JOURNALO FTHE CHEMICAL SOCIETYDALTON TRANSACTIONSInorganic ChemistryThermodynamics of Extraction Equilibria. Part 5." Correction of theMethod for the Determination of Equilibrium Constants of ExtractionProcessesBy Djordje M. PetkoviC, Chemical Dynamics Laboratory, The Boris KidriE Institute of Nuclear Sciences, P.O.Box 522,11001 -Beograd, YugoslaviaA previously established method for the determination of extraction constants has been verified for extractionprocesses in which a 1 1 solvate is formed In the case of formation of a higher solvate, the method has to becorrected so that the extraction equilibria are represented in terms of chemical equivalents. Published distributiondata on the extraction of nitric acid and uranyl nitrate with tributyl phosphate are used to illustrate the validity of themethod.IN previous Parts 1-4 of this series a method was reportedfor the determination of thermodynamic extractionconstants and its application to various extractionsystems. In Part 2 the method was described startingfrom the general extraction equilibrium (l), where M isMz+ + zA- + NSorg.+ MAz*nSorg. (1)the metal ion, A the anion, S the solvent molecule, andz the positive charge. The subscript org. denotes theorganic phase, whereas for the aqueous phase the sub-script is omitted. The expression (2) for the thermo-dynamic equilibrium constant of reaction (1) wasaMAS(org.)aMaAZanS(ory.)K =transformed into (3) where a represents the chemicalactivity. Charges and solvation numbers are omittedfrom the subscript to simplify the notation.The methodis based on the fact that equation (3), when plotted as thefunction log(l/aMaAz) = f [ a ~ ~ ~ ~ ( ~ ~ ~ : . ) l , has an inflexionpoint at ans(org.) = aMAS(org.) and reduces to log( l/aMaAZ) =log K. It was later found that consideration of thesecond derivative of equation (3) could lead to ambi-guities. Namely, the differentiation of the additional* Part 4 is ref. 4.Part 1, Dj. M. Petkovid, B. A. Kezele, and A. Lj. Ruvarac,in ' Contribution t o Co-ordination Chemistry in the Solution,'ed. Erik Hogfeldt, Swedish Natural Science Research Council,Stockholm, 1972, p. 435.condition, avLg(org.) = aMAs(org.), which has to be fulfilledat one point, is questioned. In order to avoid thisproblem it was decided to consider equation (3) in anotherway.THEORYFor simplicity, equation (3) may be restated in the generalDue to the presence of two unknown variables, form (4).y = log K + log ( ~ ~ 1 % ) (4)z and x, and their unknown inter-relation, it is not possibleto find a condition for an inflexion point if equation (4)is plotted as y = f ( x ) .It is known, however, that inextraction equilibria, if the total concentration of extractantis kept constant, z increases with decreasing x and viceversa. Furthermore, in the case of an ideal organic phase,where the activities of the free extractant and solvateapproach their concentrations, the relation between z and xcan be described by equation (5). Here, the total concen-z f n x = c ( 5 )tration of extractant, c, is kept constant.can be introduced into (4) to obtain (6).Equation (5)The second derivative of equation (6), i.e.(7), when sety = log K + log[(c - n ~ ) ~ / x ] (6)2.3n3 2.3( c - nx)z + ~2 (7)Part 2, Dj. M. Petkovid, A. Lj. Ruvarac, J . M. Konstantino-Part 3, A. Lj. Ruvarac, 2. B. Maksimovid, and R. M.Part 4, Dj. M. Petkovid and G. J. Laurence, Bull. SOC. chim.vid, and V. K. Trujid, J.C.S. Dalton, 1973, 1649.HalaSi; J . Radioanalyt. Chern., 1974, 21, 39.Fvance, 1975, 9492 J.C.S. Daltonequal to zero, is used to determine the value of the quotienton the right-hand side of equation (6) a t the inflexion point[equation (S)]. Apparently, the quotient is equal to unity(c - nx)n/x = xn-ln3n/2 (8)at the inflexion point only if n = 1, whereby equation (6)reduces to y = log K .The third derivative of equation(6), for n = 1, is not equal to zero for x = 0 . 5 ~ . Thevalidity of the method, when a monosolvate is formed inextraction equilibria, can be also numerically illustratedfor different K values and 0 < x < c.In real extraction systems the sum of the activities ofspecies in the organic phase is not a constant value. In thiscase an equilibrium-constant expression, for n = 1, has tobe written as in (9) where z is a function of x. Although they = log K + log(z/x) (9)value of the quotient z / x cannot be determined a t the in-flexion point, one may expect, on the basis of conclusionsmade from equations (6)-(8), that this quotient could alsobe equal to unity a t the inflexion point.The property ofequation (6) for n = 1 may be similar to that of equation(9) because the relation between the activities of species inthe organic phase has the same trend as that between theconcentrations of the same species. This is supported bythe treatment of experimental data on the extraction ofnitric acid with tributyl phosphate (see below).In order to extend the validity of the method to theextraction equilibria where n > 1, equation (1) must bewritten in another form giving an equilibrium-constantexpression similar to (9). This can be achieved by express-ing the chemical equilibria in terms of chemical equivalents.Thus, equation (1) is written as (10). Equations (1) andx(-MZf) 1 + zA- + z(-Sorg.) n H z [ (:M)A.3] (10)org.(10) are given for univalent anions and are slightly differentin the case of polyvalent anions.The equilibrium constant(11) of reaction (10) may also be written in logarithmicform, (12), in which, however, the numerator and thedenominator of the quotient on the right-hand side havethe same power, although n > 1. The superscript Edenotes activities on the scale of chemical equivalents. Itis evident that equation (12) belongs to the general case of(9). As in equation (9), one can expect that equation (12)will be reduced, at the inflexion point, to log(l/a&zAE) =$log K . This is supported by the treatment of distributiondata on the extraction of uranyl nitrate with tributylphosphate (see below).The value of the equilibrium constant obtained does notchange within the range of total concentration of extractantwhere the stoicheiometry n = 2 is valid.Therefore, thedistribution data on systems where n > 1 may be plottedas (13). However, the values of the activities on thelog( l/aMEaAE) = f[aES(org.)] = f[aEMAS(org.)l ( 13)5 E. Hogfeldt, ArRiv. Kemz, 1952, 5, 147.6 W. Davis, jun., Nuclear Sci. Eng., 1962, 14, 159.chemical-equivalent scale represents another problem. Forthis purpose one can use the relations between the equiva-ent and molar activities given by Hogfeldt 5 for ion-exchange equilibria. In the case of the extraction processesconsidered in this work they can be written as in equations(14)-(17).Hogfeldt also derived the indentity of theaME = aM1/' (14)aAE = aA (15)aES(org.) = anjzS(org.) (16)aEMAS(org.) = al"MAS(org.) (17)equilibrium constants when the molar or equivalent scaleis used.The position of the inflexion point can be localizedgraphically by calculating values of the gradient, Ay/Ax,of the curve y = f ( x ) and plotting them against x on thesame graph. The value of the abscissa a t the minimumwhich appears is also that of the inflexion point.RESULTS AND DISCUSSIONTreatment of Experimental Data.-Extraction of nitricacid with tributyl Phosphate (tbp). The chemical re-action (18) is considered to occur and the correspondingHf $- "031- -I- tbporg. T- HNO,*tbporg. (18)equilibrium-constant expression is the same as equation(2) but with n = 1.The standard state of the activities*/00 100Percentage of t bpFIGURE 1 Values of equilibrium constant of the extraction ofHNO, with tbp, calculated by the present method (a) fromDavis' data for different concentrations of tbp. The valuesof K f T (o), calculated by Davis,6 are also includedof the species involved in equilibrium (18) is their infinitedilution in the aqueous or in the organic phase, and theirconcentrations are expressed in mol dm-3. The distri-bution data6 have been treated by use of the functionlog(Z/a,2). = f [ ~ ~ ~ ~ ( ~ ~ ~ . ) ] and the activities of nitric acidin the aqueous phase have been taken from the liter-ature.' The equilibrium constant has been calculateddirectly from the ordinate of the inflexion point obtained.Figure 1 shows that the value of this constant, log K =-0.59, is independent of the total concentration ofY .Marcus and A. S. Kertes, ' Ion Exchange and SolventExtraction of Metal Complexes,' Wiley-Interscience, New York,1969, pp. 922-9261978 3tbp, an exceptional case in solvent-extraction chemistry.The value is in very good agreement with that obtainedby extrapolation of the product KfT, determined byDavis for various tbp concentrations, to infinite dilutionof tbp (cf. Figure 1, open circles). In the presentmethod, however, only one isotherm of the distributiondata is required for the determination of the thermo-dynamic equilibrium constant, so that the often uncertainextrapolation of the effective constants to infinite dilutioncan be avoided.The use of stoicheiometric concen-trations instead of the activities of the nitric acid solvatein the organic phase leads to the good value of thermo-dynamic constant. This can be explained providedthe total concentration of extractant, cS(tot.), is keptconstant throughout. Under these conditions CS(org.) is afunction of CMAS(org.) and is also a function ofaMAS(org.). Therefore, it is convenient to replace thequotient aS(org.)/aMAS(org.) by the often more accessibleconcentrations of the species involved.This has beenconsidered in terms of the chemical reaction (19) andExtraction of uranyl nitrate with tbp.the corresponding equilibrium constant is defined byequation (2) with n = 2. The standard state of theactivities and the concentration scale of the speciesinvolved are the same as in the case of the extraction ofHNO, with tbp.Since in this case the numerator and the denominatorof the quotient on the right-hand side of equation (3)have not the same power, equilibrium (19) has beenexpressed in terms of chemical equivalents [equation(2O)l.Consequently the equilibrium constant of re-2(&[U0J2+) + 2[N03]- + 2tbporg. *2{ (8 [UO21) "031 *tb~)org. (20)action (20) is represented by the general equation (11).Therefore, the distribution data for the [UO,] [NO,],-tbp-amsco system have been plotted according toequation (21), derived from equations (13)-( 17). Thelog( 1 /aNaU1") = f [aT(org.)] (21)activities of [UO,] [NO,], in the aqueous phase have beentaken from the literature.'The activity of free tbp has been used because activity-coefficient data for tbp in the system tbp-diluant aremore certain than the activity coefficients of the uranylnitrate solvate in the system [UO,] [N03],*2tbp-diluant.The activity coefficients of free tbp, f ~ , for the systempresented by equation (19), have been calculated accord-ing to Ryazanov by graphical integration of equation(22). A similar method for the calculation of activitycoefficients was suggested earlier by Hogfeldt for ion-exchange equilibria.The activity coefficients of tbp when [UO,][NO,], is* W.Davis, jun., J . Mrochek, and R. R. Judkins, J . Inorg.Nucleav Chem., 1970, 32, 1689.M. A. Ryazanov, Zhur.$fix.Khim., 1971, 45, 1812.not present in the organic phase, fTo, have been takenfrom Pushlenkov and Shuvalov lo and converted fromIn fT(Cunt, CTtot) ='untlnfTo(o, C T ~ ' ~ ) - Jln Keff.dCunt (22)where Keff. = cunt(org.) /aUaN2C2T(org.) (23)the mol fraction to the molarity scale l1 using the densi-ties of tbp-hexane solutions determined in our laboratory.Hexane has been assumed to simulate the amsco diluantused by Davis et aL8 The standard state of f ~ , ascalculated from equation (22), is the pure component.It has not been changed to infinite dilution because ofvery uncertain extrapolation of fT to zero tbp concen-tration. Practically, this change in the standard stateis not necessary since we require the ordinate of theinflexion point, and the abscissa, aqorg.) can be variedby a constant factor without influencing the value of theordinate.21z-c0=5 0Zc -cs) 0d-1Figure 2 shows the treatment of Davis' data'i" iI I1c (4 or u (0)T (org.) T (org.1FIGURE 2 Distribution data on the system [U0,][N0,],-15~otbp-amsco as calculated from equation (21), using the stoicheio-metric concentrations (0) or the activities (0) of free tbp.The arrows show the positions of the inflexion pointson 15% tbp by use of function (23) (open circles), andthe case in which stoicheiometric concentrations havebeen used instead of the activities of free tbp (full circles).The same value of the ordinate of the inflexion point isobtained regardless of whether the stoicheiometricconcentration or the activity of the free extractant isused.Values of the equilibrium constant of reaction (19),determined from function (21) for different constant totalconcentrations of tbp, are shown in Figure 3.Inaddition to Davis' data, unpublished data on the system[UO,] [NO,],-tbp-hexane are also included. The valuelo M. F. Pushlenkov and 0. N. Shuvalov, Radiokhimiya,R. A. Robinson and R. H. Stokes, ' Electrolyte Solutions,'1963, 5; 536.Buttenvorths, London, 1968, p. 324of K remains unchanged almost up to 0.3 mol dm-3tbp in the organic phase, while the change in value a thigher concentrations of tbp could be due to a change in2rt I ,,,.0___, O10 :5't b p ( t o t .)0FIGURE 3 Uranyl nitrate extraction constants, determined fromequation ( 2 1 ) , in hexane ( 0 ) and amsco (0) *110 .C"nt (Org )FIGURE 4 Extrapolation of the effective constants, equation(23), to infinite dilution of uranium in the organic phase for0.05 mol dm-s tbp (0) and 0.10 mol dma3 tbp (0) in hexanethe stoicheiometry of the uranyl nitrate extractionprocess. McKay found that the solvation number of[UO,] [NO,],is two when the proportion of tbp in a diluantl2 H.A. C. McKay, Proc. Internat. Conf. Peaceful Uses of13 A. Poczynailo, P. R. Danesi, and G. Scibona, J . Inorg.Atomic Energy, Geneva, 1955, vol. 7, p. 314.Nuclear Chem., 1973, 35, 3249.J.C.S. Daltonis ( 5 % (ca. 0.2 mol dm-3 tbp). Some papers dealingwith the nature of the extracted species of [U0,][N03],were reviewed by Poczynailo et aZ.13The present value of the constant of the extraction of[UO,][NO,], with tbp, log K 1.6, differs from thatobtained by Marcus,14 log K 2.25.Marcus evaluatedthe equilibrium constant of reaction (19) by extra-polating literature values of the product K ~ T , determinedin paraffin hydrocarbons, to infinite dilution of tbp. Anexplanation for the difference between these two valuescan, perhaps, be found by considering the extrapolationof KeE. values to infinite dilution. According to thepresent experimental data for a very low loading of theorganic phase and for Ctbp <0.1 mol dm-3 (Figure 4),the extrapolation of K,E. .values to infinite dilution leadsto the K value obtained by the method given. It seemsthat the product K ~ T is very sensitive to changes in thetotal concentration of tbp below 0.1 mol dm-3. This canresult from the large changes in the tbp activity co-efficients, fT, at low concentrations of tbp.1°The data presented in Figures 3 and 4 support thevalue of the extraction constant obtained by the presentmethod and justify the expression of the distributiondata in terms of chemical equivalents. Whereas thevalue of K is not changed in the region of Ctbp(tot.) up toca. 0.3 mol drn-,, on the other hand the product Kfr,as a function of the total concentration of tbp below0.1 mol dm-3, approaches the value of K at infinitedilution of tbp.I thank Professor V. A. Mikhailov, the Institute of In-organic Chemistry, Kemerovo, for his stimulating criticisns,and Dr. Aleksandar Ruvarac, the Boris KidriC Institute,for helpful discussion.[6/492 Received, 11th March, 1976114 Y . Marcus, ' Critical Evaluation of Some EquilibriumConstants Involving Organophosphorus Extractants,' Butter-worths, London, 1974, p. 61