic reaction|1,5-sigmatropic hydrogen shift]],<ref>{{cite journal
 | vauthors = Dewar MJ, Healy EF, Ruiz JM
 |year=1988
 |title=Mechanism of the 1,5-sigmatropic hydrogen shift in 1,3-pentadiene
 |journal=[[Journal of the American Chemical Society]]
 |volume=110 |issue=8 |pages=2666–2667
 |doi=10.1021/ja00216a060
|bibcode=1988JAChS.110.2666D
 }}</ref> though it is observed that it is hard to extrapolate experimental values obtained at high temperature to lower temperatures:<ref>{{cite journal | vauthors = von Doering W, Zhao X | title = Effect on kinetics by deuterium in the 1,5-hydrogen shift of a cisoid-locked 1,3(Z)-pentadiene, 2-methyl-10-methylenebicyclo[4.4.0]dec-1-ene: evidence for tunneling? | journal = Journal of the American Chemical Society | volume = 128 | issue = 28 | pages = 9080–5 | date = July 2006 | pmid = 16834382 | doi = 10.1021/ja057377v | bibcode = 2006JAChS.128.9080D }}</ref><ref>In this study the KIE is measured by sensitive [[proton NMR]]. The extrapolated KIE at 25°C is 16.6 but the margin of error is high</ref>

:[[File:KIE effect sigmatropicReaction 2006.png|400px|KIE effect sigmatropic reaction]]
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.<ref name=Kohen>{{cite journal | vauthors = Kohen A, Klinman JP | title = Hydrogen tunneling in biology | journal = Chemistry & Biology | volume = 6 | issue = 7 | pages = R191-8 | date = July 1999 | pmid = 10381408 | doi = 10.1016/S1074-5521(99)80058-1 | doi-access = free }}</ref> 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.<ref name=pollack>{{cite journal | vauthors = Wilde TC, Blotny G, Pollack RM | title = Experimental evidence for enzyme-enhanced coupled motion/quantum mechanical hydrogen tunneling by ketosteroid isomerase | journal = Journal of the American Chemical Society | volume = 130 | issue = 20 | pages = 6577–85 | date = May 2008 | pmid = 18426205 | doi = 10.1021/ja0732330 | bibcode = 2008JAChS.130.6577W }}</ref>

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.<ref name=Villa>{{cite journal
 | vauthors = Truhlar DG, Gao J, Alhambra C, Garcia-Viloca M, Corchado J, Sánchez M, Villà J
 |year=2002
 |title=The Incorporation of Quantum Effects in Enzyme Kinetics Modeling
 |journal=[[Accounts of Chemical Research]]
 |volume=35 |issue=6 |pages=341–349
 |doi=10.1021/ar0100226
|pmid=12069618
 }}</ref>

[[File:Methylamine dehydrogenase.gif|thumb|2000px|center|Mechanism of methylamine dehydrogenase, the [[quinoprotein]] converts primary amines to aldehyde and ammonia.]]

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.<ref name=KlinmanKohen>{{cite journal
 | last1= Kohen |first1=A|last2=Klinman|first2=J. P|authorlink2=Judith Klinman
 |year=1998
 |title=Enzyme Catalysis: Beyond Classical Paradigms
 |journal=[[Accounts of Chemical Research]]
 |volume=31 |issue=7 |pages=397–404
 |doi=10.1021/ar9701225
}}</ref>
[[File:Alcohol dehydrogenase.png|thumb|800px|center|Mechanism of alcohol dehydrogenase. The rate-limiting step is the proton transfer.]]
}}