J . Phys. Chem. 1989, 93, 7943-7952
7943
25Mg2+and '%I- Quadrupolar Relaxation in Aqueous MgCI, Solutions at 25 OC. 2. Relaxation at Finite MgCI, Concentrations R.P. W. J. Struis,+J. de Bleijser, and J. C. Leyte* Gorlaeus Laboratories, Department of Physical and Macromolecular Chemistry, University of Leiden, P.O. Box 9502, 2300 R A Leiden, The Netherlands (Received: October 18. 1988; In Final Form: May 31, 1989)
For MgCI2solutions, the experimentally observed 3sCl- and 2sMg2+relaxation rates are interpreted in terms of a modification of Hertz's electrostatic theory. For the respective ions, the so-called ion-water contribution to the observed relaxation rate will be estimated in accordance with the appropriate dynamical and structural properties of the water molecules hydrating the ionic species under study. Contrary to the electrostatic interpretation given in the literature, no screening length parameter is used in the present evaluation of the ion-ion contribution to the 35Cl-and the zsMg2+relaxation rate. The value of an effective polarization factor is determined from the chloride relaxation data. Comparing the results obtained for 3sCl-with 2sMg2+,it is concluded that with increasing MgCl, concentration an increasing amount of the observed magnesium relaxation rate must stem from an additional contribution. It will be demonstrated that this additional contribution may very well be induced by the encounter of the C1- ions with the [Mg(H20),12+hydration units, where during the encounter the symmetry of the hydration unit is temporarily distorted.
Introduction The study of quadrupolar relaxation of ions dissolved in aqueous solution may offer essential information on the dynamical and structural properties of solvent and solute.' From the general theory discussed by Abragam2 it follows that the quadrupolar interaction may stem from the fluctuating electric field gradients arising from the charge distributions located on the surrounding solute and solvent molecules. The quadrupolar interaction is short-ranged; however, fluctuations in the coupling may originate from dynamical processes that stretch beyond the length scale of the quadrupolar interaction. In the present study, the 25Mg2+and 35Cl- quadrupolar relaxation rates will be interpreted as functions of the MgCl, concentration in the range of 0.244 and 5.49 m ( m = mol of MgC12/kg of H 2 0 ) at 25 OC. The present study is the sequel to a previously reported N M R study,3 from here on referred to as part 1, where the 25Mg2+and 35Cl- relaxation rates (25 "C) in the limit m 0 have been interpreted in terms of a modification of Hertz's electrostatic theory. Also in the present study the electrostatic theory, independently developed by Hertz4+ and by Valiev and c o - ~ o r k e r swill , ~ be applied. In the electrostatic theory it is assumed that in an aqueous solution containing a completely dissociated electrolyte salt the relaxation mechanism for the quadrupole moment bearing ionic nucleus arises from purely electrostatic origin consisting of a contribution due to the dipoles and an additional contribution from the other ionic charges surrounding the ionic nucleus under study. As has been demonstrated in part 1, the electrostatic theory offers a quite successful tool to interpret the limiting value of the quadrupolar relaxation rate derived from the results obtained in diluted aqueous MgCI2 solutions if one adequately treats the so-called ion-water contribution in accordance with the appropriate dynamical and structural properties of the water molecules hydrating the ionic species under study. Dealing with the interpretation of the ionic relaxation rates obtained in MgCI, solutions, additional (ion-ion) contributions are expected to arise from the other ions in the solution and from changes in the water molecular properties upon increasing electrolyte concentration. Therefore, the purpose of the present study is 3-fold: (1) The first is to combine the electrostatic interpretation of the relaxation rates with the recently available information related to the dynamical and structural properties of the solvent and solute molecules, as a function of the MgC12concentration. Extensive use will be made of the results obtained in ,Hand I7ONMR,8*9quasi-elastic neutron scattering,I0 dielectric,"-'4 MD,15J6and X-rayI7 studies in MgCl, solutions,
-
Present address: Physikalisch-Chemisches Institut der Universitat Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland.
as well from the results of neutron diffraction experiments obtained in liquid NiC12/D20 solutions.I8 (2) The present estimate of the ion-water contribution to the quadrupolar interaction differs from the procedure usually applied in the l i t e r a t ~ r e . ~Furthermore, ~,~~ dealing with the ion-water contribution, in Hertz's electrostatic theory the polarization factor, P, was introduced to account for the many two-particle (dipole-dipole) cross-correlation contributions to J(O),the spectral density function at zero f r e q ~ e n c y . ~ For the continuum approximation of the polarization factor, the result, PCR,derived by Cohen and Reif,,I was generally applied. As was discussed in part 1, in the literature criticism has been raised on the applicability of PCR. In view of the uncertainty associated with PCR,in the present study the result, P = 0.43, as obtained in part 1 from the interpretation of the C1-, Br-, and Iquadrupolar relaxation rates at infinite dilution, will be used for the ion-solvent contribution. (3) In dealing with the ion-ion contribution to the quadrupolar interaction, again PCR,and the so-called screening length or shielding parameter, a,6 are important (1) N M R ofNewly Accessible Nuclei; Laszlo, P., Ed.; Academic: New York, 1983; Vol. 1 and 2. (2) Abragam, A. Principles of Nuclear Magnetism; Clarendon: Oxford, 1961 ____.
(3) Struis, R. P. W. J.; de Bleijser, J.; Leyte, J. C. J . Phys. Chem., preceding paper in this issue. (4) Hertz, H. G. Z. Elektrochem. 1961, 65, 20. (5) Hertz, H. G. Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 531. (6) Hertz, H. G. Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 688. (7) See, e g , references cited in ref 5. (8) Struis, R. P. W. J.; de Bleijser, J.; Leyte, J. C. J. Phys. Chem. 1987, 91, 1639. (9) Struis, R. P. W. J.; de Bleijser, J.; Leyte, J. C. J. Phys. Chem. 1987, 91, 6309. (10) Hewish, N. A.; Enderby, J. E.; Howells, W. S . J. Phys. C 1983, 16, 1777. (11) Harris, F. E.; OKonski, C. T. J . Phys. Chem. 1957, 61, 310. (12) Kaatze, U. Z. Phys. Chem. 1983, 135, 5 1. (13) Giese, K.; Kaatze, U.; Pottel, R. J . Phys. Chem. 1970, 74, 3718. (14) Kusalik, P. G.; Patey, G. N. J . Chem. Phys. 1983, 79, 4468. (15) Dietz, W.; R i d e , W. 0.;Heinzinger, K. Z . Nafurforsch. 1982, 37a, 1038. (16) Szlsz, Gy.1.; Dietz, W.; Heinzinger, K.; Pilinkas, G.; Radnai, T. Chem. Phys. Letr. 1982, 92, 388. (17) Pilinkas, G.; Radnai, T.; Dietz, W.; S z l s z , Gy.1.; Heinzinger, K. Z . Narurforsch. 1982, 37a, 1049. (18) Enderby, J. E.; Cummings, S.; Herdman, G. J.; Neilson, G. W.; Salmon, P. S.; Skipper, N. J . Phys. Chem. 1987, 91, 5851. Neilson, G. W. J. Phys. 1984, C7, 119. (19) Hertz, H. G.; Holz, M.; Klute, R.; Stalidis, G.; Versmold, H. Be?. Bunsen-Ges. Phys. Chem. 1974, 78, 24. (20) Helm, L.; Hertz, H. G. Z. Phys. Chem. 1981, 127, 23. (21) Cohen, M. H.; Reif, F. Solid Store Physics; Academic: New York, 1957; Vol. 5, 321.
0022-365418912093-7943$01.50/0 0 1989 American Chemical Society
7944 The Journal of Physical Chemistry, Vol. 93, No. 23, 1989
parameters within Hertz’s electrostatic theory. Here, the polarization factor is introduced to account for the many two-particle (ion-dipole) cross-correlation contributions to J ( 0 ) . The polarization factor may be related to the effects of screening of the ionic electric field gradient at the position of the relaxing nucleus by the solvent; for a one deals with the effect of screening of the ionic electric field gradient due to other ions in the solution. In and an view of the questionable applicability of PcR22*23 alternative interpretation will be discussed with which one might bypass most of the unsatisfying aspects associated with the parameters mentioned above. The parameter a will be suppressed by applying Hertz’s original theory of the ion-ion contribution to the quadrupolar relaxation rate. Then, interpreting the chloride relaxation rate in terms of the ion-water and ion-ion contribution, from the latter contribution the effective ion-ion polarization factor, P , may be obtained as a function of the MgCI2 concentration. Introducing this estimate in the interpretation of the magnesium relaxation data, one must conclude that for 25Mg2+ an additional contribution to the relaxation rate emerges. It will be shown that a small distortion of the octahedral symmetry of the magnesium hydration layer during a [Mg(H20),J2+-CI- encounter may well account for this additional effect.
Theoretical Section I . General Theory. The general theory of the relaxation of a quadrupolar nucleus by fluctuating electric field gradients has been discussed by Abragam., In the extreme narrowing limit the quadrupolar relaxation rate RQ (= 1/ TQ,,= 1/ TQ,,)is given by
Here, I is the spin of the relaxing nucleus, Q is the quadrupolar moment, e is the charge of the proton, h is Dirac’s constant, and J ( 0 ) is the spectral density function at zero frequency given in eq 2:
with m = 0, * I , or f 2 . r”,2)(t)is the mth component defined in the laboratory fixed coordinate system of the field gradient tensor at the position of the relaxing nucleus at time t . The asterisk indicates the complex conjugate. From here on the electric field gradient at the position of the studied nucleus will be referred to as the local field gradient. The integrand is the time correlation function of the electric field gradient defined in the laboratory system. In Hertz’s electrostatic theory it is assumed that in a strong electrolyte solution of finite ionic strength the relaxation mechanism arises from purely electrostatic origin consisting of a contribution to the local field gradient due to the surrounding water dipoles, d , and an additional contribution from the point charges representing the surrounding ions, A.5,6 According to Hertz, in the extreme narrowing limit the observed relaxation rate may be given by Often one refers to d and A as respectively the ion-water and ion-ion contributions to the quadrupolar relaxation rate. FQ is a constant comprising some of the molecular properties of interest as given in eq 4. Besides the parameters previously defined in FQ =
2 7 ( 2 I + 3 ) eQ(1 - ym)2 lOP(2I- 111
h
I
(4)
connection with eq 1, in the electrostatic theory the Sternheimer or antishielding factor, ym,takes into account the contribution to the electric field gradient due to deformations in the charge distribution of the electric cloud surrounding the relaxing nucleus (22) Hynes, J. T.; Wolynes, P. G . J . Chem. Phys. 1981, 75, 395. (23) Engstrom, S.; Jonsson, B.;Jonsson, B. J. Magn. Reson. 1982, 50, 1.
Struis et al. under As was noted in part 1, the term (1 - ym)differs from the term normally used in the electrostatic theory, Le., (1 + This alteration has been introduced to obtain consistency with the -ym values cited in the literature. 2. The Ion- Water Contribution in Electrolyte Solutions. The limiting behavior of the ion-water contributions to respectively the 3sCl- and the 2sMg2+relaxation rates in the zero concentration limit of MgCI, solutions have been discussed in detail in part 1. It was shown that, depending on the model of hydration appropriate for the ion under study, explicit formulas can be derived for the ion-water contribution, d. For MgCl, solutions at finite concentration these formulas will be applied by appropriately adjusting the concentration-dependent parameters in accordance to the properties of the water molecules as observed in these solutions; see subsections 2A and 2B. As will be shown, the present procedure differs with the one normally applied in the literature, which will be discussed in subsection 2C. 2A. The Ion-Water Contribution to the 35CtRelaxation Rate. At infinite dilution the ion-water contribution to the chloride relaxation rate is most successfully interpreted in terms of the so-called fully random distribution (FRD) 1nodel.4,~This hydration model assumes uniform random distribution of the centers of mass and random water dipole orientation throughout the solution. For the ion-water contribution, Hertz5 obtained ym).19320
where p is the electrical dipole moment of the water molecule, C, is the water concentration in particles per cubic centimeters, 7, is the reorientational correlation time of the water molecules, and ro is the distance of closest approach of the water dipoles and the relaxing nucleus. The polarization factor, P, is introduced to account for the many-body cross-correlation contributions to the spectral density J(0).5 Its introduction as well as its meaning in electrolyte theory have been discussed and commented on in part 1 and will be summarized in the present study in Results and Discussion, subsection 1. In MgCl, solutions eq 5 will be the basic relation with which the ion-water contribution to the chloride relaxation rate will be estimated. As mentioned in the introductory lines of this section, apart from implementing the changes in concentration-dependent parameters, like 7, and C,, also a discrimination between the various water phases distinguishable in the MgC1, solution must be i n t r o d ~ c e d . ’ ~For ~ . ~more details see Results and Discussion, subsections 2 and 3. 2B. The Ion-Water Contribution to the 2 5 M pRelaxation Rate. As was discussed in detail in ref 3, 8, and 9, in MgCl, solutions the magnesium ions exert an appreciable influence on both the dynamical and the structural properties of water molecules in its first hydration layer. For the contribution arising from the water molecules located beyond the first hydration layer, one usually applies the FRD model (eq 5), where now the distance of closest approach, ro, is replaced by the radius of the second coordination sphere, b.20*25 This estimate will also be used in the present study and will be referred to as d T D ( b ) . However, for the estimate of the contribution to the magnesium relaxation rate due to the six water dipoles located within the first hydration layer, the FRD hydration model seems less appropriate. In this case the dynamically oriented solvation (DOS) hydration model will be used.3 For the DOS hydration model one assumes a distinct first hydration layer with n, = 6 water molecules. The water dipole will be represented by a point dipole, located at the position of the water oxygen nucleus. The term distinct refers to the assumption that the radial ionoxygen pair distribution function is sharp with ro the average closest ion-oxygen distance. For each of the hydration water molecules it may be assumed that the dipole moment vector points away from the cation and that the angle @ between the water dipole moment vector and the cation-oxygen axis does not change in the (24) Sternheimer, R. M. 2.Naturforsch. 1986, 41a, 24. (25) Weingartner, H.; Hertz, H. G. Eer. Bunsen-Ges. Phys. Chem. 1977, 81, 1204.
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 1945
2sMg2+and 35C1-Quadrupolar Relaxation in MgCI2. 2 correlaton time of the quadrupolar interaction. The field gradient fluctuations in time are taken to stem from two random diffusion processes: the isotropic overall reorientation of the hydration complex and, within the complex, an internal rotation of each water dipole along its cation-oxygen axis. Estimating the ionwater contribution due to the bulk water molecules, dcRD(b)as discussed above, with the DOS hydration model, the net ion-water contribution obtained is3
where 5 is the ratio 70v/7iof the two correlation times T~~ and 7i, which characterize respectively the overall and the internal diffusion processes. AR and AT are dimensionless quenching parameters, for which explicit formulas are given in the Appendix of part 1. AR and AT account for the quenching of the total electric field gradient due to correlations between the orientations of the water molecular dipole moment vectors within the hydration layer. AR and AT refer to the radial and tangential components of the dipole moments, respectively. The parameter gQ takes into account the effect of radial ion-oxygen pair distribution.22 In contrast with the present approach the ion-water contribution at infinite dilution is usually calculated with eq 6 for 0 = 0. The tangential term is considered to be absent and small deviations of AR from zero are taken to be the origin of the ion-water contribution from the inner sphere.20.26 However, in the present study it will be assumed that in good approximation the magnesium hydration layer has an octahedral symmetry; hence AR 0. Furthermore, it is assumed that the radial ion-oxygen pair distribution function is very sharp; hence gQ 1. This distinct hydration model will be referred to as the TAN hydration model.3 With this model the more general result, eq 6 , simplifies to eq 7.
-
-
For finite MgC12concentrations the concentration dependence of 70v, q, 0, and, in d5RD(b),of T~ and Cw may be taken into account. For the parameters, A T and ro it will be assumed that these parameters are concentration-independent. 2C. The Generally Applied Procedure To Estimate the IonWater Contribution. In aqueous electrolyte solutions the ion-water contribution is usually estimated by adjusting the changes in the effective correlation time and (sometimes; see, e.g., ref 19, 20, and 26) the water particle concentration. More generally, this procedure is expressed in eq 8. The model underlying eq 8 is Ri-W ~ ( mR) H H ( C ~ J) m ) Q ( )
R&m-0)
-
x(m-0) RHH(m-0) Cw(m+O)
(8)
the FRD hydration model. In eq 8, R r ( m ) denotes the ion-water contribution to the quadrupolar relaxation rate in a, m molal salt solution. At infinite dilution one requires Rr(m-+O) = PQw(m-O). For the solvent particle concentration, Cw(m),one takes Cw(m)= (NAc)/(Mwm),with c W ( d=)NA/Mw= 3.346 X water particles/cm3. Here NA is Avogadro’s number, M , the molar weight of water, m the molal electrolyte concentration, and c the molar electrolyte concentration. RHH(m) denotes the proton relaxation rate as observed in the m molal salt solution. is thought to give an approximate The ratio RHH(m)/~HH(m+O) picture of the change of reorientation time of the water molecules as a function of the salt concentration. With the parameter ~ ( m ) , one may account for the specific influence of the studied ion on the dynamical behavior of the neighboring water molecules. For x ( m ) three conditions have to be met: (1) x(m-0) = ~ , ( m - * 0 ) / 7 where ~ ~ , rw(m-O) corresponds with the reorientational correlation time of the water molecules (26) Holz, M.; Gtinther, S.; Lutz, 0.; Nolle, A.; Schrade, P.-G. Z . Naturforsch. 1919. 34a. 944.
in the hydration sphere in the limit of infinite dilution. Usually this estimate is taken from the results obtained in ‘HN M R experiment^.^' T,O denotes the correlation time observed in the neat liquid. Whenever the pure water value is used, this will be indicated by the index O , e.g., 7w0,and the water particle concentration, C w o . (2) One assumes that x ( m r m 9 = 1 to indicate that at sufficiently high concentrations each water molecule in the solution must be close to the ion studied and thus the average reorientational correlation time in the solution is equal to the correlation time of the water close to the ion.28 (3) For a concentration m in the range 0 Im Im’, the value of x ( m ) is usually linearly interpolated between the two boundary values defined in points 1 and 2. 3. The Ion-Ion Contribution in Electrolyte Solutions. Now the ion-ion contribution to the quadrupolar relaxation rate, A, may be discussed. In solutions of strong electrolytes all the various ions may contribute to the electric field gradient at the position of the relaxing nucleus. For discussion purposes it will be assumed that the relaxing nucleus is located on an ion of type i. In the present study it will be assumed that the ion-ion contribution, Ai,arises from oppositely charged ions of type j . Some of the developments in the electrostatic theory will be briefly considered here. At the earliest stage, Hertz obtained the following r e s ~ l t : ~ . ~ 47ruj( P ) 2 ( z j e ) Z c ’ ~ c
Ai =
27a3
(9)
Here, c’ = 10-3~NA is the electrolyte concentration in particles per cubic centimeter, where c is the molar MgC12 concentration in mol of MgCl,/dm3 of solution, NA is Avogadro’s number, (z,e) is the charge of the ion of typej, uj is its stoichiometric number, a is the closest distance of approach between the ions, and 7Cis the correlation time characterizing the relative translational diffusion between the ions in question. Pdenotes the polarization factor and should not be confused with the polarization factor P previously introduced in eq 5. As was pointed out by Hertz and co-w0rkers,63~~ apparently eq 9 largely overestimates the ion-ion contribution when compared to the experimentally determined results. Hence a correction to eq 9 was introduced on the basis of the idea that the ionic field gradient contribution of an ion i already may have vanished due to the shielding effects of the surrounding ion cloud formation, whenever the ions in question have diffused from a distance a to ( a a ) , with a < a. Incorporating this effect, Hertz6 obtained
+
47r~~(p)~(z,e)~c’r,’f(a/ a) Ai =
27a3
(10)
whereflula) denotes a function of the ratio (ala). For the range x = ( a / a ) 2 1 , Ax) may be approximated by the analytical function6 x2 xz
+ 2 . 3 3 5 ~+ 0.251 + 3 . 3 3 1 ~+ 1.682
In eq 10, the prime in 7; has been introduced to indicate that 7; is the effective correlation time resulting from two different diffusion processes: first, the relative translational diffusion between the two ions in question, here characterized by the correlation time qon= a2/12D, and second, the rotational diffusion of an ion of type i around the relaxing nucleus characterized by the correlation time T ~ *= a2/3D, where D denotes the mean ionic self-diffusion coefficient, e.g., D = (Di + D j ) / 2 . Assuming that these two diffusion processes are uncorrelated, for the effective correlation time, T,’, one obtains6 7,’
= a2/3Dll
+4(~/a)~)
(12)
(27) Endom, L.; Hertz, H. G.; Thiil, B.; Zeidler, M. D. Ber. Bunsen-Ges. Phys. Chem. 1967, 71, 1008. (28) Hertz,H. G.; Stalidis, G.; Versmold, H.J . Chim. Phys. 1969, numBro spBciale Oct, 171. (29) Versmold, H.Ph.D. Thesis, Karlsruhe, 1970.
7946 The Journal of Physical Chemistry, Vol. 93, No. 23, 1989
Experimental Section For the determination of the 25Mg2+and 35Cl-relaxation rates in MgCI2 solutions, several different ,H-, I7O-,and I80-enriched stock solutions were prepared on the basis of weight, with distilled and deionized water (D,O, H2170, H2'*0) and MgC12.6H20. These solutions have also been used for the zH and I7O N M R study in MgCIz solutions previously reported by our l a b ~ r a t o r y , ~ and for details concerning the preparation of these solutions the reader is kindly referred to ref 9. With respect to the isotopic enrichments it is noted that in order to improve the accuracy of the experimentally determined 2H and I7Orelaxation rates, their concentrations have been increased slightly above their natural abundances; that is, the respective mole fractions were less than 0.7% (I7O, 180)and 1.7% (,H). These enrichments are irrelevant with respect to the dynamical and structural properties of the systems studied. For the determination of the IH relaxation rate, different MgCI2 solutions have been prepared on the basis of weight, but were not 2H, I7O, and l*O isotopically enriched. All manipulations were done in a cold room (4 "C) to minimize exchange with atmospheric humidity. Before measurement of the proton relaxation rates, the sample tubes were shaken at least five times with argon in order to remove dissolved oxygen. The N M R tubes (Wilmad 10 mm) were heated in an EDTA solution, heated in a NaHCO, solution, and stored for several days filled with distilled and deionized water. The relaxation measurements have been performed on a modified SXP spectrometer (Bruker), equipped with a 6.3-T superconducting magnet (Oxford Instruments). The temperature was maintained at 25 f 0.5 "C by fluid thermostating using Fluorinert FC-43 (3M Co.). The longitudinal IH-IH, 25Mg2+,and 35Cl- relaxation rates, (RQ,]= 1/TQ,J,were measured by the alternating phase inversion recovery method.30 For all RQ:1measurements 100 data points were collected and fitted to a single exponential by a nonlinear least-squares procedure based on the Marquardt-Levenberg alg ~ r i t h m . It ~ ~was observed that the inversion recovery curve is exponential. In view of the rather moderate sensitivity of the 25Mg2+ions, for the lower concentration MgCI2 solutions at least five different measurements have been performed, spread over several days and allowing small variations in the experimental settings. In this way an experimental reproducibility of the relaxation data on the order of 1.5% or less was obtained. For several solutions also the transverse 25Mg2+and 35Cl- relaxation rates, R,, = 1 / TQ,2,were measured by use of a spin-echo or a Carr-Purcell-Gill-Meiboom sequence.2 For 25Mg2+it was observed that the extreme narrowing limit, RQ = R Q , = ~ R,,, applies, but for 35Cl-it was found that Rcl,zwas somewhat larger (4%) than the corresponding Ra,I value. As discussed in part 1, this apparent trend is thought to be not physically significant. The relaxation rates of IH, z5Mg2+,and 35Cl-as a function of the molal ( m ) and the molar (c) MgC12concentration are presented in Table I. The proton longitudinal relaxation rates, in Table I denoted as RHH, are obtained by polynomial interpolation of the data that have been determined in our laboratory. For m = 0, 1.01, 1 S O , 2.36, 3.16,4.00, 5.39, 5.51, and 5.97 m MgC1, theobserved proton relaxation rates at 25 OC are respectively 0.283, 0.403, 0.477, 0.633, 0.824, 1.09, 1.69, 1.74, and 1.99 s-l. (Experimental reproducibility of the relaxation data is within ca. 2%.) For the polynomial interpolation, the following coefficients, ai, occurring in the expansion R H H ( m ) / R H H ( d )= x L o a i m i ,were obtained by a nonweighted least-squares procedure: a. = 1.008 f 0.024, a l = 0.340 f 0.037, a2 = 0.054 f 0.015, and a3 = 0.0097 f 0.0017, with R,,(m-O) = 0.283 s - l . The presently obtained 25Mg2+,35Cl-,and 'H-IH relaxation rates may now be compared with the data given in the literature: (30) Demco, D. E.; van Hecke, P.; Waugh, J. S. J . Magn. Reson. 1974, 16, 467. (31) Nash, J . C. Compacr Numerical Methods; Adam Hilger: Bristol, 1919.
Struis et al. TABLE I: Longitudinal uMg2+, wI-,and 'H-lH Relaxation Rates as a Function of MeCI, Concentration at 25 O C MMgCl,),
c(MgCM,"
mol/kg H 2 0 mol/dm3 0 0 0.246 0.244 0.488 0.482 0.985 0.964 I .49 1.44 2.19 2.08 2.78 2.60 3.47 3.18 3.98 3.60 4.49 4.00 4.99 4.39 5.39 4.69 5.49 4.76
RM8,s-l 4.16 4.56 4.78 5.57 6.70 8.44 10.9 14.2 18.2 23.8 32.8 45.0 50.5
* 0.036
R,,,
s-'
29.2 f 0.6b 34.5 39.9 51.9 64.7 93.5 127 184 250 339 463 613 665
RHH, s-' 0.283' 0.308 0.337 0.399 0.470 0.597 0.733 0.920 1.08 1.28 1.49 1.68 1.73
Taken from ref 9. bThe values at infinite dilution were obtained by extrapolation of the longitudinal and transverse relaxation rates as described in subsection 4 of Results and Discussion. cObtained by polynomial interpolation of experimentally determined longitudinal IH-IH relaxation rates as discussed in the Experimental Section.
35C1-. From inspection of the relaxation data observed in MgClz solutions (25 "C) in the range of 0-3 m, as shown in Figure 1 of ref 32, it is concluded that the presently derived results and those obtained by Weingartner et al. show comparable concentration dependence albeit that the latter data become increasingly smaller, up to ca. 14% at 3 m, in comparison with the results presented in Table I. 25Mg2+. Holz et a1.26determined the longitudinal 25Mg2+relaxation rates by the Fourier transform inversion recovery method in various mncentrated MgC1, solutions (25 "C) in the range of 0.1-5.5 m. It is concluded that, on the whole, the respective results agree within ca. 10%. However, it is noted that the relative uncertainties in the presently derived results (f1.5%) are ca. 3 times smaller than those reported by Holz et al. IH-IH. For the range of 0-2.2 m, the presently derived proton data agree very well with the data obtained by Endom et aL2' However, compared with the data given in Table I, beyond 2.2 m the RHH values obtained by Endom et al. become increasingly larger, up to ca. 10% at 5 m .
Results and Discussion For ease of presentation this section is divided into four subsections, 1-4. In subsection I some considerations will be given on the applicability of the continuum polarization factor as was introduced in Hertz's electrostatic theory. In subsection 2 specific information related to the dynamical and structural properties of the solute and solvent molecules in MgC12solutions will be given. In subsection 3 the observed chloride and in subsection 4 the observed magnesium relaxation rate will be interpreted in terms of the ion-water and the ion-ion contributions to the quadrupolar relaxation rate. 1. Some Considerationson the Applicability of the Continuum Polarization Factor in the Electrostatic Theory. The present discussion concerns some selected subjects related to the continuum polarization factor, P. For the sake of clarity, first some introductionary notes will be given below. As mentioned in the Theoretical Section, the quadrupolar relaxation in the extreme narrowing limit is
with m = 0, f l , or f 2 . fi;)(t) are defined as the components in the laboratory system of the field gradient tensor at the position of the relaxing nucleus. The electric field gradient at the position of the studied nucleus will be referred to as the local field gradient. k$) contains contributions from many particles. First the case (32) Weingartner, H.; Muller, C.; Hertz, H. G. J . Chem. Soc., Faraday Trans. 1 1979, 75, 2712.
25Mg2+and 35Cl- Quadrupolar Relaxation in MgCI2. 2 in which the local electric field gradient arises from the solvent dipoles may be discussed. In order to estimate the correlation I$)(t)*), it is useful to discriminate between function, (@)(O) autocorrelation and cross-correlation contributions. For the calculation of a one-particle autocorrelation function it is sufficient to consider the electric field gradient arising from a representative water dipole. For a two-particle cross-correlation function one may consider the same discrete dipole and another dipole. In Hertz's electrostatic theory the polarization factor, P , is introduced in the calculation of the many two-particle crosscorrelation functions of the local field gradient that contribute to J ( 0 ) . Here, (P2 - 1) is introduced as a constant of proportionality between the auto-correlation function for a given point dipole and the sum of two-particle cross-correlation functions of this dipole and the entire system.s For the continuum approximation of the polarization factor, the result derived by Cohen and Reif,2' eq 14, is often applied. In eq 14 t is the static dielectric
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 1941
20
15 -
10
1
constant of the dielectric medium. As discussed in part 1, in the l i t e r a t ~ r esome ~ ~ , ~criticism ~ has been directed against the applicability of PCRin the electrostatic theory when dealing with field gradient fluctuations arising from the solvent dipoles. Figure 1. Relaxation rates RM8(-+ -) and Rcl (-0-) and viscosity, 7, Therefore, also in the present interpretation the uncertainty as(-) relative to result at infinite dilution, as a function of molal MgCI2 sociated with the application of PCRwill be circumvented by using concentration (m)at 25 "C. the estimate P = 0.43 f 0.08 as obtained from the interpretation of the Cl-, Br-, and the I- quadrupolar relaxation rate at infinite dilution. This estimate is thought to apply in those cases where the ion-water contribution may be calculated with the fully random distribution (FRD) hydration model. When dealing with the local electric field gradient arising from coA / a representative ionic charge, in the electrostatic theory a polarization factor P has been introduced in the calculation of the two-particle cross-correlation function of the local electric field gradient due to this ionic charge and a water dipole. Here, ( P 2 - 1) is a constant of proportionality between the electric field gradient due to the representative point charge and the sum of the two-particle cross-correlation function of this p i n t charge and the point dipoles located in the entire system.6 For P the continuum approximation of the polarization factor derived by Cohen and Reif is normally used. Also for this case some criticism has been directed against the applicability of PcR: In the l i t e r a t ~ r e ~ ~ . ~Figure ~ 2. Orientation of the cation-oxygen axis (CO) relative to the it has been noted, e.g., that (1) PCRhad originally been derived water molecular frame (x,y,z). The water molecule lies in the xz plane. The orientation is characterized by the Eulerian angles a and 8. for quadrupole effects in solids. Contrary to solids, in aqueous electrolyte solutions, here MgC12 solutions, one has E >> c(so1id). Debye-Stokes-Einstein model. Using this model, it is expected ( 2 ) PCRdoes not show a proper limiting behavior for t >> 1; viz., that at a constant temperature Rq(m)/Rq(m+O) = q ( m ) / v P e-'. (3) Especially at relatively short ion-dipole (or ion-ion) (m-0). However, inspection of Figure 1 clearly shows that the distances, it is reasonable to expect that the polarization of the variations of the 25Mg2+and 35Cl- relaxation rates cannot be surrounding medium depends on this distance and the finite sizes accounted for in terms of variation of the viscosity. of the species involved. Inspection of eq 14 shows that PCRdoes A more satisfying approach is offered by Hertz's electrostatic not relate to such features. theory in which a more microscopic interpretation enables the Therefore, in the present study the effective value of the poincorporation of the dynamical and structural properties of the larization factor, P , will be derived from the ion-ion contribution solute and solvent molecules. At finite MgC12 concentrations at to the chloride relaxation rate. The ion-ion contribution will be least two different kinds of effects must be taken into account. obtained by subtracting the theoretically estimated ion-water (1) The ion-dipole contribution must be corrected for changes contribution from the experimentally determined chloride relaxin the dynamical and, possibly, structural properties of the suration rate. For the determination of P the interpretation of the rounding water molecules. (2) An additional contribution may chloride relaxation data is to be preferred over that of the magstem from the electrolyte ions in the solution. Starting with p i n t nesium relaxation rate, because contrary to the Mg2+ case, for 1, recent 2H and ''0N M R studies in MgC12solutions (25 0C),**9 CI- the problem is not complicated with effects associated with and the results of other experiments cited therein, show that with a structured hydration layer. respect to the nuclear relaxation, the water molecules are in fast 2. Dynamical and Structural Properties of Water and Ions exchange between two phases corresponding to the hydrated in Aqueous MgC12 Solutions. In early work the concentration magnesium ions and the remaining (bulk) water. The effects of dependence of the ionic quadrupolar relaxation rate, RQ, was chloride ions on neighboring water molecules are thought to be tentatively related to the concentration dependence of the viscosity negligibly small compared to those induced by the magnesium of electrolyte solution^.^^-^^ The basic idea stems from using the ions, and therefore these water molecules are included in the bulk water phase. It was concluded that the hydration water molecules (33) Simeral, L.; Maciel, G. E. J . Phys. Chem. 1976, 80,552. reorient anisotropically, this in contrast, e.g., to the dynamical (34) Deverall, C.; Richards, R. E. Mol. Phys. 1969, 16, 421. behavior of the water molecules in pure water. The anisotropical (35) Hall, C.; Richards, R. E.; Schulz, G. N.; Sharp, R. R. Mol. Phys. motion can be characterized by respectively the correlation times 1969. 16, 529.
I
-
7948
The Journal of Physical Chemistry, Vol. 93, No. 23, 1M9
Struis et ai.
TABLE 11: Some Parameters Characterizing the Motional Behavior of the Solute and Solvent Molecules in MgCI2 Solutions at 25 "C"
m(MgW mol/kg H 2 0 0 0.246 0.488 0.985 1.49 2.19 2.78 3.47 3.98 4.49 4.99 5.39 5.49
TOV. PS
36 39 42 48 54 67 82 108 133 166 204 24 1 252
Ti. ps 2.39b 2.42 2.45 2.51 2.57 2.66 2.73 2.81 2.87 2.93 2.99 3.04 3.05
& deg
DMg(m+o)
/DMg(m) 1' 1.09 1.16 1.32 1.50 1.85 2.28 2.99 3.70 4.60 5.67 6.70 6.99
28b 29.7 29.0 27.6 27.4 28.0 28.4 28.3 28.2 27.6 27.5 27.5 27.5
D,(m--o)lDc,(m)
rdm)/~,(m-O)
Id
IC
1.17 1.25 1.45 1.73 2.24 2.84 3.78 4.67 5.74 6.99 8.15 8.47
1.05 1.11 1.27 1.44 1.70 1.98 2.47 3.00 3.70 4.57 5.38 5.60
" Except for T~ and Dc,, all the parameter values have been taken from ref 9. bThe T~ values have been calculated with the linear relation q ( m ) = (2.39 f 0.06) + (0.12 f 0.02)m. C A t25 OC the values of DMg and T~~~~ at infinite dilution are 0.706 X cm2/s37and 1.94 ps,* respectively. d A t 25 "C one has D,(m-O) = 2.03 X cm2/s.37The polynomial interpolation formula is given in Results and Discussion, subsection 2. and T~ and the Eulerian angles a and @.The correlation time characterizes the isotropic overall diffusion of the whole magnesium hydration unit, and the correlation time T , characterizes within the hydration unit the internal diffusion of each water molecule around its cation-oxygen axis. The Eulerian angles characterize the orientation of the internal diffusion axis relative to the water molecular plane as indicated in Figure 2. From the results obtained by neutron diffraction experiments, e.g., in liquid NiCI2/D20, it is concluded that within the first cationic hydration layer the water molecular oxygen nucleus is nearest and the water deuterons have equidistant positions with respect to the cation.I8 Hence, it may be concluded that the dipole moment vector points away from the cation and for the angle a one has a = 90°. Table I1 contains the parameters T ~ T ~ and , , @,as a function of the molal MgCI, concentration. These results were obtained in a ,H and I7O N M R study in MgCI, solutions at 25 O C 9 Inspecting Table 11, one may notice that the tilt angle (3 is nearly independent of the electrolyte concentration. Therefore, the average result, /3 = 28O, will be used for all MgCI, solutions studied here. As was discussed in part I , in a reasonable approximation the correlation time T~ changes linearly with the MgC12 concentration from 2.39 ps ( m 0) to 3.08 ps (5.49 m ) . Therefore, in the present study the T~ values obtained by linear interpolation will be used, and these values are given in Table 11. On the other hand, one may observe that the overall correlation time, T ~ " increases , nonlinearly with the electrolyte concentration from 36 ps ( m 0) to 252 ps (5.49 m ) . Concerning the dynamical properties of the bulk water molecules as a function of the MgCI2 concentration at 25 "C, it is noted that for the reorientational correlation time, T,, of the water molecules in the bulk phase the results obtained in the 2H and 170N M R study in MgCI, solutions at 25 OC9 will also be used in the present study. Note that in ref 9 T , is denoted as T ~ With respect to point 2, that is, the additional contribution arising from the electrolyte ions, it is noted that the time scale relevant for this type of contribution is mainly determined by the translational diffusion of the ions involved. The ionic self-diffusion coefficients DM, and Dcl are estimated by means of polynomial interpolation of the results obtained by tracer diffusion experiments in 28Mg2+-and 36CI--enriched MgCI, solutions.36 For DMg.the details for the polynomial interpolation have already been given in ref 9. The Dcl values have been derived from the literature data2g by means of polynomial interpolation. The following coefficients, b,, occurring in the expansion Dcl(cm2/s) = Zibimi, were obtained by a nonweighted least-squares procedure: bo = 1.882 X b, = -5.457 X IO", b2 = 4.626 X lo-', and b3 = -1.406 X For the limiting value at infinite dilution one has Dcl = 2.03 X cm2/s (25 OC)." In the determination of these T~~
T~~
-
-
(36) Harris, K. R.; Hertz, H. G.; Mills, R. J . Phys. 1978, 75, 391. (37) Miller, D. G.; Rard, J. A.; Eppstein, L. B.; Albright, J. G. J . Phys. Chem. 1984, 88, 5739.
.
coefficients the limiting value for Dcl has not been used, because it was noticed that in such a case a significant smaller discrepancy was obtained between the literature and the calculated data. The values of DMg,Del, and T , are given in Table 11. It will be assumed that the main contribution to the ion-ion contribution arises from encounters of oppositely charged ions. As is suggested, e.g., by the cutoff distance of ca. 3.9 A in the Ni-CI pair distribution function obtained by neutron diffraction studies,18 it will be assumed that the magnesium hydration unit and the chloride ion may approach to at a closest distance of ca. 3.9 A. During a hydrated magnesium-chloride encounter there may be a substantial sharing of the cationic hydration water molecules between the magnesium and chloride ions. The results of the above-mentioned neutron diffraction studies confirm this type of sharing. From the geometry of the species involved it is concluded that then the cations and anions share three hydration water molecules. Concerning the lifetime of these encounters, the results of the neutron diffraction studies18 and of the study of ionic transport coefficients in MgCl, solutions37do not indicate the formation of long-lived ion pairs by inner-sphere complexation, and this will be neglected here. However, due to pair correlation effects it is not unlikely that the correlaton time associated with the [Mg(H20)6]2+-CI- may be longer than presently estimated on the basis of independently diffusing neutral particles. Fries and Patey3*concluded that the ionic charges involved may strongly influence the relative motion. Of present interest is the fact that for pairs of attracting ions the time scale related with the relative motion is larger than estimated for neutral particles or repulsing ion pairs. This conclusion was reached by numerically solving the Smoluchowski equation39 including a force term dependent upon the ion-ion potential of mean force. An experimental confirmation of this effect was obtained by Albrand et aI.@ For most of the MgC12solutions studied here this effect may be ignored because from the results obtained in the theoretical it is concluded that this effect only becomes relevant whenever the Debye screening length is relatively large, that is, for MgCI2 concentrations smaller than roughly 1 mol/dm3. 3. Interpretation of the 3sCI-Relaxation Rate Observed in MgCI, Solutions. 3A. The Ion-Water Contribution to the 35CIRelaxation Rate. In MgCI, solutions, REy(m), the ion-water (i-w) contribution to the chloride relaxation rate will be estimated according to the FRD hydration model discussed in the Theoretical Section. According to the considerations given in subsection 2, the ion-dipole contribution is expected to stem from the water molecules in fast exchange between the bulk phase and the hydration phase. It is convenient to derive an expression for R&'"(m) relative to the result obtained in the limit of infinite dilution, because in that way the expression obtains a simple form in which (38) Fries, P. H.; Patey, G. N. J . Chem. Phys. 1984, 80, 6253. (39) See, e.g., the references cited in ref 38. (40) Albrand, J. P.; Taieb, M. C.; Fries, P. H.; Belorizky, E. J . Chem. Phys. 1983, 78, 5809.
25Mg2+and 35Cl- Quadrupolar Relaxation in MgC1,. 2
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 7949
TABLE 111: Ion-Water Contribution to the %-Relaxation Rate
0 0.246 0.488 0.985 1.49 2.19 2.78 3.47 3.98 4.49 4.99 5.39 5.49
1 1.05 1.1 1 1.27 1.44 1.70 1.98 2.47 3.00 3.70 4.57 5.38 5.60
4.36 4.17 3.78 3.84 4.32 4.81 5.09 5.39 5.56 6.16 6.61 6.87
1 1.14 1.27 1.54 1.83 2.32 2.83 3.45 4.03 4.60 5.43 6.10 6.35
1 1.18 1.37 1.78 2.22 3.20 4.35 6.30 8.56 11.6 15.9 21.0 22.8
the concentration-dependent parameters become immediately apparent. Also, in this way one may obtain a relaxation rate that does not depend on the actual value of the concentration-independent parameters, e.g., 7..and P. The expression is
In eq 15 the subscripts w and hydr refer respectively to the bulk and the Mg2+ hydration water phase. f (=hm/55.56) denotes a mole fraction, where h is the number of water molecules within the first magnesium hydration layer. The parameters ro and ro' denote the distances of closest approach between CI- and the water dipole in the case that the water dipole is located in the bulk and the hydration phase, respectively. At contact separation it is estimated that r,' = ro = 3.21 A. In eq 15, s,(m) refer to the reorientational correlation time of the water molecules in the bulk water phase. In the present study, for ~ , ( m )and Thydr(m) the results derived in a 2H and I7O N M R study in MgCI, solutions the result at 25 OC will be used? For the limiting value, ~,(nrO), obtained in pure water will be used, e.g., T,O = 1.94 f 0.12 P S . ~ It is reasonable to take i h @ c equal to io*+, or to T ~ + where , lo*+ and TO+ denote the rotational correlation time of the water molecule ( H 2 0 ) in the magnesium hydration phase as observed by the ,H and I7O relaxation path, respectively. Due to the anisotropic reorientation of the hydration molecule the value of T ~ * +may differ from the io value + determined in the same MgC1, solution. However, in ref 9 it is observed that in the concentration range studied here the difference between them is not very large, e.g., less than 25%. Therefore, the values of Thy& denoted in Table + io+)/2. It is noted that 111 are obtained taking Thydr = (io*+ in the derivation of eq 15 a possible concentration dependence of the water particle concentration, C,, has not been considered here, because it may be assumed that, within the studied concentration range, the chloride ions remain surrounded by bulk and/or magnesium hydration water molecules. This assumption is supported by the neutron diffraction results in various concentrated NiCl, solutions in D20.I8 This suggests that the relevant water particle concentration, that is, the water particle concentration near the chloride ion, hardly changes. In Table 111 the calculated results denoted as {R&w(m)/Rzy(mm-O);eq 15) are presented. Inspection of Table I11 shows that in moderately concentrated MgC12 solutions the observed chloride relaxation rates are dominated by the ion-dipole contributions; at higher MgCl, concentrations the ion-ion contribution becomes equally significant. Finally, the presently derived estimates of the ion-water contribution may be compared with the results obtained by applying the usual procedure as discussed in the Theoretical Section. In Table 111, these calculated values are denoted as IR&'"(m)/REy(m-'O); eq 8). These results are calculated with eq 8, taking x(m-0) = 0.9, x(mL6) = 1, and c, m, and RHH as given in Table I. It is concluded that the ion-water contributions are equally well estimated with the usually applied procedure, eq 8. However, the results obtained with eq 15 are to be preferred in view of the implementation of recently available information
1 1.08 1.19 1.41 1.65 2.09 2.55 3.17 3.71 4.36 5.06 5.68 5.84
TABLE IV: Ion-Ion Contribution to the mol/kg HzO 0.246 0.488 0.985 1.49 2.19 2.78 3.47 3.98 4.49 4.99 5.39 5.49
wI- Relaxation Rate
R:(m)/Pcy(m-+O)
a,8,
0.04O
0.86 0.92 0.93 0.91 0.98 1.0 1.1 1.1 1.2 1.3 1.3 1.3
0.10 0.24 0.39 0.88 1.52 2.85 4.53 7.00 10.5 14.9 16.5
..
K-I,
A
3.6 2.5 1.8 1.5 1.2 1.1 1.0 0.9 0.9 0.8 0.8 0.8
P 0.027 0.030 0.030 0.029 0.032 0.034 0.036 0.039 0.041 0.043 0.046 0.047
" R g ( m )has been calculated with REi(m) = Pcy(m) - R$ (eq 15).
on the dynamical and structural properties of the water molecules in the MgCl, solution. 3B. The Ion-Ion Contribution to the 35CtRelaxation Rate. In addition to the ion-dipole contribution, RLW,also the ion-ion (i-i) rate, R E , contributes to the observe chloride relaxation rate. Theoretically, REi(m) {=PcY(m)- RZY(m); eq 15) is usually interpreted with eq 16. For the constant Fcl the following paFell 6re2(fl2c'i,'f(u/a) R#(m) = FclAcI = (16) 21a3 rameter values may be i n t r ~ d u c e d :I~= 3/2, Q = -8.02 b, e = 4.80 X esu, h = 1.055 X erg-s, and 7.. = -68.8. For P normally the continuum estimate derived by Cohen and Reif (eq 14) is used, e.g., P = PCR= 0.5. For the interionic distance a one has a = 3.9 A. Introducing in eq 16 the known values, one may evaluate the values of the screening length a as a function of the MgCI, concentration. The calculated a values are denoted in Table IV. For all the solutions studied here it is concluded that the a values lie in the range of 0.8-1.4 A. As shown by Hertz et aI.,l9 this range of a values also seems to apply for many other electrolyte solutions. However, after inspection of Table IV one may notice that the calculated a values hardly show any concentration dependence. It is noted here that this is a rather remarkable property for a shielding parameter being introduced in relation to the effects of screening of the electrostatic potential, c.q., field gradient at an ionic nucleus due to the other ions in the solution. Especially when dealing with a large concentration range, one should expect that a shows some dependence on the ionic strength of the solution on grounds analogous to those that can be given for the ionic strength dependence of the Debye screening length K-I defined in the Debye-Hiickel theory for electrolyte solution^.^^ For 1-2 strong electrolyte solutions at 25 OC one has K - ~ ( A )1.76(1/c),'/, with c in mol of salt/dm3. Although the Debye-Hiickel theory only applies to diluted electrolyte solutions, for discussion purposes K-I values have been calculated for all the MgCI, solutions studied (41,) See, e.&: Robinson, R. A.; Stokes, K.H. Electrolyte Solutions, 2nd ed.; Pitman Press: Bath, 1959.
7950
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989
Struis et ai.
TABLE V: Some Parameters Derived from the Interpretation of the 3sCl-, slBr-, and **'I- Quadrupolar Relaxation Rates in 1-1 Electrolyte Solutions at 25 "C m. mol of salt salt/icg of H ~ O y ( a / a ) cy, A P
LiCl LiBr Lil
NaCl NaBr NaI KCI
18 12 12 5 8
IO 4
KBr KI
6
RbCl
7
RbBr RbI CSCl
6
CsBr CSI NHJI NH4Br NHiI
0.0092' 0.0117 0.016 0.004 0.006 0.016 0.017 0.015 0.013 0.033 0.020 0.021 0.047 0.046 0.034 0.018 0.007 0.017
5
7 IO 6 4
5.5 5 6
1.26 1.3 1.5 0.9
1.0 1.5 1.5 1.4 1.4 2.0 1.6 1.7 2.4 2.3 2.0 1.6 1.1 1.5
0.039c 0.044 0.052 0.026 0.032 0.052 0.053 0.050 0.047 0.074 0.058 0.059 0.089 0.088 0.075 0.055 0.034 0.053
'Taken from Hertz et bCalculated taking a = 4 A. (Obtained from the equality ( P)2 = y(a/a)/6. here. The results are denoted in Table IV. Inspection of Table IV shows that, for increasing MgCI, concentration, the value of K-I decreases from 3.6 8, ( 0 . 2 4 4 ~to ) 0.81 8, ( 4 . 7 6 ~ )Note . that within the concentration range considered here, the value of K-I changes with a factor of 4 in contrast with the value of a , which on the average equals a = 1.1 f 0.2 8,. In view of the ( a l a ) dependence of 7,' andf(a/a), an increment in the value of (a/.) from, e.g., ( a / a )= 1 to 4 would lead to a 23 times smaller value of the product {T,'f(a/a)).It should be clear that such an effect cannot be ignored or remain unnoticed. In the following section it will be shown that, from the interpretation of the magnesium relaxation rates, untolerably high a values are obtained. Therefore, to circumvent the unsatisfactory application of the parameter a, in the present study an alternative interpretation of the ion-ion contribution will be considered. Here, the use of the length parameter a will be bypassed by applying the ion-ion contribution as given in eq 9 and leaving the value of the polarization factor P open. Note that the length parameter a appears in both the function f ( a / a ) and the correlation time 7:. These alterations lead to the result R#m) =
F C l ( q 21~
T ~ ~ c ' T ,
27a3
Assuming that the translational motion of the chloride and that of the hydrated magnesium ion are not correlated, for the correlation time T one estimates T , = a,2/2D.2*42 It is useful to calculate P f r o m REi(m) {=RZF - REy(eq 15)) and eq 17 by introducing the known parameter values in eq 17. The values of P a s a function of the MgCI2 concentration are presented in Table IV. It is concluded that P hardly shows a concentration dependence. The average value obtained here is ( P ) = 0.036 f 0.007. Also from the quadrupolar relaxation rate of 35Cl-, *'Br-, and I2'I- as observed by Hertz et al.I9 in various electrolyte solutions in water, one may obtain an estimate for P. Using eq 12, with P (=PCR)= 0.5, Hertz et al. determined the value of y ( a / a ) , which is equal to f ( a / a ) / { l 4 ( a / c ~ ) and ~ ] , a in various concentrated 1-1 electrolyte solutions at 25 "C. Some of their results, namely, y ( a / a ) and a,are given in Table V. For Table V only the result obtained in the highest concentrated electrolyte solution has been selected here in view of the increasing uncertainty in y ( a / a ) and a for decreasing electrolyte concentration. For the calculation of a, in their analyses the closest interionic distance, a, is taken equal to 4 A. The reinterpretation can be carried out
+
(42) Freed, J . H. J . Chem. Phys. 1978, 68, 4034.
Figure 3. Relative contributions to 35C1-relaxation rate as a function of molal MgCI2 concentration (m).Legend: (a) ion-ion; (b) ion-hydration water; (c) ion-bulk water.
in a straightforward manner: Combining eq 9 with eq 10, where in eq 9 the value of P i s not specified and T , = a2/2D and in eq 10 one has P (=PCR)= 0.5, and 7,' as given in eq 12, it follows that ( P ) 2 is equal to y ( a / a ) / 6 . The thus calculated values of P are also presented in Table V. The Pvalues shown in Table V support the estimate derived above from the interpretation of the chloride relaxation rate in MgC12solutions. From the results given in Table V one obtains an average result equal to (P) = 0.054 f 0.018. One may note that this average value is much smaller than the value P = 0.43 one deduces from the analysis of the ion-water contribution to the quadrupolar relaxation rate of 35Cl-,*IBr-, and 12'1- (part I). The presently observed difference between P and Psignifies the fact that P and Prefer to different screening effects: For the ion-water contribution the relevant electric field gradient arises from the water dipoles located near the surface of the ion under study, while in the ion-ion case the encounter of this hydrated ion with other hydrated ions has to be considered. Finally, it may be useful to summarize the various contributions to the observed chloride relaxation rate, G F ( m ) . Hereto, in Figure 3 the various contributions are shown relative to Ro,Psd(~). As is apparent from Figure 3, with increasing MgCI2 concentration the observed chloride relaxation rate is determined to an increasing extent by the ion-ion contribution, from 0% at infinite dilution up to ca.70% at 5.49 m. For the ion-water contribution it is noted that at low MgC12 concentrations (51 m ) the main contribution (275%) arises from the bulk water molecules. With increasing MgC12concentration the contribution arising from the magnesium hydration water molecules become more important and at 5.49 m is responsible for ca. 75% of the total ion-water contribution. 4 . Interpretation of the 25Mg2+Relaxation Rate Observed in MgCI, Solutions. 4A. The Ion-Water Contribution to the 25Mg2+ Relaxation Rate. The ion-water contribution, Rl;;"(m),will be estimated with eq 18, which is an extended version of the formula Rhl(m) = 2p2n,~,,(m )AT sin2 P(m) FMg{
ro8(6 + E(m))
~ C , (m ) + ~ T C L ~ Pm) T ~ (
9b5
1
(18)
derived for the infinitely diluted MgC12 solutions as discussed in the Theoretical Section, subsection 2B. Concerning the constant F M g , the following values will be used:3 I = 5 / 2 , Q = 0.22 b, y.. = -3.50, p = 1.80 X 1O-Io esu, and P = 0.43. In eq 18, the second term between the braces refers to the contribution of the bulk water molecules. At infinite dilution this contribution is rather small, that is, less than 6% of the experimentally determined magnesium relaxation rate. Therefore, for reasons of simplicity, in eq 18 the water particle concentration, Cw(m),will be replaced by its pure water value, Cwo.For the quenching parameter, AT, the limiting value at infinite dilution AT = 0.049, as obtained in subsection 4, will be applied for all the solutions studied here. The distance of closest approach between the centers of the magnesium ion and the water molecules in the hydration and in the bulk water phases and b = 3.43 8,,20*25 respectively. are estimated as r, = 2.06 8,3*20*25 The calculated ion-water contributions, denoted as RLi(eq 18), are presented in Table VI. It is concluded that from eq 18 rather moderate ion-water contributions are obtained for the whole
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 7951
25Mg2+and 35Cl- Quadrupolar Relaxation in MgCI2. 2 TABLE VI: Ion-Water Contribution to the =Mgzt Relaxation Rate
P"(m)/ P$(m)/ Go (m-4) R.,$?m-+O)
R i g q 18)/ Ri"(eq
Ri-w(eq 8)/ Ri$(eq
mol/kg H,O H20
RoMg(m-0) R!!(m-O)
P@'(m-0) PMe (m-0)
0 0.246 0.488 0.985 1.49 2.19 2.78 3.47 3.98 4.49 4.99 5.39 5.49
1 1.10 1.15 1.34 1.61 2.03 2.62 3.41 4.38 5.72 7.89 10.8 12.1
1 1.04 1.07 1.13 1.19 1.28 1.37 1.48 1.57 1.66 1.76 1.84 1.86
m(MgC12), m(MgCI2),
m(MgCI2), mol/kg H20 Rz,(m)/pMy(m-+O) 0.246 0.488 0.985 1.49 2.19 2.78 3.17 3.98 4.49 4.99 5.39 5.49
1 1.05 1.10 1.19 1.26 1.36 1.43 1.43 1.36 1.24 1.01 1.14 1.18
concentration range studied here. Table VI also contains other results obtained by applying the usual procedure as discussed in the Theoretical Section. These results, denoted as RLi(eq 8), are obtained by taking x(0) = 5.2, x ( m 3 = 1, Cw(m) = Cwo,and the concentrations c and m, and RHH as given in Table I. Inspection of Table VI shows that with eq 8 also moderate results are obtained. However, in view of the large differences in the hydration models leading to respectively eq 8 and 18, the present model is to be preferred in view of the implementation of recently available information on the dynamical and structural properties of the water molecules in the magnesium hydration layer. Despite the observation that the usual procedure as outlined in connection with eq 8 works surprisingly well, still the applicability of eq 8 may be questioned when dealing with electrolyte solutions containing structure-forming cations like MgZt. It is well-known that macroscopic quantities, e.g., the shear viscosity of the solution, do not always offer detailed and reliable information on the water dynamics on a microscopic level. The observed proton relaxation rate is an even less clear source of information due to the fact that both intra- and intermolecularly determined dipole-dipole interactions may contribute to the observed relaxation.2 4B. The Ion-Ion Contribution to the 25M2tRelaxation Rate. To start the discussion, it is noted that Helm and Hertz have interpreted, e.g., the relaxation rates of (M=) 25Mg2t, 43Ca2+, 87Sr2+,and 137Ba2+in various concentrated MBr2 and M(C104)2 solutions at 25 0C.20 From the interpretation of the observed cationic relaxation rates in terms of the ion-ion contributions as given in eq I O and 11, these authors obtained not well understood high values for the screening parameter a, e.g., cy 2 5 8, for a = 4 A. The interpretation of the present 25Mg2t relaxation data also leads to the above-mentioned a range; see Table VII. Therefore, an interpretation will be applied here analogous to the one given for the ion-ion contribution to the chloride relaxation rate. For 2SMg2+, REK(=Robd MK - Ri-" ,&eq 18)) will be interpreted by eq 19 8*FMK(P)'eZc 7' , RuK(m) = FMKAMK=
27a3
TABLE VII: Ion-Ion Contribution to the uMg2+ Relaxation Rate
(19)
with rc = U2/2D. As in the interpretation of the RE: data, also from the REg data one may determine the polarization factor, P , as a function of the MgCI2 concentration. The results are also given in Table VII. Inspection of Table VI1 shows that P little depends on the MgCI2 concentration; e.g., the average result is ( P ) = 0.19 f 0.02. Note that the result ( P ) = 0.19 is ca. 5 times larger than the estimate obtained from the interpretation of the chloride relaxation data, e.g., ( P ) = 0.036. With respect to the ion-ion contribution, neither the electrostatic theory nor, e.g., a recent derivation given for P,23indicates such a discrepancy in the respective P values. Therefore, the Pvalues obtained from the interpretation of the chloride relaxation data (Table IV) will be applied in the interpretation of the magnesium relaxation data. The following arguments may support this choice: (1) To obtain P , R6M has to be corrected for the ion-water contribution. This
0.06O 0.08 0.21 0.42 0.75 1.25 1.93 2.81 4.06 6.13 8.96 10.2
a , 8, 5.5b 4.2 4.5 4.9 4.8 4.9 4.8 4.9 5.1 5.4 6.0 6.1
P
.iR* 10, deg 5.0 3.7 4.2 4.5 4.4 4.5 4.4 4.5 4.6 4.9 5.3 5.5
0.21 0.023 0.160.013 0.17 0.016 0.19 0.019 0.18 0.018 0.19 0.019 0.18 0.017 0.19 0.018 0.19 0.019 0.20 0.022 0.22 0.026 0.23 0.028
"RZ ( m ) have been calculated with Rg,(m) = RoMy(m)- RG (eq 18). btalculated assuming that flula),eq 1 I , holds for a > a . For a the value a = 3.9 8, has been taken.
contribution requires fewer adjustable parameters in the FRD model (CI-) than in the TAN model (Mg2+). Therefore, the P values obtained from CI- may be less erroneous. (2) In view of the typical structural and dynamical properties of the water molecules within the magnesium hydration unit, it is most likely that in the present interpretation of the magnesium relaxation rate other equally significant contributions have been neglected resulting in the apparent discrepancy. For example, Helm and Hertz20 gave the following suggestion. These authors defined the excess contribution as given in eq 20. Here RzK(indirect) denotes the ion-ion contribution to the magnesium relaxation rate as can be estimated indirectly from the ion-ion contribution to the chloride relaxation rate, according to RtK(indirect)/(REi(=REPd - RLW(eq8))) = F M /2F,-l. The excess contribution was interpreted as follows. Taese authors assumed that within an octahedrally symmetric magnesium hexaaquo unit the water dipoles are radially oriented. During a cation-anion encounter both ions share three hydration water molecules, and as a result the orientations of the water dipoles may have been altered. These authors concluded that during the encounter the orientations of the respective water dipoles are shifted to a direction perpendicular to the cation-anion axis. It was thought that the neglect of this effect and of the corresponding additional ion-water contribution led to the overestimation of the ion-ion contribution, which in turn was thought to be the main source for the untolerably high values of the screening parameter, a , as derived from the cationic relaxation data.20 The concept of perturbation of the hydration structure of MgZt can be introduced in the presently used DOS hydration model. It will be assumed that during the attractive ion pair encounter the symmetry of the hydration layer shortly changes from a strict octahedral into a octahedral-like symmetry. To take this effect into account, one must consider the first term between the braces in eq 6. Equation 6 expresses the ion-water contribution in terms of the more general distinct hydration model, DOS. This term is given in eq 2 1. In eq 2 1, the symmetry of the hydration layer
dQ =
l12n&Q~ovARcos2 6
(21) ro8 is expressed by the quenching parameter AR. In the present study it is assumed that one deals with a distinct magnesium hydration unit (gQ = 1) for which the respective radial cation-oxygen pair distribution functions are very sharp; and hence, for the octahe0. drally symmetric magnesium hydration layer one has AR The above-mentioned symmetry distortion may result in a change in the value of the quenching parameter AR, from 0 to AR*. Recognizing that the distortion stems from the chloride-magnesium encounter, it seems reasonable to replace in eq 21 the overall correlation time 7, by the translational diffusional correlation time rc = a2/2D and to introduce the dependence on the chloride concentration and the integration over the entire space
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1952
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 100 ( 010 )
50
0
0
0
0
2
b
0
'
m
Figure 4. Relative contributions to 2sMg2+relaxation rate as a function of molal MgCI, concentration (m).Legend: (a) excess; (b) ion-ion (indirect); (c) ion-hydration water; (d) ion-bulk water.
by multiplying eq 21 with (2ac'a3/27). (The latter term is easily obtained from eq 9.) After the introduction of these alterations one obtains
Contrary to the procedure followed by Helm and Hertz, Ri-i Mg( i ndirect) will be estimated directly with eq 19 by introducing in eq 19 the values of the polarization factor, P , which has been derived from the interpretation of the chloride relaxation rates. One may note that with respect to the choice of the polarization factor, in part 1 a comparable situation arised when dealing with the ion-water contribution to the magnesium relaxation rate at infinite dilution. Then and also in the present case it is reasonable to apply for 35Cl-and zsMg2+the same polarization factor when dealing with a comparable type of contribution. Introducing the thus obtained values, together with the known parameter values in eq 22, one may obtain A,$ as a function of the MgClZ concentration. These results are also presented in Table VII. As one may notice, for all the MgCl, solutions studied here the calculated A,* values are rather small, suggesting relatively small symmetry distortions during the attractive ion pair encounters. To give an impression, one may express A,* > 0 in terms of AO, where AB characterizes the deviation from the radial dipole orientation within the hydration unit. Here it will be assumed that these deviations only apply for the three hydration water molecules that are shared during the encounter. For small values of A,$ one estimates that A,$ = 3 ( A ~ 9 with ) ~ AO in radians. The thus calculated A0 values (in degrees) are presented in Table VII. It is concluded that relatively small symmetry distortions as expressed by A0 = 4-6' may lead to a significant (induced ionwater) contribution to the 25Mg2+relaxation rate in finite con-
Struis et al. centrated MgC12 solutions. Of course, this resolution may also apply for the excess contribution to the quadrupolar relaxation rate of, e.g., the alkaline-earth-metal ions, (M=) 43CaZ+,*'SrZ+, and 13'Ba2+,as is apparent from the ratio (Aanion/Acation}
