Nonadditivity of secondary deuterium isotope effects on basicity of trimethylamine

Article abstract of DOI:10.1021/ja803084w

Secondary deuterium isotope effects (IEs) on basicities of isotopologues of trimethylamine have been accurately measured by an NMR titration method applicable to a mixture. Deuteration definitely increases the basicity, by ～0.021 in the ΔpK per D. The IE is attributed to the lowering of the CH stretching frequency and zero-point energy by delocalization of the nitrogen lone pair into the C-H antibonding orbital. Because this depends on the dihedral angle between the lone pair and the C-H, a further consequence is a preference for conformations with H antiperiplanar to the lone pair and D gauche. This leads to a predicted nonadditivity of IEs, which is confirmed experimentally. It is found that the decrease in basicity, per deuterium, increases with the number of deuteriums. The nonadditivity of IEs is a violation of the widely assumed Rule of the Geometric Mean.

Full text of DOI:10.1021/ja803084w

Published on Web 07/29/2008
Nonadditivity of Secondary Deuterium Isotope Effects on
Basicity of Trimethylamine
Charles L. Perrin* and Yanmei Dong
Department of Chemistry 0358, UniVersity of CaliforniasSan Diego,
La Jolla, California 92093-0358
Received April 25, 2008; E-mail: cperrin@ucsd.edu
Abstract: Secondary deuterium isotope effects (IEs) on basicities of isotopologues of trimethylamine have
been accurately measured by an NMR titration method applicable to a mixture. Deuteration definitely
increases the basicity, by ∼0.021 in the ∆pK per D. The IE is attributed to the lowering of the CH stretching
frequency and zero-point energy by delocalization of the nitrogen lone pair into the C-H antibonding orbital.
Because this depends on the dihedral angle between the lone pair and the C-H, a further consequence
is a preference for conformations with H antiperiplanar to the lone pair and D gauche. This leads to a
predicted nonadditivity of IEs, which is confirmed experimentally. It is found that the decrease in basicity,
per deuterium, increases with the number of deuteriums. The nonadditivity of IEs is a violation of the widely
assumed Rule of the Geometric Mean.
Introduction
The “simplest” cases of nonadditivity of IEs on NMR coupling
constants and nuclear shielding are intrinsic, arising from
A recent study reported secondary deuterium isotope effects
(IEs) on amine basicity and verified that R-deuteration increases
basicity.1,2 The IE was attributed to a lowered zero-point energy
(ZPE) of a CH bond when it is antiperiplanar or synperiplanar
to a nitrogen lone pair, and the conformational dependence of
IEs could be simulated computationally. We now report
evidence for the nonadditivity of secondary deuterium IEs on
amine basicity.
Conformational preference of an isotope can lead to nonad-
ditivity of IEs on chemical shifts, as in benzylic carbocations,3
1-(p-fluorophenyl)cyclopentyl cation,4 acetyl fluoride,5 and 2,3-
dimethyl-2-butyl cation.6 In principle, the conformational isotope
effect in 1,1,3,3-tetramethylcyclohexane, owing to the steric
interaction between an axial CD3 or CH2D and an axial CH3,7
should show nonadditivity if two CD3 or two CH2D are
compared. Nonadditivity of IEs on chemical shifts is also seen
in some iridium tri- and tetrahydrides,8 where it arises from a
fluxional equilibrium between isotopomers (isomers that differ
in the position of an isotope, as distinguished from isotopo-
logues, which differ in the number of isotopic substitutions).
interactions of vibrations, often bending modes.9
Conformational preference leads to nonadditivity of kinetic
IEs in solvolysis of (CH3)2CCl(CHnD3-n) (n ) 0, 1, 2, 3), where
C-H trans to the leaving chloride provides more acceleration
than C-D, but a second (and third) C-H provides a reduced
acceleration.10 We here seek the first example of nonadditivity
of IEs on a chemical equilibrium, specifically amine basicities.
This equilibrium represents a greater challenge than in solvoly-
ses, where the filled-vacant hyperconjugative interaction of a
C-H bond with the developing carbocation orbital is stronger
than the filled-filled interaction (“negative hyperconjugation”)
of a C-H bond with the nitrogen lone pair.
If the IE on amine basicity arises from the conformational
dependence of the ZPE of a CH bond, then in principle the IEs
of successive deuteriums are nonadditive. However, this non-
additivity could not be detected with CH3NH2,1 and it was
estimated that ∆∆G° per D would increase by <1 cal/mol from
CH3 to CH2D to CHD2 to CD3, which is too small to be detected
reliably. To detect nonadditivity reliably, the IE must be
magnified, as with three methyls. The isotopologues of trim-
ethylamine are then the appropriate choice. Indeed, Northcutt
and Robertson found that trimethylamine-d9 is a stronger base
than trimethylamine, with a substantial ∆pK, 0.206 at 0 °C.11
The nonadditivity of IEs on trimethylamine basicity depends
on a greater stability for the rotamer with protium anti to the
(1) Perrin, C. L.; Ohta, B. K.; Kuperman, J. J. Am. Chem. Soc. 2003,
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9
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2008 American Chemical Society
J. AM. CHEM. SOC. 2008, 130, 11143–11148 11143

A R T I C L E S
Perrin and Dong
lone pair. The first H on a methyl will thus take this position
and lead to an increase in acidity of a CHD2NH+ over CD3NH+.
An additional H, as in a CH2D, cannot take that position, so
that the increase in acidity will be less, and also less for CH3.
Thus, the IE per H will decrease as the number of H increases,
or the IE per D will increase as the number of D increases.
This is the direction of the nonadditivity that we have predicted
and endeavored to test.
no lag of one chemical shift behind the other. As a result, minute
imbalances of acidities are detectable. Moreover, because the
titration is performed on a mixture of the two acids, under
conditions guaranteed identical for both, it avoids systematic
errors due to impurities that arise in the synthesis of one of the
isotopologues but not the other. It has recently permitted
measurement of secondary deuterium IEs in carboxylic acids
and phenols.18
Success for this study can be anticipated from the preference
for equatorial deuteriums in 1,3,5,5-tetramethylhexahydropyri-
midine-2-d and 1-methylpiperidine-cis-2,6-d2,12 and from the
elegant demonstration of diastereotopic H in 1,2-dimethylpip-
eridine-1-d, owing to perturbation of the rotameric equilibrium
of the NCH2D fragment.13 Those are cases of isotopic perturba-
tion of a conformational equilibrium, whereas this study involves
a chemical reaction, with N-H bond-breaking or bond-making,
and where the conformational equilibrium affects only one side
of the equilibrium.
The titration can be performed on either the amine or the
ammonium ion. We chose to titrate the amine hydrochloride
with an anionic base, which has the advantage of maintaining
constant ionic strength and reducing medium effects on chemical
shifts. For solvents we chose DMSO-d6, with KOtBu as base,
and D2O, with NaOD as base. The IE on the basicity of
(CH3)2NH had been found to be smaller in D2O,1 owing to
hydrogen bonding to the lone pair, which reduces the n-σ*
delocalization that is responsible for the IE.1,2,12 Consequently,
DMSO offers the better prospect for detecting the nonadditivity,
but it is also subject to random variations in its water content.
Therefore, we measured IEs in both DMSO and D2O.
The key comparison is between the IE on the basicity of
(CH2D)3N relative to (CH3)3N and that on (CHD2)3N relative
to (CH2D)3N, each of these due to three additional deuteriums.
To confirm these values, the composite of these two IEs can be
measured from the comparison of (CHD2)3N with (CH3)3N, due
to six additional deuteriums. To be comprehensive, it is also
possible to measure the IE of eight deuteriums by comparing
trimethylamine with trimethylamine-d8. All of these isotopo-
logues are readily synthesized by standard procedures. We now
report not only that deuteration decreases the acidity constant
Ka of trimethylammonium ion (as had been known) but also
that the IE is nonadditive, in that the decrease, per deuterium,
increases with the number of deuteriums.
Nonadditivity is a powerful test for whether this IE might be
of inductive origin. Although protium cannot simply be more
electron-donating than deuterium,14 an alternative is an elec-
trostatic interaction between the N+ and the dipole moment of
the C-H bond, which is longer than C-D, owing to anharmo-
nicity.15 The dipole moments involved are exceedingly small,
though. Besides, the interpretation in terms of ZPEs is consistent
with the observation that the isotopomer of 1-benzyl-4-meth-
ylpiperidine-2,2,6-d3 with deuterium trans to the methyl group
is more basic.1 Nevertheless, an inductive effect ought to be
linear in the number of deuteriums, and nonadditivity would
be strong evidence against an inductive origin for these IEs.
An NMR titration method makes it possible to measure IEs
on basicities or acidities with great precision. The method
depends on isotope shifts,16 which lead to separate, resolvable,
and assignable signals for isotopologues. The procedure involves
successive additions of small aliquots of base to a mixture of
acids. The stronger acid will be deprotonated first. Its chemical
shift will then move ahead of that of the less acidic one, which
lags behind. The acidity constants Ka and chemical shifts δ of
both H and D acids can be related through eq 1,17 where δ+ or
δ0 is for the ammonium ion or amine, respectively, as measured
at the beginning or end of the titration. Therefore, a plot of the
quantity on the left vs (δH - δH0)(δD+ - δD) should be linear,
Experimental Section
1
NMR Spectroscopy. H and 13C NMR spectra were recorded
on a Varian Mercury 400 or Unity 500 spectrometer. All deuterium
decoupling experiments were recorded on a Varian Mercury 400
spectrometer with a decoupling frequency calibrated from a 2H
NMR spectrum for amine C-D (not D2O or DMSO-d6). After each
1
aliquot was added, the field homogeneity was shimmed on a H
NMR spectrum with deuterium lock, and then the cables were
switched and a 1H NMR spectrum was recorded with WALTZ
decoupling through the lock channel. The spectral window was
reduced to 700 Hz for 1H NMR at 500 MHz or to 600 Hz for
deuterium decoupling at 400 MHz, and the data were zero-filled
to increase digital resolution. Proton chemical shifts in aqueous
solutions are relative to acetonitrile (δ 2.05) as the internal standard,
or to cyclohexane (δ 1.42) in DMSO-d6. All chemical shifts were
read as Hz, to avoid roundoff error in conversion to ppm. To help
distinguish the peaks, an initial spectrum of a mixture of deuterated
species and internal standard was obtained, and then the undeu-
terated or another deuterated compound was added.
Synthesis. After several unsuccessful attempts by various
procedures that gave intolerably large contamination by
CH3(CH2D)2N, tri(methyl-d)amine hydrochloride [(CH2D)3N·HCl]
was obtained by reaction of tris(chloromethy1)amine19 with NaBD4.
This procedure was adapted from the reaction with sodium
methoxide in methanol,20 but in D2O to guard against H incorpora-
tion. Indeed, CH3(CH2D)2N did appear when D2O was replaced
by methanol or H2O. The successful synthesis is remarkable in view
of the extreme hygroscopicity and expected hydrolytic reactivity
h
d
with zero intercept. The ratio of acidity constants, Ka /Ka , can
then be evaluated as the least-squares slope of that plot.
0
0
(δH+ - δH)(δD - δD ) ) (Kah⁄Kad)(δH - δH )(δD+ - δD)
(1)
This method is capable of exquisite precision, because it is
based only on chemical-shift measurements. It does not require
accurate control of pH or volume or molarity or equivalents of
base added, as is usual in pH titrations. The method is
comparative. If there is no difference in acidities, there can be
(12) Anet, F. A. L.; Kopelevich, M. Chem Commun. 1987, 595. Forsyth,
D. A.; Hanley, J. A. J. Am. Chem. Soc. 1987, 109, 7930.
(13) Anet, F. A. L.; Kopelevich, M. J. Am. Chem. Soc. 1989, 111, 3429.
(14) Weston, R. E., Jr. Tetrahedron 1959, 6, 31.
(15) Halevi, E. A. Prog. Phys. Org. Chem. 1963, 1, 109.
(16) Batiz-Hernandez, H.; Bernheim, R. A. Prog. Nucl. Magn. Reson.
Spectrosc. 1967, 3, 63. Jameson, C. J.; Osten, H. J. Annu. Rep.
NMR Spectrosc. 1986, 17, 1. Hansen, P. E. Prog. Nucl. Magn. Reson.
Spectrosc. 1988, 20, 207. Dziembowska, T.; Hansen, P. E.; Rozwa-
dowski, Z. Prog. Nucl. Magn. Reson. Spectrosc. 2004, 45, 1.
(17) Perrin, C. L.; Fabian, M. A.; Armstrong, K. B. J. Org. Chem. 1994,
59, 5246. Perrin, C. L.; Fabian, M. A. Anal. Chem. 1996, 68, 2127.
(18) Perrin, C. L.; Dong, Y. J. Am. Chem. Soc. 2007, 129, 4490.
(19) Fluck, E.; Meiser, P. Angew. Chem., Int. Ed. Engl. 1971, 10, 653.
(20) Fluck, E.; Meiser, P. Chem. Ber. 1973, 106, 69.
9
11144 J. AM. CHEM. SOC. VOL. 130, NO. 33, 2008

Nonadditivity of Secondary Deuterium Isotope Effects
A R T I C L E S
of (ClCH2)3N: 1H NMR (D2O) δ 2.72 (t, J ) 1.78 Hz). Tri(methyl-
d2)amine hydrochloride [(CHD2)3N·HCl] was readily obtained by
an adaptation of a general procedure,21 involving ammonium
formate, formic acid, and formaldehyde-d2: 1H NMR (D2O) δ 2.70
(qn, J ) 1.78 Hz). A mixture of trimethylamine-d8 hydrochloride,
[(CHD2)(CD3)2N · HCl] and trimethylamine-d9 hydrochloride
[(CD3)3N·HCl] was obtained similarly, from ammonium-dx for-
mate-d, formic-d2 acid, anhydrous formic acid, and formaldehyde-
1
d2, with a total H content in the formates equal to 8 mol %: H
NMR (D2O) δ 2.70 (quintet, J ) 1.78 Hz), due to (CD3)2(CHD2)-
N·HCl, without interference from (CD3)3N·HCl, which is present
in large amount but invisible. Nor can appreciable CD3(CHD2)2N
be present, inasmuch as the deuterium-decoupled 1H NMR spectrum
shows no minor shoulder.
Each deuterated trimethylamine sample was extracted from
aqueous NaOH into CH2Cl2 and then acidified with 6 M aqueous
HCl and evaporated to dryness, to produce the hydrochloride salt
as a white solid. Undeuterated trimethylamine hydrochloride was
a commercial sample, used without purification.
NMR Titrations. The NMR sample prepared for each titration
contained 0.5 mL of D2O or organic solvent and appropriate
concentrations of internal standard and a pair of isotopologues of
the amine, adjusted to ensure nearly equal peak heights under
conditions of deuterium decoupling, so that signal assignments could
be maintained even if there is signal crossover during the titration.
Samples in DMSO-d6 were first acidified with 2 µL of 0.1 M
CF3COOH to ensure complete protonation, and then 5-µL aliquots
of KOtBu in DMSO-d6 were used as the base. Aqueous samples
were acidified with 5 µL of 0.1 M DCl and then titrated with as
many as 20 5- or 10-µL aliquots of NaOD in D2O.
NMR spectra were recorded after each addition. At least 10
aliquots were used for each titration. Titrations were assumed to
be complete when there was no peak movement >0.002 ppm upon
further addition of base. Small changes of chemical shifts past the
end point were detected, owing to increasing ionic strength, but
the IE values obtained were insensitive to the choice of end point.
Chemical shifts of appropriate reporter nuclei were extracted from
each spectrum, and the data were fit to eq 1.
Figure 1. NMR titration (eq 1) of a mixture of (CH2D)3NH+ and
(CH2D)3NH+.
Numbers in parentheses are errors in the last decimal place of
D
the value reported. In all cases, Ka/Ka is greater than 1,
corresponding to a deuterium IE that increases the basicity of
trimethylamine. The positive ∆pKa represents this same feature.
All titrations in D2O were repeated, and values from separate
titrations were averaged. The averages are reported in Table 3.
Some of the data in Table 3 show larger errors between titrations
than the errors in Table 2, which correspond to errors within a
titration, but the data are very reproducible. It should be noted
that the values of ∆pKa per D in Table 3 correspond to values
of ∆∆G° that range from 29.2 to 30.5 cal/mol. This reinforces
the previous conclusion that the nonadditivity of the IE in
methylamine would be <1 cal/mol.1
CH3
CHD2
The ratio Ka /Ka
, evaluated directly from titration of a
Results
mixture of trimethylamine and tri(methyl-d2)amine (third data
column), agrees well with the product (fourth data column) of
For trimethylamine, deuteration produces an upfield 1H NMR
shift, as is usual for isotope shifts.16 Moreover, the signals of
all trimethylammonium isotopologues move upfield on depro-
CH3
CH2D
CHD2
and KaCH D/Ka
. These ought to be identical,
2
Ka /Ka
and the discrepancy between them is a measure of the systematic
errors. This discrepancy is greater in DMSO, where it is
intolerably large. We attribute the error to random variations
in the water content of the DMSO-d6. The data show that the
IE is larger in DMSO, but the water content reduces the IE.
Consequently, we place less reliance on the studies in DMSO,
even though the larger IE ought to have made the nonadditivity
easier to document.
1
tonation. The H NMR signals of protio and deuterio amines
cross and recross during the titration. Because deuteration
produces an upfield intrinsic isotope shift in both the amine and
the ammonium ion, this is strong evidence that the protio
ammonium ion is more acidic.
Figure 1 shows a plot of eq 1 from the NMR titration of a
mixture of tri(methyl-d)ammonium and tri(methyl-d2)ammonium
hydrochlorides with NaOD in D2O. The blank area in the middle
is from spectra where signals crossed past each other and
obscured the exact chemical shifts. The slope is 1.1618 (
0.0004. The intercept is -0.0061 ( 0.0046, properly zero. The
correlation coefficient is 0.999999.
All IEs were measured pairwise, from mixtures of trimethy-
lamine and tri(methyl-d)amine, of tri(methyl-d)amine and tri-
(methyl-d2)amine, of trimethylamine and tri(methyl-d2)amine,
or of trimethylamine and trimethylamine-d8. Tables 1 and 2 list
the various IEs, expressed (1) as the ratio of the acidity constant
Ka of an ammonium ion to that of the corresponding ammonium
ion with 3, 3, 6, 6, 8, or 2 additional deuteriums; (2) as ∆pKa
) log10(Ka/KaD); and (3) as ∆pKa per additional deuterium.
The data in the tables are difficult to compare, because Ka/
KaD and ∆pKa correspond to the IEs from different numbers of
deuteriums. Even ∆pKa per D is difficult to comprehend,
because the values are so small and so similar. To represent
those values more clearly, ∆pKa per D from Table 3 is displayed
in Figure 2, showing pairwise comparisons of CH3, CH2D, and
CHD2 from left to right, against CH3, CH2D, CHD2, and CD3
(actually data for trimethylamine-d8) from front to back. To
exaggerate the variations, 0.021 has been subtracted from every
value, so there are very tall columns below the “floor”.
CHD2
The key result is that KaCH D/Ka
(second column of data
2
in Tables 1-3) is larger than KaCH /KaCH D (first column of data)
3
2
in both DMSO and D2O. The difference between them is
d0
d8
statistically significant. The ratio Ka /Ka (last column) is still
larger, because it is the IE of eight D, but on a per-deuterium
(21) Icke, R. N.; Wisegarver, B. B.; Alles, G. A. Organic Syntheses; Wiley:
New York, 1955; Collect. Vol. 3, p 723. Lindeke, B.; Anderson, E.;
Jenden, D. J. Biomed. Mass Spectrom. 1976, 3, 257.
basis it is intermediate between KaCH /KaCH D and KaCH D/KaCHD
3
2
2
2
CH3
CHD2
and closer to the latter. The ratio Ka /Ka
(third or fourth
9
J. AM. CHEM. SOC. VOL. 130, NO. 33, 2008 11145

A R T I C L E S
Perrin and Dong
Table 1. Isotope Effects of (CHnD3-n
) on Trimethylamine Basicities in DMSO-d6
3
CH3/CH2D
CH2D/CHD2
CH3/CHD2
CH3/CHD2a
d0/d8
d6/d8b
D
Ka/Ka
∆pKa
1.2008(10)
0.0795(4)
0.02649(12)
1.2042(8)
0.0807(3)
0.02690(9)
1.4358(11)
0.1571(3)
0.02618(5)
1.4460(15)
0.1602(5)
0.02670(8)
1.6374(19)
0.2142(5)
0.02677(6)
1.1404(16)
0.0571(6)
0.0285(3)
∆pKa per D
a Evaluated as the product of CH3/CH2D and CH2D/CHD2. b Evaluated as the ratio of d0/d8 to CH3/CHD2.
Table 2. Isotope Effects of (CHnD3-n on Trimethylamine Basicities in D2O
)
3
CH3/CH2D
CH2D/CHD2
CH3/CHD2
CH3/CHD2a
d0/d8
d6/d8b
D
Ka/Ka
1.1573(7)
1.1574(6)
0.0635(2)
0.0635(2)
0.02115(8)
0.02116(7)
1.1618(4)
1.1628(7)
0.0651(1)
0.0655(3)
0.02171(5)
0.02184(9)
1.3509(7)
1.3465(6)
0.1306(2)
0.1292(2)
0.02177(4)
0.02154(3)
1.3446(9)
1.3458(11)
0.1286(3)
0.1290(3)
0.02143(5)
0.02150(6)
1.4939(6)
1.4929(8)
0.1743(2)
0.1740(2)
0.02179(2)
0.02176(3)
1.1058(7)
1.1087(8)
0.0437(3)
0.0448(3)
0.0218(1)
0.0224(2)
Ka/KaD, repeat
∆pKa
∆pKa, repeat
∆pKa per D
∆pKa per D, repeat
a Evaluated as the product of CH3/CH2D and CH2D/CHD2. b Evaluated as the ratio of d0/d8 to CH3/CHD2.
Table 3. Averaged Isotope Effects of (CHnD3-n on Trimethylamine Basicities in D2O
)
3
CH3/CH2D
CH2D/CHD2
CH3/CHD2a
CH3/CHD2b
d0/d8
d6/d8c
D
Ka/Ka
∆pKa
1.15736(3)
0.06347(1)
0.02116(0)
1.1623(7)
0.0653(3)
0.02178(9)
1.3487(22)
0.1299(7)
0.02165(12)
1.3452(9)
0.1288(3)
0.02147(5)
1.4934(7)
0.1742(2)
0.02177(2)
1.1073(19)
0.0443(7)
0.0221(4)
∆pKa per D
a Includes data from a third titration. b Evaluated as the product of CH3/CH2D and CH2D/CHD2. c Evaluated as the ratio of d0/d8 to CH3/CHD2.
additional signals greatly complicates the NMR titration. The
signals of the more acidic CH3-containing material start down-
field but move upfield more quickly on deprotonation. Conse-
quently, those signals cross past that of the CH2D-containing
material. With so many extraneous signals, the crossovers
obscure the exact chemical shifts. Therefore, it was necessary
to eliminate the contamination.
The contamination undoubtedly arises from an impurity in
the precursors. Impurities are especially problematic, because
three deuteriums must be incorporated into (CH2D)3N. Yet it
did not matter whether the source of the deuterium was DCOOH
or NaBD4 or whether the source of the CH2 was CH2O or
-
paraformaldehyde or Na+ HOCH2SO3 or CH2(OCH3)2, any
of which might enter into a chemical exchange reaction with
the deuteride donor. Such exchange would be detrimental,
inasmuch as H- transfer would then be favored by a kinetic
isotope effect, as is seen in the hydride transfer from formate
to triarylmethyl cations.22 Surprisingly, we could not find data
on a kinetic isotope effect involving BD4-, but it is likely to be
substantial. Yet the commercial NaBD4 is claimed to be 98 atom
% D, and we could not detect BHD3- by 1H NMR. We suspect
an exchange reaction, as has been observed between (CH3)3-
NBH3 and D2O.23
Figure 2. Nonlinearity of IEs on trimethylamine basicities in D2O,
comparing (CHnD3-n)3 (n ) 3, 2, 1) to (CHnD3-n)3 (n ) 2, 1, 0), displayed
as ∆pKa per D - 0.021.
CH3
CH2D
data column) is also larger than KaCH D/KaCHD or Ka /Ka
,
2
2
We found that isotopically pure tri(methyl-d)amine hydro-
chloride could be obtained by reaction of tris(chloromethy1)amine
with NaBD4 in D2O. It may be that exchange was avoided, or
that the kinetic isotope effect was smaller with a more reactive
electrophile.
because it is the IE of six D, but on a per-deuterium basis it is
intermediate between those two (except possibly in DMSO).
Another key result is that reported in the last column of the
tables. This corresponds to the comparison between (CD3)2-
(CHD2)N and (CHD2)3N, obtained from the titrations of each
vs (CH3)3N. It represents the IE of the two additional deuteriums
in (CD3)2(CHD2)N. Per deuterium, this IE is larger than any
other. This is the tallest bar in Figure 2.
In contrast, heating a mixture of DCO2NH4, DCOOD, and
1
CD2O gave very pure (CD3)3N·HCl, with no detectable H
NMR signal. This was unsuitable for NMR titration, because
(CD3)3N is invisible in 1H NMR, and we needed (CD3)2-
(CHD2)N as an impurity. We therefore repeated the synthesis
with a proportion of H- donor that led to a mixture of
Discussion
Comments on Synthesis. Although tri(methyl-d2)amine and
tri(methyl-d3)amine were readily prepared, without contamina-
tion by higher or lower isotopologues, the synthesis of tri(meth-
yl-d)amine [(CH2D)3N] was plagued by formation of some
CH3(CH2D)2N. The amount was slight, but the presence of
(22) Stewart, R.; Toone, T. W. J. Chem. Soc., Perkin Trans. 2 1978, 1243.
(23) Davis, R. E.; Brown, A. E.; Hopmann, R.; Kibby, C. L. J. Am. Chem.
Soc. 1963, 85, 487.
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11146 J. AM. CHEM. SOC. VOL. 130, NO. 33, 2008

Nonadditivity of Secondary Deuterium Isotope Effects
A R T I C L E S
trimethylamine-d8 and -d9, without appreciable amounts of either
isotopomer of trimethylamine-d7.
gauche and anti to the lone pair, respectively (1g and 1a).
Likewise, the extra ZPEs of the conformers of (CH2D)3N,
relative to (CD3)3N, are given by 2ngEg + na(Ea + Eg), where
ng is the number of CH2DN fragments with both H gauche to
the lone pair (2g) and na is the number of CH2DN fragments
with one H gauche to the lone pair and one H anti (2a).
For further simplification, all acidity constants can be
compared to that of (CD3)3NH+. The various concentrations in
eq 2 can then be expressed in terms of Boltzmann factors
governed by the ZPEs of the various conformations, along with
their statistical factors, to obtain eqs 3 and 4. The third-power
dependence is a confirmation of the magnification of the IEs
by three methyls. For completeness, eq 5 gives the acidity
constant for (CH3)3NH+, which is a simpler expression because
it and its conjugate base are single conformers.
Origin of Isotope Effect. Before considering the nonadditivity
of IEs on amine basicity, it is necessary to recapitulate the origin
of the IE itself.1,2 The IR spectra of amines show characteristic
bands (Bohlmann bands) around 2700-2800 cm-1 24
, lower than
the 2900 cm-1 of a typical CH stretch. Upon N-protonation,
these bands revert to a typical frequency. Therefore, the ZPE
of the CH increases on protonation, but the increase is less for
CD. Consequently, a CH amine resists N-protonation and is less
basic.
The lowering of the CH stretching frequency in an amine,
relative to the protonated amine, is attributed to delocalization
of the nitrogen lone pair into the C-H antibonding orbital. This
was shown to be of stereoelectronic origin, depending on the
dihedral angle between the lone pair and the C-H.1,2,12 It
reaches a maximum when that dihedral angle is 180°, corre-
sponding to a conformation with the lone pair antiperiplanar to
the C-H.
Quantitative Origin of Nonadditivity. It is possible to express
the acidity constants Ka of each of the isotopologues of
trimethylammonium ion. For (CD3)3NH+, which is a single
conformer, this is simply [H+][(CD3)3N]/[(CD3)3NH+], and
analogously for (CH3)3NH+. For the other two isotopologues,
it is necessary to recognize that each acid and each base is a
mixture of conformers. The acidity constant for (CHD2)3NH+
is given by eq 2, where the sequence of g’s and a’s designates
how many of the three methyl fragments have their single H
gauche (1g) or anti (1a, more stable) to the lone pair. The acidity
constant for (CH2D)3NH+ is also given by eq 2, except that the
sequence of g’s and a’s designates how many of the three methyl
fragments have both H gauche to the lone pair (2g) or have
one of the H’s anti (2a, more stable). The denominators will
simplify because all conformers of an acid have the same energy,
but the denominators and the numerators have statistical factors
8, 12, 6, and 1 and 1, 6, 12, and 8 for ggg, gga, aag, and aaa
of 1 and 2, respectively.
+
Ka(CHD ) NH
[2 exp(-Eg ⁄RT) + exp(-Ea⁄RT)]3
27 exp(-3E+ ⁄RT)
2
3
)
(3)
(CD3)3NH+
K
a
(CH2D)3NH+
K
[exp(-2Eg ⁄RT) + 2 exp(-(Ea + Eg) ⁄RT)]3
27 exp(-6E+ ⁄RT)
a
)
+
Ka(CD ) NH
3
3
(4)
+
Ka(CH ) NH
[3 exp(-(Ea + 2Eg) ⁄RT)]3
27 exp(-9E+ ⁄RT)
3
3
)
(5)
(CD3)3NH+
K
a
A further simplification results from setting εg ) E+ - Eg
and εa ) E+ - Ea, leading to eqs 6-8. These expressions are
still sufficiently formidable that they verify the nonadditivity
of the IE. Linearity would require the expression in eq 7 to be
the square of that in eq 6, and would require the expression in
eq 8 to be the product of those in eqs 6 and 7, and these
requirements clearly do not hold. The nonadditivity can be traced
to the preference for conformations with H anti to the lone pair,
leading to the sums that appear in the numerators of these
equations, with εg < εa. Notice further that the temperature
dependence of the IE is quite complicated. Unfortunately, a
separation into enthalpy and entropy is far beyond the accuracy
of these measurements.
+
Ka(CHD ) NH
)
2
3
(CH2D)3NH+
[ggg] + [gga] + [aag] + [aaa]
[gggH+] + [ggaH+] + [aagH+] + [aaaH+]
[H+]
) K
a
(CHD2)3NH+
K
3
(2)
a
2
3
1
3
)
exp(εg ⁄RT) + exp(εa ⁄RT)
(6)
(CD3)3NH+
[
]
K
a
(CH2D)3NH+
K
3
a
1
3
2
3
)
exp(2εg ⁄RT) + exp((εa + εg) ⁄RT)
(CD3)3NH+
a
[
]
K
(7)
(8)
To simplify the numerators of eq 2, it is convenient to relate
the concentrations of the various conformers to their molar
ZPEs. Let the extra ZPE of a CHD2NH+ fragment, relative to
that of CD3NH+, be designated as E+, regardless of conforma-
tion. Then the extra ZPE of a protonated CHnD3-nNH+
fragment, relative to CD3NH+, is simply nE+, linear in n, the
number of H. Further, let the extra ZPE of an unprotonated
CH3N fragment (3), relative to CD3N, be 2Eg + Ea, where
separate Eg and Ea apply to H’s that are gauche or anti to the
N lone pair. Because the ZPE is lower for an H anti to the lone
pair, Eg > Ea. Furthermore, the extra ZPEs of the conformers
of (CHD2)3N, relative to (CD3)3N, are given by ngEg + naEa,
where ng and na are the numbers of CHD2N fragments with H
+
Ka(CH ) NH
3
3
) [exp((εa + 2εg) ⁄RT)]3
(CD3)3NH+
K
a
The IEs in Table 3 can be fit to eqs 6–8. From four
independent IEs, the best values of εg and εa are 3 ( 1 and 24
( 1 cm-1, respectively, or 10 and 70 cal/mol. These values
deviate slightly from a cos2 dependence of orbital overlap
on dihedral angle, which would predict that the IE of a gauche
deuterium is 25% that of an anti one. It should be noted that a
dependence stronger than cos2 was also observed in solvolysis
of (CH3)2CCl(CHnD3-n),10 and in the calculated C-D stretching
frequency in conformers of DCH2NH2. If the CH frequency of
the trimethylammonium ion is taken as 2900 cm-1, these values
of εg and εa correspond to lower frequencies, 2880 and 2740
(24) Bohlmann, F. Chem. Ber. 1958, 91, 2157.
9
J. AM. CHEM. SOC. VOL. 130, NO. 33, 2008 11147

A R T I C L E S
Perrin and Dong
cm-1, for CH gauche and anti, respectively, to the lone pair in
trimethylamine. Such values are quite reasonable.
IEs are larger in DMSO but less reliable. In water, hydrogen-
bonding to the nitrogen lone pair reduces the delocalization of
that lone pair into the C-H antibonding orbital and diminishes
the IE. The IE can be expected to be larger in the gas phase,
and the nonadditivity as well. It is not clear whether experi-
mental techniques are sufficiently accurate to detect the non-
additivity of IEs in the gas phase.
Computation might be more successful. Indeed, it should be
noted that the C-L bond lengths in (CL3)nNH3-n (n ) 1, 2, 3)
are calculated to decrease from L ) H to D to T, along with an
increased negative atomic charge at N, although the decreases
in bond lengths are not greater for the longer antiperiplanar
C-L, and no attempt was made to model the nonadditivity of
IEs.27
Significance of Nonadditivity. The nonadditivity is evidence
for the origin of the IE in the ZPE of C-H vibrations. It is
especially strong evidence that this IE is not of inductive origin,
which ought to be linear in the number of deuteriums.
The nonadditivity of IEs is a violation of the widely assumed
Rule of the Geometric Mean.25 According to that rule, the
equilibrium constants for the isotopic disproportionation equi-
libria of (CH2D)3N and (CHD2)3N (eqs 9 and 10) and of their
conjugate acids (eqs 11 and 12) must be unity (because
symmetry numbers cancel). It follows that the acidity constant
of (CH2D)3NH+ must be the geometric mean of the acidity
constants of (CH3)3NH+ and (CHD2)3NH+ (eq 13), and likewise
for (CHD2)3NH+ (eq 14). Nevertheless, the nonadditivity that
we have succeeded in detecting means that eqs 13 and 14 do
not hold, although they are close approximations.
Conclusions
Not only does deuteration decrease the acidity constant Ka
of trimethylammonium ion, but also the IE can be shown
experimentally to be nonadditive, in that the decrease, per
deuterium, increases with the number of deuteriums. The IE is
attributed to the lowering of the CH stretching frequency by
delocalization of the nitrogen lone pair into the C-H antibond-
ing orbital, which depends on the dihedral angle between the
lone pair and the C-H. The nonadditivity can further be
modeled in terms of the conformational dependence of the ZPEs
of CHnD3-nNH+ and CHnD3-nN (n ) 0, 1, 2, 3). The
nonadditivity of IEs is a violation of the widely assumed Rule
of the Geometric Mean.
2(CH2D)3N a (CH3)3N + (CHD2)3N
2(CHD2)3N a (CH2D)3N + (CD3)3N
(9)
(10)
2(CH2D)3NH+ a (CH3)3NH+ + (CHD2)3NH+
2(CHD2)3NH+ a (CH2D)3NH+ + (CD3)3NH+
(11)
(12)
(13)
(14)
+
+
+
Ka(CH D) NH ) (Ka(CH ) NH Ka(CHD ) NH
)
)
1 ⁄ 2
2
3
3
3
2 3
+
+
+
Ka(CHD ) NH ) (Ka(CH D) NH Ka(CD ) NH
1 ⁄ 2
2
3
2
3
3 3
How general is the nonadditivity of IEs in a chemical
reaction? It is quite common with kinetics in H2O-D2O
mixtures, where proton inventory studies can show whether
more than one hydrogen is transferred in the rate-limiting step.26
Of course, it is much easier to detect nonadditivity in the cases
of kinetic IEs, simply because they are larger than secondary
equilibrium IEs. Indeed, this is perhaps the first example of
nonadditive equilibrium IEs in a chemical reaction other than a
conformational equilibrium.
Acknowledgment. This research was supported by NSF Grants
CHE03-53091 and CHE07-42801. Purchase of the NMR spec-
trometers was made possible by grants from NIH and NSF. We
are grateful to Martin Saunders for discussion and encouragement
and to Vernon Anderson and Mark Ruszczycky for helpful
suggestions regarding the analysis of nonadditivity.
Supporting Information Available: Details of synthesis and
spectral characterization. This information is available free of
charge via the Internet at http://pubs.acs.org.
It would be advantageous to compare these solution results
with those obtained in the gas phase. Our data show that the
JA803084W
(25) Bigeleisen, J. J. Chem. Phys. 1955, 23, 2264.
(26) Schowen, K. B.; Schowen, R. L. Methods Enzymol. 1982, 87, 551.
Venkatasubban, K. S.; Schowen, R. L. CRC Crit. ReV. Biochem. 1984,
17, 1.
(27) Reyes, A.; Pak, M. V.; Hammes-Schiffer, S. J. Chem. Phys. 2005,
123, 064104.
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11148 J. AM. CHEM. SOC. VOL. 130, NO. 33, 2008