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Concentration Dependence of Ionic Hydration Numbers Yizhak Marcus* Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel ABSTRACT: Isothermal compressibility data of 23 aqueous electrolyte solutions at 25 °C from the literature are used to calculate their hydration numbers, which diminish as the concentration increases. Their limit at very high concentration is near the “number of adsorption sites” of water molecules on the ions, obtained by the BET method. On the contrary, hydration numbers obtained from ultrasound speed measurements yielding isentropic compressibilities cannot be valid, being much too large at infinite dilution.

d av /nm = 0.94(c /mol dm−3)−1/3

1. INTRODUCTION Hydration numbers of ions in aqueous solutions are very much operationally defined and values obtained by various methods tend not to be in good agreement. Values reported on the basis of diffraction methods, such as neutron and X-ray diffraction on electrolyte solutions,1 and those reported on the basis of computer simulations of individual ions in water1−3 yield the pair correlation functions g(r). These describe the geometric arrangement of the water molecules around the ions in a first and possibly a second hydration shell. They have little bearing on the strength of the association of the ions with the water molecules surrounding them. In particular, the computer simulations provide information on dilute solutions of ions, but not on the concentration dependence of the number of water molecules hydrating the ions. Other methods, based on thermodynamic data on aqueous electrolyte solutions, result in values that are more closely connected with this association and pertain also to its concentration dependence. Of these methods, those based on the compressibility of the solutions or on the electrostriction caused by the ionic electric field4 resulted so far mainly in values at infinite dilution. These could then be allocated to the individual ions on the basis of the additivity of the ionic values that occurs when the ions are remote from and do not interact with each other. The change of the hydration numbers of electrolytes in aqueous solutions with their concentration has received less attention, although it is now generally agreed that they must diminish when the concentration increases up to such a value at which the hydration shells overlap.5 For a typical electrolyte consisting of monatomic ions the sum of the radii of the cation and anion is ∼0.3 nm. If each of these ions were fully hydrated, with water molecules of diameter 0.278 nm, they could approach each other to a distance between their centers not smaller than ∼0.3 + 2 × 0.278 = ∼0.86 nm. However, the average distance apart of the ions of a symmetrical electrolyte of concentration c is on purely geometrical grounds:5 © 2014 American Chemical Society

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Therefore, their hydration shells must overlap already at concentrations c ≥ 1.5 mol dm−3, where dav/nm ≤ 0.82. Therefore, ions cannot maintain intact the hydration spheres which they have in dilute solutions and the values of the hydration numbers h should decrease once the hydrated ions are in contact. Thus, on purely geometrical grounds the hydration numbers of ions must diminish as their concentration increases beyond dilute solutions. To obtain hydration numbers h from compressibility data, the volume of one mole of solution containing mole fractions xW of water and xE of hydrated electrolyte can be written, following Onori,6 as V = (x W − hx E)VW + x EVEh

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where VW is the molar volume of pure water and VEh is that of the hydrated solute that binds h moles water per mole of solute. On differentiation with respect to the pressure at constant temperature and entropy,6 noting that κ = −V−1(∂V/∂P) is the general definition of the compressibility, κSV = (x W − hx E)κS WVW + x EκS EhVEh

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with the assumption that h is independent of the pressure. The adiabatic compressibility of the solution, κS, is an experimentally available quantity, obtained from the density ρ of and speed of ultrasound u in the solution: κS = 1/u 2ρ

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The quantities for pure water, its isentropic compressibility, κSW, and molar volume, VW, are known, so that three unknown quantities have to be determined: h, κSEh, and VEh. The Received: April 22, 2014 Revised: August 6, 2014 Published: August 14, 2014 10471

dx.doi.org/10.1021/jp5039255 | J. Phys. Chem. B 2014, 118, 10471−10476

The Journal of Physical Chemistry B

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Table 1. Infinite Dilution Hydration Numbers, h∞,6 and Coefficients According to Eq 10 of Hydration Numbers of Aqueous Electrolytes, h(c), at 25 °C, Obtained from Isothermal Compressibility Data in the Literature electrolyte HCl LiOH LiCl LiBr LiI LiNO3 NaOH NaF NaCl

NaBr NaI NaHCO3 Na2SO4 KCl

KBr KI K2SO4 NH4NO3 MgCl2 MgSO4 CaCl2 Ca(NO3)2 SrCl2 ZnCl2 ZnBr2 UO2(NO3)2 AlCl3 a

data ref 24 27 24 21a,b 24 21a,b 31 30 24 27 21a 24 25 28 21a 27 21a 27 35 25 27 21a 24 27 24a 27 21a 27 15 27 16 25 27 25 29 34 21a 26 35 33 32 36 34

h∞

10.4 5.3 5.3 4.2 2.1 3.1 11.2 10.3 6.1 6.1 6.1 6.1 5.1 5.1 2.9 2.9 16.8 16.8 5.1 5.1 5.1 4.0 4.0 1.8 1.8 14.7 14.7 1.0 14.5 14.5 19.1 14.5 14.5 10.0 10.0 14.5

A

B/(dm3 mol−1)

2.1 1.9 9.1 6.0 6.2 5.2 (14.8) (13.7) 10.5 8.8 6.9 7.4 6.1 7.1 6.2 6.7 5.4 6.0 (26.9) 14.6 16.4 6.8 7.0 6.9 5.9 6.3 5.3 5.9 18.9 16.9 4.0 10.7 12.6 13.4 (21.1) 17.6 10.0 12.6 21.5 (16.0) (79.2) 11.0 (25.3)

−0.14 −0.13 −0.83 −0.45 −0.44 −0.39 (−6.92) (−3.71) −1.02 −0.68 −0.62 −0.55 −0.42 −0.41 −0.42 −0.35 −0.33 −0.23 (−13.34) −1.54 −1.94 −0.74 −0.61 −0.43 −0.45 −0.34 −0.45 −0.08 −6.10 −2.21 −0.09 −0.85 −1.18 −1.09 (−7.02) −2.20 −0.96 −1.13 −3.70 (−5.05) (−58.5) −0.82 (−4.48)

C/(dm3 mol−1)2

cmax/(mol dm−3) 4.5

0.024 0.016 0.015 (1.048) (0.360) 0.035 0.035 0.020

0.018 0.017

0.054 0.021

0.020 0.033

12.5 1.0 4.1 6.3 4.5 5.0

4.5 7.4 4.4 7.3 4.2 1.0 4.5 4.2 4.3 4.2 4.6 5.9 3.8 0.6 3.8 8.2 4.4

(0.886) 0.042

(0.542) (11.7)

5.0 5.0 8.5 7.9 2.8 5.9 3.5 2.2 1.9

At 35 °C. bIncluding the data from ref 23.

the solution. It requires the input data, αP, the isobaric thermal expansibility and cP, the isobaric heat capacity per unit volume, where subscript W pertains to pure water whereas its absence pertains to the solution. The resulting hydration numbers tend to be very large, signifying problems with this approach. This point is dealt with in the Discussion, and comparisons between values obtained by the adiabatic and isothermal compressibilities as well as by other methods are shown in Table 2 below. The present paper uses instead the isothermal compressibility, κT, obtained over a range of concentrations of aqueous electrolytes at 25 °C and the expressions first presented by Passynskii9 and subsequently reiterated by Padova10 to obtain the hydration numbers of electrolytes as functions of their concentration.

application of this approach to aqueous sodium chloride solutions6 resulted in hydration numbers, h, that indeed diminish with increasing concentrations of the solute. However, Afanas’ev and co-workers7,8 pointed out that at finite electrolyte concentrations the entropy of the solution, at which κS is determined experimentally, differs from the entropy of pure water, so that eq 3 is not valid as a differentiation at constant entropy. They wrote, instead, κSV = (x W − hx E)κS WVW + (x W − hx E)αPW × TVW[αP W /cP W − αP /cP] + x EκS EhVEh

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The second term on the right-hand side permits the use of the quantities pertaining to pure water for those of the “free” water in 10472



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κT = κTW + A′c + B′c3/2 in the Supporting Information of their paper to yield essentially linear dependencies h(c) shown in Table 1. The data by Pickston et al.26 for calcium nitrate had to be converted from κT(R) to κT(c), where R is the ratio of the number of water molecules to that of salt formula units, by first conversion to molality, eq 8 and then to molarity, eq 9, using interpolated density data from Scott.23 The resulting h(c) are shown as a linear dependence in Table 1. Leyendekkers27 reported effective pressures of salt solutions in the form

2. CALCULATIONS, DATA, AND RESULTS In the approach by Passynskii9 and Padova10 it is assumed that the water molecules in the hydration shell are already maximally compressed by the huge electrical field of the ions4 and that they, as well as the ions themselves,11 cannot be further compressed by (moderately large, up to, e.g., 100 MPa) external pressures. The applied pressure operates then only on the bulk water outside the hydration spheres and compresses it, i.e., diminishes the volume of the voids between the molecules. The resulting observed compressibility of a c molar aqueous electrolyte, the ions of which are together hydrated by h moles water per mole electrolyte, is then proportional to the volume fraction of the bulk water: κT(c) = κT W(1 − hcVWh)

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c = mρ /(1 + mME)

(7)

(12)

(13)

with A″ = 137.2 and to the isothermal compressibilities of the solutions as φK = c −1(κT(c) − ρ(c)κT W /ρW ) + κT WME /ρW

(14)

From these expressions κT(c) values were extracted for several concentrations c, using density, ρ(c), values from Novotny and Söhnel.28 The resulting hydration numbers h(c) could be presented in a linear form according to eq 10 up to the cmax shown in Table 1. More recently Safarov et al.29−33 reported density data ρ(m,P) at a few salt molalities m at 25 °C and at pressures P up to 60 MPa, from which the isothermal compressibilities κT = (∂ ln ρ/ ∂P)T could be derived. Conversion of molalities to molarities with these data at 0.1 MPa with eq 8 led to hydration numbers h(c) from eq 7 expressed as eq 10 and shown in Table 1. All these values of h(c) are excessively large and do not agree with values obtained from the earlier experimental data, where available for the salts studied, for no clear reason. These values are placed in parentheses in Table 1 and are not considered to be valid. On the other hand, the data of Ghafri et al.34,35 treated in a similar manner did yield reasonable values for the divalent metal halides, but the span of molalities is too short to treat the data for NaHCO3, Na2SO4, and AlCl3 appropriately. The isothermal compressibilities κT(m) for uranyl nitrate of Apelblat36 were converted to κT(c) values with eq 9 and data for conversion from the m to the c scale,18 and the derived h(c) values could be presented as a linear expression shown in Table 1. Representative h(c) curves calculated with the coefficients of eq 10 are shown in Figure 1.

(8) (9) −1

where m is the salt molality and ME is its molar mass (kg mol ). The molar volume of completely electrostricted water at 35 °C is VWh = 0.01490 L mol−1, interpolated from published data14 for 25 and 50 °C, and κTW at 35 °C was taken from Scott.23 The resulting hydration numbers were fitted well with a correlation coefficient squared of ≥0.998 by the quadratic expression

h(c) = A + Bc + Cc 2

f (m) = a0m + a1m3/2 + a 2m2

φK = −A″Pe/[m(BT + Pe + 1)(BT + 1)]

It is, therefore, necessary to obtain κT(c) values for each electrolyte at a given temperature for the evaluation of h(c). The references where suitable data can be found for 25 °C (35 °C in some cases) are shown in Table 1, but most of the data needed to be processed to convert them to the κT(c) form. The data of Adams15 for potassium sulfate at 25 °C were in the form of the relative compression as a function of the applied pressure for several mass fractions of salt, from which the compressibilities could be calculated. The molar concentrations were calculated from the supplied densities, eqs 8 and 9 below. His data for ammonium nitrate16 were in the same form, the conversion of mass fractions via molalities to molar concentrations required available data,18 but the sodium chloride data17 could not be used, because no direct relative compression values were reported. A similar problem pertained to the data of Gibson19,20 for many electrolytes, which reported the relative compression at only one pressure (100 MPa), so that reliable isothermal compressibilities could not be calculated. The data of Scott et al.21−23 were obtained at 35 °C and reported ρ(w) and κT(w) data for nine aqueous salts over a range of mass percentages w of the salts, where ρ is the density. These were converted to κT(c) values as follows: m = [w/(100 − w)]/ME

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with the Tamman−Tait−Gibson constant BT = 300.45 MPa at 25 °C and reported the a0, a1, and a2 coefficients for 11 salts. The apparent molar compressibilities of the salts at 0.1 MPa were related to the effective pressures and molalities as

Here VWh is the mean molar volume of the compressed, completely electrostricted water, 0.01807 − 0.00347 = 0.01460 L mol−1 at 25 °C (for concentration c values in mol L−1).12 On this basis the working expression for obtaining the hydration numbers is obtained by rearrangement of eq 6 as13,14 h(c) = (1 − κT(c)/κT W )/cVWh

Pe = (BT + 1)(10 f (m) − 1)

3. DISCUSSION Theoretically, the electrostriction of the solvent around the ions of an electrolyte necessarily leads to hydration numbers h(c) that diminish with increasing electrolyte concentrations in dilute solutions:10

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with the coefficients shown in Table 1, valid up to the maximal concentration, cmax, shown. The 25 °C data by Gucker and Rubin24 for six electrolytes were in the form κT = κTW + A′c + B′c3/2, and so could be evaluated directly in terms of eq 7 and 10 with coefficients shown in Table 1. The data of Millero et al.25 for four electrolytes pertaining to seawater had to be interpolated for 25 °C from those reported as

h(c) = h∞[1 − {S V /(Vintr − V ∞)}c1/2]

(15)

Here SV > 0 is the Debye−Hückel limiting slope, Vintr is the intrinsic volume of the electrolyte and V∞ is its standard partial molar volume, and electrostriction requires that (Vintr − V∞) > 0. However, eq 15 applies only at relatively low concentrations, where the apparent molar volume changes according to the 10473



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However, the values at infinite dilution, h∞, are considerably larger, by factors from 2 to 9, than those obtained by various other methods, Table 2, and cannot be valid. They are generally also larger than coordination number sums for the salts, representing the maximal number of water molecules neighboring their ions, obtained from pair correlation functions from diffraction measurements and computer simulations.1 These water molecules are not necessarily more strongly bound to the ions than water molecules are bound to each other in the bulk. Therefore, not all of them are included in the thermodynamically13,14,48−50 and spectroscopically51,52 significant hydration numbers h. Afanas’ev and Zaitsev7 ascribed to Robinson and Stokes47 the contention that hydration numbers h are independent of the concentration. This contention, however, applied only up to moderate concentrations, as Stokes and Robinson47,48 did admit hydration numbers (“the number of water molecules that can be accommodated in the inner shell”) that are lower in concentrated electrolyte solutions than at infinite dilution. The BET method48,53 yields r values, the number of “water adsorption sites” on the ions, at such high concentrations at which the lefthand side of

Figure 1. Hydration numbers h(c) of aqueous salts at 25 °C obtained from isothermal compressibility data: LiCl (○), LiBr (▽), NaOH (◇), and Ca(NO3)2 (Δ). At the right-hand side are shown the BET parameters r (number of water binding sites per formula unit of the salt) of these salts (large filled symbols).53

(m /55.51)aW /(1 − aW ) = (c′r )−1 + [(c′ − 1)/c′r ]aW (17)

limiting law, φV = φV∞ + SVc1/2 and the slope of h(c) would be −1/2. At larger concentrations an empirical term bVc has to be added to the expression of φV. Either a more negative slope would result for bV > 0, applying to electrolytes with small ions, or a less negative one for bV < 0, applying to electrolytes with large ions.37 Such slopes do not agree with those from the compressibility data in Table 1, but a diminution of the hydration numbers with increasing electrolyte concentrations is established also from the electrostriction approach. The determination of isothermal compressibilities, κT, of aqueous electrolytes is a fairly difficult experimental procedure, generally requiring application of high pressures and a homemade piezometer. On the other hand, the determination of densities and ultrasound velocities, yielding adiabatic compressibilities, κS, via eq 4, is readily made with commercial instruments. This explains the relative proliferation of the earlier κS data38−42 and those of Afanasev and co-workers7,8,43−46 over κT ones. The latter group of authors reported hydration numbers h from the adiabatic compressibilities of aqueous salt solutions that diminish with increasing concentrations of the solute, expressed as its mole fractions, xE, as follows: h = h∞ exp( −kx E)

is linear with the activity of water, aW. Here c′ = exp(ε/RT) and ε represents the difference between the molar enthalpy of “adsorption” of water on ionic sites of the salt and the molar enthalpy of liquefaction of water. The r values represent hydration numbers and those of salts at 25 °C,53 are shown in Figure 1, and are in general agreement with the values calculated for the salts at the highest concentrations considered. Table 1 includes also the infinite dilution hydration numbers, h∞, calculated as the sum of the ionic values (prorated according to the stoichiometric coefficients). These were reported by Padova10 from the difference of the standard partial molar volumes and the intrinsic volumes, taken as the volumes in the crystal; i.e., they pertain to the electrostriction of the water in the hydration shell. These values should agree with the A coefficient of eq 10 shown in Table 1. They do so approximately, in view of the rough manner of the estimation10 of the intrinsic volumes of the electrolytes in the solution.54 Isothermal compressibilities are available at other temperatures than 25 °C25,34,35 and lead to hydration numbers diminishing with the concentration in the same manner as those at 25 °C shown in Table 1 and because the point has been made, they need not be shown here.

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Table 2. Infinite Dilution Hydration Numbers at 25 °C of Aqueous Salts, h∞, According to Various Approaches salt

adiabatic compressibilitya

isothermal compressibilityb

activity coefficientsc

NaCl NaNO3 Na2SO4 KCl KI RbCl MgCl2 MgSO4 CaCl2

23.1 26.1 58.6 21.1 22.8 20.3 32.4 39.2 30.6

6.0 4.0 17.0 5.0 2.4 5.0 14.3 19.3 14.3

3.5

1.9 2.5 1.2 13.7 12.0

colligative propertiesd

infrared spectrae

coordination no. sumf

3.9, 3.1

5

1.7, 0.4 1.7, 0.4 1.8 13, 11.0

5 4

12 12 20 12 12 12 18 14 18

12, 9.4

4 6

References 7, 43, 45, and 46. bReferences 13 and 14. cReference 48. dReferences 49 and 50. The first entry is from the freezing point depression, the second from vapor pressures at 25 °C. eReferences 51 and 52. fReference 47. a

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At finite concentrations the ions naturally interact with one another; hence contrary to the case of infinite dilution, no individual ionic hydration numbers may be derived from the data for entire electrolytes. Therefore, also, there is no point in deriving effective ionic diameters or volumes for the individual ions at such concentrations from experimental thermodynamic data. Estimates of the sizes of hydrated individual ions at finite concentrations, as Stokes radii, could eventually be made from conductivity plus transference and from diffusion data, but there are pitfalls also along this way.55

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(21) Scott, A. F.; Obenhaus, V. M.; Wilson, R. W. The Compressibility Coefficients of Solutions of Eight Alkali Halides. J. Phys. Chem. 1934, 38, 931−940. (22) Scott, A. F.; Bridger, G. L. Apparent Volumes and Apparent Compressibilities of Solutes in Solution. II. Concentrated Solutions of Lithium Chloride and Bromide. J. Phys. Chem. 1935, 39, 1031−1039. (23) Scott, A. F.; Bridger, G. L. The Apparent Volumes and Apparent Compressibilities of Solutes in Solution. J. Phys. Chem. 1936, 40, 461− 470. (24) Gucker, F. T., Jr.; Rubin, T. R. The Apparent Molal Isochoric Heat Capacity of Electrolytes. J. Am. Chem. Soc. 1935, 57, 78−82. (25) Millero, F. J.; Ward, G. K.; Lepple, F. K.; Hoff, E. V. Isothermal Compressibility of Aqueous Sodium Chloride, Magnesium Chloride, Sodium Sulfate, and Magnesium Sulfate from 0 to 45° at 1 Atm. J. Phys. Chem. 1974, 78, 1636−1643 including supplementary material.. (26) Pickston, L.; Smedley, S. I.; Woodall, G. The Compressibility and Electrical Conductivity of Concentrated Aqueous Calcium Nitrate Solutions to 6 kbar and 150 °C. J. Phys. Chem. 1977, 81, 581−584. (27) Leyendekkers, J. V. Compressibilities of Aqueous Electrolyte Solutions. Aust. J. Chem. 1981, 34, 1785−1800. (28) Novotny, P.; Söhnel, O. Densities of Binary Aqueous Solutions of 306 Inorganic Substances. J. Chem. Eng. Data 1988, 33, 49−55. (29) Safarov, J. T.; Najafov, G. N.; Shahverdiyev, A. N.; Hassel, E. (p, ρ, T) and (pS, ρS, TS) Properties and Apparent Molar Volumes Vφ of CaCl2(aq) at T = 298.15 to 398.15 K and at Pressures up to p = 60 MPa. J. Mol. Liq. 2005, 116, 165−174. (30) Safarov, J. T.; Najafov, G. N.; Shahverdiyev, A. N.; Hassel, E. (p, ρ, T) and (pS, ρS, TS) Properties and Apparent Molar Volumes Vφ of LiNO3(aq) at T = 298.15 to 398.15 K and at Pressures up to p = 60 MPa. J. Mol. Liq. 2005, 116, 157−163. (31) Safarov, J. T. (p, ρ, T) and (pS, ρS, TS) Properties and Apparent Molar Volumes Vφ of LiI(aq) at T = 298.15 to 398.15 K and at Pressures up to p = 60 MPa. J. Mol. Liq. 2006, 123, 139−145. (32) Safarov, J. T.; Jannataliyev, R. M.; Shahverdiyev, A. N.; Hassel, E. Investigation of the Thermal Properties and Apparent Molar Volumes of ZnBr2 in the Temperature Range from 298.15 to 398.15 K and at Pressures up to p = 60 MPa Using a Piezometer of Constant Volume. J. Mol. Liq. 2006, 124, 51−57. (33) Safarov, J. T.; Jannataliyev, R. M.; Shahverdiyev, A. N.; Hassel, E. Thermal Properties and Apparent Molar Volumes of ZnCl2 at High Temperatures and Pressures. J. Mol. Liq. 2006, 128, 127−133. (34) Ghafri, S. A.; Maitland, G. C.; Trusler, J. P. M. Densities of Aqueous MgCl2 (aq), CaCl 2(aq), KI(aq), NaCl(aq), KCl(aq), AlCl3(aq), and (0.9864 NaCl + 0.236 KCl)(aq) at Temperatures between (283 and 472) K and Pressures up to 68.5 MPa, and Molalities up tp 6 mol kg−1. J. Chem. Eng. Data 2012, 57, 1288−1304. (35) Ghafri, S. A.; Maitland, G. C.; Trusler, J. P. M. Densities of SrCl2(aq),_Na2SO4(aq), NaHCO3(aq), and Two Synthetic Reservoir Brines at Temperatures between (298 and 473) K and Pressures up to 68.5 MPa, and Molalities up tp 3 mol kg−1. J. Chem. Eng. Data 2013, 58, 402−4121. (36) Apelblat, A. Thermodynamic Properties of Electrolyte Solutions. Compressibility Studies of Aqueous Solutions of Uranyl Nitrate. J. Chem. Thermodyn. 2012, 47, 420−427. (37) Millero, F. J. Molar Volumes of Electrolytes. Chem. Rev. 1971, 71, 147−176. (38) Allam, D. S.; Lee, W. H. Ultrasonic Studies in Electrolyte Solutions. Part III. The Hydration Numbers of Some Acids, Alkalis, and Tetraalkylammonium Salts. J. Chem. Soc. 1966, 426−428. (39) Carplo, R.; Mehicic, M.; Borsay, F.; Petrovic, C.; Yeager, E. Investigation of Aqueous Calcium Nitrate, Zinc Nitrate, and Zinc Chloride Solutions Using Acoustic Velocity Measurements. J. Phys. Chem. 1982, 86, 4980−4987. (40) Millero, F. J.; Vinokurova, F.; Fernandez, M.; Hershey, J. P. PVT Properties of Concentrated Electrolytes. VI. The Speed of Sound and Apparent Molal Compressibilities of NaCl, Na2SO4, MgCl2, and MgSO4 Solutions from 0 to 100 °C. J. Solution Chem. 1987, 16, 269−283. (41) Rohman, N.; Mahiuddin, S. Concentration and Temperature dependence of Ultrasonic Velocity and Isentropic Compressibilities in

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Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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