Quest To Demystify Water: Ideal Solution Behaviors Are Obtained by

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B: Liquids, Chemical and Dynamical Processes in Solution, Spectroscopy in Solution

The Quest to Demystify Water. Ideal Solution Behaviors are Obtained by Adhering to the Equilibrium Mass Action Law Andreas A. Zavitsas J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b07166 • Publication Date (Web): 03 Jan 2019 Downloaded from http://pubs.acs.org on January 5, 2019

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The Journal of Physical Chemistry

The Quest to Demystify Water. Ideal Solution Behaviors are Obtained by Adhering to the Equilibrium Mass Action Law Andreas A. Zavitsas* Department of Chemistry and Biochemistry, Long Island University, 1 University Plaza, Brooklyn, New York 11201, United States

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ABSTRACT: The current belief that water solutions are not “ideal” is due to the assumption that the mass action law is not obeyed and activity coefficients are used to avoid violating an immutable law. The law is not violated when expressed by including waters of hydration as part of the reactants and as products in the formation of ion pairs by electrolytes. Mole fractions calculated by adhering to the law and recognizing effects of incomplete dissociation and amounts of water interacting strongly with solutes obey Raoult’s law to high concentrations. Another belief is that, for a solution to be “ideal,” there must be no volume or heat change on mixing the components. We demonstrate “ideality” despite volume changes and non-zero enthalpies of solution. Results with a variety of electrolytes correlate with residence times of waters near ions, viscosities, entropies of hydration, enthalpies of solution and dilution, ion ionization energies, Hofmeister series, etc. Older reports of “negative hydration,” that had been considered unreliable, are now understood; the effect is due to formation of mobile, or “fast,” water by strong hydrophobic associations. Our results are supported by findings from techniques perfected relatively recently: dielectric relaxation, mid-IR pump-probe spectroscopy, 2D-IR, atomic force microscopy, etc.


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INTRODUCTION Water solutions of electrolytes are commonly described as not being “ideal” primarily because they fail to adhere to the laws of colligative properties, except at very high dilution. Of more fundamental significance is that they also fail to adhere to the equilibrium mass action law for ion pair formation between a cation C and an anion A to form an ion pair C||A. The law requires that the equilibrium constant for the pairing reaction, K1, be independent of concentration in eq 1, where the brackets denote concentrations. Occasionally in the current literature, the subscript (aq) is added to each component. C + A ⇄ C||A

K1 = [C||A]/([C]∙[A])

(1)

In 1912, Lewis noted that K1 “varies enormously with concentration.”1 In 1915, Bates reported that for KCl solutions K1 increases by 2100% between 0.0001 N and 0.1 N.2 In 1921, Lewis and Randal wrote:3 “For those who have not examined this question closely, it may be surprising to learn how far K1 is from being a constant.” To avoid violating an immutable law they introduced “activity coefficients” to mean the observed “ion activity divided by the assumed molality.” Activity coefficients are used to this day to force K1 to be constant. No reason was provided as to why water solutions would violate a fundamental law. Activity coefficients are correction factors that expose the inadequacy of our models. We demonstrate that the mass action law is not violated and, as a direct result, there is satisfactory adherence to the laws of colligative properties. From among the colligative property laws, Lewis concluded that Raoult’s law is best for determining whether a solution is “ideal” or not, because vapor pressure measurements are made at constant temperature and pressure, unlike the other colligative properties.4 Accordingly, close adherence to Raoult’s law is the one and only criterion by which ideal behavior is defined in the present work. Lewis also noted4 that “We see therefore that in a perfect solution it is also true that the activity of the solute is proportional to its mol fraction.” For non-volatile solutes, the law requires that the ratio of the vapor pressure above the solution, p, to the vapor pressure of pure water, p0, be equal to xw, the mole fraction of water: p/p0 = xw. The experimental value of p/p0 has been described as the “activity” of water (aw). The mole fraction is conventionally calculated from molalities, m, of the solute (moles in 1,000 grams of water, 55.509 moles H2O at 25 °C) by eq 2, where i is the stoichiometric number of solute particles: 1 for non-electrolytes, 2 for 1:1 electrolytes, 3 for 2:1 electrolytes, etc. xw = 55.509/(55.509  m∙i)

(2) 3

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There are two errors in eq 2. The first is that it assumes that strong electrolytes dissociate completely into the stoichiometric number of ions. They do not. The founders of ionic dissociation Arrhenius and van’t Hoff reported the observed “degrees of dissociation” for the various aqueous electrolytic solutions they studied, although they were aware of their relation to stoichiometry. As Arrhenius recalled in 1912 “Therefore, I did not say that there is nearly total dissociation of salts in their highly diluted solutions, but I said that the salts consist of two different kinds of molecules in solutions, the one inactive and the other active. These latter conduct electricity, and the others, the inactive ones, do not.”5 Now we call the “active ones” separate positively and negatively charged species and the “inactive ones” ion pairs of various kinds or associated species acting as a single neutral particle. A 2006 review of ion-pairing6 noted that the view that ion pairs are not formed in water is maintained by some to this day “despite a plethora of experimental data to the contrary.” While the approximation of complete dissociation simplifies calculations, as was done previously,7,8 it leads to inaccurate results when there is extensive ion pairing. A 2016 review demonstrated that ion pairing is not a rare phenomenon only occurring for very particular, strongly interacting cations and anions. 9 An older suggestion that ion pairing would occur by electrostatic forces only between highly charged cations and anions, called Bjerrum-type, was augmented by Diamond to include association between large cations and large anions of small charge.10 As more data accumulated, it was noted that association is strong when water interacts with comparable energies with the cation and anion and this led to Collins’s formulation of the law of matching water affinities:11 “Oppositely charged ions in free solution form inner sphere ion pairs spontaneously only when they have equal water affinities.” The second error is that eq 2 assumes that all the water in the solution constitutes the solvent. From considerations of enthalpies of vaporization,8 the binding energy of a water molecule to other waters near room temperature is about 13.3 kcal mol. Water binds strongly to some solutes with energies greater than 13.3 kcal mol. The earliest mention of such “bound” water is in Einstein’s doctoral thesis dealing with the effect of the solute’s volume on solution viscosity.12 From the viscosity of a 1.0 w% sucrose solution in water, he reported that “The effective volume of sucrose, in a hydrodynamic sense, is significantly greater than that of the solid.” Also, “We can say, therefore, that a dissolved sugar molecule (or the molecule together with the water held bound by it) behaves in a hydrodynamic sense as a sphere.” He concluded “a 4 ACS Paragon Plus Environment

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quantity of water, whose volume is about one half of the volume of a sugar-molecule, is bound to the sugar molecule.” Einstein’s viscosity equation was stated to apply only to solutes that have molar volumes greater than the “effective volume” of water; solutions of such solutes will have viscosities greater than those of pure water. Einstein did not specify what this “effective volume” is. Liquid water exists in clusters of various sizes. The hydrodynamic, or “effective,” molar volume of water is now known not to exceed 85 cm3 (for an average of 85/18 = 4.7-water cluster) and not to be less than 54 cm3 (3-water clusters) near room temperature.13 A theoretical model of four H2O (72 cm3) described well many known properties of water.14 Infrared cavity ring down laser absorption spectroscopy showed that the trimer is the most abundant, followed by the tetramer.15 Molecular dynamics calculations of cluster distributions indicated that the tetramer is the most abundant below 300 K and the trimer above.16 Binding energies of water to various ions are available from a compilation of dissociation energies of gas phase ion-water clusters.17 At 25 °C, the binding energy of one water molecule to Li is 34.0 kcal mol; similarly for Na, 24.0; for K, 17.9; for Rb, 15.9; and for Cs, 13.7. Compared to 13.3 for water binding to other waters, some waters will bind to these ions more strongly than they bind to other waters. One example of the distribution of gas phase water-ion clusters for protonated waters is available from collision-induced dissociations.18 Fig. 1, shows the relative intensities of mass spectrometry m/z peaks for gas phase H(H2O)n clusters formed from HI solutions from n = 3 to 11. The distribution indicates a weighted average of about 7 bound waters. A second water envelope starts growing past n = 13.



70

60

50

Relative intensity

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40

30

20

10

0

3

4

5

6

7

n of clusters of

8 +

9

10

11

H(H2O)n

Figure 1. Plot of relative intensities of gas phase mass/charge peaks of H(H2O)n vs n.

RESULTS Hydration numbers. In considering interactions between solutes and the water solvent, the term “hydration number” is often used. There is confusion about what this term means. The most general definition refers to the average number of water molecules affected by interacting with solute particles so as to become in some way distinguishable from other non-interacting waters. When a water molecule is strongly bound to a solute, it becomes part of, and moves with, the solute as one particle. The number of solute particles does not change, but the number of waters available to act similarly as solvent for any additional solute decreases. The dynamic average number of waters with decreased degrees of freedom per mole of solute is denoted here by the symbol Hd. Hd stands for “hydrodesmic number” with etymology from Greek ὑδωρ (hydor, water) and δεσμος (desmos, bond). The advantage of this term is that it specifies exactly what it measures. Hd does not differentiate between waters in the first shell around the solute and those farther away. For many solutes to be treated first, the major component to the Hd values is the number of waters bound strongly to each mole of solute with energies greater than 13.3 kcal mol. Compared to undisturbed water, such waters contribute little to the vapor pressure and p/p0 decreases as a result. Justification of this is the report that “the reorientation of water 6 ACS Paragon Plus Environment

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molecules outside the dynamic hydration shells is well separated from the reorientation dynamics of the water molecules in the dynamic hydration shells.”19 Raman spectroscopy studies of O–H vibrations of aqueous alkali halides also reported that the electric field of the ions affects only water molecules near the ion.20 However, the assumption cannot be made that the vapor pressure above a solution of a non-volatile solute is always an exact measure of the amount of water not strongly bound to solute. It will be shown below that this is not always the case. Solutes that interact strongly with water, as evidenced by values of Hd greater than about 2, can also affect significantly the collective dynamics of water, thus affecting the vapor pressure and the value of Hd. Also, strongly associated ion pairs can disrupt the structure of water, creating water that is more weakly hydrogen-bonded and more mobile than pure water. This increases the vapor pressure and decreases Hd. The balance between these two opposing effects of bound and of mobile water can cause either a decrease or an increase in the mobility of water relative to pure water, affecting the self-diffusion coefficient of water21 and the vapor pressure. Both effects may be present in the same solution. When the effect of water with decreased degrees of freedom predominates, Hd will have a positive value. When the effect of mobile water predominates, Hd will have a negative value. The number of moles of solvent that would cause the observed vapor pressure is 55.509  m∙Hd. This is the amount of solvent specified by Raoult’s law. Hydrodesmic numbers are not related to hydration numbers obtained from X-ray or neutron scattering experiments, which measure the number of water molecules surrounding a solute. The larger the solute, the more water molecules will surround it. Molecular dynamics simulations obtained “hydration numbers” for cations and anions and residence times of water in their hydration shells.22 The numbers increase with the size of the ion while the residence times decrease, even though the latter are a measure of the binding energy of water to the solute. We have pointed out that such hydration numbers are related to the size of the object.23 Others subsequently shared this view: Hydration numbers obtained by diffraction measurements “have no bearing on the strength of the association of the ions with the water molecules surrounding them.”24 Hydration numbers obtained by different experimental or molecular dynamics methods vary widely because they are based on different assumptions as to what constitutes an “affected” water molecule. The measurements or calculations are valid, but they do not necessarily measure the same thing even though they may all be labeled “hydration numbers.” The confusion due to lack of clear terminology is also manifested by the report that while K and F 7 ACS Paragon Plus Environment


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have similar “hydration numbers” in their first hydration shell , the enthalpy of hydration of F is more negative by a remarkable 31-45 kcal mol.25 This is due to the fact that the number of waters that can fit around an ion is not related to the strength of the bonds formed and the enthalpy of binding. This lack of a clear definition has led to uncertainty and frustration. “The search to quantify and unify ideas on the word ‘hydration’ represents a so far unattained holy grail.”26 As also has been noted: “The individual definitions for the hydration number are not identical on the basis of different assumptions. It is not immediately obvious that one method is superior to others.”27 The question remains as to whether there is a “superior” kind of hydration numbers. Gas phase water-ion cluster binding energies for a water molecule to Cl to give Cl(H2O) have been reported by various techniques17 and give a broad spread of 15 ± 5 kcal mol; electrospray mass spectrometry gave 13.1.28 For Cl(H2O) H2O →Cl(H2O)2, 13.1 ± 1, and for Cl(H2O)2 H2O → Cl(H2O)3, 12 ± 2.17 Cl has been reported to have no water molecules strongly bound to it by dielectric relaxation spectroscopy, which determines amounts of strongly bound water to solutes;29,30 by size-exclusion chromatography, which determines the size of strongly hydrated species;31 by the seminal work of Stokes and Robinson regarding bound waters from considerations molar volumes and ionic activity coefficients,7 and by diffusion studies.32 Theoretical simulations of Cl(H2O)n clusters describe the anion not inside, but on the surface of water clusters, just as is the case with Br and I.33 Gas phase cluster binding energies of Br and I to one H2O are 12.6 and 10.2 kcal mol, respectively.17 The large monovalent anions Cl, Br, I, and NCS tend to show larger concentrations at the air-water interface than inside the bulk,34 just as other hydrophobic substances like gasoline or olive oil avoid water and float to the surface. Water does not bind to these anions more strongly than it binds to other waters and we assign Hd = 0.0 to them. This does not mean that there no energetic interactions between water and such anions, but their interactions are weaker than 13.3 kcal mol. Such ions are solvated by water but not sufficiently strongly to become part of, and move with, the solute as one particle. They have been traditionally designated as “structure breaking,” or “chaotropic.” Ions with very weak water affinities adsorb to non-polar surfaces such as polar sections of proteins and have been called “sticky” ions.35 From infrared studies of hydration behavior in HDO, anions making weaker bonds to water than water-water bonding were found as follows in order


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of decreasing binding to water: Cl, Br, I, NO3, ClO4, BF4, and PF6.25 F is hydrophilic. Also, the hydrophobic solubility of anions is reported to increase in the order Cl, Br, I, NO3, ClO4.36 Such ions prefer a hydrophobic environment and will be designated as “hydrophobic” in the present work. There are two types of hydrophobic species. The alkyl groups of amphiphiles also are hydrophobic. Because of the entropy loss on dissolution of apolar structures in water, the formation of an ordered water network around them was proposed37 and was originally called the “iceberg” effect. Raman scattering studies with multivariate curve resolution demonstrated that there is a hydrophobically enhanced water structure around alkyl groups of dissolved amphiphiles.38 The translational mobility of such water molecules was found by 2D IR to be compromised.39 The waters of the “iceberg” are waters bound more tightly to each other than those in “free” bulk water. Another measure of how strongly water interacts with ions is the Jones-Dole viscosity B coefficient.40 Einstein’s viscosity equation relates viscosity to molar volumes of solutes. Therefore, the B coefficients also are measures of molar volumes. All monoatomic ions have molar volumes smaller than the hydrodynamic molar volume of water. From Pauling’s crystal radii,41 examples of molar volumes of naked ions in cm3 are: Li, 0.545; Cs, 12.18; La, 3.84; Cl, 14.96, and I, 25.4. Ions to which water binds strongly and becomes part of the their volume will have positive B values only when at least three water molecules are bound to increase their hydrodynamic molar volume above the minimum of 54 cm3. Values of some B coefficients are:42 Li, 0.146; Na, 0.085; K, 0.009; Rb, 0.033; Cs, 0.047; Cl, 0.005; Br, 0.033; I, 0.073; and NCS,0.032.43 The equilibrium mass action law. When cations and anions with waters bound to them associate to form an ion pair of some kind, some water is released.6,44 The correct and complete description of the reaction is eq 3, not eq 1. The subscripts a, b, and c, denote the dynamic average number of H2O molecules bound to each of the ion containing entities. The number of water clusters released upon ion pair formation is d = a  b  c. The equilibrium constant for reaction 3 is given by eq 4 and the value of Ke is independent of concentration. M(H2O)a X(H2O)b ⇆ MǁX(H2O)c  (H2O)d Ke = ([MǁX(H2O)c]∙[(H2O)d])/([M(H2O)a]∙[X(H2O)b])

(3) (4) 9

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For a hypothetical example of ions with bound waters M(H2O)5 + X(H2O)3 ⇆ MǁX(H O) 2 6

+ (H2O)2, where (H2O)2 is a released water dimer. Formation of ion pairs is

considered here to be a one-collision process releasing a water dimer, not sequential one step processes releasing one water molecule at each step as initially proposed by Eigen and Wilkins.45 As noted by others, it does not appear to be necessary to postulate a multi-step process.46 Frank noted that when two strongly hydrated ions form an ion-ion pair they also form a water-water pair.47 Even though estimates of the number of water molecules released upon formation of various ion pairs are given in the relevant review of Marcus and Hefter,6 nevertheless eq 1 was used to describe the ion-pairing reaction and equilibrium. Quist and Marshall included H2O in the ion pairing reaction of NaCl and of NaBr occurring in mixtures of water and dioxane as solvent.48 However, they used the total concentration of water in the equilibrium expression and this resulted in a constant 6.4 water molecules being released upon ion pair formation with NaCl and 9.85 with NaBr. Because such high numbers are unlikely, the validity of the “complete” equilibrium equation they used has been questioned.6 However, it is not the total concentration of water that is involved in eq 3 and 4. The bulk water exchanges waters with the hydrated reactants of reaction (3) and with the products. Even if introduced in reaction (3), it would have to appear on both sides and cancel. In effect, the bulk water behaves as “spectator.” Inclusion of waters bound to ions during ion pairing has been used by Simonin et al. in modeling thermodynamic properties of LiCl and MgCl2 solutions by a variation of the mean spherical approximation (MSA) model.49 The number of bound waters per mole of solute was constant to high concentrations. Evidence that the fraction of solute existing in ion pairs is independent of concentration has been provided by narrow band THz absorption spectroscopy (2.3–2.8 THz, i.e., 76–93 cm). Aqueous solutions of alkaline earth metal chlorides, bromides and iodides in concentrations up to 3–4 M showed a linear increase of absorption with increasing concentration.50 The conclusion was: “Thus, the observed absorption changes provide no indication of effects due to denser cation/ion packing as would be expected in the presence of any concentration-dependent cooperative effect between anions and cations.” This is consistent with a constant extent of dissociation, as required by the mass action law of eq 4, but alternative explanations are possible, such as linear concentration dependence of the time-dependent polarization function as suggested by a reviewer of the manuscript.


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Commonly however, water has not been explicitly included in the equation for the reaction forming ion pairs and eq 1 is used instead.51,52 The Molar Fraction of Water. The experimental number of particles formed per mole of solute is denoted here by ie. The molar fraction of solvent, xw, available for any additional solute is given by eq 5. xw = (55.509  m∙Hd )/(55.509  m∙Hd  m∙ie)

(5)

The numerator is the amount of water acting as pure solvent and is the solvent of Raoult’s law. The denominator is the sum of the moles of solvent and of solute particles actually formed. The effect of ion-ion interactions is quantified explicitly in eq. 5 by the extent of dissociation and ion pair formation, ie. The remaining question is whether available experimental values of p/p0 contain definitive information as to whether the results obtained by eq 5 show “ideal” behavior by adhering to Raoult’s law to high concentrations and whether the extent of dissociation and the hydrodesmic number are independent of concentration or not. Application of the Equilibrium Mass Action Law. All p/p0 measurements used in the present work are at 25 °C and their provenance is in the Supporting Information. The work treats only p/p0 values reported as experimentally measured at concentrations of 0.10 molal or greater. Actual measurements of vapor pressure are unreliable below 0.10 molal, because of the very small change of pressure from that of pure water. Often reported p/p0 values for concentrations of about 0.5 m or lower are not actually measured values, but have been adjusted so that activity coefficients derived from them will smoothly converge to the assumed value of unity at infinite dilution. For isodesmic determinations, it is the osmotic coefficients of the reference compounds that have been so adjusted and this similarly affects the calculated osmotic coefficients of the solute. Doing this also allows for extrapolations to infinite dilution. The present work does not venture into such extrapolations. For NaCl solutions, values of p/p0 are available from 0.10 m up to saturation, 6.144 m (359 g in 1.000 kg H2O). The p/p0 values are queried with different hydrodesmic numbers in eq 5 to find whether there are any Hd values that result in xw values that satisfy the constancy of ie required by the mass action law and, simultaneously, demonstrate “ideal” behavior by matching all experimental p/p0 values. The procedure is illustrated in Fig. 2.


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i e, number of particles per mole


3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9

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H d=0.0

NaCl

H d=1.0

H d=2.0 H d=3.0 H d=3.7 H d=4.0

H d=5.0

H d=6.0 0

1

2

3

4 Molality

5

6

7

Figure 2. Plot of ie values required for matching exactly each experimental p/p0 of NaCl solutions with the indicated values of Hd vs. molalities. The filled circles are used to establish the constant ie = 1.77, the horizontal line.

With Hd = 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0 in eq 5 (open circles), the ie values required to match each p/p0 are different. The value of Hd = 3.7 (filled circles) is consistent with the most uniform value of ie = 1.770 ± 0.018 up to saturation and satisfies the law of mass action. Any value other than a constant Hd = 3.7 in Fig. 2 would violate the law. The values of ie, the filled circles, are not perfectly identical, but they are reasonably constant when considering the range of concentration covered and that the ionic strength, viscosity, and dielectric constant of the medium change significantly. At saturation, the amount of bound water is m∙Hd = 6.144  3.7 = 22.7 moles, or 40.9% of the total. The molar fraction of water solvent, xw, is calculated by eq 5 with ie = 1.77 and Hd = 3.7 for a Raoult’s law plot of experimental p/p0 versus xw. The result is depicted by the circles in Fig. 3, where the line is Raoult’s law. There is “ideal” behavior. The fraction of NaCl completely dissociated is 0.77, with 0.23 being in ion pairs. Each mole of NaCl produces 0.77 particles containing Na, 0.77 particles containing Cl, and 0.23 particles containing a pair of the ions. For comparison, a plot versus xw calculated by eq 2 with i = 2.0 is also shown by the triangles. 12 ACS Paragon Plus Environment

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1.00

NaCl 0.95

Experimental p/p 0

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0.90

0.85

0.80

0.75 0.75

0.80

0.85 0.90 Calculated x w

0.95

1.00

Figure 3. Plot of experimental p/p0 of NaCl solutions vs. molar fractions of free water solvent xw calculated by eq 5 from 0.10 m up to saturation at 6.144 m with a constant number of particles produced per mole ie = 1.77 and a constant hydrodesmic number Hd = 3.7 (circles). Molar fractions of water calculated by eq 2 with i = 2 (triangles). The line is Raoult’s law. For the sodium cation reported gas phase water-ion sequential binding energies are:53 for Na + H2O → Na(H2O), 24.0; for Na(H2O) + H2O → Na(H2O)2, 19.8; for Na(H2O)2 + H2O → Na(H2O)3, 15.8; for Na(H2O)3 + H2O → Na(H2O)4, 13.8; for Na(H2O)4 + H2O → Na(H2O)5, 12.3; and for Na(H2O)5 + H2O → Na(H2O)6, 10.7. Compared to 13.3 kcal mol–1 for water binding to other waters at 25 ºC, the expectation is that an average of at least three waters will bind to Na+ more strongly than to other waters and not more than five. The expectation from the positive B coefficient of Na is that more than three waters are bound to Na. Hd = 3.7 is consistent with both expectations from different methods. For 1:1 electrolytes, the mass action equilibrium constant Ke of eq 4 is given by eq 6, where m∙(ie  1) is the moles of solute particles containing only the cation and is equal to the moles of particles containing only the anion; m∙(2  ie) is the moles of associated ion pairs and is equal to the moles of released water clusters upon pair formation. Ke is independent of concentration. Ke = (2  ie/(ie  1)2

(6)



For NaCl, Ke = 0.232/0.772 = 0.0892, valid from 0.1 m up to saturation. According to the commonly used eq 1, the equilibrium constant for ion pair formation is K1 = (2  ie)/{m∙(ie  1)2}. At 0.10 m, K1 = 0.23/(0.10 × 0.772) = 3.88; at 6.144 m, K1 = 0.23/(6.144 × 0.772) = 0.0631. K1 decreases 61-fold and violates the law of mass action when the reaction for ion pair formation is written as it is in eq 1. This had necessitated the use of corrections by activity coefficients to avoid violating an inviolate law. Equation 4 removes the previously perceived necessity of doing so. Activity coefficients were invented to accommodate the behaviors of non-ideal solutions. Obviously, as Hamer and Wu noted in their compilation of osmotic and activity coefficients:54 “Activity coefficients give a measure of the deviations of real solutions from ideality and include the magnitude of all effects that lead to these deviations.” When Raoult’s law is satisfied, as it is with NaCl solutions in Fig. 3, there are no significant deviations from ideality requiring corrections. Measurements of p/p0 of water over HI solutions are available up to 10.0 m. By the procedure of Fig. 2, the most uniform value of ie = 1.99 ± 0.04 occurs with a constant Hd = 7.6 valid from 0.10 up to 3.5 m (448 g per kg H2O). HI is the strongest of the halogen acids and the ie value shows complete dissociation into two particles per mole. A Raoult’s law plot of experimental p/p0 values versus xw calculated by eq 5 is in Fig. 4. At 3.5 m, the amount of bound water is 3.5 × 7.6 = 26.60 moles, or 47.9% of the total water. At the next available measurement of 4.0 m, (m∙Hd)/55.509 is 54.8% and a significant deviation from Raoult’s law is evident. 1.00

HI

0.95

Experimental p/p 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.90

0.85 3.5 m 47.9% 0.80

4.0 m 54.8%

0.75 0.75

0.80


0.95

1.00


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Figure 4. Plot of experimental p/p0 of water over HI solutions vs. molar fractions of free water solvent xw calculated by eq 5 with ie = 1.99 and Hd = 7.6 (circles). Molar fractions of water calculated by eq 2 with i = 2 (triangles). The line is Raoult’s law. Because binding energies of water-I gas phase clusters are all below 13.1 kcal mol and the B coefficient of I is significantly negative, no part of the bound water is attributable to the anion. Hd = 7.6 is consistent with the distribution of H(H2O)n gas phase clusters of Fig. 1.

Chlorides, Bromides, Iodides and Thiocyanates. Measurements of p/p0 available for the acids, for the alkali metals and for ammonium, for the alkaline earths, and for some typical trivalent and tetravalent cations were treated as in Fig. 2 and the results of eq 5 are in Table 1. Raoult’s law plots for all solutions treated in the present work are in the Supporting Information.

Table 1. Properties of Water Solutions of Chlorides, Bromides, Iodides, and Thiocyanates. solute

ie a

Hd b

mc

%d

ge

mmax f

HCl

1.91

6.1 4.50 49.5

164

16.0

HBr

1.95

7.0 4.00 50.4

324

11.0

HI

1.99

7.6 3.50 47.9

448

10.0

LiCl

1.88

5.7 4.50 46.2

191

19.0

LiBr

1.89

6.5 4.00 46.8

347

20.0

LiI

1.91

7.4 3.00 40.0

401

3.0

NaCl

1.77

3.7 6.10 41.0

359

6.1

NaBr

1.82

4.0 6.00 43.2

618

9.0

NaI

1.89

4.5 6.00 48.6

900

12.0

NaSCN

1.86

3.7 7.00 46.7

567

18.0

KCl

1.75

2.1 4.80 18.3

358

4.8

KBr

1.76

2.3 5.50 21.5

655

5.5

KI

1.81

2.4 4.50 19.5

747

4.5

KSCN

1.76

1.3 5.00 11.7

290

5.00

RbCl

1.73

1.9 7.80 26.8

941

7.80



a

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RbBr

1.73

1.6 5.00 14.4

828

5.00

RbI

1.72

1.7 5.00 15.3 1,062

5.00

CsCl

1.68

1.8 9.50 30.8 1,600

11.0

CsBr

1.67

1.6 5.00 14.4 1,065

5.00

NH4Cl

1.77

1.6 7.40 21.3

396

7.40

NH4Br

1.78

1.8 6.10 19.8

974

6.10

NH4I

1.81

1.7 7.50 23.0

989

7.50

NH4SCN 1.76

0.7 8.20 10.5

623

8.20

MgCl2

2.58 13.6 2.00 49.0

190

5.92

MgBr2

2.72 15.1 1.75 47.6

323

5.61

MgI2

2.84 16.4 1.50 44.3

417

5.01

CaCl2

2.49 12.3 2.25 49.9

250

7.46

CaBr2

2.63 13.4 2.00 48.3

401

7.66

CaI2

2.77 14.6 1.90 50.0

482

1.90

SrCl2

2.46 11.1 2.50 50.0

396

3.52

SrBr2

2.59 12.3 2.12 46.5

526

2.12

SrI2

2.76 12.8 1.97 49.7

673

1.97

BaCl2

2.41

8.8 1.79 28.3

373

1.79

BaBr2

2.56 10.1 2.30 41.8

683

2.30

BaI2

2.67 13.5 2.00 46.1

782

2.00

AlCl3

3.19 25.0 1.12 50.3

149

1.81

ScCl3

3.16 22.7 1.18 48.2

178

1.96

CrCl3

3.16 23.9 1.00 43.1

158

1.40

YCl3

3.12 21.2 1.31 49.8

255

2.01

LaCl3

3.05 19.9 1.30 46.6

319

2.20

ThCl4

3.80 27.2 1.00 49.0

374

1.60

Extent of dissociation. bHydrodesmic number. cFrom 0.10 m up to the indicated limit of

molality adhering to Raoult’s law. dPercent of the total water bound at m. eGrams solute per kilogram of water at m. fHighest m of available p/p0 measurements.


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All of the above-mentioned expectations from binding energies of gas phase water-ion clusters of the alkali metal cations are confirmed by their Hd values. Available42 viscosity B coefficients for univalent cations in Table 1 are: K, 0.009; Rb, 0.033; Cs, 0.047; NH4, 0.008. Negative B coefficients indicate that fewer than three waters bind strongly to these cations. The viscosities of dilute solutions, up to 1.0 M at 20 °C, of KCl, KBr, KI, KSCN, RbCl, RbBr, RbI, CsCl, CsBr, NH4Cl, and NH4Br are lesser than the viscosity of water55,56 and their Hd values are smaller than 3. Viscosity data were not located for solutions of NH4I, and NH4SCN. For all other salts of Table 1, Hd is greater than 3 and the viscosities of their solutions are greater than those of water. Ab initio calculations, at the CCSD(T) level of theory, for binding energies of alkali metal cation-water clusters obtained the following values for the last n of bound water of M(H2O)n: For Liand n = 6, 13.1 kcal mol; for Na and n = 5, 11.4; for K and n = 2, 14.3; for Rb and n = 2, 12.3; and for Cs and n = 1, 13.5.57 Relative to 13.3 for water-water binding, Li should bind at least five waters, Na fewer than five, K more than two, Rb fewer than two, and Cs more than one. The theoretical results and the Hd values of Table 1 are in agreement. Consistent with the result of only 91% complete dissociation of HCl in Table 1 is a report of modeling the properties of strong electrolyte solutions with the mean spherical approximation including ion pairing (BiMSA).58 It was reported that “Surprisingly, in the case of HCl the introduction of a low degree of association permits the description to very high concentration.” Persistent ion pairing in HCl solutions was also reported more recently.59 A sodium ion magnetic resonance study reported a value of 3–4 waters bound to Na, independent of concentration.60 In fair agreement with Hd = 3.7, a dielectric relaxation study reported 4.2 ± 0.3 strongly, or irrotationally, bound waters at infinite dilution, but the value was reported as decreasing slowly with increasing concentration.61 While the observed decrease was attributed to a decrease in the number of bound waters, subsequent molecular dynamics simulations indicate that the decreases may be due to an acceleration of the collective dynamics of water with increasing solute concentration.62 Based on water’s isothermal compressibility data for LiCl solutions, a hydration number of 5.3 waters at infinite dilution was reported, decreasing with increasing concentration to 4.2 at 4.0 M.24 While the 5.3 value is in reasonable agreement with Hd = 5.7, the monotonic decrease even at concentrations as low as 0.10 M is difficult to justify in the presence of such a large excess of 17 ACS Paragon Plus Environment


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water. Also, the reported hydration number of 6.1 for NaCl is higher than that of LiCl even though the charge density of Na+ is significantly lower than that of Li+. Adiabatic compressibilities and isothermal compressibilities, as all operationally defined hydration numbers, yield quite different results24 and lead to the usual confusion about which is the superior set.27 Dielectric relaxation measurements led to the conclusion that the number of water molecules bound to Cs more strongly than to other waters, described as “irrotationally bound,” is zero.30 This is in fair agreement with the low value of Hd from p/p0 measurements of CsCl solutions. A large angle X-ray scattering study found that the influence of Rb and Cs ions on the water structure is very weak.63 The low Hd values for K, Rb, and Cs are also consistent with their known viscosity B coefficients, all slightly negative. Dielectric relaxation also found that the number of strongly bound water molecules to Mg is ~14 at infinite dilution with the value declining with increasing concentration.64 The value of ~14 is in agreement with Hd = 13.6, but the p/p0 measurements show that Hd remains constant up to 2.0 m and 49% bound water. The gas phase experimental binding energy17 for the twelfth water in Mg(H2O)11  H2O → Mg(H2O)12 is 14.3 ± 2.4 kcal mol. Compared to 13.3, more than 12 bound waters are indicated and Hd = 13.6 is consistent with this. Bush et al. reported infrared laser multiple photon dissociation spectra of Ca(H2O)n2+ clusters formed by electrospray performed on a 7 T Fourier-transform ion cyclotron. Around the ion, the inner shell contains eight bound waters, which donate a hydrogen bond to four second shell water molecules.65 The total of 12 such bound waters is in agreement with Hd = 12.3 for CaCl2 solutions. Hydrated trivalent lanthanide ions in gas phase clusters Ln(H2O)n3+ have been studied by electrospray ionization of solutions of their chlorides and the structures and reactions of the cations were determined using black body infrared radiative dissociation (BIRD) and infrared action spectroscopy.66 The gas hydrates dissociate in two ways: loss of a water molecule or loss of protonated water leaving a reduced charge metal hydroxide. Loss of neutral water is favored for large clusters, charge separation dominates at low cluster size, and both processes occur at intermediate values. Loss of neutral water becomes essentially the exclusive mode at n = 19–21 for all the hydrated lanthanides. For LaCl3 solutions, the value of Hd = 19.9 in Table 1 is in good agreement for the last bound water. 18 ACS Paragon Plus Environment

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By definition, a solution is “ideal” when the ratio of the “activity” of water (p/p0) to the mole fraction of water (xw) is 1.00. Table S1 of the Supporting Information lists such ratios with the mole fractions of water solvent calculated by eq 5 with the ie and Hd values of Table 1 for solutions of HCl, the alkali metal chlorides, and NH4Cl. For the 153 such ratios of (p/p0)/xw of Table S1, none is smaller than 0.9977 or greater than 1.0083. Table S2 of the Supporting Information provides details and demonstrates that the activity of solute particles is equal to their mole fractions and their activity coefficients, s, are essentially unity, as Lewis had established.4 Stokes and Robinson also applied a “hydration correction” n for the number of bound waters to construct an equation for matching ionic activity coefficients ±of chlorides, bromides and iodides of 1:1 and 2:1 electrolytes.7 An additional parameter was required and a parameter å was used. It was interpreted as the closest distance to which the unhydrated anion can approach that of the hydrated cation of each electrolyte. Complete dissociation was assumed. The n and å values used provided satisfactory fits to the ± values for concentrations occasionally up to ionic strengths of about 4. Data for the cesium halides could not be reconciled with any approach they tried. This is due to the fact that only 68% of CsCl dissociates completely. Fig. 5 shows that CsCl solutions are not problematic with eq 5 and adherence to the mass action law is up to 9.5 m. 1.00

0.95

CsCl 0.90 Experimental p/p 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.85

0.80

0.75 9.5 m, 1600 g kg -1

0.70

0.65 0.65

0.70

0.75


0.90

0.95

1.00

Figure 5. Plot of experimental p/p0 of CsCl solutions vs. molar fractions of free water solvent xw calculated by eq 5 with ie = 1.68 and Hd = 1.8. The line is Raoult’s law.



Page 20 of 42

The extent, if any, of ion pair formation in CsCl solutions has been examined several times previously, but with conflicting conclusions.67 The p/p0 measurements show that 32% of the salt is in ion pairs over the concentration range of 0.1 to 9.5 m and 1,600 g solute per kg of water. The n values obtained by Stokes and Robinson for the solutes of Table 1 are not widely different from the Hd values, but the range of concentration covered is much smaller. For example for HCl, HBr, and HI, n = 7.3, 8.6, and 10.6 were reported valid only up to 2.0, 1.0, and 1.0 m using their one-parameter equation, compared to 4.5, 4.0, and 3.5 m in Table 1. The Hd values have an uncertainty because an increase in Hd is somewhat compensated by a decrease in ie, as seen in Fig. 2. The extent of this uncertainty is not quantifiable exactly, but our estimate is that it may be as high as ± 5% occasionally. Twenty of the 41 entries in Table 1, have constant ie and Hd valid over all available p/p0 measurements and adhere to Raoult’s law. Twenty adhere up to the highest measured molality, where the amount of bound water does not exceed 50% of the total. The single remaining entry is CsCl, for which beyond 9.5 m and 1,600 grams of solute triple ions are likely to start forming. The existence of the species (NaINa) has been reported.68 Also, (NaClNa), (KBrK), and (RbIRb) have been observed at high concentrations as well as higher aggregates such as (MX)mM.69 At 9.0 m CsCl with 68% dissociation, the moles of bound water are 0.68 × 9.0 × 1.8 = 11.02 and of “free” water 55.51  11.02 = 44.49. Scattering experiments from many sources70 indicate values of between 6 and 8 waters surrounding Cl. Assuming an average of 7, at 9.0 m the moles of water required to surround all Cl ions are 0.68 × 9.0 × 7 = 42.8 and there is just enough “free” water available. However, at 9.5 m the amount of bound water is 11.6 moles, of “free” water is 43.9, but 45.2 moles of water are needed just to surround the free chloride anions. Fig. 5 shows the on-set of deviations from Raoult’s law past 9.5 m. Failure to adhere to Raoult’s law beyond 50% of bound water does not indicate failures of eq 5 and of the mass action law, but their validity. It has long been known that a constant number of bound waters cannot be maintained at all concentrations.7 For example with LiCl, a constant Hd = 5.7 at the highest available p/p0 measurement of 19.0 m would require 108.3 moles of water but only 55.509 are available. At 50% of the total water bound, one-half is in clusters bound to solutes and the other half to water-water clusters. The most loosely held waters in the two types of clusters would be held by the same energies. Any additional solute will bind water by removing it equally from either kind of cluster. There is no reason to expect that the law of 20 ACS Paragon Plus Environment

Page 21 of 42

mass action is invalid past the 50% point and, therefore, ie will remain constant. When p/p0 values are available beyond the 50% point, they allow the calculation of Hd that will maintain a constant ie. Because xw ≈ aw, by substituting aw for xw in eq 5 and using the established constant ie, one can solve the equation for Hd, which is the only unknown. As an example, we apply this approach to LiCl solutions, which adhere to Raoult’s law up to 4.5 m with Hd = 5.7. At 5.0 m with the established constant ie = 1.88 and the experimental p/p0 = aw = 0.7478, a decrease of Hd to 5.53 is obtained; at m = 6.0 and p/p0 = 0.6785, Hd decreases to 5.28. At high concentrations triple ions form and then this approach will fail, as shown with CsCl. Having ascribed no bound waters to Cl, Br, and I, the same Hd may be expected of their salts with a common cation. Table 1 shows that this is not the case. Stokes and Robinson also noted this conundrum, but they did not provide a specific explanation.7 The differences are caused primarily because the extents of dissociation, ie, are not the same. Collins’s law of matching water affinities indicates that the lower the difference in the affinity for water of the counter-ions, the higher is the tendency for ion pair formation and for a lower ie. Viscosity B coefficients are a measure of this affinity. The smaller the difference in B, the greater is the tendency to associate. Fig. 6 plots ie versus the differences in the B values of the counter-ions. Similar linear correlations of ie with B coefficients occur with the four sets of the alkaline earth salts and are shown in Fig. S1 and Fig. S2 of the Supporting Information. 2.00 HI 1.95 Extent of dissociation, i e

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


HBr HCl

1.90

LiI

NaI LiCl

LiBr

1.85 KI

NaBr

1.80

KBr

1.75

NaCl

KCl 1.70 0.00

0.05

0.10 0.15 0.20 Difference in B coefficient

0.25

Figure 6. Plots of extents of dissociation vs. difference in the B coefficients of the counter-ions. 21 ACS Paragon Plus Environment


The greater the extent of dissociation, the greater is the fraction of free cations to bind water, thus increasing Hd. Linear relations between Hd and ie are shown in Fig. 7. 9 8

HI

LiI

7 Hydrodesmic number, H d

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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LiBr 6

HBr

LiCl

HCl

5 NaCl

4 3

NaBr NaI

KBr

2 KCl

KI

1 0 1.70

1.75

1.80 1.85 1.90 Extent of dissociation, i e

1.95

2.00

Figure 7. Plot of hydrodesmic numbers vs. extents of dissociation for the indicated groups. If one assumes that Cl, Br, I, and the neutral ion pairs do not bind significant amounts of water, the effect of the different extents of dissociation on Hd of the cation can be approximated. The fraction of complete dissociation of NaCl solutions is 0.77. If there were 100% dissociation, Hd for Na would have the value of 3.7 of the salt divided by the fraction of 0.77 unpaired Na ions, or 4.81. Similarly for NaBr, Hd for Na would be 4.0/0.82, or 4.88; for NaI, Hd for Na would be 4.5/0.89, or 5.06; and for NaSCN, Hd for Na would be 3.7/0.86 = 4.30. The four values are similar but not identical and indicate that Cl, Br, I, NCS, and/or the ion pairs formed have some effect on the Hd of the compound. The distribution of solute particles for 2:1 electrolytes, MX2, is obtained as follows. A fraction of (ie  2.00) yields one M and two X; a fraction of (3.00  ie) yields one MX and one X. For MgCl2 solutions, ie = 2.57. A fraction of 0.57 produces 0.57 Mg2+ and 2 × 0.57 = 1.14 Cl; a fraction of 0.43 produces 0.43 MgCl and 0.43 Cl. The sum of particles is 2.57. Solutions of Amphiphiles. Amphiphilic compounds of 1:1 electrolytes exhibit different 22 ACS Paragon Plus Environment

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behaviors from the salts of the metal cations and of NH4. The results of our treatment of p/p0 measurements of solutions of alkyl-substituted ammonium chlorides and bromides are in Table 2. The non-polar alkyl substituents on nitrogen are strongly hydrophobic and associate strongly with the hydrophobic anions Cl, and Br. Measurements of p/p0 are available for some larger tetraalkylated amphiphiles, but those are known to form micelles.71

Table 2. Properties of Water Solutions of Amphiphilic Chlorides and Bromides.

a

solute

ie a

Hd b m c

%d

g/kg e mmax f

CH3NH3Cl

1.76

1.8 7.0 22.7

473

20.0 0.100

(CH3)2NH2Cl

1.72

2.2 8.0 31.7

653

17.0 0.151

(CH3)3NHCl

1.70

2.5 9.0 40.5

859

15.0 0.184

(CH3)4NCl

1.71

3.0 8.0 43.2

877

9.0 0.167

(C2H5)4NCl

1.62

5.0 5.5 49.5

912

9.0 0.376

(n-C3H7)4NCl

1.64

7.3 3.5 46.0

776

18.0 0.345

(CH3)4NBr

1.55

1.5 5.5 14.9

847

5.5 0.669

(C2H5)4NBr

1.39

3.4 8.0 49.0 1,682

12.0 2.446

(n-C3H7)4NBr

1.43

3.8 6.0 41.1 1,598

9.0 1.757

(HOCH2CH2)4NBr 1.47

0.3 5.5

6.5 1.272

Choline chlorideh

1.63

2.8 7.1 35.9

995

7.1 0.345

Choline bromidei

1.53

1.4 7.0 17.7 1,288

7.0 0.786

Tris∙HClj

1.77

1.1 8.0 15.9 1,263

8.0 0.089

3.0 1,506

Ke g

Extent of dissociation. bHydrodesmic number. cFrom 0.10 m up to the indicated limit of

molality adhering to Raoult’s law. dPercent of the total water bound at m. eGrams of solute per kg of H2O at m. fMolality at limit of available measurements. gEquilibrium constant of the mass action law, eq 4, calculated by eq 6 and valid up to m. hHOCH2CH2N(CH3)3 Cl. i

HOCH2CH2N(CH3)3 Br. jtris(hydroxymethyl)aminomethane hydrochloride.

The results of tetraalkyl ammonium salts of Table 2 demonstrate the water ordering effect, or “iceberg,” occurring around non-polar alkyl groups of amphiphiles. The tetramethyl ammonium cation, (CH3)4N, is of interest in biology because it forms the positively charged



Page 24 of 42

species in the polar head group of phospholipids.72 Comparing (CH3)4NCl to NH4Cl, the tetramethyl compound has a lower ie and a higher Hd. The higher Hd is due to the highly structured bound waters around the hydrophobic methyl groups and the lower ie is due to hydrophobic-hydrophobic association between the highly hydrophobic methyl groups and the hydrophobic chloride anion. With increasing size of the alkyl groups, ie decreases and Hd increases. The lower ie is due to increasing hydrophobic association and the higher Hd is due the greater “iceberg” effect of larger alkyl groups. Solutions of (n-C4H9)4NCl form micelles71 and we fail to find any Hd consistent with a constant ie over a wide range of concentration. The bromides of Table 2 show smaller extents of dissociation than the chlorides. Complete dissociation of tetramethylammonium bromide is only 55% compared to 71% of the chloride. Br is more hydrophobic and interacts more extensively with the hydrophobic alkyl groups. The constancy of Hd to 5.5 m indicates that there are no micelles or other aggregates up to this concentration. Polarization-resolved femtosecond infrared spectroscopy has also been used to examine the properties of tetraalkylammonium bromides in 4% D2O in H2O.72 Slowly reorienting water molecules around the hydrophobic methyl groups were found and there was no evidence that (CH3)4NBr molecules aggregate. The slow reorientation time constant was independent of concentration, indicating the constant ie required by the law of mass action. In addition, Br was reported to penetrate the highly confined waters clustering around the alkyl groups causing the reorientation of OD groups of HDO water to become very slow. Our findings are consistent with all of these results. GHz dielectric relaxation spectroscopy of tetraalkylammonium halide solutions has shown that (CH3)4N does not align the dipoles of associated water molecules, as Na does, the water being associated with the alkyl groups and not the positive charge on the nitrogen.72 Also detected was fast reorientation of mobile H2O molecules. For the tetra-n-butyl bromide, a constant ie cannot be obtained with any positive Hd. The choline ion is important in the metabolism of carbohydrates and of lipids and is a precursor for the neurotransmitter acetylcholine. For choline chloride, Hd = 2.8 is similar to 3.0 of tetramethylammonium chloride, but only 63% is completely dissociated versus 71%. The ie value of choline bromide solutions is similar to those of the other tetraalkyl bromides, but Hd is significantly lower. THz spectroscopy of choline lipids found73 that “the collective bulk-like water dynamics become less pronounced, whereas an increased amount of both very slowly reorienting (i.e., irrotational) and very rapidly reorienting (i.e., fast) water molecules appear.” In 24 ACS Paragon Plus Environment

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Table 2, choline chloride and bromide have low extents of dissociation, 1.63 and 1.53, respectively. Dielectric relaxation results with actin filaments found “hyper-mobile” waters, along with those of lower mobility that are bound to solute.74 -Amyloid fibrils can form plaques related to Alzheimer’s decease and mobile water was detected in their solutions.75

-Synuclein fibril formation is related to Parkinson’s decease.76 The hydrochloride of tris-hydroxymethyaminomethane, 2-amino-2(hydroxymethyl)propane-1,3-diol, is often used for buffering. Its ie = 1.77 ± 0.02 is the same as that of NaCl, but its Hd = 1.1 is significantly lower. As with several other entries in Table 2, adherence to Raoult’s law is maintained for more than 1,000 g solute per kg of water. Of the thirteen compounds in Table 2, four adhere to Raoult’s law over all available measurements (7th, 11th, 12th, and 13th entries). Three adhere up to the highest molality that does not result in more than 50% bound water (5th, 6th, and 8th entries). The remaining six apparently form micelles beyond the high m values listed for them. Dielectric relaxation measurements of solutions of (C2H5)4NCl, (CH3)4NBr, and (C2H5)4NBr found “cooperative relaxation of the H-bond network of ‘bulk’ water and fast reorientation of mobile H2O molecules.”77 These three amphiphiles have low extents of dissociation 1.62, 1.55, and 1.39, respectively, demonstrating greater association than the non-amphiphiles of Table 1. While there are many reports that extents of dissociation are concentration-dependent, direct experimental evidence to the contrary is available for many solutes that interact weakly with water and have small extents of dissociation. For many such solutes with positive Hd values of about 2 or smaller, one can estimate the ie values directly from only two literature values, the experimental aw and m at realistic concentrations. The extents of dissociation are given by ie = 55.509(1  aw)/m and remain essentially constant over large ranges of concentration. Details and specific examples are in Table S3 of the Supporting Information. Relations of Hd with other physical properties and methods. The present work requires adherence to the law of mass action obtained in the manner demonstrated in Fig. 2. The hydrodesmic numbers obtained are consistent with values expected from binding energies of gas phase ion-water clusters, viscosity B coefficients, and relevant theoretical calculations. Nevertheless, the question will likely arise as to whether the values of ie and Hd so obtained are simply the result of an empirical two-parameter fit to p/p0 measurements. There are already several highly parametric methods for fitting colligative property data. A fourth degree 25 ACS Paragon Plus Environment


polynomial in m can fit the experimental p/p0 values of NaCl solutions to an accuracy of one part per thousand. The five constants involved have no physical meaning. They are adjustable parameters to fit the data. However, there are two incontestable facts about such solutions: there may not be complete dissociation of the salt and some water molecules may bind very strongly with the solute and not be available to act similarly as solvent for any additional solute. The value of ie quantifies the extent of dissociation and that of Hd the dynamic average of waters with significantly decreased degrees of freedom. This information is hidden in the p/p0 measurements and our approach digs them out. In current parlance, “mining the data.” Unlike “adjustable parameters,” Hd and ie have physical meaning. The values obtained are the only two values that satisfy the equilibrium mass action law. The law is certainly not an adjustable parameter. The procedure employed here simply ensures adherence to it. The validity and relevance of the Hd values thus obtained can also be confirmed by their relation with properties apparently unrelated to colligative effects. Examples of such relations are below. The Hd values must relate in some fashion to the entropy changes accompanying hydration of metal ions, S(hyd). Fig. 8 plots experimental S(hyd) values in entropy units (eu = cal K mol) in the absolute scale assuming Sºaq(H) = 5.306 eu available from two sources78,79 versus Hd of the corresponding metal chlorides. The expectation that Hd and S(hyd), must correlate is satisfied over very wide ranges of entropies and Hd values. 0

Cs + + Rb + K

-10 -20

Na -30

+ +

Li

-40 -50

S(hyd), eu

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 42

Ba

-60

2+

Sr

2+

Ca 2+

-70

Mg

-80

2+

-90 -100

La

-110

3+

Y

3+

Sc

3+

-120 3+

-130

Al

-140

0

2

4 6 8 10 12 14 16 18 20 22 24 26 Hydrodesmic number of the chlorides, H d

Figure 8. Plot of entropies of hydrations of metal ions vs. Hd of their chloride solutions. 26 ACS Paragon Plus Environment

Page 27 of 42

Circles, ref. 78; triangles, ref. 79. The line is the best fit through all the points. Linear correlations of Hd also are obtained with available80 enthalpies of solution and of dilution of 1:1 electrolytes and these plots are in Fig. S3 and Fig. S4 of the Supporting information. Fig. S3 shows the major thermal effect of strong binding, of the order of kcal mol, of ions with water upon solution. Fig. S4 shows the smaller thermal effect, of the order of cal mol, of changes in the collective dynamics of water upon dilution. Available ionization potentials78 of metal ions correlate with Hd of their chlorides. Fig. 9 shows that only Badeviates from the straight line defined by the other members of its set. 130

3+

Al 120 110 100

Ionization energy, eV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


90

Mg 2+

Li+

80

Sc 3+

70 Ca

60 50

Na +

Sr 2+

40 K+ +

30

Y3+

Ba 2+

Rb + Cs

20

2+

La 3+

10 0 0

5

10 15 20 Hydrodesmic number, H d

25

Figure 9. Plots of ionization energies of the ions vs. Hd of the corresponding chloride solutions.

The number of waters bound more strongly to cations than to other waters has been determined by extracting metal salts of hydrophobic bases from water solutions into nitrobenzene.81 The salts examined were the dipicrylaminates, i.e., the anions of bis(2,4,6trinitrophenyl)amine, of Cs, Rb, K, Na, Li, Ba, and Ca and the tetraphenylborates of Na, Li, Ba, and Ca. The reported numbers of bound waters co-extracted with each cation into nitrobenzene with each of the two bases are plotted in Fig. 10 versus Hd of their chlorides. The dotted line denotes what would constitute perfect agreement. Two very different methods of measuring the same thing, vapor pressure and liquid-liquid extraction, agree substantially within 27 ACS Paragon Plus Environment


the uncertainties of the liquid-liquid extraction values. 16 14 Number of H2O co-extracted

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 42

Ca 2+

12

Ba

2+

10 8

+

Li 6 Na

+

4 2

+

K + Rb

0

Cs 0

+

2 4 6 8 10 12 14 16 Hydrodesmic number of the metal chlorides, H d

Figure 10. Numbers of waters co-extracted with each cation from water into nitrobenzene vs. Hd of the corresponding chlorides. Dipicrylaminates (triangles). Tetraphenylborates (circles).

Residence times of water molecules around ions have been derived from NMR selfdiffusion constants for Li, Na, and K as 39 ps, 27 ps, and 15 ps, respectively.82 Fig. S5 of the Supporting Information shows a linear correlation of the residence times plotted versus the Hd values of LiCl, NaCl, and KCl solutions. The correlation coefficient is r = 0.9979. Dielectric relaxation measurements of the reorientation time, , of bound waters relative to that of pure water, w, are available for Li, Na, K, Rb, Cs, and NH4 in 1.0 M solutions.83 Their /w values are: 2.41, 1.53, 0.9, 0.78, 0.68, and 0.72. For R4N with R = methyl, ethyl, and n-propyl, /w = 1.59, 1.96, and 2.37. Fig. 11 shows their correlations with Hd of their chlorides.


Page 29 of 42

2.6 + Li

2.4

n-Pr 4N+

w

2.2 Reorientation time ratio, /

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.0 + Et4N

1.8 Me 4N+

1.6

Na

+

1.4 1.2 K+

1.0 0.8 NH4

+ Cs

0.6 1

Rb + +

2

3 4 5 6 Hydrodesmic number, Hd

7

8

Figure 11. Plot of reorientation times of water around the indicated cations from dielectric relaxation measurements vs. Hd of solutions of their chlorides.

Pump-probe femtosecond transient vibrational spectroscopy has determined vibrational relaxation time constants of the O–H stretch of 6.0 M solutions of NaCl, NaBr, and NaI in 0.5 M HDO in D2O. The values are 2.6 ± 0.3 ps, 3.4 ± 0.3, and 3.9 ± 0.3, respectively; the value for pure water is ≈ 0.8 ps.82 The differences in the time constants were attributed to the extent of interaction of the three anions with the positive H OH dipole of water. From the extents of dissociation in Table 1, the concentration of free ions is 6 × (ie  1) molar. Fig. 12 plots the time constants versus the concentrations of free ions not in ion pairs. The good linearity confirms the validity of the ie values and the validity of eq 5. The good linearity also shows that the error bars assigned82 to the time constants were too conservative. The measurement were more accurate.



4.5 NaI 4.0 NaBr

Time constant, ps

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 42

3.5

3.0

NaCl

2.5

2.0 4.5

4.6

4.7 4.8 4.9 5.0 5.1 5.2 Molar concentration of "free" ions

5.3

Figure 12. Plot of vibrational relaxation time constants of the H–OD stretch in 6.0 M solutions vs. the molar concentration of “free” ions. Error bars are the reported uncertainties.

Spin-lattice relaxation rates of deuterium in 4.5 M D2O solutions of CsBr, RbBr, KBr, LiCl, CaCl2, and LaCl3 cover a large range of water affinities.84 The reorientational correlation times deduced are: 2.37, 2.31, 2.35, 3.58, 5.05, and 7.74 ps.83 Fig. S6 of the Supporting Information plots them versus their Hd. The linear correlation coefficient is r = 0.9963. Equivalent ionic conductivities of monoatomic cations depend on the charge and volume of the ion. The conductivities at infinite dilution, o in 10 m2 S mol, are: of Li, 39; of Na, 50; of K, 73; and of Rb and Cs, 77.55 The smallest Li binds more waters than Na, its effective volume is larger, and has the lowest conductivity. Cs and Rb are weakly hydrated, have the smallest effective volume and the greatest ease of moving through the solvent. The same trend is followed by the alkaline earth cations with the Hd values of their chlorides shown in parentheses: Mg, 53.0 (13.6); Ca, 59.5 (12.3); Sr, 59.4 (11.1), and Ba, 61.0 (8.8). The Hd values of Table 1 follow the ordering of ions in the Hofmeister series, which rates the effectiveness of various salts for precipitating proteins from aqueous solutions. The title of Hofmeister’s third publication on this subject in 1888 is “About the water withdrawing effect of salts.”85 A common ordering of increasing effectiveness of ions is as follows,36 with Hd of the chlorides of the cations in parentheses: NH4 (1.6) < Cs (1.8) < Rb (1.9) < K (2.1) < Na (3.7) 30 ACS Paragon Plus Environment

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< Li (5.7) < H (6.1) < Ca(12.3) < Mg (13.6) < Al (25.0). The Hd values and the normal Hofmeister series follow the same order. Hd measures exactly what Hofmeister called the “water withdrawing effect” of each salt. A protein in a nearly saturated solution will precipitate upon addition of a solute that binds water strongly and removes it from the solvent available for dissolving the protein. This is the simplest explanation, but Hofmeister effects are quite complex and the ordering of the normal Hofmeister series may change somewhat depending on the counter-ion and the series may even reverse depending on the pH of the protein solutions. “Negative hydration.” For the solutes treated so far, eq 5 accounts accurately for ion pairing by the use of a constant ie and of waters with decreased degrees of freedom by the use of Hd and obtains adherence to Raoult’s law over large ranges of concentration above 0.10 m. However, as noted above, this is not valid if solutes disturb significantly the collective dynamics of water or cause the formation of mobile water. Any mobile water less strongly hydrogenbonded to other waters will cause an increase in the vapor pressure, resulting in a decrease in the value of Hd obtained by eq 5. This loose water will make it appear that there is less water bound to the separate, not associated, ions than is actually the case; it can even cause negative Hd values. This occurs with several solutions and has been called “negative hydration,” which is an oxymoronic term obviously indicating something amiss. There cannot be fewer than zero bound waters. The term has been called absurd.40 The term is fanciful, but the phenomenon causing “negative hydration” is real. Samoilov described some ions as positively hydrated (Li, Na, Mg, Ca) and other ions as negatively hydrated (K, Cs, Cl, Br, I) on the basis of the free energy of exchange of water molecules next to the ion with those in the bulk.86 A more recent review reported that these terms are not now in general use and that “negative hydration” is a dead end.42 Negative Hd by eq 5 is caused primarily by the effect of the strongly associated species formed by strong hydrophobic interactions, not by that of the individual ions. The effect of strongly associated species on vapor pressure to cause “negative hydration” does not appear to have been addressed in a satisfactory and comprehensive manner previously. Examples of such “negative hydrations” are in Table 3. Most of the solutes are salts of the very weakly hydrated and hydrophobic NO3 and ClO4 anions.

Table 3. Solutions Resulting in Negative Hd Values. 31 ACS Paragon Plus Environment


a

Page 32 of 42

gc 

solute

ie

Hd

ma

mmaxb

KNO3

1.75

7.5

3.5

3.5

354

RbNO3

1.72

7.4

4.5

4.5

664

CsNO3

1.75

9.0

1.5

1.5

585

CsI

1.74

0.3

3.0

3.0

779

NH4NO3

1.69

1.5

9.0

25.9

720

AgNO3

1.69

7.5

6.0

15.0

1019

CH3NH3NO3

1.70

0.7

7.5

9.5

705

C5H14N4O3d

1.67

4.5

2.5

9.0

445

NH4ClO4

1.71

2.1

2.1

2.1

247

CH3NH3ClO4

1.67

2.3

4.0

4.0

526

(CH3)2NH2ClO4

1.58

2.6

7.0

7.5

1019

(CH3)3NHClO4

1.73

12.0

1.8

1.8

287

CH6N3Cle

1.70

2.7

4.5

12.0

432

C5H14N3ClO4f

1.78

29.0

3.0

23.0

646

(CH3)4NI

1.64

8.8

1.94

1.94

391

(C2H5)4NI

1.58

16.0

1.9

1.9

489

(n-C4H9)4NBr

1.59

6.3

5.5

27.0

1773

KTosylateg

1.82



3.5

3.5

736

From 0.10 m up to the indicated limit of molality adhering to Raoult’s law. bMolality at limit of

available measurements. cGrams solute per kilograms H2O at m. dTetramethylguanidinium nitrate. eGuanidinium chloride. fTetramethylguanidinium perchlorate. gPotassium p-toluenesulfonate.

The strength of interactions of anions with water has been studied by double-difference IR spectroscopy, showings that the strength decreases in the order Cl > Br > I > NO3 > ClO4, nitrate and perchlorate being the weakest and most hydrophobic.87 The relative tendency of anions to accumulate at the air-water interface has been determined by electrospray ionization mass spectrometry.88 Among the various anions examined, ClO4 was by far the most abundant 32 ACS Paragon Plus Environment

Page 33 of 42

at the interface and most hydrophobic, followed by BF4 and I. All entries in Table 3 have at least one weakly hydrated counter-ion and one hydrophobic, or both hydrophobic. Ample experimental evidence, cited above,72,74,75,76 has become available in recent years for the formation of such mobile, or “fast,” or “hyper-mobile” water. The results of Table 3 can hardly be explained in any other manner. One may surmise that it is the “free” nitrate and perchlorate anions that disturb the structure of water thus creating mobile water. That this is not the case is demonstrated by LiClO4 solutions, for which p/p0 measurements are available up to 4.5 m. Because of the large difference in water’s affinity for the counter-ions, there is little association and the effect of any mobile water is not significant. Equation 5 yields ie = 1.92 and Hd = 6.9 and adherence to Raoult’s law to 3.5 m (372.6 g kg, 43.5% bound water). The Hd value is not much different from 6.5 of LiBr and 7.4 of LiI solutions. LiClO4 also has the highest extent of complete dissociation, 92%. If it were the free ClO4 anions that create mobile waters and not the strongly associated pairs, a much greater diminution of Hd would have occurred with 92% of the solute being free ClO4 ions. With NaClO4 the difference in water’s affinity for the counter-ions is smaller and ie = 1.77 with Hd = 2.4 is valid over all data up to 7.0 m. The fraction of 0.23 in ion pairs forms enough mobile water to decrease the Hd from the 3.7 of NaCl. The perchlorates of K, Rb, and Cs are too insoluble to obtain p/p0 data. A typical Raoult’s law plot with a negative Hd value is in Fig. 13 for AgNO3. 1.00 AgNO3 Raoult's law x w by eq. 2 with i = 2.0 0.98

Experimental p/p 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


x w by eq. 5 with Hd = -7.5, i e = 1.69

0.96

0.94

0.92 6.0 m, 1019 g kg -1 0.90 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 Calculated x w



Page 34 of 42

Figure 13. Plot of experimental p/p0 of AgNO3 solutions vs. molar fractions of free water solvent xw calculated by eq 2 and i = 2 (triangles) and by eq 5 with ie = 1.69 and Hd = 7.5 (circles). The line is Raoult’s law. For LiNO3 solutions, ie = 1.82 with Hd = 5.0 up to 4.0 m (276 g kg). For NaNO3, ie = 1.72 with Hd = 0.0 up to 6.0 m (510 g kg). In both cases, there is a decrease in ie and in Hd relative to the corresponding chlorides. Evidently, the ion pairs with the hydrophobic nitrate generate sufficient mobile water to increase the vapor pressure and cause the declines in Hd. Purely hydrophobic association with formation of mobile water is best demonstrated with non-ionic solutes, such as the strong protein denaturant tetramethylurea. Available p/p0 measurements extend to 7.7924 m. A constant ie = 0.80 ± 0.03 occurs with Hd = , valid to 4.4757 m (520 g kg). The ie value is consistent with a fraction of 0.60 of the solute in monomeric form and 0.40 in dimers contributing 0.20 to the total particles per mole. The expected “iceberg effect” around the non-polar alkyl groups is more than counterbalanced by formation of mobile water by the strong association of the highly hydrophobic solute, resulting in the negative Hd. Solutions of tetramethylurea have been studied by dielectric relaxation spectroscopy between 0.2 GHz and 2 THz.89 “Slow” water was detected around the hydrophobic alkyl groups, but the fraction of water with fast dynamics increased markedly with increasing solute concentration from 2% in pure water to 13% at equimolar mixtures of solute and water. This was ascribed to an under-coordination of the water network and an overall weakening of the hydrogen bond network or collective dynamics of water, as also did the blue shift observed for the OD-stretch vibration in water containing some D2O. A mid-infrared pump-probe spectroscopy study of tetramethylurea found that the average hydrogen-bond strength of water decreases with increasing concentration.90 These results are consistent with the linearity of the plots we obtain with negative Hd values in Table 3 and with tetramethylurea, formation of mobile water, and negative hydration. Some of the entries of Table 3 have surprisingly large negative values of Hd. This is consistent with a report of nuclear magnetic relaxation study that found negative hydration to be a cooperative effect between many water molecules over a comparatively wide range.91 Studies of water confined in tight places have shown drastic changes in behavior compared to pure water. Confinement in the hydrophobic cavity of carbon nanotubes causes a tenfold 34 ACS Paragon Plus Environment

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speed up of water’s orientational dynamics caused by reduction in the number of hydrogen bonds.92 Water in etched channels of heights of 10, 3.8, and 1.4 nm in boron nitride crystals was probed by scanning dielectric microscopy with an atomic force microscope. The hydrogen bond contribution to the dielectric constant of water was suppressed in layers of about three water molecules.93 These results show reduction of hydrogen bonding in tight hydrophobic spaces causing the formation of mobile water. Strongly hydrophobic association causes the negative hydrations that we have examined and mobile water is likely to be similarly formed in the presence of very tightly associating hydrophobic solutes. We have applied successfully the approach exemplified here to p/p0 data of solutions of fluorides, hydroxides, nitrates, perchlorates, carboxylates, chlorates, bromates, sulfates, carbohydrates, amino acids, amides, urea and substituted ureas, various guanidinium salts, and all available data for all lanthanides. Space limitations do not allow their presentation here and these results will appear in due course.

CONCLUSIONS Adherence to the law of mass action for ion pair formation leads to “ideal” solution behavior and adherence to Raoult’s law from 0.10 m up to high concentrations. It has been thought for a long time and to this day that, for a solution to be ideal, the components must “mix in all proportions without change in volume and without any heat effect.”94 We have demonstrated that this is not so. Only the erroneous eq 2 fails to demonstrate ideal behavior, not the realistic eq 5. For example, aqueous NaCl solutions by definition are “ideal” by adhering to Raoult’ s law from 0.1 m up to saturation with eq 5 (Fig. 3), even though there is a volume change on mixing and a positive enthalpy of solution. In the large concentration ranges over which there is ideal behavior, the extent of dissociation and the hydrodesmic number are constant. As to whether there is a set of hydration numbers “superior” to Hd, any other set that correlates better with things such as waters co-extracted with cations into organic solvents, enthalpies of solution and dilution, entropies of hydration, residence and reorientation times of water, ionization energies, viscosities, conductivities, Hofmeister effects, etc., that set will be preferable.

ASSOCIATED CONTENT 35 ACS Paragon Plus Environment


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Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/…….. Three Tables: of aw/xw ratios, of s values, and of experimentally based ie values. Six plots of Hd versus other physical properties. Provenance of p/p0 values, and 73 Raoult’s law plots. AUTHOR INFORMATION *E-mail: [email protected]. Tel: (718) 488-1351 ORCD Andreas A. Zavitsas: 0000-0002-1310-8307 Notes The author declares no competing financial interest. REFERENCES (1) Lewis, G. N. The Activity of the Ions and the Degree of Dissociation of Strong Electrolytes. J. Am. Chem. Soc. 1912, 34, 1515–1529. (2) Bates, S. J. Osmotic Pressure and Concentration in Solutions of Electrolytes, and the Calculation of the Degree of Ionization. J. Am. Chem. Soc. 1915, 37, 1421–1445. (3) Lewis, G. N.; Randall, M. The Activity Coefficient of Strong Electrolytes. J. Am. Chem. Soc. 1921, 43, 1112–1154. (4) Lewis, G. N. The Osmotic Pressure of Concentrated Solutions, and the Laws of the Perfect Solution. J. Am. Chem. Soc. 1908, 30, 668–683. Raoult’s law, as originally published by F.-M. Raoult for non-electrolytes in 1887. http://gallica.bnf.fr/ark:/12148/bpt6k30607/f1429.image/ (accessed may 25, 2015). (5) Arrhenius, S. Electrolytic Dissociation. J. Am. Chem. Soc. 1912, 34, 353–364. (6) Marcus, Y.; Hefter, G. Ion Pairing. Chem. Rev. 2006, 106, 4585–4621. (7) Stokes, R. H.; Robinson, R. A. Ion Hydration and Activity in Electrolyte Solutions. J. Am. Chem. Soc. 1948, 70, 1870–1878. (8) Zavitsas, A. A. Properties of Water Solutions of Electrolytes and Nonelectrolytes. J. Phys. Chem. B 2001, 105, 7805–7817. (9) van der Vegt, N. F. A.; Haldrup, K.; Roke, S.; Zheng, J.; Lund, M.; Bakker, H. J. WaterMediated Ion Pairing: Occurrence and Relevance. Chem. Rev. 2016, 116, 7626–7641. (10) Diamond, R. M. The Aqueous Solution Behavior of Large Univalent Ions. A New Type of Ion Pairing. J. Phys. Chem. 1963, 67, 2513–2517. (11) Collins, K. Ions from the Hofmeister Series and Osmolytes: Effects on Proteins in Solution and in the Crystallization Process. Methods 2004, 34, 300–311. (12) Einstein A. Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. 1906, 19, 289–306. Correction: Ann. Phys. 1911, 34, 591–592. The writing of the thesis predates considerably its 1906 publication time. An English translation is available. http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1906_thesis.pdf/ (accessed June 3, 2008). 36 ACS Paragon Plus Environment

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TOC Graphic

1.00 0.95

Choline chloride

la

w

0.90 0.85

Ra ou lt' s

experim ental p/p 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.80 0.75 0.70 0.70

7.124 m, 995 g per kg H2O

0.75 0.80 0.85 0.90 0.95 1.00 calculated m ole fraction of w ater


Quest To Demystify Water: Ideal Solution Behaviors Are Obtained by

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