J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1875-1894 Thermodynamic Properties of 0-6 rnol kg -Aqueous Sulfuric Acid from 273.15 to 328.15 K Simon L. Clegg School of Environmental Sciences, University of East Anglia , Norwich, UK NR4 7TJ Joseph A. Rard Earth Sciences Division, Physical Sciences Department, Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA Kenneth S. Pitzer Department of Chemistry, University of California, Berkeley, CA 94720, USA Generalised equations are presented for an extended form of the Pitzer molality-based thermodynamic model, involving an ionic strength-dependent third virial coefficient. Compatibility with the established formulation is retained. Osmotic coefficients, emf measurements, degrees of dissociation of the HSO, ion, differential enth- alpies of dilution and heat capacities for aqueous H2S0, from 273.15to 328.15 K, 0-6.1 rnol kg-' and at 1 atm pressure have been critically evaluated.Treating this solution as the mixture H+-HSO,-SOz--H,O, and using hydrogen sulfate dissociation constants from the literature, the model parameters were fitted to the data yielding a self-consistent representation of activities, speciation and thermal properties together with the standard poten- tials of four electrochemical cells and standard-state heat capacities of the SO;-ion as functions of tem-perature. The model equations represent the experimental data accurately (without the use of mixture parameters OHSO4, so4 and t,hHSo4,so4,,,), and should yield values of the osmotic coefficient that are suitable for use as an isopiestic standard over this temperature and molality range.The new model will also enable improved prediction of the properties of mixed acidic sulfate systems. 1. Introduction terms of the molalities of the dissolved species, which may be Aqueous sulfuric acid is a major industrial chemical, and its ionic or neutral solute^.^ The model treats strong electrolytes thermodynamic properties have been studied extensively over as fully dissociated in solution. In addition to earlier work on many years.' Recent evaluations include those by Staples' the thermodynamics of aqueous H,S04," the model has and Rard et d3(osmotic and activity coefficients at 298.15 been used successfully in numerous geochemical applica- K), Bolsaitis and Elliott4 (partial pressures) and Zeleznik' tions.(properties of aqueous and solid phases, excluding vapour Raman spectral studies have shown that the first disso- +pressures). Sulfuric acid is also an important component of ciation of sulfuric acid (H,SO, g H + HSO,) is essentially and of complete at <40 mol kg-' (14 mol dm-3) and 298.15 K (seeatmospheric aerosols,6 notably in the ~tratosphere,~ brines. A knowledge of the sulfate-hydrogensulfate equi-also Section 3.5).14 However, this is not the case for the librium is required to calculate solubilities and partial pres- second dissociation reaction involving the hydrogensulfate sures of volatile acids such as HC1 and HNO, in acidified ion, whose dissociation must be considered explicitly : sulfate mixtures, and for relating hydrogen-ion activities in HSOi(aq,+ SO:,,) + Hi,) (1)seawater and estuarine waters to pH measurements.8 For practical applications, a treatment of aqueous H2S04 thermodynamics that readily generalises to solution mixtures is required.The Pitzer ion-interaction model9 has previously been applied to represent evaluated osmotic coefficients, emf and enthalpy data for that However, to achieve where KHso&nol kg- is the thermodynamic dissociation the accuracy that is often required, and to incorporate more constant of HSO, in solution, rn and a denote molality and recent experimental work on the osmotic coefficient, heat activity, respectively, and yi is the activity coefficient of capacity and hydrogensulfate dissociation constant, a more species i.detailed and comprehensive treatment is worthwhile. The basic model equations for osmotic (4) and ionic activ- Here we utilise an extended form of the Pitzer ion-ity coefficients contain the cation-anion (ca) interaction interaction model to represent osmotic coefficients, vapour parameters E:),fi:), and C,, ,which are functions of tem- pressure, emf, enthalpy of dilution, heat capacity and degree perature and pressure. A series of additional parameters may of dissociation data from 0 to 6.1 mol kg-', T = 273.15 to be included to describe interactions between ions and any 328.15 K. The equations and parameters presented provide neutral species present, but these are not required here.an accurate and self-consistent description of these thermody- Three terms relate to the H', SO:-pairwise interaction; namic properties, and will form the basis of an improved in the first order in molality this effect is given by the sum model for solution mixtures at moderate temperature and 1/KHSo4+ pg)so4+./3g,!w4.In second or higher order, the molality. effect of each term is different; hence there is no objection to the inclusion of all three." The limiting effective KHS04at infinite dilution is the reciprocal of this sum. Since Sg),,, and2. The Model j3g.)so4are small in comparison to 1/KH,04, their effect is to The Pitzer ion-interaction model is based upon an expression modify the limiting KHSO4 only slightly from 0.0105 to for the excess Gibbs energy of the solution in terms of an 0.010424 mol kg- '.The flf2)terms used for 2 :2 and higher extended Debye-Huckel function and a virial expansion in charged electrolytes are inappropriate for this system and would only add to the redundancies.They are therefore omitted. In practical applications, several workers have employed empirical extensions of the model to represent better experi- mental results at high ionic strength. For example, Archer,' for the system NaCl-H,O, assumed an ionic strength depen- dence of the third virial coefficient, leading to an additional parameter, Cz',).Test calculations showed that this type of extension leads to a valuable improvement in quality of fit when applied to aqueous H2S04, and it has been adopted here.(The original C,, remains from the earlier formulation, but is now designated CL:).) It is important that the extended equations for aqueous H2S04 are also applicable to solution mixtures containing other components; indeed the extensions may also prove useful for other solutes. In Appendix I are given generalised equations of the extended model for excess Gibbs energy and osmotic and activity coefficients, for a solu- tion containing an indefinite number of ionic solutes. Equa- tions for the system H,SO,-H,O are given below: ln(yH) = + m(HS04X2BH, HS04 + "L, HSO4) + m(S04X2BH, so4 + "L, SO3 + m(H)m(HS04)CL, HSO4 + m(H)m(S04)CL, so4 { + 2m(Pb)'H, Pb) + m(HS04)m(S04)~HS04, sod,H (3) ln(yHs04) = + m(HX2BH, HSO4 + "i, HSO4) + m(H)m(HS04)CL, HS04 + m(H)m(S04)CL, so4 m(S04X2@Hs04, so4 + m(H)$Hso4, so*, H) (4) ln(7s04) = 49 f m(HX2BH, so4 + "i,so*) 4-and F= { + m(H)m(Pb)@h, Pb) + m(HS04)m(S04%S0,, so4 (7) Superscript T denotes 'total'. For clarity, ion charges are omitted from the species molalities in the above equations.In some electrochemical cells (Section 3.2), PbSO, or Hg,S04 are present at low molalities. To the first order, their influ- ence is accounted for uia an increase in the ionic strength of the solution (both salts); and in the case of PbSO, only, by the unsymmetrical mixing functions QH, pb , a$,pb and WH, pb given in the terms in braces in eqn. (3), (6) and (7). In the above equations I is the ionic strength (in mol kg- '), and A, is the Debye-Hiickel constant, as recently calculated by J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Archer and Wang.16 Here we represent their A, using a Chebychev polynomial, see Appendix 11. ArcherI7 has discussed the effect of differences between his Debye-Huckel constants and those from earlier evaluations, which are greatest at extremes of temperature and pressure, on representations of solution properties using the Pitzer model. Of relevance to the present study, we note maximum differences of 1%in Debye-Huckel slopes for apparent molal enthalpy (A,/RT) and 2.5% for apparent molal heat capacity (AJR) from values tabulated by Pitzer,' which are based on the equation of Bradley and Pitzer'* for the relative permit- tivity of water.The functions B,,, Bf,, C:,, aCa,@fa and W,, contain model parameters and (in some cases) unsymmetrical mixing terms, and are defined in Appendix I, as are the molality- dependent functions Z, g'(x) and h'(x). The coefficient rca (which appears in B,, and Bf,) is normally set constant for broad ranges of electrolytes, typically to 1.4 mol-'I2 kg'/, for 2 : 2 metal sulfates" and 2.0 mol-'12 kg112 for most other valence types. However, values for individual cation-anion combinations can also be assigned, which is the approach we take here. An analogous coefficient o,,also appears in func- tions h(x)and h'(x). Model equations for apparent molal enthalpy (L+/J mol-') and heat capacity (C$/J mol-' K-') may be obtained by partial differentiation of the excess Gibbs energy expression with respect to temperature, with pressure and molality held constant: L4 = -T2 {a[G""/(n, T)l/aT)/m(H,SO4) (8) 0c;= ~$ + aL+/aT (94 = ct0-T{2a[Ge"/(n, T)]/aT + Ta2[Ge"/(n, T)]/aT2}/m(H,S04) (9b) where G'" is the excess Gibbs energy of the mixture, n, the number of kg of solvent (water) and m(H2S04) the stoichio- metric molality of sulfuric acid.The excess Gibbs energy per kg of solvent can be expressed in terms of the activity and osmotic coefficients, yielding for pure aqueous H2S04 : G'"/(n,RT) = 3m(H,SO4)[ln(y*) -k 1 -4stl (10) for ideality defined on the molality basis, where R is the gas constant (8.3144 J mol-' K-'). The stoichiometric mean activity coefficient y* and osmotic coefficient $st of H2S04 are related to the quantities in eqn.(3), (5) and (6) by: At = $Cm(H+) + m(HS0,) + m(SO:-)I/C3m(H,SO4)1 (11) Yi = 7; ~so4C~~~+~12~~~~:-~/~~C~~~2~0,)13)(12) For pure aqueous solutions of strong electrolytes the equa- tions for Lo and C$ are straightforward, and have been pre- sented many times before for the model without the additional parameter C',',).9*20*21Archer gives equations which include C',',) for thermal properties of pure aqueous solutions of 1 : 1 electr01ytes.l~ Similar equations for aqueous H2S04 would be extremely complicated, as the dissociation of HSO, varies with temperature, and so introduces extra differentials. We therefore differentiate eqn. (10) numerically to obtain the required quantities in the expressions for L@ and C$,using centred finite difference formulae incorporating either four (for a/aT) or five (for C2/8T2)terms.The step size was set to 5.0 x lOP3T,where T is the temperature of the measurement. 3. TheData Sulfuric acid and its aqueous solutions have been the subject of thermodynamic investigation for at least a century, resulting in a very large body of experimental measurements. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1877 Much early work is summarised in compilations of have generally used molar masses of H,O and H2S04 given Bichowsky and Rossini,22 and Timmermans.', Sources of by the authors of the various studies, otherwise we have used data have been compiled by Staples and Wobbeking' and the following values from the 64th edition Staples et ~1.'~Previous evaluations of the thermodynamic of the CRC Handbook,29 based upon the 1969 IUPAC properties of aqueous H,SO, include those of Giauque et recommendation^:^^ M(H,O) = 18.0152 g mol-' and u1." (0-100% acid and for solid hydrates; T < 300 K), Pitzer M(H2S04)= 98.07, g rnol-'. et a/." (0-6 mol kg-'; temperatures at, or close to, 298.15 Conversions of temperature scales to the current ITS-90 K), Staples' (0-28 mol kg- ; 298.15 K), Rard et aL3 (0.1-27.7 are tabulated by Goldberg and Weir.31 In general, the 6T mol kg-; 298.15 K) with later impr~vements,'~,'' and applicable to the experimental temperatures, for example Zeleznik' (0-100% acid and for solid hydrates; T < 350 K).-0.014 K at 25 "Cfor IPTS-48 and -0.006 K for IPTS-68,31 Vapour4iquid equilibrium in the H2S04-H,O system, with are of the same order as the accuracy of temperature control an emphasis on high temperatures, has been evaluated by which is typically 0.01-0.02 K. Also, the changes in most Gmitro and Vermeulen28 and more recently by Bolsaitis and thermodynamic properties (4,emf etc.) with temperature are Elli~tt.~ low enough that the change A, due to any temperature cor- The aim of the present study is to provide an accurate and rection, is small relative to the precision of measurement and self-consistent description of aqueous solution activities and the fit of the model. Therefore, where experimental tem-thermal properties of aqueous H2S04 from 273 to 328 K and peratures have been quoted in "C they have been converted molalities up to 6 mol kg-', within a framework (the Pitzer to absolute values simply by adding 273.15 K.Note that model) that allows ready extension to more complex mix- absolute temperatures given by Covington et uL3' and Beck tures. We have attempted to be comprehensive in our con- et u1.33,34are based upon an ice point of 273.16 K = O"C3' sideration of the available data, though we cannot claim In this instance we have subtracted 0.01 K from the tem- complete coverage. Measurements included in the present peratures given by those authors. study cover the period 1899 to the present, during which The data are discussed below.Tables 1-5 list, for each data there have been several revisions both to atomic masses and type, the concentration and temperature ranges of measure-temperature scales. Changes in atomic masses only affect ment, the numbers of experimental points, which measure- molality in the fifth significant figure, and these molalities are ments were rejected as being in error and the relative weight often quoted to only three or four figures. For consistency we assigned to each dataset. Table 1 Availability of isopiestic (iso) and direct vapour pressure (vp) data for aqueous H,SO, no. of molality/mol kg- ' T/K observations* method standard wr rejected" N ref. 1.673-21.65 298 33 (13) is0 NaOH 1.o 11 1 37 2.083-4.354 298 12 is0 NaCl 1.o 0 2 38 0.019-4.349 298 18 is0 NaCl 0.25 3 3 39 0.091-2.830 298 23 is0 KCl 0.2510.75 1 4 40 0.091-4.3 74 298 28 is0 NaCl 0.25/0.75 2 5 40 0.195-3.136 298 53 is0 KCI 0.25 0 6 41 1.918-22.63 298 20 (10) "P -1.o 1 7 48 0.073-2.871 298 13 "P -1.o 9 8 50 13.88-27.74 298 3 (0) "P ---9 115 7.326-12.58 298 9 (0) "P ---10 51 0.346-4.361 298 44 is0 NaCl 1.o 0 11 26 4.349-19.33 298-409 146 (24)' VP -0.1 2 12 47 1.133-40.78 29gd 12 (3) VP --3 13 49 2.09 1-4.3 55 298 16 is0 NaCl 1.o 0 14 43 0.141-0.1 70 298 4 is0 KCl 1.o 0 15 42 -P0.442-0.487 298 3 is0 KCl 1.o 0 16 0.189-4.175 323 44 is0 NaCl 1.o 5 17 116 1.450-4.096 273 8 is0 NaCl 0.25 2 18 55 1.033 273 1 is0 urea 0.25 0 19 55 1.14-9.56 273-373 99 (61) VP --61 20 117 95.217-23.79/ 203-250 "P ---21 52 -h4.026-4.420 298 4 is0 NaCl 1.o 0 22 N is the dataset number, referred to in the figures.Note also the work of Giauque et aLZ5who have evaluated solvent and solute activities (and other properties) of aqueous H,SO, over the entire mole fraction range. Glueckauf and Kitt,"* using a bithermal isopiestic technique, have obtained osmotic coefficients to 76 mol kg-'. Their values were not included in our calculations because they reported that it was necessary to normalise their measurements to lower molality results from other studies. Osmotic coefficients relative to the NaCl isopiestic standard were calculated using eqn. (7) and (36) of Archer.' Note the following errors in Archer's eqn. (36):lines three and four should read: Also, the coefficient b3,12 should have the value 0.06622025084.The second number in parentheses gives the number of data points within the fitted molality range 0-6.10 mol kg-'; the figure in the 'rejected' column refers only to those points within the fitted range. Molalities of the rejected data for each reference: 1.673-4.376, 5.002, 5.144;37 0.0187, 0.0456, 3.815;39 0.0909, 0.0908, 4.374;40 2.239;,* 0.073-1.282, 2.468;50 3.399, 5.490;47 0.3037, 0.2576, 0.1894, 2.1157, 3.1911;'16 1.450, 2.186 mol kg-'.55 Molalities below 1.0 rnol kg-' are given the lower weight of 0.25. Molality and temperature ranges are those of the experimental determinations. Only five values (two of which were rejected) were used here, taken from the interpolated 298.15 K isotherm in Collins' Table 3.47 Data for some other temperatures given graphically.J. A. Rard, work in progress. Vapour pressures also measured along the freezing curve of sulfuric acid tetrahydrate and hexahydrate (argued by Zhang et ~21.~~to have composition H2S04.6.5H,0). Representative data presented only in graphical form, and also as fitting equations. J. A. Rard and D. G. Archer, unpublished data. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 2 Availability of emf data for cells I-IV no. of cell molality/mol kg -' T/K observations' int.b wr rejected' N ref. I 0.1000-8.272 278-328 77 (69) 1.o 0 1 33 I 0.0073-0.096 298 5 1.o 0 2 32 I 0.01945-0.997' 298 19 19 3 59 I1 0.0050-0.050 298 2 -2 4 119 I1 0.1003-7.972 278-328 54 (47) 1.o 7 5 34 I1 0.00734.096 298 13 1.od 0 6 32 I1 0.0506-2.386 298 7 1.o 0 7 66 I1 0.0506-8.207 298 5 (4) 1.o 0 8 120 I1 0.0010-0.0 10 298 5 -5 9 121 I1 0.0050- 1.04 1 298 6 1.od 2 10 67 I1 0.1000-4.Ooo 298 7 1.o 2 11 122 I11 0.0536-3.499 298 10 1.o 2 12 123 IV 0.00104.020 273-323 25 1.o 12 13 63 'See first sentence of footnote" in Table 1.Molalities of the rejected data for each reference: 1.872, 5.767 (278 K); 0.1003, 1.872 (318 K); 0.1003, 1.872, 5.767 (328 K);34 0.02506, 0.2529;67 2.0, 4.O;lz2 0.5154, 1.036;'23 0.001 and 0.002 (all temperatures); 0.005, 0.02 (273 K).63 Emfs reported in international volts. 'Measured values (Table 1 of ref. 59) using Hamer's preferred methods (4, 5 and 6) of electrode preparation. For molalities <0.04 mol kg-', relative weights are reduced to 0.5 as the model-calculated emf is sensitive to the amount of Hg,SO, assumed to be present (see text).Table 3 Availability of enthalpy data (enthalpies of dilution) for aqueous H,SO, no. of molality"/mol kg- TIK observationsb wr rejected N ref. (> 100 Wt.%o)-O.506' 298 72 (10) 1.o 4 1 70 6.423-0.001 298 25 (24) 2.23 0 2 71 3.679-0.003 29 8 45 0.86 7 3 72 0.050-0.003d 29 8 11 0.045 6 4 73 30.860-6.07' 253 10 (1) --5 70 0.005-6.0f 303-598 ---6 75 Range given for initial molality, m,. See first sentence of footnote a of Table 1. Molalities (m,)of the rejected data for each reference: 1.508, 1.040, 0.726, 0.506;70 0.0251 (two points), 0.0125, 0.00627, 0.003 13 (two points);73 0.003 05, 0.00504, 0.00508, 0.0174 (three points), 0.0846.72 See also corrections given by Giauque et aLZ5Differential enthalpies of dilution were calculated as Adi,H= L*(m,) -L*(m,) = AAqH,O) (Table 3 of Kunzler and Giauque7').Reported mol dm-3 concentrations were converted to molality using densities compiled by Sohnel and N~votn$."~'The single data point within the fitted range was not used. Experiments carried out at 7-40 MPa, and therefore not relevant to the present study. Table 4 Availability of heat capacities for aqueous H,SO, no. of molality/mol kg- T/K observations' wr rejected' N ref. (>1OO ~t.O/o)-1.149~ 298 75 (13) 1.0 70 0.563-0.052 298 9 1.o 81 2.230-0.044' 298 13 1.0 76 1.013-0.103 298 8 0.50 80 1.013-0.103 328 8 0.50 80 1.013-0.103 313 8 0.50 80 (100 wt.%)-0.035 293 37 (20) 0.25 77 1.O 1 3-0.103 283 8 0.50 80 30.869-4.508 253 11 (2)d 70 9.2468 214-300 11 125 8.5385 230-3 19 14 125 6.9377 213-296 11 125 55.509 284-305 4 126 27.754 239-306 12 126 18.503 182-298' 12 127 18.503 244-296' 8 128 13.881 251-305 9 128 Socolik7* tabulates the specific heats (to three figures) of Savarizky (unreferenced) at 295.65, 313.15, 333.15 and 353.15 K from 6.06 to 100 wt.% H,SO,.'See first sentence of footnote 'of Table 1. Molalities of the rejected data for each reference: 1.586, 1.231;70 0.0444, 0.0713, 0.1748, 0.5515;76 0.1035, 0.1781 (T 2 298.15 K);" 0.0347.77 See also corrections given by Giauque et aLZ5'These heat capacities are on a (g H,O)-' basis.76 This is not stated in the paper, and caused Zeleznik' to reject their results as erroneous.These two data points not used. Includes supercooled solutions. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1879 Table 5 Availability of degree of dissociation data (HSO, eSO:-+ H+) for aqueous H,SO, ~ ~ ~ ~~~ molality/mol kg-' method" T/K no. of observationsb wr rejected N ref. 0.01-0.304 1 300 -10 10 1 129 0.024.00' 2 298 10 10- 2 130 0.000254.050 3 298 7 1.o 0 3 131 0.00090-2.634 4 298 13 1.o 0 4 86 0.264-42.62 2 298' 18 (9) 1.o 3 5 84 0.050-40.15" 2 273-323 38 (16) 1.010.3 2 6 14 0.052-29.26 5 298' 12 (6) 1.o 2 7 85 dissociation step at very high molalities (H,SO,= Hf + HSO,). 3.1 Vapour Pressure and Isopiestic Measurements Available osmotic coefficients and vapour pressures relevant to the present study are summarised in Table 1.Early (and less reliable) data not included here, some of it from the 19th century, have been considered by Abe136 and are also listed in bibliographies such as that of Staples and Wobbeking.' Many of the osmotic coefficient were used in the published evaluation of Rard et with newer data26*42*43 (also included in Table 1) leading to minor revisions.26 Rard and Platford, have critically assessed the osmotic coeffi- cients of H2S04 at 298.15 K to 27 mol kg-', comparing the evaluation of Robinson and Stokes44 with the later work of Rard et and of Staples., A number of serious objec- ~1.~7,~ tions were raised to the latter study, concerning the use of freezing-point depression data (see Section 3.6), the overall goodness of fit, and circularity regarding the use of CaC1, as isopiestic standard.Because of this (osmotic coefficients of aqueous CaCl, being largely determined from isopiestic equi- librium with H2S04 solutions) such data are not included here. Aqueous NaCl was the standard for most of the isopiestic .~data listed in Table 1. In the work of Rard et ~1 and Rard26 the osmotic coefficients of NaCl were calculated using the equation of Hamer and WU.~~ Since their work, Clarke and Gle~~~and Archer' ' have published substantial critical reviews of the thermodynamics of aqueous NaCl that refine its osmotic coefficient. At 298.15 K their studies agree with each other to within 0.0006 in 4, but comparisons with the equation of Hamer and Wu show systematic differences of up to 0.003. All isopiestic data for which aqueous NaCl was the standard have therefore been adjusted to Archer's values of (see also footnotes to Table l),which we consider to be the most reliable.For measurements relative to aqueous KCl the best-fit equation of Hamer and Wu45 for &,, used by Rard et ~l.,~was also adopted here. Note, however, that we have not attempted to correct for the non-ideal behaviour of water vapour, which was not considered by Hamer and WU.~~We estimate this correction to be negligible at low molalities, and no more than 0.1% at high molality. The osmotic coefficients of H2S04 for which aqueous NaOH was the standard are based upon the evaluation of Rard et uL3 of and were taken directly from their Table 1.Water vapour pressures determined by Collins,47 and Shankman and G~rdon,~'Jones49 and Grollman and Frazer" have been used to calculate osmotic coefficients. Careful attention was given to the use of values of the vapour pressures of pure water compatible with the actual tem-perature scales used in the original measurements. For the References above are restricted to studies of the variation of a as a function of concentration, and do not include those whose sole aim is to determine the infinite-dilution value of KHS04.Kerker13, recalculated r from literature data (including those of Sherrill and Noyes' 31) but many values are grossly discordant with other work, and have not been included here. Methods: 1, spectrophotometry; 2, Raman spectroscopy; 3, conductance; 4, molar volume; 5, NMR.See first sentence of footnote of Table 1. Molalities of the rejected data for each reference: 0.52, 1.03;85 2.915, 1.377, 0.264;84 0.502, 0.504.14 Concentrations in mol drn-'. 'Temperature not specified. Presumably at a 'room temperature' close to 298 K. Measurements at 273.15 and 323.15 K were given relative weights of 0.30. Data were also reported in this study for the first last three studies, corrections were made for the non-ideality of the vapour phase by use of the second virial coefficient of water vapour as tabulated by Rard and Platf~rd.,~The resulting osmotic coefficients differ by <0.0002 (Shankman and Gordon4*) and <0.0005 (Grollman and Frazer") from those given previously by Rard et aL3 Such corrections were not made for the results of Collins47 because they are impre- cise around 298.15 K.We note that the dew-point determi- nations of Hepb~rn,~' for higher molalities than used here, are of quite low precision and deviate by up to 0.04 in d, from the evaluation of Giauque et al." and are probably unreli- able. The recent vapour pressure measurements made by Zhang et dS2at temperatures below 250 K are outside the scope of the present study. However, brief comparisons indi- cate that some of the data are inconsistent with H,O chemi- cal potentials tabulated by Zeleznik' and also with water activities calculated by us.53 For further discussion of both vapour pressure and isopiestic data, see Rard et aL3 and Rard.26 Relative weights (w,) for the different data sets are given in Table 1, and are reduced for the determinations of Robinson4' (for KCl as isopiestic standard) which are more scattered than his later measurements relative to NaC1,38 and for the results of Scatchard et d4'because they did not use replicate samples.For these authors the weights were further decreased for m(H,S04) < 1.0 mol kg-', where data are more scattered, and where agreement with the more recent work of Rard26 is poor, possibly owing to insufficient time being allowed to achieve isopiestic equilibrium in the earlier study. The data of Scheffer et deviate systematically from the evaluated osmotic coefficients of Rard et d3and all other isopiestic data at low molalities and were weighted 0.25.Olynyk and Gordon54 have noted that later experi- ments by the same group agreed more closely with the work of Scatchard et ~1.:' but the results of that redetermination do not appear to have been published. The data of Platf~rd~~ at 273.15 K appear to be reliable but are quite scattered, and have also been given a reduced weight. All direct vapour pressure and isopiestic data given non- zero weights are shown in Fig. 1-3. 3.2 EMF Measurements Sources of emf data are listed in Table 2 for the following electrochemical cells : (Pt)H, 1 H,S04(m) I PbSO,, PbO,(Pt) (cell I) E = E" + (RT/2F)ln[rn(Hf)2y~m(SO~-)y,0Ju~] (13) J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 I I Ill It' I symbol N x3 04 Q5 06 11 0 15 0 16 030 0.46 0.62 0.78 Jm/mol'12 kg-'I2 Fig. 1 Stoichiometric osmotic coefficient (4s,)of aqueous H,SO, at 298.15 K and low molality, plotted against [m(H,S0,)]"2. See also Fig. 2 for measurements at higher molalities. (-) Fitted model (Section 5). Key: symbols are related to the dataset numbers (N) in Table 1. 0.86 0.90 0.94 0.98 IIIII1 0.720 F 0' 3.77 3.76 symbol N x3 O 4C 4 5a" ~6 0.7411 v 12 I I I I 0.72 1.o 1.10 1.18 Jm/mol'12 kg-'I2 symbol N F:2 x3 04 C 45 a" 06 v7 as -11 # 14 1.8 1.9 2.0 Jmlmol 'I2 kg-'I2 (Pt)H2 I H2S04(m) I Hg,S04, Hg(Pt) (cell Ir) E = E" -(RT/2F)ln[rn(H+)2yirn(SO~-)yso4] (14) Hg, Hg,SO, I H2S04(rn)I PbSO,, PbO,(Pt) (cell 111) E = E" + (RT/F)ln[rn(H+)2y~rn(SO~-)yso4/a,] (15) PbHg(amalgam), PbSO, I H2S04(m)I H,(Pt) (cell IV) E = E" + (RT/2F)ln[rn(H+)2y~rn(SO~-)ys,4] (16) where E and E" are, respectively, the measured and standard emfs (in V) of the cell and F (96484.6 C mol-I) is Faraday's constant.Note that we do not consider here emf measure- ments involving electrolyte mixtures HC1-H,S0,-H2056,57 or NaHS04-Na2S0,,58 although the model could be used to account for the ion interactions that occur.l0 Similarly, emf measurements for cells involving the Ag,SO,/Ag elec-trode are not analysed because Ag2S04 is too soluble in aqueous H2S04 to yield meaningful results for pure aqueous H,S04 solutions.Early work of Hamer59 and Harned and Hamer6' has been reported by several authors to be inconsistent with modern data.32-34 Further comparisons by one of us (J. A. R.) suggested that a partial cause of the discrepancies, for at Jmjmol 'I2 kg-'l2 1.20 1.28 1.36 I I 1 11 (b) 0.83 x3 04-0.81a" -t15 g06 v7 ~a 0.79 . 11 v 12 # 14 0.77 1.50 symbol N +l1.45 02 C x3 1.40 Q5 v7 11 1.35 v 12 fl 14 A 22 1.30 ' 2.2 2.3 2.4 2.46 Jmlmol 'I2 kg -'I2 Fig. 2 Stoichiometric osmotic coefficient (4st)of aqueous H,SO, at 298.15 K to 6 mol kg-I, plotted against [m(H2S04)]112. For each of the four graphs, the plot (and $st axis) on the left are associated with the upper molality axis.See also Fig. 1 for low molalities. (-) Fitted model (Section 5). Key: symbols are related to the dataset numbers (N)in Table 1. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1.25 -I 1 1 r I 0.7 1.1 1.5 1.9 dm/moll l2 kg-i2 Fig. 3 Stoichiometric osmotic coefficient (4$,)of aqueous H,SO, at 273.15 and 323.15 K, plotted against [m(H2S04)] 'Iz.(---) Fitted model at each temperature (Section 5). N: (0)17, (0)18, (4)19. least some of the measurements, might be systematic cyclic deviations introduced by their use of graphical smoothing and fitting equations to represent the experimental emfs, first as functions of molality and then as functions of temperature. Hamer59 reported the actual experimental data for only four molalities at 298.15 K for a comparison of a variety of types of electrode preparations.Those emfs for cell I exhibited both a large average deviation (> 1 mV) from the data of other workers, and a spread of ca. 1 mV (but 8-9 mV discrepancies for their preparation 7). The results of Hamer59 and Harned and Hamer6' are therefore not included in our analysis. The published emf data span the period 1914-1965. The change from international to absolute volts Cl.0 V (int.) = 1.OOO33 V (abs.)] occurred in 1948 and we have assumed that the results of all studies published after this year are in absolute volts, with the exception of the work of Beck et as pointed out by Covington.61 For measurements at low molalities of H2S04, dissolved PbSO, and Hg,SO, can contribute significantly to the total molality of the solution.The presence of the soluble sulfates in the cells is accounted for within the model by an increase in total ionic strength Z (assuming both salts to be fully dissociated) and sulfate molality m(SOz-). The solubility of PbSO, in aqueous H,SO, above 0.005-0.009 mol kg-' acid has been measured by Craig and Vinal at 273.15 and 298.15 K,62 and estimated for 0.001-0.02 mol kg- H2S04 (273.15-323.15 K) by Shrawder and C~wperthwaite~~ by the inter- polation and extrapolation of the measurements of earlier workers. There is reasonable agreement between the two sets of values. Equilibrium concentrations of PbS0, ,calculated from the tabulation of Shrawder and Cowperthwaite, are assumed to be present in the solutions in cells I, 111 and IV.For m(H,SO,) > 0.02 mol kg- ' the dissolved PbSO, concentra-tion is low enough (< 2.2 x 10-rnol dm -') to be neglected. Pitzer et al." note that the PbSO,/Pb electrode may be reli- able only for m(H,SO,) 2 0.005 mol kg-',therefore measure- ments for cell IV below this concentration have been rejected. Mercury@) sulfate is more soluble than PbSO, . Solubilities of 7.5 x lo-, to 1.1 x rnol kg-' at 298.15 K in 0.002 to 2.0 mol kg-' H,SO, are listed by Brown and Land,64 who combined their own measurements with interpolated values from the study of Craig et (0.001 to 3.6-4.2 mol dm-3 H,SO,, 273.15 and 301.15 K), which agree with the results of Brown and Land within experimental error.64 Test calcu- lations by us showed that the inclusion of dissolved Hg,SO, at its equilibrium molality in cells I1 and I11 yielded an improved model fit, and Hg2S04 molalities estimated by interpolation from the data of Craig et al.were therefore adopted at all temperatures. As the effect of correction for solubility of Hg,SO, on the emf was found to be significant only for m(H,SO,) < 0.04 mol kg-', the practical influence of its presence is restricted to 298.15 K. These corrections also proved quite sensitive to the concentration of Hg,SO, speci-fied. Therefore, in view of the very simple treatment of the effect of the dissolved salt, relative data weights for m(H2S04)< 0.04 rnol kg-' (for cell 11) were reduced by a factor of two.We also note that, for m(H,SO,) > 0.06 mol kg-', Hg' may be present as the complex ion Hg2(S04)(HS04)-.64 For cells of types T and 11, at 298.15 K, there are two and six data sets, respectively, that were given non-zero weights. It is possible for systematic deviations in E" (bias potential) to occur for different electrode preparations of the same type, because of subtle differences in the physical and chemical state of solid electrode material. The data were therefore examined for this by calculating E" [from eqn. (13) and (14)] for individual measurements from the observed emf and using the model to obtain the activity term. For cell I an average AEL of -0.78 mV was found between the results of Coving- ton et aL3, and Beck et For cell I1 there was agreement to within one standard deviation (in the mean value of our derived E") of the results of Beck et aL3, except in the cases of MacDougall and Blumer66 (ca.0.4 mV) and Trimble and Ebert67 (ca. 0.2 mV). To allow for these bias potentials, additional terms AEo were fitted, such that E"(experimenta1)= E"(true) + AE", for the emf data of Covington et (cell I), and MacDougall and Blumer66 and Trimble and Ebert67 (cell 11). The least-squares values of AEr were -0.78, -0.32 and -0.22 mV, respectively. Consis- tency in the variation of E" with temperature for cells I, I1 and IV was also tested, by fitting E" at each temperature indi- vidually. For cell I1 at 318.15 K 34 and cell IV at 285.65 K 63 small deviations for E" were found from the general trends (ca.0.1 mV), which were accommodated using AE" terms similar to that given above. The question of whether dissolved Hg,SO, gives rise to liquid-junction potentials within cell I1 has been examined by Dobson, who studied cells saturated with Hg,S04 using both glass and hydrogen-gas electrodes.68 There was no significant difference between the two sets of results, and it was con- cluded that there was no significant liquid-junction potential contribution to the emf in the molality range studied. Thus we made no corrections for this, although we note that the opposite view has been argued by Hamer.69 With the exception of the low-molality data for cell 11, all retained emf measurements have been given unit relative weight, and are shown in Fig.4. 3.3 Apparent Molal Enthalpies Sources of enthalpy data (differential enthalpies of dilution) are listed in Table 3. Experimental dilutions (m,-+m,) (in mol kg-') are about 30% for the work of Kunzler and Giauque," 15% or less for the determinations of Wu and Young7' and most of Groenier's but >96% for the results of Lange et and 70-80% for the other three experiments of Groenier. The most precise data are those of Wu and Young,71 who have also derived L" from their own work and that of the other authors given above. They did this by first estimating L* at low molalities using the mea- sured degrees of dissociation of Young and Blatq7, the enth- alpy of dissociation of the hydrogensulfate ion and estimates of apparent molal enthalpies, L*(H, H, SO,), (from Lo of Li,SO,) and L*(H, HSO,), and combining these generated 1882 J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 I 1 I I I II2.0 (a) cell I 278 K (+ 0.20)1.8 ’ 293 K 298K symbol N-96& 1.6 v 10 0318 K A 13 02 .,O 328 K +71.4 ­.’4 ‘s 11 (-0.20),*do’-/.’ n5 0 08 1.2 I (-0.30) x 12 I I I I I I 0 1.o 2.0 3.0 0 1.o 2.0 3.0 Jm/mol”2 kg-’” -0.68 V 0.81 \ 0.64 t ’\* 0.600.79 i0.5610.77 t x‘ 1> ‘e 1.0 2.0 3.( 0.751, . I . . I . I ,j& %, Jm/mol’/2 kg-’I2 -0.75 0.2 0.6 1.0 1.4 1.8 Jm/mol’/2 kg-’I2 1 I I I 0.73 0.3 I 273 K (e)rv I (+0.125) 285 K 0.71 (+0.0625) 0.2 *-----298 K o.69 -cell I1 (298 K) >G-*MAI 1 I I I1 0.05 0.25 0.45 0.65 A/ 310 K (-0.0625)Jm/mol’ /’ kg-’l2 0.1 */ 323K 0 J I I I I 0.07 0.09 0.11 0.13 0.15 Jm/mol’/2 kg-’IZ Fig.4 Measured emf (E/V) of cells I-IV, plotted against [m(H2S04)]1i2. (a)Cell I; (b)cell 11; (c) cell 11, 298 K; (6)cell 111, 298 K; (e) cell IV. For clarity, in parts (a),(b)and (e) the emfs are offset by fixed amounts (V), indicated by the numbers in parentheses. Fitted model (Section 5). Key: symbols are related to the dataset numbers (N)in Table 2. values with integrated hL*/h,/m obtained from the experi- other data, and when the size of the dilution is taken into mental Adi,H using a chord-area plot. account, generally agree well with Wu and Young’s tabula- All dilution enthalpies were first assessed in a preliminary tion.However, test fits with our model showed systematic way by comparison with the L* of Wu and Young71 (their deviations of their values which were larger than the experi- Table 5). The enthalpies of dilution obtained by Kunzler and mental errors. These deviations were related to differences Giauque7’ at the lowest H,SO, molalities were found to between Wu and Young’s method of analysis to obtain L* deviate systematically from the results of other workers, and values generated by our model below about 0.01 mol probably because their calorimeter was optimised for concen- kg-’. In view of the empirical nature of Wu and Young’s trated solutions.25 Those points were rejected as being in estimates of L4 in this region, the four measurements for error, as were a few others with deviations of >40 J mol-’.which deviations from the model fit were >100 J mol-were The experiments of Lange et uE.,’~ and three of the measure- rejected. ments of Groenier” involve very large dilutions and the We note that Milioto and Sim~nson~~Oak Ridgeat lowest values of m2. Groenier’s three enthalpies of dilution National Laboratory have made many enthalpy of dilution show much larger deviations from values calculated from measurements for aqueous H,SO, from 0.005 to 6 mol kg-’, Table 5 of Wu and Young71 than the comparable data of from 303 to 598 K and at pressures of 7 to 40 MPa. Because Lange et al., and were therefore rejected. In percentage terms, our evaluation is restricted to a pressure of 1 atm, their the results of Lange et al.73 appear no less accurate than results were not included in our fits.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Relative weights were assigned to each dataset as l/a2 (comparisons made with differential enthalpies of dilution calculated from Wu and Young's tabulated Lo), normalised to unit values for the measurements of Kunzler and Giauque, see Table 3. All data given non-zero weights are plotted in Fig. 5 as aL+/aJm, corrected for the difference between this quantity and the measured -AdilH/(Jml -Jm,).This cor- rection, significant to the scale of this plot only for the results of Lange et aZ.,73is also shown, as is the range of dilution (Jm,to Jm,).Data were fitted as differential enthalpies of dilution, Adi,H= Lo(rn2)-L*(m,), calculated from eqn. (8) for initial (m,)and final (m2)molalities. 3.4 Apparent Molal Heat Capacities Heat capacity measurements for H2S04-H20 are sum-marised in Table 4. For completeness we have included all the data of Giauque and c~-workers~~*~~ even though many of their determinations are outside the molality range of the present study. Abe136 considers that, of measurements made prior to about 1945, only the data of Randall and Tayl~r,'~ andBir~n~~to a lesser extent Savarizky (tabulated by Soc01ik~~)are reliable. Biron's work has been used by Craig and Vir~al'~ in their study of aqueous H2S04 related to the lead storage battery, and that of Savarizky by Giauque et 0 0.1 0.2 aL2' to estimate temperature coefficients of partial molal properties.Here we have not used the work of Savarizky, since results were reported to only three significant figures, although Savarizky's values appear to be fairly consistent with other data at lower temperatures for molalities >1.0 mol kg- '. Apparent molal heat capacities, C:/J mol-' K-', are cal- culated from measured specific heats, c,/J K-' (g)-', by the equation : C: = [1000/m(H2SO4)](~,-c;) + M(H2S04)c, (17) where c;/J g-' K-' is the specific heat of pure water at the experimental temperature and M(H2S04) = 98.07, g mol-' is the molar mass of H2S04. From eqn. (17) it is clear that at low molalities (below ca. 1 mol kg-'), where c, and ci are almost the same, the calculated Cd are very sensitive to experimental error and to the choice of ci.Heat capacities for m(H,S04) < 1.0 mol kg-' have been measured by Hovey and Hepler," Larson et Randall and Taylor,76 and Bir~n.~~ The results of the first two studies used ci taken from Ke11.82 The data of Biron" at 293.15 K are treated by Craig and Vina179 as assuming a defined c; of 1.0 cal g-' K-'. It is unclear if Biron's experimental cp are relative to this value, which would then require a transform- ation of c,/cal 8-l K-' +cJcJ293.15 K)/l.O]. However, test 5t 1 0.2b1 , I , I . , , ,j 0.36 0.56 0.76 0.96 1.16 1.2 1.6 2.0 2.4 Jm/mo1'/2 kg-lI2 ,/m/mol''2 kg-'I2 Fig. 5 Differential of the apparent molal enthalpy with respect to [m(H2S0,)]112, calculated from enthalpies of dilution at 298.15 K, and plotted against the mean of the square roots of the initial and final molalities.Horizontal lines indicate the extent of the dilution Jm,--+ Jm2. Vertical lines indicate the extent of the correction -Adi, H/[(Jm, + ,/m2)/2] to the true differential as plotted, which was estimated from the tabulated Lo of Wu and Y~ung.~' Fitted model (Section 5). The dataset numbers (N) of Table 3 are: (*) 1,(a)2, (0)(-) 3, (A)4. 1884 calculations showed that this would have a negligible effect on the apparent molal heat capacity, so C: has simply been calculated from eqn. (17) assuming ci equal to 1.0 cal g-' K-'. Randall and Taylor76 used ci = 0.9979 cal g-' K-', equivalent to a molal heat capacity of water of 17.976 cal mol-' K-' (using the atomic masses current at the time).We .~~have followed Giauque et ~1 in adjusting Randall and Taylor's data (their Table 2) by the factor (@a1 mol-' K -')/17.976 before calculating C? . The measurements of Kunzler and Giauque7O at 298.15 K, for which m(H2S04) > 1.2 rnol kg-', were considered prior to the data at lower molalities, and apparent molal heat capacities were calculated using ci = 4.1796 J g-' K-', as listed in the CRC Handbook29 and due to Osborne et rather than the value recommended by Ke11.82 At 298.15 K, these c; differ by only 0.0003 J g-' K-', and the effect of this difference upon the calculated C$ is negligible. Unit relative weights were assigned to the data of Kunzler and Giauq~e,~' Larson et aLal and Randall and Taylor.76 The apparent molal heat capacities of Hovey and Hepler" at their lowest molalities and 298.1 5 K deviate systematically from those of other workers.These data were therefore given relative weights of 0.50 at all temperatures, and the values for the two lowest molalities for T 2 298.15 K were weighted zero. At the lower temperature of 283.15 K there was very good agreement with our model, even for the most dilute solutions, so all points were retained. The very early results of Bir~n~~appear to be consistent with other data, though there are no other measurements at the same temperature for com- parison. We have cautiously assigned them a relative weight of 0.25.All data given non-zero weights are plotted in Fig. 6. The infinite-dilution value of the apparent molal heat capacity, which varies as a function of temperature, is con- sidered further in Section 3.7. 3.5 Degree of Dissociationof the Hydrogensulfate Ion Studies of the degree of dissociation, a, are listed in Table 5. Considering the variety of methods used, agreement between these different datasets is quite good. As noted by Chen and Irish,84 the derivation of ionic concentrations from Raman peak intensities is somewhat ambiguous because some of the -_ 80 -100 I-' I r 1 I I I { 100 .- 60 - symbol 0 -8o - N 1 2 TIK 298 298 Y.-I- 40 - 460 + ' L 0 r 3 4 298 298 E" -40 x . sQA -20 c, 0 5 6 7 328 313 293 V 8 283 0.5 1.0 1.5 2.0 2.5 Jrn/mo1112kg -'I2 Fig. 6 Apparent molal heat capacity of H,SO, at different tem- peratures (see key), plotted against [m(H,S0,)]1'2.Note the use of three scales, with arrows indicating the datasets to which they apply. (-) Fitted model (Section 5). Key: symbols are related to the dataset numbers (N) in Table 4 and the temperature (T)of measure-ment. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 peaks overlap. At 298.15 K, a decreases steeply from 100% at infinite dilution to about 20% at 0.4-0.5 mol kg- ',thereafter rising to a broad peak of ca. 30% from 2 to 6 mol kg-' and then decreasing slowly. Speciations determined from NMR and Raman spectral data are least reliable for dilute solutions because of a lack of sensitivity, and two of the values of Hood and Reilly" for m(H,S04) < 2 mol kg-' have been dis- carded.However, the results of Chen and Irisha4 agree fairly well with values derived from molar volumes by Lindstrom and Wirtha6 to below 1.0 mol kg-I. Chen and Irisha4 discuss several other studies of a and note that values calculated from NMR shifts and partial molal volumes are dependent to some degree on the results of the earlier Raman spectral mea- surements of Young et because of the assumptions made in interpreting the data (e.g. an assumed relation between the chemical shifts of aqueous Hf and HSO, in NMR studies). At temperatures other than 298.15 K there are only the data of Young et ~1.'~One point at 273.15 K (0.504 mol kg-') was discarded, and the remaining measurements at 273.15 and 323.15 K assigned relative weights of 0.3.All data retained at 298.15 K were given relative weights of 1.0. Speciations of sulfuric acid over a range of molality and temperature, derived from fits of the Pitzer model to available thermodynamic data, have been plotted by Holmes and Me~mer.~~In the case of the 298.15 K parametrisation of Harvie et al." the model-generated a is consistent with the data listed in Table 5 to about 3 mol kg-',13 but at higher molalities predicts a degree of dissociation that is lower than the direct experimental values. Wirth" has represented data for H,SO, up to 2.89 mol kg-' at 298.15 K with a much simpler thermodynamic model than used here. However, in that study the values of a were taken from the determinations of Young et ~1.'~and Lindstrom and Wirth,86 and were not predicted by Wirth's model.Note that even where the model is fitted simultaneously to activity, osmotic coefficient and thermal data the speciation is not fully constrained, particularly its variation with tem-perature; thus it is worthwhile including the measurements referred to in Table 5, though a low weight has been assigned to the data as a whole (Section 3.8), reflecting the uncer- tainties in the measurements and their interpretation. All values of a given non-zero weights are plotted in Fig. 7. 3.6 Freezing-temperature Depression Measurements of the freezing temperatures of solutions yield the osmotic coefficient of the solution at the freezing tem- a0*410.2 0 0.5 1.0 1.5 2.0 2.5 Jrn/mol'iz kg-'I2 Fig.7 Degree of dissociation (a) of the hydrogensulfate ion, at dif- ferent temperatures, plotted against [m(H2S0,)] 'I2. (-) Fitted model (Section 5). The dataset numbers (N)of Table 5 are: (0) 4, (+) 5, (e) 3, (0) 6, (17)7, (all at 298.15 K); and (0) 6 (273.15 K), (*) 6 (323.15 K). J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 perature, via standard equations89 involving the thermal properties of pure ice and water. The osmotic coefficients can be adjusted to some other reference temperature, usually 298.15 K, using partial molal enthalpies and heat capacities. Staples' (in his Tables 18-27) lists 155 freezing-temperature measurements from 10 ~tudies,~'-~~but rejected 80 of these data.Staples suggested that the deviation of the 75 freezing- temperature points retained in his fit from osmotic coeffi- cients derived from emf measurements was due to experimental error in the freezing temperatures. RardZ6 and Rard and Platf~rd~~ came to a similar conclusion, suggesting that the precipitating solid phase could well be dilute H,SO, rather than pure ice. A comparison of 298.15 K osmotic coefficients, calculated from the freezing temperatures and molalities tabulated by Staples,' with the present model (fitted to isopiestic and emf data) showed positive systematic deviations of the order of 0.02 in 4 below 0.1 mol kg-', though there was reasonable agreement above this molality despite considerable scatter.A recalculation by using enthalpies of Wu and Young71 and heat capacities from sources listed in Table 4 yielded essen- tially the same results as obtained by Staples, and it was apparent that no conceivable error in the thermal data could account for the differences between the freezing points and other measurements. We therefore conclude that, below about 0.1 mol kg-', all the available freezing-point depres- sion data are systematically in error. No freezing-temperature data have been included in the present evaluation. 3.7 Dissociation Constant (KHso4)of the Hydrogensulfate Ion The value of KHSO4[eqn. (2)] is such that it is difficult to determine using methods employed for weaker acids, for example emf measurements."' The double charge on the sulfate ion complicates the extrapolation of activity coeffi- cients that is required, and renders this sensitive to model assumptions, especially from uncertainty in the value of the ion size parameter in the Debye-Huckel expression.58~'0' Many determinations of KHS04 have been made, and are sum- marised in a number of revie~s.''~-~~~ By the early 1960s it was clear that, at 298.15 K, KHSO4 lay between about 0.0102 and 0.0106 mol kg-'.'02 A previous application of the Pitzer model to aqueous H2S04 involved a value of 0.0105 mol kg-'.lo Evans and Monk"' have calculated KHS04 at 298.15 K Readnour and Cobble'07 based upon the enthalpy of solu- tion of Na,SO,(,, in aqueous HCl.Recently, Dickson et ~1.''~have conducted new measure- ments of the dissociation constant of HSO, in aqueous NaCl from 323 to 523 K. In their analysis, Dickson et al. included the KHSO4estimated by Pitzer et al.," enthalpies of reaction81*108and the heat capacities of HSO, and SO:-at infinite dilution as extrapolated by Hovey and Hepler8' and Hovey et al.,lo9 respectively, and other thermal data refer- enced by them. Dickson et obtained KHSO4= 0.010865 f-0.0005 mol kg-' at 298.15 K, in satisfactory agreement with other estimates considering that their analysis is prob- ably biased toward high temperatures. They also determined A,H" = -22.8 & 0.8 kJ mol-' and Arc; = -275 +_ 17 J mol-' K-l for the dissociation reaction, both at 298.15 K.The variation of the heat capacity change (Arc;) with tem- perature can be derived from eqn. (6) of Dickson et al. :lo4 Ar C; = -1962.617 71 1 + 9.486 301 486 92T -0.012 831 099 03T2 (18) [The constants in eqn. (18) are specified so as to retain close numerical agreement with the equation of Dickson et al., and do not reflect the actual accuracy to which Ar C; can be deter- mined.] The quantity Ar Ci is related to the infinite dilution values of the apparent molal heat capacities of the ions: Ar Ci = Ci(SO:-) + C;(H+) -C;(HSO,) (19) By definition Ci(H+) is equal to zero, hence: A, Ci = Ci(SO;-) -C;(HSO,) (204 EC:" -C;(HSO,) (20b) where C$ is the infinite dilution value of the apparent molal heat capacity of the acid, on the basis of complete disso- ciation, given in eqn.(9a). We have chosen to include qo,and its variation with tem- perature, as unknowns in our model. However, values of Cp can also be obtained from data for other electrolytes, since the apparent molal properties of ions are additive at infinite dilution. Gardner et a!."' have done this, using integral enthalpies of solution, and tabulate C$" of H2S04 from 273 to 373 K (their 298.15 K value is -295.4 J mol-' K-'). The analysis by Hovey and Hepler" of their own heat capacity data, and using C:' (-282.3 J mol-' K-' at 298.15 K) obtained by the same method in an earlier paper by Hovey etfrom their own cell data and those of Nair and Nanc~llas,~~ al.,'09 yielded -17.8 J mol-' K-' for Ci(HS0;) at 298.15 Hamer'05 and Covington et a/.58 While the activity coeffi- cient equations used by Evans and Monk"' are simpler than the Pitzer model expressions, these differences should have least effect on the calculated (stoichiometric) value of KHSO4 at low molalities. The results of Evans and Monk also suggest that KHSO4x 0.0104-0.0105 for a comparable ion size term [in the Pitzer model it is 1.2, see eqn.(6) and (7)] and empiri- cal constant Q of 0.3-0.6 mol- dm3. A further re-analysis of emf data by Mussini et ~1.''~ yielded values of 0.01043 & 0.00020 and 0.01039 & O.OO0 18 mol kg-l. Within the quoted uncertainties, these agree with earlier estimates obtained by other methods.lo2 In the present study we have therefore retained KHSO4= 0.01050 mol kg-' at 298.15 K.Different estimates of the standard enthalpy change (Ar H"/kJ mol- ') for the dissociation reaction at 298.15 K are listed by Young and Irish1O2 and by Dickson et ul.'O4 and range from -20.5 to -23.8 kJ mol-'. The previous applica- tion of the Pitzer model" yielded -23.47 kJ mol-' which was assumed to remain constant with temperature as heat capacities were not included in that fit. A significantly smaller value of A, H = -17.33 & 0.29 kJ mol- was reported by K. Combining these two values gives Arc; for the disso- ciation reaction equal to -282.3 -(-17.8) = -264.5 J mol-' K-', which agrees well with the value of Dickson et ~1.''~(-275 f17 J mol-' K-') referred to above. In developing the model fits described below, we have adopted the temperature variation of KHSO4 as determined by Dickson et al.,lo4combined with our choice of the 298.15 K value of the dissociation constant of 0.01050 mol kg-', as stated above.These yield the following expression for KHSO4 as a function of temperature: log(KHS0,) = 562.694 86 -102.5154 h(T) -1.117033 x 10-,T2 + 0.247 753 8T -13 273.75/T (21) where the first constant was adjusted slightly to give agree- ment with our choice of KHSO4. Note that the above equa- tion, adapted from eqn. (6) of Dickson et al.lo4 yields 298.15 K values of ArH" and Arc; of -22.7554 kJ mol-' and -274.8772 J mol-K-l, respectively. 1886 3.8 Weights Relative weights (w,) assigned in Sections 3.1-3.5 above are internal to each dataset and do not take into account the differing magnitudes of experimental error typical of each kind of measurement, for example about 1.5 J mol-' K-' in the case of C: compared with only 1.3 x V for the emf measurements.The absolute weight given to an individual data point in the model fit is therefore set equal to the rela- tive weight multiplied by a characteristic weight (w,) for each type of data. Initially these characteristic weights were set so as to give contributions to the total sum of squared deviations approximately in the ratio 3 :2 :1 : 1 : 0.25 (4 :emf: Adi,H : C$:a). As the fit of the model was refined, the characteristic weights for each type of property were recalculated as : where y and f are observed and fitted quantities, respectively, and N is the number of points (for which wr # 0) in each dataset.Some minor adjustments to w, were later made to reflect the consistency of the individual thermodynamic properties with the overall data set, reducing the character- istic weight given to degree of dissociation data (by 50%)and increasing that for AdilH (by 25%). The characteristic weights, w, ,finally assigned to each dataset are as follows: 4, 4.3 x lo4; emf, 4.4 x lo6; AdilH,5.8 x C$,3.7 x Q, 60. The standard errors equivalent to these w, values are =~(4) 0.0015, o(E)= 1.5 x lop4 v, o(AdilH) = 13.13 J mol-', o(C$)= 1.64J K-' mol-' and o(a)= 0.041. 4. Method The sets of parameters B',o,),pi:), C::),C::),a,,and Q,, must be determined as functions of temperature for the two ionic interactions H+-SOi-and H+-HSO,, so as to minimise the total weighted sum of squared deviations for the five mea- sured properties 4, emf, Adi,H, C! and a.This was done using a generalised non-linear least-squares fitting routine (E04FDF' "), first obtaining estimates of the parameters at 298.15 K from 4 and emf data, then extending the fit to 298.15 K thermal data and finally to emf and C! measure-ments at other temperatures. The use of different temperature functionalities for the parameters was explored. Sulfate-hydrogensulfate speciation is, of course, not known a priori and is calculated for every data point for each suc- cessive set of parameter estimates.This was done by deter- mining the zero of the following function/, which describes the distribution of the total molality of hydrogen ion [=2m(H2S04)] between free Hf [m(H+)] and HSO, [m(HSO,)] : +' = m(H+)m(SO~-)T/[K;l;so4+ m(H+)] -m(HS0,) (23) where ~Z(SO,)~is the total sulfate molality [=m(H,SO,)] and K&,, is the stoichiometric dissociation product of the hydro- gensulfate ion (and not the thermodynamic constant). Because the incremental change in parameter estimates may be very small from one cycle of the calculation to the next, and the fact that enthalpies of dilution, and especially heat capacities, are calculated as numerical differentials [eqn. (8) and (9)] it was necessary to determine speciation to full machine precision (typically one part in in the fitting program.5. Results A general equation for the temperature variation of model parameter P (where P = Br:), /3::), C(pa),C',',))that was found to J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 be satisfactory is: P = q1 + (T -~,){io-~q,+ (T -T,) x [10-3q3/2 + (T -T,)10-3q4/6]) (24) where T is in K and the reference temperature, T,, is 328.15 K. The equation for the temperature variation of E"/V, the standard emf of cells I, IT and IV, is: E" = rl + 10r2(1/T-l/To)+ 10-3r3 x [T In(T)-ToIn(To)] (25) where To = 298.15 K. The equation used to represent the infinite dilution value of the apparent molal heat capacity, C$"/J mol-I K-', is: CF = ~1 + (T -TO)S~+ O.I(T -T0)'~3 (26) All parameters determined in the model fit are listed in Table 6, together with their temperature ranges of validity and the standard errors for the parameters for eqn.(25) and (26) above. Assigned values of the coefficients acaand mcaare also ~,~~given in Table 6: note that ~t varies with temperature. ~ For convenience, 298.15 K values of all parameters are listed in Table 7. Fig. 8-12 show the residuals for each dataset, and indicate the deviations of the fit from the weighted data. Note that it was not necessary to include the mixing parameters 6HSo4, so4 and ~,b~~~~,so4, to represent accurately the experi- mental data. While some 'cycling' of the residuals is evident in the fit to the emf (Fig. 9) and AdilHdata below 1.4 mol kg-' (Fig.lo), most observations are fitted substantially within experimental error. Note that the patterns of residuals for the emf data are essentially random for cell I at 328.15 K and for cells I1 and I11 at 298.15 K, and systematic trends for other temperatures and cells are G2.5 x V. As a test, to determine whether activity data and thermal properties not taken at 298.15 K were biasing the results at 298.15 K, a separate fit to osmotic and emf data at 298.15 K only was carried out. There was no decrease in the sum of squared deviations, confirming the quality of the fit. For measurements of a at 273.15 and 323.15 KI4 we note a trend towards positive deviations at low mola- lities. However, the a data are too few in number to deter- mine whether this reflects real errors in the fitted model, and 298.15 K data from the same source show a similar pattern of residuals. The residuals of the osmotic coefficients are random for the direct vapour pressure measurements [Fig.8(c)], as they are for isopiestic data at 273.15 and 323.15 K [Fig. 8(d)]. There are slightly different trends in the residuals for data at 298.15 K depending on whether NaCl [Fig. 8(a)] or KC1 [Fig. 8(b)] was used as standard. However, taken together, the results are essentially random above 0.4 mol kg-[Fig. 8(e)],which confirms the accuracy of our model fit. The standard devi- ation of A4 (unweighted) at 298.15 K is 0.0024. The standard potentials of cells I, I1 and IV at 298.15 K (Table 7) agree with results obtained earlier by Pitzer et u1.l' to within 0.62 mV (cell I) and 0.063 mV (cells I1 and IV).Gardner et aZ.' ' have calculated mean activity coefficients of H2S04 from the emf data of Covington et and third-law potentials for cell I1 from 0.1 to 4.0 mol kg-' and 273.15 to 328.15 K. These mean activity coefficients agree with values calculated using the present model to within 0.001, at 298.15 K, 0.006 at 273.15 K and 0.002 at 328.15 K. Rard et uZ.,~ and later Rard,26 have evaluated osmotic coef- ficient data at 298.15 K independently of other activity mea- surements, for the use of aqueous H2S04 as an isopiestic standard. Osmotic coefficients generated by the present model agree with the most recent values presented by Rard,26 Table 6 Fitted model parameters for aqueous H2S04 (4 P'O' P'1' C'O' ti, HSO4 H, HSO4 H, HS04 91 0.227 784 933 0.372 293 409 -0.002 800 325 20 -0.025 92 -3.786 677 18 1.50 0.216200279 18.172 894 6 q3 -0.124 645 729 0.207 494 846 0.010 1500824 0.382 383 535 94 -0.002 357 478 06 0.004 485 264 92 O.OO0 208 682 230 0.002 5 WP PCO, fl"' C'O' C'" W H,so4 If, SO4 HI so4 H, so4 c 91 0.034 892 535 1 -1.06641231 0.007 647 789 5 1 0.0 0 r 92 4.972 078 03 -74.684 042 9 -0.314698817 -0.176 776 695 W00.3 17 555 182 -2.262 689 44 -0.021 192 652 5 -0.73 1035 345 q3 0.008 225 803 4 1 -0.035 296 854 7 -0.OOO 586 708 222 0.0Y4 cell I" standard error cell 11" standard error cell III"." standard error cell IV" standard error rl 1.690 998 3.68 x 10-5 0.612 357 3 3.3 x 10-5 1.077553 6.99 x 0.3527679 6.3 x 10-5 r2 8.883 153 0.18 -7.273 884 0.20 6.844 643 9 0.38 r3 0.202 140 7 2.97 x 10-3 -0.248 245 9 3.3 x 10-3 -0.26634636 6.4 x q0* standard error S1 -286.175 2.6 s2 3.677 433 0.16 s3 -0.471 039 1 0.069 and oH,s04Both oll,llsoI are equal to 2.5 mol-'" kg1I2; the coefficient c1f1,1,so4 is set to 2.0 mol-"2 kg1'2, while alf,S04was allowed to vary with temperature ~~~according to the equation L Y ~=~2 -, 1842.843(1/T -1/298.15).As is the case for earlier the mixture parameters OHS04,S04 and IC/HS04,S04,H were found not to be needed and are set to zero. " Parameters are valid for the temperature ranges: cell I, 278.15-328.15 K; cell 11, 278.15-328.15 K; cell 111, 298.15 K; cell IV, 273.15-323.15 K; C:', 283.15-328.15 K." At 298.15 K, E" (cell I) -En (cell 11) -E" (cell 111) = 1.690998 -0.6123573 -1.077553 = 0.001088 V, whereas it should be exactly zero if all of the cell potentials were internally consistent. This suggests that the standard potential of the lead sulfate/lead dioxide electrode is known only to within about 1 mV. 1888 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 7 Model parameters and standard potentials at 298.15 K" H. so4 -0.008 386 089 24 0 NaCl (298 k)0.295 903 322 b(0) 0.400 482 398 p(1)H. so4 0.314 734 575 symbol N H, so4 0.010 192 247 4 :+j +I-0.005 657 866 56 C(0) H. so4 -0.323 662 605 92-0.409 364 246 c(1) 2.0 'H, so4 2.0 x3 2.5 OH, so4 2.5 Q5 286.2 A, 0.391 475 1.69100 A,/(RT)b 0.801 844 Z 14I I A 220.612357 AcIRb 3.836 018 1.07755 t , , Idi-0.352 768 -0.0081, I I , , I + I " Units are as follows: BE) and flLt)/kg mol-'; CL:) and C:',)/kg' mol-'; A,, z,, and o,,/kgl/z mol-'I2; C$"/J K-' mol-'; E"/V; KHsoJmol kg -'.Debye-Huckel limiting slopes were calculated from the polynomial function equation given in Appendix 11. 0.004 KCI (298K) and by Rard and Platf~rd,,~ to within +0.0032/-0.0015 for svmbol N $0 04m(H2S04) < 6.0 mol kg-'. 06Comparisons are also made with thermal data. Wu and d Young's7 tabulated Lo deviate from model calculated values 0 15 by about 300 J mol-' above 0.5 mol kg-'. However, because -0.004 I 0 16 this difference is roughly constant, and arises from differences 0 between the model and Wu and Young's graphical integra- 8 0 tion of aL*/i?Jm below 0.5 mol kg-', there is little effect on the calculated differentials of activity with respect to tem- -0.008 c1 perature.The value of C$" at 298.15 K (-286.2 f2.6 J mol-' K-') determined in the fit, is consistent with -295.4 J mol-' K-' estimated by Gardner et a!.'" from integral enthalpy of solution data for Na,SO,, and -282.3 J mol-' K-adopted by Hovey and Hepler.*O At temperatures other than 298.15 K, C$O calculated from eqn. (26) agrees with the values of Gardner et d.'" to within 33 J mol-' K-' from 283.15 to 323.15 K. Reardon and Beckie12 have also fitted the Pitzer model to thermodynamic data for aqueous H,SO, (excluding C$ and using a much smaller database) over a similar range of tem-perature and molality to that used here.A comparison of their calculated values with the data yields sums of squared deviations (measured minus calculated) that exceed those obtained using the present model by factors of 5 and 2.3 for # and Adi,H, respectively. Thus, our model better represents the properties of aqueous H2S04 by a large margin, although it is more complex than that used by Reardon and Beckie. We also believe that our model equation represents the osmotic coefficient with sufficient accuracy for it to be used as an iso- piestic standard over its range 0-6.1 mol kg-' and 273.15-328.15 K. Values of activities, degrees of dissociation and thermal 0.006properties of aqueous H2S04, calculated with the present model, are listed in Tables 8-10 for 273.15, 298.15 and 323.15 K. At 298.15 K we have included calculated relative partial 0.002+ molal enthalpies [aH,O), E(H,SO,)/J mol-'1 and heat a" 40capacities [J(H,O), J(H,SO,)/J mol-' K-'1.Values of E(H2S04) calculated by the present model average about -0.002 1.25% greater than those listed in Table 5 of Wu and with cyclic deviations between the two studies of .** . tY~ung,~' the same order below 1 mol kg-', corresponding to those in -0.006 *-I 1 ' I ' '* ' I 17 the residual plot in Fig. 10. However, our model equations represent the input values of AdilH to within about 50 J mol-' in this region, which is 0.2-0.4% of C(H,SO,).There Fig. 8 Deviations between the measured and fitted stoichiometric is no systematic difference between the relative partial molal osmotic coefficients (4st)of aqueous H,SO, at 298.15 K (a)-(c), enthalpies of water calculated in the two studies, though the 273.15 and 323.14 K (4,and all temperatures (e).(a) Aqueous NaCl maximum deviations from Wu and Young ( & 9% between 0.2 used as isopiestic standard (except NaOH for N = 1); (b) aqueous KCl used as isopiestic standard ; (c)direct vapour pressure measure- and 1.5 mol kg-' H2S04) are greater on a percentage basis ment; (d) see Table 1 for isopiestic standards. (e) All data. Keys for than for L(H,SO,). We have not tabulated apparent or rela- (u)-(d): symbols are related to the dataset numbers N in Table 1.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1889 1’ I I 1 I I I I I I-’ (a1 0.4 0 (b)-!$00.2 1 icell I 0 symbol TIK cell I> EGC X 278 .-?*X --9-+ symbol TIK 283 V 293 0 H 328 0,O 298-0.2 0 D 308 -0.41, I I , I Ii -0.4 0 1.o 2.0-0.210 1.o 2.0 0.4 I I 1 1 I (cI 0.4 X cell II cell II (298 K) symbol N0.2 3 o x + svmbol TIK 0.2 > 0 278 05 E $0 + 288 0 *6 +70 308 080 U I 318 X 328 -0.2 1 v 10-0.2 O+ ‘s 11 X I --1-0.4 -0.4 tl , , , ,-J 0 1.o 2.0 0 1.o 2.0 1 I 1 1-0.4-‘ (f) symbol TIK ’ cell III (298 K) I3 cell IV 0 285symbol N 12 0 298 273 K V 310 0 323-0.2 --0.4 -1 1 I I -0.4 0 1.o 2.0 0.05 0.01 0.15 Jm/mol’ kg-’I2 Fig.9 Deviations between measured and fitted emfs of cells I-IV. Keys give cell type, with (rounded) temperature of measurement (T)and/or dataset number (N) in Table 2. (a),(b) N = 1 for all data, except filled circles (N = 2); (c) N = 5 for all data; (f) N = 13 for all data. 298 K symbol N 75 .A I I I I I,‘ 01 *2 +3-50 04I-L ’ 2525 :o symbol N TIK X 5 328 d -25 I-50 I I I I I I 0 1.o 2.0 0 1.o 2.0 Jm/mol’’2 kg-’I2 Jmlmol ’l2 kg-’l2 Fig. 10 Deviations between measured and fitted differential enth- Fig. 11 Deviations between measured and fitted apparent molal alpies of dilution at 298.15 K, plotted against the square root of the heat capacities in H,SO,. (a)298.15 K; (b) other temperatures. Keys initial molality, m,.The dataset numbers (N) of Table 3 are: (*) 1, give the (rounded) temperature of measurement (T) and dataset 3, (44.number (N) in Table 4. (0)2, (0) d 0 318 1890 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 298 K tive partial molal heat capacities at 273.15 or 323.15 K in symbol N Tables 9 and 10, as the model is unlikely to be as well con- 03 strained with respect to these quantities at the extremes of the 04 temperature range over which it has been applied. -0.04 1 +5 07 6. Conclusions I I I I I A generalised, extended formulation of the molality-based Pitzer thermodynamic model has been presented here and applied to measured properties of aqueous H,SO,, yielding a self-consistent representation of activities, apparent molal enthalpies and heat capacities of aqueous H,SO, from 0 to 6.1 mol kg-l, for the temperature range 273.15-328.15 K.The extension to the model provides a more flexible frame- work for calculating the properties of both single- and multi- -0.1 CI I I I 1 I component electrolyte solutions at low and moderate0 1.o 2.0 molalities, and its application to sulfuric acid will lead to ,/m/rnol1/’ kg-improved calculations of thermodynamic properties of acidic Fig. 12 Deviations between measured and fitted degrees of disso-sulfate mixtures. ciation of the hydrogensulfate ion. (a) 298.15 K; (b) other tem- peratures. Keys give the (rounded) temperature of measurement (T) The work of S.L.C. was supported by a grant from the Lever- and dataset number (N)in Table 5.hulme Trust. The contributions of J.A.R. and K.S.P. were Table 8 Thermodynamic properties of aqueous H2S04 at 298.15 K (0.0001) 0.9500 0.98 13 0.982 86 497.3 -267.9 -7.5689 x lo-, 917.5 -2.895 x lo-’ 34.35 (0.0002) 0.9253 0.9712 0.967 55 890.9 -252.9 -2.6763 x 1633.6 -1.006 x 10-4 61.16 (0.0005) 0.8737 0.9493 0.928 18 1874.6 -217.2 -1.3211 x lo-’ 3341.3 -4.625 x 120.3 (0.0010) 0.8152 0.9236 0.876 21 3 150.9 -174.8 -4.0382 x lo-* 5392.5 -1.270 x lop3 181.9 (0.0020) 0.7384 0.8890 0.800 41 4 997.0 -121.2 -0.1 1087 8 074.0 -2.956 x 247.0 0.0050 0.6146 0.8325 0.667 58 8 224.4 -47.4 -0.34858 12 094 -6.719 x 313.4 0.0100 0.5145 0.7867 0.555 11 10 978 -2.11 -0.7209 1 14 980 -9.873 x lo-’ 338.9 0.0200 0.41 89 0.7440 0.447 04 13 679 27.9 -1.3468 17417 -1,160 x 346.3 0.0500 0.3098 0.6987 0.328 95 16 807 46.4 -2.7538 19 864 -9.222 x lo-’ 342.8 0.1000 0.2436 0.6759 0.265 25 18 734 50.6 -4.5207 21 243 -5.837 x 340.0 0.2000 0.1916 0.6647 0.224 99 20 299 52.5 -7.2577 22 313 -1.139 x lo-’ 341.8 0.3000 0.1672 0.6647 0.212 26 21 059 54.1 -9.3826 22 795 -2.741 x lo-’ 345.3 0.4OoO 0.1525 0.6685 0.208 14 21 530 55.8 -11.088 23 069 -5.021 x lo-’ 349.0 0.5000 0.1425 0.6744 0.207 8 1 21 857 57.6 -12.528 23 248 -7.890 x low2 352.5 0.6000 0.1353 0.68 16 0.209 47 22 100 59.3 -13.856 23 382 -0.1135 356.0 0.7000 0.1300 0.6898 0.212 26 22 292 61.1 -15.216 23 498 -0.1543 359.5 0.8000 0.1259 0.6990 0.2 15 74 22 450 62.8 -16.740 23 611 -0.2016 363.0 0.9000 0.1228 0.7088 0.219 65 22 585 64.6 -18.546 23 729 -0.2552 366.5 1.m 0.1204 0.7194 0.223 86 22 706 66.3 -20.736 23 857 -0.3150 370.0 1.2000 0.1173 0.7420 0.232 78 22 922 69.8 -26.602 24 152 -0.4507 376.8 1.4000 0.1157 0.7663 0.242 08 23 122 73.3 -34.878 24 505 -0.6012 383.3 1.6000 0.1153 0.7920 0.251 57 23 320 76.6 -45.928 24913 -0.7548 389.0 1.8000 0.1157 0.8188 0.261 13 23 522 79.8 -59.993 25 372 -0.8963 393.6 2.m 0.1169 0.8464 0.270 66 23 732 82.7 -77.234 25 875 -1.0081 396.9 2.2000 0.1186 0.8746 0.280 07 23 951 85.3 -97.777 26 418 -1.0723 398.6 2.4000 0.1209 0.9034 0.289 23 24 180 87.6 -121.74 26 996 -1.0732 398.6 2.6000 0.1237 0.9327 0.297 96 24 420 89.5 -149.28 27 607 -1.001 1 397.0 2.8000 0.1269 0.9624 0.306 09 24 671 90.9 -180.56 28 250 -0.8549 394.0 3.m 0.1306 0.9926 0.3 13 39 24 93 1 91.9 -215.79 28 924 -0.6446 390.0 3.2000 0.1347 1.0232 0.319 64 25 203 92.5 -255.18 29 629 -0.3914 385.5 3.4000 0.1393 1.0542 0.324 66 25 485 92.8 -298.88 30 364 -0.1240 381.0 3.6000 0.1443 1.0856 0.328 26 25 777 92.8 -346.94 31 126 0.1264 377.0 3.8000 0.1498 1.1172 0.330 34 26 079 92.6 -399.28 31 912 0.3331 373.9 4.m 0.1556 1.1490 0.330 85 26 391 92.3 -455.64 32 714 0.4799 371.8 4.2000 0.1620 1.1807 0.329 80 26711 91.9 -5 15.62 33 526 0.5641 370.7 4.4000 0.1687 1.2123 0.327 27 27 039 91.6 -578.70 34 340 0.5963 370.2 4.6000 0.1758 1.2436 0.323 37 27 374 91.3 -644.30 35 149 0.5977 370.2 4.8000 0.1833 1.2745 0.318 25 27 715 91.0 -711.81 35 947 0.5966 370.2 5.oooo 0.1912 1.3047 0.31209 280 60 90.7 -780.66 36 727 0.624 1 369.9 5.2000 0.1994 1.3343 0.305 05 284 08 90.4 -850.37 37 485 0.7115 369.0 5.4000 0.2080 1.3630 0.297 30 287 58 90.1 -920.51 38 220 0.8878 367.1 5.6000 0.2168 1.3908 0.289 00 291 08 89.7 -990.79 38 929 1.179 364.2 5.8000 0.2259 1.4177 0.280 3 1 294 59 89.2 -1061.0 39 613 1.608 360.0 6.oooO 0.2352 1.4437 0.271 35 298 09 88.6 -1131.0 40 272 2.194 354.5 Values in parentheses are for molalities below the lower limit for which activity data (emf, #) exist.Thermal properties L*, qH20) and QH,S04) are given in J mol-’, and C:, J(H,O) and I(H,SO,) in J mol-’ K-I. * The use of a in conjunction with y* and KHSO4allows the species activity coeficients yH ,yHS04 and ysoI to be recovered. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 9 Thermodynamic properties of aqueous H,SO, at 273.15 K molality/mol kg-Yi A* za L4 QH,O) i-W,SO,) (0.0001) 0.9569 0.9845 0.991 36 202.1 -2.8804 x 10-4 362.0 (0.0002) 0.9373 0.9770 0.983 45 351.3 -1.0150 x 633.0 (0.0005) 0.8971 0.9609 0.962 23 729.6 -5.1713 x lo-' 1303.7 (0.0010) 0.85 15 0.9420 0.932 33 1244 -1.6748 x lo-, 2 173.3 (0.0020) 0.7895 0.9 1 54 0.884 81 2 044 -5.0203 x lo-, 3 437.3 0.0050 0.6827 0.8682 0.789 95 3 621 -0.183 13 5 653.8 0.0100 0.5888 0.8260 0.697 32 5 158 -0.42698 7 528.4 0.0200 0.4923 0.7828 0.59641 6 852 -0.89264 9 330.0 0.0500 0.3746 0.7324 0.469 96 9 057 -2.0532 11 337 0.1000 0.2994 0.7042 0.392 37 10531 -3.5175 12 483 0.2000 0.2382 0.6872 0.338 22 11 740 -5.4711 13 258 0.3000 0.2089 0.6836 0.319 65 12 294 -6.5342 13 503 0.4000 0.1909 0.6847 0.313 11 12 607 -6.9415 13 570 0.5000 0.1787 0.6884 0.31209 12 800 -6.8870 13 564 0.6000 0.1698 0.6939 0.31402 12 925 -6.5613 13 532 0.7000 0.1632 0.7010 0.31764 13009 -6.1489 13 496 0.8000 0.1582 0.7093 0.322 26 13 068 -5.8226 13 472 0.9000 0.1544 0.7187 0.327 50 13 113 -5.7384 13 466 1.m 0.1516 0.7291.0.333 11 13 149 -6.0342 13 484 1.2000 0.1479 0.7523 0.344 89 13 212 -8.2298 13 593 1.4000 0.1464 0.7783 0.356 92 13 280 -13.194 13 803 1.6000 0.1465 0.8064 0.368 85 13 364 -21.546 14111 1.8000 0.1478 0.8362 0.380 5 1 13 468 -33.828 14512 2.m 0.1501 0.8676 0.391 73 13 596 -50.560 15OOO 2.2000 0.1533 0.9001 0.402 34 13 749 -72.270 15 573 2.4000 0.1574 0.9337 0.412 15 13 928 -99.470 16229 2.6000 0.1623 0.9683 0.420 93 14 133 -132.61 16964 2.8000 0.1680 1.0037 0.428 45 14 364 -172.00 17 773 3.m 0.1745 1.0400 0.434 49 14 620 -217.75 18 649 3.2000 0.1818 1.0771 0.438 84 14 900 -269.72 19 579 3.4000 0.1899 1.1 148 0.441 34 15 204 -327.5 1 20 551 3.6000 0.1988 1.1530 0.441 89 15 529 -390.50 21 550 3.8000 0.2085 1.1916 0.440 47 15 872 -457.88 22 561 4.m 0.2191 1.2302 0.437 10 16 232 -528.77 23 570 4.2000 0.2305 1.2687 0.431 88 16 605 -602.26 24 565 4.4000 0.2426 1.3069 0.424 95 16 989 -677.47 25 536 4.6000 0.2556 1.3445 0.416 50 17 381 -753.61 26 475 4.8000 0.2693 1.3813 0.406 72 17 779 -830.03 27 378 5.oooo 0.2838 1.4172 0.395 84 18 180 -906.20 28 241 5.2000 0.2989 1.4521 0.384 07 18 583 -981.76 29 063 5.4000 0.3 147 1.4858 0.371 61 18 986 -1056.5 29 846 5.6000 0.33 10 1.5183 0.358 68 19 387 -1130.2 30 591 5.8000 0.3480 1.5496 0.345 44 19 786 -1203.1 31 300 6.oooO 0.3656 1.5797 0.33206 20 181 -1275.2 31 979 'See footnotes to Table 8.performed under the auspices of the Offce of Basic Energy Definitions are given below. Equations for the activity coeffi- Sciences (Geosciences and Chemical Sciences, respectively) of cient of cation M, anion X and the osmotic coefficient follow : the US Department of Energy by the Lawrence Livermore National Laboratory (contract no. W-7405-ENG-48) and the Lawrence Berkeley Laboratory (contract no. DE-AC03-76SF00098). Appendix I The modified equation for excess Gibbs energy, from which expressions for activities and thermal and volumetric proper- ties are derived, is given below for a solution containing an indefinite number of cations, c, and anions, a.For terms involving neutral species (unchanged by the extension to the model) see reviews by Pitzerg or Clegg and Whitfield.13 Gex/(nwRT) = -(4A6Z/1.2)1n(1 +1.2JZ) +11mcma(2Bca +ZCTa) ca +c1mc mc*[ 2~ccP +1ma $,,*a c<c' 1 a<a'+11ma ma,[2Qaa, +1mc +aa*c] (~11) J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 10 Thermodynamic properties of aqueous H2S0, at 323.15 K” (0,0001) 0.9353 0.9741 0.962 80 1245.8 -1.9224 x 2 312.9 (0.0002) 0.9003 0.9588 0.931 45 2 227.8 -6.5493 x 4 045.5 (0.0005) 0.8273 0.9259 0.857 36 4515.6 -2.9190 x lo-’ 7 756.2 (0.00 10) 0.7492 0.8900 0.770 72 7 169.7 -7.9164 x lo-* 11564 (0.0020) 0.6547 0.8462 0.660 74 10529 -0.18784 15 743 0.5192 0.7839 0.500 02 15 454 -0.48 138 20 799 0.0050 0.4213 0.7401 0.386 34 18 987 -0.86032 23 763 0.0100 0.3346 0.703 1 0.292 17 22 009 -1.4129 25 930 0.0200 0.2416 0.6673 0.204 79 25 084 -2.5539 27 919 0.0500 0.1000 0.1875 0.6506 0.165 46 26 833 -4.0528 29 083 0.1458 0.6443 0.145 96 28 248 -6.680 1 30 102 0.2000 0.3000 0.1266 0.6468 0.142 92 28 960 -8.9699 30 619 0.4000 0.1151 0.6524 0.14440 29 418 -10.998 30 944 0.5000 0.1073 0.6595 0.147 65 29 748 -12.901 31 180 0.6000 0.1018 0.6676 0.151 67 30 003 -14.840 31 376 0.7000 0.0976 0.6764 0.156 02 30 212 -16.973 31 558 0.8000 0.0944 0.6858 0.160 48 30 392 -19.445 31 741 0.9000 0.09 19 0.6956 0.164 95 30 552 -22.373 31 932 1.m 0.0900 0.7058 0.169 36 30 700 -25.854 32 135 1.2000 0.0873 0.727 1 0.177 86 30 976 -34.738 32 583 1.4000 0.0857 0.7495 0.185 83 31 241 -46.440 33 082 1.6000 0.0850 0.7727 0.193 22 31 504 -61.072 33 623 1.8000 0.0849 0.7966 0.200 01 31 771 -78.614 34 196 2.m 0.0852 0.821 1 0.206 19 32 043 -99.004 34 791 2.2000 0.0860 0.8460 0.21 1 75 32 321 -122.21 35 404 2.4000 0.087 1 0.8714 0.216 66 32 604 -148.25 36 033 2.6000 0.0886 0.8971 0.220 90 32 892 -177.25 36 676 2.8000 0.0903 0.9232 0.224 47 33 186 -209.40 37 337 3.m 0.0923 0.9495 0.227 33 33 485 -244.95 38 018 3.2000 0.0946 0.9760 0.229 49 33 790 -284.14 38 719 3.4000 0.0971 1.W26 0.230 93 34 102 -327.13 39 442 3.6000 0.0998 1.0294 0.23 1 67 34419 -373.98 40 185 3.8000 0.1027 1.0562 0.231 71 34 742 -424.58 40944 4.m 0.1059 1.083 1 0.23 1 08 35 072 -478.63 41 714 4.2000 0.1093 1.1098 0.229 82 35 406 -535.60 42 485 4.4000 0.1129 1.1364 0.227 96 35 745 -594.77 43 249 4.6000 0.1167 1.1628 0.225 56 36 088 -655.26 43 995 4.8000 0.1207 1.1890 0.222 65 36 432 -716.04 44 713 5.m 0.1248 1.2149 0.2 19 30 36 777 -776.00 45 392 5.2000 0.1292 1.2404 0.215 56 37 121 -833.99 46 024 5.4000 0.1338 1.2655 0.21 148 37 462 -888.83 46 598 5.6000 0.1385 1.2902 0.207 12 37 797 -939.40 47 109 5.8000 0.1434 1.3144 0.202 52 38 126 -984.61 47 549 6.oooO 0.1485 1.3381 0.197 72 38 446 -1023.4 47915 a See footnotes to Table 8.The function 9in eqn. (AI2) and (AI3) is given by: where F= -A+[Jz/(I + 1.2J1) + (2/1.2)1n(1 + 1.2J1)I g(x) = 2[ 1 -(1 + x)exp(-x)]/x2 (A1 13) + 1C mc ma(K + ZC32) ca and two new functions are defined for the extended model: + c 1m,m,,wee, + 1 1 mama,was.(AI5)c<c’ aia‘ h(x) = (6 -[6 + x(6 + 3x + x2)]exp(-x))/x4 (AIM)In the above equations, Z is the molality based ionic strength, and summations c < c’ and a < a‘ are over all distinguishable h’(x)= exp(-x)/2 -2/44 (AI16) pairs of cations and anions, respectively. The symbol $,,., is a Note that g’(acaJI) is equivalent to Zd[g(a,, ,/Z)]/dI, and a ternary parameter for the interaction of an anion and two corresponding relationship applies for h’(mca41).Other func- distinct cations, and similarly for $,,., . The equations also tions for the standard model involve interactions between contain the following functions : pairs of ions (i,j)of like sign: CD, = 8, + e$) (A117) qj= O”,,Z) (AI18) CD~= e, + eE(q + ieE(I) (AI19) The value of parameter Oij is obtained by fitting, whereas the unsymmetrical mixing term OE(Z) and its derivative are obtained from theory,’13 and are equal to zero where the charges on ions i and j are equal in magnitude.These mixing J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 terms are given by: QE = (zi zj/4I)[J(xij) -1/2J(xii)-1/2J(xjj)] (AI20) 0: = a;./aI = -e8p + (zizj/s~2)[xij~~(xij) -(1/2)Xii J’(X,i) -(1/2)XjjJ’(Xjj)] (AI21) and J’(Xij)= aJ(xij)/axij (A122) The molality-dependent variable xijis given by: x..IJ = ~z.~.A,I”~1 J (AI23) The function J(xij) is an integral which has been evaluated numerically.The following approximating equation is used here to obtain values of J(xij)and J’(xij):’I3 J(X~~)xij/[4 + C1x? exp(~,x:)] (AT24)= with C, = 4.581, C, = -0.7237, C, = -0.0120 and C, = 0.528. More accurate (but complicated) methods are avail- able.’ However, in the present application the alternative Chebychev polynomial representation of J(xij) and J’(xij) (used by Harvie et a!., for example”) offers no improvement over eqn. (AI24). Appendix I1 Values of the Debye-Hiickel coefficient, A,, used here (at 1 atm, and different temperatures) are those determined by Archer and Wang,I6 calculated using a program supplied by Archer.’ l4 The Debye-Hiickel constants of Archer and Wang’ were obtained from least-squares equations for the relative permittivity of water as a function of temperature, pressure and water density.Since we are interested in the Debye-Hiickel limiting law slopes as a function of tem-perature only, at a single pressure of 1 atm, we decided to represent A, with one equation valid over the temperature range 234.15-373.15 K. We use a polynomial in Chebychev series form involving the normalised variable x [x = (2X -X,,, -Xmin)/(Xmax-Xmin)],where X is the temperature (in K). X,,, (373.15 K) and Xmin(234.15K) are the upper and lower limits of the fit, respectively. The polynomial, with 19 (N + 1) coefficients, is given by: A, = oh0 To(x)+ UITl(X) U2 T2(X) + u3 T3(x).. . + UN TN(X) (A111) where To(x)= 1, T,jx) = x, and for n 2 2: T,(x) = 2xT,-,(x) -T,-,(x) (AII2) The polynomial coefficients, ai,are listed in Table 11.Fitted values of A a ree with those calculated from the orig- @ .ginal program to within 0.5 x lo-’ below 250 K, and to within 0.95 x lop8 mol-1/2 kg1l2 at higher temperatures. Table 11 Chebychev polynomical coefficients for A,, 234.15 6 T/ K d 373.15 0.797 256 081 240 a,, -0.388 189392385 x 0.573 389 669 896 x 10-a,, 0.164245088592 x 0.977 632 177 788 x a,, -0.686031 972 567 x 0.489973732417 x lo-’ a13 0.283455806377 x -0.313 151 784342 x lo-’ a14 -0.115641433004 x 0.179 145 971 002 x lo-’ a,, 0.461489672579 x -0.920584241844 x aL6 -0.177069754948 x 0.443862726879 x al, 0.612464488231 x lo-’ -0.203 661 129991 x a18 -0.175689013085 x lo-’ 0.900924 147 948 x This level of precision was chosen to ensure accuracy in the calculated second differential of A, with respect to tem-perature, because it appears in the expression for the appar- ent molal heat capacity, obtained here by numerical differentiation (Section 2).The very low minimum tem-perature of fit was needed in order to use the same poly- nomial in a second study of the H2S0,-H20 system over a more extended range of temperature and comp~sition.~~ For program validation, the polynomial yields the follow- ing values: 273.15 K, A, = 0.376421452 mol-’/* kg’I2; 298.15 K, A, = 0.391 475 238 mol-1/2 kg’l2. 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