## 1. Introduction

In extraction systems with crown compounds (L), some extraction constants, such as

K_{ex} and

K_{ex±}, have been employed for evaluating their extraction-abilities and -selectivities [

1,

2,

3,

4,

5,

6]. Here, the constants

K_{ex} and

K_{ex±} have been generally defined as [MLA

_{z}]

_{org}/

P and [MLA

_{z}_{−1}^{+}]

_{org}[A

^{−}]

_{org}/

P, respectively, with

P = [M

^{z}^{+}][L]

_{org}[A

^{−}]

^{z} at

z = 1 and 2 [

1,

7,

8,

9]. The symbols M

^{z}^{+}, A

^{−}, and the subscript “org” denote a metal ion with the formal charge of

z+, a univalent pairing anion, and an organic phase, respectively. For evaluating the ability and selectivity of L for its extraction, many studies have been present [

1,

2,

3,

4,

5,

6,

7,

8,

9], but those for clarifying ionic strength (

I) dependences of the equilibrium constants seemed to be few [

10,

11]. Recently, one of the authors reported the

I and

I_{DCE} (with HNO

_{3} as an

I conditioner) dependences of the

K_{ex} and

K_{ex±} values in the silver picrate (AgPic) extraction with benzo-18-crown-6 ether (B18C6) into 1,2-dichloroethane (DCE), where

I_{DCE} refers to the

I value for the DCE phase [

12]. At the same time, conditional distribution constants (

K_{D,A} = [A

^{−}]

_{org}/[A

^{−}]) of the picrate ion Pic

^{−} (=A

^{−}) into the DCE (=org) phases have been determined [

12] and thereby distribution equilibrium potential-differences (dep; Δ

φ_{eq} as a symbol in an equation) have been evaluated [

7,

8,

12].

In the present paper, to expand such characterization [

12] for the AgPic extraction system to that for an M

^{II}Pic

_{2} extraction one, we determined at 298 K the

K_{ex},

K_{ex±}, and

K_{D,Pic} values for PbPic

_{2} extraction with B18C6 into benzene (Bz) under a co-presence of Cd(NO

_{3})

_{2} in a water phase. Then,

I and

I_{Bz} dependences of these equilibrium constants were mainly examined [

12]. Similar examinations were performed for other overall or component equilibrium-constants, such as

K_{Pb/PbL},

K_{ex,ip}, and

K_{1,Bz} (see Equations (6)–(8) for their definitions), derived from the above equilibrium constants. This study is expected to be useful for comparisons between the

K_{ex} and

K_{ex±} values, because magnitudes of their comparable values depend on

I [

10,

11] or

I_{Bz} in general. Consequently, such data relevant to

I and

I_{Bz} can make more precise comparisons between the values possible.

In addition, it had been pointed out that the presence of alkali metal and transition metal ions by high concentrations may cause significant interferences in the removal of Pb in acidic waste streams [

13]. Similarly, the M

^{z}^{+} separation with solvent extraction [

13] and membrane transport experiments has been studied [

14,

15]. However, these quantitative considerations based on any equilibrium constants have not been reported. This situation reveals the importance of these fundamental studies [

10,

11,

12] and this work as well, which can make a prediction for their separation more precise.

As well as the previous paper [

12], the dep values which were fundamentally based on the ion transfer of Pic

^{−} at the water/Bz interfaces were evaluated from the determined

K_{D,Pic} values [

16]. Moreover, the relationship between log

K_{ex±} and dep was quantitatively discussed [

7,

16].

The both M(II) ions are well-known as toxic metals to living things in nature [

17], but were employed here as simply model metal ones. Additionally, Bz was selected because a lot of data for the extraction of these M(II) ions with B18C6 or 18-crown-6 ether (18C6) is available [

1,

3,

6,

8,

18,

19,

20].

A competitive extraction between Pb(II) and Cd(II) with B18C6 into Bz had been assumed with the addition of Cd(II) in the water phase in the beginning of this study, compared with the log (

K_{ex,Pb}/

K_{ex,Cd}) value of 9.73 for the Pb(II) and Cd(II) extraction with 18C6 [

9]. However, against our plan, such an extraction behavior was not observed here.

## 2. Results and Discussion

#### 2.1. Determination of Composition of Extracted Species with Pb(II) at Some [Cd]_{t}/[Pb]_{t} Values

Determination of an M(II):L composition is based on the following

K_{ex} or

K_{ex±} definition [

1,

8,

19,

21]:

K_{ex} = [MLA

_{2}]

_{org}/

P and

K_{ex±} = [MLA

^{+}]

_{org}[A

^{−}]

_{org}/

P with

P = [M

^{2+}][L]

_{org}[A

^{−}]

^{2} at

z = 2. Taking common logarithms of both sides of these definitions and then rearranging them, we can easily obtain

with

D_{0} = [MLA

_{2}^{0}]

_{org}/[M

^{2+}],

D_{+} = [MLA

^{+}]

_{org}/[M

^{2+}] (see the

Section 2.9), and

K_{ex+} (=

K_{ex±}/

K_{D,A}) = [MLA

^{+}]

_{org}/[M

^{2+}][L]

_{org}[A

^{−}] [

1,

8,

19]. From applying the approximate that

D_{0} and

D_{+} nearly equal

D for Equations (1) and (2), respectively, the following equations were derived:

where

D is an experimental distribution ratio and defined as [Pb(II)]

_{(species analyzed by AAS measurement)}/([Pb(II)]

_{t} − [Pb(II)]

_{(species analyzed by AAS measurement)})

_{org}. In addition, [Pb(II)]

_{t} refers to a total concentration of Pb(NO

_{3})

_{2} employed. Therefore, in terms of a plot of log (

D/[A

^{−}]

^{2}) versus log [L]

_{Bz} come from Equation (1a) or that of log (

D/[A

^{−}]) from Equation (2a), we can determine the Pb(II):L compositions in the extraction systems from their slopes [

9].

Figure 1 shows such plots based on Equation (1a).

Experimentally-obtained slopes were 0.98 for the Pb(II)–B18C6 extraction system with 0 mmol·dm

^{−3} of Cd(II) (or [Cd]

_{t}/[Pb]

_{t} = 0), 1.0 for that with 0.58 of Cd(II) (or 1.06), 0.97 for that with 14 of Cd(II) (or 26.6), 0.98 for that with 48 of Cd(II) (or 88.4), and 1.0 for that with 97 of Cd(II) (or 178). From these results, we can see easily that the compositions of Pb(II):B18C6 are 1:1 for all the systems. In the present study, there was no need of employing Equation (2a). The compositions of Pb(II):Pic(−I) were speculated to be 1:2 from similarity to the systems [

3] reported before for M(II) extraction with 18C6 into Bz and from a charge balance in the Bz phases [

1,

8,

19,

21]: approximately [PbLPic

^{+}]

_{Bz} ≈ [Pic

^{−}]

_{Bz} from more-precisely 2[Pb

^{2+}]

_{Bz} + 2[PbL

^{2+}]

_{Bz} + [PbLPic

^{+}]

_{Bz} + [PbPic

^{+}]

_{Bz} ≈ [Pic

^{−}]

_{Bz} + [NO

_{3}^{−}]

_{Bz}, because it was expected that [PbLPic

^{+}]

_{Bz} >> 2[Pb

^{2+}]

_{Bz} + 2[PbL

^{2+}]

_{Bz} + [PbPic

^{+}]

_{Bz} and [Pic

^{−}]

_{Bz} >> [NO

_{3}^{−}]

_{Bz} [

9,

21].

#### 2.2. Determination of K_{ex}, K_{ex±}, and K_{D,Pic}

According to previous papers [

1,

8,

9,

22], the extraction-constant parameter (

K_{ex}^{mix}) has been proposed:

with

K_{D,A} ≈ [MLA

^{+}]

_{org}/[A

^{−}]. Using this equation, we can immediately obtain the

K_{ex} and

K_{D,A} values from a plot of log

K_{ex}^{mix} versus −log ([M

^{2+}][L]

_{org}[A

^{−}]). In addition, Equation (3) can be rewritten as:

Similarly, the

K_{ex±} value (with the

K_{ex} one; see

Table 1) can be obtained from a plot of log

K_{ex}^{mix} versus −log

P^{1/2}.

Figure 2 and

Figure 3 show examples of such plots.

From these plots, the

K_{D,Pic},

K_{ex±}, and

K_{ex} values were determined at 298 K.

Table 1 lists these extraction constants,

K_{ex} and

K_{ex±}, and the conditional distribution constants,

K_{D,Pic}, with averaged ionic strength-values (

I) for the water phase in the five [Cd]

_{t}/[Pb]

_{t} conditions. The

K_{ex} values determined with Equation (4) were equal or close to those with Equation (3). This fact raises the credibility of the values themselves and also shows the effects of Equations (3) and (4) on evaluation. The

K_{ex} and

K_{ex±} values at [Cd]

_{t}/[Pb]

_{t} = 0 were smaller than those (10

^{11.712} mol

^{−3}·dm

^{9} and 10

^{4.1} mol

^{−2}·dm

^{6} [

9]) reported before at

I = 0.0059 mol·dm

^{−3} for the PbPic

_{2} extraction with 18C6 into Bz.

#### 2.3. Dep Determination from K_{D,Pic}

From the log

K_{D,Pic} values listed in

Table 1, using the following equation and a standardized distribution constant (

K_{D,Pic}^{S}), we can easily obtain the dep (or Δ

φ_{eq}) values for the five [Cd]

_{t}/[Pb]

_{t} conditions at 298 K:

Here, the

K_{D,Pic}^{S} value is defined as the

K_{D,Pic} one at Δ

φ_{eq} = 0 V, equals antilog (Δ

φ_{Pic}^{0}′/0.05916) (=exp (Δ

φ_{Pic}^{0}′/0.02569) [

23]), and, as its common logarithmic value, −8.208 or −7.4473 is available from references [

24,

25]. In addition, the minus sign of −0.05916 (=−2.303

RT/

F ) and the symbol Δ

φ_{Pic}^{0}′ denote the formal charge of Pic

^{−} and the standard formal potential for the Pic

^{−} transfer across the water/Bz interface, respectively. We mainly employed the former value for the evaluation described below.

Table 1 lists the dep/V values evaluated from log

K_{D,Pic}^{S} = −8.208 [

24].

#### 2.4. Determination of K_{Pb/PbL}, K_{ex,ip}, K_{1,Bz}, K_{2,Bz}, and K_{D,PbL}

These constants can be evaluated from the following relations [

1,

18,

26,

27,

28].

Only for the

K_{Pb/PbL} values, they were obtained as the averages of

D/[B18C6]

_{Bz} at every [Cd]

_{t}/[Pb]

_{t} value [

27]. For the above evaluation at 298 K, 0.943 [

2] and 3.19 [

29] were used as the logarithmic values of

K_{D,B18C6} (=[B18C6]

_{Bz}/[B18C6]) and

K_{PbB18C6} (=[PbB18C6

^{2+}]/[Pb

^{2+}][B18C6]), respectively. These five logarithmic

K-values are summarized in

Table 2, together with the ionic strength-values (

I_{Bz}) for the Bz phase.

#### 2.5. Correlation between log K_{ex±} and Dep

We can obtain the following relation from the thermodynamic cycle of the PbLPic

^{+} extraction with Pic

^{−}.

Here, the log

K_{Pb/PbL}·

K_{1,Bz} term was in the range of 8.7 to 9.1 (see the data in

Table 2) and log

K_{D,Pic}^{S} (=−8.208 [

24] or −7.4473 [

25]) equals log

K_{D,Pic} at Δ

φ_{eq} = 0 V. Hence, we obtained to be −7.7 to −7.3 for the former

K_{D,Pic}^{S} value or the −6.2 to −5.8 for the latter one as the term of 2log

K_{D,Pic}^{S} + log

K_{Pb/PbL}⋅

K_{1,Bz} (see

Table 1 and

Table 2). In addition, 2

F/2.303

RT becomes 33.80 V

^{−1} at

T = 298.15 K. Rearranging Equation (11), we can immediately derive

From the regression analysis of an experimental plot in

Figure 4, the following line was obtained: log

K_{ex±} = (−5.

_{3} ± 1.

_{4}) − (27.

_{3} ± 4.

_{2})Δ

φ_{eq} at |

R| = 0.967, where the symbol

R denotes a correlation coefficient. This regression line is close to Equation (11a) which was estimated from the experimental

K values. This fact indicates the presence of dep, as similar to the results reported previously [

7,

8,

12,

16,

21,

22].

#### 2.6. I Dependences of log K_{ex} and log K_{ex,ip}

The thermodynamic extraction constant of

K_{ex} is

K_{ex}^{0} = [PbLPic

_{2}]

_{Bz}/

a_{Pb}[L]

_{Bz}(

a_{Pic})

^{2}, where

a_{Pb} and

a_{Pic} refer to activities of Pb

^{2+} and Pic

^{−} in the water phase, respectively, and it was assumed that [PbLPic

_{2}]

_{Bz} is equal to the activity in the Bz phase, because PbL

^{2+}(Pic

^{−})

_{2} is charge-less. The same is true of [B18C6]

_{Bz} too. Taking the common logarithms of both sides of the

K_{ex}^{0} definition, we can obtain

with

y_{Pb} =

a_{Pb}/[Pb

^{2+}] and

y_{Pic} =

a_{Pic}/[Pic

^{−}]. Introducing the extended Debye–Hückel (DH) equation [

30,

31] in Equation (12) and arranging it, the following equation was obtained:

Here, the DH equation was based on the mean activity coefficient and the symbol

${\xe5}_{\pm}$ denote the ion-size parameter [

30] in Å unit. Although the extended DH equation holds in the

I range of ≤0.1 mol·dm

^{−3} [

30] as you know, we approximately employed it for the condition of

I = 0.29 (see

Table 1).

Figure 5 shows curve-fittings of the plots for Equation (12a).

Its regression line was log

K_{ex} = (9.91 ± 0.03) − 6 × (0.5114)

$\sqrt{I}$/(1 + (3.4

_{0} ± 0.4

_{6})

$\sqrt{I}$) at

R = 0.980, where the coefficient

A was fixed to 0.5114 mol

^{−1/2}·dm

^{3/2} [

30] and the

${\xe5}_{\pm}$ value in water was evaluated to be 10 Å (=3.4

_{0}/0.3291) at 298 K.

Similarly, the log

K_{ex,ip} values were analyzed. Their constants were expressed as

The regression analysis of the plots yielded log K_{ex,ip} = (7.66 ± 0.03) − 6 × (0.5114)$\sqrt{I}$/(1 + (3.4_{1} ± 0.5_{1})$\sqrt{I}$) at R = 0.975 and then the a_{±} value was evaluated to be 10 Å. The accordance between Pb^{2+}–Pic^{−} distance and PbB18C6^{2+}–Pic^{−} one suggests that the former interaction between the Pb^{2+} and Pic^{−} ions in water saturated with Bz is equivalent with the latter one between PbB18C6^{2+} and Pic^{−}.

It is interesting that the evaluated

${\xe5}_{\pm}$ values are close to the sum (=11.

_{5} Å) of the ion-size parameters [

32] between Pb

^{2+} (4.5 Å) and Pic

^{−} (7 Å) for water. This

K_{ex,ip}^{0} value was well in accord with that (=10

^{7.66} mol

^{−1}·dm

^{3}) calculated from the thermodynamic cycle of

K_{ex,ip}^{0} ≈

K_{ex}^{0}K_{D,L}/

K_{PbL} (=10

^{9.91} × 10

^{0.943}/10

^{3.19}).

Considering that the

K_{ex} values are most precise ones of the some extraction constants determined here (see their errors in

Table 1), the fair dependences of log

K_{ex} on

I indicate a simple role of Cd(NO

_{3})

_{2} only as the ionic strength conditioner in the present extraction systems. In other words, the authors were not be able to clearly find out positive or negative effects of Cd(NO

_{3})

_{2} on the present Pb(II) extraction with B18C6 into Bz.

#### 2.7. I_{Bz} Dependences of log K_{1,Bz} and log K_{2,Bz}

Using

I_{Bz} and the DH limiting law [

30], both log

K_{1,Bz}^{0} and log

K_{2,Bz}^{0} can be expressed as

with

y_{PbLPic,Bz} ≈

y_{Pic,Bz} and

Rearranging Equations (14) and (15), we can obtain

Based on Equations (14a) and (15a), we prepared

Figure 6 from the data in

Table 2.

At the same time, these plots were analyzed by using the both equations. Their regression lines were log

K_{1,Bz} = (5.9

_{8} ± 0.4

_{9}) − 4 × (63 ± 89)

$\sqrt{{I}_{\mathrm{Bz}}}$ at

R = 0.449 except for the point of [Cd]

_{t}/[Pb]

_{t} = 178 and log

K_{2,Bz} = (6.8

_{1} ± 0.6

_{8}) − 2 × (282 ± 195)

$\sqrt{{I}_{\mathrm{Bz}}}$ at

R = 0.641. These lines intersected with each other at

I_{Bz}^{1/2} = 2.

_{7} × 10

^{−3} mol

^{1/2}·dm

^{−3/2}, yielding log

K_{1,Bz} = log

K_{2,Bz} = 5.3

_{1}. This fact indicates that, in the lower

I_{Bz} range less than 7.

_{1} × 10

^{−6} mol·dm

^{−3}, the

K_{2,Bz} values are larger than the

K_{1,Bz} ones. The latter values may be estimated to actually be the smaller values because of the approximation [

33] for the

K_{1,Bz} determination (see Equation (8)). Unlike the case of the CdPic

_{2}–B18C6 extraction system [

33], unfortunately, we do not have the procedure which corrects such deviations for the present extraction systems, because of a lack of adequate data used for the correction.

In addition, we tried curve-fittings to the two plots using the following equations:

with the approximation of 1 >>

$\sqrt{{I}_{\mathrm{Bz}}}$ (see

Table 2). Here, the symbols

b_{1} and

b_{2} denote empirical curve-fitting parameters [

30,

31] which were simply predicted in this study from the plot shapes (see

Figure 6). The regression analyses of the plots at 298 K gave log

K_{1,Bz} = (6.1

_{6} ± 0.8

_{0}) − 4 × (179 ± 247)

$\sqrt{{I}_{\mathrm{Bz}}}$ + (2.

_{3} ± 2.

_{7}) × 10

^{5}I_{Bz} at

R = 0.569 and log

K_{2,Bz} = (8.8

_{0} ± 0.9

_{8}) − 2 × (1625 ± 603)

$\sqrt{{I}_{\mathrm{Bz}}}$ + (7.

_{6} ± 3.

_{3}) × 10

^{5}I_{Bz} at 0.914 (see

Figure 7). Modifying these equations like the Davies one [

30,

31], their 2nd and 3rd terms became −4 × (179 ± 247)(

$\sqrt{{I}_{\mathrm{Bz}}}$ − (3.

_{2} ± 5.

_{9}) × 10

^{2}I_{Bz}) and −2 × (1625 ± 603)(

$\sqrt{{I}_{\mathrm{Bz}}}$ − (2.

_{3} ± 1.

_{3}) × 10

^{2}I_{Bz}), respectively. These

b_{1}/4

A_{Bz} and

b_{2}/2

A_{Bz} values of about 320 and 230 mol

^{−1/2}·dm

^{3/2} for the Bz phases are much larger than 0.3 [

31] for the aqueous solution at 298 K. Equation (14b) intersects Equation (15b) around

I_{Bz}^{1/2} = 3.

_{3} × 10

^{−3} mol

^{1/2}·dm

^{−3/2}, yielding log

K_{1,Bz} = log

K_{2,Bz} = 6.3

_{0}, and then their two lines equal with each other within the experimental errors (see the plots in

Figure 7).

In comparison of Equations (14a) and (15a) with Equations (14b) and (15b), the point of intersection changed from I_{Bz}^{1/2}/mol^{1/2}·dm^{−3/2} = 0.002_{7} to 0.003_{3}, while their corresponding log K_{1,Bz} (=log K_{2,Bz}) value changed from 5.3 to 6.3.

At least in the lower

I_{Bz} range less than 1.

_{1} × 10

^{−5} mol·dm

^{−3}, the

K_{2,Bz} values are larger than the

K_{1,Bz} ones. From the results of the calculation based on Equations (14a,b) and (15a,b), the relation of

K_{1,Bz} <

K_{2,Bz} holds in the range less than (0.7

_{1} − 1.

_{1}) × 10

^{−5} mol·dm

^{−3} (see above). According to the paper [

8], such a fact suggests a structural change around Pb(II) in the reaction of Pb(B18C6)Pic

^{+}_{Bz} + Pic

^{−}_{Bz} ⇌ Pb(B18C6)Pic

_{2,Bz}, such as Cd(18C6)Pic

_{2,Bz} of the Cd(II) extraction systems [

33]. Trends similar to

K_{1,Bz} <

K_{2,Bz} are observed in the reactions of Cd18C6

^{2+} with Pic

^{−}, Cl

^{−}, and Br

^{−} in the Bz phases for fixed

I_{Bz} values [

1,

8]. The higher

I_{Bz} range may lead to the formation of ion-pair complexes with other coordination structures around Pb(II), although their structures are not clear.

Table 3 shows results for the both estimated values from Equations (14a) and (15a) and those from Equations (14b) and (15b). In comparison with differences, |dif.|, in

K_{1,Bz} and

K_{2,Bz} between the experimental and estimated values, the |dif.| values estimated from Equations (14b) and (15b) were essentially smaller than those done from Equations (14a) and (15a). Especially, the former equations seem to be superior to the latter ones in the

I_{Bz} range, namely the present experimental [Cd]

_{t}/[Pb]

_{t} range, of 4 × 10

^{−7} to 8 × 10

^{−6} mol·dm

^{−3} in the cases of the prediction of

K_{2,Bz}. Unfortunately, chemical and physical meanings of

b_{1} and

b_{2} are not clear still now.

#### 2.8. I_{Bz} Dependences of log K_{ex±}^{0}′, log K_{D,Pic}^{0}′, and log K_{Pb/PbL}^{0}

The thermodynamic equilibrium constant

K_{ex±}^{0} is equal to (

y_{±,Bz})

^{2}K_{ex±}^{0}′, with

y_{±,Bz} = (

y_{PbLPic,Bz}·

y_{Pic,Bz})

^{1/2} and

K_{ex±}^{0}′ = [PbLPic

^{+}]

_{Bz}[Pic

^{−}]

_{Bz}/(

a_{Pb}[L]

_{Bz}(

a_{Pic})

^{2}). Taking the common logarithms of the both sides in this equation and rearranging it with the DH limiting law, we can easily obtain

Figure 8 shows the plot of log

K_{ex±}^{0}′ versus

I_{Bz}^{1/2} based on Equation (16). The regression analysis of this plot gave the equation of log

K_{ex±}^{0}′ = (3.1

_{1} ± 0.6

_{9}) + 2 × (315 ± 196)

$\sqrt{{I}_{\mathrm{Bz}}}$ at

R = 0.680. From this

K_{ex,±}^{0} value and the

K_{ex}^{0} one, we calculated log

K_{2,Bz}^{0} to be 6.8

_{0} ± 0.6

_{9}, being in good agreement with that (=6.8) evaluated from Equation (15a).

Similarly, the plot of log

K_{D,Pic}^{0}′ versus

I_{Bz}^{1/2} was performed in

Figure 9, where

K_{D,Pic}^{0}′ is defined as [Pic

^{−}]

_{Bz}/

a_{Pic}. This plot is based on the equation

The regression analysis yielded log K_{D,Pic}^{0}′ = (−3.0_{0} ± 0.3_{6}) + (258 ± 210)$\sqrt{{I}_{\mathrm{Bz}}}$ at R = 0.578.

In addition, the analysis was tried by using an equation similar to Equations (14b) and (15b) with 1 >>

$\sqrt{{I}_{\mathrm{Bz}}}$ Its regression line was

at

R = 0.995. Here, the latter two terms are rearranged into (1949 ± 154)(

$\sqrt{{I}_{\mathrm{Bz}}}$ − (245 ± 29)

I_{Bz}). This value, 245 mol

^{−1/2}·dm

^{3/2}, is comparable to the

b_{1}/4

A_{Bz} (=~320) and

b_{2}/2

A_{Bz} (=~230) values estimated above.

Lastly, the log

K_{Pb/PbL}^{0}′ (=log ([PbL

^{2+}]

_{Bz}/

a_{Pb}[L]

_{Bz})) values were analyzed. This constant was related with the log

K_{Pb/PbL}^{0} (=log (

a_{PbL,Bz}/

a_{Pb}[L]

_{Bz})) value by the following equation:

The log

K_{Pb/PbL}^{0}′ values were plotted against the

I_{Bz}^{1/2} ones. The regression line based on Equation (18) was log

K_{Pb/PbL}^{0}′ = (3.546 ± 0.001) + 4 × (2.1

_{6} ± 0.1

_{6})

$\sqrt{{I}_{\mathrm{Bz}}}$ at

R = 0.997, except for the two points of

I_{Bz} = 4.

_{4} × 10

^{−7} and 1.

_{7} × 10

^{−6} mol·dm

^{−3} (see

Table 2). These two log

K_{Pb/PbL}^{0}′ values excluded from the regression analysis are included in the regression line within experimental errors. However, the

A_{Bz} value is much smaller than the others. In addition, the analysis was tried by using an equation similar to Equations (14b) and (15b). However, its regression line showed the result of

A_{Bz} < 0.

By a combination with log K_{D,Pic}^{0}, the log K_{Pb/PbL}^{0} can be changed into log K_{ex2±}^{0} (=log (a_{PbL,Bz}(a_{Pic,Bz})^{2}/a_{Pb}[L]_{Bz}(a_{Pic,Bz})^{2})) = log K_{Pb/PbL}^{0} + 2log K_{D,Pic}^{0}. Thus, the log K_{ex2±}^{0} value was estimated to be −2.4_{5} ± 0.3_{7} from Equation (17) (the linear type) or −4.9_{6} ± 0.1_{3} from Equation (17a) (the f (p) = a + bp + cp^{2} type). Using log K_{ex±}^{0} = 3.1_{1} obtained from Equation (16) and log K_{1,Bz}^{0} = 5.9_{8} from Equation (14a) (the linear type), the log K_{ex2±}^{0} value was calculated to be −2.8_{7} ± 0.8_{5}. On the other hand, the log K_{ex2±}^{0} value became −3._{1} ± 1._{1} in the calculation with log K_{1,Bz}^{0} = 6.1_{6} from Equation (14b) (the f (p) = a + bp + cp^{2} type). Except for −4.9 from Equation (17a), the values calculated from the three equations agreed with each other within their calculation errors. According to the thermodynamic cycle, the relation of K_{ex±} = K_{Pb/PbL}K_{1,Bz}(K_{D,Pic})^{2} holds. From this relation, we obtained log K_{ex±}^{0} = 3.5_{4} ± 0.7_{2} (=log K_{Pb/PbL}^{0} + log K_{1,Bz}^{0} + 2log K_{D,Pic}^{0}). This value is in agreement with that (=3.1) calculated from Equation (16) within the calculation error of ±0.7. In addition, the same calculation was performed with the values obtained from the polynomial Equations (14b) and (17a). Its value was 1.2_{0} ± 0.8_{3}, being much smaller than 3.1. These results suggest that the linear-type equation is the more reliable than the polynomial-type one, from the thermodynamic points of view.

From the four experimental

A_{Bz}/mol

^{−1/2}·dm

^{3/2} values based on Equations (14a), (15a), (16), and (17), except for the value obtained from Equation (18), their average value was estimated to be 230. Consequently, this

A_{Bz} value for Bz saturated with water was about 2-times larger than that (=103.3 mol

^{−1/2}·dm

^{3/2}) calculated for pure Bz with

ε_{r} = 2.275 [

2] at 298.15 K. To agree with this conclusion, however, a reasonable reason will be required for the omission of the result of Equation (18).

#### 2.9. A Try for Estimation of Detailed Separation Factor

According to the previous papers [

8,

33], the following relations hold for given Pic

^{−} and L.

with

Here, D_{0,Pb}, D_{+,Pb}, and D_{2+,Pb} show the values of M = Pb at z = 0, 1, and 2, respectively. The same is true of the definitions for the Cd(II) (=M(II)) extraction system with B18C6.

For example, Equation (19) can be expressed as:

Assuming that [Cd

^{2+}]/[Pb

^{2+}] approximately equals [Cd

^{2+}]

_{t}/[Pb

^{2+}]

_{t}, then we can estimate the more detailed value than the separation factor. In the Pb(II) extraction with Cd(II) by B18C6 into Bz, the log (

D_{0,Pb}/

D_{0,Cd}) value was 7.09 in which the log

K_{ex} value (=9.44

_{8}) was estimated at

I = 0.095 [

33] from the regression line of

Figure 5. From Equation (25) and [Cd

^{2+}]

_{t}/[Pb

^{2+}]

_{t} = 55.

_{5} estimated from a correlation between [Cd]

_{t}/[Pb]

_{t} and

I in

Table 1, the log ([PbLPic

_{2}]

_{Bz}/[CdLPic

_{2}]

_{Bz}) value became 5.35. At least this result shows that the actual separation of Pb(II) from a test solution with an 56 excess amount of Cd(II) is possible. The same can be true of an application based on the handling for

D_{+,Pb}/

D_{+,Cd} and

D_{2+,Pb}/

D_{2+,Cd}, if the

K_{ex±,Cd} and

K_{Cd/CdL} values are determined about the CdPic

_{2} extraction with B18C6 (=L) into Bz.

#### 2.10. Relative Concentrations of the Three Species Extracted into Bz

We can immediately calculate relative concentrations of PbLPic

_{2}, PbLPic

^{+}, and PbL

^{2+} in the Bz phases from the

D_{0,Pb},

D_{+,Pb}, and

D_{2+,Pb} values, respectively [

8,

24,

30]. For example, the percentage of the relative concentration of PbLPic

_{2} can be obtained from 100

D_{0,Pb}/(

D_{0,Pb} +

D_{+,Pb} +

D_{2+,Pb}). In addition, the concentrations of PbLPic

^{+} and PbL

^{2+} were evaluated from similar equations. The thus-calculated values were: 46% for PbLPic

_{2}, 10% for PbLPic

^{+}, and 44% for PbL

^{2+} at

I = 0.0074 mol·dm

^{−3} (or [Cd]

_{t}/[Pb]

_{t} = 0); 38%, 25%, and 37% at 0.0060 (or 1.06

_{1}), respectively; 35%, 32%, and 34% at 0.048 (or 26.6

_{3}), respectively; 24%, 53%, and 23% at 0.15 (or 88.4

_{8}), respectively; and 26%, 49%, and 25% at 0.29 (or 177.

_{5}), respectively.

One can see easily that the distribution of PbLPic

_{2} and PbL

^{2+} into Bz is dominant in the lower

I or [Cd]

_{t}/[Pb]

_{t} values, while that of CdLPic

^{+} is dominant in the higher

I ones. That is, in the

I range more than 0.15 mol·dm

^{−3}, the distribution of PbLPic

^{+} with Pic

^{−} may be dominant, compared with those of both PbLPic

_{2}^{0} and PbL

^{2+} with 2Pic

^{−}. Now, the authors cannot clearly explain this result; namely, in the higher

I range, why is the univalent cationic complex more extractable to the Bz phase than the other complexes are? Conversely, can they call this phenomenon “salting out effect”? However, these data can be useful for the discussion of membrane transport phenomena with L [

34]. That is, what species mainly transfer through the membrane?

## 4. Conclusions

The thermodynamic values for K_{ex}, K_{ex±}, K_{Pb/PbL}, K_{ex,ip}, and K_{D,Pic} were determined at 298 K. The same is also true of the K_{1,Bz}^{0} and K_{2,Bz}^{0} values at I_{Bz} → 0 for the simple Bz phases. It was demonstrated that the thermodynamic relations, K_{ex}^{0} ≈ K_{ex,ip}^{0}K_{PbL}/K_{D,L}, K_{2,Bz}^{0} = K_{ex}^{0}/K_{ex±}^{0}, K_{ex2±}^{0} = K_{Pb/PbL}^{0}(K_{D,Pic}^{0})^{2}, K_{ex2±}^{0} = K_{ex±}^{0}/K_{1,Bz}^{0}, and K_{ex±}^{0} = K_{Pb/PbL}^{0}K_{1,Bz}^{0}(K_{D,Pic}^{0})^{2}, hold in the system. It seems that the linear equation is superior to the polynomial-type one for the I_{Bz} dependences of the above equilibrium constants, although the R values with the former were less than those with the latter. Consequently, these results make comparisons between the K_{ex}, K_{ex±}, or K_{1,Bz} values reported in different I or I_{org} conditions possible. However, there may be a fact that this study must be applied to the more practical extraction and separation systems. Moreover, it was clarified experimentally that log K_{ex±} is proportional to dep.

At least, the separation of Pb(II) by B18C6 into the Bz phase from the mixtures at [Cd]_{t}/[Pb]_{t} ≈ 60 was confirmed experimentally and theoretically. This condition exceeds [PbLPic_{2}]_{Bz}/[CdLPic_{2}]_{Bz} = 2.2 × 10^{5} at B18C6 (=L) and satisfies a measure (=10^{4}) of the separation factor. The K_{ex,Pb}/K_{ex,Cd} ratio at the fixed I condition can promise more precise evaluation of Pb(II) selectivity of L against Cd(II), compared with the ratio calculated at different I conditions. While, the co-presence of Cd(NO_{3})_{2} less than 180 of [Cd]_{t}/[Pb]_{t} has no clear effect to the experimental Pb(II) extraction with B18C6 into Bz. This Cd(II) salt in the present system acted only as the ionic strength conditioner in the water phases.