Isopiestic determination of the osmotic and activity coefficients of {zH2SO4+(1−z) MgSO4}(aq) at the temperatureT=298.15 K. I. Results forz=(0.85811, 0.71539, and 0.57353)

J. Chem. Thermodynamics 1997, 29, 533–555

Isopiestic determination of the osmotic and activity coefficients of {z H2 SO4 + (1 − z )MgSO4 }(aq) at the temperature T = 298.15 K. I. Results for z = (0.85811, 0.71539, and 0.57353) a Joseph A. Rard Geosciences and Environmental Technologies, Environmental Programs Directorate, Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550 , U.S.A. Isopiestic vapor-pressure measurements were made for {zH2 SO4 + (1 − z)MgSO4 }(aq) with H2 SO4 molality fractions of z = (0.85811, 0.71539, and 0.57353) at the temperature 298.15 K. Measurements extend from total molalities mT of (0.12548, 0.13563, and 0.14741) mol·kg−1 to the supersaturated molalities mT = (12.0496 , 11.0112 , and 7.20595 ) mol·kg−1, respectively. For z = (0.71539 and 0.57353) these highest molalities correspond to the greatest degrees of supersaturation that could be achieved during isothermal removal of the solvent. The corresponding water activity ranges are 0.9958 e aw e 0.2758, 0.9958 e aw e 0.3312, and 0.9958 e aw e 0.5581 for z = (0.85811, 0.71539, and 0.57353), respectively. These values of z were chosen to be separated approximately equally by 0.142 (Dz 1 1/7). The water activity curves were also found to be nearly equally spaced from each other, as was the z = 0.85811 curve from that for H2 SO4 (aq) (z = 1). Given that both H2 SO4 (aq) and MgSO4 (aq) are extensively associated electrolytes, this regularity in the mixing behavior was not anticipated. A well-defined crossover point was found at which the sums of the stoichiometric ionic molalities were equal at isopiestic equilibrium for z = (1, 0.85811, 0.71539, and 0.57353), as were the water activities and stoichiometric osmotic coefficients of these solutions. This occurred at Si ni mi = (5.9632 2 0.0036) mol·kg−1, where ni is the stoichiometric ionization number of electrolyte i. 7 1997 Academic Press Limited KEYWORDS: isopiestic measurements; vapor pressures; aqueous solutions; sulfuric acid; magnesium sulfate

1. Introduction Ternary and multicomponent aqueous solutions containing H2 SO4 are present in many natural waters and industrial solutions. For example, marine atmospheric a This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference to any specific commercial products, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.

0021–9614/97/050533 + 23 $25.00/0/ct960169

7 1997 Academic Press Limited

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aerosols are generally concentrated mixtures of H2 SO4 (aq) and (NH4 )2 SO4 (aq) containing variable amounts of sea salts.(1) Acidic mine wastes generally contain mixtures of H2 SO4 (aq) with various metal sulfates(2) and result from the aerial oxidation of metal sulfides. Acidic sulfate solutions are also frequently encountered in the hydrometallurgical extraction of ores. Rard(3) has listed most of the published e.m.f. and vapor pressure studies for ternary aqueous acidic sulfate solutions. In most cases the experimental thermodynamic results are limited to a small portion of the available composition region. The most thoroughly studied of these systems is {zH2 SO4 + (1 − z)Na2 SO4 }(aq) for which isopiestic results are available up to supersaturated molalities for 1 e z e 0 at the temperature T = 298.15 K,(3–5) and for 0.5 e z e 0 at elevated temperatures.(6,7) The thermodynamic activities for {zH2 SO4 + (1 − z)Na2 SO4 }(aq) have been analyzed by Pitzer’s equations.(5,7,8) Although some ion pairing is known to occur in Na2 SO4 (aq) solutions to form NaSO− 4 (aq) ion pairs, these ion pairs are weak enough that they can be ignored in the thermodynamic analysis with Pitzer’s equations.(9,10) Thus, the only equilibrium that need be considered in the mixtures is that for formation of HSO− 4 (aq). However, the association constants are much larger for solutions of divalent metal sulfates MSO4 (aq),(11,12) and thus it might be necessary to include specifically the presence of equilibrium amounts of the neutral divalent-metal sulfate complexes along with the HSO− 4 (aq) ion in order to obtain quantitative models for the thermodynamic properties of {zH2 SO4 + (1 − z)MSO4 }(aq). We chose {zH2 SO4 + (1 − z)MgSO4 }(aq) for study because of the importance of MgSO4 (aq) in seawater and some concentrated brines. In addition, the thermodynamic results for this system could allow better modeling of {zH2 SO4 + (1 − z)CaSO4 }(aq) where the very much lower solubilities of gypsum make the isopiestic molalities of this system too close to those of pure H2 SO4 (aq) for isopiestic measurements to be of much value. A thermodynamic analysis of {zH2 SO4 + (1 − z)MgSO4 }(aq) at T = 298.15 K using Pitzer’s equations has been reported by Harvie et al.(8) as part of their comprehensive modeling of the solubilities of ternary brines and other ternary electrolyte solutions. Their parameterization for {zH2 SO4 + (1 − z)MgSO4 }(aq) was based solely on the solubilities of Filippov and Antonova(13) and the e.m.f.s. of Harned and Sturgis.(14) However, Harned and Sturgis only studied H2 SO4 (aq) molalities of (0.01 and 0.1) mol·kg−1 with MgSO4 (aq) E 1.0 mol·kg−1. Although Filippov and Antonova also performed isopiestic measurements for this system, they only presented their results as a small plot and did not report any actual isopiestic molalities. Thus, there was relatively little thermodynamic information available when Harvie et al. performed their modeling calculations, and most of the molality regions of {zH2 SO4 + (1 − z)MgSO4 }(aq) were uncharacterized. In order to refine the thermodynamic treatment of Harvie et al.,(8) a complete series of activity measurements is required. In the present report, extensive isopiestic measurements are reported for the acid-rich region of the {zH2 SO4 + (1 − z)MgSO4 }(aq) system at T = 298.15 K; these extend from low molalities to well into the supersaturated molality region.

f and g for {zH2 SO4 + (1 − z)MgSO4 }(aq)

535

2. Experimental Water for the experiments was purified by ion exchange followed by distillation. All apparent masses were converted to masses. Assumed molar masses were 98.074 g·mol−1 for H2 SO4 , 58.443 g·mol−1 for NaCl, and 120.364 g·mol−1 for MgSO4 . The isopiestic vapor pressure measurements were performed at T = (298.15 2 0.005 ) K (IPTS-68) in our three stainless-steel isopiestic chambers which were described previously.(15) A detailed description of this method is given by Rard and Platford.(16) Air was removed from the chambers in four to six stages prior to starting each isopiestic equilibration by using the basic procedure described by Scatchard et al.(17) A solution of NaCl(aq) was used as the isopiestic reference standard up to its solubility limit, and H2 SO4 (aq) was used at higher molalities. We made a modification to the copper block of each isopiestic chamber during the series 1 experiments and before any of the other series were started. A fan-like device was added to the center of the top of the copper block which contains the sample cups. Two adjacent ‘‘fan blades’’ out of the eight were removed to make the center-of-mass of the ‘‘fans’’ off-center. This device provides some stirring of the vapor phase to assist in the transport of water vapor between the various samples (see the description by Rard(18) for more details). Figure 1 is a photograph of the modified copper block from one of the isopiestic chambers. This vapor stirring appears to improve significantly the quality of the equilibrations at low molalities, where the kinetics of the approach to isopiestic equilibrium is dominated by mass transport of the solvent. However, it is probably less important at higher molalities where heat transport is the limiting factor for equilibrium times.(16) These ‘‘fans’’ are a modified version of a device used in Albright’s laboratory as described by Mitchell et al.(19) The stock solution of H2 SO4 (aq) used as the isopiestic reference standard was prepared by mass dilution of a concentrated stock solution. This reference stock solution’s molality was determined to be (0.99932 2 0.00059) mol·kg−1 by mass titration of four samples with standardized NaOH(aq) using phenolphthalein as indicator. A value of 0.99942 mol·kg−1 was obtained from a similar analysis of the concentrated stock solution and the mass dilution results. Here and elsewhere, the reported stock solution molality uncertainties are estimated standard deviations. The mean of these two values (0.99937 mol·kg−1 ) was accepted. The NaCl(aq) isopiestic reference standard solution is the same one as described by Rard and Archer.(20) Spectroscopic and ion chromatographic analyses of this solution for impurities indicate that the material has an NaCl purity (on a dry mass basis) of 0.999415 mole fraction, with impurities of 0.000316 KCl, 0.000105 NaF, 0.000115 Na2 SO4 , and 0.000049 of other substances. The effective molar mass of 58.4566 g·mol−1 was used for calculation of molalities (see Rard and Archer for a more detailed description of this solution). Its molality was determined to be (2.92433 2 0.00082 ) mol·kg−1 based on dehydration of three samples at T = 773 K. This temperature has recently been shown to be the optimum for

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FIGURE 1. Photograph of the 3.8 cm thick copper heat-transfer block from one of the isopiestic chambers. The fan-like device is positioned above the center of the top of this block, and its back-and-forth rotation by 1p is provided by gravity as the chamber is rocked back and forth to provide mixing during the equilibrations. Up to eight tantalum metal cups containing the aqueous solutions are inserted in the gold-plated brass retainers during these equilibrations.

complete dehydration of the NaCl(s) in air, without being high enough for any significant sublimation or chloride loss to occur.(18) The MgSO4 (aq) stock solution was prepared by dissolving recrystallized Mallinckrodt Analytical Reagent MgSO4·xH2O(s) in purified water, followed by filtration through a prewashed 0.2 mm polycarbonate membrane filtering unit. A dried sample of this solution was analyzed for impurities using direct current arc optical emission spectroscopy. Impurities found (mass fractions) are 30·10−6 Ca, 10·10−6 each of Na and Al, 5·10−6 B, 3·10−6 each of Cu and Fe, and E5·10−6 Si. Twenty-seven other elements were sought but not detected. Each of the observed impurities was assumed to be present as its sulfate, except for B and Si which were assumed to be present as B2O3 and SiO2 , respectively. The purity of the MgSO4 was thus 0.99977 in mass fraction (0.999825 in mole fraction), with about half of the impurities being the chemically-similar CaSO4 , which should have nearly identical isopiestic behavior. These impurity levels are so low that the properties of our MgSO4 (aq) solution should differ insignificantly from those of absolutely pure MgSO4 (aq).

f and g for {zH2 SO4 + (1 − z)MgSO4 }(aq)

537

The molality of the MgSO4 (aq) stock solution was determined from the mass of residues from the dehydration of three samples, each containing a mass of 11 g of MgSO4 . One drop of 12 mol·dm−3 solution of H2 SO4 (aq) was added to each of the three porcelain crucibles containing solution and to an otherwise empty tare crucible. These crucibles were heated on a hotplate until the water was driven off from the solutions, and were then transferred to a box (muffle) furnace and heated to T = 773 K. The crucibles were later removed from the furnace, cooled in a desiccator for 1 h, weighed, returned to the furnace, and the process repeated for eight separate weighings on different days. The same number of weighings were subsequently carried out at T = 823 K and T = 848 K. The initial weighings yielded a slightly high molality; one of the weighings at T = 848 K yielded a slightly low molality, resulting from an anomalously high tare crucible mass. Averaging of the seven consistent weighings at T = 773 K yielded (0.45040 2 0.00022) mol·kg−1; of the eight weighings at T = 823 K, (0.45042 2 0.00022) mol·kg−1; and of the seven consistent weighings at T = 848 K, (0.45047 2 0.00016) mol·kg−1. The average of the mean molalities from all 22 ‘‘good’’ weighings is (0.450431 2 0.000078 ) mol·kg−1 where this uncertainty is an ‘‘n − 1’’ standard deviation. Differences in molalities obtained at different furnace temperatures are insignificant compared with the 1(0.03 to 0.04) per cent imprecision of the results, and indicate that dehydration was complete at these temperatures. The partial pressure of H2O in the laboratory varied from p(H2O, g) = (1.1 to 1.5) kPa during the days on which the crucibles were weighed, but there was no significant variation of the calculated molalities with p(H2O, g). The tare crucible was always weighed last, and the three sample crucibles were weighed in one order during half of the weighings at each temperature and in the reverse order during the other weighings. Calculated molalities for any particular analysis sample were consistently slightly higher when their crucible was weighed third than when they were weighed first, which indicates that there was a very slight absorption of water vapor by the anhydrous MgSO4 (s) each time the desiccator was opened. These molality differences indicate that moisture absorption caused the average of the calculated stock solution molalities to be too high by (0.000237 2 0.000127 ) mol·kg−1. The correct molality of the MgSO4 (aq) stock solution is thus {(0.450431 2 0.000078 ) − (0.000237 2 0.000127 )} mol·kg−1 = (0.45019 2 0.00015) mol·kg−1. This value was accepted for molality calculations. We did not have a sufficient amount of our H2 SO4 (aq) isopiestic reference standard solution to use for preparation of the three {zH2 SO4 + (1 − z)MgSO4 }(aq) stock solutions. A second stock solution of H2 SO4 (aq) with m 1 0.45 mol·kg−1 was therefore prepared by dilution from a highly concentrated solution. To achieve the maximum degree of internal consistency between the molalities of our isopiestic reference standard stock solution of H2 SO4 (aq) and those of our mixtures, we determined the molality of the second stock solution by isopiestic comparison against the 0.99937 mol·kg−1 reference standard stock solution. Duplicate samples of both solutions were weighed out, and were equilibrated six separate times at T = 298.15 K. Molalities of the second stock solution were back-calculated from the

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original masses of the samples, from their masses at equilibrium, and from the molality of the H2 SO4 (aq) isopiestic reference standard solution. This yielded (0.449986 2 0.000278 ) mol·kg−1 for the second stock solution. The three {zH2 SO4 + (1 − z)MgSO4 }(aq) stock solutions were prepared by mass from this second H2 SO4 (aq) stock solution and from the MgSO4 (aq) stock solution. These mixtures were prepared in the acid-rich region of {zH2 SO4 + (1 − z)MgSO4 }(aq) at values of the molality fraction of H2 SO4 (aq) of z 1 6/7, 5/7, and 4/7; in the future we hope to extend these measurements to the acid-deficient region of this system at values of z 1 3/7, 2/7, and 1/7. Solution cups were made of tantalum metal, which is highly resistant to corrosion by most aqueous electrolytes including H2 SO4 (aq). However, Rard(4) reported that prolonged contact between {zH2 SO4 + (1 − z)Na2 SO4 }(aq) and the tantalum cups resulted in the loss of one cup due to formation of a ‘‘pinhole’’ where the bottom was welded to the cylindrical portion. Tantalum owes its resistance to corrosion to the presence of a nearly insoluble and chemically-inert surface layer of Ta2O5 . This Ta2O5 has a dull smoky-gray color, in contrast to the shiny and lighter-gray tantalum metal, so the presence of an intact layer of Ta2O5 in contact with a solution can easily be seen. Prior to the present isopiestic experiments, many of the tantalum cups were heated for (2 to 3) h in air at the temperature 773 K to thicken their protective surface layers of Ta2O5 substantially. This thickened oxide layer is apparent because of its darker-gray coloration. All of the isopiestic cups containing H2 SO4 (aq) or {zH2 SO4 + (1 − z)MgSO4 }(aq) were carefully inspected after each isopiestic equilibration, and in no case was there any evidence for corrosion having occurred. It thus appears that this thermal treatment was more than adequate to prevent corrosion of the tantalum metal by acidic sulfate solutions. Isopiestic equilibration times for the experiments with molalities of solution above 12.2 mol·kg−1 generally ranged between (12 and 21) days. However, 10 day equilibrations were used at the highest molalities when only the samples with z = 0.85811 and reference standard were present. Shorter equilibration times probably would have been adequate at high molalities, but these longer times were used as a precaution in case the increasing viscosity of the solutions reduced the efficiency of mixing as the chambers were rocked. The equilibrium times were gradually increased as mT was decreased, with (75 to 104) days being used at the lowest molalities. Duplicate samples of each solution were used in all of the equilibrations. Measurements were continued well into the supersaturated regions for all three values of z, and until spontaneous crystallization occurred in both samples of {zH2 SO4 + (1 − z)MgSO4 }(aq) for the cases with z = (0.71539 and 0.57353). Based on the reported solubilities,(13) we estimate that the highest total molalities for each value of z are (1.15 to 2.4) mol·kg−1 above saturation. Tables 1 and 2 contain the experimental isopiestic equilibrium molalities using solutions of NaCl(aq) or H2 SO4 (aq), respectively, as reference standards. The uncertainties of the reported average molalities of the duplicate samples were nearly always Q0.0005·m, with most being precise to E0.0003·m, although for the lowest

TABLE 1. Isopiestic molalities mT of {zH2 SO4 + (1 − z)MgSO4 }(aq) and molality m* of the NaCl(aq) reference standard solutions and the osmotic coefficients f* of NaCl(aq) at the temperature T = 298.15 K m*/(mol·kg−1 )

f* a

Series 1 b,c 2.82067 2 0.0007 2.84313 2 0.0000 2.90136 2 0.0023 2.96185 2 0.0013 3.06192 2 0.0015 3.16077 2 0.0021 3.26564 2 0.0006 3.39130 2 0.0012 3.51461 2 0.0002 3.65337 2 0.0005 3.81707 2 0.0001 3.93808 2 0.0013 4.13818 2 0.0007 4.31791 2 0.0010 4.51690 2 0.0003 4.67167 2 0.0001

3.19391 2 0.0020 3.22773 2 0.0018 3.31010 2 0.0019 3.40503 2 0.0018 3.55313 2 0.0022 3.70132 2 0.0012 3.86434 2 0.0014 4.05725 2 0.0019 4.25115 2 0.0008 4.47155 2 0.0009 4.73462 2 0.0026 4.92820 2 0.0028 5.25579 2 0.0039 5.54995 2 0.0030 5.88155 2 0.0027 6.14034 d

1.0616 1.0639 1.0695 1.0760 1.0862 1.0965 1.1081 1.1219 1.1359 1.1521 1.1717 1.1863 1.2113 1.2341 1.2601 1.2806

3.08677 2 0.0000 2.82884 2 0.0003 2.57707 2 0.0004 2.37517 2 0.0002 2.19682 2 0.0003 1.93466 2 0.0002 1.75551 2 0.0007 1.59311 2 0.0005 1.42732 2 0.0001 1.19158 2 0.0000

Series 3 b 3.21022 2 0.0008 2.95353 2 0.0002 2.70169 2 0.0002 2.49836 2 0.0001 2.31791 2 0.0003 2.05038 2 0.0004 1.86731 2 0.0008 1.70076 2 0.0006 1.52891 2 0.0003 1.28221 2 0.0000

3.77774 2 0.0013 3.38953 2 0.0007 3.01874 2 0.0003 2.72744 2 0.0003 2.47855 2 0.0019 2.11916 2 0.0012 1.88672 2 0.0027 1.68085 2 0.0026 1.47647 2 0.0024 1.19758 2 0.0020

1.1019 1.0749 1.0499 1.0310 1.0153 0.9937 0.9805 0.9693 0.9588 0.9456

0.95230 2 0.00030 0.96215 2 0.00009 0.76949 2 0.00008 0.55723 2 0.00014 0.35196 2 0.00009

Series 4 b 1.03053 2 0.0006 1.04078 2 0.00005 0.83574 2 0.00015 0.60695 2 0.00008 0.38357 2 0.00002

0.92717 2 0.00029 0.93719 2 0.00043 0.73370 2 0.00019 0.51982 2 0.00009 0.32415 2 0.00005

0.9343 0.9347 0.9276 0.9222 0.9206

0.13563 2 0.00058

Series 5 b,e 0.14741 2 0.00060

0.12589 2 0.00035

0.9289

z = 0.85811

mT /(mol·kg−1 ) z = 0.71539

z = 0.57353

2.58055 2 0.0002 2.60216 2 0.0002 2.65642 2 0.0007 2.71946 2 0.0002 2.81661 2 0.0008 2.91451 2 0.0009 3.02338 2 0.0003 3.14604 2 0.0007 3.27511 2 0.0010 3.41544 2 0.0006 3.58097 2 0.0010 3.70362 2 0.0003 3.90861 2 0.0007 4.09173 2 0.0001 4.29473 2 0.0012 4.45350 2 0.0012

2.69633 2 0.0001 2.71867 2 0.0000 2.77512 2 0.0020 2.83700 2 0.0001 2.93491 2 0.0011 3.03366 2 0.0012 3.14079 2 0.0002 3.26567 2 0.0010 3.39218 2 0.0000 3.53282 2 0.0001 3.69773 2 0.0006 3.82004 2 0.0003 4.02371 2 0.0002 4.20539 2 0.0007 4.40783 2 0.0011 4.56489 2 0.0005

2.96704 2 0.0010 2.71101 2 0.0002 2.46179 2 0.0001 2.26119 2 0.0006 2.08498 2 0.0000 1.82887 2 0.0004 1.65307 2 0.0004 1.49650 2 0.0005 1.33682 2 0.00005 1.11078 2 0.0000 0.88387 2 0.00051 0.89265 2 0.00047 0.71205 2 0.00028 0.51410 2 0.00013 0.32434 2 0.00012 0.12548 2 0.00048 a

Osmotic coefficients of the NaCl(aq) reference standard solutions were calculated using the equation and parameters given by Archer.(21) b Different samples of NaCl(aq) or H2 SO4 (aq) and of {zH2 SO4 + (1 − z)MgSO4 }(aq) were used for each of the five series of experiments. Reported molalities for each series are given, from top to bottom in tables 1 and 2, in the actual order that the experiments were performed. c A fan-like device (see figure 1) was added to the copper block after the second from last equilibration of this first series; it provided some stirring of the vapor phase as the chamber was rocked back-and-forth. It was used during the remaining experiment of this series, and similar devices were present in all three of our chambers during the other four series of experiments. d This molality of NaCl(aq) is from a single sample since crystallization occurred in the replicate sample solution. e On day 77 or 78 of this equilibration, the ground fault interrupter was activated for the circuit that controlled the temperature sensor and heater, thus shutting off their electrical power. This caused the temperature of the constant temperature water bath to drop by over 4 K. The problem was detected on day 79, and the correct temperature of 298.15 K was reestablished in a few hours. However, the same problem reoccurred several more times. These power losses became less frequent but did not stop after the electrical power connection was moved to a different electrical outlet with a different ground fault interrupter. Replacing our newer TRONAC Precision Temperature Controller model PTC-41 with an older model 40 eliminated this problem. An additional 22 days were added to this equilibration before these samples were removed for weighing.

TABLE 2. Isopiestic molalities mT of {zH2 SO4 + (1 − z)MgSO4 }(aq) and molality m* of the H2 SO4 (aq) reference standard solutions and the osmotic coefficients f* of H2 SO4 (aq) at the temperature T = 298.15 K z = 0.85811 4.49886 2 0.0022 4.54352 2 0.0005 4.69734 2 0.0004 4.83298 2 0.0005 4.98444 2 0.0005 5.15731 2 0.0006 5.30858 2 0.0011 5.47091 2 0.0013 5.62845 2 0.0015 5.81714 2 0.0018 6.01014 2 0.0007 6.19977 2 0.0014 6.38397 2 0.0021 6.60165 2 0.0003 6.80335 2 0.0007 7.00850 2 0.0013 7.18834 2 0.0015 7.46532 2 0.0019 7.46535 2 0.0019 7.66148 2 0.0014 7.82918 2 0.0016 7.99951 2 0.0018 6.99557 2 0.0007 7.36296 2 0.0024 7.77516 2 0.0008 8.10364 2 0.0005 8.39126 2 0.0006 8.61121 2 0.0008 8.81393 2 0.0010 9.06847 2 0.0003 9.18168 2 0.0035 9.46575 2 0.0040 9.67694 2 0.0009 9.91019 2 0.0048 10.0835 2 0.004 10.2160 2 0.000 10.4735 2 0.002 10.7647 2 0.009 10.9083 2 0.004 11.2166 2 0.001 11.4375 2 0.005 11.7132 2 0.000 12.0496 2 0.009 6.09812 2 0.0017

mT /(mol·kg−1 ) z = 0.71539 4.61016 2 0.0008 4.65440 2 0.0017 4.80873 2 0.0010 4.94198 2 0.0013 5.09387 2 0.0019 5.26550 2 0.0013 5.41704 2 0.0017 5.57649 2 0.0009 5.73598 2 0.0009 5.92401 2 0.0021 6.11593 2 0.0017 6.31018 2 0.0007 6.49398 2 0.0000 6.71187 2 0.0004 6.91788 2 0.0019 7.12322 2 0.0013 7.30860 2 0.0023 7.58126 2 0.0036 7.58413 2 0.0036 7.78703 2 0.0012 7.95318 2 0.0015 8.12411 2 0.0027 7.10772 2 0.0018 7.47951 2 0.0026 7.90101 2 0.0029 8.23150 2 0.0013 8.52760 2 0.0019 8.75233 2 0.0004 8.96274 2 0.0019 9.22088 2 0.0025 9.34271 2 0.0006 9.63336 2 0.0023 9.85533 2 0.0004 10.1016 2 0.005 10.2749 2 0.004 10.4186 2 0.007 10.6858 2 0.003 11.0112 2 0.006 c

6.20542 2 0.0009

z = 0.57353 Series 2 4.71698 2 0.0001 4.76039 2 0.00005 4.91222 2 0.0007 5.04454 2 0.0012 5.19381 2 0.0011 5.36380 2 0.0009 5.51358 2 0.0003 5.67011 2 0.0002 5.82566 2 0.0021 6.01402 2 0.0002 6.20554 2 0.0005 6.39326 2 0.0012 6.57827 2 0.0020 6.79244 2 0.0036 6.99922 2 0.0011 7.20595 2 0.0010 b

m*/(mol·kg−1 ) z=1

f* a

4.39660 2 0.0001 4.44288 2 0.0002 4.59634 2 0.0006 4.73198 2 0.0007 4.88511 2 0.0004 5.05964 2 0.0005 5.21315 2 0.0014 5.37136 2 0.0011 5.53079 2 0.0005 5.71713 2 0.0018 5.90765 2 0.0013 6.09749 2 0.0031 6.28362 2 0.0009 6.49942 2 0.0031 6.70383 2 0.0021 6.90458 2 0.0011 7.07791 2 0.0011 7.35702 2 0.0005 7.35875 2 0.0011 7.55341 2 0.0025 7.71691 2 0.0028 7.87822 2 0.0050 6.98483 2 0.0009 7.25598 2 0.0008 7.66520 2 0.0013 7.98406 2 0.0027 8.27329 2 0.0007 8.48221 2 0.0028 8.67889 2 0.0003 8.92631 2 0.0021 9.03531 2 0.0045 9.31133 2 0.0042 9.51731 2 0.0045 9.74905 2 0.0040 9.91301 2 0.0034 10.0426 2 0.003 10.2859 2 0.005 10.5720 2 0.005 10.7043 2 0.004 11.0047 2 0.007 11.2077 2 0.010 11.4780 2 0.005 11.7931 2 0.011 6.00657 2 0.0046

1.2118 1.2191 1.2431 1.2640 1.2874 1.3136 1.3362 1.3589 1.3813 1.4067 1.4318 1.4560 1.4824 1.5110 1.5375 1.5630 1.5845 1.6184 1.6186 1.6416 1.6605 1.6788 1.5617 1.6062 1.6545 1.6906 1.7221 1.7441 1.7643 1.7890 1.7997 1.8259 1.8449 1.8655 1.8798 1.8908 1.9110 1.9338 1.9440 1.96655 1.9812 2.0001 2.02115 1.4445

a Osmotic coefficients of the H2 SO4 (aq) reference standard solutions up through m* = 6.09749 mol·kg−1 were calculated using the equation and parameters given by Clegg et al.,(22) and at higher molalities using the equation and parameters of Rard et al.(23) b Crystallization occurred in both of the replicate sample solutions during the next isopiestic equilibration, and they were then discarded. Interpolation of the solubilities reported by Filippov and Antonova(13) yields solubilities of mT = (10.56 2 0.06) mol·kg−1 at z = 0.85811, of mT = (8.62 2 0.04) mol·kg−1 at z = 0.71539, and of mT = (6.06 2 0.04) mol·kg−1 at z = 0.57353. Their results indicate that the coexisting solid phase should be MgSO4 ·H2O(s) (kieserite) for z = (0.85811 and 0.71539) and MgSO4·7H2O(s) (epsomite) for z = 0.57353. c Crystallization occurred in both of the replicate sample solutions during the next isopiestic equilibration. Afterward, the sample cups were capped and stored in a desiccator. The solutions were later diluted with water and returned to the isopiestic chamber for the last equilibration.

f and g for {zH2 SO4 + (1 − z)MgSO4 }(aq)

541

molality experiment results were scattered by up to 0.004·m. However, after six equilibrations had been performed with the Series 3 samples, the uncertainties of the NaCl(aq) reference standard solution molalities suddenly increased to between 0.0014·m and 0.0017·m. This is slightly larger than the usual uncertainty of our experiments. A solution sample could give a high apparent molality, for example, if some of it splattered out of its sample cup when the chambers were being degassed, and it could give either a high or a low apparent molality if it became contaminated. There was no direct evidence that either of these occurred, although the first possibility is more likely. Calculated osmotic coefficients of {zH2 SO4 + (1 − z)MgSO4 }(aq) for the previous experiments of Series 3 were fairly consistent with those for the last four experiments, as they were with the Series 4 experiments. Consequently, the results for those four experiments should be only marginally less accurate than the others. Five series of experiments were performed in which separate pairs of samples of each {zH2 SO4 + (1 − z)MgSO4 }(aq) and of reference standard were weighed out for each series. For four of these five sets of samples, multiple experiments were performed by varying the amounts of water. The molality ranges of Series 1 and 3 overlapped, and the resulting osmotic coefficients for {zH2 SO4 + (1 − z)MgSO4 }(aq) solutions were found to be completely consistent. Osmotic coefficients of {zH2 SO4 + (1 − z)MgSO4 }(aq) with NaCl(aq) as reference standard and those with H2 SO4 (aq) as reference standard were consistent to within about 0.1 per cent, which is in excellent agreement and well within the accuracy of the experiments and the uncertainties of the reference standards. The higher molality experiments with H2 SO4 (aq) as reference standard were generally performed with the molalities increased from one experiment to the next by evaporation of water. However, in two separate cases, those samples were diluted down to reinvestigate part of the molality range studied previously. Resulting values of the osmotic coefficients of {zH2 SO4 + (1 − z)MgSO4 }(aq) solutions agreed to within (0.1 to 0.2) per cent with values obtained in the earlier experiments. This reproducibility indicates that no significant chemical changes occurred due to the (potential) reaction of H2 SO4 (aq) or acidic magnesium sulfate solutions with the tantalum cups, and that no significant amounts of SO3 had volatilized from the concentrated solutions during the equilibrations.

3. Calculation of osmotic coefficients and relations among various composition scales Osmotic coefficients of the {zH2 SO4 + (1 − z)MgSO4 }(aq) solutions were calculated from the fundamental equation for isopiestic equilibrium: fS = n*m*f*/Si ni mi ,

(1)

where ni is the stoichiometric ionization number of the electrolyte i assuming complete dissociation {ni = 3 for H2 SO4 (aq) and ni = 2 for MgSO4 (aq)}, mi is

542

J. A. Rard

the stoichiometric molality of the electrolyte i, and fS is the molality-based or ‘‘practical’’ osmotic coefficient of the mixed-electrolyte solution based on the use of stoichiometric molalities. Corresponding quantities for the isopiestic reference standards NaCl(aq) or H2 SO4 (aq) are denoted with asterisks (n* = 2 for NaCl). Osmotic coefficients of the reference standards f* were calculated for NaCl(aq) from Archer’s extended Pitzer model,(21) for H2 SO4 (aq) at m* E 6.1 mol·kg−1 from the extended Pitzer model of Clegg et al.,(22) and for H2 SO4 (aq) at m* q 6.1 mol·kg−1 from the empirical equation of Rard et al.(23) The recent mole fraction thermodynamic model of Clegg and Brimblecombe(24) would also have been satisfactory as a reference standard for H2 SO4 (aq) at high molalities, but the older equation of Rard et al. was used instead, since it has a slightly better overlap with the Clegg et al. equation at m 1 6 mol·kg−1. The calculation of fS requires values of Si ni mi for the mixed electrolyte solutions, whereas the quantities reported in tables 1 and 2 are the total molalities mT . In addition, thermodynamic properties of different electrolytes or of electrolyte mixtures at different molality fractions or ionic strength fractions are frequently compared at equal ionic strengths. Thus it is useful to give the relations among these composition scales. For {zH2 SO4 + (1 − z)MgSO4 }(aq), the total molality mT is just the sum of the stoichiometric molalities of the individual electrolytes: mT = m1 + m2 ,

(2)

where 1 denotes H2 SO4 (aq) and 2 denotes MgSO4 (aq). The ‘‘osmolality’’ or sum of the ionic molalities Si ni mi is then given by: Si ni mi = 3m1 + 2m2 = (2 + z)mT ,

(3)

where z is the molality fraction of H2 SO4 (aq). Similarly, the stoichiometric ionic strength of the solution IS is given by: IS = 3m1 + 4m2 = (4 − z)mT .

(4)

Thermodynamic properties of mixed electrolyte solutions are frequently analyzed in terms of binary solution contributions at some measure of total concentration weighted by some composition fraction. The most commonly used composition fractions for electrolyte mixtures are the ionic strength fractions: y1 = 3m1 /IS = 3z/(4 − z),

(5)

y2 = 4m2 /IS = 4(1 − z)/(4 − z).

(6)

Similarly, the ‘‘osmolality’’ fractions h of the electrolytes are given by: h1 = 3m1 /(Si ni mi ) = 3z/(2 + z),

(7)

h2 = 2m2 /(Si ni mi ) = 2(1 − z)/(2 + z).

(8)

f and g for {zH2 SO4 + (1 − z)MgSO4 }(aq)

543

Values of the water activities aw can be computed from those of the osmotic coefficients fS using the equation: ln aw = −Mw·fS ·(Si ni mi ) = −Mw·fS ·(2 + z)·mT ,

(9)

−1

where Mw = 0.0180153 kg·mol is the molar mass of water. The experimental isopiestic results are presented in tables 1 and 2. Each row contains the total molalities mT of {zH2 SO4 + (1 − z)MgSO4 }(aq) and molality m* of the isopiestic reference standard that are in isopiestic equilibrium, i.e., all of which have the same value of aw . These mixtures were prepared to have approximately equally spaced values of z; somewhat surprisingly, the equilibrium values of mT also are nearly equally spaced. The molality m* of H2 SO4 (aq) in equilibrium with these solutions (table 2) is also about equally spaced from the z = 0.85811 curve. Thus the following approximate relation applies: mT 1 m* + b(1 − z),

(10)

at any constant value of aw . Furthermore, the difference in molality DmT from changing z by 11/7 under isopiestic conditions is only weakly dependent on mT and on z, ranging from DmT = (0.08 to 0.12) mol·kg−1 for mT = (1 to 7.6) mol·kg−1; this yields b = (0.56 to 0.84) mol·kg−1. Equation (10) implies that, in this range of mT values, the water activity curves for {zH2 SO4 + (1 − z)MgSO4 }(aq) are essentially parallel to each other and to the curve for H2 SO4 (aq). Figure 2 is a plot

FIGURE 2. Variation of the water activity aw of 4zH2 SO4 + (1 − z)MgSO4 5(aq) with the total molality of the solution mT at T = 298.15 K. Curves for z = (0.85811 and 0.71539) are not shown, but they are similar in their molality dependences and fall intermediate between the curves for z = 1 and z = 0.57353.

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of the values of aw as a function of mT for z = (1, 0.57353, and 0). It can readily be seen that the curves at z = 1 {pure H2 SO4 (aq)} and z = 0.57353 are indeed quite similar in their dependence on molality. Curves for z = (0.85811 and 0.71529), which are not shown, are also quite similar and fall between the z = (1 and 0.57353) curves. A mole fraction based activity coefficient of water fw in these {zH2 SO4 + (1 − z)MgSO4 }(aq) solutions is defined by: aw = fw·xw ,

(11)

where xw is the mole fraction of water in such a solution, assuming that the electrolytes are fully dissociated: xw = mw /(mw + 3m1 + 2m2 ) = mw /{mw + (2 + z)mT }.

(12) −1

The molality of water mw is defined as mw = 1/(0.0180153 kg·mol ) = 55.5084 mol·kg−1. For a solution of H2 SO4 (aq) in isopiestic equilibrium with a solution of {zH2 SO4 + (1 − z)MgSO4 }(aq): fw = f w*·x* w /xw ,

(13)

x* w = mw /(mw + 3m*).

(14)

where

Inserting equations (12) and (14) into equation (13) then yields: fw = f w* {mw + (2 + z)mT }/(mw + 3m*)

(15)

which is an exact relation for solutions of these charge types in isopiestic equilibrium. For the molality region over which the approximation given by equation (10) is applicable, equation (15) can be rewritten as: fw 1 f w* [1 − (1 − z){m* − b(2 + z)}/(mw + 3m*)].

(16)

We now consider two limiting cases for this equation. First, at the low molality limit (m* : 0): fw : f w* {1 + (1 − z) (2 + z)b/mw }.

(17)

Thus, at low molalities, a linear relation for the isopiestic molalities, equation (10), yields a quadratic dependence of fw upon z, equation (17). Secondly, at the high molality limit (m* : a): fw : f w* {(2 + z)/3},

(18)

and a linear dependence of fw upon z is obtained. However, the condition of (m* : a) can only be achieved when z is very close to 1. Because of solubility limitations at other values of z, this equation has only limited applicability. Equation (10) and equations (16) through (18) are purely empirical representations of the isopiestic results for {zH2 SO4 + (1 − z)MgSO4 }(aq). It is possible that they are also applicable to mixtures of H2 SO4 (aq) with other divalent metal salts

f and g for {zH2 SO4 + (1 − z)MgSO4 }(aq)

545

TABLE 3. Smoothed values of the stoichiometric osmotic coefficients fS for {zH2 SO4 + (1 − z)MgSO4 }(aq) at round values of the total molalities mT at the temperature 298.15 K mT

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

z=1 a

z = 0.85811

0.6744 0.7194 0.7790 0.8464 0.9180 0.9926 1.0699 1.1490 1.2280 1.3047 1.3770 1.4437 1.5111 1.5749 1.6353 1.6923 1.7460 1.7962 1.8433 1.8872 1.9281 1.9662 2.0016 2.0345

0.652 0.699 0.762 0.831 0.908 0.988 1.070 1.155 1.243 1.328 1.405 1.476 1.547 1.616 1.679 1.737 1.792 1.842 1.888 1.933 1.973 2.010 2.043 2.074

fS z = 0.71539 0.630 0.675 0.740 0.813 0.893 0.980 1.068 1.161 1.256 1.348 1.434 1.511 1.586 1.658 1.723 1.785 1.842 1.892 1.938 1.981 2.020 2.051

z = 0.57353

z=0 b

0.608 0.649 0.715 0.790 0.875 0.968 1.065 1.166 1.271 1.373 1.470 1.556 1.638 1.717

0.522 0.530 0.5791 0.6616 0.7747 0.9193 1.0982

a Osmotic coefficients of the H2 SO4 (aq) solutions up through m = 6 mol·kg−1 were calculated using the equation and parameters given by Clegg et al.,(22) and at higher molalities using the equation and parameters of Rard et al.(23) b Osmotic coefficients of the MgSO4 (aq) solutions with mT e 1.3 mol·kg−1 were calculated using the equation and parameters of Rard and Miller.(9) Values for mT Q 1.3 mol·kg−1 were obtained graphically from the unpublished results of Archer and Rard.(25)

MSO4 (aq). Equation (10) is not valid for {zH2 SO4 + (1 − z)Na2 SO4 }(aq) at T = 298.15 K.(3,4) Tables 3 and 4 contain smoothed values of fS at round values of mT and IS . Results for the mixed electrolytes were obtained by graphical smoothing of our new experimental results. Values of fS for H2 SO4 (aq) were calculated from the equation of Clegg et al.(22) for mT E 6 mol·kg−1 (IS E 18 mol·kg−1 ) and from the equation of Rard et al.(23) at higher molalities. For MgSO4 (aq), the values of fS were calculated from the equation of Rard and Miller(9) at higher molalities but, for mT Q 1.3 mol·kg−1, their values were modified by using the results of newer isopiestic measurements from our laboratory.(25) Figures 3 and 4 are plots of fS as a function of mT and of IS . As can be seen from these curves, the values of fS fall in a regular and decreasing order in going from pure H2 SO4 (aq) to pure MgSO4 (aq) at low and intermediate values of mT and of IS . However, in both cases, the order of the curves for z = (1, 0.85811, 0.71539, and 0.57353) undergoes a complete reversal at moderate to high values of mT or IS .

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TABLE 4. Smoothed values of the stoichiometric osmotic coefficients fS for {zH2 SO4 + (1 − z)MgSO4 }(aq) at round values of the stoichiometric ionic strength IS at the temperature 298.15 K z=1 a

z = 0.85811

fS z = 0.71539

0.6657 0.6870 0.7194 0.7581 0.8009 0.8464 0.8937 0.9425 0.9926 1.0438 1.1490 1.2540 1.3535 1.4437 1.5327 1.6156 1.6923 1.7631 1.8280 1.8872 1.9411 1.9901 2.0345

0.644 0.662 0.694 0.732 0.774 0.818 0.865 0.915 0.966 1.017 1.124 1.236 1.342 1.439 1.528 1.615 1.696 1.768 1.832 1.893 1.948 1.997 2.040

0.625 0.637 0.666 0.702 0.744 0.786 0.833 0.882 0.934 0.987 1.096 1.211 1.325 1.430 1.525 1.616 1.700 1.776 1.844 1.905 1.959 2.008 2.048

IS

1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30 32 34 36

z = 0.57353

z=0 b

0.606 0.612 0.637 0.671 0.710 0.752 0.796 0.846 0.898 0.952 1.065 1.185 1.308 1.424 1.530 1.625 1.717

0.548 0.522 0.519 0.530 0.551 0.5791 0.6165 0.6616 0.7143 0.7747 0.9193 1.0982

a Osmotic coefficients of the H2 SO4 (aq) solutions up through IS = 18 mol·kg−1 were calculated using the equation and parameters given by Clegg et al.(22) and at higher stoichiometric ionic strengths using the equation and parameters of Rard et al.(23) b Osmotic coefficients of the MgSO4 (aq) solutions with IS e 5.2 mol·kg−1 were calculated using the equation and parameters of Rard and Miller.(9) Values for IS Q 5.2 mol·kg−1 were obtained graphically from the unpublished results of Archer and Rard.(25)

These fS inversions occur at mT 1 (3.5 to 3.6) mol·kg−1 for z = (1, 0.85811, 0.71539, and 0.57353). Even the z = 0 curve {pure MgSO4 (aq)} crosses the other four curves but at the slightly lower mT 1 3.4 mol·kg−1. Similarly, the fS curves for z = (0.85811, 0.71539, and 0.57353) first cross each other in the range IS 1 (19.1 to 19.6) mol·kg−1. However, there are some additional crossovers at higher ionic strengths, and crossovers between the curves with z = (0.85811, 0.71539, and 0.57353) and the z = 1 curve {pure H2 SO4 (aq)} are not complete until IS 1 22 mol·kg−1. In addition, the fS curve at z = 0 does not cross the others on this plot. In neither the plot of fS against mT nor of fS against IS is there a single total molality or stoichiometric ionic strength where values of fS are identical at all of the molality fractions z = (1, 0.85811, 0.71539, and 0.57353), although they come close at mT 1 3.5 mol·kg−1. There appears to be a true crossover at a particular constant value of the ‘‘osmolality.’’ Totally by accident, an experiment was performed at almost exactly this value. The Series 3 experiment with a NaCl(aq) reference solution

f and g for {zH2 SO4 + (1 − z)MgSO4 }(aq)

547

FIGURE 3. Variation of the stoichiometric osmotic coefficient fS of {zH2 SO4 + (1 − z)MgSO4 }(aq) with the total molality mT of the solution at T = 298.15 K.

molality of 2.47855 mol·kg−1 yielded the following results for the three {zH2 SO4 + (1 − z)MgSO4 }(aq) solutions in isopiestic equilibrium: Si ni mi = 5.9591 mol·kg−1 and fS = 0.8446 at z = 0.85811; Si ni mi = 5.9653 mol·kg−1 and fS = 0.8437 at z = 0.71539; and Si ni mi = 5.9652 mol·kg−1 and fS = 0.8437 at z = 0.57353. These solutions were not equilibrated against pure H2 SO4 (aq). However, the average of the three ‘‘osmolalities’’ is Si ni mi = (5.9632 2 0.0036) mol·kg−1, for which the equation of Clegg et al.(22) for H2 SO4 (aq) yields fS = 0.8446. There is a maximum variation of Si ni mi of 0.104 per cent among these values and of 0.107 per cent among the four values of fS . Given that the values of both Si ni mi and fS are uncertain by (0.1 to 0.2) per cent, these minor differences are insignificant. Thus there is a single ‘‘osmolality’’ where both fS and aw for {zH2 SO4 + (1 − z)MgSO4 }(aq) are completely independent of the molality fraction for

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FIGURE 4. Variation of the stoichiometric osmotic coefficient fS of {zH2 SO4 + (1 − z)MgSO4 }(aq) with the stoichiometric ionic strength of the solution IS at T = 298.15 K.

1 e z e 0.57353 and, possibly, to even lower values of z. The average ‘‘osmolality’’ of Si ni mi = (5.9632 2 0.0036) mol·kg−1 corresponds to a MgSO4 (aq) molality of (2.9816 2 0.0018) mol·kg−1, for which the equation of Rard and Miller(9) yields fS = (0.9134 2 0.0032). Even with this relatively large uncertainty, this value is significantly larger than fS = (0.8437 to 0.8446) for compositions with 1 e z e 0.57353. Thus the constancy of fS and aw at Si ni mi = (5.9632 2 0.0036) mol·kg−1 should break down somewhere in the acid-deficient region. Very few other ternary systems have been studied over a sufficiently large molality range at constant z to establish whether or not the fS curves undergo crossovers like those observed for {zH2 SO4 + (1 − z)MgSO4 }(aq) at T = 298.15 K. Examination of the isopiestic results of Rard and Miller(26) for {zNaCl + (1 − z)MgCl2 }(aq) at T = 298.15 K indicates that the fS curves undergo an inversion centered around Si ni mi 1 1.25 mol·kg−1. However, for that system there is no single crossover value for Si ni mi but rather the crossovers occur over a range of Si ni mi values.

f and g for {zH2 SO4 + (1 − z)MgSO4 }(aq)

549

4. Results and discussion Scatchard’s neutral electrolyte equations provide an accurate method for analyzing the thermodynamic properties of aqueous mixed electrolyte solutions. Scatchard’s general equations as originally given(27) are complicated functions of the ionic strength, of the ‘‘osmolality’’, of the individual ionic molalities, of the ionic strength fractions of each component electrolyte, and of the osmotic coefficients of the individual constituent electrolytes at the total ionic strength of the solutions. However, the equation for the osmotic coefficient of a mixed electrolyte solution can be rewritten as a binary solution contribution together with a combination of symmetrical and asymmetrical mixing terms which depend on the ionic strength fractions and the total ionic strength. For a ternary electrolyte solution the osmotic coefficient equation may thus be rewritten(26) in the simple form: fS = h1 fS,1 + h2 fS,2 + mixing terms,

(19)

where fS,i is the stoichiometric osmotic coefficient of electrolyte i evaluated at some constant stoichiometric ionic strength IS , and fS is the stoichiometric osmotic coefficient of the mixed-electrolyte solution at the same value of IS . The binary solution mixing approximation to fS is given by the first two terms on the right-hand side of this equation: f°S = h1 fS,1 + h2 fS,2

(20)

where ‘‘osmolality’’ fraction mixing statistics are used. Figure 5 is a plot of fS against h1 , with the binary solution mixing approximation indicated by dashed lines connecting the values of fS,i for the two single-electrolyte solutions. In the ionic strength region where values of fS,i are available at h1 = 0 {pure MgSO4 (aq)}, the deviations from the binary solution mixing approximation are negative and surprisingly small. If the curve of fS,i for MgSO4 (aq) were extrapolated to even higher (supersaturated) values of IS (not shown), it is apparent that this curve would cross those for H2 SO4 (aq) and all of the mixtures between IS 1 (16 and 17) mol·kg−1. In figure 5, values of fS decrease as h1 decreases at IS = (2, 8, and 14) mol·kg−1, are nearly constant at IS = 20 mol·kg−1, and increase as h1 decreases at higher values of IS . This reversal in slope is just a result of the reversal in the relative order of the curves between IS = (19 and 22) mol·kg−1 as described in the previous section. A plot of fS against h2 for {zH2 SO4 + (1 − z)Na2 SO4 }(aq) was given by Rard,(4) and shows quite large negative deviations from the binary solution mixing approximation. In contrast, the {zH2 SO4 + (1 − z)MgSO4 }(aq) system forms nearly ideal mixtures, at least for z e 0.57353, with only very small negative deviations. Ion-pair formation is known to occur in solutions of MgSO4 (aq), and, because of the greater charge on Mg2+ than on Na+, much larger deviations might be expected in {zH2 SO4 + (1 − z)MgSO4 }(aq). This is not observed. Scatchard’s neutral-electrolyte equations(27) do not include the effects of ionic association explicitly, and their presence, if not already known, must be inferred from the sign and magnitude of the symmetrical and asymmetrical mixing terms.

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FIGURE 5. Variation of the stoichiometric osmotic coefficient fS of {zH2 SO4 + (1 − z)MgSO4 }(aq) with the ‘‘osmolality’’ fraction h1 of H2 SO4 in the solution at constant values of the stoichiometric ionic strength IS . Dashed lines represent Scatchard’s neutral-electrolyte binary-solution mixing approximation. a, IS = 32 mol·kg−1; b, IS = 26 mol·kg−1; c, IS = 20 mol·kg−1; d, IS = 14 mol·kg−1; e, IS = 8 mol·kg−1; f, IS = 2 mol·kg−1.

f and g for {zH2 SO4 + (1 − z)MgSO4 }(aq)

551

In contrast, Pitzer’s equations(28) can incorporate ionic association effects directly, and thus can provide estimates of how the speciation changes with the composition of the solution. The first dissociation of H2 SO4 (aq) is essentially complete at molalities up to about 40 mol·kg−1,(22) and thus it can be assumed to be complete at the molalities of concern to the present study. However, dissociation of the HSO− 4 (aq) ion is incomplete in solutions of H2 SO4 (aq) and in acidic sulfate solutions, and the following equilibrium must be considered explicitly when modeling their thermodynamic behavior: 2− + HSO− 4 (aq) = H (aq) + SO4 (aq).

(21)

For this reaction the thermodynamic dissociation constant is given by: 2− − + Kd (HSO− 4 ) = a(H )·a(SO4 )/a(HSO4 ).

(22)

Most modern values of Kd (HSO− 4 ) at T = 298.15 K on the molality scale range between 0.0102 and 0.0106,(22,29) and their mean of (0.0104 2 0.0002) encompasses all of these results. Recent e.m.f. measurements(30) yield Kd (HSO− 4 ) = (0.0104 2 0.0002), and conductance measurements(31) yield Kd (HSO− 4 ) = (0.0103 2 0.0003), both of which agree well with the earlier values. Clegg et al.(22) selected Kd (HSO− 4 ) = 0.0105 for their extended Pitzer model for the thermodynamics of H2 SO4 (aq). This same model predicts that the fractional extent of dissociation of the HSO− 4 (aq) ion only varies between 0.21 and 0.33 for molalities of H2 SO4 (aq) ranging from (0.05 to 6) mol·kg−1 at T = 298.15 K and that these same solutions undergo partial −1 redissociation of the HSO− 4 (aq) ion between m = (0.5 and 4.0) mol·kg . Magnesium sulfate solutions are known to undergo ion pairing to form a neutral complex MgSO4 (aq,undissoc.). Equilibrium constants are usually reported for the association reaction: Mg2+(aq) + SO2− 4 (aq) = MgSO4 (aq,undissoc.),

(23)

where Ka (MgSO4 ) = a(MgSO4 )/a(Mg2+ )·a(SO2− 4 )= 2+ )·g(SO2− {m(MgSO4 )/m(Mg2+ )·m(SO2− 4 )} {g(MgSO4 )/g(Mg 4 )}.

(24)

This thermodynamic association constant is related to the thermodynamic dissociation constant Kd (MgSO4 ) by Ka (MgSO4 ) = Kd (MgSO4 )−1. Literature values of Ka (MgSO4 ) have been summarized by Rull et al.(12) and by Yokoyama and Yamatera.(32) These values of Ka (MgSO4 ) range from 107 to 260 at T = 298.15 K, which is a very large variation. In addition, Rull et al. obtained Ka (MgSO4 ) = (20.9 2 2.8) from their own Raman spectroscopic measurements, which is considerably lower. Calculated values of Ka (MgSO4 ) are quite sensitive both to the method of measurement and to the calculation method used by the authors. Pitzer and Mayorga(33) have analyzed the thermodynamic activities of various MSO4 (aq) solutions at T = 298.15 K. Even though their model does not explicitly (2) consider the existence of MgSO4 (aq,undissoc.) ion pairs, their bMX parameter (2) is related to the thermodynamic association constant by Ka (MgSO4 ) = −2·bMX .

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(2) This yields Ka (MgSO4 ) = 74.5 using the bMX value of Pitzer and Mayorga(33) and Ka (MgSO4 ) = 65.5 using the evaluation of Rard and Miller.(9) These values are rather smaller than the ones obtained from more direct experimental thermodynamic methods, but are not as small as the value of Rull et al.(12) from Raman spectroscopy. Attempts have been made, as described below, to explain this variation in terms of the simultaneous presence of several different types of complexes. Methods based on thermodynamic or electrolyte conductance measurements are assumed to detect the overall amounts, or some type of weighted average of complexes present, whereas spectroscopic measurements may detect only one type of complex.(12,32) However, there is considerable ambiguity in these analyses, especially as to the nature of the postulated complexes. These methods are now described. Rull et al.(12) summarized some of the claims about the nature of complexes present in solutions of MgSO4 (aq). The model used to explain ultrasonic absorption measurements postulates the existence of two types of solvent-separated ion pairs in addition to contact ion pairs. Rull et al. interpreted their own Raman spectroscopic measurements as excluding the possibility of contact ion pairs, and instead proposed a four-step model in which four different types of solvent-separated ion pairs were present. In contrast, Davis and Oliver(34) did similar Raman spectroscopic measurements but concluded that contact ion pairs were formed. Yokoyama and Yamatera(32) estimated that the ratio of the formation constant for contact ion pairs to the formation constant for all types of ion pairs was about 0.3 based on analyses of vapor-pressure osmometry and electrical conductance measurements, and about 0.2 based on an analysis of spectrophotometric results for MgSO4 (aq) at T = 298.15 K. A smaller value of 0.10 was obtained for this ratio from an ultrasonic absorption study.(35) Some thermodynamic models predict that ion pairs in solutions of MgSO4 (aq) undergo redissociation with increasing molality. For example, the cluster-theory model calculations of Corti and Ferna´ndez-Prini(36) imply that redissociation begins at m 1 0.03 mol·kg−1 and is nearly complete at m 1 1 mol·kg−1. Similarly, Davis and Oliver(34) concluded from Raman spectroscopic measurements that, although the total amount of ion pairs increased regularly as the amount of substance concentration was increased, the fractional extent of association increased to a maximum of 0.13 at c 1 1 mol·dm−3 and then decreased to a constant value of 0.11 for c = (1.41 to 2.50) mol·dm−3. In contrast, other Raman spectroscopic measurements(12) from m = (0.57 to 2.87) mol·kg−1 indicate that the fractional extent of ion pair formation increases steadily from 0.044 to 0.125 with increasing molality. Pitzer and Mayorga(33) discussed the thermodynamic condition for redissociation to occur for the equilibrium described by equations (23) and (24). This involves the activity coefficient ratio: g(MgSO4 )/g(Mg2+ )·g(SO2− 4 ). They noted that the activity coefficients in the denominator of this expression could decrease rapidly enough with increasing molality so that this activity coefficient ratio divided by the stoichiometric molality would go through a minimum and then increase. This is

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equivalent to the fraction of ion-pair formation going through a maximum, followed by a decrease (redissociation) at higher molalities. As noted by Pitzer and Mayorga,(33) some theoretically-based treatments are available in which the thermodynamic activities of MgSO4 (aq) solutions are analyzed without assuming that ion-pairs are present. These models are based on avoiding the Debye–Hu¨ckel approximation of linearization of the Boltzmann distribution. Their own Pitzer model did not explicitly include the existence of these ion-pairs, although (2) they are included implicitly by use of the bMX parameter which is proportional to Ka (MgSO4 ). At the opposite extreme, ion-pairs, ion-triplets such as Mg(SO4 )2− 2 (aq) or Mg2 SO2+ 4 (aq), ion quadruplets, or even higher order ionic associations have been included in other thermodynamic models.(11,36,37) Harvie et al.(8) reported the results of an analysis of thermodynamic results for {zH2 SO4 + (1 − z)MgSO4 }(aq) using Pitzer’s equations, but found it necessary to set one of the mixing parameters and CfMX for the ‘‘binary electrolyte’’ formed from Mg2+(aq) and HSO− 4 (aq) equal to zero. This was done because of the paucity of (0) (1) , bMX , and CfMX thermodynamic results then available. Approximate values of bMX − 2+ were obtained for Mg (aq) and HSO4 (aq) interactions by Campbell et al.(38) from e.m.f. measurements with acidified synthetic seawater containing the following − ions: Na+, K+, H + , Mg2+, Ca2+, Cl−, SO2− 4 , and HSO4 . Given the complexity of that system, it is desirable to have a more direct determination of the Pitzer parameters from a system containing only the essential ions. A main goal of the present study was thus to provide extensive isopiestic results for {zH2 SO4 + (1 − z)MgSO4 }(aq) to allow a more complete analysis to be carried out with Pitzer’s equations.(28) This treatment also requires that accurate values of the Pitzer parameters be known or be evaluated for MgSO4 (aq) and for the + ‘‘binary electrolytes’’ formed from H+(aq) and HSO− 4 (aq) and from H (aq) and 2− (22) SO4 (aq). The extended Pitzer model of Clegg et al. provides accurate values of the Pitzer parameters for the latter two cases, based on a comprehensive and critical analysis of the H2 SO4 (aq) system. However, as discussed in detail above, there is much uncertainty as to the nature and extent of ionic association in MgSO4 (aq), and also as to the best way to perform the analysis with Pitzer’s equations. We have performed new isopiestic experiments at T = 298.15 K for MgSO4 (aq) with m = (0.52502 to 1.3080) mol·kg−1. Those equilibrations were done with unusually long equilibration times of (72 to 98) days, which yielded very precise osmotic coefficients. We are attempting to obtain a more satisfactory Pitzer-based model based on these results and on the published thermodynamic results for MgSO4 (aq).(25) In this report, isopiestic molalities are given for the acid-rich region of {zH2 SO4 + (1 − z)MgSO4 }(aq) at T = 298.15 K. These equilibrium molalities yield 195 values of the osmotic coefficients at total molalities ranging from about (0.125 to 0.147) mol·kg−1 to supersaturated molalities and involved over 3 years of experimental measurements. We plan to extend our measurements to the acid-deficient region of this system. (We will be moving our laboratory and all of its equipment to a different building sometime during 1997. Thus there will be a several month interruption in those measurements.) After those additional

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measurements have been performed, and after the thermodynamic analysis for the MgSO4 (aq) system has been completed, all of the isopiestic results for {zH2 SO4 + (1 − z)MgSO4 }(aq) will be analyzed with an extended version or versions of Pitzer’s equations. The detailed calculations will be presented at that time. Isopiestic molalities have been determined previously for {zH2 SO4 + (1 − z)MgSO4 }(aq) at T = 298.15 K by Filippov and Antonova(13) and by Michimoto et al.(39) However, in neither of those studies were the actual isopiestic molalities reported; results were presented solely as small plots of m1 against m2 at constant values of aw . Such plots are frequently linear or nearly linear, and the linear mixing relation is known as Zdanovskii’s rule. The study of Filippov and Antonova(13) covered a fairly wide molality range. However, the principal investigator of that study, Professor V. K. Filippov, died on 11 February 1991 so it was not possible for us to obtain the original isopiestic molalities for a comparison with our new results. Professor Y. Awakura kindly supplied us with a privately published booklet(40) that contains the actual equilibrium molalities for various isopiestic measurements from their laboratory, including those from the study of Michimoto et al.(39) These measurements were done at seven different values of aw with mT ranging from (0.7083 to 2.7049) mol·kg−1. A comparison with our results in this molality region will be made after our measurements have been extended to the acid-deficient region of {zH2 SO4 + (1 − z)MgSO4 }(aq). This work was supported under the auspices of the Office of Basic Energy Sciences (Geosciences) of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48. The author thanks Dr Donald G. Miller for helpful suggestions; Prof John G. Albright for discussions about the ‘‘fans’’ used for stirring of the vapor phase and Warren Bell for constructing them; and Mike Sharp, Bea Armstrong, Randall K. Weese, and Theresa Duewer for the spectroscopic impurity analyses. REFERENCES 1. Clegg, S. L.; Whitfield, M. Activity Coefficients in Electrolyte Solutions. (2nd edition). Pitzer, K. S.: editor. CRC Press: Boca Raton. 1991, Chap. 6. 2. Reardon, E. J.; Beckie, R. D. Geochim. Cosmochim. Acta 1987, 51, 2355–2368. 3. Rard, J. A. J. Chem. Thermodynamics 1989, 21, 539–560. 4. Rard, J. A. J. Chem. Thermodynamics 1992, 24, 45–66. 5. Hovey, J. K.; Pitzer, K. S.; Rard, J. A. J. Chem. Thermodynamics 1993, 25, 173–192. 6. Holmes, H. F.; Mesmer, R. E. J. Chem. Thermodynamics 1993, 25, 99–110. 7. Holmes, H. F.; Mesmer, R. E. J. Chem. Thermodynamics 1994, 26, 581–594. 8. Harvie, C. E.; Mo ller, N.; Weare, J. H. Geochim. Cosmochim. Acta 1984, 48, 723–751. 9. Rard, J. A.; Miller, D. G. J. Chem. Eng. Data 1981, 26, 33–38. 10. Holmes, H. F.; Mesmer, R. E. J. Solution Chem. 1986, 15, 495–517. 11. Archer, D. G.; Wood, R. H. J. Solution Chem. 1985, 14, 757–780. 12. Rull, F.; Balarew, Ch.; Alvarez, J. L.; Sobron, F.; Rodriguez, A. J. Raman Spectroscop. 1994, 25, 933–941. 13. Filippov, V. K.; Antonova, V. A. J. Appl. Chem. USSR 1978, 51, 240–244. 14. Harned, H. S.; Sturgis, R. D. J. Am. Chem. Soc. 1925, 47, 945–953.

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15. Rard, J. A. J. Solution Chem. 1985, 14, 457–471. 16. Rard, J. A.; Platford, R. F. Activity Coefficients in Electrolyte Solutions (2nd edition). Pitzer, K. S.: editor. CRC Press: Boca Raton. 1991, Chap. 5. 17. Scatchard, G.; Hamer, W. J.; Wood, S. E. J. Am. Chem. Soc. 1938, 60, 3061–3070. 18. Rard, J. A. J. Chem. Thermodynamics 1996, 28, 83–110. 19. Mitchell, J. P.; Butler, J. B.; Albright, J. G. J. Solution Chem. 1992, 21, 1115–1129. 20. Rard, J. A.; Archer, D. G. J. Chem. Eng. Data 1995, 40, 170–185. 21. Archer, D. G. J. Phys. Chem. Ref. Data 1992, 21, 793–829. 22. Clegg, S. L.; Rard, J. A.; Pitzer, K. S. J. Chem. Soc. Faraday Trans. 1994, 90, 1875–1894. 23. Rard, J. A.; Habenschuss, A.; Spedding, F. H. J. Chem. Eng. Data 1976, 21, 374–379. 24. Clegg, S. L.; Brimblecombe, P. J. Chem. Eng. Data 1995, 40, 43–64. 25. Archer, D. G.; Rard, J. A. unpublished results. 26. Rard, J. A.; Miller, D. G. J. Chem. Eng. Data 1987, 32, 85–92. 27. Scatchard, G. J. Am. Chem. Soc. 1961, 83, 2636–2642. 28. Pitzer, K. S. Activity Coefficients in Electrolyte Solutions (2nd edition). Pitzer, K. S.: editor. CRC Press: Boca Raton. 1991, Chap. 3. 29. Cabani, S.; Gianni, P. Anal. Chem. 1972, 44, 253–259. 30. Mussini, P. R.; Longhi, P.; Mussini, T.; Rondinini, S. J. Chem. Thermodynamics 1989, 21, 625–629. 31. Wu, Y. C.; Feng, D. J. Solution Chem. 1995, 24, 133–144. 32. Yokoyama, H.; Yamatera, H. Bull. Chem. Soc. Jpn. 1981, 54, 2286–2289. 33. Pitzer, K. S.; Mayorga, G. J. Solution Chem. 1974, 3, 539–546. 34. Davis, A. R.; Oliver, B. G. J. Phys. Chem. 1973, 77, 1315–1316. 35. Atkinson, G.; Petrucci, S. J. Phys. Chem. 2 1966, 70, 3122–3128. 36. Corti, H. R.; Ferna´ndez-Prini, R. J. Chem. Soc., Faraday Trans. 2 1986, 82, 921–932. 37. Gardner, A. W.; Glueckauf, E. Proc. Roy. Soc. London 1969, A313, 131–147. 38. Campbell, D. M.; Millero, F. J.; Roy, R.; Roy, L.; Lawson, M.; Vogel, K. M.; Moore, C.P. Marine Chem. 1993, 44, 221–233. 39. Michimoto, T.; Awakura, Y.; Majima, H. Denki Kagaku 1983, 51, 373–380. 40. Majima, H.; Awakura, Y.; Kawasaki, Y. Activities of Water and Solutes in the Aqueous Solution Systems H2SO4 –Mx(SO4)y and HCl–MClx . Agne Shofu Publishing Inc.: Tokyo. 1988.

(Received 22 July 1996; in final form 11 October 1996)

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