Properties of Water Solutions of Electrolytes and Nonelectrolytes - The

geochemistry and leaching of minerals,9a solubility of natural gas in deep brines,9bcosmology,9c etc. This work ...... The quantum phase-transitio...
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J. Phys. Chem. B 2001, 105, 7805-7817

7805

Properties of Water Solutions of Electrolytes and Nonelectrolytes Andreas A. Zavitsas* Department of Chemistry and Biochemistry, Long Island UniVersity, UniVersity Plaza, Brooklyn, New York 11201

J. Phys. Chem. B 2001.105:7805-7817. Downloaded from pubs.acs.org by AUCKLAND UNIV OF TECHNOLOGY on 01/28/19. For personal use only.

ReceiVed: March 20, 2001; In Final Form: May 9, 2001

Apparent large deviations of water solutions from ideal behavior are eliminated by taking account of the number of water molecules binding to solute sufficiently strongly (13.0 ( 1.5 kcal mol-1) as to be removed from the “bulk” solvent. Freezing point, boiling point, vapor pressure, and osmotic pressure measurements of electrolyte solutions of chlorides, bromides, and iodides are treated successfully, as are those of nonelectrolytes, for up to 50 wt % solute and concentrations hundreds of time greater than those over which Debye-Hu¨ckel theory applies. D-H theory focused on the behavior of ions; this work focuses on the nature of water. The nature of “free” water as solvent is changed by electrolytes. No evidence is found for ion pairs up to eutectic points. Hydration numbers (h) obtained are in agreement with many from more elaborate techniques. From freezing points, h values are as follows: H+, 6.7 ( 0.7; Li+, 6.6 ( 0.6; Na+, 3.9 ( 0.5; K+, 1.7 ( 0.5; NH4+, 1.8 ( 0.5; Mg2+, 13 ( 2; Ca2+, 12 ( 2; Sr2+, 12 ( 2; Ba2+, 10.5 ( 1.5; Al3+, 22 ( 2; Fe3+, 18 ( 2; Cl-, 0; Br-, 0; I-, 0; methanol, 1 ( 0.3; ethylene glycol, 1.8 ( 0.3; glycerol, 2 ( 0.5; glucose, 2.8 ( 0.5; sucrose, 5 ( 0.5; H2O2, 1.2 ( 0.2; NH3, 1.81 ( 0.05; and urea, 0 ( 0.5. From vapor pressures, h values are as follows: Rb+, 1.8 ( 1; Cs+, 0.6 ( 1. Cl-, Br-, and I- are solvated and not hydrated. Sequential binding energies of water to gas-phase ion-water clusters correlate with h, as do cluster distributions. Some h values show temperature dependence.

Introduction The nature of water solutions is of interest in so many areas that research in this old field remains unabated. Water tends to hydrate many solutes, and the hydration number is loosely understood to be the number of water molecules associated with solute in some distinguishable manner and is a factor in a variety of fields and important phenomena, such as the following: ion transport across biological cell membranes1 and channels2 where the size of the hydrated species relative to that of the pore or channel is of importance, as well as the ability of the hydrated species to interact with organic functional groups in the channels, the three-dimensional structure and function of biologically important molecules3 and the folding/unfolding of proteins,4 DNA intercalation reactions and the specificity of protein-DNA interactions,5 cloud nucleation6 and the formation of aerosols (both urban and sea salt) as they impact the environment,7 the protection of plants and seeds from dehydration and freezing by saccharides,8 geochemistry and leaching of minerals,9a solubility of natural gas in deep brines,9b cosmology,9c etc. This work uses measurements of freezing point depression, boiling point elevation, and vapor and osmotic pressures to obtain information about the nature of water solutions and hydration numbers. The results are compared to those obtained by recent, more elaborate experimental and theoretical approaches. Despite the bringing to bear of great intellectual and technical effort on the subject, conclusions reached from different approaches are not always consistent with each other. While much has been learned, the simple question of what the hydration numbers of various common ions are cannot be answered unequivocally. For example, for Li+, a hydration * To whom correspondence should be addressed. E-mail: [email protected].

number of 25.3 is quoted on the basis of transference measurements.10 Values of 4-6 are cited from neutron or X-ray scattering,11 and a tetrahedral arrangement is deduced from spectroscopic measurements.12 The octaaqueolithium, Li(H2O)8+, appears as a minimum by comparing theoretical and experimental infrared frequencies,13 while a hydration number of 4 is proposed from B3LYP calculations.14 Counterion extraction of Li+ from water into nitrobenzene yields a value of 6 for the number of water molecules associated with the cation that coextract into the organic layer.15 Part of the problem lies in the definition of hydration number, which is often defined in terms of the technique used. Water seems to hydrate ions in quasi-spherical shells, and scattering techniques primarily contain information about the first hydration shell,16 for which the result is sometimes described as the primary hydration number. Theoretical calculations often use the same terminology, and distinctions between first and second hydration shells are made.2b,13,17,18 Others define hydration number as the average number of water oxygens within a certain distance of the solute.19 Dielectric relaxation workers define hydration number as the average number of water molecules that are irrotationally bound to solute and unable to contribute to the solvent relaxation process.20 In conductance and diffusion studies, hydration number is the number of water molecules that lose their independent translational freedom and move with the ion.21 The conclusions reached from the different definitions are not necessarily similar. It is striking that none of the large variety of the recent work cited above makes any reference to measurements of the classical colligative properties treated here, because they might relate to hydration and to properties of water as solvent. The Debye-Hu¨ckel theory,22 based mainly on conductivity measurements and with the corrections of Onsager,23 is pre-

10.1021/jp011053l CCC: $20.00 © 2001 American Chemical Society Published on Web 07/20/2001

7806 J. Phys. Chem. B, Vol. 105, No. 32, 2001 sented as the basis of much of our understanding of water solutions of electrolytes by most current textbooks of physical chemistry. The theory postulates an ionic atmosphere of positive ions being surrounded by negative ones and vice versa and addresses the question of why the activity coefficient of ions in solution decreases with increasing concentration (an apparent less than 100% dissociation). The effects are treated quantitatively in terms of the electrophoretic effect, the asymmetry effect, and the viscous effect,24 and the theory has been supported by experimental investigations of very dilute solutions. For example, Bro¨nsted and La Mer showed that activity coefficients adhere to the D-H theory up to only 0.01 M for uni-univalent electrolyte water solutions, up to 0.003 M for unibivalent solutions, and up to 0.001 M for tri-univalent solutions;25 they also expressed reservations about aspects of the derivation of the theory, and such concerns have been expressed by others.26 It should be noted that activity coefficients derived from measurements of more concentrated solutions do not provide an explanation but simply measure deviations from ideality. Data supporting the assumptions of D-H theory “involve ionic strengths not over 0.02 and have little bearing on practical problems in analytical chemistry, in technical electrochemistry, and in biological chemistry”.27 In the same vein, the D-H theory has been described as applicable “to solutions usually so dilute that they have been uncharitably called slightly contaminated distilled water”.24 There have been efforts to extend the theory by parametric approaches,28 and such efforts are continuing.9a A widely accepted rationalization for the failure of the theory to apply to any but the most dilute solutions is the formation of ion pairs with increasing concentration, as proposed by Bjerrum29 and independently by Fuoss and by Kraus.24 Bjerrum calculated that in a 1 N solution of uniunivalent ions having a diameter of 2.8 Å, the ions are about 14% associated (NaCl would be near this category). Currently, many kinds of ion pairs are being invoked, such as contact ion pairs, solvent-separated ion pairs, doubly solvent-separated ion pairs, etc.20b,30 The fact that water forms its own hydrogenbonded structures is also thought to further complicate matters. Another complication cited is that ideal behavior would be expected only if there were no heat of solution or volume changes on mixing. A recent estimate of the current understanding of freezing point depressions is that beyond ∆t ) 3 °C, “the concentration is so high that a simple relation of concentration and freezing point depression does not exist”.31a The inability to grasp the essence of water-solute interactions was recently described as “frustrating,” since “life as we know it is water-based”.31b Freezing Point Depression Freezing point depression is expected to be a colligative property. The phenomenon is defined as one that depends on the collection of dissolved particles and not on their nature. By this definition, the freezing point depression of aqueous electrolyte solutions is not a colligative property. Expressing the relationship between freezing point depression, ∆t, and the mole fraction of solute particles, x (assuming full dissociation of strong electrolytes), in a form equivalent to the 1803 empirical relation of Henry’s law for colligative properties (∆t ) kHx), produces Figure 1 for some typical salts. It is evident that freezing point depression is very much a function of the nature of the ions involved. The straight line in the figure has been drawn with a slope of kH ) RTo2/∆Hfus ) 103.2 K, where R is the gas constant (1.987 cal K-1 mol-1), To is the freezing point of water in kelvin (273.15), and ∆Hfus is the heat of fusion of

Zavitsas

Figure 1. Freezing point depression measurements vs molar fraction of ions for the strong electrolytes indicated. The straight line is drawn as RTo2/∆Hfus ) 103.2 K-1.

water (1436 cal mol-1). This approximation for kH is commonly used for dilute solutions. The points deviate from the line in a bewildering fashion, both above and below. In contrast, organic solvents with organic solutes of similar polarity show freezing point depressions generally in agreement with Henry’s law to reasonably large concentrations. We examined freezing point depression data for water solutions of strong electrolytes and found that chlorides, bromides, and iodides show curved plots of ∆t vs x and that the curvature is always upward beyond the limited DebyeHu¨ckel domain, as shown in Figure 1 for LiCl, NaBr, and KI. We focused on such salts because this curvature is surprising. Ion pair formation or ionic assemblies with counterion atmospheres should produce downward curvatures because fewer solute particles than the stoichiometric number will be present. An upward curvature seems to imply that there is more solute, or greater than 100% dissociation, an apparent impossibility. The alternative is to conclude that there is less solvent present than the conventionally calculated stoichiometric total. Some water is missing. The Debye-Hu¨ckel theory focused primarily on the behavior of the ions. This work focuses on the water. The missing water is tied up in hydrating solute. We demonstrate that curved freezing point depression data produce straight lines when the correct hydration number is used and the corresponding amount of water is subtracted from the total in calculating mole fractions. The results are consistent with both chemical intuition and results obtained by some totally different techniques for LiCl, NaCl, NaBr, NaI, KCl, KBr, KI, CsCl, MgCl2, CaCl2, SrCl2, SrBr2, BaCl2, and BaBr2, constituting all group IA and IIA metals for which sufficient data are available, and for HCl, HBr, HI, NH4Cl, AlCl3, and FeCl3. We define hydration number as the average number of water molecules binding to solute sufficiently strongly as to become part of it. This will decrease the number of water molecules constituting “bulk” water, or free solvent, but will not change the number of solute particles. The mole fraction of solute is given by eq 1. Ms is the moles of solute (formula weight), Mw

x ) iMs/(Mw - hMs + iMs)

(1)

Properties of Water Solutions

J. Phys. Chem. B, Vol. 105, No. 32, 2001 7807

is the moles of total water, and h is the hydration number of the dissolved compound. The van’t Hoff index, i, is the number of ions constituting each strong electrolyte, 2 for NaCl, 3 for CaCl2, 4 for FeCl3, as Arrhenius pointed out in an 1887 letter to van’t Hoff.24 The term hMs is the missing or “bound” water, as distinguished from “bulk” or “free” water. Equation 1 implies 100% dissociation. The fact that water molecules tightly bound to ions are not available to act as solvent (e.g., for dissolving neutral solutes24) has been recognized for a long time.33 However, our search of the literature failed to find a quantitative application of this concept for understanding colligative properties of aqueous electrolyte solutions or for obtaining hydration numbers from them. The thermodynamic relationship between freezing point depression and mole fraction of solute is given by eq 2.33 Tf, in

1/Tf - 1/To ) -kc ln(1 - x)

(2)

kelvin, is the freezing temperature of a solution of x mole fraction of solute particles or sum of ions. The relevant quantity in this relation is (1 - x), the mole fraction of water, it being the substance freezing out. The slope is the cryoscopic constant,34 kc ) R/∆Hfus. We investigated whether a value of h exists that will produce, via eq 1, a set of x values such that the (initially curved) experimental points plot linearly, as required by eq 2, and simultaneously produce a zero intercept. We found that such an h value does exist for each compound, and we designate it to be the hydration number. The value of h that produces a zero intercept was adopted rather than the one that produces the highest correlation coefficient in a first-order regression. While these two different criteria produce similar values of h, the question of linearity might be subject to debate but the zero intercept is not. For this approach to have validity, some stringent criteria must be met. (1) Very high correlation coefficients must be obtained from the linear plots of the data passing through the origin when mole fractions are calculated by eq 1 with the correct hydration number. (2) The concentrations and temperature ranges of the linearity must be substantial, otherwise no significant progress has been made in our understanding of the phenomenon. (3) Because some empirical algorithm can generally be found to straighten out curved data, the one used must not be ad hoc. The method used satisfies all three criteria, as is demonstrated in Table 1 and Figures 2-5. The table lists the hydration numbers obtained, the concentrations and temperature ranges of the linearities they produce, and other relevant data for 28 compounds. (1) In all cases, the hydration number that produces a zero intercept also places the initially curved data on a straight line, the poorest correlation coefficient being 0.999 from firstorder regression analysis. (2) Measurements of ∆t > 40 °C plot linearly. It is noted that the concentration over which eq 2 is applicable is limited by (a) the solubility of the compound and (b) the eutectic point, because eq 2 is not valid unless the material freezing out of solution is pure water-ice. (3) Equation 1 is not an empirical algorithm but is based on a clear and simple chemical concept. Figures 2-5 show plots of eq 2 for data with NaCl, MgCl2, HCl, and LiCl. The open symbols depict the mole fraction obtained by conventional stoichiometry, i.e., disregarding the “bound” water. The filled symbols depict results with mole fractions from eq 1 with the hydration number indicated in each case. NaCl and MgCl2 are examples of linearity extending to the eutectic point. HCl is an example of onset of downward

curvature prior to the eutectic point. LiCl illustrates variations in data from different sources. Freezing point data used in this work are from standard compilations: the CRC Handbook,35 the International Critical Tables,36 and a more recent monograph.37 Because the correlation coefficients obtained are extraordinarily good and because it appears that values given in these sources may have been fitted to some smooth function with interpolations, direct experimental data were also used;38,39 no significant deterioration in the correlation coefficients was observed. Equations 1 and 2 do not involve any consideration of electrical charge of solute and should be successful with nonelectrolytes. From available data, we selected a group of compounds that can be considered as most water-like in that the ratio of hydroxy or amino groups is high relative to the number of carbon atoms: methanol, ethylene glycol (1,2ethanediol), glycerol (1,2,3-propanetriol), glucose, sucrose, urea, hydrogen peroxide, and ammonia (Table 1). Ammonia is sufficiently weak as an electrolyte (pKb ) 4.75) that the amount of ammonium hydroxide formed does not affect mole fractions significantly. Figure 6 shows a plot of eq 2 for hydrogen peroxide with x calculated by conventional stoichiometry and by eq 1 with the hydration number indicated; the plot is linear to the eutectic point. Data for ammonia show downward deviations prior to the eutectic point in such a plot, and Figure 7 shows two sets of available measurements of freezing point depressions vs reported conventional molalities. The curved line depicts freezing point depressions calculated as a function of such molalities by eqs 1 (h ) 1.8) and 2 (kc ) 1.38 × 10-3 K-1); deviations appear at ∆t > 72 °C. Figures 1-7 demonstrate that reliable hydration numbers cannot be obtained from data in a range of ∆t < 4 °C, as all compounds behave similarly in this small region. As the freezing point depression becomes greater, the determination of h becomes more reliable. Table 1 shows that the method yields linear plots of eq 2, with the proper hydration number and mole fraction from eq 1, over remarkably wide ranges of concentration, more than 13 m ions with NaBr, KI, and HBr or a range 600 times larger than that of data of uni-univalent electrolytes adhering to the Debye-Hu¨ckel theory and 2000 times larger for tri-univalent (FeCl3) electrolytes. With nonelectrolytes, linearity extends to 24 m for hydrogen peroxide and 23 m for ammonia. Expressed as weight percent of solute, the ranges of linearity extend to as high as 52 wt % for KI and 57 wt % for sucrose. Table 1 shows that there are variations in the hydration number obtained for a particular compound, depending on the source of the data and the range of ∆t. This provides an estimate of the accuracy of such values. Standard deviations of individual slopes from the linear regions are mostly below 1%, the largest being 1.4%. The linearity of plots of eq 2 over very wide ranges of concentration demonstrates that data of freezing point depressions of aqueous electrolyte solutions adhere to the requirements of a colligative property, as expressed by eq 2, when they are treated by taking proper account of the “bound” water of hydration by eq 1. Linear plots up to the limit of the eutectic point are found with NaCl, NaBr, KCl, KBr, KI, MgCl2, SrCl2, BaCl2, NH4Cl, and H2O2. Because the fraction of solute in the form of ion pairs must be concentration dependent, as originally proposed by Bjerrum,29 any significant ion pair formation with these electrolytes should have demonstrated itself by a downward curvature as the concentration increases. None is found. Ion pairing and “ionic atmospheres” were proposed as a result of observed curvatures in plots of eq 2 and are not needed in

7808 J. Phys. Chem. B, Vol. 105, No. 32, 2001

Zavitsas

TABLE 1: Freezing Point Depression. Results of Plots of Eq 2 with x from Eq 1 and h Leading to a Straight Line through the Origin. Linear Range, Eutectic Point, Linear Range of Molality, Linear Range of ∆t °C, Slope, Hydration Number, Correlation Coefficient, and Source of Data linear to wt % HCl HCl HCl HBr HI LiCl LiCl LiCl NaCl NaCl NaCl NaCl NaBr NaBr NaI KCl KCl KCl KBr KBr KBr KI KI CsCl NH4Cl NH4Cl NH4Cl MgCl2 MgCl2 MgCl2 MgCl2 CaCl2 CaCl2 CaCl2 SrCl2 SrCl2 SrBr2 BaCl2 BaCl2 BaBr2 AlCl3 FeCl3 FeCl3 CH3OH CH3OH C2H4(OH)2d C3H5(OH)3e C3H5(OH)3e C6H7O(OH)5f C6H7O(OH)5f C12H14O3(OH)8h C12H14O3(OH)8h HOOH NH3 NH3 NH3 NH3 CO(NH2)2k

12.00 12.72 19.00 35.76 39.01 14.00 14.50 15.67 23.00 23.31 23.00 23.31 17.00 40.30 23.06 19.74 18.00 19.22 32.00 31.34 31.34 40.00 52.21 20.00 13.00 19.68 19.68 21.87 20.00 21.87 16.18 22.00 18.16 24.00 24.00 26.21 33.10 21.89 22.00 46.59 22.04 24.00 24.50 38.00 24.26 40.00 40.00 31.53 30.00 24.47 42.00 57.79 45.00 28.00 25.41 25.41 28.00 32.00

eutectic point, wt %a 24.35 49.39

23.31 23.31 40.30 19.74 20.87 31.34 31.34 52.21 19.68 19.68 21.87 21.87 32.43 26.21 41.69 21.89 46.59 33.10

56.14

45.00 33.82 33.24

linear to mb

linear to ∆t

slope, ×103

h

r

ref

7.48 7.99 12.87 13.76 10.00 7.68 8.00 8.77 10.22 10.40 10.22 10.40 3.98 13.12 3.99 6.61 5.89 6.37 7.91 7.67 7.67 8.03 13.16 2.97 5.59 9.16 9.16 8.82 7.88 8.82 6.08 7.62 6.00 8.54 5.98 6.72 6.00 4.04 4.06 8.81 8.48 7.79 8.00 19.13 10.00 10.74 7.24 5.00 2.38 1.80 2.12 4.00 24.05 22.83 20.00 20.00 22.83 7.84

20.51 22.60 47.32 64.67 38.50 21.04 22.24 25.44 20.67 21.12 20.86 21.12 7.32 28.01 7.94 10.69 9.60 10.34 12.98 12.60 12.60 13.97 23.03 4.49 9.47 15.36 15.36 33.52 29.01 33.50 18.39 21.70 15.36 26.06 14.99 18.70 16.70 7.50 7.64 22.61 44.94 23.79 24.90 36.60 20.00 23.84 15.60 10.50 4.79 3.55 4.93 10.80 53.40 71.66 58.00 58.00 71.47 12.34

1.33 1.33 1.28 1.38 1.37 1.31 1.32 1.38 1.19 1.19 1.19 1.17 1.24 1.28 1.24 1.21 1.23 1.19 1.22 1.23 1.21 1.24 1.25 1.17 1.22 1.24 1.22 1.31 1.22 1.36 1.21 1.20 1.14 1.29 1.15 1.14 1.34 1.11 1.07 1.21 1.11 1.09 1.13 1.39 1.37 1.37 1.40 1.39 1.38 1.38 1.39 1.42 1.39 1.38 1.40 1.40 1.36 1.35

6.9 6.8 6.3 6.5 7.5 7.0 6.8 6.3 3.8 3.8 3.9 4.0 4.0 3.3 5.5 1.5 1.2 1.8 1.6 1.5 1.7 2.2 2.0 -0.2 2.3 1.7 1.9 12.7 14.3 12.4 15.2 12.3 13.6 11.2 13.4 13.4 12.4 10.5 11.9 9.4 21.8 18.2 17.5 0.9 1.3 1.8 1.9 2.1 2.8 2.8 6.1 5.0 1.2 1.80 1.85 1.79 1.82 -0.2

0.999 925 0.999 981 0.998 931 0.999 957 0.999 644 0.999 537 0.999 642 0.999 471 0.999 942 0.999 943 0.999 951 0.999 961 0.999 943 0.999 995 0.999 977 0.999 971 0.999 806 0.999 987 0.999 995 0.999 988 0.999 991 0.999 995 0.999 961 0.999 987 0.999 985 0.999 999 0.999 863 0.999 370 0.999 676 0.999 136 0.999 881 0.998 982 0.999 952 0.999 085 0.999 770 0.999 899 0.999 964 0.999 976 0.999 953 0.999 774 0.999 933 0.999 031 0.999 439 0.999 472 1.000 00 0.999 856 0.999 889 0.999 982 0.999 984 0.999 981 0.999 999 0.999 867 0.999 957 0.999 955 0.999 935 0.999 999 0.999 667 0.999 993

35 36 37 36c 36 35 36 38 35 36 37 38 35 36 36 36 37 38 35 36 38 35 36 35 35 36 38 36 37 38 39 35 36 37 35 36 36 36 37 36 36 35 36 35 36 35 35 36 35 40ag 35 36 36 35 36i 36j 37 35

a When specified. b Sum of molalities (m) of the ions for strong electrolytes. c One apparently erroneous value at 24.45 wt % (8.00 m) was not used. d Ethylene glycol. e Glycerol. f Glucose. g This set of data was used, despite ∆t < 4 °C, because of high accuracy. h Sucrose. i p 261. j p 255. k Urea.

the absence of curvature, especially when the theoretical slopes, or values near them, are obtained. Compounds that show downward curvatures prior to a known eutectic point are CaCl2, HCl (Figure 4), HI, FeCl3, glycerol, and ammonia (Figure 7). For electrolytes, it is tempting to ascribe these curvatures to onset of ion pair formation, but there is no proof that this is the case. An alternative explanation is more likely for both electrolytes and nonelectrolytes. Taking HCl as an example, we

calculate that there is not enough water at the eutectic point (24.35 wt % or 8.83 m HCl) to satisfy a hydration number of about 7. A constant value of h ) 7 ties up in hydration all the water at a point equal to the ratio of the molality of pure water to the hydration number, 55.51/7 ) 7.93 m HCl, i.e., before the eutectic point is reached. Linearity is actually followed to 6.43 m HCl, at which point 81% of the water is “bound”. With HI at the eutectic point, 96.2% of the water would be tied up to

Properties of Water Solutions

Figure 2. NaCl freezing point depression, plot of eq 2: open symbols, x from conventional stoichiometry, ref 38, disregarding “bound” water; filled symbols, x from eq 1 with h ) 4.0.

Figure 3. MgCl2 freezing point depression, plot of eq 2: open symbols, x from conventional stoichiometry, ref 38, disregarding “bound” water; filled symbols, x from eq 1 with h ) 12.4.

satisfy h ) 7. With CaCl2 at the eutectic point, 93% of the water would be tied up with constant h ) 12. Similarly, with FeCl3 and h ) 18, 98.9% of the water would be tied up, and with ammonia and h ) 1.8, 97.4% of the water would be tied up. As these solutions run out of water solvent, a decrease in h is expected from equilibrium considerations, with a concomitant downward curvature in plots of eq 2. Hydration numbers at the eutectic point can be calculated from freezing point measurements by solving eq 2 for x with Tf ) Teut and with kc from Table 1 and then substituting into eq 1 and solving for h. The following are found: CaCl2 (Teut ) 222.16 K) h ) 10.1, HCl (Teut ) 187.15 K) h ) 5.50, HI (Teut ) 193.04 K) h ) 6.3, FeCl3 (Teut ) 218.25 K) h ) 15.0, and ammonia (Teut ) 162.15 K) h ) 1.65. As would be expected, shortage of “free” water leads to lower hydration numbers. Among the compounds that

J. Phys. Chem. B, Vol. 105, No. 32, 2001 7809

Figure 4. HCl freezing point depression, plot of eq 2: open symbols, x from conventional stoichiometry, ref 37, disregarding “bound” water; filled symbols, x from eq 1 with h ) 6.3. The rightmost three points were not used for statistics of linearity.

Figure 5. LiCl freezing point depression, plot of eq 2: open symbols, x from conventional stoichiometry, refs 35, 36, and 38, disregarding “bound” water; filled symbols, x from eq 1 with h ) 6.6.

show linear plots to the eutectic point, MgCl2 ties up the most water, 69%. The data for AlCl3 adhere to linearity up to the highest reported concentration (2.1 m AlCl3), where 83.3% of the water is “bound” in hydration. Glycerol is the only case where curvature begins prior to a known eutectic point with only 26% of the water “bound” in hydrating solute. We conclude that this indicates onset of bonding between alcoholic OH groups. This conclusion is supported by the fact that methanol, ethylene glycol, and glycerol show onset of curvature at about the same concentration of R-OH groups: 19.1 m for methanol, 2 × 10.74 ) 21.5 for ethylene glycol, and 3 × 7.24 ) 21.7 for glycerol. Eutectic points are not reported in our data sources35-39 for the remaining compounds.

7810 J. Phys. Chem. B, Vol. 105, No. 32, 2001

Figure 6. HOOH freezing point depression, plot of eq 2: open symbols, x from conventional stoichiometry, ref 36, disregarding “bound” water; filled symbols, x from eq 1 with h ) 1.2.

Figure 7. NH3 freezing point depression vs molality: open symbols, m from conventional stoichiometry disregarding “bound” water (circles, ref 37; triangles, ref 35). The line is ∆t °C calculated from m by eq 1 (h ) 1.80) and eq 2 (kc ) 1.38 × 10-3).

The dichlorides of manganese, cobalt, nickel, copper, and zinc show the typical upward curvatures of the other compounds examined and can be treated by the method used here. The nature of water as solvent, i.e., “bulk” water in solutions, is revealed by examining the slopes obtained in plots of eq 2. For “ideal” solutions, the cryoscopic constant is kc ) 1.384 × 10-3 K-1 from thermodynamic considerations,24,33 assuming that the heat of fusion of water remains fairly constant over the temperature range involved. The mean value of kc from all 15 entries of nonelectrolytes in Table 1 is (1.383 ( 0.014) × 10-3 K-1. This demonstrates that the nature of “bulk” water, or free solvent, is not changed by very high concentrations of these nonelectrolytes; similar conclusions have been reached

Zavitsas regarding the “activity” of water for solutions of saccharides40a and polyols.40b With nonelectrolytes, the fact that the hydration number that produces a linear plot of the initially curved data through the origin also produces the theoretical slope ensures the validity of the method for these compounds. The slopes of linear plots of eq 2 for the electrolytes of Table 1 are generally smaller, as low as 1.11 × 10-3 K-1 for AlCl3 and FeCl3. The average value of kc for the 41 other electrolyte entries in Table 1 is (1.24 ( 0.06) × 10-3 K-1, about 10% lower than the theoretical value. Because cryoscopic constants (like melting or boiling points) are characteristic of different solvents, this must indicate that the nature of “bulk” water as solvent is changed by strong electrolytes. Organic chemists take advantage of this fact for “salting out” of water solutions partially miscible organic compounds. For example, the solubility of diethyl ether in pure water is 0.91 mol L-1 (25 °C), but only 0.13 mol L-1 in a 15 wt % sodium chloride solution.24 At this concentration of salt and with a hydration number of 4, we calculate that the amount of available “bulk” water decreases by 22%, while the solubility of ether decreases by 86%. It is not only a matter of less free solvent being available;24 the nature of the solvent has changed. X-ray diffraction measurements of Al3+ solutions “clearly show that drastic changes in the [bulk] water structure have occurred compared with pure water. One interesting observation is that, relative to pure water, the firstneighbor O-O distances are shortened remarkably, even in the more dilute solutions.”41 Freezing point data for NaCl show that the slope of eq 2 changes very quickly, and therefore, so does the nature of the “free” solvent. Compared to the theoretical value of kc ) 1.38 × 10-3 K-1, kc is 1.35 × 10-3 in the interval of 0.000-0.034 m ions, already beyond the Debye-Hu¨ckel domain; kc continues to decrease to 1.29 × 10-3 in the interval of 0.034-0.069 m. In the interval of 0.27-0.31 m, or -ln(1 x) ≈ 0.005, kc is about 1.2 × 10-3 K-1 and remains quite constant thereafter to the eutectic point at 10.4 m ions, or -ln(1 - x) ≈ 0.26 (see Figure 2). This region of initial downward curvature is too small to be discernible on the scale of the figures shown. Data of ∆t < 0.5 °C (1/Tf - 1/To < 0.06 × 10-4 in Figures 2-5) for electrolytes were not included in the calculations, and neither was the origin included. Hydration numbers in Table 1 are defined in terms of the compound. We conclude that it is justifiable to make the approximation that for electrolytes, h refers almost exclusively to the cation, assigning h ≈ 0 to the three halide anions, for the following reasons. Similar hydration numbers are obtained for the three different halides of each cation. For sodium, h(NaCl) ) 3.8, 3.8, 3.9, 4.0; h(NaBr) ) 4.0, 3.3; h(NaI) ) 5.5. For potassium, h(KCl) ) 1.5, 1.2, 1.8; h(KBr) ) 1.5, 1.7; h(KI) ) 2.2, 2.0. For strontium, h(SrCl2) ) 13.4, 13.4; h(SrBr2) ) 12.4. For barium, h(BaCl2) ) 9.8, 10.5, 11.9; h(BaBr2) ) 9.4. For the haloacids, h(HCl) ) 6.9, 6.8, 6.3; h(HBr) ) 6.5; h(HI) ) 7.5. Therefore, the hydration numbers of the three halides have to be quite similar, if not identical. This, coupled with the fact that h(CsCl) ≈ 0 (Table 1), indicates minimal or zero hydration of these anions. Chloride ion has been thought to exhibit “weak effects of negative hydration”.20c Far-infrared and dielectric relaxation studies led to the conclusion that for chloride and bromide ions, “the interaction with water is weak compared to alkali metal ions”.20d Halide ion-water interaction, excluding fluoride, is weaker than that of the majority of monovalent cations and comparable to water-water interactions in the bulk and in clusters,42 and “the anion’s role is small and waterwater interactions are determining when and how the cluster melts”.43a The residence times of water molecules in the

Properties of Water Solutions solvation shell around Cl- and in the bulk are found by NMR to be comparable, and this, along with other evidence, has led to the assignment h(Cl-) ) 0 in dielectric permittivity20a-c and diffusion studies.44 From purely electrostatic considerations, water is better at stabilizing cations than anions because the partial negative charge on oxygen is twice the positive charge on either hydrogen. The assignment of h ≈ 0 for chloride, bromide, and iodide does not imply that these ions are not solvated by water. In fact, there is extensive experimental evidence that they are (from clusters, infrared studies, scattering, etc.), but such solvating water molecules are not “binding to solute sufficiently strongly as to become part of it” and to be unable to act as bulk solvent. While the terms “hydration number” and “solvation number” are often used interchangeably, our definition makes a clear distinction. Not every water molecule interacting with solute, distinguishable in some way45 from those in the bulk solvent, constitutes part of the hydration number defined here. Also, “coordination” numbers with species other than water are not necessarily related to hydration number. Approximating h ≈ 0 for the halide ions examined and considering variations in hydration numbers obtained from different sources in Table 1 and the range of linear ∆t in each case, we estimate values of h and of a somewhat subjective uncertainty as follows: H+, 6.7 ( 0.7; Li+, 6.6 ( 0.6; Na+, 3.9 ( 0.5; K+, 1.7 ( 0.5; Cs+, 0.0 ( 1; NH4+, 1.8 ( 0.5; Mg2+, 13 ( 2; Ca2+, 12 ( 2; Sr2+, 12 ( 2; Ba2+, 10.5 ( 1.5; Al3+, 22 ( 2; Fe3+, 18 ( 2; methanol, 1.0 ( 0.3; ethylene glycol, 1.8 ( 0.3; glycerol, 2.0 ( 0.5; glucose, 2.8 ( 0.5; sucrose, 5.0 ( 0.5; hydrogen peroxide, 1.2 ( 0.2; ammonia, 1.81 ( 0.05; and urea, 0.0 ( 0.5. Fractional values of h can be understood in terms of the definition used. It is an average number in a dynamic process, and there is a distribution of hydrated species (see below). Hydration numbers obtained here for Li+, Na+, K+, Cs+, Ca2+, and Ba2+ are in remarkably good agreement with the numbers of water molecules found to coextract with these cations from water into an organic environment (nitrobenzene containing organic anions): 6.0 ( 0.4, 3.8 ( 0.3, 1.0 (1.346), 0.4, 14 ( 2, and 11 ( 2, respectively.15 Hydration numbers of 4.2 ( 0.3 for Na+ and 1.3 for K+ from dielectric relaxation studies20a are also in agreement and so are values of 3.8 for Na+ (under one set of assumptions) from microwave frequency studies20c and 3.9 from isothermal compressibilities.21b The h values obtained for Na+ and K+ have a direct bearing on the question of the mechanism by which K+ channel proteins are more permeable to K+ than to Na+ by factors of 103-104, despite the chemical similarities of the two ions and the smaller size of Na+.1c,2 Scattering studies indicate similar primary hydration shells. Studies by Lisy et al. of mixed water-benzene clusters showed that benzene is capable of replacing water from the first hydration shell of K+, but not from that of Na+, which retains all of its four water molecules.1c In effect, not only can K+ shed enough water to become smaller than Na(H2O)4+, but it can also free itself to interact with either the π cloud of aromatic rings in the selectivity region of the channel1c or carbonyl functions therein.47 The sodium ion is, in effect, insulated from such interactions. Values of h ) 3.9 for Na+ and h ) 1.7 for K+ are in agreement and provide good support for the reasons proposed as contributing to the high selectivities of K+ channels. Ion transport, however, is not necessarily or primarily passive. Energy is usually expended, often in intriguing fashion.48 Energy expenditure per gram of kidney tissue exceeds that of heart muscle.

J. Phys. Chem. B, Vol. 105, No. 32, 2001 7811

Figure 8. LiCl freezing point depression, plot of eq 2: open symbols, x from conventional stoichiometry, refs 35, 36, and 38, disregarding “bound” water; filled symbols, x from eq 1 with h ) 4.0 from ab initio calculations of ref 14.

The high values of h found for Mg2+, Al3+, and Fe3+ are in good agreement with the values of 12, 22, and 19, respectively, reported on the basis of diffusion coefficients of their chlorides with h(Cl-) ) 0.44 This large number of water molecules is unlikely to be accommodated directly around the cations, and X-ray diffraction has shown a highly ordered second hydration shell around Al3+.41 Also consistent with the value of h(Mg2+) are efflorescence-deliquescence measurements, in which the dodecahydrate is stable at the temperature range of the freezing point data.7d For ammonia, h ) 1.81 and the dihydrate is known. Solid NH3(H2O)2 was recently found to constitute more than 90% of the ammonia on the surface of Pluto’s satellite Charon at 50 K.9c For urea, h is essentially zero (-0.2), consistent with the conclusion reached from NMR studies of 2H spin lattice relaxation to the effect that “it is difficult to classify urea as a structure maker or breaker”;50 also, water molecules in close proximity to urea were described as “maintaining their liquid structure”.51 Infrared spectroscopy in argon matrixes has shown that hydrogen peroxide forms 1:1 cyclic complexes with water,51 consistent with h ) 1.2 ( 0.2. The above agreements between other techniques and our results are likely due to the fact that users of those techniques often define “hydration number” in ways similar to the one used here. There is broad disagreement between hydration numbers obtained in this work and values from some other techniques. For example, diffraction (neutron, X-ray) and extended X-ray absorption fine structure (EXAFS) studies yield hydration numbers of 4-8 for Li+, K+, and NH4+; 6-8 for Cs+, Mg2+, and Ca2+; 4-8 for Cl-; 4-6 for Br-; etc.11 There is current interest52 in the correct hydration number of Li+, and recent work favors a mean value of 4 from high-level ab initio calculations for Li+(aq) at infinite dilution under normal conditions at 25 °C,14 in agreement with spectroscopic evidence, as opposed to 6 from neutron scattering or 8 from other theoretical calculations in conjunction with IR spectroscopy.13 Figure 8 shows a plot of eq 2 for the LiCl freezing point data, with x obtained from eq 1 with the ab initio value of h ) 4. The straight line is the resulting first-order regression fit,

7812 J. Phys. Chem. B, Vol. 105, No. 32, 2001 constrained through the origin, to the filled symbols. A comparison with the equivalent Figure 5 (h ) 6.6) shows that the freezing point measurements between 0 and -25 °C are not in agreement with a value of h ) 4. A curve in the filled symbols is evident in Figure 8, and the slope of 1.68 × 10-3 is much too high relative to those of all other electrolytes in Table 1 and to the theoretical value. The first hydration shell of Li+ may contain four water molecules, but it was noted that a strongly structured second hydration shell has been suggested.14 Neutron diffraction experiments with LiBr solutions have been interpreted53 as indicating h(Li+) ) ca. 6 with a 10 mol % solution (6.17 m LiBr), decreasing to ca. 4 with a 33 mol % solution (27.3 m LiBr), with suggested solvent-shared ion pairs in the latter. At 10 mol % and h ) 6.6, 73.4% of the total water is “bound” in hydration. At 33 mol %, 27.3 × 6.6 ) 180 mol of H2O would be required for complete hydration and only 55.51 are available, insufficient to accommodate even an average h ) 4 for each Li+. In the absence of any “free” water, low hydration numbers must be obtained and hydrates of Li+ must solvate the anion. The maximum amount of Li+ that can be hydrated with constant h ) 6.6 is 55.51/6.6 ) 8.4 m. At 8.6 m LiCl, X-ray scattering54 indicates that solvent-shared complexes begin appearing, and at 14.9 m, such structures are quite extensive, but they were estimated not to exist below 5 m Li+. Consistent with this, there is no curvature in properly treated (eq 1 with h ) 6.6) freezing point data that are available up to 4.4 m Li+ (Table 1 and Figure 5). Cluster studies yield results consistent with those obtained from freezing point measurements. The definition of hydration number used here refers to water “binding to solute sufficiently strongly as to become part of it”. What energy is sufficient to remove a water molecule from the “bulk”? A lower limit is the heat of vaporization of water: 10.7 kcal mol-1 at 0 °C. This will provide somewhat of an underestimate of attractive forces between a water molecule and surrounding water because a molecule at the evaporative surface is not completely surrounded. Sequential energies have been measured with gas-phase ionwater clusters for reactions M(H2O)nz+ f M(H2O)n-1z+ + H2O for many cations and many n. As n increases, the exothermicity of the reaction,2b-d or the binding energy of the last water molecule,3 generally decreases. These two related thermodynamic quantities from cluster studies correlate with hydration numbers obtained from freezing points. Sequential enthalpies, ∆Hrxn, have been measured by Kebarle et al.2c,55 by van’t Hoff plots of equilibria of gas-phase ion-water clusters in a highpressure mass spectrometer (HPMS). Armentrout et al. have measured sequential binding energies by collision-induced dissociation (CID) measurements on such clusters.3 We find that the values of h from freezing points correspond to the highest n cluster value for binding with more than 13.3 kcal mol-1. Table 2 shows the relevant data for such n ) h, within the uncertainty of the latter; column 4 gives the sequential energy (all above 13.3), while column 5 gives the energy for the next higher n (all below 13.3). Thus, two quite different experimental techniques yield consistent results. Subsequent measurements56,57 have refined some of the earlier2c,55 values, and a binding energy of approximately 13 kcal mol-1 is “sufficient” in terms of our definition of hydration. A value of 13.4 kcal mol-1 reported as the energy of activation for desorption of water from pure ice at temperatures below -100 °C58 happens to be remarkably similar. Castleman et al.18 have reported the relative abundance of protonated gas-phase water clusters, H(H2O)n+. In the region

Zavitsas TABLE 2: Hydration Numbers (h) from Freezing Points and Sequential Energies from Gas-Phase Clusters for M(H2O)nz+ f M(H2O)n-1z+ + H2O Mz+ in M(H2O)nz+

h

n

H+ Li+ Na+ K+ Rb+ Cs+ Mg2+ Ca2+ Sr2+ Ba2+

6.7 ( 0.7 6.6 ( 0.6 3.9 ( 0.5 1.8 ( 0.5 1.8 ( 0.3d 0.6 ( 0.8d 13 ( 2 12 ( 2 12 ( 2 10.5 ( 1.5

6 6 4 2 2 1 11 11 12 11

energya energya,b nfn-1 n+1fn (kcal mol-1) (kcal mol-1) 16.4 13.6c 13.8 16.1 13.6 13.7 14.3 13.3 13.7 13.4

12.3 13.1 13.2 12.2 12.5 12.9 13.0 13.0 12.4

ref 7e 3a, 55a 55a 55a 55a 55a 2c 2c 2c 2c

a Values from refs 2c and 55 are enthalpies of the reaction at 298 K; values from refs 3 and 7e are binding energies. b A value of K+ ) NH4+ > Na+ > Li+; with divalent cations, the order is Ba2+ > Sr2+ ) Ca2+ > Mg2+> Be2+ (see below); with trivalent cations, the order is Al3+ > Fe3+. Cl-, Br-, and Ihave essentially identical Λo, 76.3, 78.1, and 76.8 (in 10-4 m2 S mol-1), respectively, despite their different crystal radii. Viscocities of aqueous solutions of the chlorides of many of the cations of Table 1 have been measured,63 and they follow the trend of h values. For 1.5 M solutions at 20 °C, the cation, viscocity (cP), and h are as follows: (K+, 0.98, 1.8) < (Na+, 1.13, 3.9) < (Ba2+, 1.44, 10.5) < (Sr2+, 1.52, 12) ≈ (Ca2+, 1.54, 12) < (Mg2+, 1.79, 13). Conclusions reached here from freezing points regarding hydration numbers, the absence of ion pairs, and changes in the nature of “free” water as solvent are valid for the solutions considered, for concentrations covered by linear plots of eq 2, and for the temperatures of the freezing point measurements. Although some of these conclusions may have general applicability, this is not claimed here. Boiling Point Elevation and Depression of Vapor Pressure The temperature range, ∆t, of boiling point elevations is smaller than that of freezing point depressions, and the number of available measurements beyond a ∆t of 3 °C, needed for obtaining a reliable h value, is also more limited.64 Measure-

ments of vapor pressure depressions also often are more limited in the number of concentrations studied. As a result, hydration numbers derivable from these two related properties have greater uncertainties, at least (1.0, and there is greater variation in h values obtained from different sources. Despite these limitations, examination of vapor pressure and boiling point measurements is useful. It reveals a temperature dependence of some hydration numbers, yields h values for Cs+, Rb+, and Be2+, and provides additional evidence for changes in the nature of “free” water solvent in electrolyte solutions, compared to that of pure water. The vapor pressure of solvent over a solution is expected to be a colligative property, as expressed by Raoult in 1886, eq 3.

p/po ) (1 - x)

(3)

The vapor pressure of pure water is po, and p is its vapor pressure over a solution of x molar fraction of solute at the same temperature. Even though the heat of vaporization is temperature dependent, it is not a factor in eq 3. The fractional decrease of vapor pressure should be equal to the mole fraction of remaining “free” water molecules. As with freezing point data, measurements of vapor pressures of water over concentrated solutions seldom yield straight lines that pass through the origin (1,1) in plots of Raoult’s law. Table 3 shows results with vapor pressure depression data at selected temperatures. Linearities produced

7814 J. Phys. Chem. B, Vol. 105, No. 32, 2001

Zavitsas TABLE 4: Boiling Point Elevation. Results of Plots of Eq 4, with x from Eq 1 and h Leading to Straight Line through the Origin. Linear Range of Molality, Linear Range of ∆t °C, Slope, and Hydration Number

Figure 9. NaI vapor pressure depression at 100 °C, plot of eq 3: open symbols, x from conventional stoichiometry, ref 64a, disregarding “bound” water; filled symbols, x from eq 1 with h ) 4.5.

LiCl LiBr LiBrc LiI NaCl NaI KCl KBr KI RbCl CsCl NH4Cl NH4Clc NH4I MgCl2 CaCl2 SrCl2 SrBr2 glycerol sucrose urea

linear to ma

linear to ∆t

slope, ×104

hb

10.00 10.00 9.87 10.00 13.36d 10.00 10.00 8.80 26.60d 24.34d 34.40d 20.00 19.26 18.00 6.00 15.00 15.00 9.00 6.17 5.00 2.50

6.61 7.85 6.389 5.371 8.66 7.222 5.206 4.795 18.6 13.50 19.9 9.321 9.315 9.262 4.35 15.1 11.57 6.26 3.50 3.818 1.233

1.80 1.95 1.96 1.67 1.88 1.91 1.85 1.86 2.11 1.82 1.91 1.83 1.61 1.90 1.59 1.72 1.82 1.55 2.05 2.09 1.91

4.3 5.0 4.0 3.3 3.1 4.6 2.0 2.8 1.8 1.6 1.3 0.9 1.8 1.5 13.8 7.5 5.8 9.5 0.5 4.0 1.1

a Sum of the molalities (m) of the ions for strong electrolytes. b Data from ref 64c, unless otherwise indicated. c Reference 37. d Saturated solution.

by the correct h value are satisfactory, as shown in Figure 9 for NaI. All correlation coefficients of the straight line through the origin are greater than 0.99; all slopes are 1.00 ( 0.02 (see Supporting Information). Boiling point elevation measurements also show upward curvatures when plotted according to eq 4, a van’t Hoff-type plot derived from thermodynamic considerations.24 Tb ) 373.15

1/Tb - 1/(Tb + ∆t) ) -ke ln(1 - x)

(4)

K, the boiling point of water, and ∆t is the observed boiling point elevation. The theoretical slope is the ebullioscopic constant, ke ) R/∆Hvap ) 2.05 × 10-4 K-1, where ∆Hvap is the heat of vaporization of water at the boiling point, 9708 cal mol-1. Table 4 shows the results, and Figure 10 is a typical plot. The overall behavior of the slopes is similar to that of the slopes of freezing point measurements. The average slope of the three nonelectrolytes is (2.02 ( 0.07) × 10-4, near the theoretical value of ke, and indicates that the nature of “free” water as solvent is not changed. The electrolytes generally have lower values of the slope with more scatter for an average of (1.83 ( 0.13) × 10-4, about 11% lower than the theoretical value, as found with the cryoscopic constants. Table 4 includes four cases of linearity extending fully to the saturation point. The three halides of the same cation again yield similar hydration numbers at the same temperature. Tables 3 and 4 show that h values are a function of temperature for some of the solutes. Averaging vapor pressure values of h between 0 and 30 °C in Table 3 results in the following: h(Li+) ) 6.0 ( 0.5; h(Na+) ) 3.5 ( 0.5; h(K+) ) 1.3 ( 0.4; h(NH4+) ) 1.4 ( 0.4; h(Mg2+) ) 11.7 ( 0.3; despite the precisions of these averages, the limitations in the data do not warrant an expectation of accuracy better than (1 unit of h. Values of h between 0 and 30 °C in Table 3 are lower than those obtained from freezing points at various temperatures below 0 °C (Table 1), but each is within the specified limits of uncertainty. Decreases in hydration number can be seen more clearly from vapor pressure measurements at 100 °C and from boiling point elevations. Averaging h values from Table 3 (at 100 °C) and from Table 4 (up to 120 °C) results in the following: h(Li+) )

Figure 10. NaCl boiling point elevation up to saturated solution, plot of eq 4: open symbols, x from conventional stoichiometry, ref 64c, disregarding “bound” water; filled symbols, x from eq 1 with h ) 3.2.

4.8 ( 0.7; h(Na+) ) 3.6 ( 0.7; h(K+) ) 2.1 ( 0.3; h(NH4+) ) 1.3 ( 0.4; h(Mg2+) ) 11.2 ( 1.8; h(Ca2+) ) 8.5 ( 1.0; h(Sr2+) ) 7.3 ( 1.0; h(Ba2+) ) 7.0 ( 0.4; h(Al3+) ) 14.3 ( 0.6. Some, but not all, of the hydration numbers near 100 °C in Tables 3 and 4 are clearly lower than those from freezing point measurements. Values of h for Na+, K+, and NH4+ show no clear change. Temperature dependence is most clearly seen with Al3+, where h decreases from 22 at -45 °C to 14 at 100 °C (Tables 1 and 3); hydrolysis of AlCl3 at 100 °C may contribute to this decrease. The temperature effect, however, is not necessarily major as seen with Fe3+; h decreases from 18 at -25 °C to 17 at 15 °C (Tables 1 and 3). Sucrose also has a discernible decrease from h ) 5 at -10 °C to h ) 4 near 100 °C (Tables 1, 3, and 4). Changes in h with temperature are

Properties of Water Solutions

J. Phys. Chem. B, Vol. 105, No. 32, 2001 7815

expected from thermodynamic considerations of energy changes that govern the equilibria in the competition for “monomeric” water by assemblies containing solute and by “bulk” water. A relative change of 0.2 kcal mol-1 in ∆G would produce the order of magnitude of the changes found. ∆Hvap(H2O) decreases by 1 kcal mol-1 from 0 to 100 °C, and there is no a priori reason for h to necessarily decrease with increasing temperature; this would depend on relative temperature-dependent changes in binding energies Vis-a` -Vis ∆Hvap. For example, for HCl, there is no indication of a decrease in h with increasing temperature, 6.7 ( 0.7 near -40 °C and 7.6 ( 1.0 at 25 °C (Tables 1 and 3). Vapor pressure measurements of solutions of beryllium salts at 100 °C yield h(Be2+) ) 11.5 ( 0.9, somewhat higher than h(Mg2+) ) 9.9 ( 0.2 at the same temperature (Table 3) and consistent with their relative equivalent ionic conductivities. A value of h(Rb+) ≈ 2 was expected from sequential binding enthalpies55a (Table 2). Sufficient freezing point measurements are not available, but the four h values listed in Tables 3 and 4 support this expectation, with an average of 1.8 ( 0.3. A value of h(Cs+) ≈ 1 was similarly indicated, and the freezing point data were too limited to provide an accurate determination, but the two h values from Tables 3 and 4 yield 0.6 ( 0.8, which is also consistent with 0.4 from the nitrobenzene extraction work. Distributions of ion-water gas-phase clusters have been reported by Kebarle et al.65 for several ions at 27 °C. Weighted averages of water molecules can be calculated from the distributions as follows:59 Na+, 3.1; K+, 2.0; Rb+, 1.4; Mg2+, 9.9; Ca2+, 9.5; and Sr2+, 9.4. The values of h obtained from measurements above 20 °C in Tables 3 and 4 are in fair agreement, within their scatter and the (1 uncertainty in accuracy of all h values of this section. It is significant that vapor pressure measurements for both electrolyte and nonelectrolyte solutions adhere to Raoult’s law with the same slope of unity, in contrast to freezing point depression or boiling point elevation, where the slopes from electrolytes are about 10% smaller than the theoretical value, as adhered to by the nonelectrolytes. Equation 3 for vapor pressures does not contain a term related to a physical property of water, such as ∆Hfus or ∆Hvap. The theoretical value of the slope in eqs 2 and 4 is calculated from physical properties of pure water and not of “free” water in electrolyte solutions. We do not have data to prove the conjecture that ∆Hfus and ∆Hvap may be about 10% greater for “free” water solvent in electrolyte solutions than those of pure water, but they would be different if the nature of “free” water in electrolyte solutions is altered, as proposed here and by others.41 This would account for observed slope differences between electrolyte and nonelectrolyte plots of freezing and boiling point data. Osmotic Pressure Osmotic pressure appears to have first been described in 1748 by the Abbe´ Nollet.24 The studies of van’t Hoff in 1885 established a relationship between osmotic pressure, π, and concentration for very dilute solutions, π ) cRT, where c is the concentration in mol L-1. From thermodynamic considerations,24,33 the relationship is given by eq 5. The theoretical slope

π ) -kpT ln(1 - x)

(5)

(piezoscopic constant) is kp ) R/V ) 4.555 atm K-1, where R ) 0.082 05 L atm K-1 mol-1, V is the molar volume of water, 0.018 015 L, and π is expressed in atmospheres. Osmotic pressure data producing π > 25 are needed to establish reliable

Figure 11. Sucrose osmotic pressure at 0 °C: open symbols, experimental values vs molal concentrations, ref 36, disregarding “bound” water. The line was calculated by eq 5 with the theoretical value of Tkp ) 1244.1 atm and x from eq 1 with h ) 5.

hydration numbers, and they are available for glucose and sucrose.36 The data show the typical upward curvatures in plots of π vs -ln(1 - x). The hydration number in eq 1 that produces a zero intercept also brings the experimental points to linearity and produces slopes near the theoretical value of kp. Measurements of glucose solutions at 0 °C adhere to linearity up to 3.48 m (87.9 atm) with kp ) 4.47 and h ) 2.6, consistent with h ) 2.8 ( 0.5 from freezing points. Sucrose solutions at 0 °C adhere to linearity up to 4 m (129.7 atm, all available measurements) with kp ) 4.56 and h ) 4.9, consistent with h ) 5.0 ( 0.5 from freezing points. Sucrose solutions at 30 °C adhere to linearity up to 5.0 m (193 atm, all available measurements) with kp ) 4.58 and h ) 4.4. Sucrose solutions at 57.7 °C adhere to linearity up to 5.0 m (195.5 atm) with kp ) 4.61 and h ) 3.8. Linearity to pressures up to 195 atm (2900 lb in-2) is remarkable. Correlation coefficients for all the above linear ranges of osmotic pressure data are greater than 0.999. With both glucose at 0 °C and sucrose at 57.7 °C, the highest measurement available (4.67 and 6.0 m, respectively) deviates downward from the straight line through the origin by 4.9% and 4.1%, respectively. These deviations probably indicate the onset of bonding between R-OH groups, as found with freezing point data of other alcohols. In addition, a 6 m solution of sucrose requires 2054 g to be added to 1.0 kg of water (67.3 wt % solute). With all that sugar, a sticky situation is to be expected. A form of eq 1 has been used previously for treating osmotic pressure data of sucrose solutions at 0 °C, with a hydration number of 5.66 Osmotic pressures (or their equivalent) of nonelectrolyte solutions are being used by biologists and biochemists in the “osmotic stress” method to calculate changes in the amount of “free” water in reactions of biological interest, e.g., in intercalation reactions of antibiotics with DNA.5 It has been established that the nature of “free” water solvent is not altered by sucrose. In principle, this should allow the prediction of osmotic pressures of sucrose at 0 °C by eqs 5 and 1 with h ) 5 from freezing points and the theoretical value of the piezoscopic constant. Experimental values are compared to such predictions in Figure 11. The agreement is excellent.

7816 J. Phys. Chem. B, Vol. 105, No. 32, 2001 Other Compounds Alkali metal hydroxides and fluorides are not well behaved by the criteria of the method used here. For example, the h value that produces a straight line through the origin with KF and NaOH data does not remove all the curvature from their plots. From sequential binding enthalpies of gas-phase water clusters,55b,c h(F-) ≈ 3-4 and h(HO-) g 5 can be inferred, but freezing point and vapor pressure data for KF, KOH, NaOH, and LiOH indicate that the hydration numbers of the individual ions do not necessarily produce additive effects on colligative properties. When any remaining curvatures are disregarded, h(LiOH) ) 0.7 with a slope of 1.36 × 10-4, about 10 times smaller than slopes found with the electrolytes in Table 1. From single data point calculations, h(LiOH) changes drastically from 8.1 at 0.34 m to -1.25 at 2.0 m. F- and HO- have a charge-to-size ratio considerably higher than that of Cl-, Br-, and I- and may be able to replace waters of hydration from cations, in effect forming ion pairs. This would cause the method used here to produce small or negative h values. As to the behavior of nitrates, carbonates, sulfates, phosphates, and other polyatomic anions, they often show downward curvatures and the data provide no evidence to support any one of many possible hypotheses that can be proposed. Resonance structures can be written that place a positive charge on the central atom, and these are dipolar ions (zwitterions). It is not clear how such polyatomic ions might interact with each other and how this would affect colligative properties. The nonelectrolytes treated here exclude organic compounds that, while fairly soluble in water, may introduce complications from formation of micelle-like structures. The common salts of the three halides and the nonelectrolytes that are the focus of this work constitute the simplest cases, satisfy the stringent criteria for validity of the method, and are amenable to straightforward interpretation. Conclusions The curtain of mystery surrounding colligative properties of aqueous electrolyte solutions has been partially lifted. Some widely held beliefs regarding colligative properties of aqueous solutions of electrolytes have no basis for the compounds studied here at realistic concentrations. It is not primarily interactions between ions that cause departures from theoretical linearity, nor is it ion pair formation. Large deviations disappear when the correct number of waters of hydration is subtracted from the total water solvent in calculating molar fractions. Solutions of the chlorides, bromides, and iodides of the common cations studied show 100% dissociation and the linear behavior expected of colligative properties over concentration ranges many hundreds of times larger than those adhering to the Debye-Hu¨ckelOnsager theory. The nature of “free” water as solvent is changed by even small amounts of dissolved electrolytes. No evidence is found for ion pair formation for concentrations spanning the large ranges of linearity demonstrated. Nonelectrolyte solutes are also treated successfully in the same way. Hydration numbers obtained from freezing point depression, boiling point elevation, and vapor and osmotic pressures are in agreement with many such values obtained by totally different and more elaborate techniques. They are correlated with gas-phase ion-water cluster distributions and their experimental sequential binding energies, as well as with such energies from theoretical calculations. Some hydration numbers show temperature dependence. Supporting Information Available: Table of correlation coefficients and slopes for vapor pressure depression data of

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