Transport in Aqueous Solutions of Group IIB Metal Salts (298.15 K) Part 4.4nterpretation and Prediction of Isotopic Diffusion Coefficients for Cadmium in Dilute Solutions of Cadmium Iodide BY RUSSELL PATERSON" AND LUTFULLAHJf Department of Chemistry, University of Glasgow, Glasgow G12 SQQ Receiued 6th May, 1977 A method for predicting isotopic diffusion coefficients for component ions in a self-complexed electrolyte has been developed. In an irreversible thermodynamic analysis the diffusion coefficient has been shown to be a function of the mobility coefficients of the free ion and its complexes together with the interionic coupling coefficients between labelled and unlabelled species. It is shown that these coefficients may be evaluated by an extension of the methods developed earlier to predict isothermal transport in such systems.Cadmium ion diffusion coefficients in aqueous cadmium iodide, Das, have been predicted (0.003-0.10 mol dm-3). These describe the maximum anticipated experimentally. Observed and calculated D, agree to within 3 % at 0.1 mol dm-3. In Paper 3,' measurements of the isotopic diffusion coefficients of cadmium-1 15 in aqueous cadmium iodide were reported. The variation of the diffusion coefficient of cadmium, D,,, with increasing concentration of salt was anomalous and had no parallel in the literature on isotopic diffusion of ions in aqueous media, fig. 2. and in earlier papers the isothermal transport properties of the unlabelled solution were reported and interpreted in terms of mobility and coupling coefficients between free and complexed ions in the solutions, using irreversible thermodynamic analyses.From the data already available it is obvious that the concentration dependence of D,, is one more consequence of self-complexing in this system. the diffusion coefficient of an isotopically labelled ion, a*, in an aqueous solution of salt (a, b) was represented by eqn (1). Aqueous cadmium iodide is extensively complexed In earlier discussions,4* D,, x loy3 = RT(L,JC,T- C,TL,*,/C,OCX) where C,' is the total molar concentration of ion a and C:, Cz are the concentrations of unlabelled and labelled species, respectively, (C,' = C:+C$). L, is the direct mobility coefficient of the ion (a) in the unlabelled binary solution (a, b/water). In eqn (1) D,, is expressed as cm2 s-l when RT, La, and Lza and concentrations are expressed in their usual units, J mol-l, mo12 J-1 c111-l s-l and mol dm-3, respectively.I,,, and the other mobility and coupling coefficients of dilute solutions of cadmium iodide have been predicted.2 It is observed that in solutions, below 0.1 mol dm-3, D,, passes through a maximum while L,,IC,T shows a minimum over this same concentration range. It is, therefore, obvious that the concentration dependence of D,, is largely dependent upon that of the isotope-isotope coupling term in eqn (1). It is the aim of this paper to obtain an explicit expression for this isotopic term as a function of coupling coefficients between labelled and unlabelled cadmium species t Present address : Institute of Physical Chemistry, University of Peshawar, N.W.F.P.Pakistan. 103104 TRANSPORT IN AQUEOUS SOLUTIONS in solution. Thereafter these interionic coupling coefficients may be evaluated, using the classical theories of Onsager, expressed in macroscopic irreversible thermodynamic form by Pikal.6 THEORETICAL Before developing an expression for isotopic diffusion in terms of the complex species present it is necessary to define nomenclature and symbols. As before,2 the complexes CdI,"-x are identified by the value of x, thus, for example, CdIZ- will be species 4. The free aqueous ions Cd2+ and I- will be denoted by a and b respectively. In any solution of cadmium iodide the total concentration of cadmium, CT, and iodide, Cz will be given by eqn (2) and (3). and (from the stoichiometry o f the salt Cz = 2C3. When a proportion of the normal unlabelled cadmium is removed from solution and replaced by an equal amount of isotopically labelled cadmium, the total concentra- tions of all species will remain unaltered, but each will now contain both labelled and unlabelled components.For any species i (i = a, 1,2,3,4), the total concentra- tion, Ci, will be the sum of concentrations cPand cT ; those of unlabelled and labelled respectively, eqn (4) The specific concentration of the labelled and unlabelled cadmium in each solution species are defined by p: and pf respectively, eqn (5) C,' = Ca+C,+Cz+C,+C4 (2) ct = C1+2Czf3C3+4C4+Cb (3) Ci = cP+cT i = a, 1,2,3,4. (4) p; = c;/C, and po = colCi i = a, I, 2 , 3 , 4 (5) (p"ppp = 1). It will be assumed that the labelled isotope is chemically identical to the normal bulk isotopic mixture of cadmium in the unlabelled solution.In consequence, there can be no isotopic enrichment of labelled species in any one of the complexes. This is a widely held assumption in diffusion studies and essentially equates isotopic diffusion as being equivalent to self-diffusion. In the present context it requires that the specific concentrations, p:, for all species involving cadmium are equal, and so the subscript i will be dropped in all further discussion. Similarly pp becomes po. The total concentration of labelled cadmium, Cz is therefore related to total cadmium concentration, C: by eqn (6) A The total contribution of unlabelled cadmium, Ct, is given by an expression analogous to eqn (6).A similar scheme for defining fluxes is required. Jay J b are the sums of the molar fluxes of all cadmium and all iodide species in solution respectively, eqn (7) and (8). Ja = ja +j, +J2 +j3 +j4 Jb = jb+jl f2j2 +3j3 +4j4. (7) (8) andR . PATERSON A N D LUTFULLAH 105 When a proportion of the cadmium is labelled, the total flux of any species will remain unaltered (by the assumption of the chemical identity of isotopes) but will now be the sum of labelled j : and unlabelled jp fluxes, eqn (9) ji = jP+j:, (9) The total flux of labelled cadmium species, J:, is the only flux to be measured experi- mentally thus : 4 J,* = jT (10) i = a and in any experiment J , = J,* + J," where J: is the net fiux of the unlabelled component. For a species i, the thermodynamic driving force which causes a direct flow is (-grad f i r ) ; the negative gradient of the electro-chemical potential of that species at any point in the non-equilibrium system, represented as xf, eqn (12) x f = -grad /it = -RTdln Cf/dx-RTd lny,/dx+Z,F(-d$/dx) (12) where Zi is the valency, including sign of the species i, and yr the molar activity coefficient.When the species is neutral, the valency is zero and the force is simply the negative gradient of chemical potential. When labelled and unlabelled species i are introduced with forces xp and $, respectively then :' (13) For unlabelled species the expression for xp is formally identical, except that the concentration and activity coefficient cp and yp replace those of the labelled species in eqn (13).Since the labelled and unlabelled species may be assumed to have activity coefficients which are equal to one another and to those of the total species in a local volume element and since dCi = dc? + dcf, from eqn (4), then from eqn (1 2) and (13), Under the special condition of isotopic diffusion, in which no bulk gradients of chemical potential exist and the sole source of non equilibrium is due to isotopic gradients; xi = 0 and so, from eqn (14), Equally there are no gradients of activity coefficients or of electrical potential ( ~ ) and so the forces xp and xT are simple functions of concentration gradients, defined (16) x? = -grad ,!iT = - RT d In cT/dx - RT d In yf/dx + ZiF( - dt,b/dx). cixi = cPxP+c'xf or x i = poxP+pox*. (14) pox; = -p*$. (15) by eqn (16) xr = - (RT/cT)( - dcf/dx) superscript F = O or *.IRREVERSIBLE THERMODYNAMIC ANALYSIS In Paper 2 the binary coefficients of cadmium iodide were expanded in terms of mobility and coupling coefficients of the uncomplexed and complexed species present in solution. For any species k, linear phenomenological equations were written, eqn (17). A106 TRANSPORT I N AQUEOUS SOLUTIONS Eqn (17) deals with an experimental conditions likely to be obtained in normal studies and it is assumed here and in all subsequent discussion that the Onsager Reciprocal Relations (O.R.R.) are obtained and so Iki = Zkf. If it is assumed that the experimental conditions to which eqn (17) refers are again obtained with the same solution in which now labelled cadmium isotope is present, then, from eqn (9), the sum of the flows of labelled and unlabelled isotopic species, k, are equal to the net flow, jk ; ( j , = jkq+jz).The phenomenological equations for isotopic flows will define a symmetrical 11 x 11 matrix of mobility coefficients, since five of the six flows (and six forces) of eqn (17) are now subdivided into two components. The phenomenological equations of the isotopic system are given by eqn (1 8)-(20) 4 j," = (jkiXp+?ki*$)+~k,~, ( k = a, 1, 2, 3, 4) (18) i = a and for b, which was unlabelled, From eqn (9), (18) and (19), A or, using eqn (14), Comparing terms in eqn (17) and (22), having noted that lkfxi = &@oxp+~~fp*x~, from eqn (14) then, and lki = (jki+?k*i)/po = ( j k i + + j k * i * ) / P * i and k = a, 1, 2, 3, 4 (23) (24) Ibb = 'bb (25) lbi = &i/Po = ?bi*/P* i = a, 1,2,3,4.(26) (27) zkb = (jkb+jk*b) k = a, 1, 2, 3, 4. Since the force on iodide, x b , is unaltered, under conditions when only cadmium species are labelled, then from eqn (14), (17) and (20), and From eqn (24) and (26), using the Onsager Reciprocal Relations, lkb = j,b/po = jk*b/p*. Eqn (23), (25) and (27) establish specific relationships between the mobility coefficients of the unlabelled system, eqn (17) and those of the isotopic matrix, eqn (18)-(20). We must now formulate expressions for the total flow of labelled cadmium .I,*, using eqn (lo), under the more restricted conditions of isotopic diffusion, in which no bulk chemical potential gradient exists and so all xi and j , are zero in eqn (17).R. PATERSON AND LUTFULLAH 107 Under these conditions xb, the force on iodide ion, is also zero and hence from eqn (lo), (18) and (19), Since, from eqn (14), p*xT = -pox; when xi = 0, and from eqn (23) From the analysis of the concentrations of complexes to the binary coefficients, Paper 2,2 A d where La, is the direct mobility coefficient of cadmium in the binary solution for which the phenomenological equations are : and Recalling the isotope-isotope terms in eqn (28), expansion of the summation shows that, when the Onsager Reciprocal Relations are assumed, eqn (30) is obtained.(30) Ja = Laaxa $- la bXb J b = LbaXa LbbXb. 4 4 4 4 (jki*ip*+jk*i/po) = x C (Iki*/Po P*)* k = a i = a k=a i = a From eqn (28)-(30), 4 4 J,* = RT[L,,- k=a i=a 1 (lki*/pop*)](-dp*/d.). When Cz, total cadmium concentration, is constant at all points in the solution (as it must be for isotopic diffusion) then from eqn (6), and eqn (31) becomes, dp*/dx = (l/Cz)dC:/dx (32) Eqn (33), therefore, has the form of Fick's first law of diffusion, eqn (34), The flow JZ is that measured by experiment and thus Da, is the isotopic diffusion coefficient obtained from such experiments.Comparison of eqn (33) and (34) gives, J,* = Daa( - dC,*/dx). (34) r 4 4 1 Eqn (35) and (1) reveal that expressions for isotopic diffusion coefficients in a complex or a simple dissociated electrolyte have identical forms. It may be noted that, from eqn (5), the term C,TLa,*/(C,OC,*) in eqn (1) is equal to (l/C,T)(Laa*/P*p*) making the108 TRANSPORT IN AQUEOUS SOLUTIONS identity more obvious.The isotope-isotope coupling term, which is a function of a single coupling coefficient Laas in eqn (l), is now replaced by a summation of coupling coefficients which include all possible interactions between labelled and unlabelled foriix of the cadmium containing species in the complexed solution. PREDICTION OF ISOTOPE-ISOTOPE COUPLING AND DIFFUSION COEFFICIENTS FOR CADMIUM As discussed earlier in relation to eqn (l), it is the complex variations in the function C,’L,,*/C,”C,* which determines the major trends in the concentration dependence of Daa. The direct mobility Laa/C: shows quite the inverse concentration dependence of D,, [fig. 1 of ref. (3)J In the theoretical section it was shown that the isotope-isotope coupling contribu- tion to Daa may be expressed as a summation, by comparing eqn (1) and (35), c:L,,*/C:c: = (L,,,lpop*)/c: 4 1 FikalY6 in his analysis of coupling coefficients, has used the Limiting Laws of Fuoss and Onsager to obtain a general expression for such coefficients, eqn (37), 10 2, /\/Cx = A,/p ppt [ B]l’.(37) The units of are those used in this study (moI2 J-l cm-l s-l) and concentrations are expressed in molar units. A is a constant (0.107 40), [B] is a combination of relaxation and electrophoretic terms, discussed below and I is the true ionic strength of the solution, eqn (38), (38) 11: is defined by Pikal as the ionic strength fraction, eqn (39), where the superscript f- may be either O or * for unlabelled and labelled components respectively : (39) Thus in eqn (37), I = 3 2 (cj”+cri;)Zj” ( j = a, 1,2, 3,4, b).pJ = cjzj/c (cg+cj*)zj = c;zj/zr. dppp: = JcpcE 1Zizk[/21. (40) From eqn (37), (40) and ( 5 ) , eqn (41) is obtained, which has the same mathematical form as eqn (35) The term [B] used in these equations is defined by Pikal, eqn (42) where terms in Aik* and Bo involve the relaxation and electrophoretic effects respect- ively. In eqn(42), x = &‘&‘/lZiZkl. It is assumed from the postulatedidentical chemical characteristics of labelled and unlabelled species that Aj’ = A;* where A; is the equivalent conductance of the ion at infinite dilution, and y = E/ijA;/lZjl, where the summation is over all species, labelled and unlabelled, in the solution. From eqn (39) and (4), [B] = [ x / v Aik*-(BO/2) z,&] (42)R. PATERSON AND LUTFULLAH 109 Pikal has shown that Aik* = alZiZkl where a at 298.15 K is a constant, 0.229 62.Similarly Bo is a constant, 60.495. Thus eqn (42) becomes, Combination of eqn (37) and (44) gives : (relaxation term) (- lzicil I z k c k l (ZiZk)AB,/4)I-'. (45) (electrophoretic term) It is to be noted that all terms on the right hand side of eqn (45) may be evaluated without a knowledge of the absolute concentration of isotopically labelled cadmium. Only total concentrations and equivalent conductances at infinite dilution of free and complexed ions need be known. These have been evaluated in a similar but independent analysis of the transport properties of the unlabelled salt.2 Pikal's (S.L.L.) analysis is based upon the Limiting Laws and so is precise only in very dilute solutions.Equally, since it deals with interionic coupling coefficients, there can be no satisfactory method for estimating coupling between the neutral complex (2 or 2*) and its environment of ions. This means that all Z2j* and Z2+j are indeter- minate and must be set equal to zero. This limitation was also encountered in the analysis of the unlabelled salt.2 Such omissions may cause little error however, since in dilute solutions, neutral complex amounts to no more than 5 % of the total cadmium present .z In Paper 2,2 concentrations of free and complexed ions have been obtained as a function of total salt concentration. In that paper also an optimisation procedure for estimating the 1; for complexed ions was reported. Using these data, the isotope-isotope coupling terms of eqn (36) were calculated in the concentration range C:, (0.003-0.1 mol dm-3).For clarity eqn (36) has been expanded [eqn (46)] and the individual coefficients displayed in this format in table 1. In eqn (46) the parameters lik*/pop" are represented as (jk::) 4 4 1 (~ki*/Pop*>/c~ k=a i = a = 1ICi [(aa*) + (a1 *) + (a2*) + (a3*) + (a4*) + (1 a*) + (1 1 *) + (129 - (1 3*) + (14*) + (2a*) + (21 *) + (22*) + (23") + (24*) + (3a*) +(31*) +(32*) +(33*) +(34*) + (4a*) + (41 *) + (42*) - (43*) + (44*)]. (46) Since lik*/pop* equals lki*/pop*, by the Pikal analysis, eqn (45), the matrix of coefficients is symmetrical. Electrophoretic terms in eqn (45) dominate, and so for ions of like charge lik*/pop* will be negative, since (Z,Z,) is positive. These negative terms appear at the upper left and lower right hand corners of the matrix, eqn (46).For the same reasons coefficients between ions with opposite charges are positive and these appear110 TRANSPORT IN AQUEOUS SOLUTIONS on the upper right and lower left corners, eqn (46). Representative points are shown in table 1 and a complete set of predictions over the full concentration range are given in table 2. From table 1, it is observed that in the most dilute solutions, where higher (negatively charged) complexes are not present in significant proportions, the negative contributions to (Laa*/pop*)/C~ are largely [(aa*) + 2(al*) + (1 1 *)I. Only at the highest concentrations do the negative-to-negative ion interactions, [(33 *) + 2(34*) + (44*)] contribute significantly.Positive contributions to the total isotope-isotope coupling term are solely 2[(3a*) + (31 *) + (4a*) + (41 *)] and these are important only when the higher negatively charged complexes (3 and 4) are significant. ( -La,*/pop*)/C,T increases rapidly from its value of zero at infinite dilution largely because the only significant cadmium species are Cdlf (1) and cadmium ion itself (a). As complexing increases with further TABLE 1 .-COMPONENT ISOTOPIC MOBILITY COEFFICIENTS (ik*) AS DISPLAYED AND DEFINED BY EQN (46) AT REPRESENTATIVE TOTAL CONCENTRATIONS OF CADMIUM IODIDE, C:, (mol dm-3) [as in eqn (45) each coefficient must bemultiplied by the factor x to obtain the individual -7.405 97 (-4) -1.539 19 (-4) 1.560 66 (-6) 2.273 28 (-7) - - 1.133 92 (-3) - 3.387 95 (-4) 7.823 27 (-6) 1.81425 (-6) - - 1,873 03 (- 3) - 8.736 12 (-4) 5.762 32 (- 5) 2.453 72 (- 5) - - 4.057 79 (- 3) - 3.095 45 (- 3) 7.723 56 (-4) 7.470 19 (-4) - - 6.399 45 (- 3) - 5.407 69 (- 3) 2.060 94 (- 3) 2.687 05 (-3) - zik* in units mo12 J-' cm-l s-'1.C,'= 0.003 -1.539 19 (-4) - 1.560 66 (-6) -3.165 54 (-5) - 3.484 24 (- 7) 3.484 24 (-7) - -1.551 38 (-9) 5.053 51 (-8) - -2.416 24 (- 10) - - - C,' = 0.005 -3.387 95 (-4) - 7.823 27 (-6) -1.001 97 (-4) - 2.507 70 (-6) 2.507 70 (-6) - -2.582 72 (-8) 5.79094(-7) - - 6.394 98 (-9) - - - c,T = 0.01 -8.736 12 (-4) - 5.762 32 (- 5) -4.034 15 (-4) - 2.880 00 (- 5) 2.880 00 (-5) - - 8.593 36 (- 7) 1.221 26 (-5) - - 3.901 97 (- 7) - - - C z = 0.03 - 3.095 45 (- 3) - 7.723 56 (-4) -2.337 11 (-3) - 6.323 72 (-4) 6.323 72 (-4) - - 7.000 18 (- 5) 6.090 37 (-4) - - 7.232 52 (- 5) - - - C: = 0.05 -5.407 69 (-3) - 2.060 94 (- 3) -4.519 86 (-3) - 1.875 16 (-3) 1.875 16 (-3) - - 3.050 85 (-4) 2.434 15 (-3) - -4.264 28 (-4) - - - 2.273 28 (- 7) 5.053 51 (-8) -2.416 24 (- 10) -3.733 35 (- 11) - 1.814 25 (-6) 5.790 94 (-7) - 6.394 98 (- 9) -1.571 23 (-9) - 2.453 72 (- 5) 1.221 26 (- 5) - 3.901 97 (- 7) - 1.758 45 (- 7) - 7.470 19 (- 4) 6.090 37 (-4) - 7.232 52 (- 5) -7.414 38 (- 5) - 2.687 05 (- 3) 2.434 15 (- 3) -4.26428 (-4) -5.910 51 (-4) -R .PATERSON AND LUTFULLAH 111 TABLE 1 .-corztd. C z = 0.07 -9.177 09 (-3) -7.930 51 (-3) - 3.744 54 (-3) -7.930 50 (-3) -6.774 58 (-3) - 3.494 12 (- 3) - - - - 3.744 54 (-3) 3.494 12 (-3) - -6.802 83 (-4) 5.811 05 (-3) 5.398 14 (-3) - - 1.135 48 (-3) c: = 0.09 -1.243 73 (-2) -1.069 78 (-2) - 5.755 54 (-3) - 1.069 78 (-2) -9.091 26 (-3) - 5.358 42 (-3) 5.755 54 (-3) 5.358 42 (-3) - - 1.154 53 (-3) 1.007 95 (-2) 9.341 12 (-3) - -2.180 00 (- 3) - - - - c: = 0.10 - 1.425 31 (-2) -1.217 49 (-2) - 6.871 61 (-3) - 1.217 49 (-2) -1.027 27 (-2) - 6.359 90 (-3) 6.871 61 (-3) 6.359 90 (-3) - - 1.419 32 (-3) 1.263 72 (- 2) 1.164 22 (-2) - -2.818 65 (-3) - - - - 5.811 05 (-3) 5.398 14 (-3) - 1.135 48 (-3) - 1.878 39 (- 3) - 1.007 95 (-2) 9.341 12 (-3) -2.180 70 (-3) - 4.080 39 (- 3) - 1.263 72 (- 2) 1.164 22 (-2) -2.818 65 (-3) -5.544 04 (-3) The individual isotopic mobility coefficients, (ik*), are represented with the power of ten bracketed, thus - 6.399 45 (-3) equals -6.399 45 x loe3.The isotope-isotope term, (l/Cz)(Laa*/pop*),.of eqn (46) is obtained by summation of the coefficients of each matrix divided by the correspondmg value of C,T. increase in concentration of total salt, the positive-to-negative ion coupling interactions become significant. This causes (-Laa*/pop*)/Cz to pass through a maximum and subsequently decrease, see fig. 1, tables 1 and 2. TABLE 2.-PREDICTED ISOTOPIC DIFFUSION COEFFICIENTS FOR CADMIUM IN AQUEOUS CADMIUM IODIDE mol dm'3 0.003 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.09 0.10 -3.585 7 - 3.772 6 -3.779 1 -3.202 7 -2.451 0 - 1.727 0 - 1.073 8 -0.489 8 0.036 19 0.949 86 1.354 55 2.738 2 2.757 9 2.790 3 2.858 7 2.932 8 2.999 3 3.055 1 3.101 5 3.140 4 3.200 6 3.224 3 7.677 7.772 7.854 7.882 7.878 7.863 7.840 7.810 7.776 7.698 7.657 Values of (1 /Ca Laa*/pop* were obtained from eqn (45) and (46), using concentrations of individual ions in solution and their corresponding equivalent conductances at infinite dilution, A:, as obtained in Paper 2.' (The optimised values of A; for the complex ions were A;, 33.51 ; A;, 39.70 and A:, 68.67 cm2 s1-' equiv-I). The coefficients LaaIC,T given in the third column of this table are those predicted previously where LaalC,' = 2LaallV in the notation of ref.(2). The dimensions of Laa and Laa* are moI2 J-' cm-I S-I.I12 TRANSPORT IN AQUEOUS SOLUTIONS In terms of these Limiting Law predictions the additional positive contributions [(33*) + 2(34*) + (44*)] never overcome this downward trend. Above 0.2 mol dm-3 the experimental data increase once more and pass through a second maximum, fig.1. Such concentrations are far beyond the range of classical theory and it is not surprising that this analysis cannot reproduce them. I 9 - 2.0 ! c. 2 0.4 0.6 dC FIG. 1.-Isotope-isotope coupling terms in eqn (1) and (35) as a function of the square root of rnolarity of cadmium iodide, C,T. Experimental values obtained in Paper 3 and calculated points, from table 2, are represented as and 0 respectively. In Paper 2 the binary coefficients for cadmium iodide were predicted using a similar analysis and calculated values of L,,/C,T obtained. (These were represented as Laa/Ca in that paper.2) This intrinsic mobility coefficient passes through a minimum in dilute solutions, while D,, shows an initial rise to a maximum in the same range.Since both terms in eqn (1) have now been predicted, using the same optimised values for A:, it was of interest to predict the cadmium ion diffusion coefficient, D,,, purely by these theories. The results are given in fig. 2 and table 2. Calculated diffusion coefficients are shown to increase from infinite dilution and pass through a maximum as inferred from experimental data.l Although the method is limited to dilute solutions, the predicted diffusion coefficient is only 2.5 % lower than observed at 0.1 mol dm-3. This method of calculation does not provide an insight into the basic causes of the initial maximum in D,,, shown in fig. 2. From eqn (45) of this text and eqn (20) of Paper 2, it is easily shown that Zik = Iik/p0p* i # k. (47) From eqn (47), (29) and (35) D,, may be expressed as a summationR.PATERSON AND LUTFULLAH 113 where ai equals C,/C,', the proportion of total cadmium present as complex i. dil have the form of diffusion coefficients and are defined by eqn (49) The functions dir are the isotopic diffusion coefficients of the complexes (i). These would be obtained by hypothetical experiments in which complex i alone was labelled and was no longer in dynamic equilibrium with the other cadmium species in solution. 8.2 i 'm N 7.8 ---. \o 2 X ce < 7: 4 7.0 I 1 I I I I 0,2 0; 4 0:6 FIG. 2.--Isotopic diffusion coefficients Daa x lo6 (cm2 s-l) for lr5Cd2+ in aqueous cadmium iodide as a function of the square root of molarity of cadmium iodide, C,T. Measured coefficients obtained previously and those calculated here (table 2) are represented as 0 and 0 respectively.D,, is, therefore, the weighted sum of these diffusion coefficients. At infinite dilution & = d i = RTIIP/IZ,IH2 and from eqn (45) and eqn (20)2 it is easily shown that I I1 (relaxation term) Eqn (50) contains no electrophoretic terms, showing that for a complexed electrolyte, as for dissociated ones, isotopic diffusion is affected only by relaxation contributions. Using limiting equivalent conductances estimated previously,2 d& dTl, d3\ and d& are 7.12 x 8.92 x 10.57 x and 9.14 x respectively in units of cm2 s-I. Each of the complexed ions has a larger diffusion coefficient at infinite dilution than free cadmium ion itself. As concentration is increased and complexation becomes significant (table 2)2 the first summation in eqn (50) increases progressively.Relaxation terms, although zero at infinite dilution, make an increasingly negative contribution, causing D,, to describe a maximum. The numerical output of this calculation is of course identical to that presented in fig. 2 and table 2 and so tabulation of the terms in eqn (50) is not presented.114 TRANSPORT I N AQUEOUS SOLUTIONS What may be concluded from these limiting law predictions is that although all possible coupling interactions between ions have been considered, coupliiig coefficients between chemically different species cancel in eqn (49, reducing the thirty basic mobility coefficients to only eight. These in turn may be grouped to define the four isotopic diffusion coefficients of cadmium and its complexes, each making a propor- tional contribution to D,,. This calculation, taken with those earlier predictions of binary electrolyte transport in cadmium iodide solutions,2 shows that the classical concepts of transport may be applied with considerable success to the predictions of both isotopic diffusion coefficients, transport numbers, conductance and salt diffusion in this complexed electrolyte. We are grateful to the Ministry of Education of the Government of Pakistan for a Research Scholarship to Lutfullah during the period of this work. R. Paterson and Lutfullah, Paper 3, J.C.S. Furaday I, 1978, 74, 93. R. Paterson, J. Anderson, S . S. Anderson and Lutfullah, J.C.S. Faruduy I, 1977, 73, 1773, Paper 2. R. Paterson, J. Anderson and S. S. Anderson, J.C.S. Faraday I, 1977, 73, 1763, Paper 1. J. Anderson and R. Paterson, J.C.S. Furaduy I, 1975, 71, 1335. S. Liukkonen, Acra Polytechnica Scund. (Chemistry including Metallurgy Series), No. 11 3, Helsinki, 1973. M. J. Pikal, J. Phys. Chem., 1971, 75, 3124. 0. Kedem and A. Essig, J. Gen. Physiol., 1965, 48, 1047. R. Paterson, Furaday Disc. Chem. SOC., 1978, 64, in press. (PAPER 7/770)