Pb(II) Extraction with Benzo-18-Crown-6 Ether into Benzene under the Co-Presence of Cd(II) Nitrate in Water

Abstract

Extraction of Pb(II) with picrate ion [Pic⁻] and 0, 0.58, 15, 48, or 97 mmol·dm⁻³ Cd(NO₃)₂ by benzo-18-crown-6 ether (B18C6; L as its symbol) into benzene (Bz) was studied. Three kinds of extraction constants, K_ex, K_ex±, and K_Pb/PbL (or K_ex2±), were determined at 298 K: these constants were defined as [PbLPic₂]_Bz/P, [PbLPic⁺]_Bz[Pic⁻]_Bz/P, and [PbL²⁺]_Bz/[Pb²⁺][L]_Bz (or [PbL²⁺]_Bz([Pic⁻]_Bz)²/P), respectively. The symbol P shows [Pb²⁺][L]_Bz[Pic⁻]² and the subscript “Bz” denotes the Bz phase, Bz saturated with water. Simultaneously, conditional distribution constants, K_D,Pic (=[Pic⁻]_Bz/[Pic⁻]), of Pic⁻ with distribution equilibrium-potential differences (dep) were determined. Then, based on the above four constants and others, the component equilibrium constants of K_1,Bz (=[PbLPic⁺]_Bz/[PbL²⁺]_Bz[Pic⁻]_Bz), K_2,Bz (=[PbLPic₂]_Bz/[PbLPic⁺]_Bz[Pic⁻]_Bz), and K_D,PbL (=[PbL²⁺]_Bz/[PbL²⁺]) were obtained. Using these constants, the Pb(II) extraction with B18C6 under the co-presence of Cd(II) in the water phase was characterized. In such a characterization, I and I_Bz dependences on the constants were mainly discussed, where their symbols denote the ionic strength of the water phase and that of the Bz one, respectively.

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 HNO₃ 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; Δφ_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^IIPic₂ extraction one, we determined at 298 K the K_ex, K_ex±, and K_D,Pic values for PbPic₂ extraction with B18C6 into benzene (Bz) under a co-presence of Cd(NO₃)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 = [MLA₂]_org/P and K_ex± = [MLA⁺]_org[A⁻]_org/P with P = [M²⁺][L]_org[A⁻]² at z = 2. Taking common logarithms of both sides of these definitions and then rearranging them, we can easily obtain

log (D₀/[A⁻]²) = log K_ex + log [L]_org

(1)

and log (D₊/[A⁻]) = log K_ex+ + log [L]_org

(2)

with D₀ = [MLA₂⁰]_org/[M²⁺], D₊ = [MLA⁺]_org/[M²⁺] (see the Section 2.9), and K_ex+ (=K_ex±/K_D,A) = [MLA⁺]_org/[M²⁺][L]_org[A⁻] [1,8,19]. From applying the approximate that D₀ and D₊ nearly equal D for Equations (1) and (2), respectively, the following equations were derived:

log (D/[A⁻]²) ≈ log K_ex + log [L]_org

(1a)

and log (D/[A⁻]) ≈ log K_ex+ + log [L]_org

(2a)

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(NO₃)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[Pb²⁺]_Bz + 2[PbL²⁺]_Bz + [PbLPic⁺]_Bz + [PbPic⁺]_Bz ≈ [Pic⁻]_Bz + [NO₃⁻]_Bz, because it was expected that [PbLPic⁺]_Bz >> 2[Pb²⁺]_Bz + 2[PbL²⁺]_Bz + [PbPic⁺]_Bz and [Pic⁻]_Bz >> [NO₃⁻]_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:

log K_ex^mix = log {([MLA₂]_org + [MLA⁺]_org + [ML²⁺]_org + …)/P}
≈ log {K_ex + (K_D,A/[M²⁺][L]_org[A⁻])}

(3)

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 ([M²⁺][L]_org[A⁻]). In addition, Equation (3) can be rewritten as:

$$\log K_{{\mathrm{ex}}^{\mathrm{mix}}}\approx \log \left(K_{\mathrm{ex}}+\sqrt{K_{\mathrm{ex}\pm }/P}\right)$$

(4)

Similarly, the K_ex± value (with the K_ex one; see

Table 1. Basic data for the Pb(II) extraction by B18C6 from the water phase with co-presence of Cd(NO₃)₂ into Bz at 298 K.

[Cd]t/[Pb]t 1	I2/mol dm−3	log Kex [] 3	log Kex± 3 (log yPic 4)	log KD,Pic (Δφeq 5/V)
0	0.0074	9.715 ± 0.006 (9.70 ± 0.01)	2.66 ± 0.25 (−0.04)	−3.21 ± 0.14 (−0.30)
1.06	0.0060	9.68 ± 0.02 (9.61 ± 0.04)	3.90 ± 0.23 (−0.03)	−2.44 ± 0.13 (−0.34)
26.6	0.048	9.58 ± 0.02 (9.51 ± 0.03)	3.97 ± 0.25 (−0.07)	−2.37 ± 0.16 (−0.35)
88.5	0.15	9.39 ± 0.03 (9.24 ± 0.05)	3.95 ± 0.19 (−0.10)	−2.60 ± 0.15 (−0.33)
178	0.29	9.31 ± 0.02 (9.21 ± 0.03)	3.68 ± 0.16 (−0.12)	−2.65 ± 0.09 (−0.33)

¹ [Pb(NO₃)₂]_t = 5.48 × 10⁻⁴ mol·dm⁻³. ² Averaged ionic strength for the water phase. ³ Values determined from Equation (4). ⁴ Logarithmic activity coefficient of Pic⁻ in water, calculated from the I value. ⁵ Dep values calculated from Equation (5).

) 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·dm⁹ and 10^4.1 mol⁻²·dm⁶ [9]) reported before at I = 0.0059 mol·dm⁻³ for the PbPic₂ 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 Δφ_eq) values for the five [Cd]_t/[Pb]_t conditions at 298 K:

Δφ_eq = −0.05916(log K_D,Pic − log K_D,Pic^S) = Δφ_Pic⁰′ − 0.05916log K_D,Pic

(5)

Here, the K_D,Pic^S value is defined as the K_D,Pic one at Δφ_eq = 0 V, equals antilog (Δφ_Pic⁰′/0.05916) (=exp (Δφ_Pic⁰′/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 (=−2.303RT/F ) and the symbol Δφ_Pic⁰′ 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].

log K_Pb/PbL = log ([PbL²⁺]_Bz/[Pb²⁺][L]_Bz) ≈ log (D/[L]_Bz),

(6)

log K_ex,ip = log ([PbLPic₂]_Bz/[PbL²⁺][Pic⁻]²) = log (K_exK_D,L/K_PbL),

(7)

log K_1.Bz = log ([PbLPic⁺]_Bz/[PbL²⁺]_Bz[Pic⁻]_Bz) ≈ log {K_ex±/K_Pb/PbL(K_D,Pic)²}

(8)

= log (K_ex±/K_ex2±),

(8a)

log K_2,Bz = log ([PbLPic₂]_Bz/[PbLPic⁺]_Bz[Pic⁻]_Bz) = log (K_ex/K_ex±),

(9)

and log K_D,PbL = log ([PbL²⁺]_Bz/[PbL²⁺]) ≈ log (K_Pb/PbLK_D,L/K_PbL).

(10)

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 (=[PbB18C6²⁺]/[Pb²⁺][B18C6]), respectively. These five logarithmic K-values are summarized in

Table 2. Some equilibrium constants obtained from the Pb(II) extraction experiments by L = B18C6 from the water phase with co-presence of Cd(NO₃)₂ into Bz at 298 K.

[Cd]t/[Pb]t	log KPb/PbL (log yPb 1)	log Kex,ip	log K1,Bz (IBz 2/10−6)	log K2,Bz	log KD,PbL
0	3.42 ± 0.05 (−0.16)	7.47	5.7 ± 0.3 (0.44)	7.1 ± 0.2	1.18
1.06	3.42 ± 0.08 (−0.14)	7.143	5.4 ± 0.3 (2.6)	5.8 ± 0.2	1.17
26.6	3.23 ± 0.12 (−0.34)	7.34	5.5 ± 0.4 (2.7)	5.6 ± 0.2	0.98
88.5	3.13 ± 0.21 (−0.50)	7.15	6.0 ± 0.4 (1.7)	5.4 ± 0.2	0.88
178	2.95 ± 0.28 (−0.62)	7.06	6.0 ± 0.3 (7.8)	5.6 ± 0.2	0.71

¹ Logarithmic activity coefficient of Pb²⁺ in water, calculated from the averaged I value. ² Averaged ionic strength for the Bz phase.

, 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⁻.

log K_ex± = 2log K_D,Pic + log K_Pb/PbL + log K_1,Bz
= 2log K_D,Pic^S − 2(F/2.303RT)Δφ_eq + log K_Pb/PbL·K_1,Bz

(11)

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 Δφ_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, 2F/2.303RT becomes 33.80 V⁻¹ at T = 298.15 K. Rearranging Equation (11), we can immediately derive

log K_ex± ≈ (−7.7 to −7.3) − 33.80Δφ_Equation

(11a)

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)Δφ_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_ex⁰ = [PbLPic₂]_Bz/a_Pb[L]_Bz(a_Pic)², where a_Pb and a_Pic refer to activities of Pb²⁺ and Pic⁻ in the water phase, respectively, and it was assumed that [PbLPic₂]_Bz is equal to the activity in the Bz phase, because PbL²⁺(Pic⁻)₂ is charge-less. The same is true of [B18C6]_Bz too. Taking the common logarithms of both sides of the K_ex⁰ definition, we can obtain

log K_ex⁰ = log K_ex − log {y_Pb(y_Pic)²}

(12)

with y_Pb = a_Pb/[Pb²⁺] 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:

$$\log K_{\mathrm{ex}}=\log K_{{\mathrm{ex}}^0}-6A\sqrt{I}/(1+B{å}_{\pm }\sqrt{I})$$

(12a)

Here, the DH equation was based on the mean activity coefficient and the symbol ${å}_{\pm }$ 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 + (3.4₀ ± 0.4₆)$\sqrt{I}$) at R = 0.980, where the coefficient A was fixed to 0.5114 mol^−1/2·dm^3/2 [30] and the ${å}_{\pm }$ value in water was evaluated to be 10 Å (=3.4₀/0.3291) at 298 K.

Similarly, the log K_ex,ip values were analyzed. Their constants were expressed as

$$\log K_{\mathrm{ex},\mathrm{ip}}=\log K_{\mathrm{ex},{\mathrm{ip}}^0}-6A\sqrt{I}/(1+B{å}_{\pm }\sqrt{I})$$

(13)

with K_ex,ip⁰ = [PbLPic₂]_Bz/a_PbL(a_Pic)² = K_ex,ip/y_PbL(y_Pic)²

(13a)

The regression analysis of the plots yielded log K_ex,ip = (7.66 ± 0.03) − 6 × (0.5114)$\sqrt{I}$/(1 + (3.4₁ ± 0.5₁)$\sqrt{I}$) at R = 0.975 and then the a_± value was evaluated to be 10 Å. The accordance between Pb²⁺–Pic⁻ distance and PbB18C6²⁺–Pic⁻ one suggests that the former interaction between the Pb²⁺ and Pic⁻ ions in water saturated with Bz is equivalent with the latter one between PbB18C6²⁺ and Pic⁻.

It is interesting that the evaluated ${å}_{\pm }$ values are close to the sum (=11.₅ Å) of the ion-size parameters [32] between Pb²⁺ (4.5 Å) and Pic⁻ (7 Å) for water. This K_ex,ip⁰ value was well in accord with that (=10^7.66 mol⁻¹·dm³) calculated from the thermodynamic cycle of K_ex,ip⁰ ≈ K_ex⁰K_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(NO₃)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(NO₃)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,Bz⁰ and log K_2,Bz⁰ can be expressed as

$$\log K_{1,{\mathrm{Bz}}^0}\approx \log K_{1,\mathrm{Bz}}-\log y_{\mathrm{PbL},\mathrm{Bz}}=\log K_{1,\mathrm{Bz}}+4A_{Bz}\sqrt{I_{\mathrm{Bz}}}$$

(14)

with y_PbLPic,Bz ≈ y_Pic,Bz and

$$\log K_{2,{\mathrm{Bz}}^0}=\log K_{2,\mathrm{Bz}}-\log \left(y_{\mathrm{PbLPic},\mathrm{Bz}}·y_{\mathrm{Pic},\mathrm{Bz}}\right)=\log K_{2,\mathrm{Bz}}+2A_{Bz}\sqrt{I_{\mathrm{Bz}}}.$$

(15)

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

$$\log K_{1,\mathrm{Bz}}\approx \log K_{1,{\mathrm{Bz}}^0}-4A_{Bz}\sqrt{I_{\mathrm{Bz}}}$$

(14a)

$$\mathrm{and}\log K_{2,\mathrm{Bz}}=\log K_{2,{\mathrm{Bz}}^0}-2A_{Bz}\sqrt{I_{\mathrm{Bz}}}$$

(15a)

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 = (5.9₈ ± 0.4₉) − 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 = (6.8₁ ± 0.6₈) − 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 = 5.3₁. 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 CdPic₂–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:

$$\log K_{1,\mathrm{Bz}}\approx \log K_{1,{\mathrm{Bz}}^0}-4A_{\mathrm{Bz}}\sqrt{I_{\mathrm{Bz}}}+b_1{I}_{\mathrm{Bz}}$$

(14b)

$$\mathrm{and}\log K_{2,\mathrm{Bz}}=\log K_{2,{\mathrm{Bz}}^0}-2A_{\mathrm{Bz}}\sqrt{I_{\mathrm{Bz}}}+b_2{I}_{\mathrm{Bz}}$$

(15b)

with the approximation of 1 >> $\sqrt{I_{\mathrm{Bz}}}$ (see Table 2). Here, the symbols b₁ and b₂ 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 = (6.1₆ ± 0.8₀) − 4 × (179 ± 247)$\sqrt{I_{\mathrm{Bz}}}$ + (2.₃ ± 2.₇) × 10⁵I_Bz at R = 0.569 and log K_2,Bz = (8.8₀ ± 0.9₈) − 2 × (1625 ± 603)$\sqrt{I_{\mathrm{Bz}}}$ + (7.₆ ± 3.₃) × 10⁵I_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) × 10²I_Bz) and −2 × (1625 ± 603)($\sqrt{I_{\mathrm{Bz}}}$ − (2.₃ ± 1.₃) × 10²I_Bz), respectively. These b₁/4A_Bz and b₂/2A_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 = 6.3₀, 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 = 0.002₇ to 0.003₃, 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 (0.7₁ − 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 Cd18C6²⁺ 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. Comparison between Equations (14a) and (15a) and Equations (14b) and (15b) in the re-production of the experimental K_1,Bz and K_2,Bz values ¹ at 298 K.

[Cd]t/[Pb]t	log K1,Bz				log K2,Bz
	Equation (14a)	|Dif.| 2	Equation (14b)	|Dif.| 2	Equation (15a)	|Dif.| 2	Equation (15b)	|Dif.| 2
0	5.81	0.2	6.16	0.2	6.44	0.6	6.98	0.1
1.06	5.58	0.2	5.61	0.2	5.91	0.1	5.54	0.2
26.6	5.56	0.1	5.61	0.1	5.87	0.3	5.50	0.1
88.5	5.66	0.4	5.62	0.4	6.08	0.6	5.87	0.4
178	5.28	0.8	5.96	0.1	5.24	0.4	5.62	0.0

¹ See Table 2 for these values. ² Absolute value for the difference between the experimental K_1,Bz or K_2,Bz value and their estimated one.

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 b₁ and b₂ are not clear still now.

2.8. I_Bz Dependences of log K_ex±⁰′, log K_D,Pic⁰′, and log K_Pb/PbL⁰

The thermodynamic equilibrium constant K_ex±⁰ is equal to (y_±,Bz)²K_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

$$\log K_{\mathrm{ex}{\pm }^{0\prime }}=\log (K_{\mathrm{ex}\pm }/y_{\mathrm{Pb}}{\left(y_{\mathrm{Pic}}\right)}^2)=\log K_{\mathrm{ex}{\pm }^0}+2A_{\mathrm{Bz}}\sqrt{I_{\mathrm{Bz}}}$$

(16)

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′ = (3.1₁ ± 0.6₉) + 2 × (315 ± 196)$\sqrt{I_{\mathrm{Bz}}}$ at R = 0.680. From this K_ex,±⁰ value and the K_ex⁰ one, we calculated log K_2,Bz⁰ to be 6.8₀ ± 0.6₉, being in good agreement with that (=6.8) evaluated from Equation (15a).

Similarly, the plot of log K_D,Pic⁰′ versus I_Bz^1/2 was performed in Figure 9, where K_D,Pic⁰′ is defined as [Pic⁻]_Bz/a_Pic. This plot is based on the equation

$$\log K_{\mathrm{D},{\mathrm{Pic}}^{0\prime }}=\log (K_{D,Pic}/y_{\mathrm{Pic}})=\log K_{\mathrm{D},{\mathrm{Pic}}^0}+A_{\mathrm{Bz}}\sqrt{I_{\mathrm{Bz}}}$$

(17)

The regression analysis yielded log K_D,Pic⁰′ = (−3.0₀ ± 0.3₆) + (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

$$\log K_{\mathrm{D},{\mathrm{Pic}}^{0\prime }}=\left(-{4.2}_5\pm {0.1}_2\right)+\left(1949\pm 154\right)\sqrt{I_{\mathrm{Bz}}}+\left(-{4.7}_6\pm {0.4}_2\right)\times {10}^5{I}_{\mathrm{Bz}}$$

(17a)

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 b₁/4A_Bz (=~320) and b₂/2A_Bz (=~230) values estimated above.

Lastly, the log K_Pb/PbL⁰′ (=log ([PbL²⁺]_Bz/a_Pb[L]_Bz)) values were analyzed. This constant was related with the log K_Pb/PbL⁰ (=log (a_PbL,Bz/a_Pb[L]_Bz)) value by the following equation:

$$\log K_{\mathrm{Pb}/{\mathrm{PbL}}^{0\prime }}=\log \left(K_{\mathrm{Pb}/\mathrm{PbL}}/y_{\mathrm{Pb}}\right)=\log K_{\mathrm{Pb}/{\mathrm{PbL}}^0}+4A_{\mathrm{Bz}}\sqrt{I_{\mathrm{Bz}}}$$

(18)

The log K_Pb/PbL⁰′ values were plotted against the I_Bz^1/2 ones. The regression line based on Equation (18) was log K_Pb/PbL⁰′ = (3.546 ± 0.001) + 4 × (2.1₆ ± 0.1₆)$\sqrt{I_{\mathrm{Bz}}}$ 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/PbL⁰′ 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,Pic⁰, the log K_Pb/PbL⁰ 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/PbL⁰ + 2log K_D,Pic⁰. Thus, the log K_ex2±⁰ value was estimated to be −2.4₅ ± 0.3₇ from Equation (17) (the linear type) or −4.9₆ ± 0.1₃ from Equation (17a) (the f (p) = a + bp + cp² type). Using log K_ex±⁰ = 3.1₁ obtained from Equation (16) and log K_1,Bz⁰ = 5.9₈ from Equation (14a) (the linear type), the log K_ex2±⁰ value was calculated to be −2.8₇ ± 0.8₅. On the other hand, the log K_ex2±⁰ value became −3.₁ ± 1.₁ in the calculation with log K_1,Bz⁰ = 6.1₆ from Equation (14b) (the f (p) = a + bp + cp² 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/PbLK_1,Bz(K_D,Pic)² holds. From this relation, we obtained log K_ex±⁰ = 3.5₄ ± 0.7₂ (=log K_Pb/PbL⁰ + log K_1,Bz⁰ + 2log K_D,Pic⁰). 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 1.2₀ ± 0.8₃, 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.

log (K_ex,Pb/K_ex,Cd) = log (D_0,Pb/D_0,Cd),

(19)

log (K_ex±,Pb/K_ex±,Cd) = log (D_+,Pb/D_+,Cd),

(20)

and log (K_Pb/PbL/K_Cd/CdL) = log (D_2+,Pb/D_2+,Cd)

(21)

with

D_0,Pb = [PbLPic₂]_Bz/[Pb²⁺] = K_ex[L]_Bz[Pic⁻]²,

(22)

D_+,Pb = [PbLPic⁺]_Bz/[Pb²⁺] = K_ex±[L]_Bz[Pic⁻]/K_D,Pic,

(23)

and D_2+,Pb = [PbL²⁺]_Bz/[Pb²⁺] = K_Pb/PbL[L]_Bz.

(24)

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:

log (D_0,Pb/D_0,Cd) = log ([PbLPic₂]_Bz/[CdLPic₂]_Bz) + log ([Cd²⁺]/[Pb²⁺])

(25)

Assuming that [Cd²⁺]/[Pb²⁺] approximately equals [Cd²⁺]_t/[Pb²⁺]_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 (=9.44₈) was estimated at I = 0.095 [33] from the regression line of Figure 5. From Equation (25) and [Cd²⁺]_t/[Pb²⁺]_t = 55.₅ estimated from a correlation between [Cd]_t/[Pb]_t and I in Table 1, the log ([PbLPic₂]_Bz/[CdLPic₂]_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 CdPic₂ extraction with B18C6 (=L) into Bz.

2.10. Relative Concentrations of the Three Species Extracted into Bz

We can immediately calculate relative concentrations of PbLPic₂, PbLPic⁺, and PbL²⁺ 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 PbLPic₂ can be obtained from 100D_0,Pb/(D_0,Pb + D_+,Pb + D_2+,Pb). In addition, the concentrations of PbLPic⁺ and PbL²⁺ were evaluated from similar equations. The thus-calculated values were: 46% for PbLPic₂, 10% for PbLPic⁺, and 44% for PbL²⁺ at I = 0.0074 mol·dm⁻³ (or [Cd]_t/[Pb]_t = 0); 38%, 25%, and 37% at 0.0060 (or 1.06₁), respectively; 35%, 32%, and 34% at 0.048 (or 26.6₃), respectively; 24%, 53%, and 23% at 0.15 (or 88.4₈), respectively; and 26%, 49%, and 25% at 0.29 (or 177.₅), respectively.

One can see easily that the distribution of PbLPic₂ and PbL²⁺ 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 PbLPic₂⁰ and PbL²⁺ 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?

3. Materials and Methods

3.1. Materials

Purities of the reagents Pb(NO₃)₂ (Wako, 99.9%) and Cd(NO₃)₂·4H₂O (Kanto, Guaranteed pure reagent (GR), >98.0%) were determined by an EDTA titration with Na₂EDTA·2H₂O (Dojin, Kumamoto in Japan, >99.5%): their purities obtained were 98.3% for Pb(II) and 96.6–98.7% for Cd(II). A basic aqueous solution (pH > 10) of picric acid, HPic·mH₂O, (Wako, GR, >99.5%: added water 15–25%) was analyzed at 355 or 356 nm by using a Hitachi UV–Visible spectrophotometer (type U-2001) (Hitachi High-Technologies Corporation, Tokyo, Japan) and then its concentration was determined with the calibration curve (ε₃₅₆ = 1.45 × 10⁴ cm⁻¹·mol⁻¹·dm³ [35]) for Pic⁻. Using a calibration curve (ε₂₇₃ = 2.50 × 10³·cm⁻¹ mol⁻¹·dm³ [6]) of B18C6 at 273 nm, a concentration of an aqueous solution with its ether (Tokyo Chemical Industry, Co. Ltd., Tokyo, Japan, >98.0% and others) was determined spectrophotometrically. The diluent Bz (Wako Pure Chemical Industries, Ltd., Osaka, Japan, or Kanto Chemical Co., Ltd., Tokyo, Japan) was washed three times with pure water and then saturated with water. Other chemicals were of the GR grades. Pure water was prepared as follows: a tap water was distilled once with a stainless-steel still and then passed through the Autopure system (Yamato/Millipore, type WT 101 UV) (Tokyo, Japan).

3.2. Extraction Procedures

Aqueous solution containing 5.47₈ × 10⁻⁴ mol·dm⁻³ Pb(NO₃)₂, 1.27₂ × 10⁻³ HPic, x Cd(NO₃)₂, 0.019₈ HNO₃, and y B18C6 was prepared and mixed with the equal volume (10 or 12 cm³) of Bz saturated with water in a stoppered glass tube of about 30 cm³. Here, x was fixed at 0 mol·dm⁻³, 5.81₀ × 10⁻⁴, 0.0145₉, 0.0484₇, or 0.0972₆ and, at a fixed x, y was changed in the ranges of 6.5 × 10⁻⁶ to 1.6 × 10⁻⁴ mol·dm⁻³ (see circle in Figure 1), 3.9 × 10⁻⁶ to 1.9 × 10⁻⁴ (square), 1.3 × 10⁻⁵ to 4.4 × 10⁻⁴ (diamond), 3.9 × 10⁻⁶ to 2.6 × 10⁻⁴ (full circle), or 3.7 × 10⁻⁵ to 7.4 × 10⁻⁴ (triangle), respectively. The glass tubes with some kinds of the L concentrations were agitated for 2 min. by hands and then mechanically shaken for 2 h in a water bath thermostated at 25 ± 0.3 °C. After it, the mixtures were centrifuged. The Bz phases were separated at 25 °C, transferred into the other tubes, and some cubic centimeters of 0.1 mol·dm⁻³ HNO₃ were added to them. These mixtures in the tubes were handled with the same manner as that described above. The Pb(II) amounts of the acidic water phases, into which the Pb(II) species normally-extracted into Bz was back-extracted, were determined at 283.3 nm by the atomic absorption spectrophotometer (Hitachi, type Z-6100) (Hitachi High-Technologies Corporation, Tokyo, Japan) with an air–C₂H₂ flame. At the same time, the amounts of Cd(II) in all the acidic phases were atomic-absorption-spectrophotometrically measured at 228.8 nm, but its element was not detected.

3.3. Extraction Model Employed for the Analysis of the System

Since significant amounts of Cd(II) in the Bz phases were not detected with the AAS measurements, the following extraction model was employed for the analysis of the present system: (1) Pb²⁺ + L ⇌ PbL²⁺ [29] (K_CdL = 0.89 mol⁻¹·dm³ [18] was omitted; (2) Pb²⁺ + Pic⁻ ⇌ PbPic⁺ [21]; (3) Cd²⁺ + Pic⁻⇌ CdPic⁺ [1]; and (4) H⁺ + Pic⁻ ⇌ HPic [33] in the water phase; (5) Pic⁻ ⇌ Pic⁻_Bz; (6) HPic ⇌ HPic_Bz [36]; (7) L ⇌ L_Bz [2]; and (8) PbL²⁺ ⇌ PbL²⁺_Bz between the water and Bz phases; and (9) PbL²⁺_Bz + Pic⁻_Bz ⇌ PbLPic⁺_Bz; and (10) PbLPic⁺_Bz + Pic⁻_Bz ⇌ PbLPic_2,Bz in the Bz phase. Except for the Processes (5), (8)–(10), the equilibrium constants of the above processes at 298 K were available from References [1,2,18,21,29,36,37]. Analytic method of the extraction system based on this model was essentially the same as that reported before [1,9] (see Section 2.2).

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,Bz⁰ and K_2,Bz⁰ values at I_Bz → 0 for the simple Bz phases. It was demonstrated that the thermodynamic relations, K_ex⁰ ≈ K_ex,ip⁰K_PbL/K_D,L, K_2,Bz⁰ = K_ex⁰/K_ex±⁰, K_ex2±⁰ = K_Pb/PbL⁰(K_D,Pic⁰)², K_ex2±⁰ = K_ex±⁰/K_1,Bz⁰, and K_ex±⁰ = K_Pb/PbL⁰K_1,Bz⁰(K_D,Pic⁰)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 [PbLPic₂]_Bz/[CdLPic₂]_Bz = 2.2 × 10⁵ at B18C6 (=L) and satisfies a measure (=10⁴) 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(NO₃)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.