CdI2 Extraction with 18-Crown-6 Ether into Various Diluents: Classification of Extracted Cd(II) Complex Ions Based on the HSAB Principle

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1. Introduction

It is well known that crown ethers (L) extract Cd(II) and Pb(II) salts, such as metal picrates (MPic_{2}) [1] [2] [3] [4] [5], the former chloride [6], and bromides [1] [6], into various diluents. Similar extraction behaviors into benzene (Bz) and nitrobenzene (NB) have been reported for Ca(II), Sr(II), and Ba(II) picrates with L [7] [8] . In these studies, the distribution equilibrium potentials (dep or Δϕ_{eq}) for monovalent anions (A^{−}) between the water and diluent bulk phases and the ion-pair formation for ML^{2+} and MLA^{+} in the diluent phases saturated with water have been examined and clarified, respectively [1] - [6] [8] . For the latter [1] [2] [4] [6] [8], the reactivities of CdL^{2+} and CdLA^{+} with A^{−} = Cl^{−}, Br^{−}, and picrate ion Pic^{−} in various organic (org) phases have been quantitatively discussed at L = 18-crown-6 ether (18C6). The complex ions composed of a soft Cd^{2+} and hard L, Cd18C6^{2+} and CdB18C6^{2+}, have been classified in terms of the HSAB rule [9] as the hard acids in water [10], where B18C6 refers to benzo-18C6. This classification would make the studies on reactivity of the Cd(II) complexes and properties of the diluent molecules in the extraction interesting. However, there were few comprehensive studies for the M(II) extraction systems with L and various diluents [11] .

In the present paper, by doing extraction experiments of CdI_{2} with 18C6 into ten diluents, we determined extraction constants, K_{ex} and K_{ex}_{±}, and their related equilibrium constants, K_{D,I} and K_{Cd}_{/CdL}, [4] [5] at 298 K. Here, K_{ex}, K_{ex}_{±}, K_{D,I}, and K_{Cd}_{/CdL} were defined as [CdLI_{2}]_{org}/P, [CdLI^{+}]_{org}[I^{−}]_{org}/P with P = [Cd^{2+}][L]_{org}[I^{−}]^{2}, [I^{−}]_{org}/[I^{−}], and [CdL^{2+}]_{org}/[Cd^{2+}][L]_{org} [1] - [6], respectively. From these values and the thermodynamic relations, K_{1,org} and K_{2,org} values were evaluated:
${K}_{\text{1},\text{org}}={K}_{\text{ex}}{}_{\pm}/\left({K}_{\text{Cd}/\text{CdL}}{K}_{\text{D},\text{I}}^{2}\right)$ and K_{2,org} = K_{ex}/K_{ex}_{±} [4] [5] (see the Section 2.4). Using these evaluated K_{1,org} and K_{2,org} values, reaction properties of CdLA^{+} and CdL^{2+} with mainly A^{−} = I^{−}, Br^{−}, and Cl^{−} in the org or diluent phases were also classified based on the HSAB principle [9] [10] [11] . Moreover, molar volumes (V/cm^{3}∙mol^{-}^{1}) of the ion-pair complex CdLI_{2} and complex ion CdL^{2+} were determined at 298 K with the plots based on the regular solution theory (RST) [1] [2] [3] [4] [6] and then their comparable sizes were estimated from these V. On the basis of these data, the HSAB acidic and structural properties of the Cd(II) complexes with 18C6 were discussed independently.

2. Results and Discussion

2.1. Composition Determination of Cd(II) Species Extracted into Various Diluents

According to previous papers [1] - [6], the following equation was employed for the determination of the composition of Cd(II) species extracted into the org phases.

$\mathrm{log}\left(D/{\left[{\text{A}}^{-}\right]}^{\text{2}}\right)\approx \mathrm{log}{\left[\text{L}\right]}_{\text{org}}+\mathrm{log}{K}_{\text{ex}}$ (1)

with D defined as [Cd(II)]_{org}/([Cd(II)]_{t} − [Cd(II)]_{org}) at A^{−} = I^{−} and L = 18C6. This equation was derived approximately from the definition of K_{ex} [7] described in the introduction. Here, the symbols D, [Cd(II)]_{org}, and [Cd(II)]_{t} denote an experimental distribution ratio for Cd(II), a measurement concentration of all the Cd(II) species extracted into the org phase determined by AAS, and a total concentration of CdI_{2} included in the water phase at the beginning of the extraction experiment, respectively. When slopes obtained from the plots of log (D/[A^{−}]^{2}) vs. log [L]_{org} are unity, they would mean that the extracted species have the composition of Cd(II): L = 1:1 [1] - [7] .

The experimental slopes were 0.95 at correlation coefficient (R) = 0.813 for the NB system, 1.03 at 0.989 for 1,2-dichloroethane (DCE), 0.97 at 0.939 for o-dichlorobenzene (oDCBz), 1.03 at 0.940 and 0.96 at 0.754 for dichloromethane (DCM), 1.07 at 0.883 for chlorobenzene (CBz), 1.08 at 0.959 for bromobenzene (BBz), 1.02 at 0.769 for chloroform (CF), 1.01 at 0.871 for Bz, 0.90 at 0.827 for toluene (TE), and 1.05 at 0.924 for m-xylene (mX). Here, the R values were obtained from the regression lines determined with the log(D/[I^{−}]^{2}) vs. log[18C6]_{org} plots. Also, the composition of I(−I) was speculated from the formal charge of Cd(II). This speculation was based on the experimental data plots of the log(D/[L]_{Bz}) vs. log [Pic^{-}] with the slope of two [7] . These results indicated that the complexes composed of Cd(II):18C6:I(−I) = 1:1:2 were extracted into the employed ten diluents.

2.2. Determination of K_{D,I}, K_{ex}_{±}, and K_{ex} by Using the Parameter
${K}_{ex}^{mix}$

For the determination of K_{D,I}, K_{ex}_{±}, and K_{ex}, we employed the parameter

${K}_{\text{ex}}^{\text{mix}}={\left[\text{Cd}\left(\text{II}\right)\right]}_{\text{org}}/P$ (2)

as similar to the previous papers [1] - [6] . Therefore, we can determine the K_{D,I} and K_{ex} values from the plot of
$\mathrm{log}{K}_{\text{ex}}^{\text{mix}}$ vs. −log([Cd^{2+}][L]_{org}[I^{−}]) based on

${K}_{\text{ex}}^{\text{mix}}\approx {K}_{\text{ex}}+{K}_{\text{D},\text{I}}/\left(\left[{\text{Cd}}^{\text{2}+}\right]{\left[\text{L}\right]}_{\text{org}}\left[{\text{I}}^{-}\right]\right)$ (2a)

while can do the K_{ex}_{±} and K_{ex} ones from that of
$\mathrm{log}{K}_{\text{ex}}^{\text{mix}}$ vs. −logP^{1/2} on

${K}_{\text{ex}}^{\text{mix}}\approx {K}_{\text{ex}}+{\left({\left[{\text{CdLI}}^{+}\right]}_{\text{org}}{\left[{\text{I}}^{-}\right]}_{\text{org}}/{P}^{\text{2}}\right)}^{1/2}={K}_{\text{ex}}+{\left({K}_{\text{ex}\pm}/P\right)}^{1/2}$ (2b)

at L = 18C6; see the Section 3.3 for the detailed derivation of Equations (2)-(2b). Figure 1 and Figure 2 show examples of these plots for the present Cd(II) extraction systems and logarithmic values of these equilibrium constants were listed in Table 1. The K_{ex} values determined with Equation (2a) in the 5 diluent systems (Table 1) were in accordance with those with Equation (2b) within their experimental errors.

From the thermodynamic relation of K_{ex} = K_{CdL}K_{ex,ip}/K_{D,L} [1] - [7], the K_{ex,ip} values were evaluated at the same time (see Table 1). Here, the symbols K_{CdL}, K_{ex,ip}, and K_{D,L} denote a complex formation constant (=[CdL^{2+}]/[Cd^{2+}][L]) [12] for CdL^{2+} in water, an ion-pair extraction constant (=[CdLI_{2}]_{org}/[CdL^{2+}][I^{−}]^{2}) of

Figure 1. Plots of
$\mathrm{log}{K}_{\text{ex}}^{\text{mix}}$ vs. −log([Cd^{2+}][L]_{org}[I^{−}]) based on Equation (2a) at L = 18C6. CdI_{2} was extracted with L into NB (square), DCE (circle), BBz (triangle), and CF (diamond).

Table 1. Logarithmic values of K_{D,I}, K_{ex}_{±}, K_{ex}, and K_{ex,ip} for the CdI_{2} extraction with 18C6 into various diluents at 298 K.

a. Abbreviations of the diluents were NB: nitrobenzene; DCE: 1,2-dichloroethane; oDCBz: o-dichlorobenzene; DCM: dichloromethane; CBu: 1-chlorobutane; CBz: chlorobenzene; BBz: bromobenzene; CF: chloroform; Bz: benzene; TE: toluene; mX: m-xylene; b. Ionic strength for the water phases; c. Values determined with Equation (2b); d. Calculated from logK_{ex,ip} = logK_{ex} − logK_{D,18C6}- logK_{Cd18C6} = logK_{ex} - logK_{D,18C6} + 0.05. See ref. [12] ; e. Not determined with Equation (2b).

Figure 2. Plots of
$\mathrm{log}{K}_{\text{ex}}^{\text{mix}}$ vs. −logP^{1/2} based on Equation (2b). CdI_{2} was extracted with 18C6 into NB (square), DCE (circle), BBz (triangle), and CF (diamond).

CdLI_{2} into the org phase, and a distribution constant (=[L]_{org}/[L]) [13] [14] of L into the org phase, respectively. As described below, the K_{ex,ip} and K_{D,L} values are employed for the RST plot [1] - [6] [13] .

2.3. Estimation of Dep for Some Diluent Systems

The relation between dep or Δϕ_{eq} and K_{D,A} has been reported for these extraction systems [1] - [6] [8] .

$\Delta {\varphi}_{\text{eq}}=\left(\text{2}.\text{3}0\text{3}RT/{z}_{\text{A}}F\right)\left(\mathrm{log}{K}_{\text{D},\text{A}}-\mathrm{log}{K}_{\text{D},\text{A}}^{\text{S}}\right)$ (3)

$=\left({0.0591}_{6}/{z}_{\text{A}}\right)\left(\mathrm{log}{K}_{\text{D},\text{A}}-\mathrm{log}{K}_{\text{D},\text{A}}^{\text{S}}\right)$ (3a)

at T = 298 K, where the symbols z_{A} and
${K}_{\text{D},\text{A}}^{\text{S}}$ denote the formal charge of A^{−} with its sign and a standard distribution constant at dep = 0 V for an A^{-} transfer across the interface between the water and org bulk phases, respectively. The value of K_{D,A} called a conditional distribution constant of A^{-} into the org phase changed in depending on species of M(II), L, and diluent molecule [1] - [6] [8] .

Estimated dep values at z_{I} = -1 were −0.01_{0} V for the NB system, −0.06_{4} for DCE, and 0.02_{4} and 0.02_{1} for DCM. Here, the following
$\mathrm{log}{K}_{\text{D},\text{I}}^{\text{S}}$ values at 298 K were used for these calculations: −4.0 [15] for NB, −4.56 [15] for DCE, and −3.790 [16] for DCM. The dep presences were clarified at least for these diluents systems, as similar to the results [1] - [6] [8] reported previously.

2.4. Determination of K_{1,org} and K_{2,org}

Referring to the previous papers [1] - [6] [8] [17], the K_{1,org} and K_{2,org} values were obtained from

$\begin{array}{c}{K}_{\text{1},\text{org}}={\left[{\text{CdLA}}^{+}\right]}_{\text{org}}/{\left[{\text{CdL}}^{\text{2}+}\right]}_{\text{org}}{\left[{\text{A}}^{-}\right]}_{\text{org}}\\ ={K}_{\text{ex}\pm}/{K}_{\text{ex2}\pm}\approx {K}_{\text{ex}\pm}/\left({K}_{\text{Cd}/\text{CdL}}{K}_{\text{D},\text{A}}^{\text{2}}\right)\end{array}$ (4)

and

${K}_{\text{2},\text{org}}={\left[{\text{CdLA}}_{\text{2}}\right]}_{\text{org}}/{\left[{\text{CdLA}}^{+}\right]}_{\text{org}}{\left[{\text{A}}^{-}\right]}_{\text{org}}={K}_{\text{ex}}/{K}_{\text{ex}\pm}$ (5)

for a given ionic strength (I_{org}) in the org phase. Here, the equilibrium constant K_{Cd}_{/CdL} has been assumed to be equal to D/[L]_{org} [3] [5] [6] [17] . The thus-calculated values are listed in Table 2, together with the K_{Cd}_{/CdL} and their corresponding I_{org} values.

Figure 3 shows the K_{1,org} and K_{2,org}values with ten kinds of the diluents described in Table 2. The x-axis indicates the decrease of the diluent’s polarities from No. 1 (NB) to 10 (mX). Except for the DCE and mX systems, there was the relation [3] [5] [6] [17] of K_{1,org} ³ K_{2,org}. This trend seems to be similar to that [18] [19] of the complex formation for CdA_{2} in water with A^{-} = Cl^{-}, Br^{-}, and I^{-}. For the two systems, some structural changes around Cd(II) in the second-step reaction,
${\text{CdLI}}_{\text{org}}^{+}+{\text{I}}_{\text{org}}^{-}\rightleftharpoons {\text{I-CdLI}}_{\text{org}}$ at L = 18C6, could be suggested [3] [5] [6] .

Figure 3. LogK_{1,org} (triangle) and logK_{2,org} (circle) values with ten kinds of diluents described in Table 2.

Table 2. Logarithmic values of K_{Cd}_{/CdL}, K_{1,org}, and K_{2,org} for the CdI_{2} extraction with L = 18C6 into various diluents at 298 K.

a.^{.}See the footnote a in Table 1; b. Average values calculated from experimental D/[L]_{org} ones; c. Calculated from Equation (4); d. Calculated from Equation (5); e. The values were employed for the plots of Figure 3.

2.5. Determination of K_{ex2±}, K_{D,CdLI}, and K_{D,CdL} and Their Characterization

The extraction constant K_{ex2±} {see Equation (4)} and the two conditional distribution constants, K_{D,CdLI} and K_{D,CdL}, were calculated from the following thermodynamic relations [1] [3] [6] .

${K}_{\text{ex2}\pm}={\left[{\text{CdL}}^{\text{2}+}\right]}_{\text{org}}{\left[{\text{I}}^{-}\right]}_{\text{org}}^{2}/P\approx {K}_{\text{Cd}/\text{CdL}}{K}_{\text{D},\text{I}}^{2}$ (6)

${K}_{\text{D},\text{CdLA}}={\left[{\text{CdLI}}^{+}\right]}_{\text{org}}/\left[{\text{CdLI}}^{+}\right]\approx {K}_{\text{ex},\text{ip}}/\left({K}_{\text{1}}{K}_{\text{2},\text{org}}{K}_{\text{D},\text{I}}\right)$ (7)

and

${K}_{\text{D},\text{CdL}}={\left[{\text{CdL}}^{\text{2}+}\right]}_{\text{org}}/\left[{\text{CdL}}^{\text{2}+}\right]={K}_{\text{ex},\text{ip}}/\left({\beta}_{\text{2},\text{org}}{K}_{\text{D},\text{I}}^{2}\right)$ (8)

with K_{1} = [CdLI^{+}]/[CdL^{2+}][I^{-}], called the ion-pair formation constant for water, and β_{2,org} = K_{1,org}K_{2,org}, which is an overall ion-pair formation constant for the org phase. As described in Equation (3a), K_{D,CdLI} and K_{D,CdL} are expressed as functions of Δϕ_{eq} and called the conditional distribution constants:
$\mathrm{log}{K}_{\text{D},\text{CdLI}}=\mathrm{log}{K}_{\text{D},\text{CdLI}}^{\text{S}}+\left(\Delta {\varphi}_{\text{eq}}/0.0{\text{591}}_{\text{6}}\right)$ {=(standard distribution constant at Δϕ_{eq} = 0 V) + (Δϕ_{eq} term)} at z_{CdLI} = +1 and
$\mathrm{log}{K}_{\text{D},\text{CdL}}=\mathrm{log}{K}_{\text{D},\text{CdL}}^{\text{S}}+\left(\text{2}\Delta {\varphi}_{\text{eq}}/0.0{\text{591}}_{\text{6}}\right)$ at z_{CdL} = +2 and 298 K.

Assuming that the relation of
${K}_{\text{1}}^{0}={K}_{\text{CdI}}^{0}{K}_{\text{CdLBr}}^{0}/{K}_{\text{CdBr}}^{0}$ (=308 × 10^{−0.25}/118 = 10^{0.17} mol^{-}^{1}∙dm^{3} [1] [6] [10] [19] at I ® 0 & 298 K) holds, the K_{1} values in Equation (7) were estimated approximately from the experimental I (Table 1) and
${K}_{\text{1}}^{0}$ (≈y_{II+}K_{1}) values. Here, the
${K}_{\text{CdA}}^{0}$ and
${K}_{\text{CdLBr}}^{0}$ refer to an ion-pair (or a complex) formation constant [19] of Cd^{2+} with A^{-} (=I^{-} & Br^{-}) and that [10] of Cd18C6^{2+} with Br^{-} in water at I ® 0, respectively. The activity coefficient (y_{II}_{+}) of CdL^{2+} in water was evaluated from the Davies equation [20] . These calculated values, with the K_{D,18C6} values available from references [13] [14] were listed in Table 3. The logK_{ex2±} values were energetically (−ΔG˚/2.303RT = logK) the

Table 3. Logarithmic values of K_{ex2}_{±}, K_{D,CdLI}, K_{D,CdL}, and K_{D,L} for the CdI_{2} extraction with L = 18C6 into various diluents at 298 K.

a. See the footnote a in Table 1; b. Calculated from Equation (6); c. Calculated from Equation (7); d. Calculated from Equation (8); e. Refs. [13] & [14] ; f. The values were employed for the plots of Figure 4.

smallest in the three extraction constants determined: logK_{ex2±} < logK_{ex}_{±} < logK_{ex} (see Table 2 for K_{ex} & K_{ex}_{±}). Equations (7) and (8) are related with pseudo-RST plots described in the Section 2.8.

As shown in Figure 4, the K_{D,j} values were in the order j = I^{-} (<18C6) < Cd18C6^{2+} < Cd(18C6)I^{+}. This order is basically different from that [3] for the CdPic_{2}-B18C6 extraction system: j = Pic^{-} {
^{0}} < Cd(B18C6)Pic
^{+} < CdB18C6
^{2+}. Equations (7) and (8) predict that a difference between K
_{D,CdLA} and K
_{D,CdL} is proportional to that between
${K}_{1}^{-1}$ and (K
_{1,org}K
_{D,A})
^{-}
^{1}. The relation of
${K}_{1}^{-1}>{\left({K}_{\text{1},\text{org}}{K}_{\text{D},\text{I}}\right)}^{-\text{1}}$ can cause the K
_{D,j} order of j = Cd18C6
^{2+} < Cd(18C6)I
^{+}, while that of
${K}_{1}^{-1}<{\left({K}_{\text{1},\text{org}}{K}_{\text{D},\text{Pic}}\right)}^{-\text{1}}$ can do that of CdB18C6
^{2+} > Cd(B18C6)Pic
^{+}. These experimental results of the CdI
_{2}-18C6 extraction systems were in the -logK
_{1} range of -0.0
_{3} to 0.0
_{3} and in the -log(K
_{1,org}K
_{D,I}) one of −2.6
_{3} to −2.0
_{4}. On the other hand, the values of the CdPic
_{2}-B18C6 systems were evaluated to be in the −logK
_{1} range of −4.60 to −4.39 (see Appendix for the calculation of logK
_{1} averaged in the Bz system) and in the −log(K
_{1,org}K
_{D,Pic}) one of −2.7 to −0.1 [3] . These experimental orders are in good agreement with the orders predicted above.

2.6. For Relative Concentrations of CdLI_{2}, CdLI^{+}, and CdL^{2+} Extracted into the Diluents

We have defined distribution ratios D_{0}, D_{+}, and D_{2+} as described below [3] [5] [6] [17] . Using the experimental data sets of [L]_{org} and [I^{-}], these values were calculated from

${D}_{0}={\left[{\text{CdLI}}_{\text{2}}\right]}_{\text{org}}/\left[{\text{Cd}}^{\text{2}+}\right]={K}_{\text{ex}}{\left[\text{L}\right]}_{\text{org}}{\left[{\text{I}}^{-}\right]}^{\text{2}}$ (9)

${D}_{+}={\left[{\text{CdLI}}^{+}\right]}_{\text{org}}/\left[{\text{Cd}}^{\text{2}+}\right]={K}_{\text{ex}\pm}{\left[\text{L}\right]}_{\text{org}}\left[{\text{I}}^{-}\right]/{K}_{\text{D},\text{I}}$ (10)

Figure 4. Variation of logK_{D,j} with kinds of diluents. Here, j = I^{−} (full circle), L (full diamond), CdL^{2+} (square), and CdLI^{+} (triangle) at L = 18C6. For the DCM system, the logK_{D,I} value of −4.2 in Table 1 was used for this plot.

and

${D}_{\text{2}+}={\left[{\text{CdL}}^{\text{2}+}\right]}_{\text{org}}/\left[{\text{Cd}}^{\text{2}+}\right]\approx {K}_{\text{Cd}/\text{CdL}}{\left[\text{L}\right]}_{\text{org}}$ (11)

at each experimental point. Here, the K_{ex}, K_{ex}_{±}, K_{D,I}, and K_{Cd}_{/CdL} values at L = 18C6 in Table 1 and Table 2 were used for the calculations. From the three equations, we can calculate relative concentrations (or molar fractions), such as f_{0}/% = 100D_{0}/D_{t} and f_{+} = 100D_{+}/D_{t} with D_{t} = D_{0} + D_{+} + D_{2+} [5] . The mean values of f_{0}, f_{+}, and f_{2+}were listed in Table A1 of the Appendix, where the symbols f_{0}, f_{+}, and f_{2+} (=100D_{2+}/D_{t}) denote the relative concentrations of CdLI_{2}, CdLI^{+}, and CdL^{2+}, respectively.

As can be seen from Figure 5 and Table A1, the f_{+} values were the largest in the extraction into the many diluents, except for the values of the DCE and mX systems. Especially, the f_{+} values exceeded 50% in the NB, oDCBz, DCM, BBz, CF, and Bz systems. These behaviors in Figure 5 can be explained as follows. Considering a homogeneous reaction defined as K_{1,org}/K_{2,org}.
$\left(={\left[{\text{CdLI}}^{+}\right]}_{\text{org}}^{2}/{\left[{\text{CdL}}^{\text{2}+}\right]}_{\text{org}}{\left[{\text{CdLI}}_{\text{2}}\right]}_{\text{org}}\right)$, we can evaluate the formation of CdLI^{+} or CdLI_{2} which is dominant about the reaction of
${\text{CdL}}_{\text{org}}^{2+}+{\text{CdLI}}_{\text{2},\text{org}}\rightleftharpoons {\text{2CdLI}}_{\text{org}}^{+}$ .

From the K_{1,org} and K_{2,org} values in Table 2, the log (K_{1,org}/K_{2,org}) values were calculated to be negative (namely K_{1,org} < K_{2,org}) for the org = DCE and mX systems, while their values to be positive (namely K_{1,org} > K_{2,org}) for the other systems. Therefore, we can easily see that the formation of CdLI_{2} is dominant,
${\text{CdL}}_{\text{org}}^{2+}+{\text{CdLI}}_{\text{2},\text{org}}$, in the DCE and mX phases, while that of CdLI^{+} is dominant, 2CdLI^{+}_{org}, in the other diluents. The diluent dependence of the f values in Figure 5 reflects mainly the difference between K_{1,org} and K_{2,org} (see the Section 2.4). Considering these phenomena from ion-pair-formation point of view [3] [6], the systems dominant for the distribution of CdLI^{+} can be a major case in the present extraction systems.

Figure 5. Variation of the relative concentrations of CdLI_{2} (circle), CdLI^{+} (triangle), and CdL^{2+} (square) with kinds of diluents at L = 18C6. Their concentrations are expressed by f_{0}, f_{+}, and f_{2+}, respectively.

2.7. Classification of the Acidity of CdL^{2+} and CdLA^{+} in the Org Phases Based on the HSAB Rule

According to our previous paper [10], the complex ions Cd18C6^{2+} and CdB18C6^{2+} in water have been classified as the hard acids in their reactions with A^{−} = Cl^{−}, Br^{−}, (I^{−},) or Pic^{−}. As standards of the HSAB classification, we assumed that 1) trends in the hardness and softness of the anions A^{−} in the org phases are the same as those [9] [10] in water. That is, I^{−} and Br^{−} are soft bases [9], while Cl^{−} and Pic^{−} are hard bases [9] [10] . 2) The reactions with the halogen ions are primarily employed for the classification. Only when one of the reactions with the three halogen ions lack, the reaction of Pic^{−} was used for it. In the classification, 3) we neglected effects of the I_{org} values on K_{1,org}, K_{2,org}, and β_{2,org}, because the I_{org} values were in the lower ranges [1] [2] [3] [6] : see Table 2 as an example.

For example, the K_{1,NB} and β_{2,NB} (=K_{1,NB}K_{2,NB}) values were in the order Pic^{−} < I^{−} < Cl^{−} (see Table A2 in Appendix). These orders suggested that the Cd18C6^{2+} is a borderline acid in the NB phase, because the order between the hard and soft bases is random. The K_{1,oDCBz} values were in the order Br^{−} < I^{−} < Cl^{−}, while the β_{2,oDCBz} ones were Cl^{−} < I^{−} < Br^{−} (see Table A2). The former order suggested that Cd18C6^{2+} in the oDCBz phase is a hard acid. On the other hand, the latter one indicated that Cd18C6^{2+} is a soft acid. This discrepancy in the classification between K_{1,org} and β_{2,org} can reflect the soft acidity of the intermediate ion-pair complex ion, Cd(18C6)A^{+}; namely the effect of K_{2,org}. A similar trend was observed in the Bz systems: they were classified as the hard acid from K_{1,Bz} (Br^{−} < I^{−} < Cl^{−}) and as the borderline acid from β_{2,Bz} (I^{−} < Cl^{−} < Br^{−}). The Cd18C6^{2+} ions in the other diluents were classified as the soft acids for the DCE, DCM, CBz, BBz, and CF systems, the borderline acid for CBu, and the hard acid for mX and TE: see Table A2 in Appendix. In these systems, the HSAB classifications by K_{1,org} were in accordance with those by β_{2,org}.

On the basis of the above results, it could be considered that Cd18C6^{2+} in water almost changes from the hard acid to the soft or borderline acids in the extraction into the org phases. This indicated that the hardness and softness of Cd18C6^{2+}might be changed with species of the diluents, according to the criteria of the A^{−} basicity.

The following measure can be also considered for the HSAB classification of Cd(18C6)A^{+} in the each phase, because there were no data for the reactions, such as
${\text{CdLCl}}_{\text{org}}^{+}+{\text{Br}}_{\text{org}}^{-}\to {\text{Br-CdLCl}}_{\text{org}}$ and
${\text{CdLCl}}_{\text{org}}^{+}+{\text{I}}_{\text{org}}^{-}\to {\text{I-CdLCl}}_{\text{org}}$ . The ratio of K_{2,org}(A)/K_{2,org}(Cl) = [CdLA_{2}]_{org}[CdLCl^{+}]_{org}[Cl^{−}]_{org}/([CdLCl_{2}]_{org}[CdLA^{+}]_{org}[A^{−}]_{org}) at L = 18C6 was proposed and simply expressed as K_{2,org}(A/Cl). Fixing the ([CdLCl^{+}]_{org}[Cl^{−}]_{org}/[CdLA^{+}]_{org}[A^{−}]_{org}) term or both [CdLCl^{+}]_{org}[Cl^{−}]_{org} and [CdLA^{+}]_{org}[A^{−}]_{org} terms at unity, the ratio virtually can become the [CdLA_{2}]_{org}/[CdLCl_{2}]_{org} ratio. Hence, we considered that if the logK_{2,org}(A/Cl) value is positive, the formation of CdLA_{2} in the org phase becomes dominant and if it is negative, that of CdLCl_{2} does dominant. The former case means the softer complex ion, while the latter one does the harder ion. So, this K_{2,org}(A/Cl) value gives us a criteria for evaluating the HSAB acidity of Cd(18C6)A^{+} in the org phases (water). Consequently, the order of K_{2,org} among A^{−} yields the magnitude in the formation of CdLA_{2} in the org phase under the assumption for the above ratio.

As an example, the logK_{2,oDCBz}(A/Cl) values were in the order A^{−} = Cl^{−} = 1.0 < I^{−} < Br^{−} (see Table A2), suggesting that Cd(18C6)A^{+} in the oDCBz phase is the soft acid. Similarly, the Cd(18C6)A^{+} in the other diluents were classified as the soft acid for the org = NB, DCE, DCM, CBz, BBz, CF, TE, and mX systems, the borderline acid for CBu {logK_{2,CBu}(A/Cl): 1.0 = Cl < Br < Pic} and Bz (I < 1.0 = Cl < Br), and not the hard acid, except for water {logK_{2}(A/Cl): Br < 1.0 = Cl < Pic}. From the above, all Cd(18C6)A^{+} change from the hard acids in water to the soft and borderline ones in the org phases.

Thus, the changes of the diluents (or the org phases) are reflected into the HSAB acidities of these complex ions in the extraction of Cd18C6^{2+} and Cd(18C6)A^{+}. In other words, this means that the HSAB acidity of the complex ion or the ion-pair cation varies with the kinds of the diluents, if the HSAB basicity of the A^{−} can be considered to be the standard. It can be seen that it is easier for the monovalent CdLA^{+} to become the soft acid than for the divalent CdL^{2+} to do it with the extraction into the diluents. This can be supported by the fact that Cd(18C6)A^{+} in the 9 diluents among the 11 ones is classified as the soft acids, compared with Cd18C6^{2+} in the 4 diluents done as the hard acids (Table A2). We can see it particularly from this comparison that the six diluents, DCE, oDCBz, DCM, CBz, BBz, and CF, are the higher effect than the others in softening the acidity of the complex ions. It is interesting that these diluents contain the Cl- or Br-group(s) in their molecules, though CBu, Cl-CH_{2}CH_{2}CH_{2}CH_{3}, does not clearly show its effect.

2.8. Comparisons of Molar Volumes among the Ion-Pair Complexes

We obtained the regression line from the RST plot [1] [2] [3] [6] [13] of logK_{ex,ip} vs. logK_{D,18C6} for the present Cd(II) extraction systems, except for the points of the NB and CF ones [3] [13] : logK_{ex,ip} = (0.7_{5} ± 0.2_{1})logK_{D,18C6} + (6.8_{0} ± 0.2_{5}) at R = 0.800. Also, using V_{18C6} = 214 ± 47 cm^{3} mol^{-}^{1} [13] reported by Takeda, the V_{CdLI2} value was calculated to be 160 ± 57 from the slope of the RST plot. Adding the data of previous papers, the V_{j} values became in order V_{CdLI2} ≤ V_{CdLPic2} (=171 cm^{3} mol^{-}^{1} [2] ) ≤ V_{L} [13] ≤ V_{CdLBr2} (=248 [1] ) < V_{CdLCl2} (=398 [6] ) at L = 18C6. At least, there is a tendency in the order of V_{CdLA2} among A = Cl, Br, and I.

In general, the RST plot for the M(II) extraction system is expressed as logK_{ex,ip} = (V_{MLA2}/V_{L})logK_{D,L} + C + log β_{2} in the form of a linear equation, where the constant C shows solute-solvent (or non-electrostatic) interactions term with cohesive energy densities [1] [2] [3] [6] [13] . From the thermodynamic relation of Equation (8), we can derive the following equation:

$\begin{array}{c}\mathrm{log}{K}_{\text{D},\text{CdL}}=\left({V}_{\text{MLA2}}/{V}_{\text{L}}\right)\mathrm{log}{K}_{\text{D},\text{L}}+C+\mathrm{log}{\beta}_{\text{2}}-\mathrm{log}\left({\beta}_{\text{2},\text{org}}{K}_{\text{D},\text{A}}^{2}\right)\\ =\left({V}_{\text{MLA2}}/{V}_{\text{L}}\right)\mathrm{log}{K}_{\text{D},\text{L}}+{C}^{\prime}\end{array}$ (12)

with
${C}^{\prime}=C+\mathrm{log}\left({\beta}_{\text{2}}/{\beta}_{\text{2},\text{org}}{K}_{\text{D},\text{A}}^{2}\right)$ . Hence, one can see that the C’ term includes the β_{2}/β_{2,org} term corresponding to the ion-ion interactions in addition to the solute-solvent interactions term C. The plot of logK_{D,CdL} vs. logK_{D,L} for the CdI_{2}-18C6 extraction systems is shown in Figure 6. Its regression line was logK_{D,CdL} = (0.5_{6} ± 0.1_{5})logK_{D,L} + (2.4_{7} ± 0.1_{7}) at R = 0.774, where the data of the NB and CF systems were added in the estimation, because of the plot for the ionic species. This slope was somewhat smaller than that (»0.8) of the RST plot. If this difference reflects a difference in V_{j} between j = CdLI_{2} and CdL^{2+}, then the ratio between the slopes can directly express that between V_{j}. So, the ratio of slope(CdLI_{2})/slope(CdL^{2+}) (=1.3) is equivalent to V_{CdLI2}/V_{CdL} at a fixed V_{L}. Therefore, the V_{CdL} value was estimated to be 120 ± 64 cm^{3}∙mol^{-}^{1} from the V_{CdLI2 }one (=160). This value was smallest in the V_{j} with j = CdLCl_{2}, CdLBr_{2}, CdLI_{2}, and CdLPic_{2}. This is in good agreement with the image that the size of CdL^{2+} is smaller than those of CdLA_{2}.

The same trend as above can be seen in a plot of logK_{D,CdLI} vs. logK_{D,L} (see Table 3 for their data): the V_{CdLI} value was 155 ± 46 cm^{3}∙mol^{-}^{1} at L = 18C6. Similarly, V_{CdLBr}/cm^{3}∙mol^{-}^{1} was estimated to be 225 ± 55 from the slope (=1.0_{5} ± 0.1_{1} [1] ) of the logK_{D,CdLBr} vs. logK_{D,L} plot reported previously. These values satisfy the following relations: V_{CdLI2} ≥ V_{CdLI} ≥ V_{CdL} and V_{CdLBr2} ≥ V_{CdLBr} ≥ V_{CdL}.

2.9. Estimation of Apparent Sizes for the Cd(II) Complexes

From the V_{j} data, we can evaluate apparent sizes of Cd(18C6)A_{2} or Cd18C6^{2+}. Assuming
${V}_{j}=\text{4\pi}{R}_{j}^{3}/\text{3}$, namely that shapes of the ion pairs and complex ion are close to spheres, we can easily calculate apparent radii (R_{j}) from the V_{j}. Their

Figure 6. Pseudo-RST plot of logK_{D,CdL} vs. logK_{D,L} at L = 18C6.

R_{j} values were 5.4 Å for j = CdLCl_{2}, 4.6 for CdLBr_{2}, 4.0 for CdLI_{2}, 4.1 for CdLPic_{2}, and 3.6 for CdL^{2+} at L = 18C6. As similar to the results of V_{CdL} (see the Section 2.8), the R_{CdL} value was smallest of the R_{j} ones.

The R_{Cd18C6} value (=3.6 Å) was larger than the following data of bond lengths [21] ; the DFT study of [Cd(18C6)(OH_{2})_{2}]^{2+}, in which 18C6 acts as a tridentate ligand, has reported that the Cd-O and Cd-OH_{2} bond lengths were 2.40 Å and 2.34, respectively. This fact suggested that the R_{Cd18C6} value expresses the hydration structure around Cd18C6^{2+}. Regarding this, it is demonstrated from the Karl-Fischer titration that Ca18C6^{2+} is present in the NB phase as Ca18C6^{2+}∙4.7H_{2}O [22], where the ion size (=0.95 Å) of the six coordinated Cd^{2+} is close to that (=1.00) of the Ca^{2+} [23] . Besides, the suggestion is supported by the facts that Cd-A bond lengths (d_{Cd}_{-A}, see below for its values) in CdLA_{2} crystals [24] [25] with L = 18C6 are in the order d_{Cd-Cl} < d_{Cd}_{-Br} < d_{Cd}_{-I}, while the R_{j} values are in that j = CdLCl_{2} > CdLBr_{2} > CdLI_{2}. Additionally, it was shown that d_{Cd}_{-Pic} is apparently close to d_{Cd}_{-I}.

Also, the bond lengths d_{Cd-Cl} and d_{Cd}_{-O} in a Cd(18C6)Cl_{2} crystal have been reported to be 2.364 Å and 2.752, respectively [24] . The same trend is also observed in Cd(18C6)Br_{2} and Cd(18C6)I_{2} crystals [25] : d_{Cd}_{-Br} = 2.506 Å and d_{Cd}_{-O} = 2.752 for CdLBr_{2} and d_{Cd}_{-I} = 2.692 and d_{Cd}_{-O} = 2.768 for CdLI_{2}. Interestingly, the three d_{Cd}_{-O} values have been almost constant among the crystals. These results suggested that CdLCl_{2}, CdLBr_{2}, and CdLI_{2} with L = 18C6 are close to solvent-separated or -shared ion pairs, such as CdL(OH_{2})_{x}A_{2}, in phases. If this suggestion is correct, then both the R_{j} and V_{j} values can strongly reflect the structural properties of the complexes “in the water phase”. On the basis of the above results, the V_{j} values obtained in the section 2.8 and those reported before seem to be self-consistent.

3. Experimental

3.1. Chemicals

Commercial CdI_{2} {guaranteed pure reagent (GR): >99.0%, Kanto Chemical, Japan} and Cd(NO_{3})_{2}×4H_{2}O (GR: >98.0%, Kanto Chemical) were used: their purities were determined by the chelatometric titration with di-Na(I) salt of EDTA [1] [6] . Here, this nitrate was employed for the preparation of calibration curves in the AAS measurement. The crown ether, 18C6 (GR: 98.0%), was purchased from Tokyo Chemical Industry (Japan) and its solutions were prepared by weighed amounts. The ten commercial diluents were of GR grades: NB (>99.5%), oDCBz (>99.0%), DCE (>99.5%), DCM (>99.5%), BBz (>98.0%), CF (>99.0%), Bz (> 99.5%) and mX (>98.0%) were purchased from Kanto Chemical and CBz (>99%) and TE (>99.5%) done from Wako Pure Chemical Industries, Japan [1] [2] [3] [4] [6] . These diluents were washed three times with pure water and stored in the state saturated with water [1] [2] [4] [6] . Other chemicals were of GR grades. A tap water was distilled once and then deionized by passing through Autopure System (Yamato/Millipore, type WT 101 UV).

3.2. Extraction Procedure

Basic operations and equipment were the same as those described before [1] - [6] . That is, the operations were constructed of original Cd(II) extraction, its back one, and Cd(II) analyses with the AAS measurements at 228.8 nm. The calibration curves of Cd(NO_{3})_{2} in the aqueous 0.1 mol∙dm^{−3} HNO_{3} solutions were employed for the AAS determination of Cd(II). Here, differences in the calibration curve between pure water and the aqueous HNO_{3} solution were experimentally negligible. So, the back extraction was operated with pure water instead of 0.1 mol∙dm^{−3} HNO_{3} [1] - [6] as the back extraction phase, because the Cd(II) amounts in the latter acidic solutions analyzed by the AAS deviated largely.

In the extraction experiments, the [CdI_{2}]_{t} values were in the range of 0.0029 - 0.0081 mol∙dm^{−3} and total concentrations of 18C6 in the water phases were in the ranges of (0.5_{6} - 2.1) * 10^{−5} mol∙dm^{−3} for the NB system, (0.1_{1} - 5.5) * 10^{−4} for DCE, (0.2_{5} - 2.5) * 10^{−5} for oDCBz, (0.01_{3} - 1.1) * 10^{−4} and (0.2_{5} - 4.1) * 10^{−5} for DCM, (0.2_{5} - 4.1) * 10^{−5} for CBz, (0.8_{2} - 2.5) * 10^{−5} for BBz, (0.2_{5} - 2.5) * 10^{−5} for CF and Bz, (0.2_{5} - 7.4) * 10^{−5} for TE, and (1.4 - 7.4) * 10^{−5} for mX. The water phases containing these CdI_{2} and 18C6 were mixed with equal volumes of the diluents or org phases.

3.3. Extraction Equilibrium Model and Its Data Handlings

The following extraction model [4] was employed for the analysis of the present extraction system with L = 18C6: 1) Cd^{2+} + L ⇌ CdL^{2+} [12] and 2) Cd^{2+} + I^{−} ⇌ CdI^{+} [19] in the water phase; 3)
${\text{I}}^{-}\rightleftharpoons {\text{I}}_{\text{org}}^{-}$, 4)
${\text{CdLI}}^{+}\rightleftharpoons {\text{CdLI}}_{\text{org}}^{+}$, 5)
${\text{CdL}}^{\text{2}+}\rightleftharpoons {\text{CdL}}_{\text{org}}^{2+}$, and 6) L ⇌ L_{org} [13] [14] between the water and org phases; 7)
${\text{CdL}}_{\text{org}}^{2+}+{\text{I}}_{\text{org}}^{-}\rightleftharpoons {\text{CdLI}}_{\text{org}}^{+}$ and 8)
${\text{CdLI}}_{\text{org}}^{+}+{\text{I}}_{\text{org}}^{-}\rightleftharpoons {\text{CdLI}}_{\text{2},\text{org}}$ in the org phase. Except for the processes 3)-5), 7), and 8), the equilibrium constants of the above processes at 298 K were available from the references [12] [13] [14] [19] .

Data analyses of the extraction equilibria based on this model were essentially the same as those reported before [1] [2] [4] . The parameter ${K}_{\text{ex}}^{\text{mix}}$ has been defined as

$\begin{array}{l}{K}_{\text{ex}}^{\text{mix}}={\left[\text{Cd}\left(\text{II}\right)\right]}_{\text{org}}/P\\ =\left({\left[{\text{CdLI}}_{\text{2}}\right]}_{\text{org}}+{\left[{\text{CdLI}}^{+}\right]}_{\text{org}}+{\left[{\text{CdL}}^{\text{2}+}\right]}_{\text{org}}+{\left[{\text{CdI}}^{+}\right]}_{\text{org}}+{\left[{\text{CdI}}_{\text{2}}\right]}_{\text{org}}+\cdots \right)/P\end{array}$ (13)

Assuming that

${\left[{\text{CdLI}}_{\text{2}}\right]}_{\text{org}}+{\left[{\text{CdLI}}^{+}\right]}_{\text{org}}\gg {\left[{\text{CdL}}^{\text{2}+}\right]}_{\text{org}}+{\left[{\text{CdI}}^{+}\right]}_{\text{org}}+{\left[{\text{CdI}}_{\text{2}}\right]}_{\text{org}}+\cdots $

this equation can be rearranged into

${K}_{\text{ex}}^{\text{mix}}\approx {K}_{\text{ex}}+{K}_{\text{D},\text{I}}/\left(\left[{\text{Cd}}^{\text{2}+}\right]{\left[\text{L}\right]}_{\text{org}}\left[{\text{I}}^{-}\right]\right)$ (2a)

in the case of [CdLI^{+}]_{org} » [I^{−}]_{org} which was approximately derived from the charge balance equation for the org phase [1] - [6] . At least, the conditions of
${\left[{\text{CdLI}}_{\text{2}}\right]}_{\text{org}}+{\left[{\text{CdLI}}^{+}\right]}_{\text{org}}+{\left[{\text{CdL}}^{\text{2}+}\right]}_{\text{org}}\gg {\left[{\text{CdI}}^{+}\right]}_{\text{org}}+{\left[{\text{CdI}}_{\text{2}}\right]}_{\text{org}}+\cdots $ were checked by blank experiments of the CdI_{2} extraction without L = 18C6. On the other hand, in the case of [CdLI^{+}]_{org}/P » K_{ex}_{±}/[I^{−}]_{org}, we can immediately obtain

${K}_{\text{ex}}^{\text{mix}}\approx {K}_{\text{ex}}+{\left({\left[{\text{CdLI}}^{+}\right]}_{\text{org}}{\left[{\text{I}}^{-}\right]}_{\text{org}}/{P}^{\text{2}}\right)}^{1/2}={K}_{\text{ex}}+{\left({K}_{\text{ex}\pm}/P\right)}^{1/2}$ (2b)

The parameter
${K}_{\text{ex}}^{\text{mix}}$ was calculated from the experimental [Cd(II)]_{org}, [Cd^{2+}], [18C6]_{org}, and [I^{−}], where the latter three concentrations were determined with a successive approximation procedure, using the equilibrium constants of the processes 1), 2), and 6) [2] [4] . When a negative value for K_{ex} had been obtained from the analysis with Equation (2b), its analysis was performed again by fixing the K_{ex} value to that determined by the analysis with Equation (2a) [1] - [6] (see the footnotes c & e in Table 1).

4. Conclusions

The ion-pair formation in the 11 diluents saturated with water was classified in terms of the HSAB principle, although the hardness and softness of the simple A^{-} in the diluents were assumed to be the same as those in water. This classification mainly showed us the two results. 1) CdL^{2+} and CdLA^{+} with L = 18C6 and A^{-} = Cl^{-}, Br^{-}, and I^{-} change from the hard acids in water to almost the soft or borderline acids in the extraction into the org phases at least. 2) The charge effects on CdLA^{+} and CdL^{2+} in the org phases are remarkable. Namely, CdLA^{+} softens more its acidity than CdL^{2+} does in the extraction. Especially, DCE, oDCBz, DCM, CBz, BBz, and CF have the higher ability to soften the HSAB acidity of the complex ions.

The presence of dep was also observed in the CdI_{2}-18C6 extraction into NB, DCE, and DCM. The relation of f_{+} < f_{0} simply reflects that of K_{1,org} < K_{2,org}, about which the structural changes around Cd(II) were suggested, while the relation of f_{+} > f_{0} does that of K_{1,org} > K_{2,org}.

The molar volumes V_{j} obtained from the RST plots indicated the size-dependence on the Cd(18C6)A_{2} (=j) ion pairs. Additionally, the V_{Cd18C6} value was evaluated from the pseudo-RST plots and then was the smallest of the V_{j} ones of all the Cd(18C6)A_{2} examined. At the same time, it was demonstrated that the apparent radii R_{j}, estimated from the V_{j} values, reflects inversely the bond lengths of Cd-A with A^{-} = Cl^{-}, Br^{-}, and I^{-} in the crystallographic and DFT studies. These V_{j} and R_{j} results proved validities for the analyses of the RST and pseudo-RST plots about such extraction systems and thereby indicated a possibility that the two plots give the structural information about some complexes, although it is unclear which of org or water phase is the corresponding phase.

Acknowledgements

Y. K. and Y. I. thank Mr. Quan Jin for his experimental support with the comparison between the AAS calibration curve with pure water and that with the acidic solution.

Appendix

The following Table A1 is supplementary data for the discussion in the Section 2.6. The numbers of Table A1 express the diluents in Figure 5.

In the following Table A2 are listed basic data for the HSAB classification. The data were used for consideration in the Section 2.7. The Pic with parenthesis in Table A2 was not employed for it.

Table A1. Relative concentrations^{a}, f_{0} for CdLI_{2}, f_{+} for CdLI^{+}, and f_{2+} for CdL^{2+}, in the CdI_{2} extraction with L = 18C6 into various diluents at 298 K.

a. Definition: f_{0} = 100D_{0}/D_{t}; f_{+} = 100D_{+}/D_{t}; f_{2+} = 100D_{2+}/D_{t} with D_{t} = D_{0} + D_{+} + D_{2+}. All the values were mean ones; b. See the footnote a in Table 1; c. The values were employed for the plots of Figure 5.

Table A2. Orders of the K_{1,org}, K_{2,org}(A/Cl), and β_{2,org} values^{a} at 298 K for the HSAB classification.

a. See Refs. [1] [2] & [6] for the numerical data; b. See the footnote a in Table 1; c. Refs. [10] & [26] ; d. See the Section 2.5 for its definition; e. Ratio between the ion-pair formation constant for A^{-} in water, defined as [CdLA_{2}]/[CdLA^{+}][A^{-}], & that for Cl^{-}; f. Overall ion-pair formation constant for water defined as [CdLA_{2}]/[CdL^{2+}][A^{-}]^{2}.

The evaluation of log{K_{1}(average)} for the CdPic_{2} extraction with B18C6 into Bz was as follows. Using
${K}_{1}^{0}$ = 6.4 × 10^{4} mol^{−1}∙dm^{3} [26] at 298 K and the Davies equation [20], we calculated its value from the thermodynamic relation of

$\mathrm{log}{K}_{1}^{0}\left\{=\mathrm{log}{K}_{\text{1}}+\mathrm{log}\left({y}_{\text{CdLA}}/{y}_{\text{II}+}{y}_{\text{A}}\right)\right\}\approx \mathrm{log}{K}_{\text{1}}-\mathrm{log}{y}_{\text{II}+}$

where the activity coefficient ratio y_{CdLA}/y_{A} » 1. Rearranging this equation and then introducing the Davies equation in it, the following equation can be easily obtained at 298 K:

$\mathrm{log}{K}_{1}\approx \mathrm{log}{K}_{1}^{0}-{0.511}_{4}\times {2}^{2}\times \left\{{I}^{1/2}/\left(1+{I}^{1/2}\right)-0.3I\right\}$ (A1)

So, introducing
${K}_{1}^{0}$ and I = 0.095 mol∙dm^{−3} [3] in this equation without the ion-size parameter of CdL^{2+}, we obtained immediately logK_{1} » 4.39. Here the I value was an average one for the Bz extraction system [3] . Similarly, the log{K_{1}(average)} values calculated from Equation (A1) with the I data [3] at 298 K were 4.57 for the NB system with B18C6 and CdPic_{2}, 4.60 for DCE, 4.56 for oDCBz, DCM, CF, and mX, 4.55 for CBu and dibutylether, 4.49 for CBz and TE, and 4.53 for BBz.

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