Determination of Distribution Equilibrium-Potential Differences Based on Extraction with Several Crown Ethers by Nitrobenzene, 1,2-Dichloroethane and Dichloromethane

Extraction constants (Kex± & Kex) were determined at 298 K for the extraction of sodium picrate (NaPic) by nitrobenzene (NB), 1,2-dichloroethane (DCE) and dichloromethane using 3m-crown-m ethers and their benzo-derivatives (m = 5, 6; abbreviated as L) together with the determination of conditional distribution constants (KD,Pic) of picrate ion, Pic, into these diluents. The K1,org (= [NaLPic]org/[NaL]org[Pic]org) values at the organic (org) phases, such as NB & DCE, were calculated from the relation Kex/Kex± = K1,org. Distribution equilibrium-potential differences (Δφeq) at extraction equilibria were evaluated from the equation Δφeq = −0.05916×(log KD,Pic − constant) at 298 K. Correlations of the above equilibrium constants, particularly Kex±, with Δφeq were examined. Furthermore, the standard formal potentials for Na transfers across the interfaces were briefly evaluated from calculated Δφeq, [Na] and [Pic]org. The above extraction systems were characterized by K1,org and the complex formation constants of Na with L in the org phases.


Introduction
In our previous papers (Kudo & Takeuchi, 2014;Kudo & Katsuta, 2015), distribution equilibrium-potential difference (dep, Δφ eq as a symbol) has been reported through studies of extraction of some metal picrates {AgPic & MPic 2 with M = Ca(II), Sr(II) & Ba(II)} by several diluents, such as 1,2-dichloroethane (DCE) and nitrobenzene (NB), using crown ethers (L), such as 18-crown-6 ether (18C6) and benzo-18C6 (B18C6).In these studies, the dep values have been experimentally determined using the equation log K D,A =−{Δφ eq −(standard formal potential)}/0.05916 at 298 K, where K D,A is a conditional distribution constant of A − into an organic (org) phase under the condition of Δφ eq ≠ 0 V (Kudo & Takeuchi, 2014;Kudo et al., 2014;Kudo et al., 2015).Additionally, it has been proved that such Δφ eq values are essentially equal to equilibrium potentials that are calculated electrochemically from charge balance equations of ionic species in the org phases (Kudo & Katsuta, 2015;Takeda et al., 1995a).These facts have shown that the Δφ eq values are related to the equilibrium interfacial potentials determined by electrochemical measurements at liquid/liquid interfaces (Markin & Volkov, 1989).rings will make the presence of dep in the extraction systems surer.
In the present work, we tried to expand such examples to the NaPic extraction systems with 15-crown-5 ether (15C5), 18C6, benzo-15C5 (B15C5) and B18C6.NB, DCE and dichloromethane (DCM) were selected in this study as diluents for the experiments, because of their polarities (Lide, 1995).The Δφ eq values were determined at 298 K, using the two procedures that their values were obtained from the K D,Pic values (Kudo & Takeuchi, 2014;Kudo & Katsuta, 2015;Kudo et al., 2016;Kudo et al., 2014;Kudo et al., 2015) and the charge balance equations (Kudo & Katsuta, 2015) which were proposed for ionic species present in the org phases.From these results, a dependence of log K ex± or log K ex on −Δφ eq was examined.Basically, the former values were proportional to the dep.Also, we discuss on the approximate method for determining standard formal potentials (Δφ Na 0 ′) for Na + transfers across the water/DCE, water/NB and water/DCM interfaces.Further, these systems were characterized based on the ion-pair and complex formation in the org phases.On the NaPic extraction with some L and the less-polar diluent, benzene (Kudo et al, 2016a), a similar study has been already reported.

Chemicals
Commercial NaPic monohydrate (> 95.0%, extra pure, Kanto) was used for the present extraction experiments (Kudo et al., 2011).Its purity was determined by an AAS measurement for Na(I), a spectrophotometric one for Pic(−I) and a Karl-Fischer titration for a water content.From these measurements, the ratio of Na(I):Pic(−I) was determined to be 1:1.06 and the content of water in the commercial NaPic, NaPic⋅nH 2 O, to be 6.232 ± 0.003% which corresponds to n = 0.9263.Also, the water contents of commercial 15C5 (98%, Acros), B15C5 (>96.0%,Tokyo Chemical Industries) and 18C6 (98%, Aldrich) were determined by the titration to be 0.155 ± 0.000 1 %, 0.003 ± 0.000 0 and 0.062 ± 0.000 3 , respectively.The concentrations of these ethers were calculated as 100% purities from their weighed amounts.Nitrobenzene (guaranteed pure reagent, Kanto) was washed three times with water before use (Kudo & Takeuchi, 2014;Kudo & Katsuta, 2015;Kudo et al., 2015).Tap water was distilled once and then purified through Autopure system (type WT101-UV, Yamato/Millipore) (Kudo et al., 2014a).Such water was used for preparing all aqueous solutions and washing NB employed in this study.

Extraction Experiments
Aqueous solutions of NaPic {(3.4-7.0)× 10 −4 mol dm −3 } and those of L were mixed with NB saturated with water at equal volumes (10 or 12 cm 3 ) in stoppered glass tubes of about 30 cm 3 .Total concentrations of the L solutions were (0.35-5.1) × 10 −4 mol dm −3 for L = 15C5, 0.0029-0.031for B15C5, (0.10-6.2) × 10 −4 for 18C6 and (0.10-6.4) × 10 −4 for B18C6.The glass tubes were shaken vigorously by hand and then agitated at 298 K (25 ± 0.4 °C) for 2 h in a mechanical shaker (Kudo et al., 2016).After these operations, the two phases in the tubes were centrifuged and then the NB phases were separated by pipettes.The determination of total amounts of Na(I) extracted into the NB phase was done by the following two procedures.
Procedure (A).This procedure (Kudo & Takeuchi, 2014;Kudo & Katsuta, 2015;Kudo et al., 2015;Kudo et al., 2011) was employed for the 15C5, 18C6 and B18C6 systems.Amounts of water were added into the separated NB phases, then these mixtures were shaken by hand, and thereby the extracted species were back-extracted into these water phases.Such back-extraction experiments were repeated three times.The concentrations of all the Na(I) species back-extracted were determined at 589.0 nm by AAS measurements, for which a Hitachi atomic absorption spectrophotometer (type Z-6100) was used with an air-acetylene flame (Kudo et al., 2014;Kudo et al., 2015).
Procedure (B).The Na(I) concentrations of the separated water phases were directly determined by the AAS measurements, although the experimental accuracy might be lowered, compared with Procedure (A).This procedure was employed for only the B15C5 system.There is reason why the back-extraction experiments with B15C5 did not finish with three times.
In these experiments, blank ones without L were performed.A content of the blanks can correspond to [NaPic] NB + [Na + ] NB .The pH values were measured at 298 K for the firstly-separated water phases using a Horiba pH/ion meter (type F-23) equipped with a pH electrode (Laqua, model 9615-10D) (Kudo et al., 2014).

Data Analysis
The extraction data of the DCM and DCE systems were the same as those (Takeda & Takagi, 1994;Takeda et al., 1995;Takeda et al., 1998;Takeda et al., 2002) reported before by one (Y.K.) of us and his co-workers.The analytic procedure employed was essentially similar to that reported previously (Kudo et al., 2016;Kudo et al., 2014;Kudo et al., 2016a;Kudo et al., 2011a).The concentrations at equilibrium, [Na + ], [L] NB and [Pic − ] (see Appendix A), were calculated from experimental extraction data by a successive approximation method (Kudo et al., 2016;Kudo et al., 2014;Kudo et al., 2016a).The parameter, K ex mix (Kudo et al., 2016;Kudo et al., 2014), was defined as ([NaLPic] NB + [NaL + ] NB )/P.Here, the P value was estimated from the three concentrations at equilibrium and the numerator, [NaLPic] NB + [NaL + ] NB , of K ex mix has been calculated from {the total Na(I) concentration in the NB phase determined by AAS measurements} − {blank value without L}.

Determination of a Composition of Extracted Complexes and of Their K ex± , K D,A , K ex and K 1,org Values
Plots of log (D Na /[Pic − ]) versus log [L] org for the determination of composition of species extracted into the org phase yield straight lines with slopes (a) and intercepts (b) in general (Kudo & Katsuta, 2015;Kudo et al., 2014;Kudo et al., 2016a;Kudo et al., 2011a): see Appendix B. In this case, the b values correspond to the log K ex ones (Kudo et al., 2011a).Experimental lines obtained from these plots were the lines with a = 0.95 and b = 4.20 at a correlation coefficient (r ) = 0.999 for L = 15C5, 0.98 and 3.57 at r = 0.999 8 for B15C5, 0.91 and 3.58 at r = 0.The a values being in the range of less than 0.90 indicate that the extracted ion-pair complexes are dissociable in the org phases (Kudo et al., 2011a) (see the smaller log K 1,org values in Table 1 for their four systems).Similarly, the a values less than 0.90 for the plots have been obtained in the NaMnO 4 extraction with B18C6 into DCE, B15C5 or B18C6 into NB (Kudo et al., 2011a).Figure 1 shows examples of the plots for the four NB systems.Compositions of Na(I):L which equals 1:1 for extracted species were proved to the cases that the slopes of the plots were in the range of 0.90 to 1.10 ( Kudo & Takeuchi, 2014;Kudo & Katsuta, 2015;Kudo et al., 2016;Kudo et al., 2014;Kudo et al., 2015;Kudo et al., 2016a;Kudo et al., 2011a).On the other hand, the ratios of Na(I):Pic(−I) were assumed to be 1:1 from the electroneutrality between NaL(I) and Pic(−I).Therefore, another plot was tried for the 18C6 and B18C6 extraction into DCE and the B15C5 and B18C6 one into NB.Lide, 1995; Takeda, 2002.b Values obtained from the plots of log K ex mix vs. −(1/2)log P. In addition to these data, the K ex & K 1 values have been published in supplementary materials of Ref. Kudo et al., 2016a.c Values obtained from the plots of log K ex mix vs. −log ([Na + ][L] org ).d Ionic strength value, I or I org , which was averaged for the water or org phase.e Values calculated from the relation log K 1,org = log (K ex c /K ex± ). f Values calculated from the relation log K NaL,org = log K ex± − log K D,Na ⋅K D,Pic .g Values calculated from the equation log K D,Na = (Δφ eq − Δφ Na 0 ′)/0.05916= Δφ eq /0.05916 + log K D,Na S at 298 K. h Refs.Iwachido et al., 1982;Iwachido et al., 1977. i Values estimated from Δφ Na 0 ′ = 0.33 9 V. See the section 3.7 in the text.j Ref. Kolthoff, 1981.k See Ref. Kikuchi &  Sakamoto, 2000.l The log K ex values were calculated from the relation in the footnote e. m Values at 293 K. See Ref. Lide,  1995.n Ref. Kudo et al., 2012.o Values calculated from the K NaL,NB ones polarographically-determined, using the common logarithmic form of Eq. ( 5).See the footnote n. p Values at I NB = 0.05 mol dm −3 (Bu 4 N + BPh 4 − ).See Ref. Kudo  et al., 2012.q Values obtained from an average & its standard deviation of several experimental K ex mix values. -2 The y axis of the B15C5 system is 2log D. The slope for the B18C6 (= L) system was less than 0.90 From the above result, the composition of the B15C5 extraction system into NB was determined by employing the plot of 2log D Na versus log [L] org (see Appendix B) (Kudo et al., 2011a).Its slope (a′) and intercept (b′) were 1.05 and 1.71, respectively (Fig. 1).Here, the b′ value corresponds to the log K ex± one (Kudo et al., 2011a).Similarly, the compositions and the apparent log K ex± values were a′ = 1.06 and b′ = 2.19 for the B18C6 extraction-system into NB, 1.08 and 0.34 for the 18C6 one and 0.98 and −1.12 for the B18C6 one into DCE, respectively.These a′ values in the 2log D Na plots consequently showed mainly the extraction of the species with the Na(I): L composition of 1:1 (Kudo et al., 2011a), that is, the distribution of NaL + into the org phase with Pic − as a conuter anion.
Thus, the extraction of NaLPic or NaL + with Pic − were proved in all the extraction systems.Next, we determined the K D,A values (Kudo & Takeuchi, 2014;Kudo & Katsuta, 2015;Kudo et al., 2016;Kudo et al., 2014;Kudo et al., 2015;Kudo et al., 2011a), together with the K ex ones, by using the relation of Here, K ex mix (see the section 2.3 for its definition) nearly equals ([MLA] org + [ML + ] org )/P (see the introduction for P) with the charge balance equation, for the org phase.Figure 2 shows examples of the plots for the 15C5 and B18C6 extraction into NB.The thus-obtained log K ex values were in agreement with the values (Takeda & Takagi, 1994;Takeda et al., 1995;Takeda et al., 1998;Takeda et al., 2002) reported before in figures, except for the 18C6 extraction into DCE {log K ex = 4.13 (Takeda et al., 1998)}.For the NB extraction system with Pic − and L, we were not able to find out suitable literature values of K ex .
The log K ex± order was DCM < DCE << NB for L = 18C6 and B18C6, while DCE < DCM << NB for 15C5 and B15C5 (Table 1).Essentially, these orders seem to reflect that reported by Danesi et al. for the NaPic extraction with dibenzo-18C6 into mixtures between NB and toluene (TE): log K ex± = 0.28 for a 30%NB-70%TE phase with the dielectric constant (ε r ) of 10.6, 0.65 or 0.79 for 50%NB-50%TE with ε r of 15.6 and 1.78 or 2.03 for 100%NB with ε r of 35.6 at 295 K (Danesi et al., 1975).An increase in polarity of diluent essentially causes an increase in log K 1,org −1 , facilitating the dissociation of NaL + Pic − in the org phases, except for L = 18C6 (see Table 1).Here, the ε r values of pure DCM, DCE and NB are 8.93, 10.36 at 298 K (10.42 at 293 K) and 35.6 at 293 K, respectively (Table 1) (Lide, 1995;Takeda, 2002).The above results indicate that the interaction between NaL + and Pic − in the org phase is mainly electrostatic, although the diluents are saturated with water, as described next.

Trends of the Ion-Pair or Complex Formation in the Org Phases
In Table 1, the log K 1,org orders, 18C6 < B18C6 < 15C5 ≤ B15C5, at org = DCM and DCE can be briefly explained in terms of the combined parameter among the basicity of donor O atoms (parameter B), Na + selectivity (M′) and molar volume (V): see Appendix C (Kudo et al., 2016a) for figures relevant to these parameters.The L's order due to the combined parameter is 18C6 (sum = B + M′ + V = 2 + 0 + 1 = 3) < B18C6 (3.8) < 15C5 (4) < B15C5 (4.8).Thus, the combined parameter, B + M′ + V, reproduces well the above log K 1,org orders.The parameters of B & V reflect effects on the benzo-group substitution, while the parameter M′ does that on a size-fitting of the central Na + to the cavity of L. These agreements suggest that such L's characteristics compositely contribute to the NaLPic formation in the org phase (Kudo et al., 2016a).
However, the above parameter was not held for the log K 1,NB values.The log K 1,NB order was B15C5 ≤ B18C6 < 15C5 ≤ 18C6.The L's order due to the combined parameter, B + S, is B15C5 = B18C6 (1.8) < 15C5 = 18C6 (4).The combined parameter, B + S, reproduces the log K 1,NB order.Similarly, the parameter S reflects an effect of flexibility on the L's ring (Kudo et al., 2016a).These facts suggest that the flexibility of L is dominant for the ion-pair formation in the NB phase.On the other hand, the size effect based on the benzo-group substitution can be dominant for that in the DCM and DCE phases.
The log K NaL,DCE order, L = B15C5 < 15C5 < B18C6 < 18C6 (see Table 1), can be mainly explained in terms of the combined effect among the flexibility of L (parameter S), number of donor O atoms (N) and cavity size (C).This combined parameter, C + N + S, shows the order B15C5 (sum = 7) < 15C5 (8) < B18C6 (9) < 18C6 (10).This order well agreed with the log K NaL,DCE order.Individual relations in this order were B15C5 < 15C5 and B18C6 < 18C6 in the substitution (positive changes in B &/or S); 15C5 << 18C6 and B15C5 << B18C6 in the size change (those in C, N &/or V).Additionally, the parameter N should relate to contribution of the Na(I)-O bonds to the K NaL,DCE value.The K NaL,DCE relation of B18C6 < 18C6 was in accord with that reported before, although the reported values {10 9.71 mol −1 dm 3 for L = 18C6 & 10 9.43 for B18C6 at 298 K (Kikuchi & Sakamoto, 2000)} were much larger than the present ones.As well as the log K NaL,DCE order, we can explain the log K NaL,DCM one using the combined parameter, B + N : B15C5 (sum = 5.8) < B18C6 (6.8) < 15C5 (7) < 18C6 (8).
Using the combined parameter, C + N + S, the same discussion can be essentially true of the log K NaL,NB values, except for the relation 18C6 ≤ B18C6.The K NaL,NB values (Kudo et al., 2012) reported by ion-transfer polarographic measurements were close to or somewhat larger than the values determined here (see Table 1).This log K NaL,NB order is well reflected by the parameter, C + N + S.

Determination of Δφ eq and a Consideration for Its Content
According to the previous papers (Kudo & Katsuta, 2015;Kudo et al., 2016;Markin & Volkov, 1989), an approximate value of Δφ eq has been obtained from comparing the experimental 0.05916×log K D,A value with a standard formal potential (Δφ A 0 ′) by using the following equation: (3) in a V unit at 298 K. Here, the Δφ A 0 ′ value is available from Refs.Kudo & Katsuta, 2015;Markin & Volkov, 1989;Kudo et al., 2011 (see the section 3.6) and the K D,A one is determined experimentally (see the section 3.1 & Table 1).The Δφ Pic 0 ′ values are −0.040V at the water/DCM interface (Danil de Namor et al., 1989), −0.0598 at the water/DCE one (Kudo et al., 2011) and 0.0030 at the water/NB one (Kudo et al., 2011).Table 2 summarizes the thus-obtained Δφ eq values.Also, the same values were calculated from an equation {see Eq. ( 6) below} which was derived from Eq. ( 2) with Eq. ( 3) (Kudo & Katsuta, 2015;Takeda et al., 1995a) and then they were symbolized as Δφ eq,cb .The Δφ eq data have been published in the supplementary materials of Ref. Kudo et al., 2016a.b The Δφ eq values calculated from the charge balance equation, Eq. ( 2) or ( 6).c The Δφ eq values calculated from the equation, Δφ eq = Δφ Na 0 ′ + 0.05916×log K D,Na , with the log K D,Na values of the footnote g in Table 1.
For the B18C6-DCM and -DCE systems, the Δφ eq values based on the K D,Pic values were different from the Δφ eq,cb ones based on the charge balance (cb) equations over experimental errors.The same may be true of the 15C5-NB system based on the K D,Na value (see the footnote b in Table 2).On the other hand, the former values were in accord with the latter ones for the other systems.These tendencies were in agreement with those (Kudo & Takeuchi, 2014) reported before for the extraction of alkaline-earth metal picrates by 18C6 or B18C6.The Δφ eq orders, obtained from the K D,Pic values, were 18C6 < 15C6 < B18C6 < B15C5 for the DCM system, 18C6 < B18C6 < 15C5 < B15C5 for DCE and B15C5 < B18C6 < 15C5 < 18C6 for NB.Also, the Δφ eq,cb orders were 18C6 < 15C5 < B18C6 < B15C5 for DCM, 18C6 < B18C6 < 15C5 ≤ B15C5 for DCE and B15C5 < 18C6 ≤ B18C6 ≤ 15C5 for NB.
Except for a position of 18C6 in the dep order of the NB systems, both orders agreed with each other.Furthermore, the Δφ eq values which were calculated from the log K D,Na values due to the footnote o in Table 1 were in the order L = B15C5 ≤ 18C6 ≤ B18C6 < 15C5 for the NB system, agreeing with the Δφ eq,cb order.Essentially, the above results indicate that the Δφ eq values, determined approximately, reflect the Δφ eq,cb ones.One can see intuitively that the Δφ eq,cb values are more-precisely calculated from the extraction experiments (Kudo & Katsuta, 2015).

Distribution Constants of the Neutral Ion-Pair Complexes into the Org Phases
The distribution constants (K D,MLA ) of NaLPic 0 into the org phases were calculated from a thermodynamic relation (Kudo et al., 2016): (4) where the symbols K ML , K 1 and K D,L refer to the complex formation constant, the ion-pair formation one in water and the distribution constant of L into the org phase, respectively.The log (K NaL /mol −1 dm 3 ) values at 298 K have been reported to be 0.70 for L = 15C5, 0.45 for B15C5, 0.73 for 18C6 and 0.81 for B18C6 (Katsuta & Takeda, 2003) and also the log K D,L values which are available in Refs.Kudo et al., 2011a& Iwachido et al., 1982 are listed in Table 1.Moreover, Table 1 summarizes the log K 1 values which were calculated from the log K 1 0 (log K 1 at I → 0) values with the averaged I values.From the above data, the log K D,NaLPic values were evaluated as follows: 2.87 for L = 15C5, 3.01 for B15C5, 1.88 for 18C6 and 2.47 for B18C6 in the DCM system; 2.47 for 15C5, 2.61 for B15C5, 1.16 for 18C6 and 2.06 for B18C6 in DCE; 3.82 for 15C5, 3.75 for B15C5, 3.68 for 18C6 and 3.71 for B18C6 in NB (Kudo et al., 2016a).
These log K D,NaLPic orders were 18C6 < B18C6 < 15C5 < B15C5 at DCM and DCE and 18C6 ≤ B18C6 ≤ B15C5 < 15C5 at NB.It is interesting that the log K D,NaLPic orders at the former two diluents is the same as that predicted from the combined parameter of B + M′ + V: see the section 3.2.In addition to B + M′ + V, the order at NB can suggest a contribution of the parameter S to the 15C5 derivatives in particular.

On the Simple Evaluation of the Complexation-Ability of L Based on K ex±
When the salt MA is fixed, from the thermodynamic relation (Kudo et al., 2012) (5) it is obvious that the magnitude of log K ML,org S (= log K ML,org at Δφ eq = 0 V, see the section 3.6) is directly reflected to that of log K ex± .For examples, 1. 1 of log {K ex± (15C5)/K ex± (B15C5)} was reflected to 1. 1 of log (K Na15C5,DCE /K NaB15C5,DCE ).Similarly, −2.0 3 of log {K ex± (15C5)/K ex± (18C6)} to −2.0 3 of log (K Na15C5,DCE /-K Na18C6,DCE ).Additionally, −1.5 2 ± 0.4 3 of log {K ex± (B15C5)/K ex± (B18C6)} for the NaMnO 4 extraction with B15C5 and B18C6 into DCE corresponds to −2. 0 of log (K NaB15C5,DCE /K NaB18C6,DCE ) in this study; −1.5 2 (= log K NaB15C5,DCE − log K NaB18C6,DCE = 5.6 7 − 7.1 9 ) which was re-calculated from the K D,MnO4, K D,Na , and K ex± values in Refs.Kudo et al., 2011& Kudo et al., 2011a.The both values agreed with each other within the experimental errors.The above analysis based on Eq. ( 5) can more simplify an evaluation of the complexation-ability of L in the DCE phase saturated with water.The same is basically true of the DCM and NB extraction systems.In particular, the relation like Eq. ( 5) becomes important for the extraction systems with polar diluents.
As a more-exact plot (Kudo et al., 2016), we plotted log (K ex± /K D,Na ) against −Δφ eq .The results were log (K ex± /K D,Na ) {= 16.90(−Δφ eq ) + log (K D,A S ⋅K ML,org S )} = 37. 4 (−Δφ eq ) + 6. 3 at r = 0.997 for org = DCM and 42. 9 (−Δφ eq ) + 6. 9 at 0.997 for DCE.Both the r values were improved, but the slopes were much larger than the theoretical one (= 17 V −1 ) at 298 K.The latter facts suggest an effect of the deviation of the log K NaL,org values in changing from L = 15C5 to B18C6.The log K NaL,org values distributed from 3.4 to 5.42 for org = DCM and from 4 to 7.26 for DCE (see Table 1).
On the other hand, the plots of log K ex versus −Δφ eq for the NaPic extraction of all L into the same diluents did not show the same correlations as those in Fig. 3.These results suggest that deviations among the log K 1,org values are large in Eq. (4a).The following results support this suggestion.The log K 1,org orders were 18C6 < B18C6 ≤ 15C5 ≤ B15C5 for org = DCM, 18C6 < B18C6 < 15C5 ≤ B15C5 for DCE and B15C5 ≤ B18C6 < 15C5 ≤ 18C6 for NB (Table 1).
Due to the reversal in order between the log K 1,org and 16.90(−Δφ eq ) values, a sum of both values can become roughly a constant in the relation, log K ex = 16.90(−Δφeq ) + log (K D,M ⋅K ML,org S ) + log K D,A S + log K 1,org ≈ log (K D,M ⋅K ML,org S ) + log K D,A S + constant.The constant terms were in the ranges of 3.20-4.6for the DCM system, 3.06-5.0for DCE and 3.1-4.7 for NB.Consequently, the equation becomes log K ex ≈ constant within ±1.(2).Dividing the both sides of Eq. ( 6) by [M + ] and rearranging it, the equation becomes

Brief Estimation of
we easily obtain (8) Introducing the determined Δφ eq , I org and [M + ] values in this equation, we can immediately calculate Δφ M 0 ′.In the K ML [L] org /K D,L terms, the maximum K NaL /K D,L value in Eq. ( 7) was 10 1.73 (= 54) mol −1 dm 3 for the 18C6 extraction into NB, while the minimum value was 10 −1.97 (= 0.011) for the B15C5 extraction into DCM.Therefore, the above assumption can hold on Eq. ( 7), because the [L] org (K NaL /K D,L ) terms become much smaller than unity in the experimental [L] org ranges (for example, see the [L] org ranges in Fig. 1).
Thus, the standard formal potentials for the Na + transfers were calculated at the number of data = 4. Their Δφ Na 0 ′ values at 298 K were 0.33 9 ± 0.07 6 V for the DCM extraction-systems, 0.26 1 ± 0.09 9 for DCE and 0.20 6 ± 0.07 7 for NB.The value at the transfer across the water/NB interface became 0.24 5 ± 0.04 3 V, neglecting the Δφ Na 0 ′ data for the B15C5 extraction, where its I NB value was largely different from those of the other L extraction into NB (see Table 1).These results indicate that Na + is easy to transfer from the water phase into the org phase in the order of org = DCM ≤ DCE ≤ NB.The value averaged for the DCE extraction systems agreed with the values (0.30 & 0.36 V) (Kudo et al., 2011;Kudo & Takeuchi, 2013), which have been reported at the Na + transfer across the water/DCE interface by one (Y.K.) of the authors, within experimental errors.Also, the maximum value (= 0.28-0.29 V) obtained from the deviation for the NB systems was somewhat smaller than Δφ Na 0 ′ = 0.31 V reported before (Kudo et al., 2011).These facts suggest that the above method is intrinsically effective for the determination of the Δφ M 0 ′ values and at the same time the value for the DCM system is valid.For the DCM systems, the log K D,Na values at Δφ eq ≠ 0 V in Table 1 were estimated from the Δφ Na 0 ′ value (= 0.34 V) described above.

Conclusion
For the DCM and DCE extraction systems with similar polarities, the log K ex± values were proportional to the −Δφ eq ones.The K ex± values estimated previously for the NB system were 18-times larger in maximum than those experimentally-determined here.The differences between the Δφ eq and Δφ eq,cb values were found out in several cases, but they were less than 0.04 V.However, this study strongly ensures the dep determination with the K D,A value obtained from the extraction experiment.Also, the procedure based on Eq. ( 8) can be used for the Δφ M 0 ′ determination, although improvements of the precision of the Δφ M 0 ′ values, actually the Δφ eq ones, will be required for its procedure.For the DCM and DCE systems, the benzo-group substitution on L caused an increase in K 1,org , while for the NB system the increase in the ring-size of L did that in K 1,NB .Also, the increase in the ring size or the donor oxygen-atoms caused that in K NaL,org for all the diluent systems employed.Lastly, the authors were convinced that the presence of dep was confirmed also in the NaPic extraction systems with some L.
999 7 for 18C6 and 0.92 and 3.19 at r = 0.999 for B18C6 in the extraction into org = DCM; those with a = 0.92 and b = 4.29 at r = 0.998 for 15C5, 1.01 and 3.77 at r = 0.993 for B15C5, 0.61 and 2.92 at r = 0.991 for 18C6 and 0.86 and 3.20 at r = 0.989 for B18C6 in the extraction into DCE; those with a = 0.95 and b = 6.46 at r = 0.783 for 15C5, 0.26 and 4.00 at r = 0.917 for B15C5, 0.93 and 6.83 at r = 0.865 for 18C6 and 0.57 and 4.76 at r = 0.875 for B18C6 in the extraction into NB.

Figure 2 .
Figure 2. Plots of log K ex mix vs. −log ([Na + ][L] NB ) for the 15C5 (= L) and B18C6 extraction systems.Open circles and full diamonds show the plots for the systems with L = 15C5 and B18C6, respectively, and are in accord with the symbols in Fig. 1

Figure 3 .
Figure 3. Plots of log K ex± vs. −Δφ eq for the three diluents systems composed of the four L. The Δφ eq values calculated from the experimental K D,Pic ones were employed for the x axis Figure 3 shows plots of log K ex± versus −Δφ eq for the NaPic extraction of all employed L into DCM, DCE and NB.The log K ex± values monotonically increased with an increase in −Δφ eq , except for the NB system.These lines were a = 20.7 ± 4. 9 and b = 0.6 1 ± 0.6 3 at r = 0.948 for the DCM system and 26. 2 ± 2. 5 and 0.8 5 ± 0.2 4 at 0.991 for DCE (Fig.3).These trends can be predicted from the modified Eq. (5a), Figure3shows plots of log K ex± versus −Δφ eq for the NaPic extraction of all employed L into DCM, DCE and NB.The log K ex± values monotonically increased with an increase in −Δφ eq , except for the NB system.These lines were a = 20.7 ± 4. 9 and b = 0.6 1 ± 0.6 3 at r = 0.948 for the DCM system and 26. 2 ± 2. 5 and 0.8 5 ± 0.2 4 at 0.991 for DCE (Fig.3).These trends can be predicted from the modified Eq. (5a), log K ex± = 16.90(−Δφeq ) + log (K D,M ⋅K ML,org S ) + log K D,A S .(5d) Here, 16.90 V −1 and the term, log (K D,M ⋅K ML,org S ) + log K D,A S , at 298 K correspond to slopes and intercepts for the plots, respectively.If differences in log (K D,M ⋅K ML,org S

Table 1 .
Various equilibrium constants for the NaPic extraction with L into DCM, DCE and NB at 298 K (Kikuchi et al., 2001))ar concentrations of water dissolved in their diluents have been reported to be f w = 0.0078 or [H 2 O] org = 0.128 mol dm −3 for org = DCM, 0.0082 or 0.127 for DCE and 0.0149 or 0.178 for NB.These f w values were approximately reduced to [H 2 O] DCM ≈ 0.12 mol dm −3 , [H 2 O] DCE ≈ 0.10 and [H 2 O] NB ≈ 0.148, respectively.Thus, the [H 2 O] org values estimated from the f w ones(Iwachido et al., 1982)are close to those(Kikuchi et al., 2001)determined experimentally.Obviously, the three [H 2 O] org values were much larger than the experimental [NaLPic] org and [NaL + ] org (or [Pic − ] org ) ones.

Table 2 .
Comparison of Δφ eq values estimated from the K D,Pic values a with those from the charge balance equation b for the NaPic extraction with L into DCM, DCE and NB at 298 K a