J. Phys. Chem. 1987,91, 1639-1645
1639
Dynamic Behavlor and Some of the Molecular Properties of Water Molecules in Pure Water and in MgCI, Solutions R. P. W. J. Struis, J. de Bleijser, and J. C. Leyte* Gorlaeus Laboratories, Department of Physical and Macromolecular Chemistry, University of Leiden, 2300 R A Leiden, The Netherlands (Received: June 10, 1986; In Final Form: October 22, 1986)
From the study of the relaxation rates of 2H, I7O, and 'H due to 170in pure water the values of the interaction constants were derived and it is demonstrated that on a picosecond time scale water reorients isotropically for temperatures between -10 and 53 OC. In MgC12 solutions it is shown that in the cationic hydration shell the water molecule reorients anisotropically and that the effective correlation times and coupling constants of the relaxation rates are significantlychanged. The anisotropic motion of the hydration water is interpreted in concordance with recent neutron diffraction results. The concentration and temperature dependence of these effects are discussed.
1. Introduction The study of the nuclear relaxation rates of the nuclei 2H, I7O, and 'H in pure water and electrolyte solutions provides information about the reorientational correlation time T of water molecules. In pure liquid water one has the simple case that the extreme narrowing limit applies. The relaxation rates of the quadrupolar nuclei ZHand 170and the dipolar coupling or 'H due to 1 7 0 (lH--I7O) are dominated by intramolecularly determined interactions. In these cases the relaxation rate R = (l/Tl = 1/T2) is the product of an interaction constant and the correlation time 7 . The correlation time T is characteristic for the orientational correlation loss of the interaction tensor of the nucleus under study.I Therefore to determine T the interaction constant must be known. Unfortunately, in pure water and electrolyte solutions the quadrupolar coupling constants of the nuclei 2H and I7Oare not available from other sources and the interpretation of the dipolar 'H-lH relaxation is complicated by an intermolecular contribution to the relaxation process. As was pointed out by Lankhorst et al.? the study of the 170-enhancedproton relaxation rate is extremely useful. This dipolar relaxation process is almost completely intramolecularly determined and the interaction constant, which is inversely proportional to the sixth power of the intramolecular proton-oxygen distance rHo,can be obtained from recent neutron scattering experiments in liquid water. In this way one obtains a reliable value for the isotropic reorientational correlation time in liquid water, and as a result reliable estimates of the ZHand 170quadrupolar interaction constants can be derived. If the small value of the asymmetry parameter qD is neglected the 2H and the (I7O induced) proton interactions have almost the same angular dependence and as a consequence rH0 equals T~ irrespective of the type of molecular motion. This, in combination with an empirical relation of the quadrupole coupling constant of 2H and 170and the value for rHOfrom neutron scattering, allows a detailed analysis of the 'H, 2H, and 170 relaxation rates in water and electrolyte s o l ~ t i o n s . ~Thus - ~ the adequacy of the isotropic model for the molecular reorientation was demonstrated for water at room temperature, Le., on a time scale of a few picosecond^.^ The equality of rH0and rDmay be invalidated if the symmetry of the reorientation accidentally magnifies the relative importance of qD For the case of anisotropic rotation diffusion it was shown4that for anisotropies up to ( D , , / D l ) = 2 the equality of rH0and rDholds within 4% for all orientations of the diffusion tensor of the water molecule. This shows that for the small anisotropies expected at worst in liquid water the equality of rH0and T~ should obtain. In this report a relation (1) Abragam, A. Principles of Nuclear Magnetism; Clarendon: Oxford, 1961. ( 2 ) Lankhorst, D.; Schriever, J.; Leyte, J. C. Ber. Bumenges. Phys. Chem. 1982, 86,215. ( 3 ) van der Maarel, J. R. C.; Lankhorst, D.; de Bleijser, J.; Leyte, J. C. J . Phys. Chem. 1986, 90, 1470. (4) van der Maarel, J. R C.; Lankhorst, D.; de Bleijser, J.; Leyte, J. C. Chem. Phys. Lett. 1985, 122, 541.
0022-3654/87 /2091-1639$01.50/0 , I
,
between the 2H quadrupole coupling constant xD and the intramolecular distance rH0 will be used to circumvent the a priori use of the equality rH0= T ~ The . relation in question results from fitting experimental and theoretical data from the literature to a theoretically suggested relation. The results confirm the isotropic behavior of water molecules on a picosecond time scale within present experimental accuracy. The influence of the temperature on T , xD,and rHO has now been determined in the range of -10 to 53 O C . In electrolyte solutions additional complications arise. Univalent ions exert a moderate influence on the dynamical and intramolecular properties of the neighboring water molecule^.^ For ions with a high charge density, (e.g., Mg2+)there effects may be expected to be more important. Following procedures often used in the interpretation of relaxation measurements in electrolyte solutions we will consider the water molecules to be in fast exchange between distinguishable phases, corresponding to the hydrated cations and the bulk water. The influence of the C1ions on the water dynamics, in contrast with cations like Mg2+ ions, is negligibly small as is shown, e.g., by NMR experiment^.^,^ Therefore the water molecules neighboring the C1- ions will be included in the bulk water phase. The observed relaxation rate of a nucleus in water will be the average over the relaxation rates in the two phases, weighted with the fraction of water molecules in these phases. On identifying the properties of the bulk water with those of pure water, the relaxation rate in the hydration phase may be calculated. One may obtain information on the dynamical properties of the hydration water molecules by combining the 2H and 170relaxation rates in the hydration phase. For further interpretation of the motional behavior in the hydration phase rotation diffusion models will be discussed and a connection will be sought with the orientation of the water molecules with respect to the ion as suggested by neutron diffraction experiments.6
2. Experimental Methods For the determination of the 170-enhanced proton relaxation rates in pure water and in two (1 and 4 m)MgC12solutions, several different isotropically enriched solutions were prepared, on the basis of weight, with distilled and deionized water, oxygen isotopically enriched water, and MgCl2-6H2O. Distilled water was deionized and filtered by a Milli-Q water purification system (Millipore Corp.). Oxygen isotopically enriched water was obtained from Monsanto Research Corp., Miamisburg, containing 9.9 (weight)% 0-16, 51.1% 0-17, and 39.0% 0-18. The analytical reagent MgC12.6H20 was obtained from Baker. All manipulations were done in a cold room (4 "C) to minimize exchange with atmospheric humidity. The N M R tubes (Wilmad 10 mm) were heated in an EDTA solution, heated in a NaHC03 solution, and stored for several days filled with distilled and deionized water. ~~
( 5 ) Hertz, H. G. Water, a Comprehensiue Treatise; Franks, Ed.; Plenum:
New York, 1973;Vol. 3. (6) Neilson, G. W. J . Phys. 1984, C7, 119.
0 1987 American Chemical Society
1640 The Journal of Physical Chemistry, Vol. 91, No. 6, 1987
Struis et al.
TABLE I: Isotope Effect Corrected Longitudinal Relaxation Rates (Rxu.e, s-l) of 'H, *H, and I'O in Pure Water and in 1 and 4 m MgCI, Solutions, at Various Isotopic Compositions' and Temperatures T , 'C MMgCM, mol/kg of water PIS X -10 5 14 25 36 53 * * 0 H 0.853 0.503 0.386 0.283 0.227 0.159 * * 0 D 5.90 3.55 2.77 1.96 1.52 1.07 0
0 0 0.97 0.97 0.97 0.99 1.oo 1.01 1.03 4.00 4.00 4.00 4.01 4.02 4.03
*
*
0.148 0.311
0.107 0.226
* * *
0.159 0.231 0.307 0.461
* *
0
*
H H H
*
0
* 0.115 0.167 0.223 0.334
* *
D H H H H H D
*
*
0
0.116 0.174 4.293
0.084 0.126 0.212
H H H
433 0.996 1.14 1.336 8.82 610 1.57 1.66 1.77 2.00
256 0.587 0.664 0.790 4.93 339 0.923
200 0.447 0.509 0.603 3.88 266 0.704
1.03 1.16
141 0.324 0.369 0.441 2.78 189 0.514 0.539 0.572 0.642 1.09 1 6.25 429 1.21 1.27 1.40
0.787 0.879
110 0.262 0.294 0.336 2.15 148 0.389 0.414 0.439
77.3 0.179 0.206 0.241 1.51 104 0.284 0.298 0.314 0.350
"An isotope fraction denoted by (*) means that the mole fractions of the oxygen and the deuteron isotopes are in the order of twice the natural abundance values. The isotope effects on the relaxation rates are neglected. The spin-lattice relaxation rates were measured by the alternating phase inversion recovery m e t h ~ d . Measurements ~ have been performed at different field strengths of 2.1 and 6.3 T. The experiments at 2.1 T were done on a home-built spectrometer equipped with a 2.1-T electromagnet (Bruker). The temperature was maintained within f0.5 O C by fluid thermostating using Fluorinert FC-43 (3M Co.). Measurements at 6.3 T were performed on a modified SXP spectrometer (Bruker), equipped with a 6.3 T superconducting magnet (Oxford Instruments). The temperature was maintained within f l "C by using a Bruker B-VT 1000 gas thermostat. The solutions were shaken at least four times with argon before measuring the proton relaxation rates, in order to remove dissolved gaseous oxygen. For all relaxation measurements 100 data points were collected and fitted to a single exponential by a nonlinear least-squares procedure based on the Marquardt-Levenberg algorithm.* The results ( R D o w c , and RHobdF)are presented in Table I. The estimated accuracy (=experimental reproducibility) in the 2H and I7O relaxation rates is ca. I%, in IH ca. 2%. The superscript c in Pbsd,c refers to a (small) viscosity correction for the oxygen isotope effect on the reoriental correlation time. A linear correction is applied, based on the (small) isotope effect on the viscosity.2 In the remaining text the superscript c will be neglected. The "0-induced proton relaxation rate RH0 can be obtained from the observed proton relaxation rate RHObsdas a function of the I7O mole fraction ~ 1 7 according , to
TABLE 11: Relaxation Rates R H Hand RH0 Calculated with F,q 1
0 0
-10 5 14 25 36 53 -10 5 14 25 36 53 25
0
0 0 0 1 1 1 1 1 1 4
0.913 f 0.029 0.518 f 0.026 0.396 f 0.009 0.276 f 0.001 0.216 f 0.010 0.152 f 0.008 1.442 f 0.013 0.785 f 0.016 0.596 f 0.010 0.432 f 0.008 0.336 f 0.002 0.234 f 0.008 1.055 f 0.014
0.856 f 0.006 0.505 f 0.005 0.387 f 0.002 0.283 f 0.001 0.228 f 0.002 0.158 f 0.002 1.337 f 0.003 0.792 f 0.005 0.605 f 0.003 0.442 f 0.002 0.336 f 0.001 0.243 f 0.002 1.089 f 0.003
0.9995 0.9988 0.9998 0.9999 0.9990 0.9988 0.9999 0.9996 0.9997 0.9995 0.9999 0.9983 0.9998
Correlation coefficient obtained with the linear least-squares procedure. Here x (=e2qQ/h)is the quadrupolar coupling constant, Q is the nuclear quadrupole moment, eq is the main component of the electric field gradient in its principal axis system, and 7 is the asymmetry parameter. The correlation times 7 are effective correlation times because the nature of the reorientational motion has not yet been specified. The intramolecular part of the IH is relaxation rate due to the IH-l7O dipolar coupling, RHOintra, (4)
In both the pure water and the electrolyte solutions a relation between RHObsdand pi7 was observed. The slope RH0 and the intercept RHH were obtained with a linear least-squares procedure. The results are presented in Table 11. The results, in pure water at T = 25 OC, equal within experimental error the results of Lankhorst et a].* 3. Molecular Reorientation in Water In the extreme narrowing limit the relaxation rates of 2H and I7O may be expressed according to eq 2 and 3.
(7) Demco, D. E.; van Hecke, P.; Waugh, J. S . J . Magn. Reson. 1974, 16, 467. (8) Nash, J. C. Compact Numerical Methods; Adam Hilger: Bristol, 1979.
To determine the correlation times in the eq 2-4, the quantities rHo6,xo2(1 sg2/3), and xo2(1 4- qO2/3) must be known. The proton-xygen distance rHOwill be taken to be rHO= 0.98 f 0.01 8, in accordance with recent neutron scattering9 and diffraction experiments,I0 M C simulations," and MD calculations.12 The parameters xD and xo are correlated by an empirical relation which was proposed by Poplett13 for solid hydrates, ice, and water vapor xo = (38.214 f 1.293)X~- (1650 f 302) (5)
+
with X D and xo in kHz. A correlation of xDand the value of rHOis expected theoreti ~ a l l y ' ~and - ~ ' it has been observed e ~ p e r i m e n t a l l y . ~ l -From ~~ (9) Powles, J. G. Mol. Phys. 1981, 42, 757. (IO) Thiessen, W. E.; Narten, A. H. J . Chem. Phys. 1982, 77, 2656. ( 1 1 ) Reimers, J. R.; Watts, R. 0. Chem. Phys. 1984, 91, 201. (12) Chen, S.-H.; Toukan, K.; Price, D. L.; Teixeira, J. Phys. Rev. Lett. 1969, 53, 1360. (13) Poplett, I. J. F. J . Magn. Reson. 1982, 50, 397.
The Journal of Physical Chemistry, Vol. 91, No. 6,1987
Properties of Water Molecules
zI
TABLE 111: Experimentally Determined and Theoretically Calculated Values of xo as a Function of
-
X
c
// Y
1641
I
source
run, A
yn, kHz
ref
(CH2COOH)2 (CHzCOOH)2 KH(CHZC00)20 KOH.Hz0 Ba(OH)2.H20 BeS04.4H20 LiOH.H20 LiOH.H20 D 2 0 ice (Ih) D 2 0 ice (IX) H D O (gas) AIClp.6DzO LiSO4*H2O LiSO4.H20 calculation
1.020 f 0.03 1.020 f 0.03 1.152 f 0.005 1.020 f 0.02 1.000 f 0.02 1.000 f 0.02 1.000 f 0.02 1.010 f 0.02 1.014 f 0.015 0.983 f 0.02 0.9572 f 0.005 1.040 f 0.04 0.990 f 0.03 1.000 f 0.05 0.956 0.960 0.980
172 f 2 177 f 2 -64 f 2 152.7 f 1.4 176.2 f 0.7 194.1 f 0.3 173.6 170.3 214.3 f 1.8 220.2 f 2.0 307.95 f 0.14 111.1 236.6 239.8 303 318 230 187 170 151 114 54 32 29 -32 -62
21 21 21 13 13 13 13 13 38 38 38 38 38 38
Figure 1. Orientation of the D,axis of the axially symmetric diffusion tensor relative to the water molecular frame (x, y , 2). The orientation is characterized by the Eulerian angles a and
1 .ooo
1.010 1.020 1.03 1 1.070 1.100 1.100 1.150 1.225
b.
,, 300.
,
21 21 21 21 21 21 21 21 21 21 21 21
OUncertainties in the experimental determined values of rHOand are given in the cited literature.
xD
TABLE IV. Correlation Times T ~ * rHO, , and T~ in Pure Water as a Function of Temperature, Calculated with Eq 2-6 Taking rHO = 0.98
A
T, OC -10 5 14 25 36 53 b\,
-100
L
ld
10
Figure 2. Deuteron quadrupolar coupling constant
1.2
Lo&)
xD as a function of
the intramolecular distance T H O . The symbols employed denote the experimental (0)and the theoretical ( 0 )values cited in the literature. The line (- -) denotes the values calculated with eq 6, taking A = 594 (kHz.A3) and B = -390 (kHz/A).
-
the experimental values for solid hydrates, ice 11, ice IX, and water vapor, together with the theoretical results obtained by Mayas,21 the constants in the theoretical expression, eq 6 were determined. A XD = - + BrHO (6) rHO
With a weighted least-squares fitting procedure this relation was found to reproduce the experimental data quite well with A = 595 f 1 1 (kHz.A3) and B = -390 f 9 (kHz/A). The data were (14) Chiba, T. J. Chem. Phys. 1964, 41, 1352. (15) Weissmann, M. J. Chem. Phys. 1966, 44, 422. (16) Lindgren, J.; Tegenfeldt, J. J. Mol. Struct. 1974, 20, 335. (17) Ranberg, B. J.; Ermler, W. C.; Shavitt, I. J. Chem. Phys. 1976,65, 4072. (18) Mayas, L.; Plato, M.; Winscom, C. J.; Mobius, K. Mol. Phys. 1978, 36, 753. (19) Davidson, E. R.; Morokuma, K. Chem. Phys. Lett. 1984, I l l , 7. (20) Cummins, P. L.;Bacslay, G. B.; Hush, N. S.; Halle, B.; Engstrom, S . J . Chem. Phys. 1985,82, 2002. (21) Mayas, L. Ph.D. Thesis, Freie Universitlt, Berlin. (22) McGrath, J. W. J. Chem. Phys. 1968, 48 5549. (23) Goren, S . D. J. Chem. Phys. 1974, 60, 1892. (24) Edmonds, D. T.; Goren, S. D.; White, A. A. L. J . Mugn. Reson. 1977, 27, 35.
ps 6.09 3.66 2.86 2.02 1.57 1.11
TD*,
ps 6.23 3.68 2.87 2.03 1.58 1.11
70,
THO,
ps
6.50 3.69 2.82 1.96 1.54 1.08
(TD*/THO)
0.94 0.99 1.02 1.03 1.02 1.03
weighted with the reciprocal value of the squared uncertainty in rHo. (For the theoretical results the uncertainties in rHOof 0.03 A have been taken arbitrarily.) The data are presented in Table I11 and in Figure 2 xDvs. rHOis shown. The small bond length difference for H 2 0 and H D O of ca. 0.006 A as observed by gas-phase electron d i f f r a c t i ~ nwas ~ ~ neglected. Now the correlation times may be calculated from the eq 2-6 if the relaxation rates and the asymmetry parameters are known. For RD and Ro, the experimental values may be used directly. from RHOobSd a 2% correction is applied for To calculate RHOintra the intermolecular contribution according to Lankhorst et al.? (RHOintra/RHOobSd) = 0.98. In liquid water the values of the asymmetry parameters vD and vo are not known. One may expect that the values of qD and vo lie intermediate between the gas-phase values (vD = 0.135; vo = 0.75) and the ice-phase values (vD = 0.12; vo = 0.93). The gas values will be used here. The influence of the value of vD on the results are negligible. For vo the influence is small. Assuming that, like the value of T H O , in liquid water the value of vo lies intermediate between the gas and ice value (e.g., vo = 0.83) the value of the correlation time ro decreases with ca. 3.5% relative to the value one obtains using vo = vo(gas). The correlation times, as determined from eq 2-6 a r e presented in Table IV. Due to the experimental conditions RD, and therefore rD, refers to HDO molecules. To allow direct comparison of the was corrected for a small isotope effect to correlation times iD obtain rD*which refers to H 2 0 dynamics with2 rD(HDO)/rD*(H,O) = 1.05 f 0.02 (7) (25) Shibata, S.;Bartell, L. S. J. Chem. Phys. 1965, 42, 1150.
1642 The Journal of Physical Chemistry, Vol. 91, No. 6,1987 TABLE V Parameters rHoIxo, and xo and Correlation Times T~~ and T~ in Pure Water as a Function of Temperature, Calculated with Ea 2-6 Taking T ~ =* T ~ ~ ' ~ T, O C -10 5 14 25 36 53
THO.
8,
0.985 0.981 0.979 0.978 0.978 0.978
XD, kHz
237 248 252 254 253 253
OThe estimated relative errors in spectively 1, 3, 7, 6, and 14%.
XO,
MHz
7.4 7.8 8 .o 8.1 8.0 8.0 THO,
THO,
PS
6.70 3.70 2.80 1.94 1.52 1.07
xD, xo, i H Oand ,
70, PS
7.02 3.71 2.79 1.93 1.52 1.07 TO
are re-
From the results in Table IV it is concluded that the three correlation times are equal within experimental uncertainty. The fact that T~ equals TD* and THO leads to the conclusion that the reorientational process for water molecules in pure water reaches spherical symmetry (isotropy) on a picosecond time scale. Contrary to earlier work4 the equality T~~ = TD* has been avoided here because this equality is only a priori valid if q D = 0. As pointed it was highly improbable that the use of THO = T?* would lead to misleading conclusions and the procedure used in this report confirms this. The neutron scattering result on rHOrefers to 21 O C . It is seen that this value for rHo leads to equal correlation times for the other temperatures as well. The implicitly assumed lack of temperature dependence of rHOin liquid water may be confirmed by the following reasoning. From the equality of the three correlation times at the temperature at which rHOis known it is concluded that the improbable combination of anisotropic reorientation and a symmetry type for the motion which magnifies the normally negligible effect of qD does not occur in pure liquid water. Thus T ~ = * THO may reasonably be used a priori for the other temperatures. Then, from eq 2-6 one may calculate THO, TO, THO, xD, and xo for all temperatures investigated. These results are displayed in Table V. It is seen that the quadrupole coupling constants xD and xo and distance rHo hardly change within the studied temperature range of -10-53 OC, with average values (rHO)= 0.98 f 0.01 A, (xD) = 250 f 7 kHz, and (xo) = 7.9 f 0.3 MHz, respectively. The insensitivity to temperature has also been noted, e.g., by Hindman et a1.26 They related the dielectric to the rotational correlation time and obtained constant values for XD ( ~ 2 5 8 . kHz, 6 with qD = 0.1) and xo (=7.85 MHz, with qo = 0.94) in the temperature range of 5-60 "C. M C simulations of Reimers and Watts" for liquid water, e.g., show that rHOchanges negligibly (0.978-0.980 A) between 0 and 77 OC.
4. Molecular Reorientation in MgClz Solutions 4.1. The Two-Phase Model. For the interpretation of the relaxation rates of water molecules in MgCI2 solutions the water molecules are considered to be in fast exchange between the cationic hydration shell and the bulk water. This assumption seems reasonable because the residence time of the hydration water of magnesium ions is of the order of microseconds2' and the relaxation times of the water nuclei are in the range of seconds (IH) to milliseconds ("0). For the chloride ions MD simulations2* and magnetic resonance dataSestimate the residence time to be of the order of picoseconds. This is comparable with the reorientational time scale of water molecules in pure water. With respect to the distinguishability of the water phases it is noted that the chloride ions appear to have little influence on the dynamical properties of the surrounding water molecules. This can be concluded, e.g., from the small changes in the proton relaxation rates29in KCI (26) Hindman, J. C.; Zielen, A. J.; Svirmickas, A,; Wood, M. J . Chem. Phys. 1971, 54, 621. (27) Neely, J. W. Connick, R. E. J . Am. Chem. SOC.1970, 92, 3476. (28) Impey, R. W.; Madden, P. A.; McDonald, I. R. J. Phys. Chem. 1983, 87., 507 - 1.
(29) Endom, L.; Hertz, H. G.; Thul, B : Zeidler, M. Ber. Bunsenges. Phys. Chem. 1967, 71, 1008
Struis et al. solutions relative to the results in MgC12solutions of comparable ionic strength. For the cationic hydration site only the first hydration layer will be considered, although quasi-elastic neutron scattering exp e r i m e n t ~indicate ~~ the existence of a second layer which, at low concentration, shows diffusional properties different from the bulk water. (This conclusion has recently been questioned3]). The observation suggests that more than two phases have to be introduced, or alternatively the cationic hydration number in the two-phase model must be interpreted as an effective number. However, within a reasonable range the choice of the hydration number is not crucial for the nature of the conclusions reached here. The hydration number of the first layer is taken to be six, as is supported by experimental N M R observation in the slow exchange limit and by structural studies in 1.1 m MgC12.32 The relaxation rate Rxobsdof the nucleus x (D; 0; HO = 'H due to I7O) observed in the MgC12 solution is then partitioned according to
Rxobsd= fR,+
+ (1 - JR,"
(8)
Here, f = (6m/55.56) is the mole fraction of water molecules hydrating the magnesium ions and m is the electrolyte concentration on the aquamolality scale. The superscripts + and o refer to the hydration and bulk water phase, respectively. 4.2. Motional Models f o r the Hydration Water. The bulk water will be assumed to behave as pure water. On a picosecond scale the reorientation is isotropic and spherical rotation diffusion is an adequate model to interpret the N M R results. For the hydration phase a single correlation time turns out to be insufficient and it will be assumed that the reorientation is still a rotational diffusion process but with a possibly reduced symmetry. In the present analysis the contribution of the overall tumbling of the cationic hexaaquo complex will also be considered. The overall tumbling takes place on a time scale of lo-" s as observed from proton relaxation data in solutions containing divalent paramagnetic cation^.^ To start with, the formalism for an axially symmetric diffusion tensor will be applied. This tensor is characterized by two diffusion constants D , and D2 = D3 and by two Eulerian angles a and p, which describe the orientation Q of the D , axis relative to the water molecule frame ( x , y , z ) (see Figure 1). The axially symmetric diffusion of the hydration water will result in effective correlation times 7,' of water nucleus x. The values of TD+ and T ~ may + differ mutually as a result of the different orientations of the interaction tensors involved. In gaseous water, the main component of the quadrupole interaction tensor at the 2H nucleus lies almost along the 0-D bond. (In this study the orientation of the 0-D axis will be used.) The main component at the 170nucleus is oriented perpendicular to the water molecular plane. Under extreme narrowing conditions the effective correlation time will be given by33
k=-2
with T,-'
= 602
+ m2(D1- D2)
The Wigner matrix elements DE; describe the transformation of the principal axis system of the interaction tensor to the principal axis system of the diffusion tensor, characterized by the orientation Qx. The lattice part of the coupling Hamiltonian, Le., the interaction tensor, will be described by the irreducible elements In the principal axis system of the dipolar interaction tensor, one
a2).
(30) Hewish, N. A.; Enderby, J. E.; Howells, W. S. J. Phys. C 1983, 16, 1777. (31) Friedman, H. L. Chem. Scr. 1985, 25, 42.
(32) Elinkas, C.; Radnai, T.; Dietz, W.; S z l s z , Gy. I.; Heinzinger, K. 2. Naturforsch. 1982, 37a, 1049. (33) Mulder, C. W. R.; Schriever, J.; Leyte, J. C. J. Phys. Chem. 1983, 87, 2336.
The Journal of Physical Chemistry, Vol. 91, No. 6,1987 1643
Properties of Water Molecules has F$2) = 60k(2/3)1/2rH0-3.In the principal axis system of the 2H and I7O interaction, one has V$') = eq(3/2)1/2 The asymmetry parameter 9 is defined as 9 = (Vyy- VXJ/V22,34 For ease of reference eq 9 may be condensed
::id$r5;/2.
TABLE VI: Relaxation Rates R,+ in the Hydration Phase, Relative to the Relaxation Rate R x 0in the Bulk Water Phase, as a Function of the MeCl, Concentration and the TemDerature
"
mol/kg of water T, OC
where amxare normalized geometrical constants. In view of the fast exchange between the hydrogen nuclei in water the average over both hydrogen positions will be taken. Rephrased in practical terms, the effective correlation time T ~ + is a weighted average of the three correlation times T ~ T, ~ and , 72, which are combinations of the diffusion constants D , and D2. The weighting factors are calculable for a given nucleus x in the water molecule if the orientation of the principal axis system of the interaction tensor is known relative to the axially symmetric diffusion tensor. For the interpretation of the correlation times 7, three mathematically equivalent models35 are available. Model 1: Axially symmetric diffusion: T,-, = 6 0 , (m)2(Dll - 0 , ) . (Le., D I = DIland D2 = D3 = D l ) . Model 2: Axially symmetric diffusion in combination with an isotropic overall tumbling of the hexaaquo complex (characterized by the diffusion constant Do,):T,-' = 6(Dov + 0 , ) + "(D,, - 0,) (Le., D , = Dll+ Do, and D2 = D3 = D , + Do"). Model 3: Isotropic overall tumbling in combination with one internal rotational diffusion ( 0 , )axis which will be taken along the cation-xygen axis here, 7,' = 6D0, + (m)2Di(Le., D1 = Do,, D, and D2 = D3 = Do,). According to models 1 and 2 the distribution function of the solvent protons relative to the cation should be broad and it should spread nearly symmetrically around the peaked distribution function of oxygen. It is clear by now that the results of neutron diffraction studies6 indicate peaked distributions for both oxygen and the protons, with oxygen closest to the cation. For a coherent interpretation of the diffraction data and the N M R results model 3 is left. The overall rotation refers to the hydrated cation and the internal rotation axis represents the motion of a water molecule within the hydrated sphere. The orientation of this axis implies a proton distribution function with respect to the cation and this orientation is therefore an important criterion for the consistency or otherwise of the N M R and the diffraction results. As mentioned in the Introduction a comparison of the effective rotational correlation times of the 2H and 1H-170relaxation rates in the hydration phase generally yields no information concerning the nature of the diffusion, because T ~ 1:~T ~ +* + . This information can be extracted from the I7O and 2H relaxation rates in the hydration phase. To this end the theoretical value of the ratio ( T ~ + / T ~ * + )which , is a function of the Eulerian angles a and @, and of t (=1 Di/Dov),is compared with the experimentally determined ratio, which one obtains from the deuteron and oxygen relaxation rates in the hydration phase. For the theoretical ratio of ( T ~ + / T ~ * + )one has
1 1 1 1 1 1 4
m(MgC1A mol/kg of water 1 1 1 1 1 1 4
TD*+
-
+ 6aI0(5 + ()-I + 3a2°(1 + 20-l aoD+ 6ulD(5 + t)-' + 3U2D(1 + 25)-' aoo
(12)
From eq 12 one obtains a quadratic equation in t . The root(s) of the equation can be calculated and after reinsertion of a real root in, e.g., T ~ one + obtains the absolute values of the diffusion constants Do, and D,. A noncomplex root is considered to be physically acceptable, if two conditions are met: (1) Do, and Di > 0 and (2) calculated correlation times ~"0' and THH+ must be consistent with the experimental data. 4.3, Hydration Water Results. First the determination of the molecular parameters rHo+,xD+,xo+ and the effective correlation times of the hydration water molecules will be discussed. Sub(34) Spiess, H. W . N M R Basic Principles and Progress; Diehl, Ed.; Springer Verlag: West Berlin, 1978; Vol. 15. (35) Schriever, J.; Westra, S. W. T. Leyte, J. C. Ber. Bunsenges. Phys. Chem. 1977, 81, 281.
i 0.8 f 0.4 f 0.3 f 0.7 f 0.9 f 0.1
6.3 6.2 6.2 5.4 6.0 7.6
f 0.2 4.6 f 0.2 4.0 f 0.2 f 0.1 4.7 f 0.2 4.1 f 0.2 f 0.1 4.9 f 0.2 4.2 f 0.2
f 0.1 4.8 f 0.2 4.2 f 0.2 f 0.2 4.8 f 0.2 4.2 f 0.2 f 0.1 6.1 f 0.1 5.7 f 0.1
T , OC -10 5 14 25 36 53 25
Pg
PDHO
0.88 0.80 0.83 0.78 0.79 0.80 0.81
f 0.06 f 0.08 f 0.06
f 0.04 f 0.07 f 0.10 f 0.02
0.86 f 0.04 0.87 f 0.05 0.86 f 0.05 0.85 f 0.06 0.87 f 0.05 0.87 f 0.05 0.94 f 0.02
TABLE VIII: Parameters THO+, xD+, and xo+ and the Correlation Times To+ and THO+ ( = T ~ * + ) in the Hydration Phase as a Function of the MgCl, Concentration and the Temperaturea
+
-70'-
x = HO x = H H x=D x=O f 0.5 6.2 f 0.1 5.6 f 0.2 4.8 f 0.2
6.4 5.8 5.7 6.2 6.1 6.0 7.5
TABLE VII: Relative Ratios p s o and pg as a Function of the MgClz Concentration and the Temperature
+
+
-10 5 14 25 36 53 25
dMgC1d3 mol/kg of water 1 1 1 1 1 1 4
T, OC -10 5 14 25 36 53 25
THO+,
A 0.995 0.997 0.993 0.996 0.996 0.995 0.994
XO',
XO',
kHz 216 210 220 213 213 215 217
MHz 6.6 6.4 6.8 6.5 6.5 6.6 6.6
THO',
70'~~
ps
PS
45.2 23.6 17.3 13.5 10.4 7.1 16.1
39.0 20.6 14.6 11.4 9.0 6.2 15.0
"The estimated relative errors in THO+, xO+, XO+. THO+, and 70' are respectively 1, 4, 8, 6, and 16%. *The values of T ~ +are calculated taking qo+ = 0.93.
sequently the motional behavior will be analyzed by using the results of section 4.2. The relaxation rates R,+ (x = D , 0, and HO) in the hydration phase are calculated with the two-phase model (eq 8). The values of the ratios (R,+/R,O) are presented in Table VI. Two characteristic properties of the hydration phase are obtainable by comparing the ratio (R,+/R,O) of 2H with IH-l7O and 2H with 170,respectively. PRO
( ~(vD+)~/~)(~Ho+)~ = RD+/RDO - ( x D + ) ~ + (13) RHO+/RHOO (xD")'(~ + ( 7 ~ " ) ~ / 3 ) ( r ~ 0 ' ) '
P$ =
R0+/Ro0 RD+/RDO
+ (ao+)2/3)(~D0)2(1+ ( 7 ~ " ) ~ / 3 ) 7 0 ' (xOo)2(1
+ (70°)2/3)(xD')2(1 + (7D+)2/3)7D*'
(14)
In eq 13 it is assumed that the equality T ~ = * T ~ , , , which is valid in the bulk phase, also obtains in the hydration phase. The experimentally determined ratios pEo and p g are presented in Table VII. It is noted that the experimental ratios p i o are significantly less than 1. For the t e m p e r a t u r e r a n g e -10 to 53 OC, the average value, the result for 4 m MgCl, included, is 0.8 1 0.03. In view of the small value of vD, changes in vD upon hydration are neglected. Then, pEo # 1 implies that (XD2rHo6) must be different in the hydration and bulk phases. The experimental values for p g are also significantly less than one. For the 1 m MgCl, solution the average value for all temperatures studied is found to be 0.86 A 0.01. The main cause for this effect should + T ~ * + ,indicating be the occurrence of unequal values of T ~ and
*
1644 The Journal of Physical Chemistry, Vol. 91, No. 6, 1987
anisotropic reorientation of the water molecules in the hydration phase. From the discussion above it is clear that it is worthwhile to xo+, and 70' from eq 2-6 calculate xD+,rHO+,THO+ (=q,*+), and 8 and the experimental relaxation rates. The results are presented in Table VIII. In the derivation of the parameters of interest in the hydration phase it is assumed that RHOlnte'*+ = 0.02RHo+. This is a reasonable assumption because within the octahedral magnesium hydration complex the smallest intermolecular 1H-170distance is ca. 3.3 A, as was estimated from the Ni-H and Ni-0 distances obtained by neutron diffraction experiments in NiCl, solutions. Assuming that the intermolecular distances within the hydration complex are effectively constant, the intermolecular IH-I'O interaction will be modulated in time in the order of the overall correlation time 70vof the hydration complex. Taking 70v= 38 ps (25 "C) one estimates RHOlnte',+ = 0.029 s-.l, which is ca. 0.018 times RHO' (25 'C). For all temperatures and electrolyte solutions constant values are obtained for rHO+,xD+,and xo+ with average values of respectively (rHo+)= 0.995 & (xD+) = 215 kHz, and ( ~ 0 ' )= 6.6 MHz. It is interesting to note that the values of these parameters are very similar to the values one obtains for iceous water, viz.* rHO= 1.01 A, xD= 214 kHz, and xo = 6.4 MHz. Therefore the ice-phase value of the asymmetry parameter qo+ of 0.93 is used in the evaluation of the correlation time 70' (for oD the difference between the ice and vapor values is immaterial here). It is noted that the differences of the parameters THO+, xD+,and xo+ relative to the corresponding values obtained in pure water become more pronounced (larger) if, in the applied two-phase model R,O,the relaxation rates of the water nuclei in the bulk water phase are larger than the corresponding relaxation rates obtained in pure water. The change in rHOupon hydration has also been noticed, e.g., in recent MD calculations of 1.1 m CaC1236in liquid water. In the simulation the average rHOdistance in the cationic phase was found to be increased by ca. 0.018 A, relative to the MD result for pure water of 0.9755 A. However, no change was observed in the CI- hydration phase. In the MgC12solutions an appreciable increase of the effective correlation time THO+ is observed relative to the correlation time THO0 in the bulk phase (see Tables V and VIII). The ratio (7~0+/7~0 for" )1 m MgC1, is found to be independent of temperature, with average value 6.6 f 0.3. In the 4 m solution the ratio is 8.3 f 0.7 ( T = 25 "C). Comparison with Results Cited in the Literature. Hertz et al.5929obtained from 'H-'H relaxation, in the limit of infinite dilute MgC1, solution at T = 25 "C, the following results: r+/ro= 5.2, with 70 equal to 2.5 ps. In the present work it is concluded, using ]H-I7O relaxation in 1 m MgCI,, 7 + / 7 O = 6.6 f 0.3, with 7' = 1.94 f 0.12 ps. As is clear the results were derived for different electrolyte concentrations, that is, for m 0, using a polynomial expansion in m (Hertz et al.) and for m = 1, using eq 8 (this study). A comparison of the results seems reasonable, because the observed relaxation rates change, in good approximation, linearly with the electrolyte concentration up to ca. 1.5 m.29 Although the results look alike, the underlying details differ. As is known, intra- and intermolecular interactions contribute to the observed proton relaxation rates RHHobsd and RHOobsd.For both the proton relaxation rates the correlation time 7 O may be derived from the intramolecular determined interaction RIntra,provided that the intermolecular contribution is known and that a reliable is available. Hertz et value of the interaction contant of Rlntra al. theoretically estimated that in pure liquid water the intermolecular contribution to RHHobsd is ca. 38%. However, this result may be questioned. From the study of I70-enhanced 'H relaxation in water Lankhorst et al.' concluded that this contribution to R H H o b s d is considerably underestimated (viz. 57%). This study confirms the conclusion reached by Lankhorst et ai. In pure water for temperatures between -10 and 53 "C the intermolecular
-
(36)Probst, M. M.; Bopp, B.; Heinzinger, K.; Rode, B. M . Chem. Phys. Let?. 1984, 106, 317.
Struis et al.
0
10
20
30
LO
B
Figure 3. Physically acceptable roots [ for cationic hydration water in 1 m MgClz solutions at T = 25 "C, as a function of the tilting angle 6 ( a = 90'). The lines have been drawn as an aid to the eye.
determined 'H-IH interaction is found to contribute for ca. 56 (f2)% to the observed relaxation rates. The results derived from the "0-enhanced IH relaxation rates in water and in electrolyte contributes at worst solutions is to be preferred because RHOinter less than 2% to RHOobsd and the interaction constant of RHOintra can be obtained from other experiments and from the study of the ,H and IH-l7O relaxation rates. Moreover, in this way one is able to take into account the possible influences of electrolyte ions on the properties of water molecules. This study shows that decreases ca. upon hydration the interaction constant of RHOintra 10%. Regarding the dynamical behavior of water molecules in electrolyte solutions Hertz and other authors37hinted at the anisotropic reorientation of the magnesium hydration water. In this study the anisotropic reorientation will be specified. Now the anisotropic reorientation of the hydration water may be discussed. As mentioned in the Introduction, a connection will be sought with the results of neutron diffraction experiments. From the experiments on NiCI2 in liquid D206 the following information is obtained. The experimental gNi(r)curve refers to water molecules relative to the cations. For the hydration water two maxima are found. Comparison of the areas under both peaks led to the conclusion that the oxygen nucleus is nearest and the two protons are positioned equidistantly to the cation. The rather small line width of both peaks imply that the distribution of the orientation of the hydration water, relative to the cation, is limited. Hence the librational and reorientational diffusion of the hydration water along the cation-oxygen axis is appreciably restricted. In view of these observations the effective correlation times of the deuteron and oxygen relaxation rates in the magnesium hydration shells will be interpreted in terms of an overall (isotropic) reorientation of the hydrated cation, together with an internal reorientation of the water molecule within the hydrated sphere (model 3 of section 4.2). One has D , = Do, + Di and D, = D, = D,, where D, and Di are the diffusion constants of the overall and the internal reorientation, respectively. The orientation of the D1 axis relative to the water molecule frame is characterized by the Eulerian angles a and 0 (Figure 1). In concordance with the neutron diffraction results, a = 90' and the tilting angle p will be variable. Combination of the experimental and theoretical results of the effective correlation times 7D*+and ro+(eq 12) lead, per value of p, to a quadratic relation in ( ( = D I / D , ) . The experimentally determined correlation times -obtained for the hydration water in 1 m MgC12 at T = 25 'C will be interpreted; that is, i D *=+ 13.5 and T o + = 11.4 ps. For the asymmetry parameters in the hydration phase the values qD+ = 0.12 and qof = 0.93 are used. It is found that only real roots for 6 are obtained for 0 5 P IP(max), with P(max) = 33.7'. As may be expected, (37) Connick, R. F.; Wuthrich, K. J . Chem. Phys. 1969, 51, 4506. ( 3 8 ) Weiss, A,; Weiden, N. Advances in Nuclear Quadrupole Resonance; Smith, J. A. S., Ed.; Heyden: London, 1980; Vol. 4.
J . Phys. Chem. 1987, 91, 1645-1648 (ps'
Within this concentration range the value of increases almost linearly with the electrolyte concentration from 0' (f20') to 42' (f8'). For 1 m NiCI, one estimates that p is of the order of 32' (*IO0).
I
t
4olI
*
t
0
I
1645
10
20
I
I
30
LO
B
Figure 4. Physically acceptable correlation times T~~ and T , for cationic hydration water in 1 m MgC12 solutions at T = 25 'C, as a function of the tilting angle fl (a = 90°). The symbols 0 and 0 denote the corre-
lation times T, and T,, respectively. The lines have been drawn as an aid to the eye.
the range of p values is found to lie intermediate between the dipole (a = 90'; ?! , = 0') and the lone pair (a = 90'; /3 = 55') configuration. Applying the conditions for the physical acceptability of a real root ( (section 4.2), only one acceptable root, (, per value of fl is obtained. In Figure 3 the acceptable roots ( and in Figure 4 the corresponding values of the correlation times T, (= 1/6D0,) and T , (= 1/6D,) are presented as a function of the tilting angle fl. As is shown in Figure 3 rather moderate values of ( are found. The value of ( ranges intermediate between 1.6 and 6.0. The value of the correlation time T~~ is found to range between ca. 16 (@ =0 ' ) and 32 ps (for /3 = p(max)). Now additional information concerning the value of T,, may be incorporated. The overall tumbling time T,, can be estimated theoretically by Debye's equation for the rotational diffusion of a sphere with a volume equal to the volume of the hydrated cation. For an infinite dilute solution one obtains T~~ = 38 ps. Experimentally one can estimate T,,,,, e.g., from proton N M R experiments in water solutions containing divalent paramagnetic cations with an ionic radius comparable to Mg2+ ions. For Cu2+-containing solutions one obtains T , = 25 ps.' From Figure 4 it is seen that with the additional condition 25 I~ ~ " ( p5s )38 only a highly restricted range of values apply for the tilting angle 6, with fl = p(max). For 1 m MgCI, one obtains 33O 5 @ I33.7' with T , = 7.4 f 1.0 ps. The value of the tilting angle /3 agrees rather well with the result one can estimate from the neutron diffraction experiments on NiC12 in liquid D,0.9 From these experiments it is concluded that the tilting angle changes with the electrolyte concentration m for 0.1 S m (molal) I1.46.
5. Conclusions In pure water the values of the interaction constants and the correlation times of the 2H, I7O, and 'H (due to I7O) relaxation rates were determined in the temperature range of -10-53 OC. The interaction constants of 2H, " 0 , lH-170 are respectively related to the quadrupolar constants xD and xo and the intramolecular distance rHo. The interaction parameters xD,xo; and rHOwere calculated by using two emperical relations between xD and xo and between xD and rHO,respectively, together with the observed equality of the deuteron and the I70-induced proton correlation times. The interaction constants were found to be temperature-independent. From the observed equality of the , T~~ it is concluded that the isotropic correlation times T ~ * T, ~and model of water reorientation is adequate within the studied temperature range. From the study of the relaxation rates in 1 and 4 m MgC12 solutions it is concluded that in the magnesium hydration shell the values of the interaction parameters are significantly changed. Relative to the pure water results both the xD and xo decrease with ca. 15%, while the value of rHo increases with ca. 1.5%. The values of the parameters xD,xo, and rHOare found to be insensitive to the electrolyte concentration and the temperature. For the water reorientation in the hydration shell the values of the effective correlation times are considerably enlarged and the hydration water is found to reorient anisotropically, with T ~ < + T**+. The value of the ratio ( T ~ + / T ~ * +changes ) with the electrolyte concentration but is insensitive to the temperature. The anisotropic reorientation of the hydration water, as is depicted in the deuteron and oxygen correlation times, has been interpreted in terms of the overall rotation of the hydrated cation and an internal rotation of the water molecule within the hydrated sphere along the cation-oxygen axis. In concordance with the neutron diffraction results, the symmetry axis of the molecular diffusion tensor may be consistently taken to lie in the bisextrix plane of the water molecule, albeit that only a restricted range of values apply for the angle between this axis and the axis bisecting the water plane.
Acknowledgment. This work has been carried out under the auspices of the Netherlands Organization for the Advancement of Pure Science (Z.W.O.). Registry No. H 2 0 , 7732-18-5; MgCI2, 7786-30-3.
Diffusion of Isomeric Polycyclic Aromatic Hydrocarbons in Compressed Propane Michal Roth; Joette L. Steger,*and Milos V. Novotny* Department of Chemistry, Indiana University, Bloomington, Indiana 47405 (Received: June 20, 1986; In Final Form: November 21, 1986)
Limiting interdiffusion coefficients of ten model polycyclic aromatic hydrocarbons in compressed propane (384.35 K, 103.4 bar, 0.3771 g ~ m - were ~ ) measured by the Taylor dispersion technique. The Stokes-Einstein coefficients were further calculated from these interdiffusion coefficients. The temperature dependencies of the interdiffusion coefficients of anthracene and phenanthrene at constant density of propane were investigated within the range of 358.15-393.15 K. The differences between the values for isomeric aromatic hydrocarbons are discussed.
Introduction Various studies of both fundamental and applied nature necessitate knowledge of the interdiffusion coefficients of large 'Present address: Institute of Analytical Chemistry, Czechoslovak Aca-
demy of Sciences, 61 142 Brno, Czechoslovakia.
'Present address: Radian Corp., 3200 East Chapel Hill Rq & Nelson Hwy, Research Triangle Park,NC 27709.
0022-3654/87/209 1- 1645$01.50/0
molecules in supercritical or near-critical fluids. Information on such Parameters is, for example, needed in the design of supercritical fluid extractors and the supercritical fluid chromatographic systems. In spite of a rapidly growing technological importance of such fluids, the literature diffusion data are Scarce and, often, inaccurate' Among various techniques for measuring the interdiffusion 0 1987 American Chemical Society
