J . Chem. SOC., Faraday Trans. I, 1989, 85(9), 2649-2664 Experimental Activity Coefficients in Aqueous Mixed Solutions of KCl and KF at 25 "C compared to Monte Carlo Simulations and Mean Spherical Approximation Calculations? Torben S. Serensen,* Jergen B. Jensen and Peter Sloth Fysisk- Kemisk Institut and Center for Modelering, Ikke-lineare systemers Dynamik og Irreversibel Termodynamik (MIDIT), Technical University of Denmark, Bldg. 206, DK-2800, Lyngby, Denmark A system of three different ions in solution has been studied experimentally and theoretically. Mean molar ionic activity coefficients have been measured in pure and mixed, aqueous solutions of KF and KCI at 25 "C and 1 atm by means of valinomycin, LaF, and Ag/AgCl electrodes. More than 200 independent electrometric measurements were considered.The ionic strength varied from 0.0005 to 4 rnol dm-3. The activity coefficients were closer to unity for pure KF than for KF in equimolar mixture with KCl at the same ionic strength for ionic strengths higher than 1 mol dm-3. The activity coefficients for KCI in pure and mixed solutions could not be statistically separated up to 4 rnol dm-3. The Harned coefficients are estimated to be Ok0.025 dm3 mol-' for KCI and 0.055+0.025 dm3 mol-' for KF. The Debye-Huckel limiting law is obeyed within 1.5% in the region from 0.0005 to 0.01 rnol drnp3, indicating that the ions involved are small. Comparison with calculations for the primitive electrolyte model using the Kirkwood-Buff equations and the generalized DHX theory has shown, that the expcrimental data are approximately fitted using diameters of 2.9, 2.9 and 3.4A for the K+, C1-, and F- ions, respectively.The same values fit approximately the data in the mean spherical approximation (MSA). The MSA calculations demonstrate the validity of Harned's rule. From the latter theory we also obtain single-ion activity coefficients. Monte Carlo (MC) simulations of single-ion activity coefficients have been performed for the primitive, electrolyte model for the above-mentione$ ionic diameters and in addition for a diameter of the F- ion equal to 3.7 A. Widom's test-particle method is used in conjunction with the extrapolation procedure suggested by Sloth and Smensen. The MSA as well as the MC calculations support earlier suggestions, that the single-ion activity coefficient of F- in KF-KCI mixed solutions is almost independent of the salt ratio.The same is approximately torue for the C1- ion. Increasing the diameter of the F- ion from 3.4 to 3.7 A does not alter this conclusion, but the separation between single-ion activity coefficients becomes larger. Bagg and Rechnitzl found the interesting experimental result, that the single-ion activity coefficient of the fluoride ion for mixtures of trace concentrations of NaF in NaCl up to 1 rnol kg-' or of trace concentrations of KF in KX solutions up to 4 mol kg-' (X = C1, Br, and I) were the same as in pure NaF of KF solutions at the same ionic strength. Similar results were also obtained by Leyendekkers,* who studied trace activities of F- in NaF-NaCI and NaF-KCl mixed solutions. Although such electrochemical measurements of single-ion activities based on the use of salt bridges are always t Paper presented at the Third International IUPAC Symposium on Solubility Phenomena.held at the University of Surrey, 23-26 August, 1988. 2649valinomycin membrane aq. KCl-KF solution with electrode (K+) KCl fraction = X and ionic The valinomycin and LaF, electrodes are both close approximations to ideal ion- selective membranes for K+ and F- ions, respectively.1*2*19.20 We have used the electrodes F23 12K and F1052F manufactured by Radiometer A/S. The Ag/AgCl electrodes were high-precision electrodes ( & 10 pV) of the thermal-electrolytic kindly lent to us from the pH-calibration department of Radiometer A/S. A detailed description of the performance of these electrodes have been given elsewhere.'.21 The cells were built up as a water-jacketed Pyrex glass cell with lid. Through holes in the lid, the electrodes were immersed in ca. 100 cm3 of the solution. The temperature was kept at 25.00+0.05 "C by means of circulating water from a thermostat. The e.m.f. values were registered on a pH-meter (PHM64, Radiometer A/S) with a precision of t 1 atm = 101 325 Pa. Ag/AgClT. S. Sorensen, J. B. Jensen and P. Sloth 265 1 k 0.1 mV [old data from ref. (8)] and on a Radiometer PHM84 with the same measuring precision (new data). The potentials were followed on an REC 80 recorder from Radiometer, until constant potentials were obtained (within 0.1 mV). Normally, that took, 2-4 h depending on the concentration.The old data' were grouped in 15 different series with KC1 fractions X = 0, 0.1,0.2, 0.5, 0.8, 0.9 and 1. In each series, the ionic strength was varied from 0.001 to 0.2 mol dm-,. Additional measurements were performed in more concentrated solutions up to 4 mol dm-3 for X = 0 and 1 (pure salts) and up to 2 mol dmP3 for X = 0.5. Each series was prepared and selected at random, and measurements were made independently. In ref. (8), standard potential differences and Debye-Huckel limiting slopes were determined for each series from polynomial fits of the e.m.f. values (corrected for the simple Nernst concentration terms) against h. There seemed to be no systematic variation of the standard potentials or the limiting slopes with X , which is a test of the lack of interference of the foreign ions on each of the three ion-selective electrodes.However, we found a slow variation of the normal potentials of the valinomycin and LaF, electrodes over a timescale of several months.' Thus, it is necessary to find new normal potentials after a few experimental series. The activity coefficients have to be calculated with the 'local ' normal potential difference for the series. Afterwards, however, activity coefficients may be pooled together in order to improve the precision of the final mean data. The procedure followed in order to find the normal potential of a given experimental series was the same in the new experiments as described earlier. The e.m.f. values of the cells were corrected for the Nernst concentration term and plotted against the square root of the ionic strength.The most significant polynomials in h at the 95% level of significance were found, and the normal potentials were determined by extrapolation to In the earlier experiments,' the limiting slope for all the series pooled together was statistically indistinguishable from the Debye-Huckel value for the KC1 measurements (cell 1). However, the limiting slope for the KF data (cell 2) was ca. 10% too low. Nevertheless, with the new data added we shall see, that lny+(KF) is also compatible with the limiting law. The old experiments consisted of 56 measurements of y+(KCl) on cell 1, and 46 measurements of y+(KF) on cell 2. The new experiments-added 73 independent measurements on cell (f) and 82 independent measurements on cell (2).The analysis presented here is therefore based upon 129 measurements on cell (1) and 128 on cell (2). The chemicals used were Merck (pro analysis) KF and KCl. They were dried at 110 "C for at least 24 h and then used without further purification. They were cooled in desiccators with silica gel at least 30 min before weighing with a precision *0.05 mg. KF especially is hygroscopic. Solutions with I 2 0.1 mol dm-3 were made independently by weight (k 0.05 mg). The rest was made by careful dilution of stock solutions. The water used was distilled and deionized water with a conductivity < 1.0 pS at 25 "C. The Monte Carlo calculations were exploiting Metropolis sampling on the primitive electrolyte model (charged hard spheres in a dielectric continuum) as described e a ~ l i e r .~ ~ - ~ ~ The new features are that there are one cation and two anions which may all be of different sizes. Furthermore, 64 test particles of each type are inserted in the central box in a regular lattice. The interactions of the test particles with the 'real ions' are averaged in order to calculate the excess chemical potential of each ion according to the method of Wid~m."*'~ The real ions pass freely through the test ions, however. Thus, the presence of the test particles does not disturb the Metropolis Markov chain. In order to extrapolate to an infinite number of particles ( N = a), we have proved that the obtained values for the excess chemical potentials for each ion should be plotted against the cube root of 1 / N and extrapolated linearly to zero.This was proven in ref. (16) (fig. 1 and table 3). fi = 0.2652 Aqueous Mixed Solutions of KCl and KF Statistical Treatment of Experimental Data and MC Data There was a considerable scatter in the lny+ values at very low concentrations. However, if we collect all measuring points for Iny+(KCl) and 0.0005 < I / m ~ l d m - ~ < 0.01 without regard to X (66 data points), only a fir%-9egree1 polynomial in $ is significant at the 95 O/O level with a slope of - 1.22f0.10 dm? mol-?. Similarly, the 70 independent measurements for lny+(KF) in the same concentration range without regard to <gave ? straight line at the 95 % level of significance with a slope equal to - 1.10 0.07 dmz mol-2. Both values are cFmpatjble with the limiting-law value from the Debye-Hiickel theory - 1.1779 dm.mo1-p.26 When all the 136 data points were pooled, without regard to X and to the type of electrolyte, we still obtain:d a stlraight line at the 95 YO level of significance, and the slope was - 1.160 f 0.025 dm? mol-5, which is only ca. 1.5 YO from the limiting-law value. The reason why the limiting law is followed in the whole region from 0.005 to 0.01 mol dmP3 is that the diameters of the hydrated ions for this specific system are extraordinarily small (see later). Thus, there seems to be no reason to believe that the limiting law should not be fulfilled for the present pure and mixed solution:, andlwe proceed by enforcing the initial slope of the lny+ us. fi plots to be - 1.1779 dmz mol-5 for lny+(KCl, X = l), lny+(KCl, X = O S ) , lny,(KF, X = 0) and lny+(KF, X = 0.5) us. fi.- From the data points, we then determine the highest polynomials which are significant at the 95 % level. We use orthogonal Forsythe polynomial^^^ for the ‘ free polynomials ’ for which the coefficient of each new orthogonal polynomial of one higher degree can be tested without any covariance by the Student’s After having determined the significant degree and the error belts of the polynomial, we determine the least-square polynomial of the same degree subject to the restrictions, that it should go through (0, 0), which the free polynomials do within the uncertainty, and that they should have the correct limiting slope. We observed only a very weak statistical tendency at intermediary concentrations for the lny+(KCl) values to move towards the lny+(KF) values with admixture of KF.Therefore, we have preferred to pool together dl data for Iny+(KCl) - for X = 1 and X = 0.5. - We obtained the following results : p = $ ( I in mol dmP3) (3) (0 < p d 2.0) (4) lny+(KCl, - pure and X = 0.5) = - l.l779p+ 1.0958~~-0.532823p3+0.10659p4 Number of experiments: 13 (previous, X = 1) + 21 (new, X = 1) + 17 (previous, X = 0.5) + 52 (new, X = 0.5). lny,(pure KF) = - l.l779p+ 1.5956~~- 1.5179p3+0.78008p4-0.142732~~ (0 < p < 2.0) ( 5 ) Number of experiments : 10 (previous) + 30 (new). lny+(KF, - X = 0.5) = - l.l779p+ 1.0935~~-0.47354p3+0.083062p4(0 < p < 1.9) (6) Number of experiments: 12 (previous) + 52 (new). The results are shown in fig.1-3 together with the best least-square polynomials (full curves) and some Mean Spherical Approximation calculations (dashed lines). Experimental points with I < 0.01 mol dm-3 have been omitted on these figures, since the great amount of low-concentration data cannot be properly represented. However, they have not been dropped from the least-squares calculations. Fig. 4 exhibits the error belts inside which the true (smoothed) values of -In y+ - (KF)T. S. Sorensen. J. B. Jensen and P. Sloth 0.6 0.7k A 2653 I " " " " " " " ' * " ' ~ . p 0.1 0.5 1.0 1.5 2.0 I /mol* dm- 2 Fig. 1. The negative natural logarithm of the mean ionic, molar activity coefficient of KCI, -Iny+ ure and mixed aqueous solutions of KCI us. the square root of the molar ionic strength, (KC1), at 25 "C and atmospheric pressure.a, fraction of KCI = X = 1. A, X = 0.5. ., coincident data points (X = 1 and 0.5). Solid curye: polynomial [eqn (4)]. Dashed curve: MSA calculation for X = 1 with diameters 2.9 and 2.9 A for K' and C1-. All measurements below the vertical dashed line have been omitted from the figure, but not in calculating the least-square polynomial [eqn (4)]. 1 1 1 Fig. 2. Aqueous solutions of KF at 25 "C and atmospheric pressure. Salt fraction of K F = X,, = 1 . Plot of -Iny+(KF) us. fi. Filled circles: experiments. All experimental points below vertical dashed line have been omitted from the plot, but not in calculating the least-square polynomial [eqn (5)]. Solid curve: polynomial [eqn (5)].oDashed curve : MSA calculation with diameters 3.4 and 2.9A for F- and K'. lie within a probability of 66 % for pure KF as yell as for equimolar KCl-KF mixtures.The two curves clearly separate above = 1 molTdmi, and the separation at higher ionic strengths is much more pronounced and unequivocal than for KCl. The KF curves move in the direction of the combined KCl curve with admixture of KC1. The Monte Carlo calculations were also subjected to statistical analysis. The values of the negative logarithms of the single-ion activity coefficients (- lny,) were plotted against the inverse cube root of the number of particles. In all cases investigated, straight lines only were significant at the 95% significance level.2654 Aqueous Mixed Solutions of KCl and KF Fig. 3. Aqueous solutions of equimolar mixtures of KF and KCI (X = 0.5) at 25 "C and atmospheric pressure.Plot of -In y,(KF) us. k. Filled circles : experiments. All experimental values below the vertical dashed line have been omitted from the plot, but not in calculating the least-square polynomial [eqn (ti)]. Solid curve : polynomial [eqn (6)]. 0.7 i 1 1 3 1 2 /mol! dm- 2 Fig. 4. Plots of the smoothed polynomial values of -Iny+(KF) us. b for (a) pure (XKp = 1 ) and (b) mixed (XKV = 0.5) aqueous KF solutions at 25 "C and atmospheric pressure. The error belts are shown, within which the 'true' values of -Iny+(KF) lie with 66% probability. A clear separation of the two curves is seen- above I = 1 mol dm-3. MSA and MC Calculations From fig. 1 and 2 we learn, that the experimental data for pure KCl and pure KF solutioons are fitted quite well by MSA calculations assuming a contact distance equal to 2.90 A between K+ and C1- and 3.15 A between K' and F-.The fit is valid up to ca. 2 mol dm-3. Furthermore, an equal division of the 'primitive model effective ionic radii' has been assumed for K+ and C1-, since the limiting conductivities of those two ions areT. S. Sorensen, J. B. Jensen and P . Sloth 2655 I ' I I I I I I I I I Fig. 5. Plots of -Iny+(KCI) [(a), (b) and (e)] and of -Iny+(KF) [(c), ( d ) and (f)] us. A',, and XKc,, respectively, at constant total ionic strength (4, 1 rnol dm-3; B, 0.5 rnol dmP3) calculated by the MSA (solid lines). Bjerrum length &, = 7.135 A (water, 25 "C), diameters 2.9, 2.9 and 3.4 A for K', C1- and F-, respectively. Dashed lines are similar M C test particle calculations for a total ionic strength equal to 1 mol dm-3.The MSA calculations exhibit clearly the Harned linearity of the 'artificial' KCl-KF system. The MSA calculations, as well as the MC calculations, verify the empirical rule of thumb, that the two trace activity coefficients are nearly, but not completely, identical. approximately equal. Thus, the effective ionic diameters are 2.9, 2.9 and 3 . 4 i for K+, C1- and F-, respectively. Next, we consider mixtures. We want to test Harned's rule' which can be written in the form lny+(KCl, - fixed I ) = lny,(pure KCl, Z)-aKc.(l - X ) (7) lny+(KF, - fixed I ) = lny,(pure KF, I)-a,,X. (8) The (non-dimensionalized) Harned coefficients are aKCl and aKF. As first derived by Glueckauf er al.,29 the Harned coefficients with dimension dm3mol-l (or kg solvent mol-') may vary with ionic strength, but their sum must be independent of ionic strength in order for the system of equations (7) and (8) to satisfy the Maxwell equations of cross- differentiation of free energy.This means, that the sum of the non-dimensional Harned coefficients must be proportional to the total ionic strength: aKCl + aKF = constant x I. (9) Fig. 5 and 6 demonstrate the Harned linearity of MSA calculations of Iny+ for mixtures of KCl and K F at constant ionic strength. The strict linearity is observed-at total ionic strengths equal to 0.5, 1 .O, 1.5 and 2.0 mol dmP3. In all cases, the logarithms of the trace activity coefficients of one salt in great excess of the other tend almost to meet each other, approximately in the middle between the values for the two pure salts.The tendency for trace activity coefficients to be almost, but not exactly, equal has long been known as an empirical rule of thumb for many electrolyte mixture^.^2656 Aqueous Mixed Solutions of KCl and KF 0.50 0.40 I I I 1 I 1 I I I I 0 0.2 0.4 0.6 0.8 1 .o XKCI ~ ~ X K F Fig. 6. Plots of -lny+(KCl) [(a) and (b)] and of -Iny+(KF) [(c) and ( d ) ] us. A', and XKC,, respectively, at constant total ionic strength (A, solid linei: 1.5 mol dmP3; B, dashed lines: 2 mol dm-:) calculated by the MSA. Bjerrum length AH = 7.135 A (water, 25 "C), diameters 2.9,2.9 and 3.4 A for K+, C1- and F-, respectively. The Harned linearity is clearly exhibited, and the two trace activity coefficients are nearly, but not completely, identical.Table 1. MSA values for the dimensionless Harned coefficients for KF-KCl" total I/mol dmP3 &KCl ~ K F ~ K C I +a,, 0.5 - 0.0294 0.304 0.00 10 1 .o - 0.0268 0.0280 0.00 12 1.0 (MCb) -0.050 & 0.0 1 1 0.008 f 0.01 7 - 0.042 & 0.020 1.5 - 0.0256 0.0270 0.0014 2.0 -0.0255 0.0260 0.0005 " Diametets: 2.9,2.9 and 3.4 A for K', C1- and F-, respectively; Bjerrum length: AR = e:/(4nekT) = 7.1355 A. MC values calculated from table 2. Table 1 shows calculated dimensionless Harned coefficitpts from the MSA theory. All calculations are performed for di9meters 2.9,2.9 and 3.4 A for K+, C1- and F- and with a Bjerrum length equal to 7.135 A (ca. 25 "C in water). The Bjerrum length is given by: AB = ei/4nek, T (10) where e, is the elementary charge, k, is Boltzmann's constant, T the absolute temperature and E the dielectric permittivity of the solvent.From table 1, it is obvious, that the dimensionless Harned coefficients vary slowly with the total ionic strength. The sum (last column) does not seem to be proportional to the total ionic strength. However, the sum is only 2-5 O/O of the value of the individual Harned coefficients. Thus, it might be argued that the sum should really be zero if not for some inevitable lack of consistency in the MSA approximation. Then, the data are compatible with eqn (9) with the constant = 0. In order to check the MSA calculations, we have also made a direct determination of the single-ion activities in a 1 mol dmW3 mixture of KC1 and KF by Monte Carlo simulations using Widom's test-particle formalism.W? have made two sets of calculations, one with the diameter of F- set equal to 3.7 A and one with the diameterT. S. Smensen, J. B. Jensen and P. Sloth 2657 Table 2. Monte Carlo results for lnyi for KF-KCl mixtures with dF = 3.4Au Nb N(K+) N(CI-) N(F-) -Iny, -InY,, - In Y , no. config./ lo6 32 16 64 32 80 40 100 50 150 75 216 108 350 175 512 256 1000 500 c o - 100 50 140 70 160 80 216 108 360 180 512 256 c o - 64 32 80 40 100 50 130 65 150 75 216 108 350 175 512 256 c o - 0 X = 0, pure KF, d, = 2.9 A 0 16 0.2096 - 0. I950 0 32 0.2530 - 0.2435 0 40 0.2709 0.2566 0 50 0.2763 - 0.2686 0.291 1 0 75 0.2986 - 0 108 0.3 141 0.3 127 0 175 0.3421 0.3161 0 256 0.3682 - 0.3513 0 500 0.3679 - 0.3544 0.45 13 - 0.4345 - + 0.0079 0.0072 - - - - - X = 0.5, KF : KCl = 1.1 , dK = 2.9 A, d,, = 2.9 A 25 25 0.308 1 0.3404 0.2742 35 35 0331 1 0.3597 0.295 1 40 40 0.3340 0.3657 0.2978 54 54 0.347 1 0.3743 0.3109 90 90 0.3565 0.403 1 0.3333 128 128 0.3827 0.4047 0.3345 - - 0.469 1 0.501 3 0.4255 f 0.0 134 f 0.0095 f 0.0089 0 0 X = 1, pure KCl, d, = 2.9 A, d,, = 2.9 A 32 40 50 65 75 108 175 256 - 0 0 0 0 0 0 0 0 - 0.3082 03093 0.3237 0.3259 0.3379 0.3367 0.3495 0.3509 0.3562 0.3576 0.3723 0.3764 0.3977 0.3965 0.4 1 73 0.4032 0.5101 k0.0037 1 .o 0.5 0.5 0.5 0.5 0.5 0.5 0.67 0.5 - 0.5 0.5 0.5 0.5 0.5 0.5 - - 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 - a Total ionic strength, I = 1 mol dm-3; Bjerrum length, N = co by linear regression us.cube root of I / N . set to 3.4 A. In both cases, the diameters of K+ and C1- 0 EL, = 7.1355 A.Extrapolations to were set to 2.9 A and the Bjerrum length to 7.135 A. The former set was made to have a better separation between the activity coefficients and to test the sensitivity of the conclusions drawn to changes in the diameter of the larger ion. Tables 2 and 3 show the result of the MC simulations for differential total numbers of ions ( N ) in the central box. As proven in two earlier ~ a p e r s , ~ ~ . ~ ~ the extrapolation to the thermodynamic limit should be made by means of straight-line regressions of lny, us. the cube root of (l/N). Examples are shown in fig. 7. Fig. 8 and 9 show the 'spectra' of lny, and lny, obtained from the MC simulations for the two sets at 1 mol dm-3 and ca. 25 "C. For comparison, the MSA values for the single-ion activity coefficients are also shown.The correspondence between the two methods is quite good, and the most important feature is, that not only is lny, approximately independent of composition (even to the limit of trace activities), but the same feature is exhibited by lny,,. Furthermore, lny, seems to follow the dominant anion in the mixture, but the K+ ion has to strike a compromise in the case of an equal2658 Aqueous Mixed Solutions of KCI and K F Table 3. Monte Carlo results for Iny, for KF-KCI mixtures with d , = 3.7 A - 80 100 150 216 3 50 512 1000 co 130 260 390 520 650 so 64 80 100 I40 160 216 360 512 co I04 I30 260 390 520 650 M 40 50 75 108 175 256 500 - 65 I30 195 260 325 - 32 40 50 70 80 108 180 256 - 52 65 I30 195 260 325 - X = 0, pure KF, d , = 2.9 A - - 40 0.23 10 0.21 13 50 0.2544 0.2240 - 75 0.2637 0.2477 - I08 0.2865 0.2647 I75 0.2988 0.2896 - 256 0.3095 0.3064 - 500 0.3456 0.3289 - .- - 0.4 149 0.4 174 +0.0104 & 0.0028 - - - - - - - __ - X = 1/13, KF:KCI = 12: 1, d , = 2.9A, d(,l = 2.9A 5 60 0.2678 0.3521 0.238 I 10 120 0.3007 0.3846 0.2670 15 180 0.3 148 0.4020 0.2930 20 240 0.3206 04223 0.2952 25 300 0.3438 0.4327 0.3208 - 0.4327 0.5409 0.4203 k0.0152 +0.0119 Ifr0.0186 - 0 0 X=0.5,KF:KCI= l : l , d K = 2 .9 A , d ( . , = 2 . 9 A 16 16 0.2690 0.308 1 0.20 18 20 20 0.2786 0.3248 0.2171 25 25 0.2940 0.3406 0.2299 35 35 0.3 156 0.35 1 I 0.2434 0.2546 40 40 0.3 192 0.3671 54 54 0.3274 0.3748 0.2749 90 90 0.36 I6 0.4034 0.2970 I28 128 0.3661 0.4247 0.303 1 0.4692 0.5299 0.41 12 k 0.007 I k 0.0082 k 0.0060 - - 0 0 X = 12/13, KF:KCI = 1 12, d , = 2.9 A, d,., = 2.9 A 48 4 0.3292 0.3398 0.2345 60 5 0.3455 0.349 1 0.241 5 I20 10 0.3759 0.3956 0.2879 180 15 0.3874 0.4024 0.2972 240 20 0.4042 0.4194 0.3072 0.3 160 300 25 0.41 75 0.4 I22 .- 0.5097 0.5 168 0.4161 k 0.0082 0.01 30 f 0.0077 - no.config./ 10" 0.5 0.5 0.5 0.5 0.5 0.5 0.5 __ 0.78 0.5 0.5 0.5 0.5 - 0.5 0.64 0.6 0.5 0.55 0.5 0.5 0.5 - 1.535 0.5 0.5 0.5 0.5 0.5 - a Total ionic strength, I = 1 mol dm-3; Bjerrum length, & = 7.1355 A. by linear regression us. cube root of l / N . Extrapolation to N = M mixture of the two salts. These conclusions seem to be independent of the precise magnitude of the larger ion. Table 4 exhibits a comparison between mean ionic activity coefficients and Harned coefficients for the ionic system with d , = 3.7 A calculated by MC and MSA.It is observed, that although the correspondence between MSA and MC is quite good for the mean activity coefficients, the values are not exactly the same, and the Harned coefficients are very sensitive to the method of calculation.T. S. Sorensen, J . B. Jensen and P. Sloth 0 0 0 0 0 2659 0.05 0.10 0.15 0.20 0.25 Fig. 7. MonJe Carlo test particle simulations. Bjerrum length AB = 7.135 A; ionic diameters 2.9, 2.9 and 3.7 A for K', C1- and F-. Total ionic strength = 1 mol dmP3. Linear plots of the negative natural logarithm of the single ion activity coefficients of K+, C1- and F- us. the inverse cube root of the total number of particles in the simulation.The extrapolated values are close to the MSA values. At least one half million of Metropolis configurations per simulation. Periodic boundary conditions and minimum image cut-off of the configurational energies. 1 N - 3 Table 4. Mean activity and Harned coefficients from Monte Carlo and MSA" - - - 0 - 1 -0.487f0.011 -0.4734 -0.025+0.012 -0.0444 13 $ - 0.4995 & 0.0054 - 0.4923 -0.021 f0.013 -0.0442 - 13 l2 -0.5133f0.0056 -0.51 10 0.042 f 0.087 - 0.0442 1 -0.5101 f0.0037 -0.5144 - - - 0 -0.4161 f0.0040 -0.4152 - - 1 0.13 0.21 0.048 1 - 13 -0.426f 0.016 - 0.41 89 2 - 0.4402 0.0060 - 0.439 1 0.048f0.014 0.0478 0.0507 & 0.0069 0.0475 - l 2 -0.4629 f 0.0049 - 0.4590 1.3 1 a d, = d,, = 2.9 A, d, = 3.7 A.0.38 0.40 0.42 0.44 0.16 0.46 0.50 0.52 0.54 0.56 0.58 Fig.8. 'Spectrum' of negative natural logarithms of activity coefficients obtained by MC test particle simulations (66 % error belts OF extrapolated values) and by the MSA method (dashed lines) with a 'large' F- ion (diameter 3.7 A) at 1 mol dmP3 total ionic strength and 25 "C. The single- ion activity coefficients of F- and C1- are almost independent of the composition of the mixtures, even close to trace conditions. In solutions with two ions (perhaps with a trace of a third), the activity coefficients of the two ions are almost identical also for unequally sized ions. For an 0.40 0.45 0.50 1 I 1 - 1 1 ' 1 I 1 I 1 I 1 - I < ICn I I I - - 0 1 - 1 -!- ln * I1 - G; s; I ' V U l I U 31 I 1 - , 2 2 r I I I b zi mI I m=l-l h l N 0 q a. 2' K t r; + I Yl II 1; II + I GYg a" I I I I 1 ,KCI I I 1.T F- K + mi ., K l , . ,. 2!, , t I I I KCI 'a -1: $ $2 E $ 3 L; 'x 11' pzzi7 I I - 11- - 1 I I I I I I I I I 1 , $0 I 0.40 0.45 0.50 Fig. 9. Similar 'spectrum' as in fig. 8, but with a smaller F- ion (3.4 A). The qualitative features are the same as in fig. 8, but the separations of the values are less pronounced.T. S. Sorensen, J. B. Jensen and P. Sloth 266 1 Discussion It is far from evident, that the primitive model is a good model for electrolytes having concentrations of the order of 1 mol drnp3. One might anticipate, that the structure of the solvent, specific ionic interactions, dielectric saturation and lowering of the dielectric constant with increased salt concentration all contribute to modify the results given here.Furthermore, as a McMillan-Mayer theory, the primitive model yields activity coefficients at the given osmotic pressure rather than at atmospheric pressure, which leads to a significant correction at concentrations higher than ca. 1 mol dmP3. However, a great amount of ignorance as to the precise correction to carry out would be manifest, if one tried to perform all these corrections. As an alternative, we can follow the ‘Golden Rule of Physical Chemistry’ stating that, in complex systems, it is better in comparison with experiments to ignore the sophisticated features in models rather than to correct for a part of these features in a more or less rigorous manner. This is so, since the ignored corrections very often tend to counteract each other, so that they partly cancel.In this spirit, Ebeling and Scherwinski14 demonstrated recently, that the primitive model by means of simple MSA calculations, was very efficient in mimicking the real behaviour of 1 : 1 electrolytes (alkali-metal halides) in water at 25 “C up to at least 1 mol dm-3. The effective ionic radii seemed to be quite realistic when compared to crystallographic radii and to the expected amount of hydration of the different ions. Similar results were obtained much earlier by Triolo et al.30 However, the contact distances quoted in the two above-mentioned papers differ considerably, and we think that the distances from the latter paper are less reliable, since a fit was made up to 2 mol dmP3 without any corrections, and since the less exact Percus-Yevick equation was used for the hard-sphere contribution.The inaccuracies also showed up in a bad fit at moderate concentrations. In contrast, Ebeling and Scherwinski fitted only up to 1 mol dm-3 and used a generalized Carnahan-Starling approximation for unequal hard spheres. Ebeling and Scherwinski attempted a least-squares determination of the individual ionic radii by fitting to mean ionic activities of all combinations of Li+, Na+, K+, Rb+ and Cs+ with C1-, Br- and I-. This ought to be an impossible task, and so it proved, since the authors had to fix one of the 8 radii ‘to obtain a stable fit’. Finally, the extra- thermodynamic assumption made was that the iodide ion was not hydrated, and that the effective radius of this ion was equal to the Pauling radius.A table of 8 radii was then constructed, which yields approximately correct activity coefficient curves for 15 alkali- metal halides. The radii for I$+ and for C1- were found to be 1.36 * 0.04and 1.78 * 0.04 A, where we have used 1.45 A for both of these ions. We think it is more natural to ascribe similar radii to K+ and C1- in the framework of the primitive model, since the limiting conductivity in water is almost equal for these two ions. In a primitive model, only the size of the ion, the absolute value of the ionic charge and the viscosity of the surrounding medium should be of importance for the conductivity. In addition, one might envisage a considerable ‘dielectric drag’ on such small ions due to the finite relaxation time of the reorientation of the water m01ecules.~~-~~ However, this drag is also equal for equally sized and charged cations and anions. Th: difference in the contact distance between K’ and C1- (3.14 A compared to our 2.90 A) may be ascribed to some difference in the underlying experimental data.Ebeling and Scherwinski have used the compilations of Hamer and WU.~’ We discussed in ref. (8) the discrepancy between our data for KCl and the data compiled by Hamer and Wu (transformed from molal to molar values) and concluded, that our data for KC1 should be considered to be more reliable, even with the reduced number of data points in ref. (8) compared to the present study. This was so, because the majority of data in the Hamer and Wu compilation was made with cells with liquid junctions, and because the extended Debye-Hiickel equation was used in the ‘dilute end’ of the cell to calculate2662 Aqueous Mixed Solutions of KCI and KF Iny+(KCl) in the ‘concentrated end’.The great increase in the number of experiments made in this paper has not changed the mean values of Iny+(KCl) - very much, but the uncertainty belts have shrinked considerably. Other primitive model values for the contact distance in KCI solutions have been stated by Triolo et al. (osmotic coefficients at 25 0C),3(’ by Moller (osmotic coefficients at 0 “C), ref. (38) fig. I 15, p. 41 7, and by Onsager and Fuoss (concentration dependence gf conductance data), ref. (38), p. 451. The values obtained were 2.80, ca. 2.0 and 3.07 A, rFspectively. If we disregard the very low value from the theory of Moller, the value 2.9 A obtained in this study is quite compatible with the other values.It is interesting to compare the contact distance obtained for KCI in aqueous solutions to the ion-ion distance calculated from crystals. Adams has given a survey over ion hard- sphere dimensions estimated in various ways from solid-state structural ~hemistry.~’ The best evaluations are probably derived from electron density maps for ionic crystaJs obtained by X-ray diffraction. The radius of the K+ and the C1- ions are i.49 and 1.64 A, respectively. Thus, the ‘contact distance’ without hydration is 3. I3 Ad This is very close to the value in solution given by Ebeling and Scherwinski (3.14 A). Thus, KC1 seems from such studies to be little or very weakly hydrated, but this might be the result of cancellation of different effects neglected in the primitive model.One should note, however, that it is quite possible to have a contact distance between two unhydrated ions in solution which is somewhat less than the distance in crystals, since the ion pairs in crystals are pulled apart by their other neighbours. Fo! KF, we fit a contact distance equal to ca. 3.15 A. This is not far from the value 3.08 A obtained by Triolo et al.30 from osmotic coefficients. Thus, the effective size of the F- ion in aqueous solution seems to be greater than the size of the C1- ion. The contact dista5ce without hydration in a KF crystal estimated from electron density maps is only 2.65 A.39 Thus, the only plausible explanation for the difference seems to be, that the F- ion is ?trongly hydrated.This is not surprising, when one takes the small size of this ion (1.16 A from electron density maps) and its hydrogen-bond forming ability into account. We have not found any significant deviation between MSA calculations and MC test particle simulations at I mol dm-3 using the same radii. Also, the qualitative behaviour of mixtures of KF and KCI at 1 mol dm-3 total ionic strength is the same (fig. 5). The quantitative relations (Harned coefficients) are quite different, however. The conclusion is that MSA is not quite good enough to calculate activity coefficients in mixtures, at least in cases with very small differences between the lny+ us. h curves of the two salts, see table 1 . The rule of thumb, mentioned in the monograph of Robinson and Stokes,9 that the trace activity coefficient of one salt in excess of another almost has the same value as the trace activity coefficient of the second salt in excess of the first, seems to be verified by the MSA calculations as well as the MC calculations, see fig.5, 6, 8 and 9. However, we also obtain a deeper look into the variations of the single-ion activity coefficients with the degree of mixture between two salts with a common ion. The single-ion activity coefficients of the two counterions to the common-ion (Cl- and F-) seem to be very nearly independent of the salt fraction at constant ionic strength even to the limit of trace conditions. The activity coefficient for K+ at I mol dm-3 is very near to the activity coefficient of the counter-ion in purc solutions evenjor differing ionic radii (KF with diameter of the F- ion equal to 3.4 A as well as 3.7 A).However, in a 1 : I mixture the activity coefficient of K’ has to strike a compromise between the single-ion activity coefficients of the two counterions. Passing to the trace limit, the activity coefficient of the K+ ion moves towards the activity of the counterion in excess. The two trace activities will be identical, if the single-ion activities of the two counterions are unaffected by the mixing, and if the single-ion activities of the two ions in a pure (or nearly pure) solution are the same. The MC simulations with diameters 2.9, 2.9 and 3.4A for K+, C1- and F- (fig. 5,T. S. Surensen, J . B. Jensen and P. Sloth 2663 Table 5.Mean activity and Harned coefficients from experimental KF data ~ - Iny+(pure KF) -lny+(KF, X = 0.5) I/mol dm-3 [polynomial, eqn (5)] [polynomial, eqn (6)] ~ K F - 1 .o 0.463&0.010 0.475 t0.010 0.024 & 0.028 1.5 0.476 0.01 0 0.485f0.010 0.0 19 & 0.028 2.0 0.455 &- 0.01 5 0.486 & 0.01 5 0.062 f 0.042 3 .O 0.345 f 0.025 0.473 k 0.020 0.256 & 0.064 4.0 0.203 & 0.030 0.441 f 0.020 0.477 k 0.072 _ _ _ _ _ _ ~ dashed curves) show some variation in the mean ionic activity coefficients with the salt fraction. The values of lny+(KCl) should vary more than the values of lny+(KF). Contrarily, the experiments-seem to indicate, that In y+(KCl) is almost unaffected, whereas there is a clear variation in lny+(KF) (fig. 1 4 ) . Thus, either we have not yet hit the best effective ionic radii, or the physkal properties of real mixtures are too subtle to be completely explained by primitive model calculations. One might do a lot of fitting with the MSA model, but in view of the different Harned coefficients obtained from MSA and from MC, this does not seem to be worth while.We cannot test the Harned linearity in the experimental data, since we have only measured pure salts and 1 : 1 mixtures. However, the MSA calculations as well as the MC simulations exhibit clear-cut Harned linearity. Thus, assuming Harned's rule to hold, we may calculate the dimensionless Harned coefficient czKF at various total ionic strengths (table 5). The Harned coefficient aKC, we have assumed to be zero at all ionic strengths by pooling together the data for lny,(pure KCl) and lny+(KCl, X = 0.5).The Harned coefficients aKF found have a considerable uncertainty. They cannot really be statistically distinguished from zero before above 2 mol dmV3. In order for the Maxwell conditions to be fulfilled, a,, should be strictly proportional to I when aKCl is zero, see eqn (9). The experimental data are not in conflict with such a proportionality, since there is a considerable and significant increase in aKF in passing from I = 2 mol dm-3 to 4 mol dmP3. The proportionality constant is 0.055 0.025 dm3 mol-1 when estimated as the average of the values at the five concentrations stated in table 5 . Thus, from the present study we extract the following estimates for the Harned coefficients: a K c , / I = 0 0.025 dm3 mol-1 (1 1) aKF/I = 0.055 0.025 dm3 mol-'.(12) The uncertainties on the two Harned coefficients have been taken as equal. With the above very large uncertainties, the experimental Harned coefficients are not completely incompatible with the MSA Harned coefficients (table 1). (The MSA Harned coefficients should be preferred to MC Harned coefficients, since experimental data have been fitted to MSA curves and not to MC curves.) To measure Harned coefficients in the KF-KCl system with a precision significantly better than obtained in this paper would require at least 1000 independent data points, since the precision of each single measurement cannot be improved with the present electrodes. We thank Teknologistyrelsen (The Danish National Agency of Technology) for economic support for the present study.We are grateful to Dr Hans Bjarne Kristensen and Dr Hans Boye Nielsen, the pH Calibration Department, Radiometer A/S, for the delivery of high-precision Ag-AgC1 electrodes.2664 Aqueous Mixed Solutions of KC1 and KF References 1 J. Bagg and G. A. Rechnitz, Anal. Chem., 1973, 45, 1069. 2 J. V. Leyendekkers, Anal. Chem., 1971, 43, 1835. 3 T. S. Serrensen and K. F. Jensen, J. Chem. Soc., Faraday Trans. 2, 1975, 71, 1805. 4 N. A. 0sterberg, J. B. Jensen and T. S. Serrensen, Acta Chem. Scand., Part A , 1978, 32, 721. 5 N. 0. Osterberg, J. B. Jensen, T. S. Serrensen and L. D. Caspersen, Actu Chem. 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