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Rotation of Water Molecules and Its Relation with the Chemistry and Physics of Liquid Water J. H. Sluyters* and M. Sluyters-Rehbach Frederik Hendriklaan 27, Zeist, 3708 VA, Netherlands ReceiVed: July 27, 2009; ReVised Manuscript ReceiVed: October 2, 2009
The literature values of the limiting ionic conductivities of H+, OH-, K+, Cl-, Ag+, and Na+ in water between 0 and 156 °C are analyzed as for the two possible mechanisms of conduction, i.e., controlled by an activation process or by the rotation of the water molecules. Plots of the data versus T1/2 give straight lines for H+ and OH-, which supports the rotation control mechanism for these ions. The other ions give curved plots and therefore are investigated in terms of the activation control mechanism. A remarkable phenomenon is discovered, namely, that except for H+, the graphs for the other ions on extrapolation to lower temperatures have a common intersection point, T01/2, with the abscissa corresponding to T0 ) 243.4 K, i.e., -30 °C. This seems to indicate the presence of a virtual phase transition at about -30 °C, foreboding itself at higher temperatures. Below this temperature the supercooled water does not allow ions to migrate. Also, diffusion of solutes is found to cease, and dissociation constants drop to zero. The values of many physical properties of water appear also to approach zero at -30 °C, viz. the self-diffusion coeffient, reciprocal dielectric relaxation time, and solubilities of sparingly soluble salts. From data on the fluidity (reciprocal viscosity) and selfionization constant it follows that the transition temperature of supercooled D2O is 9° higher than of H2O. From the nearly quadratic shape of the several temperature dependencies it is inferred that the phase transition in question possibly is of some higher order. Implications for the transport number of protons to be expected in supercooled water are finally discussed. 1. Introduction
TABLE 1: Limiting Ionic Conductivities (from refs 6 and 7)
Several indications for restricted rotation of water molecules in the liquid exist. Most of them derive from optical spectra1 and will not be discussed here. Electrochemical arguments, related to the abnormally high mobility of protons in water, are exhaustively treated in ref 2. The first indication from electrochemistry is the fact that the ionic conductivity of H+ in H2O is 21/2 times greater than that of D+ in D2O.3,4 Further evidence comes from the perfect agreement between experiment and theory on the collective mobilities of protons and deuterons in mixtures of H2O and D2O.5 From that work it also follows that single water molecules are the rotating entities and not some aggregate. We decided to make an attempt to further proove the mechanism proposed in ref 5 via the temperature dependence of the limiting ionic conductivity λ0 of H+, originally published in ref 6 and also available in ref 7 and where also the data on other ions are reported in an exceptionally large temperature range. Although at first instance the latter were not of primary interest, their incorporation in the present study turned out to be most fortunate for the surprising finding of a particular behavior of supercooled water at temperatures far below 0 °C. The data to be used are summarized in Table 1. In Figure 1 the limiting ionic conductivities in Table 1 have been plotted versus the square root of the absolute temperature for reasons to be explained below. Two observations are striking at first sight. (i) Apart from H+ and OH- the plots for the ions form a bundle of curves that on extrapolation to low temperature have one common intersection with the T1/2 axis at a value T01/2 ) 15.6 K1/2, which corresponds to T0 ) 243.4 K. (ii) The plots for H+ and OH- are linear. The one for OH- intersects the T1/2 axis at exactly the same T01/2 value as the bundle of curves.
t/°C T1/2/K1/2 H+ OHK+ ClAg+ Na+
0 16.52 240 105 40.4 41.1 32.9 26.0
18 17.05 314 172 64.6 65.5 54.3 43.5
25 17.26 350 192 74.5 75.5 63.5 50.9
50 17.97 465 284 115 116 101 82.0
75 18.65 565 360 159 160 143 116
100 19.31 644 439 206 207 188 155
128 20.02 722 525 263 264 245 203
156 20.72 777 592 317 318 299 249
Strikingly, only the plot for H+ intersects at a lower value of T01/2 ) 14.95 K1/2. These phenomena point to different mechanisms of the respective conductivities as well as a remarkable halt down of the mobility of the ions at the temperatures of the intersections. In the section to follow we discuss these findings concerning the conductivities. Thereafter, we consider a number of other processes and properties that provide additional and supporting evidence. 2. Activation- and Rotation-Controlled Conductivity 2.1. Activation Control. Because ionic conductivity concerns a rate process,2 the limiting ionic conductivity of all the ions in refs 6 and 7 except for H+ and OH-, which show anomalous behavior in Figure 1 both qualitatively and quantitatively, can be plotted as an Arrhenius plot in order to inspect for the heat of activation ∆Hact. The lower curve in Figure 2, as an example, gives the result for Na+, where the left-hand side relates to the experimental data and the right-hand side to the extrapolated part at temperatures below 273 K in Figure 1. Of course, the intersection with the T1/2 axis in Figure 1 transforms into a vertical asymptote at 1000/T0 ) 1000/15.62 ) 4.11 K-1 in Figure 2.
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Figure 1. Limiting ionic conductivity of the ions selected for the present work from,6,7 plotted versus T1/2. Meaning of the symbols: (0) H+, (9) OH-, (O) Cl-, (2) Ag+, and (∆) Na+.
Sluyters and Sluyters-Rehbach
Figure 3. Apparent values of ∆Hact calculated (a) from the slopes of Figure 2 [larger symbols] and (b) by applying eqs 3 and 6 [smaller symbols]. Meaning of symbols as in Figure 2.
λ0 ) 0 for T ) T0 it follows that
λ0(T) ) a(T2 - T20) + b(T - T0)
(1)
hence, applying the Arrhenius equation
∆Hact(T) ) -Rd[ln λ0(T)]/d[1/T]
(2)
∆Hact(T) ) RT2(2aT + b)/[a(T2 - T20) + b(T - T0)]
(3)
Figure 2. Arrhenius plots of the limiting ionic conductivities of (0) H+, (9) OH-, and (∆) Na+. The smaller dots indicate the plots calculated with eqs 5 and 1, respectively.
Here, it can be concluded that the curvature of the Arrhenius plot already observed in the experimentally accessible temperature range is just an advent to the more strongly curved part at the low-temperature side. This conclusion is rather special, as curved Arrhenius plots are not uncommon but are usually only vaguely explained in terms of a mechanistic complication and not as a property of the solvent. If λ0 is plotted against T, for the ions that migrate under activation control we find that to a high degree of precision they obey a quadratic temperature dependence. Thus, writing
λ (T) ) aT + bT + c 0
and
2
In Figure 3 the upper curve shows as an example ∆Hact(T) calculated according to eq 3, with a and b derived from the experimental λ0(T) values for Na+ in Table 1. The right-hand side of eq 3 is a general expression applicable not only to the heat of activation of activation-controlled transport in water but also to the enthalpy of equilibria to be discussed further on, provided that the equilibrium constant of interest can be described by a quadratic function of the temperature, as is often found to be the case. 2.2. Rotation Control. According to the generally accepted model for the H+ conduction in water,2 the proton jumps from a hydronium ion to a water molecule, when the latter has rotated into a favorable orientation. Rotation of a water molecule about the bisector of the well-known H-O-H 105° angle does not change its orientation toward the H3O+ proton donor molecule. The two other angles, perpendicular to the bisector, are equivalent, so that the time dependence of the availability for the acceptance of a proton for our purpose can be described with one angular frequency ω only. Classically the energy of a rotating molecule with moment of inertia I and angular frequency ω about a fixed axis is given by 1/2Iω2 and equals, on average, 1/2kT as required by the equipartition principle. Thus
ω ) (kT/I)1/2 +
(4)
As for H conduction it can be assumed that per revolution of the receiving water molecule there is a fixed temperature-
Chemistry and Physics of Liquid Water independent probability 0 < P < 1 for a proton to make a successful jump from an H3O+ ion to a neighboring H2O molecule. If so, its limiting ionic conductivity should be proportional to T1/2. In Figure 1, where the data of Table 1 are plotted versus T1/2, it can be seen that a linear relationship is indeed obtained. However, the straight line has an intercept with the T1/2 axis at T01/2 ) 14.95 K1/2, corresponding to T0 ) 223.5 K, in contradiction with eq 4, which predicts this to occur at 0 K. The difference may be due to the rotations not being free but hindered by some energy barrier that allows rotation only if 1/2kT exceeds the height of that barrier. If so, the barrier height for the water rotation can be calculated to be 0.5k × 223.5 ) 154.3 × 10-23 J ) 929.3 J mol-1. For OH- conduction, however, a proton is necessarily donated by an adjoining water molecule, suggesting that the rotation of the latter is rate determining and eq 4 would equally apply. The data in Figure 1 show a straight line also for OH- but with an intercept at T01/2 ) 15.6 K1/2, corresponding to T0 ) 243.4 K. From this result the barrier height for the rotation can be calculated to be 1/2k × 243.4 ) 168.0 × 10-23 J ) 1011.9 J mol-1. Of course, identical water molecules cannot have different energy barriers for rotation, so an explanation has to be found for the peculiar behavior of hopping protons. Next to a possible tunnelling mechanism, a proton zero-point energy could serve as an acceptable explanation, for if so such a proton as a consequence could dispose of its zero-point energy Ezp with which the required rotation energy 1/2Iω2 ) 1/2kT will be lowered. It might even be conjectured that from this temperature difference the Ezp could be calculated to amount to 1/2k(243.4 - 223.5) ) 13.7 × 10-23 J or 82.6 J mol-1. Hence, it follows that the rotational energy barrier height for the water molecule rotation amounts to 1011.9 J mol-1. Making this comparison is not completely fair because a dipolar water molecule situated next to a H3O+ ion could behave differently from one adjoining an OH- ion. Also, it should be realized that a free rotator cannot have a zero-point energy. However, a restricted rotator behaves partly as an oscillator, increasingly so the more it is restricted.8 In the literature2,9 values of the limiting ionic conductivity of H+ have been used as excess values. These were obtained by correcting the original data as in Table 1 for H3O+ migration by subtracting the limiting ionic conductivity of the Na+ ion. Subsequently, these excess values were used for the calculation of the heat of activation, ∆Hact, that appeared to be strongly temperature dependent. We found that such a correction spoils the linearity of the λ0H+ and λ0OH- plots in Figure 1. Clearly subtraction of the full limiting value λ0Na+ is an overcorrection. Also, it should be realized that the surroundings of a proton that rapidly and frequently hops between water molecules will be slow to adjust. Moreover, it is not obvious why migration and rotational modes of transport should be independent and additive. Although it is of course inappropriate to construct Arrhenius plots for a rotation-controlled process for which an activation enthalpy is an ill-defined quantity (the rotation barrier height is the more relevant property), we nevertheless decided to perform such an analysis just to make a comparison with the literature.2,9 The linear dependence of λ0 on T1/2 allows a generally valid derivation of an expression for the apparent ∆Hact. For the limiting ionic conductivity of both H+ and OH- it follows from Figure 1 that
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λ0(T) ) B(T1/2 - T1/2 0 )
(5)
where T0 is the temperature at the relevant intersection and B the slope in Figure 1. Substitution of this expression into eq 2 then gives
1 ∆Hact(T) ) RT3/2 / (T1/2 - T1/2 0 ) 2
(6)
The Arrhenius plots for the conduction of H+ (upper) and OH(middle) are also shown in Figure 2. Again, on the left the experimentally accessible temperature ranges occur, and on the right are the two asymptotes at 1000/T0 ) 4.475 and 4.11, respectively. In this way also here the curvatures in the accessible temperature ranges can be understood, although they have a lesser physical meaning. Nevertheless, eq 6 proved useful to construct the temperature dependence of ∆Hact(T) in Figure 3, where the lower curve pertains to H+ and the middle one to OH-. Comparison of Figure 3 with refs 2 and 9 shows that the present temperature dependence of ∆Hact(T) for H+ is considerably weaker, owing to our omitting subtraction of a Na+ migration. An analysis of the conduction by OH- ions seems to be nonexistent in the literature, and its mechanism of being rotation controlled has not been discovered earlier. In Figure 1 the H+ data points at the high-temperature end show a departure from linearity toward lower values of λ0H+, down to -11%. The thermal expansion of water, which causes the distance between molecules to increase by 0.8% at 75 °C to 3.5% at 156 °C relative to 0 °C, could have a decreasing effect on the probability 0 < P < 1 of a successful protontunneling event per revolution. Strikingly, such a decrease is totally absent in the OH- plot, which suggests that a proton released from a water molecule and next falling into a negatively charged proton hole does not tunnel but rather proceeds via a classical mechanism that is less dependent on distance. 3. Some Further Chemical Processes One might wonder whether the occurrence of a “dead stop temperature” of -30 °C as found in Figure 1 is a more general phenomenon. A search of the literature for limiting equivalent electrical conductivities and diffusion constants gives abundant evidence that this is indeed so for widely differing solutes. In Figure 4, by way of examples, limiting molar conductivities are shown of aqueous solutions of KCl and LiCl10 and of limiting ionic conductivities of trimethylammonium and bis-2methoxyethylammonium ions.11 Figure 5 shows diffusion coefficients of benzene and mannitol12 versus the temperature in degrees Celcius. Also, chemical equilibria are affected by the nearby phase transition. In Figures 6 and 7 some examples demonstrate the extrapolation of several equilibrium constants from the experimentally accessible temperature range13–15 toward lower temperature, all suggesting a decrease to zero in the neighborhood of -30 °C or, in other words, their reaction enthalpies ∆Η° tending to infinity. Normally a variation of an equilibrium constant with temperature is explained by the difference of the heat capacities of the participating reactants and their temperature dependences. For the present examples in water, however, the observed variation is far too large to be explained in such a way. Clearly it is the change in the nature of the solvent, owing to the progressive slowing down of the rotation of the water molecules with decreasing temperature, that causes these effects.
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Figure 4. Limiting molar conductivity of (b) KCl and (O) LiCl10 and limiting ionic conductivity of (0) trimethylammonium and (9) bis-2methoxyethylammonium11 vs temperature.
Figure 5. Diffusion coefficients of (b) benzene and (O) mannitol12 in water vs temperature.
In Figure 6 the second acid dissociation constants of phenolsulfonic acid and phosphoric acid13 and the equilibrium constants of the reactions CO2 + H2O T H+ + HCO-313 and NH4+ T H+ + NH314 are drawn as a function of the temperature, and in Figure 7 the dissociation constants of acetic, propionic, and n-butyric acids are shown.15 Curiously, the curves of acetic acid and propionic acid are very close to congruent. Also, the curve for formic acid15 (not shown) has exactly this shape, its dissociation constant being 10 times higher than that of acetic acid at all temperatures. It is striking that butyric acid at low temperature tends to follow acetic acid and at high temperature approaches the behavior of propionic acid. One might wonder where the curves in Figure 7 could lead at the high-temperature side. Linear extrapolation suggests a common intersection with the temperature axis at about 250 °C for the three simplest acids. Extrapolations that are not based on some theory should be considered with caution, their reliability depending on the range and curvature. Figures 6 and 7 should therefore be considered as speculative and at best predictive. On the other hand, the
Sluyters and Sluyters-Rehbach
Figure 6. Second acid ionization constants of (b) phenolsulfonic and (0) phosphoric acid,13 and the equilibrium constants (9) of CO2 + H2O T H+ + HCO3- 13 and (O) of NH4+ T H+ + NH3.14
Figure 7. Temperature dependence of the dissociation constants of (b) acetic acid, (O) n-butyric acid, and (9) propionic acid (from ref 15).
data in Table 1 on the limiting ionic conductivity of OH- have a quality that allows an accurate least-squares analysis of the linear dependence on T1/2 in Figure 1. The “dead stop” temperature then is found to be -30.0 °C. Moreover, we inspected the behavior of all the limiting ionic conductivities reported in refs 6 and 7 (but for H+) and found all the intersection points on the temperature axis to occur within the range from -28.5 to -31.0 °C. As for the origin and nature of the energy barrier for the rotation of a water molecule, 1011.9 J mol-1, its electrical and electromagnetic interaction with the surrounding bulk water phase is the only possibility. Surprisingly, this value does not appear to depend on the presence of a nearby solute molecule or ion and the charge of the latter. There is not any specific effect of the identity of the solutes; only the H+ ion behaves differently.
Chemistry and Physics of Liquid Water
Figure 8. Fluidity of (0) light and (9) heavy water as a function of the temperature, data from refs 16–18, together with the square root of the self-ionization constant of (O) water19 and (b) heavy water20 versus the temperature.
4. Physics of Supercooled Water If indeed the supercooled water below 243.4 K behaves as a new phase, impenetrable to ions and molecules, it might be expected that the phase transition also shows up in other, physical, properties of liquid water on extrapolation to low temperatures. As a first attempt we made a plot of the water fluidity Φ, i.e., the reciprocal value of the viscosity η of pure water, data taken from refs 16–18, versus the temperature t. The result is shown in Figure 8, the intersection with the abscissa occurring at -31 °C, indeed nicely close to the value obtained from Figures 1 and 4-7. Thus, next to ionic and molecular transport the flow of supercooled water below the transition temperature comes to a halt as well. Interestingly, the same behavior is found for heavy water, be it with the intercept 9° higher, as is also shown in Figure 8, using viscosity data from ref 18. This indicates that the rotations in D2O have a 37 J mol-1 higher barrier height, presumably owing to its stronger hydrogen bonds. In Figure 8 we also represent the square root of the water self-ionization constant KW,19 which in fact is equal to the equilibrium activities aH+ ) aOH- in pure water, together with the analogous data for heavy water.20 The “dead stop” temperatures here are -31 and -22 °C, respectively, in fair agreement with the values obtained from the fluidities. In Figure 9 the self-diffusion coefficient of water, taken from refs 21–25, is shown upon extrapolation to lower temperatures to drop to zero at -31 °C together with the reciprocal dielectric relaxation time τ-1 of water26 with the familiar shape. More recently27 with time-resolved HD Optical Kerr Effect experiments τ-1 versus temperature data were obtained down to -19 °C, which are congruent with those in ref 26. Also, the solubilities of the salts PbBr2 and PbI2 in water reported in ref 28 are shown in Figure 10 to extrapolate to zero at t ) -30.5 °C. Finally, in Figure 11 we plotted the molar conductivities of 0, 10 -3, 1, and 3 M KCl solutions from ref 29. Clearly these only slightly curved plots extrapolate to ever lower values of T0, suggesting the occurrence of a kind of freezing-point depression from which a heat of fusion could be calculated. To
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Figure 9. Temperature dependence of the self-diffusion coefficient of water, obtained from tracer studies with (b) H2O18 and (O) HDO from ref 21, (9) HDO from ref 22, (0) HDO from ref 23, and (9) data from NMR measurements combined with various tracer studies.24 In the supercooled region data obtained by a tracer study with (×) HTO from ref 25 are added. (2) Reciprocal dielectric relaxation time of water versus the temperature from ref 26 and (∆) similar data in ref 27 downscaled with a factor 28.2.
Figure 10. Solubility of (O) PbBr2 and (b) PbΙ2 in water from ref 28 versus the temperature.
that end, however, conductivity data obtained at low temperatures are essential to attain the necessary accuracy. This depression is as yet the only accessible property of the new phase. The quantities depicted in the Figures 4-11 appear to depend on the temperature in a nearly purely quadratic way, which points to the nature of the phase transition possibly to be of some higher order. There is an analogy with the rotating NH4+ and NO3- groups in their ionic crystal lattices, where the transition also occurs gradually within an extended temperature range.30 The limiting cation transport number t+ of HCl in water is reported in the literature31 to depend on the temperature between 5 and 50 °C. This dependence can be understood from Figure 1 by calculating λ0H+/(λ0H+ + λ0Cl-) ) t+, leading to the behavior
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Figure 11. Molar conductivities29 of (b) 0, (O) 0.01, (9) 1, and (0) 3 M KCl versus the temperature.
Figure 12. Limiting cation transport number t+ of hydrochloric acid as a function of temperature: (() experimental data,31 (s) theoretical prediction [see text].
of t+ drawn in Figure 12. Including also the temperature region below 5 °C, we find that t+ increases further to become unity at -30 °C. At that temperature and below, apparently protons are the unique particles that could be transported in the supercooled water phase. However, this cannot be true because all other processes considered above provide evidence that below -30 °C the water molecules do not rotate anymore, so that the proton conduction mechanism becomes inactivated. Also, because the water phase at t < -30 °C is inaccessable to all anions, electroneutrality demands the absence of protons as well. Therefore, any extrapolation below -30 °C, both in Figure 1 and Figure 12 (indicated by the dashed lines), has no physical meaning. 5. Conclusion There may be more properties of liquid water showing the same behavior on decreasing the temperature down to -30 °C. As a warning we report the water vapor pressure32 as a deceptive choice, which indeed at first sight on extrapolation seems to drop to zero at -30 °C. This would suggest that at low
Sluyters and Sluyters-Rehbach temperature supercooled water is the stable phase with respect to ice. This can be easily proved to be an artifact by showing that the ClausiussClapeyron equation is perfectly obeyed also at such low temperatures. In conclusion, it can be stated that the various phenomena and data presented here together provide a compelling case for the presence of a phase transition in water that on cooling under atmospheric pressure becomes completed at about -30 °C, at which temperature water molecule rotation comes to a halt. Although the analyses all rely on extrapolations to unattainable regimes of supercooled water with the systems in question, the presence of a phase transition is actually felt in higher, practical temperature regions, where it explains why “constants” there appear to be not constant at all. The question may rise how our observations compare to the numerous experimental studies on the peculiarities of supercooled water, reviewed, for instance, in ref 33. Among many other phenomena, in some of these studies evidence for a liquid-liquid transition, both in supercooled water34 and heavy water,35 is reported also. At present such a comparison is beyond our scope, and the references we mention are far from being exhaustive. In many of the experimental studies the supercooling is conditioned by some way of confining the water, e.g., by depositing it on a cooled foreign substrate or occlusion in a nanoporous system.36 The work presented here only involves bulk water systems and may therefore be a valuable contribution in answering the question whether bulk water may be compared with occluded water. To our knowledge, a special role of the temperature of -30 °C has not been reported before. It would be most useful if the present work could be linked to molecular dynamics computer simulations,37 taking also the rotation of the water molecules into consideration. Also, supplementing the data in the various graphs with experiments in D2O and at the lowest possible temperatures would be valuable for further experimental support. For instance, extension of Table 1 with λ0(Τ) values of D+ in D2O would be highly desirable. Insertion of such data into Figure 1 should present a linear dependence of λ0D+(Τ) on T1/2, intersecting with the T1/2 axis also at T ) 223.5 K just like H+ in H2O. If this would not be so, the ratio λ0Η+/λ0D+ would be temperature dependent and consequently its experimental value 21/2 at 25 °C only accidental. The present results make it clear that it can be most fruitful, besides an analysis based on the integrated van‘t Hoff equation for equilibria and the Arrhenius equation for rate processes, also to simply inspect plots of the equilibrium and rate constants proper versus the temperature. References and Notes (1) Hawley Cartwright, C. Phys. ReV. 1936, 49, 470. (2) Bockris, J. O’M.; Reddy, A. K. N. In Modern Electrochemistry; Plenum Press: New York, 1970; pp 470-486. (3) (a) Bernal, J. D.; Fowler, R. H. J. Chem. Phys. 1933, 1, 515. (b) Conway, B. E.; Bockris, J. O’M.; Linton, H. J. Chem. Phys. 1956, 24, 834. (4) Karlstrom, G. J. Phys. Chem. 1988, 92, 1318. (5) Sluyters, J. H.; van Duijneveldt, F. B. J. Electroanal. Chem. 1996, 413, 37. (6) Johnston, J. J. Am. Chem. Soc. 1909, 31, 1010. (7) In Handbook of Chemistry and Physics, 55th ed.; Weast, R. C., Ed.; CRC Press Inc.: Cleveland, OH, 1974; Table D-132. (8) Eyring, H.; Walter, J.; Kimball, G. E. Quantum Chemistry; Wiley: New York, 1944; p 77. (9) (a) Hu¨ckel, E. Z. Elektrochem. 1928, 34, 546. (b) Glasstone, S.; Laidler, K. J.; Eyring, H. The theory of rate processes; McGraw-Hill: New York, 1941; p 562 ff. (c) Reference 2: Table 5.6, p 473. (10) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolyte Solutions; Reinhold: New York, 1958; Table 6-8-5, p 234. (11) Lobo, S. T.; Robertson, R. E. Can. J. Chem. 1977, 55, 3851. (12) Tominaga, T.; Matsumoto, S. Bull. Chem. Soc. Jpn. 1990, 63, 533. (13) Reference 10: Table 15-13-1A, p 760.
Chemistry and Physics of Liquid Water (14) Reference 10: Table 15-13-4A, p 763. (15) Reference 10: Table 15-6-1A), p 755. (16) Reference 7: Tables F-47 and F-49. (17) Hallett, J. Proc. Phys. Soc. (London) 1963, 82, 1046. (18) Hardy, R. C.; Cottington, R. L. J. Chem. Phys. 1949, 17, 509. (19) Reference 10: Table 15-2-1A, p 754. (20) Reference 7: Table D, p 131. (21) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1955; p 13. (22) Easteal, A. J.; Price, W. E.; Woolf, L. A. J. Chem. Soc., Faraday Trans. 1 1989, 85, 1091. (23) Mills, R. J. Phys. Chem. 1973, 77, 685. (24) Holz, H.; Heil, S. R.; Sacco, A. Phys. Chem. Chem. Phys. 2000, 2, 4740. (25) Pruppacher, H. R. J. Chem. Phys. 1972, 56, 101. (26) Reference 10, p 12. (27) Torre, R.; Bartolini, P.; Righini, R. Nature 2004, 428, 296. (28) Clever, H. L.; Johnston, F. J. J. Phys. Chem. Ref. Data 1980, 9, 751.
J. Phys. Chem. B, Vol. 114, No. 2, 2010 869 (29) Lobo, V. M. M. Handbook of electrolyte solutions; Elsevier: Amsterdam, 1989; p 833. (30) Evans, R. C. An Introduction to Crystal Chemistry; Cambridge University Press: New York, 1948; p 269. (31) Reference 10: Table 11-9-1A, p 723. (32) Reference 7: Table D159. (33) Debenedetti, P. G. J. Phys.: Condens. Matter 2003, 15, R1669– R1726. (34) (a) Mishima, O.; Stanley, H. E. Nature 1998, 306, 329. (b) Meyer, M.; Stanley, H. E. J. Phys. Chem. B 1999, 103, 9728. (35) Mishima, O. Phys. ReV. Lett. 2000, 85, 334. (36) Chen, S.-H.; Mallamace, F.; Mou, C.; Broccio, M.; Corsaro, C.; Faraone, A.; Liu, L. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 12974. (37) Poole, P. H.; Sciortino, F.; Essmann, U.; Stanley, H. E. Nature 1992, 360, 324.
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