Recently one of the authors has reported an expression by an interfacial potential difference (Δφ_eq) of an extraction constant on silver picrate extraction with crown ethers (L) into 1,2-dichloroethane (DCE) or dichloromethane (DCM) [1]. In this study, its extraction constant has been defined as [AgL⁺]_o[Pic⁻]_o/([Ag⁺][L]_o[Pic⁻]), in addition to the well-known definition of K_ex = [AgL⁺Pic⁻]_o/([Ag⁺][L]_o[Pic⁻]), where the subscript “o” and Pic− denote an organic (o) phase, such as DCE and DCM, and picrate ion, respectively. An introduction of Δφ_eq in extraction experiments also gave an answer for a problem of the deviation between the electrochemically-determined K_D,A values and extraction-experimentally-determined values [1,2]. Here, the symbol K_D,A refers to an individual distribution constant (= [A⁻]_o/[A⁻]) of A⁻ into the o phase. Similar problems have been observed in the extraction of divalent metal salts, such as CdPic₂, PbPic₂ and CaPic₂, by L into various diluents [3-5].

In the present paper, we determined at 298 K the K_D,Pic values for the extraction of alkaline-earth metal picrates (MPic₂: M = Ca, Sr, Ba) by 18-crown-6 ether (18C6) or benzo-18C6 (B18C6) into nitrobenzene (NB) which shows the higher polarity. Then, the Δφ_eq values were evaluated from differences between the K_D,Pic values electrochemicallydetermined and those determined by the present extractionexperiments. Here, the electrochemically-determined constant was expressed as K_D,Pic^S, showing the equilibrium constant standardized at Δφ_eq = 0 V [1,6,7]. Moreover, the functional expressions of K_ex±, K_ex2± and K_ex by Δφ_eq were examined; the symbols, K_ex±, K_ex2± and K_ex, refer to [MLA⁺]_o[A⁻]_o/([M²⁺][L]_o[A⁻]2), [ML²⁺]_o[A⁻]_o ²/([M²⁺][L]_o[A⁻]²) and [MLA₂]_o/([M²⁺][L]_o[A⁻]²), respectively [3,4,8]. On the basis of the above values determined, the MIIPic₂ extraction system with L into NB was characterized.

Derivation of a potential difference at the water/o interface

Using properties of electrochemical potentials [9,10], we had reported relations between Δφ_eq and the constants expressing overall extraction equilibria, such as M⁺ + L_o + A⁻MLA_o and M⁺ + L_o+ A⁻ML⁺_o + A⁻_o [1]. The same handling [1,11] was applied for the present extraction equilibria. For example, the authors will apply it to the process:

M²⁺ + L_o + 2A⁻MLA⁺_o+ A⁻_o.      (1)

This process was expressed by as

     (2)

μ_M ⁰ + RTln a_M + 2Fφ_M + μ_L,o ⁰ + RTln a_L,o + 2(μ_A ⁰ + RTln a_A − Fφ_A)

= μML_A,o ⁰ + RTln aML_A,o + FφML_A,o + μ_A,o ⁰ + RTln a_A,o− Fφ_A,o.      (2a)

Rearranging this equation for the K_ex± definition, then we obtained

RTln K_ex± 0 + μML_A,o ⁰ + μ_A,o ⁰ − μ_M ⁰ − μ_L,o ⁰ − 2μ_A⁰

= F{(2φ_M − φ_MLA,o) − (2φ_A − φ_A,o)}.

Therefore, the following equations were derived:

Δφ_eq = Δφ_ex± ⁰ + (2.3RT/F)log K_ex± ⁰      (3)

with Δφ_eq = (2φ_M − φ_MLA,o) − (2φ_A − φ_A,o), Δφ_ex± ⁰ = (μML_A,o ⁰ + μ_A,o ⁰ − μ_M ⁰ − μ_L,o ⁰ − 2μ_A⁰)/F and K_ex± 0 = aML_A,o a_A,o/{a_Ma_L,o(a_A)²}. Here,φj,α and aj,α denote an inner potential for species j in the phase α (= o) and an activity of j in the α phase, respectively; the symbols without α mean those to the water (w) phase, although there is an exception to this rule. From Equation (3), the interfacial potential difference Δφ_eq at an equilibrium was defined [1,11]; in principle, Δφ_eq has been defined as φ(w phase) − φ(o phase). Then, rearranging Equation (3) in a molar concentration unit, we immediately obtain

Δφ_eq = Δφ_ex± ⁰′ + (2.3RT/F)log K_ex±      (3a)

with Δφ_ex± ⁰ ′ = Δφ_ex± ⁰ + (2.3RT/F)log [y_MLA,oy_A,o/{y_M(y_A)2}] and K_ex± = [MLA⁺]_o[A⁻]_o/([M²⁺][L]_o[A⁻]2). Here, yj,α refers to an activity coefficient of the ionic species j {= MLA(I), A(−I), M(II)} in the α phase; the symbol y without α shows the coefficient for the w phase; Δφ_k ^0′ means a standard formal potential. Similarly, equilibrium constants of other processes were expressed as functions of potential differences. These results are listed in Table 1. The condition of Δφ_eq = 0 V was applied for some processes from their properties: namely, Δφ_eq essentially becomes zero, when all species relevant to the inner potentials are present in a single phase [1,9].

Process	Symbol	Relationa
Overall
M2+ + Lo + 2A- MLA2,o	Kex	Log Kex = -2fΔfex0¢
M2+ + Lo + 2A- MLA+o + A-o	Kex±	Log Kex± = f(Δfeq - Δfex±0¢)
M2+ + Lo + 2A- ML2+o + 2A-o	Kex2±	Log Kex2± = 2f(Δfeq - Δfex2±0¢)
Component
M2+ M2+o	KD,M	Log KD,M = 2f(Δfeq - ΔfM0¢)
ML2+ ML2+o	KD,ML	Log KD,ML = 2f(Δfeq - ΔfML0¢)
A- A-o	KD,A	Log KD,A = -f(Δfeq - ΔfA0¢)
M2+o + Lo ML2+o	KML.org	Log KML.org = -2fΔfML.org0¢
M2+ + LML2+	KML	Log KML = -2fΔfML.w0¢b
LLo	KD,L	Log KD,L= -fΔfL0¢
ML2+o + A-o MLA+o	K1,org	Log K1,org = -fΔf1,org0¢
MLA+o + A-o MLA2,o	K2,org	Log K2,org = -fΔf2,org0¢
ML2+o + 2A-o MLA2,o	bip,orgc	Log bip,org = -2fΔfip,org0¢

Table 1: Relations between the potential differences, Δφ_eq, Δφ_k ^0′ or Δφ_j^0′, and Log K_k values in an _extraction system.

Using thermodynamic cycles and the various equilibrium constants in Table 1, we can express the overall extraction processes [1]. Thereby, it becomes possible that we express the overall extraction constants as functions of some formal potentials with Δφ_eq. As an example, K_ex2± (see Introduction for its definition) is expressed as K_D,M(K_D,A)²K_ML,org. Taking logarithms of both sides in this equation and rearranging it based on the corresponding relations in Table 1, we easily obtain

Δφ_ex2± ^0′ = Δφ_M ^0′ − Δφ_A ^0′ + Δφ_ML,org ^0′ + Δφ_eq      (4)

from Δφ_eq − Δφ_ex2± ^0′ = (Δφ_eq − ΔφM 0′) − (Δφ_eq − Δφ_A ^0′) − Δφ_ML,org ^0′. The same was true of K_ex and K_ex±, where the condition of Δφ_eq = 0 V was satisfied for K_ex, since all species relevant to the inner potentials were present in the single phase [1,9]. Table 2 summarizes these results. According to the previous paper [1], when the log K_D,A values are determined experimentally and the Δφ_A ^0′ ones are available, we immediately can calculate the Δφ_eq values from the relation in Table 1.

Overall process & Its cyclea	Relationb
M2+ + Lo + 2A- MLA2,o Kex = KD,M(KD,A)2KML,org bip,org	Dfex0¢ = DfM0¢ - DfA0¢ + DfML,org0¢ + Dfip,org0¢
M2+ + Lo + 2A- MLA+o + A-o (a) Kex± = KD,M(KD,A)2KML,org K1,org (b) Kex± = KML KD,ML K1,org(KD,A)2/KD,L	(a) Dfex±0¢ = 2DfM0¢ - 2DfA0¢ + 2DfML,org0¢ + Df1,org00¢ + Dfeq (b) Dfex±0¢ = 2DfML,w0¢ + 2DfML0¢ + Df1,org00¢ - 2DfA0¢ - DfL0¢ + Dfeq
M2+ + Lo + 2A- ML2+o + 2A-o Kex2± = KD,M(KD,A)2KML,org	Dfex2±0¢ = DfM0¢ - DfA0¢ + DfML,org0¢ + Dfeq

Table 2: Some examples on relations between the Δφ_k ^0′ values and the potential differences based on component equilibrium constants.

For an analytical handling of extraction processes

The extraction-constant parameter, K_ex ^mix, has been employed for the determination of K_D,A and K_ex [3-5]:

log K_ex ^mix = log {([MLA₂]_o + [MLA⁺]_o)/([M²⁺][L]_o[A⁻]²)}

= log {K_ex + K_D,A/([M²⁺][L]_o[A⁻])}      (5)

under the condition of [A⁻]_o ≈ [MLA⁺]_o (>> 2[M²⁺]_o + 2[ML²⁺]_o). A regression analysis to the plot of log K_ex mix versus −log ([M²⁺][L]_o[A⁻]) yielded the K_D,A and K_ex values [3-5]. Equation (5) can be also rearranged as

log K_ex ^mix = log {K_ex + (K_ex±/[M²⁺][L]_o)1/2[A⁻]⁻¹}.      (5a)

Then, this equation makes it possible that one obtains the K_ex± value from the plot of log K_ex mix versus −log {([M²⁺][L]_o)1/2[A⁻]}. In this study, the regression analyses with Equation (5a) were performed at a fixed condition of the K_ex value which was determined in terms of the analysis of Equation (5) and accordingly the thus-obtained K_ex± value was checked by calculating it from each experimental point (Table 3).

Table 3: Fundamental equilibrium constants for the extraction of alkaline-earth metal picrates by L into nitrobenzene at 298 K.

Evaluation of stepwise ion-pair formation constants for MLA₂ in the o phase

Stepwise ion-pair formation constants for MLA₂ in the watersaturated o phase for given I_org,av values were evaluated from the following relations.

K_1,org = [MLA⁺]_o/[ML²⁺]_o[A⁻]_o = K_ex±/K_ex2±      (6)

K_2,org = [MLA₂]_o/[MLA⁺]_o[A⁻]_o = K_ex/K_ex±      (7)

Here, the symbol, I_org,av, was defined as (ΣI_org)/N with a number (N ) of run and ionic strength (Iorg) for the o phase. Table 3 lists the five equilibrium constants determined with the above procedures. The K_ex2± ^S values which were available from references [8] were actually used as K_ex2± in the K_1,NB-calculation with Equation (6). Strictly speaking, there is a difference between actual contents calculated from Equations (6) and (7). See Appendix for this details.

Chemicals

Purities of commercial Ca(NO₃)₂•4H₂O {Kanto Chemical Co. (Kanto), guaranteed reagent (GR)}, Sr(NO₃)₂ (Kanto, GR) and Ba(NO₃)₂ {Wako Pure Chemical Industries (Wako), GR} were checked by a chelatometric titration with disodium salt of EDTA. Also, a purity of commercially-available picric acid, HPic, with amount of 10-15%(w/w) water (Wako, GR) was checked by an acid-base titration [1,5]. Commercial crown ethers, 18C6 (99%, Acros) and B18C6 (98%, Aldrich), were dried at room temperature for >20 h under a reduced pressure. Their purities were checked by measurements of the melting points: 39_.7-40_.1 or 37_.2-39_.9 °C for 18C6; 42_.3-42_.8 for B18C6. Additionally, their water contents were determined by a Karl-Fischer titration: 0.462₈%(w/w) for 18C6 and 0.410₇ for B18C6. Nitrobenzene (Kanto, GR) was washed three-times with water and then kept at a water-saturated condition. Other chemicals were of GR grades and used without further purifications. A tap water was distilled once with a still of the stainless steel and then purified by passing through the Autopure system (type WT101 UV, Yamato/Millipore). Thus purified water was employed for the present work.

Extraction procedures

Alkaline-earth metal nitrates M(NO₃)₂, HPic and L were mixed with 0.002 mol dm⁻³ HNO₃ in a stoppered glass-tube of about 30cm³ and then the same volume of NB was added in its solution. Their total concentrations were [Ca(NO₃)₂]_t = 0.0012 mol dm⁻³, [HPic]_t = 0.0024 and [18C6]_t = (0.70-7.0) × 10⁻⁴; [Sr(NO₃)₂]_t = 6.8 × 10⁻⁴, [HPic]_t = 0.0014 and [18C6]_t = (0.86-6.4) × 10⁻⁴; [Ba(NO₃)₂]_t = 8.8 × 10⁻⁴, [HPic]_t = 0.0018 and [18C6]_t = (0.071-2.6) × 10⁻³ and [Ca(NO₃)₂]_t = 8.0 × 10⁻⁴, [HPic]_t = 1.6 × 10⁻³ and [B18C6]_t = (0.080-2.4) × 10⁻³; [Sr(NO₃)₂]_t = 4.1 × 10⁻⁴, [HPic]_t = 8.2 × 10⁻⁴ and [B18C6]_t = (0.082-2.5) × 10⁻³; [Ba(NO₃)₂] t = 3.5 × 10⁻⁴, [HPic]_t = 7.1 × 10⁻⁴ and [B18C6]_t = (0.032-2.0) × 10⁻³. The thus-prepared glass tube was shaken for 1 minute by hand and was agitated at 298 ± 0.2 K for 2 h in a water bath (Iwaki, type WTE-24) equipped with a driver unit (Iwaki, SHK driver) and a thermoregulator (Iwaki, type CTR-100). After this operation, its mixture was centrifuged with a Kokusan centrifuge (type 7163-4.8.20) for 7 minutes.

A portion of the separated NB phase was transferred into another stoppered glass-tube and then 0.1 mol dm³ HNO₃ was added in this tube. By shaking the tube, all M(II) species in the NB phase were bacK_extracted into the HNO₃ solution. If necessary, the operation for this back extraction was repeated. An amount of all the M(II) species in the aqueous HNO₃ solution was determined by a Hitachi polarized Zeeman atomic absorption spectrophotometer (type Z-6100) with a hollow cathode lamp of Ca (type 10-020, Mito-rika Co. under the license of Hitachi, Ltd.; measured wavelength: 422.7 nm) or Sr (type 10-038, Mito-rika Co.; 460.7 nm). A calibration-curve method was employed for the determination of the M(II) concentration by AAS. For the Ba(II) determination, a >0.1 mol dm⁻³ NaOH solution was added in the back-extracted solution with Pic− and then its Ba(II) solution was measured at 355 nm based on the Pic− absorption and 298 K by a spectrophotometer (Hitachi, type U-2001). The Ba(II) concentration was determined with a calibration curve which had been prepared at 355 nm. On the other hand, the pH value in the separated w phase was measured at 298 K with the same electrode and pH/ion meter [1,4,5].

Determination of K_ex mix

We used here the same extraction model as that [5] for subanalysis described previously. Its component equilibria are 1) M²⁺ + A^- MA⁺ [12], 2)* A^- A^- _o, 3)* M²⁺ M²⁺_o, 4) MLA⁺ _o + A^- _o MLA_2,o, 5) ML²⁺ ML²⁺_o, 6)* L Lo, 7)* M²⁺ + L ML²⁺, 8)* M²⁺_o + Lo ML²⁺ _o, 9)* H+ + A^- HA [4], 10)* HA HA_o [4], 11)* HA_o H⁺_o + A^- _o [13], 12)* H+ H⁺ _o [6], 13)* X− X^- _o [11] and 14)* H⁺ _o + X^-_o HX_o [11], where the symbol HX shows a strong acid, such as HNO₃ and HCl, in water. In particular, the two reactions, 7) and 8), indirectly yielded the process 5). Also, the parentheses with the asterisks show that their equilibrium constants at 298 K have been already determined by several methods (see below for some values).

The equilibrium constants for the reactions 1) and 9) were estimated taking account of the ionic strength, I, for the w phases in a successive approximation [3,5]. Here, the formation constants (K_MA) for MA⁺ = MPic+ (M = Sr, Ba) in the w phase were determined by the same method as that [4] reported before: as the log K_MPic values at 298 K and I → 0 mol dm⁻³, 2.18 for M = Sr and 2.08 for Ba were obtained. While the K_CaPic value was estimated with the value available from reference [12].

The procedure for the calculation of K_ex^mix values by the successive approximation was essentially the same as that [3] described previously (see the theoretical section). Plots of log (D_expl./[Pic⁻]²) versus log [L]_NB gave a straight line with a slope of 0.56 and an intercept of 7.9 for the CaPic₂-18C6 system, 0.91 and 11.1 (≈ log K_ex) for SrPic₂-18C6, 0.62 and 9.4 for BaPic₂-18C6, 0.38 and 5.9 for CaPic₂-B18C6, 0.65 and 8.5 for SrPic₂-B18C6 and 0.44 and 7.9 for BaPic₂-B18C6. Here, the distribution ratio Dexpl. is defined as [AAS-analyzed M(II)]_NB/{[M(NO₃)₂]_t − [AASanalyzed M(II)]_NB} and the intercept corresponds to the log K_ex value only when the slope is about unity [3-5]. Except for the SrPic₂-18C6 system, the compositions of M(II):L:Pic(−I) were assumed to be 1:1:2 in the determination of K_D,Pic, K_ex and K_ex± [3-5]. The slope’s values less than unity indicate dissociations of MLPic₂ in the NB phases [4]. Figures 1 and 2 show the plots for the BaPic₂-18C6 extraction system based on Equations (5) and (5a), respectively, yielding the K_D,Pic, K_ex and K_ex± values (Table 3). The less correlation coefficient (R) of the plot in Figure 2 may reflect the defect in the sub-analytical extraction model [4,5], that is, the absence of the ion-pair formation for MLPic₂ in the w phase. Also, the other extraction systems yielded similar plots, from which we got similarly these three kinds of values (Tables 3).

Figure 1: Plot of log K_ex^mix versus log ([Ba²⁺][L]_NB[Pic⁻]) for the extraction of BaPic₂ by L = 18C6 into NB. A broken line shows a regression one with R = 0.965 analyzed by Equation (5). Its analysis yielded K_D,Pic = 0.20 ± 0.02 and K_ex/ mol⁻³ dm⁹ = (5.6 ± 1.0) × 10¹⁰.

Figure 2: Plot of K_ex^mix versus log {([Ba²⁺][L]_NB)1/2[Pic⁻]} for the extraction of BaPic₂ by L = 18C6 into NB. A broken line shows a regression one with R = 0.850 analyzed by Equation (5a). Its analysis yielded K_ex/ mol⁻² dm⁶ = (2.21 ± 0.56) × 10⁷ at K_ex/ mol⁻³ dm⁹ fixed in 5.6 × 10¹⁰.

In the determination of K_D,Pic, K_ex and K_ex± for the BaPic₂-L extraction system, it was assumed that a total amount, [Ba(II)]_NB,t, of Ba(II) in the NB phase nearly equals the half of that, [Pic⁻]_NB,t, of Pic− in the NB one. This assumption was derived as follows. A charge balance equation for the NB phase was essentially

2[Ba²⁺]_NB + 2[BaL²⁺]_NB + [BaLPic+]_NB + [H⁺]_NB = [Pic⁻]_NB + [X⁻]_NB,      (8)

where the distribution of BaPic+ into the NB phase was neglected, because its data were not available. When [H⁺]_NB ≈ [X⁻]_NB holds, Equation (8) becomes 2[Ba²⁺]_NB + 2[BaL²⁺]_NB + [BaLPic⁺]_NB ≈ [Pic⁻]_NB. Rearranging this equation and adding 2[BaLPic₂]_NB in its both sides, we can immediately obtain

[Pic⁻]_NB + 2[BaLPic₂]_NB = [Pic⁻]_NB,t

≈ 2[Ba²⁺]_NB + 2[BaL²⁺]_NB + [BaLPic⁺]_NB + 2[BaLPic₂]_NB.      (8a)

Hence, when the condition of 2[BaLPic₂]_NB > [BaLPic⁺]_NB (> 2[BaL²⁺]_NB + 2[Ba²⁺]_NB) holds, the half of the left hand side of Equation (8a) approximately becomes [Ba²⁺]_NB + [BaL²⁺]_NB + [BaLPic⁺]_NB + [BaLPic₂]_NB, namely [Ba(II)]_NB,t. We were able to determine spectrophotometrically the ([Pic⁻]_NB + 2[BaLPic₂]_NB) value at least by the back extraction experiments.

Tendencies of K_D,Pic, K_ex, K_ex± and K_n,NB at n = 1, 2

As can be seen from Table 3, the log K_D,Pic values are different from each other in spite of the same definition. These are in the orders of Ca < Sr < Ba for a given L. Also, the orders are B18C6 < 18C6 for a given M(II). Thus, these K_D,Pic orders are influenced by sizes [14,15] of M²⁺ and L, not cavity sizes of L; molar volumes of L were reported to be 214 cm³ mol−1 for L = 18C6 and 252 for B18C6 [15].

The values of both log K_ex and log K_ex± were in the orders of Ca < Sr < Ba (Table 3). These tendencies are similar to those for log K_D,M S and log K_ML,NB (see below for these values). Such facts suggest the presence of these equilibrium constants in the thermodynamic cycles (Table 2). Also, two procedures for evaluating the log K_ex± values in Table 3 well agreed within calculation errors. These facts support that the regression analyses based on Equation (5a) are essentially valid under the conditions of constant K_ex values.

Orders of the log K_1,NB values were Ca > Sr > Ba for the both L, when we neglected differences in I_NB among the extraction systems (see Table 3 for I_NB). On the other hand, the log K_2,NB values were Ca < Sr > Ba. These differences suggest that sizes of M(II) are more-effectively reflected to stability of the 1st-step ion-pair formation than to that of the 2^nd-step formation. In other words, these results seem to be due to differences in a size and/or charges, such as the formal and net charges, between ML²⁺ and MLPic⁺ as reaction species in NB saturated with water. Also, such effects may be reduced in the more-bulky SrPic₂- and BaPic₂-B18C6 systems.

As another explanation for the K_2,NB orders, it can be considered that the I_NB values of the Ca(II) system are largest of all the systems. The I_NB orders were of M = Ca > Sr < Ba for L = 18C6 and Ca > Sr ≥ Ba for B18C6 (Table 3). The highest I_NB values for the Ca(II) systems may cause the lowest K_2,NB values. While their values were less effective for the K_1,NB values, because K_1,NB is constituted by the concentrations of all the ionic species.

If the 1^st-step ion-pair formation is assumed to be

ML²⁺•hH₂O_NB + Pic⁻_NB MLPic+•mH₂O_NB + pH₂O_NB      (9)

with p = h − m, then K_1,NB can be expressed as K_1,NB′/([H₂O]_NB)p with K_1,NB′ = [MLPic⁺]_NB([H₂O]_NB)p/[ML²⁺]_NB[Pic⁻]_NB (as an expression without a hydrated H₂O). Here, the hydration of Pic− in the NB phase was neglected [16]. From Table 3, the larger the h values [17] are, the larger the K_1,NB values become. These facts suggest that the m values are about a constant and thereby the p values are proportional to the h ones. Taking logarithms of the both sides of the equation, K_1,NB = K_1,NB′/ ([H₂O]_NB)p, we can immediately obtain the equation, log K_1,NB = log K_1,NB′ − p × log [H₂O]_NB. Under the conditions of [H₂O]_NB (= 0.178 mol dm⁻³ at 298 K [17,18]) < 1 and p > 0, the equation becomes log K_1,NB = log K_1,NB′ + 0.750p. Therefore, the log K_1,NB values basically increase with an increase in p and consequently can increase with that in h (Table 3). The above results suggest the existence of the reaction (9) with H₂O molecules in NB phase, as reported before on the process of M⁺ + LNB + A^- ML+ NB + A^- NB [8].

Calculation of the Δφ_eq values from the experimental log K_D,Pic values

Using the relation of log K_D,A = −(Δφ_eq − Δφ_A ⁰)/0.05916 at 298 K (Table 1), we calculated the Δφ_eq values from the log K_D,Pic and Δφ_Pic^0′ones, where the Δφ_Pic0′value of 3.0 × 10⁻³ V [6] at the w/NB interface was employed. Thus obtained values are summarized in Table 4. The Δφ_eq range for the 18C6 system was a little smaller than that for B18C6 one. From a comparison with the log K_ex± values in Table 3, the smaller the Δφ_eq values are, the larger the log K_ex± ones become. This trend is similar to that reported previously for the extraction of AgPic by B18C6 and benzo-15-crown-5 ether B15C5 into DCE or DCM [1].

Table 4: Δφ_eq and the log K_ex± and log K_ex2± values at 298 K standardized by their potentials.

Determination of the log K_ex± ^S and log K_ex2± ^S values

The Δφ_ex± ⁰ ′ values were evaluated from the Δφ_eq and log K_ex± values (Tables 3&4) using log K_ex± = (Δφ_eq − Δφ_ex± ^0′)/0.05916 listed in Table 1. After these evaluations, the log K_ex± ^S values were calculated from the same equation at the condition of Δφ_eq = 0 V. The same values were evaluated from Δφ_ex ^{0 ′}= 2φ_M^0′ − 2Δφ_A^0′ + 2Δφ_ML,org ^0′ + Δφ_1,org ^0′ + Δφ_eq in Table 2 and then log K_ex± ^S = −Δφ_ex± ⁰ ′/0.05916. Here, we calculated the Δφ_M^0′ values from log K_D,M S = −11.799 for M = Ca, −11.562 for Sr and −10.818 for Ba [19] and similarly the Δφ_ML,NB ^0′ values from log K_ML,NB = 11.2 for ML²⁺ = Ca(18C6)2+, 13.1 for Sr(18C6)2+, 13.4 for Ba(18C6)2+, log K_ML,NB = 9.43 for Ca(B18C6)2+, 11.1 for Sr(B18C6)2+ and 11.6 for Ba(B18C6)2+ [20]. Also, the Δφ_1,NB^0′ values were calculated from the relation of log K_1,NB = −Δφ_1,NB^0′/0.05916 (Table 1) and the log K_1,NB values in Table 3. Similar evaluations were performed for the log K_ex2± ^S values using the relation among the potentials listed in Table 2.

As can be seen from Table 4, the log K_ex± ^S values calculated from the relation in Table 1 are equal or close to those calculated from that in Table 2. Especially, a little larger deviations for the BaPic₂-L systems may be due to the approximation of [Ba(II)]_NB,t ≈ [Pic⁻]_NB,t (see Materials and Methods). On the other hand, the log K_ex2± ^S values calculated from the relation in Table 2 are very small, compared to those [8] reported before, although the order in magnitude of the calculated values is the same as that of the reported ones [8]. A correlation between these two orders was expressed by the following equation: log K_ex2± ^S(calcd) = (1.54 ± 0.05)log K_ex2± ^S(found) − (2.62 ± 0.07) at R = 0.998. Also, differences between the log K_ex2± ^S(calcd) and log K_ex2± ^S(found) values

were in the range of 1.2 to 3.9 (Table 4). (i) These differences can be due to experimental errors of data, because estimated fractions, [ML²⁺]_NB/ [ AAS-analyzed M(II)]_NB, were in the ranges of 0.0012-0.043% for M = Ca, 0.024-0.33 for Sr and 0.05-3.1 for Ba. That is, these values indicate that the amounts of ML²⁺ in the NB phases are negligible, compared to those of all species with M(II) in the phases and accordingly the K_ex2± evaluations may cause the larger errors. (ii) Or the difference can come from the fact that the original extraction model does not take account of the overall process, M²⁺ + LNB + 2Pic− ML²⁺ NB + 2Pic⁻_NB. While the overall process, M²⁺ + LNB + 2Pic⁻ MLPic+ NB + Pic⁻_NB, has been included in the model [3-5] {see Equation (5a)}. However, it is unclear whether the two above facts, (i) and (ii), cause the negative errors of log K_ex2 ^S or not.

Figure 3 shows a plot of log K_ex± versus log K_ex± ^S(found) for all the MPic₂-L systems. Here, the former logarithmic values are listed in Table 3, while the latter values in Table 4. The plot gave a good correlation between both the values: log K_ex± = (0.81 ± 0.03)log K_ex± ^S(found) + (2.07 ± 0.13) at R = 0.998. This fact indicates that the experimental log K_ex± values clearly reflect the log K_ex± ^S ones at Δφ_eq = 0. The result is the same as that [1] reported before for the AgPic extraction by B15C5 and B18C6 into DCE or DCM. On the other hand, in this study, we were not able to obtain the log K_ex2± values which are comparable to the above log K_ex± ones.

Figure 3: Plot of log K_ex± versus log K_ex±^S for all the MPic2-L extraction system. The regression analysis gave the straight line (- − -) of log K_ex± = 0.81 × log K_ex±^S + 2.07 with R = 0.998. The label log K_ex±^S on the abscissa is identical to log K_ex±^S(found) in Table 4.

Accordance of Δφ_eq with the distribution of ML²⁺ into NB

The individual distribution constants (K_D,ML) of ML²⁺ into the NB phase can be evaluated from the other thermodynamic cycle, K_ex± = K_MLK_D,MLK_1,org(K_D,A)2/K_D,L (Table 2). Here, K_ML and K_D,L (Table 1) are defined as [ML²⁺]/[M²⁺][L] and [L]_o/[L] [15], respectively, and these values at 298 K were available from references; log KM18C6 = 0.48 for M = Ca, 2.72 for Sr and 3.87 for Ba [4]; log KMB18C6 = 0.48 for M = Ca, 2.41 for Sr and 2.90 for Ba [21,22]; log K_D,L = −1.00 [18] for L = 18C6 and 1.57 [20] for B18C6. Using the logarithmic form of the above equation, we obtained log K_D,M18C6 = 0.88 for M = Ca, −0.17 for Sr and −0.99 for Ba and log K_D,MB18C6 = 2.62 for M = Ca, 1.44 for Sr and 1.61 for Ba. There was a tendency that these values increase with an increase in the log K_D,ML S ones: log K_D,ML = (1.61 0.50)log K_D,ML S + (3.67 ± 0.92) at R = 0.848. Here, the log K_D,ML values [20] determined in terms of ion-transfer polarography were employed as log K_D,ML S. Also, the Δφ_eq values can be easily evaluated from a modified form, log K_D,ML = 2(Δφ_eq/0.05916) + log K_D,ML S, of the relation in Table 1: Δφ_eq = 0.088 V for the CaPic₂-18C6 system, 0.060 for SrPic₂-18C6, 0.047 for BaPic₂-18C6, 0.12 for CaPic₂- B18C6, 0.081 for SrPic₂-B18C6 and 0.071 for BaPic₂-B18C6 at 298 K. These Δφ_eq values were in good agreement with those listed in Table 4. This fact indicates that the expression of K_ex± and K_D,Pic by Δφ_eq in the MPic₂ extraction systems with L into NB does not conflict with the data [20] obtained from the electrochemical measurements. Additionally, the Δφ_ex± ^0′ values calculated from the relation in Table 1 were well reproduced by the Δφ-relation (b) in Table 2.

Expressions of the extraction constants by Δφ_eq were extended into K_ex±, K_ex2± and K_ex of the MIIPic₂ extraction systems with L, in addition to K_ex± and K_ex of the AgPic-L systems [1]. These expressions were summarized in Tables 1 and 2. However, the matters for precision of the values in the analyses have been present for the determination of K_ex2± ^S. Also, there may be self-inconsistency for the estimation of the Kn,NB values. It was demonstrated that the log K_ex± values well reflect the log K_ex± ^S ones. So, one can see markedly the relation between K_ex± obtained from an extraction experiment and K_ex± ^S from an electrochemical one. This result fundamentally enables us to discuss the extraction-ability and -selectivity of L against M²⁺ from both the values. Moreover, it was shown that the expressions of K_ex±, K_D,ML and K_D,Pic by Δφ_eq do not conflict with data obtained from the electrochemical measurements.

The authors thank Mr. Shinichi Nakajima for his experimental assistance in the KMPic determination.

In practice, we have calculated the values of

log K_ex± − log K_ex2± ^S = f(Δφ_eq − Δφ_ex± ⁰ ′) − log K_ex2± ^S (A1)

from Equation (6) and the relation in Table 1. Rearranging this equation, the following equation can be easily obtained

log K_ex± − log K_ex2± ^S = log K_ex± ^S − log K_ex2± ^S + fΔφ_eq. (A2)

Also, Equation (7) gave

log K_ex − log K_ex± = log K_ex ^S − log K_ex± = log K_ex ^S − f(Δφ_eq − Δφex± 0′)

= log K_ex ^S − log K_ex± ^S − fΔφ_eq (A3)

Log K_1,NB and log K_2,NB in Table 3 are equal to Equations (A2) and (A3), respectively. As examples, the log K_1,NB and log K_2,NB values for the SrPic₂-B18C6 system were calculated from these equations to be 4.73 and 5.07, respectively. These values were in good accord with those listed in Table 3.

Why do Equations (A2) and (A3), the functions expressing log K_1,NB and log K_2,NB, contain Δφ_eq? This question comes from the fact that log Kn,NB (n = 1, 2) in Table 1 were derived from the condition of Δφ_eq = 0 V. Also, the above results are self-consistent to the fact that all the log Kn,NB values in Table 3 are conditional equilibrium-constants, such as Kn,NB fixed in an I_NB value. Unfortunately, the authors cannot now explain these inconsistencies.