On an Expression of Extraction Constants without the Interfacial Equilibrium-Potential Differences for the Extraction of Univalent and Divalent Metal Picrates by Crown Ethers into 1 , 2-Dichloroethane and Nitrobenzene

An idea on interfacial equilibrium-potential differences (∆φeq) which are generated for the extraction of univalent metal picrate (MPic) and divalent ones (MPic2) by crown ethers (L) into high-polar diluents was improved. These potentials were clarified with some experimental extractiondata reported before on the M = Ag(I), Ca(II), Sr(II) and Ba(II) extraction with 18-crown-6 ether (18C6) and benzo-18C6 into 1,2-dichloroethane (DCE) and nitrobenzene (NB). Consequently, it was demonstrated that the ∆φeq values from the extraction-experimentally obtained logKD,Pic ones are in agreement with or close to those calculated from charge balance equations in many cases, where the symbol, KD,Pic, denotes an individual distribution constant of Pic− into the DCE or NB phase. Also, it was experimentally shown that extraction constants based on the overall extraction equilibria do not virtually contain the ∆φeq terms in their functional expressions.

To clarify a reason for such differences, the authors have applied the idea [8] of an interfacial potential difference (∆φ eq ) at extraction equilibrium to an expression of log K D,A , namely ( ) [7], where the negative sign being in the front of f, which denotes F/RT, comes from the electrical charge of A − .In addition to this, extraction constants, K ex± and K ex2± , have been electrochemically expressed as ( ) ± and 1 z = and at ex±, ex2± and 2 [3] [7].Here, 0 A Δφ ′ and 0 Δ k φ ′ refer to standard formal potentials for the single distribution of A − into the diluent or organic (o or org) phase and the formal potentials for the overall equilibrium, respectively.Also, K ex± and K ex2± have been defined experimentally by extraction as [ ] ( ) [3]   and [ ] ( ) , respectively.
On the other hand, from the thermodynamic points of view, these extraction constants are resolved into ( ) D,M D,A ML,org 1,org ex for 2 z = [7], D,M D,A ML,org K K K for 1 [3] and for 2 z = [7].Here, the component equilibrium constants, K ML,org (complex   formation in the o phase) and K 1,org (1st-step ion-pair formation in the o one), do not contain the ∆φ eq terms in their expressions, because the constants are of homogeneous systems that all species relevant to the reaction are present in the single o phase [3] [7]; namely no interface is involved in these processes.Similarly, the distribution constant of M z+ has been expressed with K D,M (see Equation (3) at 1 z = in the Section 2.1) [3].Therefore, since K D,M and K D,A are present in the term, the both terms must cancel out mutually the ∆φ eq ones.Thereby, the extraction constants virtually lose the ∆φ eq terms on their functional expressions.Thus, the above expression, such as ( ) , has caused contradictions on the thermodynamic cycles [3] [7].Furthermore, such contradictions can cause discrepancies in 0 Δ k φ ′ between experimentally-evaluated values and theoretically-reproduced ones [7].
In the present paper, in order to solve the above two contradictions, namely the differences of K D,A caused by experimental conditions of extraction and the contradiction based on the thermodynamic cycles [3] [7], we proposed another expression without ∆φ eq of the extraction constants, K ex± and K ex2± .In course of clarifying this expression, some experimentally-determined constants [3] [7], such as K ex± , an individual distribution constant (K D,ML ) of the complex ion ML 2+ into the NB phase and that of AgL + into DCE, were also reproduced by calculation.Here, the AgPic and MPic 2 (M = Ca, Sr & Ba) extraction with L = 18C6 and/or B18C6 [3] [7] were employed as model systems.Also, a meaning of the ∆φ eq values [3] [7] & [8] which were calculated from the logK D,A ones determined by the extraction experiments was discussed based on an electroneutrality-point of view [8] for the o phases.Moreover, the thus-obtained expressions for the extraction constants were applied to other types of extraction systems with o = DCE and NB.

Theory
for the o phase.The concentrations of M + and A − in the o phase were modified as by using electrochemical equations [6] [8] such as ( ) and ( ) see Appendix B in ref [6] for a detailed derivation from electrochemical potentials to this equation.Here, 0 Δ j φ ′ and [j] o /[j] denote a standard formal potential of species j {=M(I), A(−I) & ML(I); see the introduction and section 3.3} and the individual distribution constant (K D,j ) of j between the two phases, respectively.At least, the 0 Δ j φ ′ values are available from references for M = Ag(I) [9], Ca(II) [10], Sr(II) [10] and Ba(II) [10] and A = Pic(−I) [11] into the DCE and NB phases.Additionally, the M exp , Δ a fφ Accordingly, the following equation is derived.
( ) The data of [ML + ] o ≤ 0 were neglected in a further computation.(ii) Case of the M(II) extraction with L. Similarly, we can consider the following stepwise extraction-equilibria [6] [12] at the same time: ( ) As described above, this equation was modified to [8] ( ) Defining as ( ) eq exp Δ f x φ = and then rearranging Equation ( 8), we easily obtain the cubic equation with ( ) ( ) and ( ) We can exactly solve this equation for x based on the mathematical formula [13].Its real solution is where p b a ′ ′ = and q c a ′ ′ = . Therefore, we can similarly obtain the ∆φ eq value from the combination of Equations ( 6) and (10).
The b′ values were evaluated from the relation, values were directly determined by AAS measurements in the extraction experiments [2] [7] and also we were able to calculate the other values in r + from the experimental data [7].

On Expressions of the Extraction Constants without ∆φeq
According to previous papers, the two of the three extraction constants have been defined as for the M I A-L extraction system [3] and for the M II A 2 -L extraction one [7].Here, logK ex± (or logK ex2± ) equals These two kinds of extraction constants contain the ∆φ eq terms as parameters in their functional expressions [3] [7].On the other hand, logK ex has been expressed as without ∆φ eq and spontaneously became an expression electrochemically-standardized at eq Δ 0 V In the above functions, some contradictions have been observed in the former cases: see Appendix in ref. [7].As an example similar to that described in the introduction, the relation, give a function without ∆φ eq , because the resulting component equilibrium-constant K 2,org does not relate with ∆φ eq [7]; namely K 2,org and K ex are the constants at eq Δ 0 V φ = .However, using the above definition [3] [7], the same term, and then the ∆φ eq term does not disappear, where ( ) . The same is also true of the result of ( ) These two facts obviously have the contradiction with respect to ∆φ eq .In order to cancel such contradictions, we assume here that the two extraction constants are functions without ∆φ eq , as well as that of K ex [3] [7].Accordingly, the constants are defined as That is, by our traditional sense, it is proposed here that complicated equilibrium constants, such as K ex , K ex± and K ex2± , do not contain the ∆φ eq terms in their functions.This means that these constants are ordinarily defined without ∆φ eq or under the condition of eq Δ 0 V φ = and thereby are electrochemically-standardized as . Table 1 lists new (or traditional) expressions of such extraction constants composed of some component equilibrium constants based on thermodynamic cycles.
The relations in Table 1 shows that the individual distribution process of A − [12] cancels out that of a cation [14], such as M + , R 4 N + , M 2+ and ML 2+ , in ∆φ eq .As an example, the thermodynamic relation for M(II can be rearranged into ( ) ( ) Table 1.Relations between K ex± or K ex2± and its component equilibrium constants and their corresponding a k = ex±, ex2±, ML,org, ML,w, & 1,org, where the symbol "w" shows a water phase; b Thermodynamic cycle; c Ref. [3]; d Ref. [7].
Therefore, the relation (c) in Table 1 is immediately obtained.From Equations ( 2) and ( 8), one should obviously see that ∆φ eq of K D,M equals that of K D,A in the extraction system of Equation (13).Also, we can rewrite Equation (13) Consequently, Equation ( 14) or (13) does not contain the ∆φ eq term and is virtually expressed with only the standard formal potentials (at eq Δ 0 V φ = ) as Equation (13a).The thermodynamic relations are also satisfied with the expressions such as Equations ( 11) and (12).The same is true of the other relations in Table 1.

On a Meaning of ∆φeq Estimated from log KD,A
Table 2(a) lists fundamental data [3] for the extraction of AgPic by B18C6 into DCE.The ∆φ eq values were calculated from Equation ( 4) and the experimental log Here, ( ) [11] at 298 K was employed in the calculation.; c Unit: mol dm −3 ; d Ref. [3]; e Values re-calculated from the same data as that reported before.See ref. [3]; f Additionally determined values which were calculated from the same data as that reported before.See ref. [3]; g Data obtained from additional extraction experiments.Experimental conditions and data analyses are essentially the same as those reported on ref. [3].For only the data no.2, the w phases were prepared with about 0.1 mol dm −3   Also, we estimated ∆φ eq,av from Equation (6) with Equation ( 5), where ∆φ eq,av denotes an average value for each run.
The both values, expressed as Δφ in Table 2(b), agreed well within experimental errors.
Average I values of the extraction systems in Table 2(a) were 0.0036 mol⋅dm −3 for the no.1A [3], 0.0028 for 1B, 0.0027 for 1C and 0.097 for 2; I denotes the ionic strength of the water phase in the extraction.Except for the data no.2, we can handle other three data on the average, because experimental conditions [3] of the data are essentially the same (see the footnote g in Table 2(a) for no.So the following values were obtained at 298 K and L = B18C6: logK ex± = 0.
in the same I DCE range.The symbol, I DCE , refers to the average ionic strength of the DCE phase; the same is true of I NB (see Table 3).Table 3(a) summarizes the fundamental data [7] for the extraction of MPic 2 (M = Ca, Sr & Ba) by 18C6 and B18C6 into NB.
The ∆φ eq values were calculated from Equation ( 4) with the logK D,Pic values in Δφ ones from Equation ( 6) with Equation (10).On the other hand, the former values are larger than the latter ones for the B18C6 extraction systems.

P4 av
Δφ values of the B18C6 systems, the above results indicate that the interfacial equilibrium-potential differences, ∆φ eq , based on Equation ( 4) are essentially the same as those based on Equation ( 6).The differences between ( )

P3 eq
Δφ and and Equation (7) or (8).In other words, the condition of   3(a) corresponding to them were employed accordingly.
As similar to  [7].This fact indicates that Equation ( 12) satisfies indirectly the thermodynamic cycle of (f).
The above calculation results for the AgPic and MPic 2 extraction with L indicate that the assumption of Equations (11) and ( 12) without ∆φ eq is essentially valid.In other words, the overall extraction constants, K ex± and K ex2± , must be expressed rationally as functions without ∆φ eq .

For Applications to Other Extraction Systems
The above handling based on As examples, thermodynamic points of view suggest the following cycles for the above equilibria: ( ) . This does not contradict the fact [14] that the determination of ( )  Δφ ′ value in references.

Conclusion
It was demonstrated that the ∆φ eq values calculated from the experimental logK D,Pic ones are in agreement with or close to those more-accurately done from the charge balance equations for the species with M(I) in the DCE phase and with M(II) in the NB one, except for some cases.This demonstration indicates that the plots of values and then the first-approximated ∆φ eq ones.These results will give an answer to how one explain the differences in K D,A among extraction experiments of various MA or MA 2 by various L. Also, clarified that the assumption of Equations ( 11) and ( 12) is valid for the AgPic and MPic 2 extraction with 18C6 and/or B18C6.This eliminated the contradictions [3] [7] due to ∆φ eq from the thermodynamic cycles.Moreover, the present work indicates a possibility that the proposed handling can be applied to various extraction systems with neutral ligands at least.Similarly, the K ex± values have been evaluated from the other arranged form of Equation (A15), [ ] ( ) for 0,1 N = at 1 z = or for 1,2 at 2 [3] [7].
]; see Appendix II for the K D,A determination.Defining as

5 mol⋅dm − 3 (
3 1 ± 0.1 4 and logK D,Pic = −2.54± 0.07; I DCE range of (0.40 -1.1) × 10 −see the data in systems can be due to those in the charge balance equation between extraction experiments (see Appendix II) and electrochemical (or theoretical)

2.1. ∆φeq Values Derived from Charge Balance Equations for the o Phase (
i) Case of the M(I) extraction with L. For the extraction equilibrium,

Table 3
[11]ones reported previously.From Equation (6) with Equation (10), the ∆φ eq,av values were estimated in the same manner.The above findings are listed in Table3(b).

Table 2 (a).
We obtained the log K ex± values of the AgPic extraction with B18C6 into DCE from the relation (a) in Table1The K D,AgL calculation can be an indirect proof of K ex± without ∆φ eq .First, the log K D,AgL values (namely 3.2.Experimental Proof of Kex ± and Kex2 ± without ∆φeq

Table 2 (
(11)re in good accordance with the values listed inTable 2(a).Thus the log K D,AgL values can be well reproduced.From the results of The logK ex± values for the M(II)-B18C6 extraction into NB were calculated from the relation (c) in Table 1.± values are in accordance with the values in Table 3(a); the logK ex± values in Table 3(a) have been determined by the procedure [2][7]described in Appendix II.This accordance indicates that Equation(11)without ∆φ eq is satisfied.In this calculation, 1A and 1B were close to those in Table2(a).

Table 1 .
Here, the adopted

Table 1
can be also applied to the practical extraction equilibria of D,T cancels out K D,A in (E13c), where T − denotes another anion.That is, E15), one can handle them in the same manner as that described above for the AgPic and MPic 2 extraction with L, respectively.We can easily see that the K D,H and K D,Pu values cancel out the K D,Cl one in (E16c).That is,