J. Chem. SOC., Faraday Trans. 1, 1986,82, 247-253 Multicomponent Diffusion of Double Salts Sodium Hydrogen Sulphate in Aqueous Solution Betty Wiens and Derek G. Leaist* Department of Chemistry, The University of Western Ontario, London, Onturio, Canada N6A 5B7 Diffusion coefficients for the double salt sodium hydrogen sulphate in aqueous solution have been determined at 25 "C from infinite dilution to 2 mol dm-3 using limiting ionic conductances, the Harned restricted-diffusion method and Gouy optical data reported previously for aqueous sodium sulphate-sulphuric acid mixtures. Because the total flux of hydrogen ions exceeds the flux of less-mobile sodium co-ions, diffusion of aqueous NaHSO, forms a ternary mixture of NaHS0,-H,SO, in front of the diffusion boundary while a ternary mixture of NaHS0,-Na,SO, remains behind.Diffusion in aqueous solutions of NaHSO, can be described by a coupled flow of H,SO, in addition to the flow of NaHSO,. At concentrations below 0.1 mol dm-3 the coupled flow of H,SO, exceeds the flow of NaHSO,. Expressions are developed to predict ternary transport coefficients for diffusion of double salts. Although a solution of a double salt [such as AlK(SO,),-H,O] at thermodynamic equilibrium is a two-component system,l? diffusion of a double salt is a multicomponent pro~ess.~ In a solution of salt MNX for example, the equilibrium concentrations of M+, N+ and X2- are all identical and the composition of the system is defined by a single concentration variable. However, if a gradient in concentration of the salt is formed, co-ions M+ and N+, which have different mobilities, may be expected to diffuse at different rates.If M+ is the more mobile co-ion the extra flow of M+ will lead to a coupled flow of M2X in addition to the main flow of MNX. Diffusion would then resolve an initial binary solution of MNX into a ternary mixture of MNX + M2X in front of the diffusion zone and a ternary mixture of MNX+N,X behind it. These considerations apply to transport of numerous important electrolytes including acid salts of polyprotic acids and double salts of mineralogical or biological significance. No multicomponent diffusivities have been reported for these salts. This research was undertaken to measure diffusion coefficients for the double salt sodium hydrogen sulphate in aqueous solution. It is shown that diffusion coefficients for this salt can also be obtained from ternary diffusion coefficients reported previously for aqueous sodium sulphate-sulphuric acid mixtures.*~ The results for sodium hydrogen sulphate are compared with binary diffusion coefficients for aqueous sodium sulphate5-' and sulphuric acid.*.Experimental Diffusion coefficients were determined at 25 "C using a simplified version of Harned's restricted-diffusion lo In this technique salt fluxes are followed by monitoring changes in electrical conductance near the top and bottom of short diffusion columns. The conductimetric method allowed measurements to be made at low concentrations ( < 0.1 mol dm-3) where bisulphate is appreciably dissociated. Initial concentration gradients were formed by injecting NaHSO, solutions into the lower end of cylindrical diffusion channels filled with dilute NaHSO, solutions.At the 247248 Multicomponent Diflusion of NaHSO, same average cell compositions separate experiments were performed with initial gradients in the concentration of H2S0,. This was accomplished by introducing NaHS0,-H,SO, solutions into cells filled with NaHS0,-Na,SO, solutions; equimolar amounts of H,SO, and Na,SO, were added to each cell to ensure that the average cell composition corresponded to NaHSO, + H20. Eigenvalue analysisll of the conductance versus time data from the NaHS0,- and H,SO,-gradient experiments gave ternary diffusion coefficients for NaHSO,(c,~H,SO,(c, = 0)-H,O. Details of the equipment and procedure have been reported.9* l1 All materials were reagent grade.Na,SO, was dried in a vacuum oven at 130 "C. Concentrations of stock solutions of NaHSO, and H2S0, were determined by poten- tiometric titration against standardized NaOH. Solutions were made up in calibrated volumetric flasks using twice-distilled, deionized water. Results Transport equations Diffusion in aqueous sodium hydrogen sulphate solutions entails transport of Na+ and rapidly interconverting HSO;, HS and SO:- species HSO, = H+ + SO:- K2(25 OC)l2? l3 = 0.0105 mol dm+. Diffusion of the salt may be described using flow equations3? l4 for each of the four solute species j i = - C & k ~ & . (1) Here j , is the molar flux density of species i, V,i& is the gradient in electrochemical potential of species k, and lik are ionic Onsager coefficients.Owing to the constraints imposed by local chemical equilibrium 4 k=1 - only two solute fluxes are independent. Isothermal diffusion in aqueous NaHSO, solutions is therefore a ternary', process. Because the single ion potentials appearing in eqn (1) are difficult to measure it is more convenient to analyse diffusion of NaHSO, in terms of fluxes Ji of neutral electrolyte components rather than fluxes j i of charged species. Owing to the exceptional mobility15 of H+, the total flux of H+ (free H+ plus bound Hi as HSO;) may be expected to exceed the flux of Na+ co-ion. Since any extra flux of H+ accompanying the main NaHSO, flux would constitute coupled flow of H,SO,, we will describe diffusion in aqueous NaHSO, solution in terms of fluxes of neutral NaHSO,(I) and H,SO,(2) components.In this representation the flux of the H,SO, component equals one-half of the total H+ flux in excess of the Na+ flux. J2 = +&+ -&a+)P (4) The flux of Na+ and the flux of the NaHSO, component are identical. J1 = jNa+. If a denotes the degree of bisulphate dissociation, a = [SOi-]/([HSO,] + [SO:-])B. Wiens and D. G . Leaist 249 the remaining species’ fluxes are given as follows: jH+ = d1+(1+a)J, (7) JHSOT = (1 - a) (4 + J,) (8) Jsoi- = a( J1+ J,). (9) Interacting flows of neutral NaHSO, and H,SO, components are related to measurable concentration gradients by an extension of Fick’s equation16 - J1 = Dll VC, + D12 VC, - J, = D,, VC, + D,, VC, (10) (1 1) in which Di, are ternary diffusion coefficients and ci are component concentrations in moles per unit volume.Diffusion Coefficients Ternary diffusion coefficients for NaHSO,(c,)-H,SO,(c, = 0 j H , O were determined conductimetrically at four NaHSO, concentrations ranging from 0.005-0.1 mol dm-3. The results are listed in table 1. Experimental precision is quoted as & two standard deviations estimated from runs performed in triplicate with different cells. Wendt4t5 has used the Gouy optical method to measure precise ternary diffusion coefficients for Na,SO,(O.S mol dm-3)-H,S0,(0.5 mol dm-3FH,0 and for Na,SO,( 1 .O rnol dm-3)-H,S0,( 1 .O mol dmM3)-H,0 at 25 “C. Bearing in mind the equilibrium Na,SO, + H,SO, = 2NaHS0, + H20, these solutions are identical in composition to NaHSO,( 1 .O mol dm-3)-H,S0,(0.0 rnol dmP3)-H,O and NaHS0,(2.0 mol dm-3)- H,SO,(O.O mol dmP3)-H,O, respectively.The optical data for equimolar Na,SO,-H,SO, mixtures should be closely related to diffusion coefficients for NaHSO,. The relationship can be established by noting that an aqueous solution of composition NaHSO,(c,)- H,SO,(c,) is identical to a solution of composition Na,SO,(c~)-H,SO,(c~) provided c; = c,/2 and ci = (c,/2) + c,. Concentrations c: are thus linear combinations of ci 2 where A , , = 0.5, A,, = 0, A,, = 0.5, and A,, = 1. It is easily shownll that ternary diffusion coefficients Di, for NaHSO,(c,)-H,SO,(c,) mixtures are obtained from diffusion coefficients Dik for Na,SO,(c~)-H,SO,(c~) mixtures by the linear transformation D = A-’D’A which provides D,, = D;,+D’,, (13) D,, = 2D;, (14) D2, = (-Oil - Di2 + Oil + Dk2)/2 (15) D,, = - D;, -I- D;,.(16) Ternary diffusion coefficients for NaHSO,(cl~H,SO,(c, = 0) mixtures that were derived from Wendt’s data for Na,SO,-H,SO, mixtures by use of eqn (13)-(16) are listed in table 1. Discussion It is evident from table 1 that values of D,, for dilute NaHSO, solutions are large and positive. Since D,, measures the coupled flow of H,SO, generated per unit gradient in concentration of NaHSO,, diffusion of aqueous NaHSO, is indeed a ternary process.250 Multicomponent Difusion of NaHSO, Table 1. Ternary diffusion coefficients for NaHSO,(c,)-H,SO,(c, = O)+H,O at 25 "C cJmol dm-3 4 1 a m2 s-' D2 1 /lop9 m2 s-l 0.000" 0.005b O.OIOb 0.050b 0.1 OOb 1 .00OC 2.0ooc 1.00 0.571 0.81 0.56+0.01 0.73 0.55 f 0.07 0.55 0.69 f 0.03 0.48 0.75 f 0.09 - 0.93 0.56 - - 1.476 -1.41 f0.12 - 1.21 f0.05 - 1.15k0.02 - 1.05 f 0.02 -0.35 - 0.06 1.710 1.43 &O.11 1.37 f0.15 0.91 k0.05 0.72 0.01 0.08 0.14 D22 / m2 s-l 4.899 4.27 & 0.04 4.01 k0.12 3.60 & 0.04 3.43 * 0.08 1.45 0.88 ________. ._ a Limiting value calculated from eqn (27)-(30). measurements on Na2S0,-H2S0, mixture^.^^ Conductimetric data. Derived from optical At 0.005 mol dm-3 NaHSO,, for example, D,, is twice as large as Dll, the diffusivity of the NaHSO, component. Although this solution contains no added H,SO,, the gradient in concentration of the NaHSO, component produces a parallel flow of H,SO, (- D,, Vc,) that is twice as large as the initial flow of the NaHSO, component (- D,, Vc,).This behaviour stands in sharp contrast to the diffusional properties of chemically inert in which a gradient in solute 1 is unable to generate coupled flow of solute 2 in a solution that is free of added solute 2, i.e. D,, -+ 0 as c,/c, + 0. At high NaHSO, concentrations where dissociation of bisulphate is less important, Na+ and HSO; are the major transporting species for the NaHSO, component. In this region the gradient in concentration of NaHSO, is incapable of producing significant additional flow of H+ ; values of D,, are therefore small and diffusion of NaHSO, is more nearly a binary process. When sulphuric acid diffuses in water, a relatively large electric field is induced in order to slow down highly mobile H+ and avoid significant charge separation. In aqueous NaHSO, solutions the electric field produced by diffusion of H,SO, has the additional effect of driving countertransport of Na+.For this reason values of D,, are negative. Limiting Diffusion Coefficients A detailed interpretation of diffusion of aqueous NaHSO, solutions is difficult because it involves two interacting solute flows consisting of Na+, HSO;, H+ and SO:- in proportions that vary with composition. Ternary diffusivities for the system are thus complicated weighted averages over the diffusivities of various species. To develop a model for diffusion of NaHSO, and other double salts we will make the dilute solution approximationls? l9 &k = 6,kEiDi/RT (17) where 6ik is the Kronecker delta, R is the gas constant, Tis the temperature, and ti and Di are the concentration and diffusion coefficients for species i.Diffusion coefficients for the solute components can then be estimated using the identity,, where Vik and Cim are stoichiometry coefficients which denote the number of moles of constituent ion i per mole of component k and species m, respectively, and ,uik = api/ac, is the derivative of the chemical potential of solute component i with respect to theB. Wiens and D. G. Leaist 25 1 concentration of solute component k. Superscript T indicates the transpose of a matrix. We will number the components, constituent ions, and species as follows component constituent ions species 1 NaHSO, 1 Na+ 2 H,SO, 2 H+ 3 HSO; which provides 1 0 0 P = O 1 0 i 0 0 1 Degrees of dissociation and derivatives Pik used from the exmessions 1 Naf 2 Hi 3 HSO, 4 so;- -;).1. to predict values of D i , were evaluated Pl = P? + RT ln [(I - 00 C l h + c2)A 931 f l 2 = Pi + RT In [(aci + (1 +a) c2) (1 -a) (C1-k c,)92.?3]. (22) (23) Corrections were made for departures from ideal solution thermodynamics using activity coefficients for the species estimated from the relation20 (24) I = (1+2a)(c,+c,). (25) l n j i = - 1.17~?Z~’~/(l +I1/,) where I denotes the ionic strength In fig. 1, ternary diffusion coefficients predicted for dilute NaHSO, solutions are compared with experimental results. Agreement is close, generally within +0.2 x m2 s-l. For D,,, however, predicted values are too low by ca. 65 x m2 s-l at concentrations above 0.05 mol dm-3. The calculations employed limiting diffusion coefficients for each solute species: D,(Na+) = 1.334 x D,(H+) = 9.3 15 x m2 s-l.These values were obtained from published limiting molar ionic cond~ctances~~~ 21 using the relation At high concentrations the effects of non-zero off-diagonal lik values are primarily responsible for discrepancies between calculated and observed Dik values. At infinite dilution the analysis of diffusion in NaHSO, solutions is simplified by complete dissociation of bisulphate. Furthermore, departures from the dilute solution model vanish. In this limit eqn (17)-(26) provide the following exact expressions for the ternary diffusion coefficients of aqueous NaHSO, D3(HS0,) = 1.363 x and D4(SOi-) = 1.065 x Di = RTil;/(zifl2. (26) Dl D4 D, i- D, i- 40, D, + D, + 40, DYl = 6 D1(D4 - D 2 ) DY2 = 2 = 0.511 x m2 s-l = - 1.476 x 1 0-9 m2 s-I252 Mult icomponen t Diflusion of N a H SO, I 1 I - 2 I-== 0.0 0.1 0.2 0.3 c,Y2/rno1112 drn-Y2 Fig.1. Comparison of observed and predicted ternary diffusion coefficients for NaHSO,(c,)- H,SO,(c, = 0)-H,O at 25 "C: 0, 0, conductimetric data; B, limiting values calculated from eqn (27)-(30); solid line, values predicted by eqn (17)-(26). = 1.710 x m2 s-l D,(D, - 0,) DZ1 = 3 D, + D, +4D, 2 0 , D2 - D, D, + 5D2 D, Do = = 4.899 x 10 -9 m2 s-l. D, + D, -+ 4 0 , 22 As shown in fig. 1 the measured ternary diffusivities extrapolate correctly to these limits. Although dilute aqueous NaHSO, is extensively dissociated to sulphate and rapidly diffusing hydrogen ions, the diffusivity of the NaHSO, component is surprisingly small, ca.one-half as large as the binary diffusivity for aqueous Na,SO, solution~.~-~ This apparent discrepancy can be understood by recalling that diffusion of NaHSO, generates flow of sulphuric acid. Because the electric field induced by diffusion of the acid drives countertransport of Na+, the ternary diffusivity of the NaHSO, is much smaller than experience with binary diffusion would suggest. By contrast, the ternary diffusivity of H,SO, in dilute NaHSO, solutions is almost twice as large as the binary diffusivitys*g of H,SO,. In binary solutions, however, hydrogen ions are constrained by electroneutrality to travel at the same rate as less mobile bisulphate ions, whereas hydrogen ions, and thus sulphuric acid, can diffuse more rapidly in NaHSO, solutions at the expense of counterflow of Na+.The limiting cross-coefficient Oil measures the flux of H,SO, generated by a gradient in fully dissociated NaHSO,. As mentioned earlier, the flux of H+ in excess of the flux of Na+ along a gradient in concentration of NaHSO, is responsible for the coupled flow of H2S0,. From eqn (29) we see that the limiting value for D2, is directly proportional to D, - D,, i.e. the diffusivity of H+ in excess of the diffusivity of Na+. This result stresses the importance of the difference in co-ion diffusivities in determining the multicomponent nature of diffusion of a double salt.B. Wiens and D. G. Leaist 253 Conclusions Isothermal diffusion in a solution of a double salt is a ternary process. A conductimetric procedure suitable for measuring diffusion coefficients of these salts has been successfully tested on aqueous sodium hydrogen sulphate solution.Previously published data for equimolar Na,SO,-H,SO, are a source of additional information on diffusion of NaHSO,. The experimental results together with limiting diffusion coefficients derived from ionic conductances provide ternary diffusion coefficients for the salt at 25 "C from infinite dilution up to 2 mol dmP3. In this system the large mobility of H+ relative to the mobility of Na+ leads to coupled flow of H,SO, whenever NaHSO, diffuses. Below 0.1 mol dm-3 the coupled flow of H,SO, produced by diffusion of NaHSO, exceeds the parent flow of NaHSO,. The authors gratefully acknowledge financial support by the Natural Sciences and Engineering Research Council of Canada.References 1 K. S. Pitzer and J. C. Peiper, J. Phys. Chem., 1980, 84, 2396. 2 G. Scatchard and R. C . Breckenridge, J . Phys. Chem., 1954, 58, 596. 3 W. H. Stockmayer, J . Chem. Phys., 1960, 33, 1291. 4 R. P. Wendt, J . Phys. Chem., 1965, 69, 1227. 5 R. P. Wendt, J . Phys. Chem., 1962,66, 1279. 6 H. S. Harned and C. A. Blake, J . Am. Chem. Soc., 1951, 73, 5880. 7 J. A. Rard and D. G. Miller. J. Solution Chem., 1979, 8, 755. 8 M. R. Savino and V. Vitagliano, Ric. Sci., 1962, 2, 341. 9 D. G. Leaist, Can. J . Chem., 1984, 62, 1692. 10 D. G. Leaist, J . Chem. Soc., Faraday Trans. I , 1984, 80, 3041. 11 R. A. Noulty and D. G. 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