Isotope effects in liquid water by infrared spectroscopy

Article abstract of DOI:10.1063/1.1448286

The heavy and light liquid water (H₂O-D₂O) mixtures were studied by Fourier transform infrared attenuated total reflectance (FTIR-ATR) spectroscopy. The deformation bands of liquid water clearly indicate the presence of the three typ

Full text of DOI:10.1063/1.1448286

Isotope effects in liquid water by infrared spectroscopy
Jean-Joseph Max and Camille Chapados
Citation: J. Chem. Phys. 116, 4626 (2002); doi: 10.1063/1.1448286
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 116, NUMBER 11
15 MARCH 2002
Isotope effects in liquid water by infrared spectroscopy
a)
b)
Jean-Joseph Max and Camille Chapados
D e´ partement de Chimie-Biologie, Universit e´ du Qu e´ bec a` Trois-Rivi e` res, Trois-Rivi e` res, Qu e´ bec,
Canada G9A 5H7
͑
Received 10 September 2001; accepted 11 December 2001͒
The light and heavy liquid water (H O–D O) mixtures in the 0–1 molar fraction were studied in
2
2
the mid-infrared by Fourier transform infrared attenuated total reflectance ͑FTIR-ATR͒
spectroscopy. Five principal factors were retrieved by factor analysis ͑FA͒. When D O is mixed with
2
H O, the HDO formed because of the hopping nature of the proton ͑H or D͒ results in three types
2
of molecules in equilibrium. Because of the nearest-neighbor interactions, the three molecules give
rise to nine species. Some of the species evolve concomitantly with other species giving the five
principal factors observed. We present the spectra of these factors with their abundances. The
calculated probability of the species present at different molar fractions which when the concomitant
species are combined gives the observed abundances. To appreciate clearly the difference between
the principal spectra, a Gaussian simulation of the bands was made. Because of the numerous
components that make up the stretch bands, they are not very sensitive to changes in composition
of the solutions; nevertheless, they do indicate the presence of new entities other than the pure
species. The deformation bands, more sensitive to such changes than the stretch bands, clearly
indicate the presence of the three types of molecules as well as of intermediate species. These bands
are sensitive to the two hydrogen bonds on the oxygen atom that a reference molecule makes with
its nearest-neighbors, but not to the hydrogen bonds that the nearest-neighbors make with the next
nearest neighbors. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1448286͔
I. INTRODUCTION
first glance. HDO abounds in a H O–D O mixture. There-
2
2
fore, the HDO spectrum was obtained by subtracting 25% of
each spectrum of H O and of D O from the spectrum of an
Although many organic substances are completely mis-
cible in water, they rarely make ideal mixtures. Alcohol ͑R–
OH͒ is a good example. The high molecular weight alcohols
2
2
2
equimolar H O–D O mixture. The resulting HDO spectrum
2
2
shows sigmoids in the deformation regions of the parent
molecules, suggesting that either the spectral subtraction was
not adequate or that other species are present in the mixture.
Since heavy water and deuterated compounds are used ex-
͑7 C and more͒ are not soluble in water; as the number of
carbons is decreased, solubility increases, becoming more
and more ideal. An illustration of this is found in an IR study
of 1-propanol and water where we have shown that these
substances do not form ideal mixtures even though they are
5
–7
tensively in the study of biological compounds, and since
8
1
D O is an important economic commodity, it is worth ex-
2
completely miscible. The lowest molecular weight alcohol
amining the H O–D O system again. The purpose of this
2
2
is water ͑H–OH͒, an ideal solution difficult to study. To ob-
serve the interaction of HOH with HOH, we choose its clos-
est parent, heavy water ͑DOD͒. Since the object of this paper
is to study aqueous ideal mixtures, we choose H O and D O
paper is to expose the difficulties we encountered in such a
study and to describe the techniques and methods we devised
to overcome them.
2
2
IR spectroscopy is a technique permitting the different
species present in solution to be identified and their abun-
dances to be determined.9 However, because of the pres-
ence of strong hydrogen bonds that shift the IR bands from
their free molecule ͑gas phase͒ positions, the absorptivity of
the vibrational modes are enhanced ͑see Ref. 22 and refer-
ences therein͒. Hence, OH stretch vibrations are strong ab-
sorbers, making IR transmission measurements of aqueous
mixtures so as to bring other problems inherent to this ideal
mixture system to the surface.
–21
The hydrogen atoms in a water molecule are not fixed to
one molecule but can migrate to neighboring molecules. The
same is true for the deuterium in heavy water. Consequently,
the H or D atoms in a mixture of light and heavy water will
switch with the atoms of neighboring molecules to form a
third species, HDO.2 The study of H O–D O mixtures is
–4
2
2
solutions difficult to obtain.23 Light water (H O) is then a
not as easy as it first seems because HDO cannot be isolated.
We cannot buy a bottle of HDO as we can for H O and D O.
2
very poor IR solvent because it is a strong absorber over
most of the range from the near infrared to the far infrared.
Heavy water is often used to replace ordinary water
when it absorbs in regions of interest to the solute ͑e.g.,
amide A, B, and I bands of proteins͒ and when the isotopic
substitution has no effect on the solute other than the band
displacement resulting from the exchange of the labile
protons.24 The IR study in light and heavy water of chloro-
2
2
Even though HDO cannot be isolated chemically, can it be
observed spectroscopically? This would appear easy to do at
a͒Current address: Scientech R&D, 247 Thibeau, Cap-de-la-Madeleine,
͑QC͒, Canada G8Y 6X9.
b͒
Author to whom correspondence should be addressed. Electronic mail:
Camille Chapados@uqtr.ca
0021-9606/2002/116(11)/4626/17/$19.00
4626
© 2002 American Institute of Physics
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J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
Isotope effects in liquid water by infrared spectroscopy
4627
phyll photosystem II and glycine are examples of the use of
proteins and amino acids, respectively ͑see Refs. 6, 7, and
references therein͒. In IR and neutron diffraction studies, iso-
topic substitution is often used in aqueous solutions to dis-
tinguish the OH groups of the solute from those of the sol-
II. EXPERIMENTAL AND DATA TREATMENT
A. Chemicals and solutions
Heavy water ͑Aldrich Chemical Co., purity Ͼ99.9 at %
D͒ was used without further purification. Deionized water
was used for light water. Light and heavy water were used to
prepare the mixtures. Each sample was prepared by weighing
the proper quantities in a 5.0 ml volumetric flask. A series of
2
5,26
vent, or to determine the hydration shells.
However, the
disproportionation effect in the H O–D O mixtures, which
2
2
comes from the particular influences that deuterium and hy-
6
3 samples ranging from pure light water to pure heavy wa-
drogen have on oxygen, is the cause of the differences be-
ter was prepared, the composition of which is given in Table
I. This table gives the species concentrations, the D mole
fraction ͑␹ ϵ͓D O͔/(͓H O͔ϩ͓D O͔), where the concen-
tween the two molecules.2
7–31
Therefore, the use of isotopic
2
6
substitution in water used as a solvent could result in no-
ticeable perturbations in its molecular vibrations.
D
2
2
2
trations are initial concentrations͒, and the measurement tem-
peratures.
Other than a shift in the IR bands, heavy water (D O)
2
behaves similarly to light water in the infrared. In H O–D O
2
2
mixtures, however, the formation of HDO complicates the
situation. With such handicaps, the logical position would be
to find an alternative to such a solvent. The problem is that
there is none because water is the natural solvent of all bio-
logical molecules and is present in all biological events. It is
also omnipresent in our world and every substance on earth
comes in contact with it.
Raman spectroscopy is a way to overcome the difficulty
due to the strong IR absorbance of liquid water, given that
the Raman signal of the OH stretch vibrations are weaker
than the corresponding IR absorption.32 On the other hand,
attenuated total reflectance ͑ATR͒ configuration allows the
B. IR measurements
The IR measurements were obtained using a model 510P
Nicolet FTIR spectrometer with a TGS detector. Two KBr
windows isolated the measurement chamber from the out-
side. The samples were contained in a Circle cell ͑Spectra-
Tech, Inc.͒ equipped with a ZnSe crystal rod ͑8 cm long͒ in
an ATR configuration ͑the beam is incident at an angle of 45°
with the rod’s axis and makes 11 internal reflections of which
9 are in contact with the liquid sample͒. The spectral range of
Ϫ1
this system is 6500–670 cm . The spectra were taken under
nitrogen flow to ensure low CO and water vapor residues in
2
complete mid-IR spectra of aqueous systems to be obtained
the spectrometer. Each spectrum represents an accumulation
of 500 scans at 4 cm resolution. The measurements were
with high precision and reproducibility.7
,9–19,21,33
We have
Ϫ1
made at 27.1Ϯ1.2 °C. The cell was carefully dried before
each measurement. Since model 510P is a single-beam spec-
trometer, a background was taken with the cell empty before
measuring each sample.
developed an adequate quantitative method using ATR that
provides consistently reliable results as long as the following
basic requirements are met: ͑1͒ a proper crystal is used
whose refractive index is sufficiently different from that of
the solution; ͑2͒ the IR beam is at a proper angle of inci-
dence; ͑3͒ the ATR crystal is of adequate length. When these
1
0
The IR measurements consisted in obtaining the ATR
background intensity, R , and the ATR sample intensity, R.
0
The ratio of R/R is the intensity I for the spectral range
0
conditions are satisfied, the IR-ATR spectra reflect the sys-
tem’s chemical changes.10 When we used this technique to
being studied. Thereafter, the 3023 data points, ͓I (
˜
␯) vs
˜␯
Ϫ1
͑in cm ͔͒ of each spectrum were transferred to a spread-
study alkali halide aqueous solutions as well as several acid–
sheet program for numerical analysis. The intensities I were
then transformed to absorbances, log(1/I) ͑abbreviated in
some cases as a.u.͒. A small baseline shift ͑less than 0.01
a.u.͒ was made to obtain a null mean absorbance in the
7
,11–19,21
base titrations,
we found that water forms stable
clusters with the ions when a binary salt ͑NaCl, KCl͒, a
strong acid ͑HCl͒ or a strong base ͑NaOH͒ is dissolved in
11–14,17,19,21
it.
With this technique, we detected the formation
Ϫ1
23
5
700–5500 cm region, where light and heavy water do
of complexes in aqueous phosphoric and sulfuric
not absorb.
1
5,16,18
acids;
some of these complexes had previously been
deduced only from thermodynamic measurements.
In previous studies, we applied principal factor analysis
FA͒ to a series of IR spectra to determine the abundances of
C. Factor analysis „FA…
͑
Factor analysis of a spectral data set is a process by
which the principal factors of an evolving system can be
identified and their concentrations obtained from derived
multiplying factors ͑MFs͒. An evolving system is a system
that is modified when a parameter such as temperature, pres-
sure, concentration, or in this case the molar faction, is var-
ied. However, when two or more species evolve simulta-
neously with the varying parameter, FA cannot separate them
because the relative concentrations of two or more species
remain constant. It may then happen that the principal factors
retrieved by FA contain more than one molecular species.
Such a situation was observed in aqueous solutions of pro-
the species. This type of analysis is suited to cases where
some of the species can easily be obtained from a set of
experimental spectra. Other methods of applying FA can be
used that usually result in abstract spectra; these can then
be transformed into real spectra by more demanding
3
4
2
0,35
methods.
The purpose of the present study is to apply the tech-
nique mentioned above to analyze the mid-IR spectra of
H O–D O mixtures to determine: ͑i͒ the number of species;
2
2
͑
ii͒ their abundances; ͑iii͒ their compositions; and ͑iv͒ the
hydrogen bonding network.
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4628
J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
J.-J. Max and C. Chapados
TABLE I. Concentrations, molar fraction, measurement temperatures, and
final MFs retrieved of the H O–D O mixtures.
2
2
͓
D O͔ ͓H O͔
Final MFs
2
2
Ϯ0.4% Ϯ0.4%
M͒ ͑M͒
T
͑°C͒
(
a)
H O HDOЈ HOD° HDOЉ D O
͑
␹
2
2
D
0
0
0
1
1
2
2
3
4
4
4
5
6
6
6
7
8
8
8
9
9
.00 54.971 0.0000* 27.14 1.001 0.000 0.000 0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
.629 54.271 0.0115 27.31 0.954 0.046 0.002 0.000
.961 54.071 0.0175 27.48 0.928 0.071 0.002 0.000
.462 53.563 0.0266* 27.92 0.893 0.105 0.004 0.000
.991 53.095 0.0361 28.04 0.853 0.140 0.008 0.000
.465 52.542 0.0448* 28.23 0.830 0.156 0.016 0.000
.965 51.996 0.0539 27.27 0.794 0.194 0.015 0.000
.510 51.625 0.0637 27.42 0.763 0.211 0.028 0.001 Ϫ0.002
.022 51.152 0.0729 27.42 0.736 0.234 0.031 0.001 0.000
.506 50.693 0.0816 27.87 0.707 0.258 0.036 0.001 Ϫ0.001
.983 50.079 0.0905* 28.07 0.681 0.270 0.046 0.003 Ϫ0.001
.524 49.476 0.1004 28.20 0.644 0.305 0.046 0.005 Ϫ0.001
.027 48.933 0.1097 28.29 0.617 0.318 0.057 0.005
.458 48.715 0.1171 26.29 0.611 0.315 0.069 0.006
.968 48.157 0.1264 26.50 0.577 0.328 0.078 0.007
.519 47.863 0.1358* 26.77 0.559 0.342 0.089 0.008
.008 47.143 0.1452 27.22 0.530 0.362 0.094 0.010
.475 46.677 0.1537 27.22 0.508 0.372 0.105 0.010
.961 46.131 0.1627 27.37 0.485 0.393 0.105 0.015
FIG. 1. ATR-IR spectra of H O–D O mixtures. Shown are 18 out of the 63
2
2
0.000
0.000
0.000
0.000
0.000
0.000
0.000
spectra taken. The sample compositions are given in Table I.
panol and in aqueous solutions of sucrose, where some hy-
drated species could not be sorted from the principal factors
obtained by FA.1
,36
.525 45.530 0.1730 27.30 0.463 0.386 0.131 0.014 Ϫ0.002
.959 45.115 0.1808* 27.37 0.444 0.396 0.136 0.019 Ϫ0.002
FA was performed with the spreadsheet program using
1
0.909 44.223 0.1979 26.25 0.412 0.407 0.152 0.023
0.000
0.000
0.002
0.003
0.003
0.004
0.005
0.005
0.011
0.013
0.028
0.045
0.062
0.091
0.125
0.170
0.230
0.275
0.310
0.346
0.383
0.415
0.446
0.487
0.533
0.579
0.604
0.631
0.659
0.693
0.718
0.753
0.789
0.813
0.848
0.873
0.922
0.929
0.954
0.978
0.984
0.993
15,16
the following iterative procedure:
͑1͒ H O and D O were
2 2
11.921 43.592 0.2147 26.53 0.379 0.408 0.176 0.030
used first as principal spectra along with HDO obtained from
1
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
2.970 42.120 0.2354 26.64 0.342 0.414 0.196 0.040
3.975 41.128 0.2536* 26.80 0.304 0.419 0.222 0.040
4.952 39.797 0.2731 27.47 0.271 0.419 0.238 0.053
5.973 39.798 0.2895 27.43 0.250 0.415 0.252 0.065
7.072 37.887 0.3106 27.41 0.219 0.406 0.274 0.084
7.918 37.223 0.3250 27.44 0.202 0.396 0.289 0.094
9.045 35.987 0.3466 26.50 0.184 0.376 0.309 0.110
0.189 34.704 0.3678* 26.70 0.165 0.362 0.324 0.124
2.441 32.528 0.4083 27.02 0.117 0.332 0.350 0.163
5.011 29.971 0.4549* 27.30 0.081 0.294 0.364 0.205
7.326 27.686 0.4967 27.38 0.062 0.250 0.373 0.250
9.850 25.619 0.5381* 27.44 0.038 0.212 0.368 0.293
2.181 22.755 0.5858 27.22 0.028 0.166 0.348 0.345
4.519 20.596 0.6263* 27.29 0.010 0.137 0.329 0.370
7.066 17.698 0.6768 27.29 0.007 0.092 0.288 0.409
9.323 15.618 0.7157 27.43 0.005 0.064 0.248 0.421
0.269 14.467 0.7357* 27.44 0.000 0.060 0.236 0.428
a ͑1:1͒ H O and D O mixture; ͑2͒ a set of realistic MFs were
2
2
chosen with which to multiply the principal spectra and form
a sum ͓⌺(MFϫprincipal spectra)͔; ͑3͒ the sums were com-
pared to the experimental spectra and the resulting residues
minimized in accordance with a least-squares fit procedure;
͑4͒ the presence of nonzero residues serve to identify another
principal spectrum that is then added to the previous princi-
pal spectra and the procedure is repeated from step ͑1͒ until
the residues are minimized. With the number of factors in the
system determined, these final MFs are multiplied by the
sample principal spectra concentrations to give the species
concentrations. The presence of negative or inadequate con-
centrations is taken as a sign that the principal spectra are not
orthogonal; an adequate procedure is used to render them
orthogonal. The final MFs were then obtained and the or-
thogonal principal spectra ͑eigenspectra͒ and final concentra-
tions determined.
1.351 13.342 0.7561 27.51 0.002 0.046 0.211
0.431
2.403 12.181 0.7768 27.58 0.000 0.036 0.191 0.425
3.618 11.167 0.7962 27.60 0.001 0.029 0.159 0.428
4.206 10.605 0.8065 26.93 0.001 0.023 0.159 0.411
5.427 9.482 0.8273* 27.15 0.001 0.016 0.128 0.409
6.678 8.271 0.8495 27.47 0.000 0.012 0.110
0.375
7.691 7.302 0.8672 27.46 0.000 0.011 0.086 0.361
7.963 6.725 0.8770 27.46 0.000 0.008 0.075 0.348
8.838 6.088 0.8892* 27.58 0.000 0.007 0.056 0.341
9.239 5.605 0.8978 26.58 0.000 0.002 0.060 0.315
9.783 5.030 0.9082 26.63 0.000 0.005 0.046 0.289
0.384 4.445 0.9189 26.82 0.001 0.004 0.035 0.266
1.006 3.887 0.9292* 26.91 0.000 0.002 0.031 0.239
1.504 3.349 0.9389 27.16 0.000 0.002 0.021 0.208
2.130 2.777 0.9494 27.11 0.000 0.005 0.0138 0.173
III. RESULTS AND DISCUSSION
A. IR-ATR spectra
Of the 63 experimental IR-ATR spectra taken of the
H O–D O mixtures, 18 are presented in Fig. 1 ͑identified by
2
2
in Table I͒. The large number of solutions was necessary to
*
obtain several spectra at low and high D molar fractions
where only two water molecular types are prevalent. The
series of spectra presents regular evolving characteristics: the
2.555 2.223 0.9594* 27.10 0.000 0.003 0.011
0.152
3.104 1.675 0.9694 27.13 0.000 0.001 0.008 0.120
3.699 1.108 0.9798 26.04 0.000 0.000 0.007 0.079
3.784 0.830 0.9848
*
26.23 0.000 0.000 0.003 0.063
H O spectrum ͑Fig. 1͒ shows the broad intense ␯ band of
2
1
,
3
Ϫ1
4.040 0.568 0.9896 26.29 0.000 0.000 0.003 0.042
4.424 0.250 0.9954 26.38 0.000 0.000 0.001 0.019
4.403 0.142 0.9974 26.70 0.000 0.000 0.001 0.010
4.520 0.000 1.0000* 26.67 0.000 0.000 0.000 0.000
water from 3700 to 2800 cm , the weak broad combination
Ϫ1
Ϫ1
band at 2115 cm , the ␯ band at 1638 cm , and the strong
2
Ϫ1
absorption that starts near 1100 cm due to the libration
band situated near 670 cm . The counterpart D O bands are
situated, respectively, at 2700–2200, 1550, 1205 cm , and
Ϫ1
2
a␹Dϵ͓D O͔/͓H O͔ϩ͓D O͔; the labeled values ͑*͒ are the experimental
spectra shown in Fig. 1.
Ϫ1
2
2
2
Ϫ1
near 800 cm
.
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J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
Isotope effects in liquid water by infrared spectroscopy
4629
As H O is replaced by D O in the mixture, the 3700–
2
2
Ϫ1
2
800 cm band decreases to the profit of the 2750–2200
cm one. The H O deformation band at 1638 cm de-
creases to the profit of the 1450 cm band until D and H are
Ϫ1
Ϫ1
2
Ϫ1
in equal concentrations, whereupon this band also decreases
Ϫ1
Ϫ1
to the profit of the 1205 cm band. The 1450 cm band is
assigned to the deformation mode of the newly formed mol-
2
,4
ecule, HDO (␦HOD). The stretch bands of this molecule
appear where the parent molecules absorb.
In the mixture spectra ͑Fig. 1͒, isosbestic points are ob-
Ϫ1
served at 2750 and 1250 cm , and quasi-isosbestic points
Ϫ1
are seen near 2180 and 1150 cm . The isosbestic points
indicate a regular transformation of the water bands into deu-
terated ones, although the quasi-isosbestic points suggest that
other species may be present.
B. Simple spectral analysis
Because the variation of the intensity and band maxima
are often used in the spectral analysis of mixtures, we ana-
lyzed these parameters in the H O–D O mixture spectra.
2
2
1. Integrated intensity of O–H and O–D stretch bands
The number of O–H valence bonds in the sample is
proportional to the product of the sample molar concentra-
tion with the probability of finding a hydrogen atom attached
FIG. 2. Spectral features of the experimental spectra. ͑A͒, ␯OH ͑⌬͒ and ␯OD
O͒ integrated intensity. Band maxima of ͑B͒ ␯OH(⌬) and ␯OD(O) , ͑C͒
␦HOH , ͑D͒ ␦HOD , and ͑E͒ ␦DOD
͑
to an oxygen atom (P ). Therefore, from Eq. ͑A2͒ ͑see Ap-
H
.
pendix A͒, if the integrated intensity of these vibrations is
independent of neighboring water molecules, the integrated
intensity of the O–H stretch band will be proportional to ␹D .
The same situation exists for the O–D stretch vibration. For
all experimental spectra, we calculated the integrated inten-
important in this spectral region. For D O, ␯
of ␦DOD
2
max
Ϫ1
decreases linearly in steps from 1218 to 1206 cm , a varia-
Ϫ1
Ϫ1
sity in the 3700–2750 and 2750–2160 cm ranges for ␯OH
tion of 12 cm ͓Fig. 2͑E͔͒.
and ␯OD , respectively. The results given in Fig. 2͑A͒ show
that both integrated intensities are linear functions of ␹D .
This indicates that the integrated ␯OH and ␯OD intensities are
not affected by neighboring isotopic water molecules.
The situation of HDO is more complex. The band sand-
Ϫ1
wiched between the ␦
cm ͒, which cannot be assigned to H O or D O, is assigned
͑1638 cm ͒ and ␦DOD ͑1207
HOH
Ϫ1
2
2
to ␦HOD ͑Fig. 1͒. The ␯
ϭ0.5 is 1446 cm . At low and high ␹ , the ␯
of ␦HOD , which occurs at ␹D
max
Ϫ1
positions
D
max
are not reliable because the intensities are weak and compo-
nents of the other species are important. The ␯max are reliable
2. Band maxima variations
Figure 2͑B͒ shows the position of ␯OH and ␯OD band
only between ␹ ϭ0.1 and 0.9. At lower and higher values,
D
Ϫ1
maxima as a function of ␹ . Both maxima vary linearly,
␯max increases from 1447 to 1455 cm for a variation of 8
D
Ϫ1
Ϫ1
over the range 3340–3406 cm in the first case, and over
cm on both sides.
Ϫ1
the range 2505–2473 cm in the second case. These linear
The OH and OD stretch bands and the three clearly vis-
ible deformation bands of H O, HDO, and D O indicate the
relationships suggest the presence of two OH species and
two OD species. This could implicate (2ϫ2ϭ)4 species, but
indicates that at least three species are present should one
combine equally with OH and OD to form one species dif-
ferent from that of pure H O and pure D O.
2
2
presence of these three molecules. For H O and D O, the 13
2
2
Ϫ1
and 12 cm variation of ␯max of the deformation bands
would indicate the presence of intermediate species of H O
2
and D O between HDO and H O on one side and D O on
2
2
2
2
2
Figures 2͑C͒, 2͑D͒, and 2͑E͒ show the position as a func-
the other side. The very weak variation of ␯max of ␦HOD with
␹D and the band being situated between the other two may
not be indicative of supplementary species.
tion of ␹ of the HOH, HOD, and DOD deformation bands,
D
respectively. Between ␹ ϭ0 and 0.5, ␯
of ␦HOH increases
D
max
Ϫ1
linearly from 1639 to 1651 cm ͑steps are due to the finite
Although crude, this analysis would suggest the presence
of five spectroscopically different entities. To sort them, we
proceeded in sequence: first, the HDO spectrum was ob-
tained from the H O–D O equimolar solution spectrum; sec-
Ϫ1
Ϫ1
spectral sampling of 1.92 cm giving a resolution of 4 cm
Ϫ1
following Shannon’s theorem͒, a variation of 13 cm ͓Fig.
2
͑C͔͒. Between ␹ ϭ0.5 and 0.8, ␯
of ␦HOH remains con-
D
max
2
2
stant. Thereafter the position of this band cannot be deter-
mined exactly because its intensity is weak ͑less than 4% of
that of pure water͒, and other bands become increasingly
ond, FA was made on all solution spectra with three factors
to retrieve the supplementary species; third, FA was made
with five principal factors.
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4630
J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
J.-J. Max and C. Chapados
C. Factor analysis using three principal factors
1. Determination of a third specie
Pure H O and pure D O are obviously two principal
2
2
factors in the series and are obtainable in pure form. The
third species, HDO, cannot be obtained in pure liquid form
since it exists solely in equilibrium with H O and D O, as
2
2
described by the equation,2
,37
H OϩD Oꢀ2HDO.
͑1͒
͑2͒
2
2
The thermodynamic equilibrium constants, K , is
e
HDO͔2
H O͔͓D O͔
͓
K ϭ
e
,
͓
2
2
where K values found in the literature at 25 °C ranged from
e
2
7–31
3
.76 to 3.94.
The ‘‘pure’’ HDO spectrum had been obtained previ-
ously by subtracting 0.25 H O spectrum and 0.25 D O spec-
2
2
2
trum from an equimolar H O–D O solution spectrum. Be-
2
2
cause it presents sigmo ¨ı dal features near the deformation
bands of the parent molecules, the resulting spectrum cannot
be considered exact. Such features are not due to optical
effects produced by the ATR configuration since they are still
present in the real and imaginary parts of the refractive index
2
spectra ͑n and k͒. The sigmo ¨ı dal features indicate that too
much of the H O and D O spectra have been subtracted from
2
2
the equimolar spectrum. Bearing this in mind and using the
1
,7,11–19,36
condition of no negative band previously used,
we
obtained the HDO spectrum by subtracting 0.18ϫpure H O
2
spectrum and 0.18ϫpure D O spectrum from the equimolar
2
H O–D O spectrum. The resulting spectrum normalized to
2
2
55.13 M ͑Ref. 38͒ was identified as that of HDO and is
shown in Fig. 3͑A͒ with those of H O and D O.
2
2
2. Multiplying factors (MF)
The spectra of H O, HDO, and D O shown in Fig. 3͑A͒
2
2
are orthogonal since each spectrum is of a pure molecular
species. These spectra served to determine the MFs of the
different experimental spectra ͑Fig. 1͒ using FA. The results,
given as a function of ␹ in Figs. 3͑B͒, show a regular pat-
D
tern for the three species, indicating that the species chosen
yield coherent results.
FIG. 3. FA results with three eigenspectra: ͑A͒ ATR-IR spectra of H O
2
͑
bottom͒, HDO ͑middle͒, and D O ͑top͒; ͑B͒ MFs retrieved; ͑C͒ Residues of
2
3. Residues
the difference between experimental spectra ͑Fig. 1͒ and calculated spectra
͑
⌺ MFϫprincipal spectrum͒ at ␹ Ͻ0.5 and ␹ Ͼ0.5. ͑␹ is the molar frac-
D D
The residues obtained from the difference between ex-
perimental and calculated spectra were divided into two re-
tion͒.
gions in Fig. 3͑C͒. ␹ Ͻ0.5 and ␹ Ͼ0.5. The residues are
D
D
not zero, and show a regular pattern near the OH and OD
stretch bands and near the HOH, HOD, and DOD deforma-
tion bands. These results suggest the presence of other spe-
cies in the solutions.
We verified that introducing a fourth principal factor into
FA did not decrease the residues significantly, but adding a
fifth one did. We then concluded that at least five principal
factors were necessary to perform FA on the 63 experimental
spectra.
D. Factor analysis using five principal spectra
Pure H O and D O are obviously the two first principal
2
2
factors. Three other principal factors must be determined.
1. Determination of the three remaining factors
A first step in the analysis was undertaken to determine
the variation path of the five MFs retrieved by FA. The com-
position of the three unknown factors could then be deter-
mined.
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J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
Isotope effects in liquid water by infrared spectroscopy
4631
P
TABLE II. Multiplying factors ͑MFs͒ of the five experimental spectra S
used to obtain the matrix P ͑see text͒.
Since the equimolar H O–D O mixture contains more
than three species, we chose to start with the extreme differ-
ences in concentration—those samples should be the least
exp
2
2
␹D
0.000 0.054 0.497 0.949 1.000
1.000 0.794 0.062 0.000 0.000
complicated to study. In the ␹ Ͻ0.05 mixture, the relative
D
1
2
3
st principal factor
nd principal factor
rd principal factor
molar amount of D O at equilibrium is expected to be less
2
0.000 0.194 0.250 0.005 0.000 ϵ P
0.000 0.015 0.373 0.014 0.000
0.000 0.000 0.250 0.173 0.000
0.000 0.000 0.062 0.813 0.993
2
than (0.05) ϭ0.0025 ͑see Appendix A͒; the solution would
contain principally H O with around ͓2ϫ0.05ϫ(1Ϫ0.05)
4th principal factor
5th principal factor
2
Ϸ͔0.10 molar ratio of HDO. The first unknown factor con-
3
9
tains H O and HDO, a species that we call HDOЈ.
2
To determine the composition of HDOЈ, we choose the
experimental spectrum at ␹ ϭ0.027. The operation of maxi-
D
evolving patterns, indicating that FA proceeded correctly. In
Fig. 4͑B͒, the continuous lines are obtained from calculations
discussed in Sec. III E 2.
mizing the subtraction of the pure water spectrum from the
experimental result was monitored near 1638 cmϪ1 subject to
the constraint that there be neither a negative band nor a
sigmoid absorption in this region. The MFs determined for
The residues from the differences between experimental
and calculated spectra are given in Fig. 4͑C͒. The integrated
squared intensity of the difference spectra were less than
one-30th that obtained with three principal factors ͑Fig. 3͒.
The remaining residues do not show as regular a pattern as in
the previous case. In Fig. 4͑D͒, we compare the integrated
squared intensity of the residues with the mean temperature
difference at which the sample spectra were recorded. Since
the two curves are correlated, we can relate the remaining
residues ͓Fig. 4͑C͔͒ to a temperature variation between the
spectra. Furthermore, the amplitude and shape of the resi-
dues are similar to those reported for a temperature variation
of 1.2 °C,41 which is the temperature stability of our system.
We concluded from the above FA that five principal fac-
tors are present in the H O–D O mixtures: H O; HDOЈ;
pure H O and HDOЈ were 0.892 and 0.108, respectively.
2
Similarly, the experimental spectrum at ␹ ϭ0.985 was as-
D
sumed to contain only two species: D O and the second un-
2
known factor that we call HDOЉ. The subtraction of the
heavy water spectrum from the experimental one, which was
Ϫ1
monitored near 1205 cm , gave 0.930 and 0.070 for the
MFs of pure D O and HDOЉ, respectively.
2
The third unknown factor, which we named HDO°, was
obtained from the mixture at ␹ ϭ0.497 ͑Ref. 40͒ by sub-
D
tracting the spectra of the four other factors determined pre-
viously from the experimental spectrum, monitoring the
maximum extent of the subtraction in the three deformation
regions. The MFs subtracted were 0.062, 0.25, 0.25, and
2
2
2
0
.062 for H O, HDOЈ, HDOЉ, and D O, respectively. The
2
2
HDO°, HDOЉ, and D O. No other principal factors could be
2
MF of HDO° is therefore 0.376.
observed given the limit of the present experimental preci-
sion. The nature and composition of the five principal factors
can now be determined.
The small MF values of HDOЈ and HDOЉ ͑0.108 and
.070, respectively͒ resulted in a low signal to noise ratio for
0
the corresponding eigenspectra. To compensate for this, we
translated the information gained with two easily identified
principal factors to more concentrated solutions than initially
utilized. The abundances of the two factors being higher pro-
vides a better signal to noise ratio. This is presented in the
next section.
3. Composition of the five principal factors
The composition of the five eigenspectra of Fig. 4͑A͒
was calculated with Eq. ͑3͒. The composition of the samples
P
that gave the experimental spectra S was multiplied by the
exp
Ϫ1
inverse matrix P to obtain the composition corresponding
2. Principal factors with high signal to noise ratios
to the eigenspectra. Spectrum a is that of water; spectrum b
is that of HDOЈ composed of a H O–HDO 1:1 mixture ͑the
2
The precision of FA was increased by minimizing the
residues using matrix multiplication with the five eigenspec-
tra previously determined and with the experimental spectra
Ϫ1
absence of ␦
band near 1205 cm indicates that D O is
D O
2
2
not present͒; spectrum d is that of HDOЉ made up of a
D O–HDO 1:1 mixture ͑the absence of ␦ band near 1640
H O
2
at ␹ ϭ0.000, 0.054, 0.497, 0.949, and 1.000 ͑Table I͒. The
2
D
P
cmϪ1 indicates that H O is not present͒; and spectrum e is
eigenspectra matrix S was obtained from the matrix of the
five experimental spectra S from the relation,
2
P
exp
that of D O. The situation of spectrum c ͑HDO°͒ is more
2
complex: the IR bands centered near 1640, 1455, and 1205
cm indicate the presence of the three molecular types of
SPϭSP ϫPϪ1
,
͑3͒
exp
Ϫ1
where P is the matrix of the MFs of the five experimental
water: H O, HDO, and D O in the ratio 1:4:1. The resulting
2
2
spectra ͑at ␹ ϭ0.000, 0.054, 0.497, 0.949, and 1.000͒ ob-
tained from the five principal spectra determined in the pre-
ceding section. The P matrix is given in Table II.
compositions of the five principal factors are reported in
Table III. No situation exists where a ‘‘pure’’ HDO spectrum
could be retrieved because H O and D O molecules are
D
2
2
The five principal spectra obtained with Eq. ͑3͒ and nor-
malized to the pure water concentration ͑55.13 M͒ are shown
in Fig. 4͑A͒. With these principal spectra, FA was then ap-
plied to the 63 experimental spectra. The MFs retrieved are
given in Fig. 4͑B͒ and in Table I. The vertical lines in the
figure indicate the molar fractions at which the eigenspectra
were determined. The MFs of the five species show regular
present in the HDO° factor, which are slightly different from
that of ‘‘pure’’ H O and D O, and which evolve simulta-
neously with ‘‘pure’’ HDO. Note that we used Eq. ͑1͒ with
2
2
K ϭ4.0 for calculating the HDO composition in the HDOЈ,
e
HDO°, and HDOЉ spectra because the disproportionation ef-
fects which influence the constant are very small and can be
neglected ͑see Appendix A͒.
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4632
J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
J.-J. Max and C. Chapados
E. Interpretation of the five species retrieved
The results of FA on the series of 63 IR spectra of mix-
tures of H O and D O indicate that five principal factors are
2
2
present in the samples ͑Fig. 4͒. Two of these factors are pure
H O and D O, labeled H Oa and D Oe, respectively. The a
2
2
2
2
and e tags indicate the evolving factors; the others are b, c,
and d. Most of molecules in liquid water have 4 hydrogen
bonds with 4 neighboring water molecules of the same
type4
2,43
͑see Appendix B͒. The spectra in Fig. 4͑A͒ indicate
that pure water (H Oa) and factors b and c contain H O
2
2
molecules because the bands observed at 1638, 1649, and
Ϫ1
1
654 cm , respectively, can only be assigned to ␦
. Fac-
H O
2
tors c, d, and e contain D O molecules because the bands
2
Ϫ1
observed at 1214, 1212, and 1207 cm , respectively, can
only be assigned to ␦
. We first address the question of
D O
2
evaluating the difference in the H O molecular environment
2
that could induce a perturbation of the H O deformation
2
Ϫ1
band near 1638 cm
.
1
. Environmental influence on the H O deformation
2
vibration
The oxygen of water can make four bonds with H and/or
D atoms: two covalent bonds and two hydrogen bonds. The
five situations possible are illustrated at the bottom of Fig. 5,
where valence and hydrogen bonds are not distinguished.
Depending on the neighboring molecules, several situations
exist. These are schematically illustrated in Fig. 5 ͑top͒ by a
two-dimensional water network where each water molecule
(
H O, HOD, and D O͒ is represented in a square box. Each
2 2
box is linked to four different boxes through four hydrogen
bonds ͑dotted lines͒. The boxes are identified by a column
letter and a line number, as in a crossword puzzle. For ex-
ample, the situation in pure liquid water is depicted by box
C2 because the nearest-neighbors to the H O (H Oa) in the
2
2
box are 4 H O molecules. The two hydrogen atoms of H Oa
2
2
are hydrogen bonded to the oxygen atoms of two neighbor-
ing H O molecules ͑D2 and C3͒ while the oxygen atom of
2
H Oa is hydrogen bonded to two hydrogen atoms of two
2
other neighboring H O molecules ͑B2 and C1͒.
2
When few HDO molecules are present in the solution, as
in E3, the H and D of an HDO molecule are hydrogen
bonded to the oxygen atoms of H O in F3 and E4, respec-
2
tively. The perturbation on H O in E4 is of first order be-
2
cause its oxygen is now hydrogen bonded to both a H and a
D atom. The oxygen in HDO is hydrogen bonded to two
nearest-neighbor H O molecules in D3 and E2 via two hy-
2
drogen atoms. The perturbation on the D3 molecule is con-
sidered second order because the hydrogen atoms are always
hydrogen bonded to an oxygen atom of a neighboring water
molecule.
FIG. 4. FA results using five eigenspectra. ͑A͒, ATR-IR spectra of: H O, a
2. Probability evaluation of the different cases
2
͑
bottom͒, b, c, d, and D O, e. Each spectrum is shifted 1 a.u. from the
2
While Fig. 5 illustrates only part of the situation, Table
IV lists all possible combinations that a reference water mol-
preceding one. ͑B͒ MFs retrieved ͑symbols͒. Full lines come from the the-
oretical calculations ͑see text͒. Light vertical lines indicate where the prin-
cipal spectra were determined. ͑C͒ Residues of the difference between ex-
perimental ͑Fig. 1͒ and calculated spectra using five principal factors at
ecule can have with four neighboring water molecules ͑H O,
2
4
HDO, or D O͒. Table IV enumerates the (2 ϭ) 16 possible
2
␹DϽ0.5 and ␹ Ͼ0.5. Same presentation as in Fig. 3. ͑D͒ Integrated inten-
D
combinations of 4 ordered elements with H and D that a
reference HOH molecule situated at the center can make
sity of the residues ͑O, bottom trace͒ and temperature variation of the spec-
tra ͑ϫ, top trace͒.
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J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
Isotope effects in liquid water by infrared spectroscopy
4633
TABLE III. Composition of the five principal factors in H O–D O mixtures.
2
2
Initial concentration
Concentration at equilibriuma
Spectrum
H O
D O
H O
HDO
D O
Molar ratiosa
2
2
2
2
͓Fig. 4͑A͔͒
͑M͒
͑M͒
͑M͒
͑M͒
͑M͒
H O:HDO:D O
2
2
a
55.13
0.00
55.13
27.51
9.19
0.00
27.65
36.65
27.60
0.00
0.00
0.00
1:0:0
1:1:0
1:4:1
0:1:1
0:0:1
H O
2
b
HDOЈ
c
HDO°
d
41.33
27.56
13.80
0.00
13.83
27.42
41.53
55.13
9.14
0.00
27.72
55.13
HDOЉ
e
0.00
D O
2
a
The evaluation of HDO abundance was obtained using Eq. ͑1͒ and K ϭ4.00.
e
with the two water molecules H-bonded to its oxygen atom.
One OH valence bond of this molecule is directed toward the
top of the table, making a hydrogen bond with an oxygen
atom of a water molecule ͑OH , OHD, or OD ͒. Similarly,
the other OH valence bond is directed toward the bottom of
the table.
Listed to the left and right of the reference HOH mol-
ecule in Table IV are two columns containing the different
combination of HOH, DOD, HOD, and DOH molecules
whose H or D makes a hydrogen bond with the oxygen of
the reference molecule. Depending on the hydrogen bond
made with the oxygen atom of the reference molecule, three
different situations may be distinguished with the letters a, b,
and c ͓it will be shown later that the letters a, b, and c refer
to the spectra in Fig. 4͑A͔͒: a, an H of the molecules in both
left and right columns are hydrogen bonded to the reference
molecule O ͑combination #1, 3, 13, and 15; e.g., C2, E5, and
C4 in Fig. 5͒; b, an H from the left and a D from the right or
vice versa are hydrogen bonded to the reference molecule O
similar molecules. HOH and DOD have a C2v symmetry,
while that of HOD is C . Therefore, each situation involving
s
an HOH or a DOD molecule H-, or D-bonded to the oxygen
of the center water molecule must be counted twice, since
the equivalence of their two possible molecular orientations
renders them indistinguishable. The degeneracy numbers are
listed in Table IV. As an example, in combination 1 two
water molecules are involved; each is twice degenerate, for a
total degeneracy of 4. The disproportionation effect, in which
H and D are not considered equivalent ͑in terms of probabil-
ity͒, has been invoked by some authors in their studies of
2
2
2
7–30
H O–D O mixtures.
We demonstrate in Appendix A
2
2
that this phenomenon can be neglected in the present circum-
stances.
The probability of each of the three different species of
HOH in each sample may now be calculated. The species
H Oa is encountered in combination 1, 3, 13, and 15 ͑Table
2
IV͒. Therefore, we have
͑
combination #2, 4, 5, 7, 9, 11, 14, and 16; e.g., E4 and E8 in
P
H O
2
2
2
Fig. 5͒; c, a D of the molecules in both left and right columns
are hydrogen bonded to the reference molecule O ͑combina-
tion #6, 8, 10, and 12; e.g., F10 in Fig. 5͒. The three different
situations in the case of a central HOH molecule are indi-
P
ϭ
͑4P ϩ4P ϫPHDOϩP
͒,
͑4͒
H Oa
H O
H O
HDO
2
2
2
4
where P
is the probability for one water molecule to be
H Oa
2
cated in the column entitled H O ͑1͒ at the right of Table IV.
of type H2Oa, and where PH O and PHDO are given by Eqs.
2
2
Similarly, the columns headed HOD ͑2͒ and D O ͑3͒ are for
͑A3͒ and ͑A4͒. In Eq. ͑4͒, the first term P
represents the
H O
2
2
the HOD reference water molecule, giving species b, c, and
d, and the DOD reference water molecule participating in the
spectra c, d, and e in Fig. 4. Therefore, for each of the three
central molecules ͑HOH, HOD, or DOD͒, three possibilities
were identified, resulting in (3ϫ3ϭ) 9 different species. For
probability that the reference molecule ͑the center molecule
in Table IV͒ is a H O molecule. The dividing factor 4 was
2
introduced to take into account the 2ϫ2 number of indepen-
dent permutations of the right and left atoms ͑H or D͒ of the
right and left side water molecules. The numerical factors
inside the parenthesis ͑4, 4, and 1͒ are the degeneracy num-
bers ͑Table IV͒.
H O these are 1a, 1b, and 1c ͑C2, F3, and F10 in Fig. 5͒.
2
Similarly, for HDO these are 2b, 2c, and 2d ͑F5, F6, and F9͒,
and for D O these are 3c, 3d, and 3e ͑E9, G6, and G8͒. Four
2
Equation ͑4͒ reduces to
of the nine species evolve simultaneously with other species,
giving five principal factors ͑a, b, c, d, and e͒. This will be
demonstrated below.
The abundances of the nine different species identified
above are related to their probabilities identified in Table IV.
The degeneracy of each of the 16 combinations depicted in
Table IV has to be determined. Any one water molecule
P
H O
2
P
ϭ
͑2P ϩPHDO͒2.
͑5͒
H Oa
H O
2
2
4
The probabilities for one water molecule to be of the
type H Ob, H Oc were evaluated in the same way from data
2
2
͑HOH, HOD or DOD͒ cannot be distinguished from other
in Table IV,
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4634
J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
J.-J. Max and C. Chapados
FIG. 5. Top: Two-dimensional representation of the hydrogen bond network in the H O–D O mixtures. Each square contains one water molecule ͑with H or
2
2
D͒. Full and dotted lines represent covalent and hydrogen bonds, respectively. Bottom: The five principal factors where two bonds are covalent bonds and the
other two are H-bonds ͑see text͒.
P
PHDO
H O
2
PHDObϭP
PHDOcϭP
PHDOdϭP
,
,
,
͑8͒
͑9͒
P
ϭ
͓4P ϫP ϩPHDOϫ͑2ϪPHDO͔͒
͑6͒
H Oa
H Ob
H O
D O
2
2
2
2
PH O
2
2
and
PHDO
H Ob
P
2
H O
2
P
H O
2
P
H Oc
ϭ
͑2P ϩPHDO͒2.
D O
2
͑7͒
2
4
PHDO
The six remaining probabilities were deduced from Eqs. ͑5͒
to ͑7͒ and from Table IV,
͑10͒
H Oc
2
P
H O
2
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J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
Isotope effects in liquid water by infrared spectroscopy
4635
TABLE IV. Relation between a reference HOH molecule and neighboring molecules in H O–D O mixtures.
2
2
aH–O–H is the reference molecule ͑1͒ which can be replaced by HOD ͑2͒ and DOD ͑3͒ depending on the
situation considered.
b
One H of the central molecule is H-bonded to an O of either OH , OHD, or OD . The H or D of the
2
2
neighboring molecule does not influence the number of species ͑see text͒.
The numbers refer to the reference molecule and the letters a, b, c, d, and e to the evolving factors. Vg, the
c
species 1b and 2b evolve simultaneously as a function of ␹D and cannot be separated by FA.
and
rors are on the theoretical calculated lines. The experimental
errors correspond to the symbol sizes in the figure
͑ϳϮ0.003͒.
P
P
D O
2
P
P
P
ϭP
,
͑11͒
͑12͒
͑13͒
D Oc
H Oa
Using Eqs. ͑4͒, ͑6͒, and ͑8͒ together with Eqs. ͑A3͒–
2
2
H O
2
͑A5͒, and neglecting the reported disproportionation ͑i.e., ␣
ϭ␤ϭ␥ϭ1͒, one obtains
P
D O
2
ϭP
D Od
,
PHDObϭP .
H Ob
2
͑14͒
H Ob
2
2
P
H O
2
In the same way, it is easily demonstrated that
ϭP
P
D O
2
P
,
D Oc
2
͑15͒
͑16͒
H Oc
ϭP
.
2
D Oe
H Oc
2
2
P
H O
2
PHDOdϭP
,
D Od
2
The numerical values computed with Eqs. ͑5͒–͑13͒ are
and
plotted as functions of ␹ in Fig. 6. Adding the abundances
D
of all species with the same letter ͑aϭ1a; bϭ1bϩ2b; c
ϭ1cϩ2cϩ3c; dϭ2dϩ3d; and eϭ3e͒, the total relative
amount of species b, c, and d were computed using Eqs.
PHDOcϭ4P
.
͑17͒
H Oc
2
Therefore, species 1b and 2b remain in the same 1:1
͑6͒–͑12͒ and plotted in Fig. 4͑B͒ ͑full lines͒ along with the
proportion whatever may be the value of ␹ . The same situ-
D
experimental results. The experimental results with their er-
ation prevails for species 2d and 3d. Finally, species 1c, 2c,
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4636
J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
J.-J. Max and C. Chapados
deuterium atoms, i.e., heavy water ͑specie 3e in Fig. 5, e.g.,
B9, B10, C8, and G8͒.
The probability model confirms that principal factor b
͓
Fig. 4͑A͔͒ is related to a 1:1 H O–HDO mixture. In the
2
same way, principal factor d ͓Fig. 4͑A͔͒ is related to a 1:1
D O–HDO mixture. Finally, principal factor c ͓Fig. 4͑A͔͒ is
2
related to a 1:4:1 H O-HDO–D O mixture. This agrees with
2
2
the composition that was ascribed to the five principal factors
in Sec. III D 3 ͑Table III͒. The different types of water mol-
ecules that are included in principal factors b, c, and d are
not necessarily hydrogen bonded to one another ͑Fig. 5͒.
IR spectroscopy allows a direct measurement of the spe-
cies present in liquid solutions, as opposed to other methods,
such as vapor pressure measurements30 where solution com-
position is determined by indirect measurements. Five of the
nine species anticipated were detected in the present experi-
ments. With variations of temperature and/or pressure, the
disproportionation effect could be made to yield values of ␣,
␤
, and ␥ different from 1 ͑see Appendix A͒. Equations ͑14͒–
17͒ would not be valid in that case, and all nine water spe-
cies in the H O–D O mixtures could be sorted.
͑
2
2
F. Thermodynamic equilibrium of H O–D O liquid
2
2
mixtures
Equation ͑1͒ represents chemical equilibrium in the gas
phase. For the liquid phase, with the five principal factors
FIG. 6. Probability of different species in the H O–D O mixtures. ͑A͒ D O
2
2
2
species: D Oc; D Od; and D Oe; ͑B͒ HDO species: HDOb; HDOc; and
2
2
2
obtained by FA of the H O–D O mixtures, we calculated the
HDOd; ͑C͒ H O species: H Oa; H Ob; and H Oc.
2
2
2
2
2
2
total amount of H O, D O, and HDO; this permitted us to
2
2
evaluate the equilibrium constant K ͓Eq. ͑2͔͒ for different
e
␹D . These results are plotted in Fig. 7͑A͒ ͑᭺ symbols͒,
where the upper and lower curves from the median symbols
are the error limits when an error level of Ϯ0.003 is applied
and 3c remain in the 1:4:1 proportion independently of the
value of ␹ . These properties fully explain the reason why
D
only five principal factors were retrieved by FA when nine
species are present in the H O–D O mixtures. FA sort as a
on all MFs. The mean value of K was found to be 4.16
e
2
2
Ϯ0.30 in the 0.1Ͻ␹ Ͻ0.9 range. Some concentrations were
D
single factor the species that evolve simultaneously.
Furthermore, the results obtained by FA on the 63
H O–D O mixture spectra are well supported by a simple
not obtained with sufficient precision outside these limits.
The constant is, within experimental error, the same as that
2
2
27–30
obtained by vapor measurements.
However, since we
probabilistic model. The nine water species are included in
the five principal factors. This relation can be explained as
follows: each oxygen atom forms four bonds with the hydro-
gen or deuterium atoms, two covalent bonds and two hydro-
gen bonds. Principal factor a is related to an oxygen atom
with four bonds to hydrogen atoms, i.e., water ͑1a in Fig. 5,
e.g., B2, C3, D4, and E5͒. Principal factor b is related to an
oxygen atom with three bonds to hydrogen atoms and one
bond to a deuterium atom ͑1b and 2b in Fig. 5, e.g., B4, E2,
E3, and F5͒. Species 1b and 2b are not necessarily hydrogen
bonded to one another ͑e.g., B4͒, but evolve simultaneously
know that there are more than three species in the solutions,
this result has limited value other than for purposes of com-
parison.
The five principal species retrieved by FA from the
mid-IR spectra of H O–D O mixtures, which are designated
2
2
by the letters a to e ͓Fig. 4͑A͔͒, can be summarized as fol-
lows: a, H O (OH ); b, HDOЈ (OH D); c, HDO°
2
4
3
(
OH D ); d, HDOЉ (OHD ); and e, D O (OD ). The sym-
2 2 3 2 4
bols in parentheses are the effective principal factors ob-
served with IR. The notation OX ͑where XϭH or D͒ indi-
4
cates that two bonds are covalent and two are hydrogen
͓they have the same probability, Eq. ͑14͔͒. Principal factor c
bonds. For example, the principal factor b, OH D, corre-
3
is related to an oxygen atom with two bonds to hydrogen
atoms and two bonds to deuterium atoms ͑1c, 2c, and 3c in
Fig. 5, e.g., B5, B7, C5, and G3͒. Species 1c, 2c, and 3c are
not necessarily hydrogen bonded to one another, but evolve
simultaneously ͓Eq. ͑15͒ and ͑17͔͒. The concentration of
specie 2c is four times that of specie 1c ͓Eq. ͑17͔͒. Principal
factor d is related to an oxygen atom with one bond to a
hydrogen atom and three bonds to deuterium atoms ͑species
sponds to two situations: ͑1͒ H–O–H ͑two OH covalent
bonds͒ with two hydrogen bonds on the oxygen ͑one O¯H
and one O¯D͒; ͑2͒ H–O–D ͑one OH and one OD covalent
bonds͒ with two hydrogen bonds on the oxygen ͑two O¯H͒.
In liquid water, proton hopping ͑or tunneling͒ occurs at a
very fast rate,44 so that the OH D grouping frequently passes
3
from one situation to the other. Furthermore, due to the in-
dividual molecular rotation and proton hopping the three fol-
lowing equilibria take place in a H O–D O mixture:
2d and 3d in Fig. 5, e.g., B8, D8, D9, and G10͒. Species 2d
2
2
and 3d evolve simultaneously ͓Eq. ͑16͔͒. Finally, principal
factor e is related to an oxygen atom with four bonds to
OH ϩOH D ꢀ2OH D,
͑18͒
4
2
2
3
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J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
Isotope effects in liquid water by infrared spectroscopy
4637
portionation is not observed on the mixture spectra, and jus-
tifies the approximation, ␣ϭ␤ϭ␥ϭ1, used in the present
study.
OH D
is an average description where all possible
i
(4Ϫi)
forms have almost equal probability because they change
rapidly from one form to the other. For instance, OH D rep-
3
resents from time to time H O–(H,D) or HDO–(H) , each
2
2
one is transforming into the other by exchanging covalent
4
4
bonds to hydrogen-bonds at the rate of ϳ1 ps. In OH D
3
grouping, there is an equal possibility that the two covalent
bonds give either H O and HDO molecules ͑see details be-
2
low for OH D grouping͒. This is why the composition of b
2
2
species was found to be 1:1 H O:HDO. Furthermore, from
2
Eq. ͑6͒, ͑A3͒–͑A5͒ and considering ␣ϭ␤ϭ␥ϭ1, one ob-
tains
3
P
ϭ2␹ ͑1Ϫ␹ ͒ .
͑24͒
H Ob
D
D
2
Equation ͑24͒ together with Eq. ͑14͒ gives the total probabil-
ity of species b,
3
P ϭ4␹ ͑1Ϫ␹ ͒ .
͑25͒
b
D
D
Equation ͑25͒ exactly represents the probability of
OH D grouping, that is the probability for an oxygen to be
3
surrounded by three H and one D atoms. The transformation
from H O–(H,D) into HDO–(H) through proton hopping
2
2
in a chain of neighboring water molecules ͑‘‘proton
wires’’͒4
5,46
at a rate in the order of 1 ps does not permit the
FIG. 7. Experimental ͑᭺͒ and theoretical values of the equilibrium con-
stants ͑A͒ for the equilibrium Eq. ͑1͒; ͑B͒ equilibrium Eq. ͑18͒; ͑C͒ equi-
librium Eq. ͑19͒; and ͑D͒ equilibrium Eq. ͑20͒ ͑see text͒. In ͑A͒, the high
and low continuous lines from the central terms indicate the error margin.
spectral separation between both molecular species. This fast
proton hopping could be partly responsible for the broadness
of the OH stretch bands in liquid water ͑1 ps corresponds
Ϫ1
roughly to a bandwidth of 33 cm ͒. The same analysis can
be applied to the five OH D
species, showing that
i
(4Ϫi)
4
OH DϩOHD ꢀ2OH D ,
͑19͒
͑20͒
P ϭ͑1Ϫ␹ ͒ ,
͑26͒
͑27͒
͑28͒
3
3
2
2
a
D
2
D
2
OH D ϩOD ꢀ2OHD ,
2
2
4
3
P ϭ6␹ ͑1Ϫ␹ ͒ ,
c
D
whose equilibrium constants are
3
D
P ϭ4␹ ͑1Ϫ␹ ͒,
d
D
OH D͔2
3
͓
and
K ϭ
,
͑21͒
͑22͒
͑23͒
1
͓
͓
͓
OH ͔͓OH D ͔
4 2 2
4
P ϭ␹ .
͑29͒
e
D
2
͓
OH D ͔
2 2
K ϭ
2
,
The numerical results obtained with Eqs. ͑25͒–͑29͒ cor-
OH D͔͓OHD ͔
3
3
respond exactly to the results obtained with Eqs. ͑5͒–͑13͒ for
the five spectroscopic factor abundances. However, the
OHD ͔2
͓
3
K ϭ
2
.
OH D
representation indicates that the oxygen atom en-
OH D ͔͓OD ͔
i
(4Ϫi)
2
2
4
vironments ͑i.e., the four atoms bonded to it: two covalent
bonds and two hydrogen bonds͒ produce the spectroscopic
species or entities which justify the equilibrium Eqs. ͑18͒–
With the principal spectra normalized to the same con-
centration ͑55.13 M͒, the MFs are transformed into concen-
trations that can be substituted into Eqs. ͑21͒–͑23͒ to evalu-
ate the three constants. The results are given in Figs. 7͑B͒,
͑20͒. For example let H1, H2, D1, and D2 of the OH D2
2
grouping be the four atoms that surround the oxygen. Two
atoms make the covalent bonds which determine the mol-
ecule that produces the bending vibration. The six combina-
tions possible for both covalent bonds are OH1H2, OH1D1,
OH1D2, OH2D1, OH2D2, and OD1D2. The first molecule is
H O, the four middle ones are HDO, and the last one is D O
7
2
͑C͒, and 7͑D͒, where the mean constant values are 2.65,
.25, and 2.65 for K , K , and K , respectively. The devia-
1
2
3
tion points calculated where the concentrations of the species
are low are not reliable ͑at the limits of the curves͒. The
experimental values with their experimental errors fall on the
theoretical ones ͑horizontal lines͒ calculated from the prob-
ability equations ͓Eqs. ͑5͒–͑13͔͒. This indicates that dispro-
2
2
giving the ratio 1:4:1 that was found for the spectroscopic
species c.
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4638
J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
J.-J. Max and C. Chapados
FIG. 8. Gaussian fitting in the OH and OD stretch regions of the principal
FIG. 9. Gaussian fitting of the principal spectra in the combination and
spectra: ͑a͒ H O(ϵOH ); ͑b͒ HDOЈ(ϵOH D); ͑c͒ HDO°(ϵOH D ); ͑d͒
deformation: ͑a͒ H O(ϵOH ); ͑b͒ HDOЈ(ϵOH D); ͑c͒ HDO°
2
4
3
2
2
2
4
3
HDOЉ(ϵOHD ); ͑e͒ D O(ϵOD ); ͑top͒ residues. Note that for OX
(ϵOH D ); ͑d͒ HDOЉ(ϵOHD ); ͑e͒ D O(ϵOD ); ͑top͒ residues. Same
3
2
4
4
2 2 3 2 4
͑where XϭH or D͒ two bonds are covalent bonds and two are hydrogen
note as in Fig. 8.
bonds.
were previously defined.͔ The position and assignment of the
bands are given in Table V.
G. Band simulation and band assignments:
Decomposition with Gaussian profiles
The low values of the residues given at the top of Figs. 8
and 9 are a sign that the spectrum simulations were reason-
ably good. While it is also clear that there is room left for
improvement, this would require better quality spectra and
many additional components ͑their number, band shapes, and
bandwidths͒. We do not have access to this information at
present for liquid water.
In an effort to obtain some spectroscopic evidence of the
nine species in the H O–D O mixture solutions, we pro-
2
2
ceeded with the band simulation of the principal spectra. We
started by fitting the band contours with Gaussian profiles.
Although the Gaussian profile is not the best one to simulate
IR spectra, it worked reasonably well in the case of
H O–D O. We also tried the Voigt and the Lorentzian pro-
Nonetheless, the H O and D O spectra contain the same
2
2
2
2
number of components with the bands of D O shifted rela-
2
files, as well as products of the Gaussian and Lorentzian
profiles. These proved no more useful than the Gaussian pro-
file because the experimental bands are wide, intense, and
composed of many components that are not all separated.
Using all the spectral features as indicative of the presence of
tive to that of H O due to the isotopic displacement. The
2
intermediate species b contains the spectra of HDO and
H O, but with the components slightly displaced compared
2
to the parent molecules. The same is true of species d that
contains the spectra of HDO and D O. The species c con-
2
components as well as the bands observed in the far-IR spec-
tra of liquid water and heavy water,47 we determined the
tains the spectrum of HDO, D2O, and H2O with the compo-
nents displaced from that of the four other species. We found
no component that could be assigned to species other than
the five observed.
number of components and their positions in the H O–D O
2
2
principal spectra.
The simulation of the five principal spectra in the
H O–D O mixtures is presented in Figs. 8 and 9 for the
2
2
Ϫ1
Ϫ1
IV. CONCLUSION
regions 5500–2000 cm and 2600–600 cm , respectively.
Spectra a, b, c, d, and e are for H O (wOH ), HDOЈ
The molecules H O, HDO, and D O are present in the
2
4
2
2
(
(
wOH D), HDO° (wOH D ), HDOЉ (wOHD ), D O
H O–D O solutions where they form a complex mixture.
3
2
2
3
2
2 2
wOD ), respectively. ͓The OX ͑where XϭH or D͒ terms
Each water molecule has two covalent bonds and can make
4
4
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J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
Isotope effects in liquid water by infrared spectroscopy
4639
TABLE V. Position ͑in cmϪ1͒, full width at half height ͑FWHH in cmϪ1͒, and intensities ͑in ATR-IR abs. units͒ of the IR Gaussian components of liquid H O,
2
HDOЈ, HDO°, HDOЉ, and D O.
2
a: H O
b: HDOЈ
c: HDOo
d: HDOЉ
e: D O
2
2
Position FWHH Int. Position FWHH Int. Position FWHH Int. Position FWHH Int. Position FWHH Int.
␯3
ϩ␯ ϩ␯ H O
5160
240
0.015
5160
240
0.009
2
T1
2
2
3
3
␯3
␯3
␯3 D O
5015
250
0.006
5050
600
0.003
2
␯2 H O
4920
4640
4350
4000
240
240
310
310
0.005
0.002
0.002
0.010
4940
4640
4350
4000
240
240
310
310
0.013
0.005
0.006
0.008
4880
250
0.008
2
␯2Ϫ␯T1 H O
ϩ␯L,T H O
ϩ␯L1 H O
2
4410
4020
310
310
0.004
0.009
2
2
3
3
880
700
310
105
0.012
0.010
3
␯3
␯2ϩ␯T1 D O
3833
3660
3420
170
200
360
0.015
0.004
0.005
2
ϩ␯L2 H O
3730
310
0.280
3735
310
0.018
3760
310
0.013
2
3
␯2 D O
2
␯1
␯3
␯3
ϩ␯L2 H O
3620
3528
105
135
0.280
0.544
3618
3522
105
141
0.215
0.490
3601
3518
105
105
0.221
0.194
3599
3529
105
105
0.073
0.062
2
ϩ␯T2 H O
2
ϩ␯L1 D O;␯
2
OH
␯OH
3426
191
0.880
3421
3300
3135
198
196
135
0.650
0.190
0.063
␯3
H O, ␯
3389
3222
195
195
1.560
1.490
3389
3236
198
198
1.330
0.950
2
OH
3
300
3215
120
196
140
130
0.862
0.075
0.149
␯1
H O
2
3
␯3
␯1
␯3
ϩ␯LT D O
3070
2820
270
200
0.006
0.021
2
Ϫ␯T1 H O
3079
2974
135
135
0.309
0.183
3079
2974
135
135
0.140
0.150
2
Ϫ␯L2 H O
2
2
␯2 HOD
2965
2826
210
120
0.125
0.018
2955
2840
165
125
0.061
0.025
␯1
␯3
␯3
␯3
␯3
␯1
␯3
Ϫ␯L2 H O
2840
2685
135
135
0.067
0.027
2840
2662
135
135
0.075
0.060
2
ϩ␯L2 D O
2
Ϫ␯L1 H O
2
ϩ␯T1 D O
2684
2621
55
140
0.030
0.533
2678
2618
81
81
0.415
0.363
2
ϩ␯T2 D O
2620
135
164
0.395
1.372
2
Ϫ␯L1 H O
2583
2460
2340
115
135
135
0.021
0.333
0.023
2
D O
2546
141
1.290
2
␯OD
2509.5
2351
159
0.980 2491.5
2494
153
1.256
␯2
␯1
␯1
␯2
␯1
ϩ␯LT H O
2
ϩ␯T1 D O
2476
2393
81
141
0.603
1.640
2
D O
2397
0.075
140
115
0.400 2385.7
130
95
0.705
0.160
2
ϩ␯L1 H O
135
156
2
Ϫ␯T2 D O
2277
0.160
2260
2290
81
0.191
2
2288
0.052
␯3
␯1
␯2
␯3
␯1
Ϫ␯L2 D O
2232
2161
81
100
0.180
0.090
2
Ϫ␯L2 D O
2170
2114
90
250
0.020
0.065
2155
2071
1844
90
0.037
0.047
0.076
2
ϩ␯L2ϩ␯T2 H O
2115
1815
350
170
0.140
0.065
2117
350
250
0.105
0.100
2
Ϫ␯L1 D O
180
260
2060
1840
150
190
0.039
0.022
2
Ϫ␯LT D O
2
1815
1844
280
0.107
␯2
␯2
ϩ␯T1 H O
2
ϩ␯LT D O
1740
1555
230
235
0.012
0.105
2
1721
95
0.021
0.088
␯2
␯2
␯2
␯2
␯2
␯2
␯2
␯2
ϩ␯T2 H O
1710
1638
85
81
0.183
0.900
1721
1649
85
80
0.076
0.560
1700
1654
1560
1583
1515
1446
1407
65
65
250
65
67
76
0.058
0.213
0.060
0.052
0.100
2
H O
2
ϩ␯L2ϩ␯T2 D O
1567
1535
160
2
Ϫ␯T2 H O
1551
85
0.220
1542
1455
85
75
0.200
0.610
2
ϩ␯T HOD
HOD
Ϫ␯T HOD
65
70
65
0.012
0.525
0.047
0.540 1460.5
65
0.050
0.275
0.023
1410
1368
1267
Ϫ␯T1 H O
1446
180
0.190
2
1379
140
420
0.185
0.260
1362
250
210
0.156
␯2
␯2
␯2
␯2
␯2
␯2
ϩ␯T1 D O
1344
1265
1207
180
52
55
0.046
0.112
0.957
2
ϩ␯T2 D O
1264
1214
48
55
48
50
0.064
0.556
2
D O
0.250 1211.6
2
Ϫ␯L2 H O
1190
380
0.277
1157
2
Ϫ␯T2 D O
1156
48
0.050
1153
1085
48
0.090
0.155
1154
1110
52
100
0.115
0.130
2
Ϫ␯T1 D O
2
1086
210
100
0.235
0.055
210
970
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4640
J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
J.-J. Max and C. Chapados
TABLE V. ͑Continued.͒
a: H O
b: HDOЈ
c: HDOo
d: HDOЉ
e: D O
2
2
Position FWHH Int. Position FWHH Int. Position FWHH Int. Position FWHH Int. Position FWHH Int.
␯2
␯2
Ϫ␯L1 H O
950
831
686
170
120
344
0.169
0.180
2.470
2
Ϫ␯L2 D O
930
838
115
350
0.050
0.165
926
785
360
100
0.235
0.039
2
8
13
90
260
341
0.278
1.800
␯LT H O
␯2
839
613
240
341
0.160
2
Ϫ␯L1 D O
2
719
180
341
0.480
0.810
a
␯L1 H O
2
5
500
␯LT D O
520
449
315
177
50
200
340
208
140
80
0.370
1.240
0.790
0.830
0.180
2
a
␯L1 D O
2
a
␯L2
395
183
50
300
148
80
0.850
0.860
0.170
a
␯t1
␯t2
a
aZelsmann ͑Ref. 47͒ from fitted components obtained from transmission measurements at 0 °C.
four H-bonds: two through the H or D and two with the
neighbor makes with the third neighbors. In liquid water as
observed by IR, the focus point is on the oxygen atom
around which the protons ͑H or D͒ turn as in a merry-go-
oxygen. On the oxygen of a reference molecule ͑H O, HDO,
2
and D O͒ are four bonds: two covalent and two H-bonds.
2
Because of the hopping nature of the proton ͑and proton-D͒,
the bonds are continually changing. The proximity of neigh-
boring molecules causes nine different combination or spe-
cies in the mixtures. These are: ͑1͒ H O–(H) ; ͑2͒
round. The model of liquid water is made with OX ͑with
4
XϭH or D͒. With the oxygen atom two X make two cova-
lent bonds and two X make two hydrogen bonds. These four
bonds are continually exchanging at a fast rate.
2
2
H O–(H,D); ͑3͒ H O–(D) ; ͑4͒ HDO–(H) ; ͑5͒ HDO–
The nine different species present in the H O–D O so-
2
2
2
2
2
2
͑
͑
H, D͒; ͑6͒ HDO–(D) ; ͑7͒ D O–(H) ; ͑8͒ D O–(H,D);
lutions were modeled using a simple probabilistic model.
Combining the species that evolve simultaneously gave the
five principal factors that were spectroscopically identified.
This result provides a better description of the liquid water
situation than the three species previously identified. The five
principal factors identified by their IR spectra and their abun-
dances provide a good understanding of the species present
in the solutions. This is important for those working with
H O–D O mixtures in the preparation of biological and
2
2
2
2
9͒ D O–(D) . The abundances of the species depend on
2
2
the initial concentrations of the parent molecules.
From the IR-ATR spectra of the 63 solutions that we
took, five principal factors were retrieved by FA giving their
spectra and their abundances. The four remaining species
that could not be sorted evolve simultaneously with one or
two others within one of the five principal factors. Species
that evolve simultaneously cannot be sorted by FA. Of the
nine species, seven are grouped into three spectroscopically
different entities. These entities plus the two parent mol-
ecules ͑H O and D O͒ yield five principal irreducible or-
2
2
chemical samples, and for those who use infrared spectros-
copy as an analytical tool for the quantification of heavy
water in the nuclear industry.
2
2
thogonal factors. These are ͑a͒ H O–(H) ; ͑b͒ H O–(H,D)
2
2
2
and HDO–(H) in the 1:1 ratio; ͑c͒ H O–(D) , HDO–͑H,
2
2
2
APPENDIX A: DISPROPORTIONATION EFFECTS
D͒, and D O–(H) in the ratio 1:4:1; ͑d͒ HDO–(D) and
2
2
2
D O–(H,D) in the ratio 1:1; and ͑e͒ D O–(D) .
2
2
2
Disproportionation effects have been reported in
An increase in the disproportionation between the spe-
cies in the H O–D O solutions would lead to a better sepa-
27–30
H O–D O mixtures with the equilibrium Eq. ͑1͒.
These
2
2
2
2
effects were determined by comparing the deviation of the
ration of all nine species. This could be obtained with better
quality spectra through higher resolution and better tempera-
ture stability than the ones presented here. A temperature
and/or pressure study might also retrieve all nine species. No
hydrates in the sense of sequestering a particular arrange-
ment were observed. As well, no cluster of pure HDO was
observed. Consequently, the spectrum of HDO with some of
the parent molecules could be observed and retrieved, but
not the pure HDO spectrum.
The OH:OD stretch bands of light and heavy water are
not very sensitive to the presence of HOD, but deformation
bands are. These were found to be sensitive to the two hy-
drogen bonds on the oxygen atom of the reference molecule,
but less sensitive to the hydrogen bond that the second
equilibrium constant K ͓Eq. ͑2͔͒ from the ideal value of 4,
e
the value that would be obtained if both H and D atoms were
found covalently linked to an oxygen atom with equal prob-
ability. In this Appendix, we will evaluate the effects of the
reported disproprotionation on the concentration of the three
types of water molecules in the H O–D O mixtures. Let us
2
2
consider one liter of a mixture initially made up of n moles
of H O and m moles of D O. The D molar fraction is given
2
2
by ␹ ϭ͓D O͔/(͓H O͔ϩ͓D O͔), so that ␹ ϭm/(nϩm).
D
2
2
2
D
Let P and P be the probability that an atom attached to a
D
H
given oxygen atom be deuterium or hydrogen, respectively.
We then have
PDϭ␹
D
͑A1͒
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J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
Isotope effects in liquid water by infrared spectroscopy
4641
and
At equilibrium, the solution is made up of H O, HDO or
2
D O. Therefore,
2
P ϭ1Ϫ␹ .
͑A2͒
H
D
2
2
P
ϩPHDOϩP ϭ␣͑1Ϫ␹ ͒ ϩ2␤␹ ͑1Ϫ␹ ͒ϩ␥␹ ϭ1.
D O D D D
D
2
Therefore, the probability for a water molecule of the mix-
ture at equilibrium to be H O, HDO or D O is given by
H O
2
͑A8͒
2
2
2
P
ϭ␣͑1Ϫ␹ ͒ ,
͑A3͒
Let x be the number of moles of HDO obtained in a liter of
H O
D
2
the mixture at equilibrium. The number of remaining H O
2
PHDOϭ2␤␹ ͑1Ϫ␹ ͒,
͑A4͒
͑A5͒
D
D
and D O moles are nϪx/2 and mϪx/2. Hence, we have
2
2
P
ϭ␥␹ ,
D
D O
x
x
2
nϪ
mϪ
2
2
nϪm
where ␣, ␤, and ␥ are correction functions of ␹D that have
been introduced to take into account the disproprotionation
in Eq. ͑1͒. Without it, ␣ϭ␤ϭ␥ϭ1. The equilibrium con-
stant is then
P
ϪP
ϭ
D O
2
Ϫ
ϭ
ϭ1Ϫ2␹D . ͑A9͒
H O
2
nϩm nϩm nϩm
Combining Eqs. ͑A3͒, ͑A5͒, and ͑A9͒ one obtains
2
2
HDO͔2
͑PHDO͒2
␥␹ ϭ␣͑1Ϫ␹ ͒ Ϫ͑1Ϫ2␹ ͒.
͑A10͒
Combining Eqs. ͑A3͒, ͑A4͒, ͑A8͒, and ͑A10͒, one obtains
␤␹ ϭ1Ϫ␣͑1Ϫ␹ ͒. ͑A11͒
D
D
͓
D
K ϭ
e
ϭ
.
͑A6͒
͓
H O͔͓D O͔
P
P
2 2
2
2
H O D O
Using Eqs. ͑A3͒–͑A5͒, Eq. ͑A6͒ may be written as follows
excluding ␹ ϭ0 and 1͒:
D
D
͑
D
Substituting Eqs. ͑A10͒ and ͑A11͒ into Eq. ͑A7͒ and solving
for ␣, one obtains
2
2
D
2
2
D
2
4
␤ ␹ ͑1Ϫ␹ ͒ ϭK ␣␥␹ ͑1Ϫ␹ ͒ .
͑A7͒
D
e
D
2
Ke
Ke
Ke
1
ϩ
ͩ
1Ϫ
ͪ
͑1Ϫ2␹ ͒Ϫ
ͱ
ͫ
1ϩ
ͩ
1Ϫ
ͪ
͑1Ϫ2␹D͒
ͬ
Ϫ4͑1Ϫ␹ ͒2
ͩ
1Ϫ
ͪ
D
D
4
4
4
␣
ϭ
.
͑A12͒
͑A14͒
Ke
2
2
͑1Ϫ␹ ͒
ͩ
1Ϫ
ͪ
D
4
Functions ␤ and ␥ are obtained by substituting Eq. ͑A12͒
into Eqs. ͑A11͒ and ͑A10͒. A numerical example of the cor-
rection functions ␣, ␤, and ␥ is presented in Fig. 10 for the
and
4
lim ␣ϭ
D
.
2
7
Ke
case K ϭ3.74. This example indicates that the correction
␹
→1
e
functions are very close to unity, justifying the approxima-
tion obtained by using the direct probability relations (␣
ϭ␤ϭ␥ϭ1). By studying the limiting values of the function
In the same way,
4
lim ␥ϭ
Ke
͑A15͒
͑A16͒
͑A17͒
␣
when ␹ approaches 0 and 1, one obtains
D
␹
→0
D
lim ␣ϭ1
D
͑A13͒
␹
→0
and
lim ␥ϭ1.
␹
→1
D
Finally,
lim ␤ϭ lim ␤ϭ1.
␹
→0
␹
→1
D
D
The extreme value of ␤ is obtained when ␹ ϭ0.5 by
D
2
␤extremeϭ␤͑␹Dϭ0.5͒ϭ
.
͑A18͒
4
1
ϩ
ͱK
e
From the above results, we calculated the concentration
deviations for the different types of water molecule at the
maximum deviation of HDO for which ␹ ϭ0.5, yielding ␣
ϭ␥ϭ1.017 and ␤ϭ0.983. These results indicate a concen-
FIG. 10. Correction functions ␣, ␤, and ␥ when the disproportionation func-
tion Ke is 3.74 ͑see text͒.
D
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4642
J. Chem. Phys., Vol. 116, No. 11, 15 March 2002
J.-J. Max and C. Chapados
1
1
7
8
tration variation of 1.7% compared to no disproportionation.
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2
19
D O, and 0.5ϫ0.017ϭ0.008 for HDO. These deviations are
2
2
2
2
0
1
2
within our experimental errors. A temperature stability better
than 0.05 K would be needed to observe the disproportion-
ation effect in H O–D O mixtures.
2
2
23
2
2
2
4
5
6
APPENDIX B: ‘‘FREE’’ OH IN AQUEOUS SOLUTIONS
͑
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5
4
8
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of the structure of acetonitrile–water mixtures was made.50
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approximately constant throughout the solubility range ͑un-
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in the aqueous solutions. Although this important issue can-
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than 1% of the water molecules have less than four hydrogen
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͑
2
2
2
3
7
8
9
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Ϫ0.00682ϫc ͑initial D O͒.
2
39
Because species H Ob and HDOb are present simultaneously, they were
2
identified as H O•HDO when first observed as a spectroscopic species.
2
However, as shown in Sec. III E 2, species H2Ob and HDOb are always
present in the same proportion, but they are not necessarily hydrogen
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