Technical Supplement 7A
A Chemist’s View of Isotope Effects
Or
Calculating Isotope Effects to Study Reaction Mechanisms
June 20, 2003
From: Dr. Arthur Fry (with some ghostwritten editorial changes)
To: Mr. Polychaete,
Since you asked, I decided to write you this little discourse about kinetic isotope
effects (KIEs). The first part presents a fundamental equation for KIEs. This Bigeleisen
equation was first published in 1949 but still provides a good starting point for
understanding detailed isotope effects involved when chemical bonds change. The second
part of this letter what really happens when Fractionation Frank is on the loose at the
heart of the chemical reactions, at the activated complex. The Bigeleisen equation help
pin down the action, identifying which atoms are actually involved in bond cleavage and
bond formation as substrate converts to product.
1. The Bigeleisen Equation
The whole approach to studying organic reactions with isotopes is based on the
theory of absolute reaction rates. This theory discusses kinetics of reactions in terms of a
pseudo equilibrium between reactants and an activated complex. In this theory, the
activated complex is a molecule that sits at the top of the energy barrier between reactants
and products. The activated complex is just like any other molecule, except that one
aspect of the bond in this molecule has changed. Specifically, one of the vibrational
frequencies of the bond has been converted into a translational frequency, putting the
molecule on the path to becoming a final product. So, thinking about changes in bonds,
we focus in on vibrations and vibrational frequencies. The vibrational frequencies are
calculated from bond force constants and the masses of the bonded atoms. For labeled,
isotope-substituted molecules, the bonding force constants do not change, but the masses
do, and so do the vibrational frequencies and vibrational energies. Vibrational
frequencies can be measured in simple molecules via infrared spectroscopy of pure
compounds. Using this input, you and your computer can calculate these vibrational
frequencies for different types of simple and more complex molecules. These vibrational
frequencies are then used to calculate the KIEs, and a famous early isotope effect
calculation of this sort was the “Bigeleisen Equation”.
I should note that today there are much more sophisticated methods for these
calculations. Calculations of kinetic isotope effects are relatively simple computer
calculations, as are calculations for vibrational frequencies and isotope dependencies of
these vibrational frequencies. The calculations for vibrational frequencies of complex
molecules are the hardest part, exceedingly more complex than the isotope effect
calculations themselves. You can imagine the difficulties by envisioning multiple springs
all linked together – how does one accurately calculate the vibrational frequencies of the
whole molecule? So, if you want the most up-to-date guides to these calculations, you
should probably consult the internet. That said, let’s look at the “easy” part of the old
basic calculations.
The version of the Bigeleisen equation I will use is this one:
k1/k2 = (1L‡/2L‡) [1 + in G(ui)ui - in G(ui‡)ui‡]
terms:
1) k1 and k2 are the kinetic reaction rates for molecules containing the light isotope and
the heavy isotope, respectively, with the lighter isotope expected to react faster so that k1
> k2 and k1/k2 >1,
2) 1‡ and 2‡ are reaction coordinate terms that apply to the activated complex through
which molecules pass as bonds are broken or formed, with 1 and 2 referring to the light
and heavy isotopes, respectively. The "" is actually the pseudo vibrational frequency
along the reaction coordinate - the path that leads from reactions to products. The "" is a
vibrational frequency that “goes missing” if a bond breaks, and is "partially there", sort
of, if a bond forms. The “L” subscripts are often omitted, but also reflect progress of the
reaction.
3) The first term inside the brackets is unity – the second and third add to or subtract from
it. The second term inside the brackets refers to the reactants, and the third term (carrying
the double dagger symbol, ‡) refers to the activated complex. G(ui) is a an energy term
from quantum mechanics and combines the numerical function (G) and the energy value
of the “i”th vibrational bond frequency ui.
ui is the difference in vibrational energies of a bond for the light and heavy isotopes,
with vibrational frequencies being the measured variables used to estimate the vibrational
energies,
5) G(ui‡)ui‡, as definitions 3) and 4) above, but this time for the activated complex
denoted by (‡).
6) Concerning the summation terms inside the brackets in the above equation, the
following gives more detail than you might want to know. The summation term refers to
summing over all the vibrational states. Summation is based on realizing that every
molecule that contains n atoms has 3n degrees of freedom corresponding to three kinds of
bond energies, the rotational, translational and vibrational energies. (With the exception
of the hydrogen isotope calculations where translational effects are important (and where
isotopes sometimes cheat and “tunnel” through energy barriers using their magic
quantum powers, rather than bothering to go up and over the energy hill), the isotope
effect calculations focus only on the vibrational energies, but for completeness, all the
energy states are included in the Bigeleisen equation). That said, three of the n are
rotations, and 3 are translations. All the rest are vibrations, so there are 3n-6 vibrations
(except for linear molecules for which there are only two rotational degrees of freedom,
resulting in 3n-5 vibrations instead of 3n-6). For the activated complex, the number for
vibrations is 3n-7, with the missing vibration assigned to the developing action in the
breaking or formation of the bond. The activated complex has only 3n-7 real vibrational
frequencies, not the usual 3n-6, because the "vibration along the reaction coordinate" is
not there anymore. So, in the end, the upshot of all this is that when there is an isotope
dependent effect in the reaction, the first summation in the brackets will be larger than the
second summation, because there are fewer vibrational energy terms in the second
summation where one of the vibrational frequencies is changed into a translational
frequency.
Summarizing to this point, the focus is on two basic parts to the Bigeleisen
equation, the part concerning the substrate and the part concerning the activated complex
(or product in equilibrium calculations). Comparison of energies in the substrate vs.
activated complex gives the predicted isotope effect, and the approach is based on a
quantum mechanical view of reactions, with reaction rates derived from various
“partition functions”, the summation terms above. The calculation is especially based on
differences in vibrational energies of bonds. The vibrational energies are the prominent
energies differing between bonds with light and heavy isotopes, at least for the heavier,
non-hydrogen elements.
Let’s conclude this presentation of the Bigeleisen equation with a few last notes
about certain terms. First, there is a "reaction coordinate" term (1L‡/2L‡) in front of the
bracketed term. This is often taken as the “reduced mass ratio” of, for example, a
12
C-to-12C bond vs. a 12C-to-13C bond in a simple bond breaking process =
Square root of [(12*13/25)*(12*12/24)] = 1.0198.
The “reduced mass” calculation comes from the physics of springs that vibrate (like
slinkies), or “simple harmonic oscillators” (or “SHO”s; Criss 1999). We chemists
borrowed this nifty equation from the physicists.
For other cases, the “reaction coordinate” involves an estimate for the isotope
dependence of the vibration in the reactant that becomes a translation in the activated
complex. Evaluation of this term was always one of the weak spots in the move from
"qualitative" to "semi-quantitative" in using the Bigeleisen equation. Bigeleisen has
always claimed that this term is always greater than one, regardless of the nature of the
reaction--I've always had some reservations about that for bond formation cases, but I've
generally accepted it.
Inside the brackets, you add to unity, a summation partition function term
(containing 3n-6 vibrational frequencies; the first G term) for the reactions and subtract a
similar summation partition function term for the activated complex (the second G term
containing 3n-7 vibrational frequencies, with the missing one being the one that becomes
the translation). Note that the G(u) terms are the vibrational frequencies expressed in
energy terms, and the u terms are the isotope shifts (energy terms) for each of those
frequencies.
2. Applications to Studies of Reactions Mechanisms
With a general understanding of the Bigeleisen equation developed above, now
let’s consider how this complex equation can be applied to a fundamental problem facing
organic chemists, specifically what is the structure of the activated complex in reactions.
One can deduce this hard-to-observe activated complex by KIE studies as follows.
First consider the vibrations of a complex molecule, the G(u) terms in the
Bigeleisen equation above, and the isotope dependence of each of them = the u terms.
If there is no labeling, all u terms are zero, so the term in brackets is just unity (and the
term in front of the brackets is also unity (no labeling, so no "labeling effect") so the ratio
of rate constants is unity by definition.
Now, put a label somewhere in the molecule. Each of the vibrations of the
molecule includes every atom in the molecule, but an excellent first approximation is that
all vibrations involve mainly a single bond with minor effects extending to nearby bonds,
and practically negligible effects on more distant bonds. ("Secondary" isotope effect
arise from these more distant effects, and are important only with deuterium and tritium
studies. Here think in terms of a bunch of interconnected springs. That is, stretch one, and
the whole molecule wiggles, but the wiggle is most where you stretched it).
Now consider two cases. (1) If the label is at a position in the molecule where
"nothing is happening" in the reaction (mechanism) as reactants form from products, one
of the vibrational frequencies in the first sumation term will be missing in the second G
term, so there is no contribution from the second G term. For that particular frequency, u
= 0 (by definition--it is missing), so it contributes nothing to the sum, neither positive nor
negative. For that specific frequency in the first G term, by definition, u = 0, so "its
term" makes no contribution to the first G term either. All the other 3n-7 frequencies in
the first summation term will have positive contributions, but for each of them there will
be a corresponding IDENTICAL negative term from the second summation, so the term
in brackets is also unity. "Nothing is happening" at a labeled position (by definition) so
the term in front of the brackets for the one vibrational frequency that becomes a
translation is also unity, and there is no isotope effect and the rate constant ratio = unity.
(2) If the label is at the position in the molecule where the reaction is "happening", that
frequency will have a u and a u term in the first G term, but it is "gone" (u = 0) (by
definition) in the second G term. It has become a translation and is "handled" by the term
in front of the brackets. But all the other 3n-7 vibrations have positive G(u)u terms in
the first G term are exactly matched by negative G(u)u terms in the second G term.
Therefore the term in the brackets will be unity plus (just) one, positive summation
G(u)u term, so that in the end, the rate constant will be greater than unity. Notes: 1) All
G(u)u terms are, by their nature, positive, and 2) there are complications on this point in
intramolecular cases, which I have studied a lot but will not go into.
Now, having waded through these mechanics, let’s think about an example where
you wanted to study not the reactants or the products, but the transition complex which is
very evanescent and hard to observe by any known means, except via isotopes. If you
apply the steps outlined in the above paragraph to the same molecule but now labeled in a
different position, then you will have a kinetic isotope effect when and only when there
are bonding changes in going from reactants to the activated complexes. By switching
the position of the label around in a molecule (chemists do this for fun and fame, to show
they are worth their salt as organic chemists), you can successively identify all the atomic
positions in a molecule which undergo bonding changes in going from reactants to
activated complexes by whether or not (qualitative argument only needed) isotope effects
are observed with the molecule labeled at each of those positions. In principle, also one
can derive all of the structural features of the reaction's activated complex by focusing on
the magnitudes of isotope effects, and on the comparisons of observed and calculated
isotope effects for different reactant and activated complex models.
As it turns out, it is often fairly easy to match experiment and calculation for more
than one model for a molecule labeled at only one position. But if you label the molecule
successively at several different positions, the number of models that match experiment
and calculation decreases drastically. Ideally, with enough data there should be only one
acceptable model, the correct one. So, you unravel nature’s secrets with isotopes, the
most intimate reaction mechanisms via the “successive labeling” technique.
I have no idea about whether this will be of any help to you, but it was fun to "go
back over it again".
Love, Dad
P.S. You should consult some of the references below for more details; especially the
Bigeleisen and Wolfsberg review paper gives derivations and a calculated example – pay
special attention therein to equation II.21 and pp. 65-73.
Further Reading
Anderson, T.F. and M.A. Arthur. 1983. Stable isotopes of oxygen and carbon and their
application to sedimentologic and paleaoenvironmental problems, pp. 1-1 – 1-151. In:
Arthur, M.A., T.F. Anderson, I.R. Kaplan, J. Veizer, and L.S. Land (eds). 1983. Stable
Isotopes in Sedimentary Geology. SEPM Short Course #10, Dallas. Society of Economic
Paleontologists and Mineralogists.
Bigeleisen, J. 1949. The validity of the use of tracers to follow chemical reactions.
Science 110:14-16.
Bigeleisen, J. 1949. The relative reaction velocities of isotopic molecules. Journal of
Chemical Physics 17:675-678.
Bigeleisen, J. and M.G. Mayer. 1947. Calculation of equilibrium constants for isotopic
exchange reactions. Journal of Chemical Physics 15:261-267.
Bigeleisen, J. and M. Wolfsberg. 1958. Theoretical and experimental aspects of isotope
effects in chemical reactions, pp. 15-76. In: I. Prigogine, Advances in Chemical Physics,
v. 1.
Criss, R.E. 1999. Principles of Stable Isotope Distribution. Oxford University Press.
Fry, A. 1964. Application of the successive labeling technique to some carbon, nitrogen
and chlorine isotope effect studies of organic reaction mechanisms. Pure and Applied
Chemistry 8:409-419.
Hasan, T., L.B. Sims and A. Fry. 1983. Heavy atom isotope effect studies of elimination
reaction mechanisms. 1. A kinetic and carbon-14 kinetic isotope effect study of the basepromoted dehydrochlorination of substituted 1-phenylethyl-2-14C chlorides. Journal of
the American Chemical Society 105: 3967-3975.
Sims, L.B., A. Fry, D.E. Lewis, and L.T. Netherton. 1983. Interplay of experiment and
theory in isotope effect research, pp. 261-266. In W.P. Duncan and A.B Susan (eds),
Synthesis and Applications of Isotopically Labeled Compounds, Proceedings of an
International Symposium. Elsevier.
Urey, H.C. 1947. The thermodynamic properties of isotopic substances. Journal of the
Chemical Society (London), Part 1:562-581.