J. CHEM. SOC. DALTON TRANS. 1983 59 1Thermodynamics of Extraction Equilibria. Part 6." A Further Study of theMethod for Determination of Equilibrium Constants of ExtractionProcessesDjordje M. PetkovicChemical Dynamics Laboratory, The Boris Kidric Institute of Nuclear Sciences, P.O. Box 522,1 7001 Beograd, YugoslaviaThe equilibrium constant ( K ) expression of the extraction of an inorganic compound (M) from theaqueous phase by an organic solvent (S) forming a solvate (M-nS), rearranged in the form-log aM = y = log K + log asn/aMs (a = chemical activity of species), has been used to determine K.The method is based on the appearance of an inflexion point in a plot of the experimentaldata as y = f(CMS), where c denotes analytical concentration. The quotient asn/aMs at the inflexionpoint can be substituted by an expression containing the total concentrations of the solvent and thecompound extracted.An inorganic compound (M) is usually extracted from theaqueous phase by an organic solvent (S) forming a solvate(Mans) in the organic phase.Such an extraction process ispresented by equation (1). In order to simplify the writing ofchemical formulae an extra notation of the existence of speciesin the aqueous or organic phase is avoided; M is alwaysrelated to the aqueous phase and S and M*nS to the organicphase. M is also used to denote ions of an inorganic com-pound as well as the compound itself; the distinction is un-important here.the equilibrium constantexpression (2), rearranged in the form (3), was the start-ing point of a method for experimental determination of K.Here a is the chemical activity of the species which are markedwith appropriate subscripts (the subscript M*nS is simpli-fied to MS).The method is based on the use of an inflexionIn Parts 1 and 2 of this seriesdissociation of strong acids,6 have shown that the ordinatevalues of inflexion points approach the values of equilibriumconstants determined by other methods. However, the con-clusion drawn from the mathematical treatment of idealextraction systems and the assumption that it could be ex-tended to real extraction systems was an over-simplifica-tion.In this paper a mathematical procedure using the appear-ance of an inflexion point to evaluate the quotient C I , " / U ~ ~is reported.Therefore, the previous findings3 have to beunderstood as a special case of the general solution ofequation (3) given in this work.TheoreticalThe first derivative of equation (3), that is equation (6),contains the expression dln as/daMs which can be evaluatedpoint in the sigmoid curve, obtained by plotting equation(3) as y = f(CMS) where c is the analytical concentrationof the various species, to evaluate the quotient as"/aMs. Nothaving found a mathematical solution for the quotient atthe inflexion point in previous work, it was assumed thatas"/aMs = cS"/CMS and mass balance equation (4) could beintr~duced.~ Equation (3) then becomes equation (5). Thesecond derivative of equation (S), equalized to zero, showsthat the inflexion point exists where cs = CMS = &(tot.)when n = 1 and the system approaches ideality in the organicphase.Only in this case, equation ( 5 ) is reduced to In ( l / a M ) =In K at the inflexion point. In the case when n differs fromunity we suggest that the extraction equilibrium (1) is pre-sented in the form of chemical equivalent^.^The experimental data on the extraction of nitric acid4and uranyl nitrate ' with tri-n-butyl phosphate, as well as on* Part 5 is ref. 3.separately via the Gibbs-Duhem equation (7). The applicationof the Gibbs-Duhem equation to a two-phase extractionsystem is based on the form of the Gibbs function for a multi-phase system.'*t Corresponding formulae for a system E con-sisting of several phases are obtained by summation over allthe phases, that is dGz = 2 2 j.tardnar, when temperature andpressure are constant.Here, 2 denotes summation over thecomponents and 2 denotes summation over the phases.Since a~ = f(aMs), as = F(uM~), and dy/daM, = - dln U M /daMs, equation (7) gives an expression in the desired form,equation (8). Introducing equation (8) into equation (6) weobtain equation (9) which, with the mass balance equations(4) and (lo), leads to equation (11). Here, cs(tot.) is kepta iiat According to Lewis and Randal (' Thermodynamics,' revised byK. S. Pitzer and L. Brewer, int. student edn., McGraw-Hill, NewYork, 1961, p. 218), when two phases in equilibrium are present,for example the two liquid phases obtained by mixing diethyl etherand water, the system resembles a single phase of pure substance5920.10 0.15 0.20 0.25 0.30J.CHEM. SOC. DALTON TRANS. 1983(9)constant but the total concentration of the species subjectedto the extraction, CM(tot.), is a variable. Further differentiationleads to the second derivative, equation (12). When it is equal-ized to zero to fulfil the condition for the inflexion point andrearranged, equation (13) is obtained. Here, the asteriskindicates that these values are obtained from the secondderivative expression equalized to zero.(1 3) ndcL, to t . ) dlna;, = -ncM(tot.) - CS(tot.)Following the same pattern as in equations (3)-(13), butdifferentiating equation (3) with respect to as and whenaMS = f(a,), equation (14) is obtained which determines thevalue of as at the inflexion point.dln a; = - dln c$(tot.) (14)Equations (13) and (14) determine, for a given cS(tot.), uMsand as values at the inflexion point of a sigmoid curve lyingin the space (y, as, aMS).For different cSCtot.) values, but for thesame extraction system, a family of sigmoid curves is obtained.The inflexion points of the sigmoid curves also form a curvein the same system that allows the integration of equations(13) and (14). By a further rearrangement equation (15) isobtained, where P represents an integration constant.Evaluation of the integration constant is based on theassumption that the chemical activities of the species in theorganic phase are equal to their stoicheiometric concen-trations.Then, the second derivative of equation (5) equalizedto zero gives equation (16), derived previ~usly.~ Together withequation (15), it leads to equation (17). Here, the superscript(id.) denotes the ideal case. Finally, equation (18) is obtainedwhich expresses the in terms of the total analyticalconcentrations of the extractant and species undergoing theextraction process. In practice, the logarithm (base 10) ofequation (2) can be taken and the expression for the quotient(u;)~/u',, introduced from equation (18) to calculate K as logK = log (l/uL) - log [(a',)"/aLs].3rThe complicated equation (18) can be simplified to the form(19) where the term B is easily recognised from equation (1 8).It is interesting to note that equation (19) is reduced toa&& = B for n = 1.However, the quotient B can beapproximately equattd to unity if ckctot.) does not differ muchIn practice, the value of the quotient B can be determinedwithout difficulty. c;,,,,., is calculated by equation (lo), whereCL is taken from the extraction isotherm (cMS, cM) for cLs, theabscissa value of the inflexion point. CL~:;;!) is calculated byequation (10) in the same way but c:kd.) is evaluated byequation (20), obtained from the second derivative of equationfrom C.M{k!) and if CM(tot.) 9 Cs(tot.).( 5 ) equated to zero. The aMS value, needed to plot equation (3)as y = f(aMs), can be calculated from the experimental dataon cMS and cM by means of graphical integrati~n.~*~ How-ever, it may be avoided by presenting equation (3) as y =f(CMS) provided that the total concentration of the extractant ismaintained constant throughout. Then, cs is a function ofcMS and as is also a function of aMs.Therefore, we may replaceQMS by cMS. For the systems previously investigated 4*5 thedifferences between the ordinate values of the inflexion pointsobtained by plotting the data as y = f(aMs) or y = f(cMS) arewithin the error in the location of the inflexion point. TheuM values, necessary to obtain y, as it is defined in equation(3), are given in the literature.Results and DiscussionStarting from experimental data on the extraction of uranylnitrate with tri-n-butyl phosphate the value of B for fivetotal concentrations of the extractant in the range 0 .1 - 0 . 3mol dm-3 was calculated. It was found that B = 1.00 f 0.05for all extractant concentrations, justifying the substitution oJ. CHEM. SOC. DALTON TRANS. 1983 593quotient B with unity in equation (19). Naturally, the value ofB has to be calculated for every extraction system. Theextraction constant was determined, using the same experi-mental data, according to equations (3) and (19) as log K =2.23 f 0.11. It is in very good agreement with the value 2.25critically evaluated by Marcus." The lower value, log K =1.65 f 0.21, obtained previously is the consequence of themathematical treatment of equation (S), where activities ofthe species in the organic phase are equalized with their con-centrations, and the assumption that it can be extended toreal extraction systems.The Figureillustrates a graphical presentation of previousex-perimental data as log (l/aM) = f(cMs).The lines are the bestfit of the data obtained by the least-squares procedure on thethird degree polynomial. The extraction constant values havebeen calculated using equation (3) where y has been equalizedwith the ordinate value of the inflexion points and the valueof as2/aMs has been evaluated by equation (18) for the cor-responding total concentrations of tri-n-butyl phosphate.I have presented a new approach for evaluation of thethermodynamic extraction constants based on the use ofdistribution data belonging to the middle parts of the extrac-tion isotherms. In this range, determination of concentrationsof different species in the aqueous and organic phase is morecertain than in very dilute solutions.The method does notdemand any extrapolation of the apparent equilibriumconstants to infinite dilution, but the values of thermodynamicextraction constants obtained are only trustworthy if activi-ties, that are available, represent the quantities needed toevaluate y in equation (3). However, it has no influence onthe general validity of the method presented in equations(3)-(18).AcknowledgementsI am indebted to Dr. M. M. Kopecni and Dr. S. K. Milonjic,from the same Laboratory, for careful reading of the manu-script and valuable discussions.References1 Part 1, Dj. M.' ContributionPetkovic, B. A. Kezele, and A. Lj. Ruvarac, into Co-ordination Chemistry in Solution.' ed.Erik Hogfeldt, Swedish Natural Science - Research Council,Stockholm, 1972, p. 435.2 Part 2, Dj. M. Petkovic, A. Lj. Ruvarac, J. M. Konstantinovic,and V. K. Trujic, J. Chem. SOC., Dalton Trans, 1973, 1649.3 Dj. M. Petkovic, J. Chem. SOC., Dalton Trans., 1978, 1.4 Dj. M. Petkovic, Bull. SOC. Chim. Beograd, 1978,43,529.5 Dj. M. Petkovic, Bull. SOC. Chim. Beograd, 1980, 45,281.6 Dj. M. Petkovic and A. Lj. Ruvarac, Bull. SOC. Chim. Beograd,7 E. A. Guggenheim, ' Thermodynamics,' North-Holland,8 E. Hogfeldt, Ark. Kemi, 1952, 5, 147.9 M. A. Ryazanov, Zh, Fit. Khim., 1971,45, 1812.1980,45,549.Amsterdam, 1967, p. 25.10 Y. Marcus, ' Critical Evaluation of Some Equilibrium Con-stants Involving Organophosphorus Extractants,' Butterworths,London, 1974.Received 30th December 198 1 ; Paper 1 11 98