Isotope effect on the translational and rotational motion in liquid water and ammoniaEdme H. Hardy, Astrid Zygar, Manfred D. Zeidler, Manfred Holz, and Frank D. Sacher

Citation: The Journal of Chemical Physics 114, 3174 (2001); doi: 10.1063/1.1340584 View online: http://dx.doi.org/10.1063/1.1340584 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/114/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Isotopes Substitution Effects on the Spin Tunneling in the Molecular Nanomagnet Fe8 Studied by 57FeNMR AIP Conf. Proc. 850, 1147 (2006); 10.1063/1.2355109 Isotope effects associated with tunneling and double proton transfer in the hydrogen bonds of benzoic acid J. Chem. Phys. 120, 11107 (2004); 10.1063/1.1738644 Deuterium isotope effect in the phase memory time of triplet pyrimidine in benzene J. Chem. Phys. 111, 4629 (1999); 10.1063/1.479224 A study of the molecular motion in glucose/water mixtures using deuterium nuclear magnetic resonance J. Chem. Phys. 110, 3472 (1999); 10.1063/1.478215 Self-diffusion in CD 4 and ND 3 : With notes on the dynamic isotope effect in liquids J. Chem. Phys. 110, 3037 (1999); 10.1063/1.477898

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JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 7 15 FEBRUARY 2001

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Isotope effect on the translational and rotational motionin liquid water and ammonia

Edme H. Hardy,a) Astrid Zygar, and Manfred D. ZeidlerInstitut fur Physikalische Chemie, RWTH Aachen, D-52056 Aachen, Germany

Manfred Holz and Frank D. SacherInstitut fur Physikalische Chemie, Universita¨t Karlsruhe, D-76128 Karlsruhe, Germany

~Received 11 October 2000; accepted 21 November 2000!

The dynamic isotope effects~IE! on the translational and rotational motion in liquid ammonia andwater are reassessed by NMR measurements. For H2O/D2O the translational and rotational IE areclearly distinct. At 298 K, 23% and 30% are obtained, respectively. Both effects as well as theslopes of the temperature dependencies increase with decreasing temperature. For NH3/ND3 arotational IE of 37% was observed at 298 K. A small increase to 40% at 222 K could be ascertained.The translational IE is about 15% at room temperature and exhibits a stronger temperaturedependence. It is suggested that the observed deviations of the IE’s from the square root of mass andsquare root of moments of inertia laws are caused by translation-rotation coupling as well asquantum effects. The experimental data obtained in the present paper are also of importance for thecorrect interpretation of all kinds of experiments on water and ammonia, where isotopicsubstitutions are involved. ©2001 American Institute of Physics.@DOI: 10.1063/1.1340584#

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I. INTRODUCTION

At first sight the dependence of molecular motionsthe isotopic mass in a neat liquid seems to be a comptively simple physical–chemical problem. However, thfundamental problem is far from being solved. Thereseveral classical theoretical approaches to the problem onamic isotope effects~IE!,1–4 where mass-independent intemolecular potentials are assumed. The classical theoryFriedman,3 published more than 35 years ago, yields for tisotope effect on the translational motion the square-roomass law,

D trans~m1!/D trans~m2!5~m2 /m1!1/2, ~1!

whereD trans(mi), i 51,2, are the self-diffusion coefficientof chemical identical liquids, however, with two differenmolecular massesm1,2 due to two different isotopic compositions. The definition of the self-diffusion coefficient as itegral of the velocity time-correlation function~tcf! is used inFriedmans derivation. Equation~1! is derived for liquidswithout coupling of translational and rotational motions. Tresult is identical to Graham’s Law for dilute gases.

The rotational motion may also be characterized by crelation times, which are integrals of appropriate tcf’s. Fthe simplest case of diffusive rotational motion of an isotpic rotator, described by a diffusion coefficientD rot ,Friedman3 derived a similar expression, namely the ‘‘squaroot of moment of inertia law,’’

D rot~ I 1!/D rot~ I 2!5~ I 2 /I 1!1/2, ~2!

whereI i , i 51,2, are the moments of inertia of the two istopically different molecules. Again, Eq.~2! assumes the ab

a!Present address: Laboratorium fu¨r Physikalische Chemie, ETH-ZentrumCH-8092 Zurich, Switzerland.

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sence of translation-rotation coupling. For a symmetricasymmetric rotator, the description of rotations is morevolved, especially if it is not diffusive.

If we denote the ‘‘normal’’ ~full-protonated! liquid asliquid 1, the deuterated liquid as liquid 2, and the principaxes bya, b, c, we then can use the following abbreviations:

~m2 /m1!1/25mr1/2, ~ I 2i /I 1i !

1/25~ I r1/2! i , i 5a,b,c,

D trans~m1!/D trans~m2!5Dr21~ trans!,

D rot~m1!/D rot~m2!5Dr21~rot!,

and for the viscosity

h2 /h15h r .

However, in particular from experimental results of dnamic isotope effects on the translational motion it has bshown that in many cases Eq.~1! failed ~for a more detaileddiscussion see, e.g., Ref. 5!. The reasons for the failure oEqs. ~1! and ~2! can be of manifold nature. Chandler hashown that the translational6 and rotational7 diffusion inrough hard spheres is influenced by translation-rotatcoupling.8 Other authors prefer a description of the transtional diffusion in liquids without translation-rotationcoupling.9 The translation-rotation coupling generally dpends on details of the intermolecular potential3 with theconsequence that a complex theory is required for a succful prediction of the dynamic isotope effects for a given liuid. It has been pointed out that also quantum effects~e.g.,different zero point energies! can have considerable influence, since they can lead to a mass dependence of themolecular potential~see, e.g., Ref. 10!. These quantum ef-

4 © 2001 American Institute of Physics

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3175J. Chem. Phys., Vol. 114, No. 7, 15 February 2001 Isotope effect on molecular motion

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fects are in particular expected, when in hydrogen-bonliquids the proton in the OH-group is replaced by a deutewith its double mass.

Several years ago, in consideration of the absencedefinite theories, we thought that an extension of the expmental database would be the best way to contribute toadvance in the understanding of dynamic isotope effectneat liquids. Thus, we first studied the self-diffusion of sevisotopically labeled methanol species,5,11 finding that the iso-tope effect on the translational diffusion, describedDr

21(trans), was close to the square root of the ratio ofmoment of inertia (I r

1/2) i around distinct molecular axesiinstead to the small effect on the square root of mass. Fotwo main axes perpendicular to the C–O vectorI r

1/2 issmaller than for the C–O axis, and the smaller value agrquite well with most of the experimental results.11 Only inthe case of CH3OD/CH3OH the ratioDr

21(trans) gets closeto I r

1/2 for the C–O axis.@In this connection we point out thain Refs. 5 and 11 an error occurred. For the pCH3OD/CH3OH the value of (I r

1/2)c is 1.039 instead of 1.082as given in the two papers, wherec refers to an axis perpendicular to the C–O vector in methanol.# This exceptionalbehavior of the pair CH3OD/CH3OH could already be regarded as an indication that the replacement of H by D inhydroxyl group represents a special case among the posisotopic substitutions. In the limited temperature ranwhere we could perform measurements which were preenough for dynamic isotope effect studies, we foundmethanol a weak temperature dependence of the dynamfor the translational motion, which slightly increased widecreasing temperature. In the extension of our experimework we later studied the dynamic IE on the viscosity atranslational molecular motion for eleven further simple mlecular liquids12 in a limited temperature range. Again wobserved for these liquids~with the exception of benzene!within experimental error limits a proximity ofDr

21(trans)and (I r

1/2) i for distinct axesi of rotation and a very weaktemperature dependence. The IE on the viscosity typicdecreases by less than 1% in going from 15 to 45 °C.

In general, the experimentally found relation betweDr

21(trans) and a (I r1/2) i was at a first sight surprising with

regard to Eqs.~1! and ~2!, however, for the case of stroncoupling of translational and rotational molecular motion tproximity of Dr

21(trans) and one of the (I r1/2) i ’s has been

predicted already in theoretical approaches.1–3 This was thereason why we wanted to study simultaneously both theon translation and rotation. Therefore we combined in acent paper12 the usual measurement of the dynamic IEviscosity and on translational motion for the first time sytematically with17O and14N spin-lattice relaxation studiesThis intramolecular nuclear quadrupole relaxation allowsobservation of the IE on reorientational correlation timeswas found12 that, with exception of one liquid, for all othemolecular liquids investigated at 25 °C the IE on the rotional motion was only slightly above the IE on the transtional motion. It could be concluded that indeed for the tmolecular liquids a strong translation-rotation coupling eists at ambient temperatures.

Recently we also studied the isotope effect on the an

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tropic rotational motion in benzene13,14and acetonitrile/watermixtures15 using multinuclear relaxation measurements. Fneat acetonitrile an anisotropy of the IE was clearly oserved. For the rather free rotation about the symmetry aan IE of about 25% was determined. The IE for rotatioabout axes perpendicular to the symmetry axis was measto 6.2~5!%. No temperature dependence could be detecbetween 233 and 348 K. We also obtained first results forrotational isotope effect in liquid ammonia from15N and2Hrelaxation measurements.16

In the discussion of the dynamic IE, its temperature dpendence is of importance, since according to the classtheories Dr

21 should be temperature independent. It wthought that a temperature dependence could occurH-bonded liquids, where such a dependence had beentected, e.g., for the IE on the viscosity of water17 and onDr

21(trans) of water, methanol, HF, and ammonia~see Ref.18 and literature cited therein!. However, in their recent paper Buchhauseret al.18 found that the dynamic IE on thetranslational motion of CH4/CD4 showed almost noT-dependence in the temperature range 275 K,T,450 K,but at temperatures below 275 K the authors measuresurprisingly strong increase of the IE also for this apoliquid and concluded that more complex mechanisms, in pticular collective phenomena, are responsible for the detions from the classical theories.

In our opinion, in the future the problem of the madependence of molecular dynamics in the liquid state willmore and more tackled by computer simulations~for waterand heavy water, see, e.g., Refs. 10, 19–21!.

Bearing in mind the above described facts, we thougthat it is worthwhile to perform a systematic comparatistudy on the temperature dependence of the IE on trantional motion~via self-diffusion studies! and rotational mo-tion ~via NMR relaxation studies! of the two importanthydrogen-bonded liquids water and ammonia. For H2O/D2Oa number of separate studies on the mass dependencmolecular dynamics have been performed several yeago,22–24 mostly at higher pressures and over a wide teperature range and most of these data have been comand discussed in a review article.25 As can be seen there, thmeasurements of the dynamic IE’s are connected with elimits, which are comparatively high in relation to the rquired accuracy for a reliable detection of the typically smdynamic isotope effects. In particular the conclusion derivfrom these data, that at normal pressure the three Inamely, on the17O spin-lattice relaxation, on self-diffusioncoefficients and on the viscosity of water are identical, waleast at 25 °C in contradiction to our own experience andother literature data. Our unpublished measurement of thon the17O spin-lattice relaxation yielded at 25 °C the resDr

21(rot)51.30, in good agreement with a value of 1.2found in the literature.22 On the other hand it is well knownthat at 25 °C for H2O/D2O both IE’s on self-diffusion and onthe viscosity have the same value of 1.23~see, e.g., Ref. 4!,which means thatDr

21(rot) andDr21(trans) are not equal

With possibly the exception of ethanol, such a clear diffence in the directionDr

21(rot).Dr21(trans) (5h r) has

never been found in our previous studies for other liquid12

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3176 J. Chem. Phys., Vol. 114, No. 7, 15 February 2001 Hardy et al.

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and therefore water, the most interesting H-bonded liqumight be a representative of a special case with respect tomass dependence of the molecular dynamics. Also formonia, measurements of the translational26 and rotational27

IE exist. Again, the error limits of these older measuremeare comparatively high. At high pressures, recent measments of the translational IE~Ref. 18! exist.

Thus, by trying to improve the measuring accuracy copared to previous studies, we want to investigate inpresent work a possibly systematic difference between thon the translational and rotational motion of water as a fution of temperature. Since ammonia, an isoelectronic sstance to water and thus regarded as waterlike, mightshow similarities in the dynamic isotope effects, we acompare the translational and rotational IE for ammonia afunction of temperature.

II. EXPERIMENT

A. Materials

Heavy water~.99.95% D! was obtained from FlukaNeu-Ulm, and used without further purification. Normal wter was drawn from an ion exchanger and bidistilled befthe measurements. NH3 was obtained from Aldrich, Germany, and ND3 ~99%! from Merck, Canada.

B. Measurement of the translational self-diffusioncoefficients

The measurement of self-diffusion coefficients pformed by the NMR pulsed gradient spin-echo~PGSE! tech-nique on normal liquids by1H NMR and on fully deuteratedliquids by 2H NMR has been done in exactly the same wand on the same equipment as described in our prevpaper and therefore for details we refer to that work.12

We emphasize here once again that for accurate andliable self-diffusion studies the use of a temperature gradifree thermostatting proved to be crucial and thereforeavoided the usual gas-thermostatting and applied as inprevious work a proton-free temperature bath liquid~GaldenD 20, Ausimont, Milan! resulting for water in an overalaccuracy of the self-diffusion coefficients of<1%. For theammonia samples, kept under vapor pressure in sealed twalled glass tubes, the measuring accuracy was lo~62 %!. Thus, the accuracy forDr

21(trans) is<2% for wa-ter and64% for ammonia.

C. Measurement of 17O and 14N relaxation rates

The longitudinal relaxation ratesR were measured withthe inversion recovery sequence (RD-p-t-p/2-ACQ) on aBruker AM 250 FT NMR spectrometer. The relaxation delRD was at least five to six times the relaxation timeT1

51/R. The intensities were fitted to

I ~t!5I max~12a exp$2Rt%!

with 17 to 26 differentt values. Measurements were peformed with natural abundance of17O and14N. The signal tonoise ratio was improved by accumulating 64 scans. Anponential line broadening that doubled the line width w

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applied. The temperature was controlled with a modifiBruker B-VT 1000 unit and checked with a methanol NMthermometer.28 The accuracy was about 0.5 K.

Special care was taken to keep the conditions of msurement identical for the deuterated and protonated samIn particular the measurements at each temperature wdone alternately on both samples. After changing the teperature an equilibration delay of at least half an hour wallowed. To check the reproducibility each measurement wperformed at least six times. The standard deviations forrelaxation rates as well as the reproducibility were typicaless than 1%.

Relaxation measurements on water were performedambient pressure, on ammonia at saturated vapor press

III. RESULTS AND DISCUSSION

A. HÕD isotope effect on the translational motion

The translational self-diffusion coefficients of light anheavy ammonia and water, obtained directly from the PGmeasurements, are shown in Fig. 1. For light water, thesults of our recent measurements published in a differcontext29 are also used. Although the molecular massessimilar, the translational diffusion of ammonia is one ordof magnitude faster than that of water.

For ammonia, the temperature range was too smallthe experimental uncertainty too high to obtain a reasonafit of the temperature dependence. Older measurementsbeen performed in a larger temperature range~200–298 K!by O’Reilly et al.26 Parameters of a fit to the Arrhenius Laare given with activation energies of 8.7 and 9.7 kJ/molNH3 and ND3, respectively.

The temperature dependence for water could not beted to an Arrhenius Law. A good description of the transtional self-diffusion coefficients is obtained by a fractionapower law30

D trans5D fpl~~T2Ts!/Ts!g. ~3!

FIG. 1. Translational self-diffusion coefficients of H2O, D2O (l), andNH3 , ND3 (3) obtained from PGSE NMR measurements. The fitsdescribed in the text. PGSE NMR measurements on supercooled H2O andD2O (L) by Priceet al. ~Refs. 31 and 32! are also included in the figure

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3177J. Chem. Phys., Vol. 114, No. 7, 15 February 2001 Isotope effect on molecular motion

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The parameters areg52.19(10), Ts5209.4(43) K, D fpl

515.0(10) 1029 m2 s21 for light and g52.01(10), Ts

5222.5(40) K,D fpl516.32(53) 1029 m2 s21 for heavy wa-ter. The parameters for light water differ slightly from thoin Ref. 29, where a larger temperature range is consideusing results from the literature between 273 K and 373

Also shown in Fig. 1 are the translational self-diffusiocoefficients for supercooled light and heavy water measurecently by Priceet al.31,32

The translational IE’s calculated from the current resuin Fig. 1 ~from the individual data and from the fits! areshown in Fig. 2. Both liquids show a considerable IE, tone of water being about 5% higher than that of ammonAlso both translational IE’s depend strongly on temperatuFor water, it increases from 18% at 328 K to 26% at 288For ammonia, the IE increases from about 16% at 298 Kabout 22% at 278 K. The isotope effect calculated fromfit parameters given by O’Reillyet al. increases from 13% a298 K to 37% at 200 K.26

B. HÕD isotope effect on the rotational motion

The relevant interaction for the relaxation of a quadpolar nucleus~spin I>1) is the intramolecular interaction othe electric quadrupole moment of the nucleus (eQ) with theelectric field gradient at the site of the nucleus (eq5Vzz).The strength of the coupling is given by the nuclear quadpole coupling constante2qQ/h. The efficiency of the reori-entational dynamics for the relaxation is given by specdensities. They are the Fourier transformations of tcf’s ofsecond rank Wigner rotation matrix elements describingreorientation of the molecule. In the simplest case~isotropicreorientation, fast reorientation compared to the NMR fquency!, the spectral densities reduce to one correlation ttc . The relaxation is monoexponential with the rate,33

R53p2~2I 13!

10I 2~2I 21!S e2qQ

hD 2S 11

h2

3D tc . ~4!

FIG. 2. Translational isotope effect~IE! for the pairs H2O, D2O and NH3,ND3, respectively, obtained from the current PGSE NMR measurem~directly, and, for water, form the fits!.

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For Vzz>Vxx>Vyy , the asymmetry parameter is definedh5(Vxx2Vyy)/Vzz. In the case of diffusive reorientationamotion, the correlation time,

tc5E0

`

dt^P2~e~0!e~ t !!& ~5!

is related to the rotational diffusion coefficientD rot by

D rot51

6tc

. ~6!

P2 is the second order Legendre polynomial,e(t) a unitvector in the molecule at the timet. The brackets denote thensemble average. The rotational diffusion coefficientstained from the measured relaxation rates and Eqs.~4! and~6! are shown in Fig. 3. The coupling constant and asymmtry parameter for17O in liquid water, 8.9 MHz and 0.8, respectively, were taken from Ref. 34. The corresponding vues used for14N in liquid ammonia are23.7 MHz and0.15.35,36The coupling constants are expected to be equalthe protonated and deuterated molecules.37,38

At the same temperature, ammonia reorients abouttimes faster than water, although the moments of inertiaammonia are higher, see Table I. Thex-test of Wallach andHuntress compares the correlation timetc to the correlationtime of a free rotator of the same moment of inertia atsame temperature,39

tsFIG. 3. Rotational diffusion coefficients for the pairs H2O, D2O and NH3,ND3 obtained from NMR relaxation measurements on17O and14N, respec-tively. The fits are described in the text.

TABLE I. Results of the ‘‘square root laws’’ for the pairs NH3 , ND3 andH2O, D2O. Gas phase moments of inertia can be obtained, e.g., fromcrowave spectroscopy or molecular mechanics calculations. In the precontext, precise absolute values~that depend on the type of geometry use!are not necessary.

NH3 MD3 Ratio1/2 H2O D2O Ratio1/2

m/g mol21 17 20 1.085 18 20 1.054I a/10249 kg m2 282 544 1.39 101 182 1.35I b/10249 kg m2 282 544 1.39 193 385 1.41I c/10249 kg m2 449 898 1.41 302 578 1.38

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3178 J. Chem. Phys., Vol. 114, No. 7, 15 February 2001 Hardy et al.

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18D rotS kT

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.

Gillen and Noggle give the following rough distinction:40

inertial region:x,3, intermediate region: 3,x,5, diffu-sion region: 5,x. For ammonia,x varies from 3 at thehighest temperature to 7 at the lowest. Thus the reorientais in the intermediate region, the influence of intermolecuinteractions becoming bigger with decreasing temperaturethe calculation,I a5I b was used, as the relaxation of14N inammonia is caused only by rotations perpendicular tosymmetry axis. For water, rotations about all three axes lto relaxation, so a mean value ofI 52310247 kg m2 wasused to calculatex. It increases from 10 at the boiling pointo 100 at the melting point. The influence of intermolecuinteractions as well as their increase with decreasing tperature are markedly stronger than for ammonia. Howethe fact that the correlation times are one to two ordersmagnitude higher than for the free rotator does not necesily imply that the reorientational dynamics in water is diffusive.

The rotational diffusion coefficients obtained for liquammonia between 222 K and 303 K were fitted to an Arrhius Law,

D rot5D0 exp~2Ea /kT!.

The parameters areEa57.13(4) kJ/mol,D052.00(4) 1013

rad2 s21 for light andEa57.48(3) kJ/mol,D051.70(3) 1013

rad2 s21 for heavy ammonia. For water between 278 K a358 K, the rotational diffusion coefficients could not be dscribed by an Arrhenius Law. Shown in Fig. 3 is the fit tofractional-power law@see Eq.~3! ~Ref. 30!#. The parametersare g51.892(37), Ts5225.6(16) K, D fpl59.57(16) 1011

rad2 s21 for light andg51.767(33),Ts5235.7(13) K,D fpl

59.084(76) 1011 rad2 s21 for heavy water. The fit to theVFT equation,41–43

D rot5DVFT exp~2EVFT /k~T2Tg!!

is of about the same quality~weighted sum of squared residuals divided by number of degrees of freedom! and theparameters EVFT55.49(22) kJ/mol, Tg5139.3(35) K,DVFT57.13(54) 1012 rad2 s21 for light andEVFT54.39(16)kJ/mol,Tg5162.3(26) K,DVFT54.20(26) 1012 rad2 s21 forheavy water were obtained.

The rotational IE defined asD rot( light)/D rot(heavy) isshown in Fig. 4~calculated from the individual data anfrom the fits!. It follows from Eqs.~4! and ~6! that

D rot~ light!/D rot~heavy!5R~heavy!/R~ light!.

Both liquids show a large isotope effect in the rotationmotion. In liquid ammonia, the isotope effect is nearly costant from 303 K to 250 K at 37%. With further decreasitemperature, a slight increase of the isotope effect to ab40% at 222 K could be detected. This detail could beserved because of the great care taken, described in thperimental section. In an older work, Atkinset al.27 per-formed the same measurements between 303 K and 25and obtained a larger, temperature independent value42%.

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In liquid water, the isotope effect shows a very strotemperature dependence. It increases from about 22% aboiling point to 38% near the melting point of heavy wateThis is at variance with an older result of 0% by Garret al.,44 the reason for the absence of an isotope effectheir measurements is not clear. The value of 28~2!% mea-sured by Lankhorstet al.22 at 298 K is in good agreemenwith the present result of 30~3!%. The high pressure measurements by Lang and Lu¨demann23 on water emulsionsmentioned in the introduction extrapolated to 0.1 MPasimilar to our results.

C. Comparison and discussion of the isotope effects

The IE’s on the translational and rotational motion tgether with the IE on the viscosity, the square root of mand the square roots of moments of inertia are compiledFigs. 5 and 6 for water and ammonia. The IE’s on the vcosity were taken from the compilations in Refs. 17, 45, a46 for water and from Ref. 47 for ammonia. For water, tIE on the correlation timet1 obtained from dielectric relax-ation measurements48 and the translational IE for supercooled water obtained from the fits to the data of Pret al.31,32 are also shown.

For both liquids, the IE’s on translation and rotation aclearly distinct. For water the IE on the viscosity is in excelent agreement with the results from the self-diffusion mesurements. For ammonia, the results still agree withinlarger uncertainty for both measurements. The rotationain ammonia is higher by about 20% than the translationalIn water, the difference is about 5%.

The rotational IE in water is in excellent agreement wthe IE obtained in the group of Barthel from dielectrmeasurements.48 This agreement is not obvious, as NMR rlaxation is related to single particle second order Legenpolynomial tcf’s, see Eqs.~4! and ~5!, whereas dielectricrelaxation relates to the collective tcf of first order Legendpolynomials. In the case of rotational diffusion, a relatianalogous to Eq.~6! holds for the dielectric correlation timeD rot51/(2tc,diel). Again, the fact that the IE’s obtained from

FIG. 4. Rotational isotope effect for the pairs H2O, D2O and NH3, ND3

obtained from NMR relaxation measurements on17O and14N, respectively~directly from the measurements and from the fits!.

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3179J. Chem. Phys., Vol. 114, No. 7, 15 February 2001 Isotope effect on molecular motion

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NMR and dielectric relaxation measurements are equalexpected for rotational diffusion, does not necessarily signthat the reorientational motion of water is indeed diffusiv

The translational IE for supercooled water was obtainby fitting the data of Priceet al.31,32 ~see Fig. 1! to the frac-tional power law Eq.~3!. The results in the range from 278to 298 K exceed the current results by about 5%. The reafor this deviation is not clear.

Inspection of Figs. 5 and 6 shows that for ammonia awater in the normal liquid range the IE’s on rotation atranslation are clearly different but closer than predicted

FIG. 5. Dynamic isotope effects for the pair H2O, D2O. 1: rotational IEfrom NMR relaxation measurements;l: translational IE from PGSE NMRmeasurements~current work!; s: IE from viscosity~Refs. 17, 45, and 46!.Solid lines, IE’s from the fits to the data of the current work; dashed lin‘‘square root of mass’’ law~bottom line! and ‘‘square roots of moments oinertia’’ laws ~top lines!; dotted line, IE from the relaxation timet1 obtainedfrom dielectric relaxation measurements~Ref. 48!; dashed–dotted linetranslational IE from the fits to the PGSE NMR measurements on sucooled H2O and D2O by Priceet al. ~Refs. 31 and 32!.

FIG. 6. Dynamic isotope effects for the pair NH3 and ND3. 1, rotational IEfrom NMR relaxation measurements;3, translational IE from PGSE NMRmeasurements;s, IE from viscosity~Ref. 47!. Dashed lines, ‘‘square rooof mass’’ law~bottom line! and ‘‘square roots of moments of inertia’’ law~top lines!.

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the ‘‘square root laws’’ Eqs.~1! and~2!. If in the condensedphase translation-rotation coupling exists, it seems plausthat the IE’s deviate from Eqs.~1! and~2! to become similar.

As Figs. 5 and 6 also clearly show, the values of the IEdepend strongly on the temperature. For the rotational IEtemperatures near the melting point, values in the vicinitythe ‘‘square root law’’ Eq.~2! are reached. For the supecooled liquids, values above the values for the square roof moments of inertia seem possible. In the recent PGNMR study of deeply supercooled light31 and heavy32 waterby Priceet al., even forDr

21(trans) a value as high as abo1.6 has been obtained at 244 K, see Fig. 5. These resuggest that the IE’s on the translational and rotational mtion of water diverge towards infinity at low temperatureThe most probable reason for this divergence will be dcussed below.

A temperature dependence of the IE is not obtainedthe classical treatment. It is however to be expected ifintermolecular interactions depend on the isotopic maThat this is the case for water and ammonia is already incated by the different melting points for the normal and deterated species. The relevant quantities for the temperadependence of the IE’s are the activation energies for amnia and the singularity temperaturesTs for water.

The rotational IE for ammonia calculated from thArrhenius laws again follows an Arrhenius equation with tactivation energyEa5(7.1327.48) kJ/mol520.35 kJ/mol.The translational IE obtained from the fit given by O’Reilet al. has three times larger activation energy of21.0 kJ/mol.

The IE’s for water obtained by the application of thfractional-power laws diverge towards infinity at the singlarity temperatures for heavy water. At this temperatures,calculated self-diffusion coefficients for heavy water azero, whereas for light water with lower singularity tempertures they are still positive. The ratio of two fractional-powequations is not a fractional-power equation, even for eqexponents. The difference of the singularity temperaturesheavy and light water which we derive for the rotation(235.72225.6) K510(3) K. The difference of the singularity temperatures obtained here for the translational sdiffusion coefficients is~222.52209.4! K513~8! K. As fortranslation and rotation different exponents are obtain~about 2.0 and 1.8, respectively!, the singularity temperaturemay not be compared directly. The difference howeversimilar. Other singularity temperature differences foundthe literature are~2302223! K57 K for the rotation23 and~2262223! K53 K for the translation.24 The agreement isreasonable, considering again that the values from Refwere obtained for emulsions by extrapolation of higpressure data and that the values from Ref. 24 were obtaby combining measured data with data from the literatuThe difference of singularity temperatures obtained fromfits of the translational self-diffusion coefficients in supecooled light and heavy water measured by Priceet al.31,32 is~222.22216.6! K56~4! K. It should be noted that the singularity temperatures obtained by ‘‘least square fitting’’ of thdata to a fractional-power law depend on the weightingthe residuals in the weighted sum of squared residuals. M

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programs use an equal weight for all residuals as defaulthe absence of individual uncertainties the use of a consrelative uncertainty might be more realistic. This weightihas been used for all of the presented fits. In Refs. 31 andslightly different data analysis were used, resulting in eqsingularity temperatures for light and heavy water.

The difference of singularity temperatures is close todifference of temperatures at which light and heavy wahave maximum density, namely~284.32277.1! K57.2 K.This has already been observed in Ref. 24. Vedamuet al.49 claim that the properties of light and heavy watshould be compared by shifting the temperature@e.g.,D rot(H2O,T)/D rot(D2O,T17.2 K)] and that comparison aidentical temperatures results in misleading isotope effeIndeed, the question arises if the ratiosDr

21 are ‘‘good quan-tities’’ at low temperatures~the same might be true also foother liquids when approaching their glass temperatur!.This point of view has also been adopted in Ref. 48. In R24 it is stated that the translational and rotational IE~whichare found to be equal! vanish if the temperature shift is applied. If the temperature shift is applied to the present dthe rotational IE ranges from 10% at the boiling point to 5at the melting point. An almost identical decrease of thewith decreasing temperature has been obtained by applthe temperature shift to the dielectric relaxation data.48 Thetranslational IE calculated by shifting the temperatureD2O in the fractional-power law is about 2.5%, decreasalso slightly with decreasing temperature.

The procedure of comparison at different temperatuhowever raises several questions. How should the tempture shift be included in theories based on statistical mechics? How should a rotational IE close to the square roomass law and a translational IE far below be interpretWhat is the meaning of the opposite temperature depdence?

It might be more sensible to consider the IE’s calculawithout temperature shift and study the different factors tlead to the IE. As stated above, deviations of the IE’s frEqs.~1! and~2! originate from translation-rotation couplingmass dependent potentials and possibly further quantumfects which may not be included in effective classical pottials.

In Ref. 19 results of molecular dynamics simulationsliquid H2O and D2O ~and T2O) are presented. A mass indpendent SPC/E model is used, so that deviations from E~1! and~2! result either from translation-rotation couplingfrom numerical imperfections. At the simulation temperatuof 298 K a translational IE of 11% is calculated. This libetween 5.4% for the square root of mass law and 23%termined experimentally. Such a result suggests thatranslation-rotation coupling exists in the simulation but thquantum effects need to be considered to reproduce theperimental IE. Also calculated in Ref. 19 are IE’s for throtational dynamics characterized by correlation timestained by integrals over 10 different tcf’s for the reoriention. The observed IE’s scatter between 5% and 43%detailed discussion of the results for the various correlatimes associated with the reorientational motion is beyothe scope of this article. None of the results is directly co

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parable to the experimental results as in the case of reation by anisotropic reorientation of an asymmetric tensseveral tcf’s of second rank Wigner rotation matrix elemeare involved. In the case of isotropic diffusive reorientatiall IE’s should be equal, which obviously is not the casethe simulated SPC/E waters.

A molecular dynamics study of liquid water and heawater taking into account quantum effects is reported in R10. The classical central-force potential is modified toeffective quasi-classical pair potential which is different fH2O and D2O. Significant quantum effects are observedambient condition which ‘‘vanish with increasing temperture less drastically than generally assumed.’’ At a simution temperature of 298 K a translational IE of 29% is cculated. This is even higher than the experimentadetermined 23%~and close to the rotational IE of 30%!. At263 K, 126% are obtained for the translational IE, whichsignificantly higher than the 39% measured in Ref. 24. Tresults of this calculation show that quantum effects increthe translational IE and reproduce the direction of the teperature dependence. An increase is also to be expectethe rotational IE.

Conceptually promising is the approach ofab initio mo-lecular dynamics simulations. Separate studies exist whthe translational self-diffusion coefficients of heavy20 andlight21 water were determined. The accuracy of this methis however not yet high enough to study dynamic IE’s, epecially if different functionals are used~as it is the case inthe two references mentioned!.

We suggest the following rationalization of the dynamIE’s in water and ammonia. Translation-rotation couplicontributes to a lowering of the rotational IE and an increaof the translational IE compared to the square root laws. Tdeviation is not necessarily equal for translation arotation.3 Both IE’s are additionally increased by quantueffects, the influence of which becomes more pronouncelower temperatures. For ammonia, the temperature dedence of the translational IE is markedly stronger than thathe rotational IE. This could be explained by several factoPossibly the reorientational motion is less affected by intmolecular interactions, as it is suggested by the result ofx-test. Atkinset al.27 use the terms conditional inertial rotation and ‘‘gaslike’’ liquid in context with ammonia. Thesmall temperature dependence of the rotational IE in amnia can also be explained by the occurrence of two effewith opposite temperature dependence. Quantum effectsto dynamic isotope effects that increase with decreasing tperature. On the other hand, the number of hydrogen boincreases also with decreasing temperature. This mightto a stronger translation-rotation coupling, which increathe translational IE but decreases the rotational IE.

IV. CONCLUSION

The large and temperature dependent dynamic isoteffects in ammonia and water indicate the importancetranslation-rotation coupling and quantum effects in theliquids. The influence of isotopic substitution on the moleclar dynamics can conveniently be extracted from NMR msurements and can be compared to the results of molec

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3181J. Chem. Phys., Vol. 114, No. 7, 15 February 2001 Isotope effect on molecular motion

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dynamics simulations. It is advisable to study both the tralational and rotational dynamics, although further expemental and theoretical efforts are desired to characterizegenerally complicated reorientational dynamics.

For an accurate interpretation of experiments involvdifferent isotopes~e.g., NMR experiments, neutron scatteing, or tracer experiments! dynamic isotope effects have tbe considered. The data presented in this work shoulduseful for this purpose.

ACKNOWLEDGMENTS

Financial support by the Fonds der Chemischen Indtrie is gratefully acknowledged. One of the authors~M.H.! isgrateful for financial support by the ‘‘Deutsche Forschungemeinschaft’’~DFG! ~Project No. Ho 805/6-1!. We thankDr. S. R. Heil for performing a number of translational sediffusion measurements on light and heavy water.

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