J.C.S. Faraday I, 1980,76,2552-2557Electrolyte Diffusion at Very Low Concentrations inIonized WaterBY PENTTI PASSINIEMI," SIMO LIUKKONEN AND ZOLTAN NOSZTICZIUS-/-Helsinki University of Technology, Department of Chemistry,SF-02150 Espoo 15, FinlandReceived 6th November, 1979The diffusional behaviour of an electrolyte Mv +Xv- inwater is studied up to concentrations wherethe effect of the ionization must be taken into account. The mutual diffusion in this system changesdue to the dissociation of water under the decreasing electrolyte concentration, first to multi-component diffusion and then, at concentrations < lo-* mol dm-3, to tracer diffusions of the electro-lyte ions.By applying Nernst-Planck equations the approximate values of the proper difiusion coefficientsare calculated.These rather simple calculations confirm the phenomena which were observedexperimentally by Mills at low concentrations in the system H20+ MgBrz at 298.15 K.Very little attention has been paid to transport phenomena in very dilute aqueoussolutions of electrolytes where the ionization of water can no longer be neglected.Measurements in these solutions are often very difficult and, from the theoreticalpoint of view, the reference velocity may give rise to questions.However, there is one very interesting series of measurements from 1962, in whichthe diffusion has been studied up to a concentration of 3 x 10-8moldm-3 usingradioactive 82Br in the analysis. In these experiments of Mills,l magnesium bromidediffused out of a capillary into pure water.The results of this series were analysedtheoretically by Woolf.2As the treatment of the transport processes in very dilute aqueous electrolytesolutions raises some fundamental questions we want to consider this problem anew.Qualitatively the diffusion in such a system is clear as has already been discussed byMills' and Woolf et aL2* In concentrated binary solutions a mutual diffusionprevails but, on the other hand, in very dilute solutions (< mol dm-3) thereexist one or more tracer diffusions. Between these extreme cases or in the inter-mediate region the concentrations of H+ and OH- ions due to the water are com-parable with the concentration of the electrolyte. Then there is essentially a multi-component system and coupled diffusions appear.To get a closer and more quantitative description of these diffusion phenomenawe will apply Nernst-Planck equations to the ionic flows.The component flowswith proper diffusion coefficients are then estimated in terms of the ionic molar con-ductivities at infinite dilution.THEORYThe isothermal-isobaric system to be considered is composed of asociated electrolyte Mv,Xv- in partially ionized water. The electrolytest Present address : Technical University of Budapest, Budapest, Hungary.2552fully dis-dissociaeP. PASSINIEMI, S . LIUKKONEN AND 2. NOSZTICZIUS 2553according to eqn (1) and (2) :M,+X,- + v+MZ+ +v-X"-- (1)H 2 0 + H++OH-. (2)Kw = C H C o H = mo12 dm-6. (3)The ionic product of water is at 25°CFor simplicity, the symbol for an ionic species is written without charge, e.g., Hinstead of H+, etc.In studying the diffusion of the electrolytes the approximations of Nernst-Planckequations are applied to the ionic flows in the concentration range to be discussed.The flows are 4,-ji = Di(VCi +ziFCiV4/RT) i = H, OH, M, X,where j i , Di, Ci and zi are, respectively, the flow, the diffusion coefficient, the con-centration and the charge of the ionic species i.4 is the so-called diffusion potential.Di is defined by(4)Di = RTA?/(z?P), ( 5 )in which A? is the ionic molar conductivity at infinite dilution. We assume that theseconstant A? values can also be applied to ionic conductivities at very dilute con-centrations.In order to describe the diffusion in the most general case we start with the ternarysysteminstead of the binary one defined by eqn (1) and (2).* In Nernst-Planck equationsof this system V4, VC, and VC,,, respectively, are eliminated by the conditions ofthe zero electric current densityH2O + H,z-,X(l) + M(OH)z+(2) (6)X _ _ziFji = 0, (7)i = Hthe constancy of the ionic product of water [eqn (3)] in the formand the electroneutrality in the formX c zivc, = 0, (9)i = HIn eqn (7) and (9) the sum includes all the ions, i.e., H, OH, M and X.The ionic flows are thusX-jk = D, C (~ik-(ZiZkCk/X)[Di-(KDOH+ ~ , ) / ( l +K)I}VCi k = M, X, (10)(11)i = MX-j, = [D,Ai/(1 +K)I 1 Zi(1 -(CHIX)[(KDO,+ ~ d - ( 1 +K)DiI}VCii = MI = H,OH and AH = - 1 , AOH = K.* It is necessary to make clear the difference between the total concentration of the hydrogen ionsand the concentration C1 of the component Hlz-1X [in eqn (6)].It means in this treatment thatCH # 12-1 CI. The same is also true for the hydroxyl ions2554 DIFFUSION AT VERY LOW CONCENTRATIONSThe sum in eqn (10) and (11) is taken only over the ions M and X. &k in eqn (lo)is a Kronecker delta and K and x in eqn (10) and (1 1) are, respectively, given byIf we choose for the independent flows jM and j , we can interpret the coefficientsof VCM and VC, in eqn (10) to be the four diffusion coefficients of the ternary system(6). These diffusion coefficients are concentration dependent and they include theeffect of the dissociation of water. Thus the flows areMUTUAL DIFFUSION CM, C, CH, COH ; Z+CM = -2-c~With these conditions we have&/v+ =jx/v- = JMy+xv- ; j H = j o H =I 0,and the approximate mutual diffusion coefficientequals the limiting value of N e r n ~ t .~DMly+Xv- = [(.+ -z-)DMDXl/(z+DM-z-DX)TRACER DIFFUSION CM, Cx < CH, C ~ HIn this case we see from eqn (14) thatThere are two independent tracer flowsDkf = 6 k i D k i, k = M, x. (1 7)(184( W- j M = -JM(OH)z+ = DMvcM.-jx = -JH,.-IX = D X V C Xin which the tracer diffusion coefficients DM and Dx are given in eqn (5).case both j , and joH approach zero.In thisINTERMEDIATE REGION CM and Cx comparable to CH and C,,We now have ternary diffusion in system (6) with non-zero cross-diffusion coefficients.There are also non-zero concentration gradients for H and OH ions (pH is notspatially constant).THE SYSTEM H20+MgBr2We have chosen this system as an example to which we will apply the abovetreatment.The system is the only one for which there are experimental data fromthe binary (mutual) diffusion to the tracer diffusion. In calculations the followingvalues were used for @' :6 A"(H+) = 349.81 ; A"(0H-) = 197.8; A"(Mg2+) =106.10; j103(Br-) = 78.14 cm2 Q-' mol-'P . PASSINIEMI, S . LIUKKONEN AND Z . NOSZTICZIUS 2555The diffusion coefficients thus obtained are given in table 1. They have beencalculated in the whole concentration region at the point which corresponds topH = 7.00. C in table 1 is the concentration of MgBr, at this point.TABLE 1 .-ESTIMATED DIFFUSION COEFFICIENTS IN THE SYSTEM H20 + MgBrz AT 298.15 KC" Dllb D12b D21b D22b, DBrb DMgb DBrMg?10-310-410-510-710-91 o-6lo-*10-l05.1885.1825.1254.6523.0872.2232.0962.082- 7.852 - 1.054- 7.837 - 1.052- 7.693 - 1.033- 6.496 - 0.872-2.542 -0.341- 0.359 - 0.048- 0.037 - 0.005- 0.003 - 0.0013.370 1.2623.365 1.2623.3162.9101.5690.8280.719 2.077 0.7090.706 2.082 0.706~~ ~ ~~~~~~a Units : mol drr3 ; units : lop9 m2 s-l.The mutual diffusion coefficient DMgBr2 is determined from eqn (16) but it can alsobe obtained by combining Dki, under the assumptions of the mutual diffusion caseabove, as follows :DMgBrz = *(2Dll + O12) = (2D21 + D22)- (19)In eqn (19) subscript 1 denotes, according to system (6), the component HBr and 2the component Mg(OH),. At concentrations z mol dm-3, DI2 and DZl arenearly zero.If we assume that the concentration gradients VC, and VC,, are alsozero we can combine Dki to find the approximate values for the tracer diffusioncoefficients DMg and D,, as follows :These values are given in table 1.The mutual diffusion coefficient DMgBr2 can be determined by labelling, e. g.,Br ion (or Mg ion) in MgBr, and by applying the diaphragm-cell or open-endedcapillary method. The measurements for the mutual diffusion coefficient can bemade as given in table 1 up to concentrations z mol dm-3 and from these resultsthe limiting value of Nernst can be reached with the usual extrapolation.On the other hand, the limiting values of the tracer diffusion coefficients DMg(Mg Iabelled in MgBr,) and DBr (Br labelled in MgBr,) are obtained with two differentmethods.Firstly, under the conditions of the usual tracer- (or better the self-)diffusion experiment where the total concentration of the electrolyte is constant.The limiting values can almost be found at concentration z mol dm-3. Thedissociation of water has no effect in this case. Secondly, the same limiting valueDBr is achieved by measuring the diffusion of Br-labelled MgBr, or of any otherelectrolyte containing Br ion into pure water at concentrations < mol dm-3.Correspondingly, D,, is obtained with Mg-labelled MgBr, or with any other electro-lyte containing Mg2+-ion.The four diffusion coefficients in the intermediate region are also, in principle,measurable with the open-ended capillary method by applying, e.g., the ideas ofToukubo and Nakanishi.'DBr *(2D11 +D12) ; DMg z 2D21i-0222556 DIFFUSION AT VERY LOW CONCENTRATIONSDISCUSSIONTo complete our analysis of the diffusion in the system H20 + MgBr, we have alsocalculated the diffusion coefficients in the systems H20 + HBr and H 2 0 + Mg(OH),.In both cases the treatment above shows that there is only one independent flow in thewhole concentration region.In the former system the flow of Br- is given by-jBr = -JHBr = DBr(l + ( c H B r / x > ( l + K)-l[(KDO€l + DH) -+K)DBrl)VCHBr (20)and in the latter system the flow of Mg2+ is given by-jMg = -JMg(OH)2 = +(4cMg(OH,,/x)(1 + K)-l+ DH) - (l + K)DMg]}vcM~(OH)2- (21)In eqn (20) and (21) we have only one measurable diffusion coefficient in the wholeconcentration region.These coefficients could be determined, e.g.? with the open-ended capillary method by using the labelled Br- (or Mg2+) ion for analysing thechanges in the electrolyte concentration. At higher concentrations (up to zi moldm-3) the value measured approaches the limiting value of the mutual diffusioncoefficient. At concentrations under mol dm-3 this mutual diffusion coefficient,i 3.0 iI1 .o]-10 - 8 - 6 -4log (C/mol dm-3)FIG. 1.-Calculated diffusion coefficients at very low concentrations. The full lines (a), (c) and ( d )are, respectively, the Onsager limiting laws of the aqueous binary diffusion for HBr, Mg(OH), andMgBr,. The full lines (6) and (e) are the Onsager limiting laws for the tracer diffusion of Br- andMg2+ in aqueous MgBr2.The broken lines (f) and (9) describe the change of the binary diffusioncoefficient in aqueous HBr and Mg(OH)2, respectively. The dotted lines (h) and (i) give the sums(Dll+0.5 D12) and (2021+022) in the system H20+MgBr2 with respect to the concentration atpH = 7.00 [see also the text after eqn (19)]P . PASSINIEMI, S . LIUKKONEN AND Z . NOSZTICZIUS 2557however, ought to change quite rapidly according to eqn (20) and (21) and in a solu-tion concentration z mol dm-3 the values of the tracer diffusion coefficients(DBr and DMg) could be reached. This behaviour can be seen in fig. 1 where thediffusion coefficients in the system H,O+MgBr, are also shown.There one seeshow MgBr, is divided into HBr and Mg(OH), as its concentration decreases. Notethat the traditional extrapolation in mutual diffusion coefficients is made from experi-mental results in the concentration range of z 10-3-5 x mol dm-3 to zeroelectrolyte concentration. The limits thus obtained do not contain the effect dueto the water dissociation.Eventhe small maximum in the diffusion coefficient could be explained by the pH-changedue to the possible adsorption of Mg-ions on plexiglass. BeneS and KopiEka *have shown that especially divalent cations are adsorbed readily on plexiglass invery dilute solutions. Unfortunately, Mills measured only the diffusion in the systemH,O + MgBr, where the interpretation of his diffusion coefficients in the region105-10-8 mol dm-3 is not unambiguous [cf.eqn (10) above].If we compare the diffusion coefficients given in our table 1 with those in table ofWoolf’s paper there are quite large deviations. Woolf used the method of Wendtfor the approximate treatment in system (6). However, in principle that method isanalogous with the Nernst-Planck equations. O In Woolf ’s paper the treatment ofdiffusion is divided into three regions. It seems to us that the connection of themutual and the tracer diffusion regions with the intermediate region is not clear.In the present paper the gradual change from the mutual to the ternary diffusionand further to the tracer diffusion is obvious.We want to emphasize that the Nernst-Planck equations are also written here tohydrogen- and hydroxyl-ions.This means the electric conductivity of the systemalso includes the part due to the dissociated water as given in eqn (12b). Therefore,if the concentration of the electrolyte MX goes to zero the transport numbers ofMZ+- and Xz--ions also go to zero. This corresponds to the result in molten saltsbut now the non-measurable transport numbers of H+ and OH- in pure water donot equal one and zero but are related to their ionic conductivities.To test the above derivations for the diffusion coefficients one would need diffusionmeasurements in very dilute solutions especially in systems of the form H20 + H,,-,Xand H 2 0 + M(OH),+.Our calculations confirm the main points of Mills’ experimental data.lWe acknowledge scholarships made available by the cooperation between theP. P. thanks the Finnish Cultural Technical University of Budapest and Helsinki.Foundation and S . L. the Finnish Academy for the scholarships.R. Mills, J. Phys. Chem., 1962, 66, 2716.L. A. Woolf, J. Phys. Chem., 1972,76, 1166.L. A. Woolf, D. G. Miller and L. J. Gosting, J. Amer. Chem. SOC., 1962, 84, 317.W. Nernst, 2. phys. Chem., 1888, 2, 613 ; 1889, 4, 129.M. Planck, Ann. phys., 1890, 40, 561.R. A. Robinson and R. H. Stokes, Electrolyte SoZutions (Butterworths, London, 2nd edn,1959), p. 463. ’ K. Toukubo and K. Nakanishi, J. Phys. Chem., 1974,78,2281.* P. Bene’s and K. KopiEka, J. Inorg. Nuclear Chem., 1976, 38, 2043.R. P. Wendt, J . Phys. Chem., 1965, 69, 1227.lo H.-J. Schonert, 2. phys. Chem. (Frankfurt), 1967, 54, 245.l 1 C. Sinistri, J. Phys. Chem., 1962, 66, 1600.(PAPER 9/1776