J . Chem. SOC., Faraday Trans. I , 1986, 82, 2283-2299 Solvent Structural Constant and Solvation Behaviour Applied to the Description of Aqueous Electrolytes at 25-300 OC William L. Marshall Chemistry Division, Oak Ridge National Laboratorj?, Oak Ridge, Tennessee 37831, U.S.A. From the application of a solvent structural constant and calculations of ' free' water remaining after solute solvation, water activities are described at temperatures up to 300 "C for NaCl solutions and at 25 "C for other electrolyte solutions. Ionization constants for NaCl are also obtained. The concentration of free (non-solvating) water is proportional to the activity of water (derived from vapour pressures) raised to a power B, which is a temperature independent constant (6.67) believed to represent a structural property of the solvent.The solute-solvent mass-action equilibrium appears to be predominantly a function of the density of water in the immediate vicinity of each solvated solute species, rather than of the bulk concentration of free water. Relative solvation numbers obtained for univalent ions quantitatively agree with expectations. Thus a model with all parameters defined to have chemical significance describes reasonably well the activity of water over aqueous electrolyte solutions. ____ ~ _ _ __ _ _ ~ _ _ _ _ Previously, application of a complete constant (P), which includes the solvent as a reactant, has described ionization behaviour of many aqueous electrolytes over wide ranges of temperature, pressure and concentration of ~ a t e r .l - ~ This constant was applied only where the concentration of electrolyte approached zero ; therefore the concentration or density of pure water was always used. This concentration could be varied by hydrostatic pressure or dilution with an assumed ' inert' soivent (dioxane), thus providing a value of K" that was isothermally a constant. In the present study, the behaviour of aqueous electrolytes at moderate to high concentrations is evaluated through a consideration of their extents of solvation and through the activity of water [a(H,O)] (derived from vapour pressures) which is related to the concentration of free water [C'(H,O)] in the solution. At a given molarity ( M ) of electrolyte, the concentration of ' free' (non-solvating) water [C'(H,O)] should equal the analytical concentration of total water [C(H,O)] minus the concentrations removed by solvating each of the electrolyte species.Thus for NaCl C'(H,O) = C(H,0)-jC(NaC1"~jH20)-pC(Na+~pH,0)-qC(C1--qH,0) ( I ) wherej, p , and q are average solvation numbers and C is the molarity of the designated species. For each electrolyte, an ionization constant ( K ) that is needed to obtain the fraction of ionization (xi, a value ofj, and a term n (which equals p + q -j, the average increase in solvation number in the ionization process) can be applied to a description of a(H,O) as a function of C'(H,O). A question arises, however, in the evaluation of K in the electrolyte solution given the acceptance of the behaviour of a complete constant. The complete constant is expressed as = K/[H,OIn 76-2 2283 (3)2284 Solution of Aqueous Electrolytes where the brackets signify absolute concentration (in mol dmP3) and y+ is the mean molar activity coefficient of NaCl.In the previous studies, at constant temperature and at the approach to infinite dilution of electrolyte, the concentration of H,O was varied either by hydrostatic pressure or by the dilution of H,O by 'inert' dioxane.lPs The value of K" was found to remain a constant, with K and [H,O] varying in a manner to yield a constant value of n. The question is whether [H,O] in eqn (3) represents the concentration (or density) of free water in the immediate vicinity (the microscopic concentration) of each electrolyte species (concept 1) or the bulk free water concentration, which should equal C'(H,O) (concept 2).At infinite dilution of electrolyte these two concentrations are identical. Up to a moderately high range of electrolyte concentration at saturated vapour pressures, the microscopic concentration of water should be relatively constant and, if accepted for application, provides a nearly constant value of K in accordance with eqn (3). This microscopic concentration is assumed to equal approximately the concentration or density of pure water at the given temperature and total, relatively constant, pressure. [In solvent4ectrolyte equilibria adhering to eqn (3), the microscopic concentration can be changed significantly only by large changes in hydrostatic pressure or by dilution with an inert solvent, as observed in the many studies at the approach to infinite dilution of ele~trolyte.l-~ Decreases in vapour pressure are thus assumed to cause insignificant changes in the microscopic concentration of H,O.] Conversely, if the bulk concentration of free water is the important variable in eqn (3), then this concentration decreases significantly with increasing electrolyte concentration owing to both the volume occupied by the solvated electrolyte species and to the water removed by solvation.Its concentration is expressed by eqn (1). Accordingly, under this condition K would decrease sharply with increasing electrolyte concentration in order to maintain K" as a constant. Evaluation of the fits of the activity of water over electrolyte solutions by both of these approaches has been applied in this study to determine whether concept 1 or concept 2 should be favoured.Customarily for a stoichiometric equi-valent electrolyte, the ionization constant can be expressed : from which x, the fraction of electrolyte ionized, may be calculated by a quadratic equation. For comparison of fits to the subsequent eqn (6), the value of K is taken either as the usual constant, independent of electrolyte concentration {where [H,O] in eqn (3) is taken as a constant (concept l)), or as a variable (concept 2). The introduction of x into eqn (1) and rearrangement yields the following expression for C'(H,O) for an equi-valent, incompletely ionized, electrolyte, K = x2y$M/( 1 -x) (4) C'(H,O) = C(H,O) - M(nx +j). ( 5 ) An equation that appears to represent a(H,O) up to high solute concentrations may be written as a(H,O) = [C'(H,O)/C(H,O-~U~~)]~'~ (6) where C(H,O-pure) equals 55.51 x density (in g ~ m - ~ ) and B is believed to represent a structural term for the solvent relating to a solvent structural unit.Thus when a solvent vaporizes, its liquid structure must be destroyed to form perhaps B separate moles of vapour from each mole of (free solvent) structural unit, and a mass-action expression can be applied. The following equations describe this behaviour : [H20], (in liquid) =+ B H,O (in vapour) (7)W. L. Marshall 2285 where Ef(H20)] (vapour) is the vapour fugacity (f), [(H,O),] (liquid) is the free concen- tration of the structural unit of solvent, which equals C'(H,O)/B, and K(vaporization) is taken to be a constant at low to moderate electrolyte concentrations.By normalizing eqn (8) to the concentration of solvent at infinite dilution and to the corresponding fugacity, and substituting a(H,O) for f(H,O)/f(H,O-pure), we obtain, [a(H2O>IR K'(vaporization) = [C'(H,O)/C(H,O-pure)] ' (9) At a given temperature at saturated vapour pressure, values of a(H,O) and C'(H,O) can be varied only by the dilution of the solvent by solutes or other solvents. K'(vaporization) is unity for pure solvent and, if it remains a constant upon dilution of solvent, then eqn (9) is simply equivalent to eqn (6). Constants and Data Used The objective was to test adherence to the model (with a comparison of fits by the use of concepts 1 and 2) by electrolytes for which values of the conventional K and of n could be obtained or estimated and for which very accurate values of a(H,O) and activity coefficients were available.Previous papersl-s have listed and illustrated several electrolytes that adhere to a straight-line relationship of log K us. log C'(H,O) in dioxane-water solutions, but always at the approach to infinite dilution of electrolyte. By extrapolating these straight lines to log C(H,O-pure), values of Kat this limit are obtained. The relevant values of n are obtained from the slopes. Values of K for the strong acids were approximated as that for NaCl, and their values of n were set equal to that for acetic acid in dioxane-water solutions.2 The activities of water, or the osmotic coefficients (+), and the mean molal activity coefficients ( y + ) [which were converted to molar activity coefficients b+) through application of4ensityl were collected from the critically evaluated tables at 25 "C given by Robinson and Stokes.loa Some more recent absolute vapour pressure measurements at 25 "C for NaCl-H,O solutionsll compared favourably with those compiled by Robinson and Stokes.(These latter measurements show graphically the difficulty in obtaining directly measured solvent activities at 25 "C to much better than 0.5% .) For NaC1-H,O solutions at temperatures from 25 to 300 "C values were obtained from the activity measurements of Liu and Lindsay1,* l3 and the compiled tables of Pitzer, Peiper and Busey.14 Since the model is developed on the basis of absolute concentration, tables of densities given e l ~ e w h e r e ~ ~ - ~ ~ were used for conversions from molality to molarity, and these separate sets of densities were fitted to quadratic equations for easy application. Acceptance of Concept 1 over Concept 2 Computer programs were written to evaluate solvent activities by concept 1 (C(H,O-pure) is used for [H,O] in eqn (3) whereby K" and K are both constants up to moderate electrolyte concentrations} and by concept 2, where C'(H,O) from eqn (1) is incorporated into eqn (3). The non-linear least-squares (NLLS) program of LietzkelR was used for the evaluations in these and all subsequent least-squares evaluations.The NLLS determination for a constant value of Kand values o f j and B by concept 1 was relatively easy. However, for concept 2, where C'(H,O) is used, K must vary with C'(H,O) in order to maintain the required constant value of K".This behaviour was incorporated into an iterative procedure together with a consecutive series of values for B in order to obtain the best- fit values of B and j for a given set of electrolyte solutions. The fits by concept 2 were somewhat poorer over a molality range 0-2 mol kg-I than those obtained by concept 1 [0.32% (2) versus 0.0604 (1) average deviations in calculated a(H,O)]. The fits when2286 Solution ojq Aqueous Elec frolytes extended to much greater molalities than 2 mol kg-l were vastly poorer with concept 2 than those by application of concept 1. With the much better fits by concept 1, there seemed to be no further reason to continue with concept 2. Therefore, it was concluded that the concentration of solvent water to be applied in solvent-solute equilibrium is that concentration (or density) of solvent in the immediate vicinity of the solute species, which is the microscopic concentration and not the bulk concentration. Over moderate ranges of electrolyte concentration, it thus would appear that K remains constant with the use of this microscopic water concentration for [H,O] in eqn (3).[Again, large changes in hydrostatic pressure or dilution with 'inert' dioxane will sharply change the microscopic concentration and accordingly K. The present, separate evaluations are all made isothermally at saturated vapour pressure, and therefore under these conditions (concept 1) K remains essentially a constant.] All further discussions in this paper consider only concept 1 as the most likely or predominant description of behaviour.Aqueous Sodium Chloride Behaviour, 25-300 "C Simultaneous Calculation for K, j and B up to 300 "C By ming published values of n (25-800 OC),l? 2 j 4-69 8. with interpolations for intermediate temperatures, simultaneously determined best fit values of K, j , and B were obtained by application of eqn (4)-(6) for describing the activities of water over aqueous NaCl solutions at temperatures from 25-300 "C. For these particular calculations, the range of molality of NaCl was limited to 2 mol kg-l so that values of K, j , B and n would not be expected to change significantly with electrolyte concentration at constant temperature. Moreover, y + was set equal to unity, as discussed later, since there were insignificant differences in fit, and obtained values of K and j with the use of y+ were not reasonable.Table l ( a ) gives the simultaneously determined values of logx, the solvation number of the neutral ion-pair (jZrn) and the structural constant (B). Included also are the values of n used, the sum (n +j2m), which should equal the sum of the average solvation numbers of the Na+ and C1- ions, and the percentage average deviations in fit from the published smoothed activities of water. It should be noted that the fit to the mathematically smoothed table values of Pitzer et al.14 allows an almost perfect fit at each temperature. This fit, however, may not reflect the inherent uncertainty in the 'raw' data evaluated in tabulating or calculating the smoothed values.The fits described in this study in reality simply compare calculated values of a(H,O) by the present model with the particular model of Pitzer et al. applied at 25-300 "C and, for the later comparisons, of Robinson and Stokes at 25 "C used to obtain their particular smoothed table values.loa To compare the many sets of original raw data at this time would have been prohibitive. Fig. 1 shows the best-fit values of log K,j,,, and B, and of the resulting (n +j,,) [table 1 (a)] plotted against temperature. The curves drawn represent the smoothed values given in table 1 (b) and discussed below. We observe that: (i) the separate values of log K make a relatively smooth curve, showing the usually observed maximum for aqueous electrolyte constants in the vicinity of 50-100 "C; (ii) j,, decreases with increasing temperature and approaches the constant value of 1.8 at 40&800 "C as observed previously;g (iii) B is essentially a constant, with an average value of 6.60k0.15; and (iv) the sum (n +j2m) is almost constant with an average value of 1 1.3, which is very close to the constant value of 12.0 given earlier for the range 400-800 0C.9 The S-shape of an assumed plot exactly through a set of best-fit values (log K, j2,, or B) or of (n +j2m) as a function of temperature is most certainly due to the fitting not of the 'raw data', but to the Pitzer et al.mathematical model that calculates a(H,O) as a function of both the temperature and the concentration of NaCl, as mentioned above.W. L. Marshall 2287 Table 1.Simultaneously obtained values of log K, j and B together with smoothed values for the ionization-solvation behaviour of NaCl at 25-300 "Ca (a) simultaneously obtained average n deviation T/"C usedb log K j,, B (n+kl> E% a(H2O)I" 25 6.43 0.693 ( f 0.034) 4.79 (f 0.19) 6.63 (k 0.03) 50 6.98 0.946 (k0.058) 3.58 (k0.35) 6.37 (f0.17) 100 7.87 1.004 (k0.071) 2.52 (20.46) 6.28 (k0.24) 150 8.55 0.784 (f0.048) 2.52 (fO.37) 6.50 (k0.19) 200 9.16 0.644 (k0.046) 2.38 (k0.37) 6.78 (k0.19) 250 9.66 0.416 (k0.038) 2.72 (f0.30) 7.19 (rfi0.14) 300 10.00 0.176 (f0.034) 2.07 (10.24) 6.46 (kO.10) average 6.60 ( & 0.15) ~ ~_________ ~ ~~ _ _ ~ ~ ~ ~ (b) smoothed values (B = 6.67) average deviation log K" T/"C log K j,, PA a(H,O>IC (n+jz,) ( c a W d i 1.22 10.56 10.39 11.07 11.54 12.38 12.07 11.32 0.002 0.003 0.004 0.004 0.005 0.005 0.005 0.004% from literatureb log K log K" 0 25 50 100 150 200 250 300 0.352" 0.724 0.823 0.864 0.801 0.640 0.446 0.220 5.85" 4.78 4.20 3.30 2.70 2.30 2.00 1.90 average : - 11.70" - 9.90 - - 0.005 11.21 - 10.48 1.179f - 10.00 0.004 11.19 -11.32 1.043 -11.13 0.005 11.17 -12.72 0.882 -12.70 0.0 10 11.25 -13.79 - 0.020 11.46 - 14.78 - - 0.01 7 11.66 -15.46 - 0.040 11.90 -15.96 0.210 -16.00 0.014% 11.44 - - a 0-2 mol kg-l NaCl; K based on molarity. The values in parentheses represent one standard deviation.Average deviations from reported smoothed values (an almost perfect fit). They do not reflect a fit to the 'raw' data (see text). From log K(2nd col), n(part a ; 5.85, extrapolated for 0 "C), and densities of water, with use of eqn (3); Extrapolated values at 0 "C.f Slightly revised from 1.155.l. 2i References in text. Fig. 2 shows representative plots of the deviations in fit, expressed as A log[a(H,O)], at 25 "C (part a) and 300 "C (part b) in solving forj,, versus a series of preselected values of B, with values of K at the two temperatures from table 1 (b). These plots are shown to emphasize the sharp minima in the deviations of fit for B at or near 6.67. A (universal) value of 6.67 was selected for the solvent constant for application up to moderate con- centrations since it was close to the average value found from the similar treatment of the series of electrolytes listed in table 2. Also, in the evaluation of a vaporization property, AE/(PAV), of liquids at the approach to the critical temperature, this property approaches asymptotically a constant value of 6.67 f 0.02 for water.lg The quantitative similarity seemed striking, and it was hypothesized that there might be a correlative relationship with respect to solvent structure. In another series of calculations, B was held constant at 6.67 and other values of log K and j,, were obtained simultaneously.The smoothed values of log K and of j,, from these latter calculations are given in table l(b), with percentage deviations in fit for a(H,O) by their use over molalities of 0-2 mol kg-l NaCl. Included also are extrapolated values for log K andj,, at 0 "C. From the values of log Kin table 1 (b) and of n in table 1, complete ionization constants (A") were calculated as defined by eqn (3) and where2288 Solution of Aqueous Electrolytes I I 1 I T 4 1 .o 0 .5 OD -- 0 f (b) 1 2 '\ 1 2 2 4 B 10 0 100 200 30 0 4 00 T/O c Fig. 1. Simultaneously determined values of (a) log K(based on molarity), (b)j2m and (c) B, and ( d ) the sum (n +j2m); aqueous sodium chloride, 0-2 mol kg-l, 25-300 "C. Triangles indicate results from ref. (9). the concentration of NaCl approaches infinite dilution. Again, at this limit concepts 1 and 2 are identical. These calculated values of log K" are included in table 1 (b) together with published 4-69 8* both of log K and of log K" obtained by extrapolation of values in dioxane-water solutions to pure water at saturated vapour pressure. Fig. 3 shows a plot of the presently calculated values of log K" and those from the literature at temperatures from 0 to 800 "C.The agreement is excellent, with the excep- tion of small divergences at 25 and 50 "C. The extrapolated value of log K at 25 "C from logarithms of the ionization constants in dioxane-water solutions is 1.179 (based on molarity) whereas the best-fit smoothed value from the present study is 0.724 as given in table 1 (b). Although this difference might appear to be seriously large, it should be recognized that ionization constants of 5, 15 or greater are very difficult, if not impossible, to obtain accurately by experiment. In this light, the moderate differences observed only at 25 and 50 "C should not be considered too seriously. However, any other reason for these differences is not obvious since the literature values at these two temperatures were obtained by extrapolation just like those at the higher temperatures.W.L. Marshall 2289 0.0010 0.0008 0.0006 * m .- 5 0.0010 e, 2 2 0 . 0 0 0 8 0.0006 0 . 0 0 0 t 0.0002 I \ ' I ' I ' I ' I ' I 1 1 1 1 1 1 1 1 1 1 1 1 1 - 0 2 4 6 8 10 12 B Fig. 2. The rerage deviation [A log a(H,O)] versus B for the solvent-ionization equilibria fit t 1 the observed activities of water over 0-2 mol kg-l sodium chloride solutions, 25 "C (part a> and 300 "C (part b). Solvation Number of the Ion Pair, NaCl" The values ofj,, given in table 1 (b) were plotted as logj,, against l/T(K) to produce a (pseudo) van't Hoff type of plot as shown in fig. 4. An approximately straight line is obtained that provides an 'enthalpy' (AH) of solvation or formation of 1.20 kcal mo1-l.j- This low value for AH is reasonable for a very weak interaction of water with the dipole of the ion pair. Thus, at moderately low salt concentrations, ca.4.78 water molecules would appear to associate on average with NaCl" at 25 "C; this value decreasing to ca. 1.9 at 300 "C and approaching a constant value of ca. 1.8 at higher temperature^.^ The sum of the average (primary) numbers of water molecules attached to the sodium and chloride ions, however, appears to remain constant from 25 to at least 800 "C. This proposal, given previously for the behaviour at 40&800 "C9 would now appear to be supported experimentally in the present study over the much larger range of temperature from 0 to at least 800 "C.Note that the value ofj,, at 0 "C, given in table 1 (b) and obtained by straight-line extrapolation as shown in fig. 4, is 5.85. This value is close to 6, which is certainly a t 1 cal z 4.18 J.2290 Solution of Aqueous Electrolytes Table 2. Ionization constants and solvation numbers for aqueous salts used to describe the activity of water at 25 "Ca _ _ _ _ _ _ _ _ _ _ ~ _ _ _ ~ _ _ ~ ~ ~ ~ ~~ ~ ~~ salt LiCl NaCl KC1 RbCl CSCl LiBr RbBr CsBr LiI RbI HC1 HBr HI (from elsewhere) Kh nc fit, (from fit) &2 mol dmP3 jSm (% deviation)" 10.0 6.40 15.1f 6.43 7.91 6.24 6.03 6.24 5.34 6.24 10.0 6.40 10.3 6.24 10.0 6.40 8.35 6.41 10.0 6.20 15.09 7.69 15.09 7.69 15.09 7.69 5.50 0.05 4.12f 0.05 3.70 0.05 3.53 0.03 3.00 0.02 5.59 0.1 1 2.85 0.05 2.09 0.04 5.00 0.07 2.43 0.03 4.5 1 0.07 4.78 0.13 4.78 0.09 average = 0.067; a &2 mol kg-' salt, B = 6.67, activity coefficients set at unity.Ionization constant, molar units (text). Increase in average solvation number upon ionization (text). Average solvation number of ion pair from data over a 0-2 moldmP3 range; standard errors in jzm are 0.02-0.04. Average ;< deviation in a(H,O) calculated from reported smoothed values. f Values of 5.30 and 4.78 for K and j Z m , respectively, obtained directly from activities (table lb). 9 Estimated (see text). reasonable number for a primary coordination shell around a small-sized solute species. The extrapolated value of n at 0 "C is also 5.85. The sum is 11.70 and closely agrees with the sums at higher temperatures given in table 1.It might appear that water molecules at the lowest temperature of liquid-phase existence (for pure water) solvate to essentially the maximum amount as a primary shell around the NaCI" ion pair. At higher temperatures, because of the weak NaCl"-H,O interactions, solvated water on the ion pair is removed rather easily. In contrast, H,O in the primary shells of Na+ and C1- ions appears to be held strongly over a wide range of temperature because of the electrical charge of the ions, which is present on the neutral ion pair only as a dipole separation. Concentration versus Activity for Behaviour of Vapour At temperatures up to 100 "C, the water vapour pressure is sufficiently low that the system can be treated as ideal. At higher temperatures there is a moderately large divergence from ideality.Since, in this description, concentration is used to describe the behaviour of 'free' water in the liquid phase, it was of interest to test whether a concentration ratio [rather than activity used in applying eqn (6)] of the water vapour would provide an acceptable description. Fits in determining K , j,, and B either simultaneously or separately therefore were made using a vapour concentration ratio for comparison with the use of activity. Vapour concentrations are easily calculated from the reported equilibrium vapour pressures12y l3 and the steam tables.,O The vapour concentration ratio (Cr) was taken to equal C(vap)/C(vap-pure) where C(vap) is the saturated concentration of water vapour over the solution and C(vap-pure)W.L. Marshall 229 1 -9 -10 -11 -12 h - u 2" -13 v 5 -14 -15 w -16 -17 18 TI0C 800 400 200 100 25 0 I 0 ,'- - / 1 .O 1.5 2.0 2.5 3.0 3.5 K/ T Fig. 3. Log K" for aqueous sodium chloride uersus l/T(K), 0-800 "C. (a), Results from this study. (O), Results from: ref. (9) (300-800°C); ref. (4) (300°C); ref. ( 6 ) (100°C); ref. (8) (50 "C); ref. (l), (2) and (3b) (25 "C). 0.80 0.75 0.70 0.65 0.60 e 0.55 .-. TI" C 300 200 100 50 25 0 M 2 0.50 :::I / , , , , , , 4 0.30 0.2 5 1.5 2 .o 2.5 3.0 3.5 4 .O KI T Fig. 4. Logj,, for NaCl" uersus l/T(K); 0-2 mol kg-I NaCl, 6300 "C. Logj,, = - 0.192 + 262/T. AH (solvation) = 1.2 kcal mol-l.2292 Solution of Aqueous Electrolytes is that over pure water at the same temperature. By substituting C, for activity on the left-hand side of eqn (6), the comparisons at 200 and 300 "C showed that only the use of activity [in eqn (6)] as given by the values of Liu and Lindsayl2, l3 and those compiled by Pitzer et al.14 produced consistent behaviour with the other measurements together with the good fits described above. Although solving simultaneously for j,,, K, and B with the use of C, [substituted for activity in eqn (6)] gave relatively good fits (but not so good as with activity), the values of K obtained differed widely from literature values. The simultaneously obtained values of B changed sharply with temperature and at 300 "C differed greatly from 6.67.Best-fit values for j,, did not allow the consistent behaviour obtained by the fits to activity. When for a molality range of 0-2 mol kg-l NaCl B was taken as 6.67, and either the literature or present values of K were used in solving for jZm, the average deviations in fit at 300 "C increased from 0.01 by use of activity to 0.5% with concentration. Thus, in the description given, the concentration of 'free' liquid solution water on the right-hand side of eqn (6) allows the good fits, but activity [in eqn (6)] is necessary for the best description of vapour.In our previous studies of ionization equilibria at the approach to infinite dilution of electrolyte, we had shown that concentration of the solvent water from 25 "C to supercritical temperatures, but always at densities higher than about 0.3 g ~ r n - ~ , provided very good fits to a model of mass-action solvation equilibria. The present observations appear to confirm experimentally (by the model) that activity of solvent is the proper quantity to use in gas-like regions.However, upon reaching liquid, or high density, regions the use of concentration greatly simplifies the description, and its use under these conditions would appear to be significant. Other Electrolytes and Extension to Concentrated Solutions Values o f j at 25 "C were calculated over six ranges of electrolyte molality for several salts and acids with the values of K and n given in table 2 and also with the assumption that K was equal to infinity. Attempts to obtain values of K, j and B simultaneously or of K and j over the 0-2 mol kg-l range for electrolytes other than NaCl were not satisfactory in that, while the fits were good, the best-fit values for the constants ( K , j and B or K andj) varied widely from consistent behaviour and reasonable expectations.As mentioned above, solving simultaneously for B andjzm for the 13 electrolytes at 25 "C did, however, produce an average value of B close to 6.67. It was concluded that the smoothed activities of NaCl represent the most extensively studied and evaluated set of activity measurements and presumably the most accurate, thus allowing the simultaneous attainment of relatively reasonable values of K, j , and B by the fitting procedure described above. For the other electrolytes, independently determined values of K and n were needed for obtaining reasonable values o f j . The determined values o f j over the 0-2 mol kg-* range (izm) and 0-full range OF) for the several 1 : 1 electrolytes by this procedure, together with the fits, are given in tables 2 and 3.Because the value of j for a given electrolyte depended partly upon the range of electrolyte concentration used for its calculation, the separately determined values were plotted against molality of electrolyte and extrapolated to infinite dilution to obtain j,. These values are included in table 3. In fitting over the entire range of molality, how- ever, an average best-fit value for B was found to be 5.50, which is perhaps reasonable if one considers that a high concentration of solute will certainly partly destroy solvent structure. When n for NaCl behaviour was set equal to (1 1.3-j,,) for the fits over the full range of molality at 25 to 300 OC, and with B kept equal to 6.67, best-fit values of j , were unreasonable and sometimes negative, with slightly poorer fits than by the procedure taken above. For this reason, the particular approach with the assumption of a constant value for n was used, although some ambiguity exists.However, attainment of a bestW. L. Marshall 2293 Table 3. The extrapolated solvation numbers for the ion pair at infinite dilution (j,,) and average values over a full range of electrolyte concentration OF) at 25 "C" full of over full range salt J o j , molality ("4 deviation)b range fit using j , LiCl NaCl KCl RbCl CSCl LiBr RbBr CsBr LiI RbI HCl HBr HI 4.82 4.90 4.54 4.14 3.66 4.50 3.72 3.22 4.09 3.16 3.68 3.37 4.0 1 3.57 3.06 1.94 1.89 1.52 3.77 1.06 0.53 3.45 0.82 2.34 3.36 3.35 0-6.0 C6.0 0 4 .8 0-5.0 c 6 . 0 0-6.0 0-5.0 0-5.0 G3.0 0-5.0 C6.0 0-3.0 b3.0 average 1.09 0.38 0.1 1 0.06 0.09 1.48 1.01 0.03 0.25 0.03 0.14 0.32 0.34 0.41 a B = 5.50, y , = 1; standard errors i n j are 0.02-0.04 % deviation in a(H,O-calcd) from reported smoothed values. Average fit for an average solvation number G when B is near 5.5, as described next, supports a decrease in the value of B at the high molalities. With K assumed to be infinity, x becomes unity and (nx+j) in eqn ( 5 ) can be replaced by an average (overall) solvation number (G), with no distinction made for solvated ions or ion pairs. Fits to a(H,O) in obtaining B and G simultaneously were made over the full range of molalities (producing GF) of the aqueous electrolytes listed in table 2.As expected, these values of G, fell between those of (n+jF) and j , while the average best- fit value of B was still in the vicinity of 5.5. The introduction of activity coefficients did not significantly change the fits or determined values o f j , GF or B, and therefore the activity coefficients were taken to be unity for the calculations in table 3, as discussed below. Consideration of a Debye-Huckel Contribution and Electrolyte Activity Coefficients Since the description of the water activities to high concentrations of electrolyte is described predominantly by the simple single term of eqn (6), the addition of an extended Debye-Huckel term (DHT) to this model can only be significant at moderately low concentrations. In order, therefore, to test its application in the descriptions, a DHT was divided by (1 + m2) that allowed the decrease in the Debye-Huckel contribution (DHC) to moderately low levels at high molalities (m) of electrolyte. The simple form of this denominator was taken after best-fit evaluations were made of several functions as discussed elsewhere.21 The introduction of DHC as an additional term in eqn (6), however, gave poorer fits to the measurements. Also, solving simultaneously for K, j and B [for NaCl solutions (25-300 "C)] gave values of K that were far removed from literature values.The addition of DHT alone (rather than DHC) sharply decreased the fit. The value of DHT becomes large at high ionic strengths and requires an offsetting term that eqn ( 5 ) and (6) cannot satisfy by best-fit adjustments in the values of K, j or B.2294 Solution of Aqueous Electrolytes - 0.0°2 t 1 NaCl/mol dm-3 Fig. 5.Deviation plot for the log activity of water at 25 "C over sodium chloride solutions from 0 concentration to saturation (5.3 mol dm-3) by the solvation model using activity Coefficients (y*) and solving simultaneously for B and jk. K = 15.1, n = 6.428, B = 5.455 and j , = 2.083. The introduction of electrolyte activity coefficients into eqn (2) or (4) did not significantly change the fits when values of K and n were used for the calculations ofj. However, when the measurements were fitted simultaneously for j and K, with activity coefficients, the resulting values of Kwere far removed from literature values. Nevertheless, when NaCl solutions at 25 "C were fitted over the entire range of concentration, the introduction of activity coefficients did moderately improve the fit as shown in fig.5 [0.05 % deviation cf. 0.13 % without activity coefficients]. Therefore it is difficult to decide unambiguously upon the significance of the electrolyte activity coefficients in this mass-action solvation model except that they do not make a large contribution. Simultaneous calculations for j , K and B, with activity coefficients, do not give reasonable values of j and K, and values of B are temperature and salt dependent. Lastly, a combination of both activity coefficients and DHC or DHT did not produce a better description or realistic values of K. In contrast, the model with activity coefficients taken as unity provides chemically reasonable quantities as shown in table I and fig.1 and a good, overall fit to the measured activities. Discussion Sodium Chloride Solutions At infinite dilution for NaCl at 25 "C, j o equals 4.90 (table 3) with n equal to 6.43 (table 2); the total solvation number for the two ions (n +j,) is therefore 1 1.33. Previously, j , was estimated to equal between 2 and 4 at 25 "C19 and 1.8 at 40&800 0C.9 The value of n increases from 6.43 at 25 "C to 10.2 at 400-800 0 C . 1 ~ 2 ~ 4 - 6 ~ 8 ~ 9 As noted above, if it is reasoned that water of solvation is strongly held by Na+ and Cl- ions even as the temperature rises, but weakly by the dipolar NaC1" ion pair, thenj, could decrease, n increase, and (p+ q) remain equal to 11 or 12 over the range 25-800 "C.Fig. 1 shows the decrease with increasing temperature in the simultaneously calculated values ofjBm, and the approximate constance of (n +jZm) over the 0-300 "C range. By accepting a valueW. L. Marshall 2295 of 4.9 forj, at 25 "C, @+q) would equal 11.3 at this temperature and be approximately the same at 400-800 "C (10.2+ 1.8 = 12). The present observations as described in table 1 and fig. 1 and 3 correlate extremely well with the earlier studies. At 25 "C there is excellent agreement of calculated activities of water with reported values (fig. 5) for NaC1-H,O solutions over the entire range of molality [0-6 mol kg-l (near saturation)] obtained by using the simultaneously determined values of B (5.455) andj, (2.083) together with values of y+ . The larger number of significant figures given for a(H,O) of NaCl solutions than forthose of other salt implies greater accuracy.Activity coefficients appear to be significant for NaCl when applied at high concentrations, but not for the other salts at 25 "C, possibly owing to the presumed lower accuracies for the other values. Nevertheless, by setting y+ equal to unity and again solving simultaneously for B and j F , the average deviation Tor NaCl(0-6.0 mol kg-l) is still low at 0.13 "/o in a(H,O). The effect of using y , - in the calculations seems to be small. Additivity of Obtained Solvation Numbers Table 4 compares differences in total solvation numbers (n+j,) between the several electrolytes for testing the consistency of the model in obtaining relative solvation numbers of ions.Column 2 shows that the several differences in each of the several sets are relatively close together, and in column 3 the differences are averaged. Recently Biggin et al.', have reported a coordination number of 5.5 & 0.4 for the C1- ion in NaCl solutions (obtained by neutron scattering structural studies for a molality of 5.32 at ambient temperature). If the solvation number ( q ) of C1- is assigned a value of 5.66 from the present study, based on (n+jo) for NaCl of 11.33 where 11.33/2 equals 5.66, and the values of (n+j,) for LiCI, NaCl and KCl are used in obtaining relative values ofp(Li+), p(Na+) and p(K+), then relative solvation numbers for the other cations and anions can be calculated, again by difference. The complete set of values are given in table 4, and they generally show an expected decrease in relative solvation number with an increase in ionic size for both the cations and the anions separately.The overall consistency would appear to be surprisingly good. Free Water and Solubility Calculated concentrations of free water by eqn (9, with K, n andj,, from table 2, and with B and y* equal to 6.67 and unity, respectively, are plotted against M(sa1t) in fig. 6 for the several salts except those of lithium. Extrapolations to C'(H,O) equal to zero provide values of M(sa1t) in the vicinity of the solubility for each salt. There is very close agreement for NaC1, RbBr and RbI. If the rationale is significant, strong divergences for the other salts must occur at C'(H,O) below ca. 15 mol dm-3.The high solubilities of the lithium salts do not justify this reasoning, and it can only be conjectured that C'(H,O) reaching 'zero' is meaningful with respect to solubility for the other salts. Conclusions There are many experimental and theoretical studies aimed toward describing concen- trated aqueous electrolyte behaviour. The majority of studies are concerned with calculation of activity coefficients and, except for those ions that strongly complex, do not include ionization (or association) constants or solvation numbers. The present study provides chemical insight into the actual behaviour of aqueous electrolytes and thereby shows that their behaviour may be described predominantly by a chemical model, with a chemical significance for all terms used. The correlative behaviour of additional parameters and terms, while providing better fits at high concentrations to the reported smoothed values (including any inaccuracies introduced2296 Solution of Aqueous Electrolytes Table 4.Relative solvation differences and numbers for ions at 25 "Ca relative differences ionic in total solvation electrolyte solvation average number, pair numberb difference 25 "C HC1-LiC1 HBr-LiBr HI-LiI HCI-RbBr HBr-RbBr HI-RbI HCl-CsCl HBr-CsBr LiCl -RbC1 LiBr-RbBr LiI-RbI RbC1-CsCl RbBr-CsBr LiCl-LiBr RbC1-RbBr CsC1-CsBr LiCI-LiI RbC1-RbI LiBr-LiI RbBr-RbI ~~~ 0.06 0.07 1.11 0.90 1.01 1.1 1 1.38 1.35 0.84 0.94 1.14 0.48 0.34 0.32 0.42 0.28 0.72 1.02 0.40 0.60 ~~ ~~ H+ 5.97 Lif 5.56 0.41 (H+-Li+) Na' 5.66 Ki 5.12 1.39 (H+ - Rb+) 1.36 (H' - CS') Rb+ 4.59' Cs+ 4.40' 0.97 (Li+-Rb+) - Cl- 5.66* Br- 5.32 0.41(Rb+-Csf) I- 4.80' 0.34 (ClP - Br-) 0.87 (C1- - I-) 0.50 (Br ~ -I-) a Values for n andj, from tables 2 and 3 ; ionic solvation numbers based on q(C1F) = 5.66 and (n+j,) for LiCl, NaCl and KC1 and averages in column 3 ; T = 25 "C.Number represents ( n +j,) (1 st electrolyte) minus (n +j,) (2nd electrolyte), values from table 2. Averaged from two values from column 3. Arbitrarily assigned a value of 5.66. by smoothing), would effectively reduce the quantitative significance of the chemically defined constants in the model. The earlier model of Robinson and Stokes10bt23 includes solvation numbers, but not association constants. GlueckauP* emphasized that solvation numbers would decrease with increasing concentration of electrolyte, which is the observed behaviour in this study with respect to average solvation numbers.(The observed average decrease evidently reflects a lessening reliance on concept 1 as the range of electrolyte concentration is extended since concept 1 assumes a constant microscopic concentration of solvent, thereby assuming a constant solvation number for each species. However, full reliance on concept 2 is not indicated, as discussed above.) The Robinson-Stokes-Glueckauf model describes activity coefficients in terms of solvation numbers of ions, with application of Debye-Huckel theory, but without allowance for some association to form ion pairs.W. L. Marshall 2297 I I I I I 60 m I c 40 e 0 E 0- u . - z 20 0 salt concentration/mol dm-3 Fig.6. The approach of the free water concentration [C'(H,O)] to zero at the solubility limit for several selected salts; solvation model with B = 6.67 and j z m ; T = 25 "C. Solubilities: (O), NaCl; (a), KC1; (O), RbCl; (a), CsC1; (m), RbBr; (B), CsBr; (A), RbI; and (---), extrapolated values. The electrostatic-interaction model of Pitzer et aZ.25-28 has been applied widely for fitting aqueous electrolyte behaviour over a broad range of temperature to obtain activity coefficients and solution thermodynamic properties. Helgeson et al. 29 9 30 have presented an electrostatic-interaction model also for application to aqueous electrolytes at higher temperatures. In another approach, the Lietzke-St~ughton-Fuoss~~ two-structure model for aqueous electrolytes demonstrates good accuracy in calculating electrolyte activity coefficients by decreasing the Debye-Huckel contribution as the concentration of electrolyte increases.The present model incorporates solvation numbers, an ionization constant, and a universal structural constant (B) relating the activity of water to the concentration of free (non-solvating) water in the liquid phase. It must be noted that, for a given electrolyte, values of K and of n can be obtained from independent and completely different types of studies, e.g. from interpretations of conductance rneas~rements.~-~ If B is takefl to have a universal value of 6.67 (for water), then there is only one necessary parameter (j) for fitting the activities. Since j is believed to be the average solvation number of the ion pair, then it too can, in principle, be obtained independently for application with eqn (6), thus providing a description of solvent activity requiring no chemically undefined parameters. There does not seem to be any inconsistency in this approach and the earlier approaches taken for descriptions at infinite dilution.lP9 Perhaps it would appear that acceptance of concept 1 is not consistent with application of a complete constant at infinite dilution of electrolyte, where C(H,O) varying as the nth power allows K" to remain a constant. However, at infinite dilution concept 1 equals concept 2, and there is no contradiction. The mass-action equilibria between solvent and electrolyte species of finite concentration with respect to the microscopic concentration of solvent would seem to be reasonable chemical behaviour (as borne out by better fits compared to application of concept 2).Again, the microscopic concentration can be easily changed by hydrostatic pressure or by dilution with an 'inert' solvent in the same manner as C(H,O) was changed at the limit of infinite dilution. The apparently constant value of the defined solvent structural constant B at moderately low electrolyte concentrations over the wide range of temperature (25-300 "C)2298 Solution of Aqueous Electrolytes might be considered inconsistent since some may feel that water structure changes markedly with widely changing temperature. Narten et al.,32 however, have shown by X-ray diffraction that for H 2 0 the average number of nearest (primary) neighbours remains constant at 4.4 from 4 to 200 "C.Thus 'structure', as related to the number of nearest neighbours, is independent of temperature over this range. The conclusions in the present paper agree with Narten et al. By accepting the conclusion of Narten et al. a coordination number for secondary neighbours must also remain constant with changing temperature (from 4 to 200 "C). The value of B is defined to represent a coordination number for a solvent molecule in a unit structural cell, and this number may represent both nearest neighbour and other near neighbours. As temperature increases, the liquid density decreases because of increasing kinetic energy that increases intermolecular distances. It does not necessarily follow that the liquid structure must change.The presumed evidence from this indicates that an average coordination number, representing ' bonds ' destroyed in the vaporization (including nearest and possibly some secondary neighbours) of a water molecule remains essentially a constant with increasing temperature, and may remain so even as the critical temperature is ap~r0ached.l~ A structural constant for H 2 0 of 6.7 was first given by this author in 197033 and equations representing successive solvent-solute mass-action equilibria in 1 972.3b The present paper describes the application to concentrated solutions from extensions, revisions and refinements of that earlier work. This paper thus presents a chemical model using mass-action equilibria that closely describes a particular experimental behaviour of aqueous electrolyte systems.The model produces a greater insight into the chemical equilibria that are necessarily involved in these systems. The observations describe the activity of water to be proportional to the concentration of free (non-solvating) water raised to the reciprocal power of the structural constant, rather than to unity as might be generally assumed. The model produces ionization constants (for NaC1) and average solvation numbers for electrolytes in excellent agreement with expectations and values obtained independently, and provides possible insights into the structural nature of the solvent and its behaviour with temperature. Another paper describes the behaviour of aqueous electrolytes up to 300 "C by a modification of Raoult's law in accordance with the same rationale.21 I thank R.H. Busey, M. F. Holmes, M. H. Lietzke, H. F. McDufEe, R. E. Mesmer and J. E. Ricci for their helpful comments. This work was sponsored by the Division of Chemical Sciences, Office of Basic Energy Sciences, U.S. Department of Energy, under contract DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc. References 1 W. L. Marshall and A. S. Quist, Proc. Natl. Acad. Sci., USA, 1967, 58, 901. 2 A. S. Quist and W. L. Marshall, J. Phys. Chem., 1968, 72, 1536. 3 W. L. Marshall, J . Phys. Chem., (a) 1970, 74, 346; (b) 1972, 76, 720. 4 L. B. Yeatts and W. L. Marshall, J. Phys. Chem., 1972, 76, 1053. 5 L. A. Dunn and W. L. Marshall, J . Phys. Chem., 1969, 73, 723. 6 L. B. Yeatts, L. A. Dunn and W. L. Marshall, J . Phys. Chem., 1971, 75, 1099. 7 W. L. Marshall, Rec. Chem. Prog., 1969, 30, 61. 8 T. H. Leong and L. A. Dunn, J . Phys. Chem., 1972,76,2294. 9 A. S. Quist and W. L. Marshall: J. Phys. Chem., 1968, 72, 684. 10 R. .4. Robinson and R. H. Stokes, Electrolyte Solutions (Butterworths, London, 2nd edn, 1965); (a) 11 C. N. Pepela and P. J. Dunlop, J . Chem. Thermodyn., 1972, 4, 255. 12 C. Liu and W. T. Lindsay Jr, J . Phys. Chem., 1970, 74, 341. 13 C. Liu and W. T. Lindsay Jr, J . Solution Chew., 1972, 1, 45. 14 K. S. Pitzer, J. C. Peiper and R. H. Busey, J . Phys. Chem. Ref. Data, 1984. 13, 1. 15 International Critical Tables, ed. E. W. Washburn (McGraw-Hill, New York, 1928), vol. 111. appendices 8.3-8.10; (b) pp. 238-252.W. L. Marshall 2299 16 R. W. Potter and D. L. Brown, Geological Survey Bulletin 1421-C (U.S. Government Printing Office, 17 P. S. Z. Rogers and K. S. Pitzer, J . Phys. Chem. Ref. Data, 1982, 11, 15. 18 M. H. Lietzke, A Generalized Least Squares Program for the IBM 7090 Computer, Oak Ridge National Laboratory Report ORNL-3259 (April 1962). 19 W. L. Marshall, J . Phys. Chem., 1985,89,4128. 20 International Formulation Committee 1984 Steam Tables of the International Association for the Properties of Steam, 1984. 21 W. L. Marshall, J . Solution Chem., 1986, 15. 439. 22 S. Biggin, J. E. Enderby, R. L. Hahn, and A. H. Narten, J . Phys. Chem., 1984, 88, 3634. 23 R. H. Stokes and R. A. Robinson, J . Am. Chem. Soc., 1970, 70, 1870. 24 E. Gleuckauf, Trans. Faraday Soc., 1955, 51, 1235. 25 K. S. Pitzer, J . Phys. Chem., 1973, 77, 263. 26 K. S. Pitzer and G. Mayorga, J . Phys. Chem., 1973, 77, 2300. 27 K. S. Pitzer and G. Mayorga, J . Solution Chem., 1974, 3, 539. 28 K. S. Pitzer and J. J. Kim, J. Am. Chem. Soc., 1974, 96, 5701. 29 H. C. Helgeson and D. H. Kirkham, Am. J . Sci., 1974, 274, 1089; 1199; 1976, 276, 97. 30 H. C. Helgeson, D. H. Kirkham, and G. C. Flowers, Am. J . Sci., 1249, 281, 1249. 31 M. H. Lietzke, R. W. Stoughton, and R. M. Fuoss, Proc. Natl Acad. Sci. US, 1968, 59, 39. 32 A. H. Narten, M. D. Danford, and H. A. Levy, Discuss. Faraday Soc., 1967, 43, 97. 33 W. L. Marshall, Complete Equilibrium Constants andh'ew Relationships to Electrolyte-Solvent Equilibria, Published Book of Abstracts, 160th Natl. Meeting of the American Chemical Society, Abstract 91, Div. of Phys. Chem. (American Chemical Society, Chicago, 1970). Washington 1977). Paper 5/61 1 ; Received 1 Ith April, 1985