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

inorganics

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

Yoshihiro Kudo *, Tsubasa Nakamori and Chiya Numako

Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan; [email protected] (T.N.); [email protected] (C.N.) * Correspondence: [email protected]; Tel.: +81-43-290-2786

Received: 18 May 2018; Accepted: 31 July 2018; Published: 9 August 2018

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, Kex, Kex±, and KPb/PbL (or Kex2±), were determined at 298 + − 2+ 2+ K: these constants were deﬁned as [PbLPic2]Bz/P, [PbLPic ]Bz[Pic ]Bz/P, and [PbL ]Bz/[Pb ][L]Bz 2+ − 2 2+ − 2 (or [PbL ]Bz([Pic ]Bz) /P), respectively. The symbol P shows [Pb ][L]Bz[Pic ] and the subscript “Bz” denotes the Bz phase, Bz saturated with water. Simultaneously, conditional distribution − − − constants, KD,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 + 2+ − + − constants of K1,Bz (=[PbLPic ]Bz/[PbL ]Bz[Pic ]Bz), K2,Bz (=[PbLPic2]Bz/[PbLPic ]Bz[Pic ]Bz), 2+ 2+ and KD,PbL (=[PbL ]Bz/[PbL ]) 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 IBz 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.

Keywords: extraction constants; conditional distribution constants; distribution equilibrium potential; ion-pair formation constants; ionic strength; lead picrate; cadmium nitrate; benzo-18-crown-6 ether; benzene

1. Introduction

In extraction systems with crown compounds (L), some extraction constants, such as Kex and Kex±, have been employed for evaluating their extraction-abilities and -selectivities [1–6]. Here, the constants + − Kex and Kex± have been generally deﬁned as [MLAz]org/P and [MLAz−1 ]org[A ]org/P, respectively, z+ − z z+ − with P = [M ][L]org[A ] at z = 1 and 2 [1,7–9]. The symbols M ,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 [1–9], 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 IDCE (with HNO3 as an I conditioner) dependences of the Kex and Kex± values in the silver picrate (AgPic) extraction with benzo-18-crown-6 ether (B18C6) into 1,2-dichloroethane (DCE), where IDCE refers to the I value for the DCE phase [12]. − − − At the same time, conditional distribution constants (KD,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; ∆ϕeq 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 II that for an M Pic2 extraction one, we determined at 298 K the Kex, Kex±, and KD,Pic values for PbPic2

Inorganics 2018, 6, 77; doi:10.3390/inorganics6030077 www.mdpi.com/journal/inorganics Inorganics 2018, 6, 77 2 of 16

extraction with B18C6 into benzene (Bz) under a co-presence of Cd(NO3)2 in a water phase. Then, I and IBz dependences of these equilibrium constants were mainly examined [12]. Similar examinations were performed for other overall or component equilibrium-constants, such as KPb/PbL, Kex,ip, and K1,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 Kex and Kex± values, because magnitudes of their comparable values depend on I [10,11] or IBz in general. Consequently, such data relevant to I and IBz 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 Mz+ 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 [10–12] 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 KD,Pic values [16]. Moreover, the relationship between log Kex± 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 [1,3,6,8,18–20]. 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 (Kex,Pb/Kex,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 Kex or Kex± definition [1,8,19,21]: + − 2+ − 2 Kex = [MLA2]org/P and Kex± = [MLA ]org[A ]org/P with P = [M ][L]org[A ] at z = 2. Taking common logarithms of both sides of these definitions and then rearranging them, we can easily obtain

− 2 log (D0/[A ] ) = log Kex + log [L]org (1)

− and log (D+/[A ]) = log Kex+ + log [L]org (2)

0 2+ + 2+ with D0 = [MLA2 ]org/[M ], D+ = [MLA ]org/[M ] (see the Section 2.9), and Kex+ (=Kex±/KD,A) = + 2+ − [MLA ]org/[M ][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:

− 2 log (D/[A ] ) ≈ log Kex + log [L]org (1a)

− and log (D/[A ]) ≈ log Kex+ + 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 − 2 of Pb(NO3)2 employed. Therefore, in terms of a plot of log (D/[A ] ) 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]. Figure1 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 Inorganics 2018, 6, 77 3 of 16 no needInorganics of employing 2018, 6, x FOR Equation PEER REVIEW (2a). The compositions of Pb(II):Pic(−I) were speculated to be 1:23 fromof 16 similarity to the systems [3] reported before for M(II) extraction with 18C6 into Bz and from a charge + Inorganics18C6 into2018 , Bz6, x FORand PEERfrom REVIEW a charge balance in the Bz phases+ [1,8,1−9,21]: approximately [PbLPic3 of2+] 16Bz ≈ balance in the Bz phases [1,8,19,21]: approximately [PbLPic ]Bz ≈ [Pic ]Bz from more-precisely 2[Pb ]Bz + [Pic2+ −]Bz from more+ -precisely 2[Pb+ 2+]Bz + 2[PbL− 2+]Bz + [PbLPic− +]Bz + [PbPic+]Bz ≈ [Pic−]Bz + [NO3−]Bz, because+ 2[PbL ]Bz + [PbLPic ]Bz + [PbPic ]Bz ≈ [Pic ]Bz + [NO3 ]Bz, because it was expected that [PbLPic ]Bz >> + 2+ 2+ + − − + 18it2+C6 was into expected Bz and2+ thatfrom [PbLPic a charge+ ]Bz >>balance 2[Pb −]inBz +the 2[PbL Bz phases]Bz− + [PbPic [1,8,19]Bz,2 1]:and approximately [Pic ]Bz >> [NO [PbLPic3 ]Bz [9,21].]Bz ≈ 2[Pb ]Bz + 2[PbL ]Bz + [PbPic ]Bz and [Pic ]Bz >> [NO3 ]Bz [9,21]. [Pic−]Bz from more-precisely 2[Pb2+]Bz + 2[PbL2+]Bz + [PbLPic+]Bz + [PbPic+]Bz ≈ [Pic−]Bz + [NO3−]Bz, because + 2+ 2+ + − − it was expected that [PbLPic ]Bz >> 2[Pb6 ]Bz + 2[PbL ]Bz + [PbPic ]Bz and [Pic ]Bz >> [NO3 ]Bz [9,21].

5.5 ) 6 6 dm -2 5.5 5 ) 6 /mol 2 ] - dm -2 4.5 /[Pic 5 D /mol 2 ] - log ( 4.5 4 /[Pic D

log ( 3.5 4 -6 -5.5 -5 -4.5 -4 -3.5 -3

log ([B18C6] /mol dm-3) Bz 3.5 -6 -5.5 -5 -4.5 -4 -3.5 -3 FigureFigure 1. Plots1. Plots for for composition composition determination determinatiolog ([B18C6]n based based/mol dm onon-3) EquationEquation (1a)(1a) underunder thethe conditions of of Bz [Cd]t[Cd]/[Pb]t/[Pb]t =t 0 = (open 0 (open circle), circle), 1.06 1.06 (square), (square), 26.6 26.6 (diamond), (diamond), 88.588.5 (full(full circle), and 178 178 (triangle). (triangle). Figure 1. Plots for composition determination based on Equation (1a) under the conditions of 2.2. Determination of Kex, Kex±, and KD,Pic 2.2. Determination[Cd]t/[Pb]t = 0 of (open Kex,K circle),ex±, and1.06 K(square),D,Pic 26.6 (diamond), 88.5 (full circle), and 178 (triangle). mix AccordingAccording to previous to previous papers papers [1,8,9,22 [1,8,9,], the22 extraction-constant], the extraction-constant parameter parameter (K mix) has(Kex been) has proposed: been 2.2. Determination of Kex, Kex±, and KD,Pic ex proposed:

mix + 2+ exmix According tolog previouslogKex Kexmix= papers= log log {([MLA {([MLA [1,8,9,22]22]orgorg], ++the [MLA [MLA extraction+]org]org + [ML+-constant [ML2+]org +]org …)/ parameter+P .} . . )/P }(K ) has been proposed: 2+ − (3) ≈ log {Kex + (KD,A/[M ][L]org[A ])} (3) ≈ log {Kex + (KD,A/[M2+][L]org[A−])} log Kexmix = log {([MLA2]org + [MLA+]org + [ML2+]org + …)/P} + + −− withwithKD,A K≈D,A[MLA ≈ [MLA]org]org/[A/[A ]. Using this equation,equation, we cancan immediatelyimmediately obtainobtain thethe KKexex and KD,A valuesvalues(3) ex D,A 2+ org − mixmix ≈ log {K2+2+ + (K /[M− − ][L] [A ])} fromfrom a plot a plot of log of Klogex Kex versus versus− −loglog ([M ([M ][L]orgorg[A[A]). ]).In Inaddition, addition, Equation Equation (3) can (3) canbe rewritten be rewritten as: as: with KD,A ≈ [MLA+]org/[A−]. Using this equation, we can immediately obtain the Kex and KD,A values exmix ex / (4) mix log K 2+ ≈ log (K− + p ± ) from a plot of log Kex versus −log ([M ][L]org[A ]). In addition, Equation (3) can be rewritten as: log Kexmix ≈ log Kex + Kex±/P (4) Similarly, the Kex± value (with the Kex one; see Tableex 1) can be obtained from a plot of log Kexmix mix �𝐾𝐾 𝑃𝑃 1/2 log Kex ≈ log (Kex + ±/ ) (4) versus −log P . Figures 2 and 3 show examples of such plots. mix Similarly, the Kex± value (with the Kex one; see Table1) can be obtained from a plot of log Kex Similarly, the Kex± value (with the Kex one; see Tableex 1) can be obtained from a plot of log Kexmix versus −log P1/2. Figures2 and3 show10 examples of such�𝐾𝐾 plots.𝑃𝑃 versus −log P1/2. Figures 2 and 3 show examples of such plots.

10 ) 9 dm -3 ) 9

/mol 9.5 dm mix -3 ex K

/mol 9.5 log ( mix ex K log (

9 10 10.5 11 11.5 12 12.5

-log ([Pb2+][L] [Pic-]/mol3 dm-9) 9 Bz 10 10.5 11 11.5 12 12.5

2+ - 3 -9 mix -log ([Pb2+ ][L] [Pic− ]/mol dm ) Figure 2. Plot of log Kex versus − log ([Pb ][L]Bz[PicBz ]) with L = B18C6 at [Cd]t/[Pb]t = 88.5. The line

is based on Equation (3). mix 2+ − K mix − 2+ − FigureFigure 2. 2.Plot Plot of of loglog Kex versusversus − loglog ([Pb ([Pb][L]Bz][L][PicBz])[Pic with]) L with= B18C6 L = at B18C6 [Cd]t/[Pb] at [Cd]t = 88.5.t/[Pb] Thet =line 88.5. Theis linebased is on based Equation on Equation (3). (3).

Inorganics 2018, 6, 77 4 of 16 Inorganics 2018, 6, x FOR PEER REVIEW 4 of 16

10 ) 9 dm -3

/mol 9.5 mix ex K log (

9 6.6 6.8 7 7.2 7.4 7.6 7.8

-log (P 1/2/mol2 dm-6)

mix − 1/2 FigureFigure 3. 3. PlotPlot of of log log KKexmixex versusversus −log Plog1/2 withP B18C6with B18C6 at [Cd] att/[Pb] [Cd]t =t/[Pb] 88.5. tThe= 88.5. line is The based line on is Equation based on (4).Equation (4).

FromFrom these plots, the KKD,PicD,Pic,, KKexex±,± and, and KKex exvavalueslues were were determined determined at at 298 298 K. K. Table Table 1 lists thesethese extractionextraction constants,constants,K Kexexand andK exK±ex±,, and and the the conditional conditional distribution distribution constants, constants,KD,Pic K,D,Pic with, with averaged averaged ionic ionicstrength-values strength-values (I) for the(I) waterfor the phase water in the phase five [Cd] in tthe/[Pb] fivet conditions. [Cd]t/[Pb] Thet conditions.Kex values determinedThe Kex values with determinedEquation (4) with were Equation equal orclose (4) were to those equal with or Equationclose to (3).those This with fact Equation raises the (3). credibility This fact of theraises values the credibilitythemselves of and the also values shows themselves the effects and of Equationsalso shows (3) the and effects (4) on of evaluation. Equations The (3) andKex and(4) onKex evaluation.± values at 11.712 −3 9 4.111.712 −2 −3 6 9 4.1 −2 6 The[Cd] Kt/[Pb]ex andt =Kex± 0 were values smaller at [Cd] thant/[Pb] thoset = 0 (10were smallermol thandm thoseand 10 (10 mol mol·dm dm[9 ])and reported 10 mol before·dm at −3 −3 [9])I = 0.0059reported mol before·dm atfor I = the 0.0059 PbPic mol2 extraction·dm for with the PbPic 18C62 into extraction Bz. with 18C6 into Bz.

Table 1. BasicBasic data data for for the the Pb(II) Pb(II) extraction extraction by by B18C6 B18C6 from from the water phase with co co-presence-presence of Cd(NOCd(NO33))22 intointo Bz Bz at at 298 298 K. K.

3 loglogK Kex loglog K ex± 3 log logKD,PicK [Cd]t/[Pb]1 t 1 I2 2/mol dm−3 −3 ex ex± D,Pic [Cd]t/[Pb]t I /mol dm 3 3 44 5 5 [][] (log(log yPic )) (Δφ(eq∆ϕeq/V)/V) 9.715 ± ±0.006 0.006 2.62.66 ±± 0.20.25 −3.2−13.2 ± 0.1±40.1 0 0.0074 6 5 1 4 0 0.0074 (9.70 ± 0.01) (−0.04) (−0.3 ) (9.70 ± 0.01) (−0.04) (−0.30) 0 9.68 ± 0.02 3.9 ± 0.2 −2.4 ± 0.1 1.06 0.0060 9.68 ± 0.02 3.900 ± 0.233 −2.44 ± 40.13 3 1.06 0.0060 (9.61 ± 0.04) (−0.03) (−0.34) (9.61 ± 0.04) (−0.03) (−0.34) 9.58 ± 0.02 3.97 ± 0.25 −2.37 ± 0.16 26.6 0.048 9.58 ± 0.02 3.97 ± 0.25 −2.37 ± 0.16 26.6 0.048 (9.51 ± 0.03) (−0.07) (−0.35) (9.399.51± ±0.03 0.03) 3.9(−0.07± 0.1) (−0.3−2.65)± 0.1 88.5 0.15 5 9 0 5 (9.249.39± ±0.05) 0.03 3.9(5− ±0.10) 0.19 −2.60 (±− 0.10.335) 88.5 0.15 (9.319.24± ±0.02 0.05) 3.6(−0.10± 0.1) (−0.3−2.653)± 0.09 178 0.29 8 6 (9.21 ± 0.03) (−0.12) (−0.33) 9.31 ± 0.02 3.68 ± 0.16 −2.65 ± 0.09 1 178 −4 −0.293 2 3 [Pb(NO3)2]t = 5.48 × 10 mol·dm . Averaged(9.21 ionic ± strength 0.03) for the(−0.12 water) phase.(−0.33)Values determined from Equation (4). 4 Logarithmic activity coefﬁcient of Pic− in water, calculated from the I value. 5 Dep values calculated 1from [Pb(NO Equation3)2]t = (5).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

2.3. Depvalues Determination calculated from from Equation KD,Pic (5). From the log K values listed in Table1, using the following equation and a standardized 2.3. Dep DeterminationD,Pic from KD,Pic S distribution constant (KD,Pic ), we can easily obtain the dep (or ∆ϕeq) values for the ﬁve [Cd]t/[Pb]t conditionsFrom the at 298 log K: KD,Pic values listed in Table 1, using the following equation and a standardized distribution constant (KD,PicS), we can easily obtain the dep (or Δφeq) values for the five [Cd]t/[Pb]t S 00 conditions at 298∆ϕ K:eq = −0.05916(log KD,Pic − log KD,Pic ) = ∆ϕPic − 0.05916log KD,Pic (5)

Δφeq = −0.05916(log KD,Pic − log KD,PicS) = ΔφPic0′ − 0.05916log KD,Pic (5)

Inorganics 2018, 6, 77 5 of 16

S 00 Here, the KD,Pic value is deﬁned as the KD,Pic one at ∆ϕeq = 0 V, equals antilog (∆ϕPic /0.05916) 00 (=exp (∆ϕPic /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 00 − − ∆ϕPic 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 S described below. Table1 lists the dep/V values evaluated from log KD,Pic = −8.208 [24].

2.4. Determination of KPb/PbL,Kex,ip,K1,Bz,K2,Bz, and KD,PbL These constants can be evaluated from the following relations [1,18,26–28].

2+ 2+ log KPb/PbL = log ([PbL ]Bz/[Pb ][L]Bz) ≈ log (D/[L]Bz), (6)

2+ − 2 log Kex,ip = log ([PbLPic2]Bz/[PbL ][Pic ] ) = log (KexKD,L/KPbL), (7) + 2+ − 2 log K1.Bz = log ([PbLPic ]Bz/[PbL ]Bz[Pic ]Bz) ≈ log {Kex±/KPb/PbL(KD,Pic) } (8)

= log (Kex±/Kex2±), (8a) + − log K2,Bz = log ([PbLPic2]Bz/[PbLPic ]Bz[Pic ]Bz) = log (Kex/Kex±), (9) 2+ 2+ and log KD,PbL = log ([PbL ]Bz/[PbL ]) ≈ log (KPb/PbLKD,L/KPbL). (10)

Only for the KPb/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 2+ 2+ logarithmic values of KD,B18C6 (=[B18C6]Bz/[B18C6]) and KPbB18C6 (=[PbB18C6 ]/[Pb ][B18C6]), respectively. These ﬁve logarithmic K-values are summarized in Table2, together with the ionic strength-values (IBz) for the Bz phase.

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.

log KPb/PbL log K1,Bz [Cd]t/[Pb]t 1 log Kex,ip 2 −6 log K2,Bz log KD,PbL (log yPb ) (IBz /10 ) 3.42 ± 0.05 5.7 ± 0.3 0 7.47 7.1 ± 0.2 1.18 (−0.16) (0.44) 3.42 ± 0.08 5.4 ± 0.3 1.06 7.143 5.8 ± 0.2 1.17 (−0.14) (2.6)

3.23 ± 0.12 5.5 ± 0.4 26.6 7.34 5.6 ± 0.2 0.98 (−0.34) (2.7)

3.13 ± 0.21 6.0 ± 0.4 88.5 7.15 5.4 ± 0.2 0.88 (−0.50) (1.7)

2.95 ± 0.28 6.0 ± 0.3 178 7.06 5.6 ± 0.2 0.71 (−0.62) (7.8) 1 Logarithmic activity coefﬁcient of Pb2+ in water, calculated from the averaged I value. 2 Averaged ionic strength for the Bz phase.

2.5. Correlation between log Kex± and Dep We can obtain the following relation from the thermodynamic cycle of the PbLPic+ extraction with Pic−. log Kex± = 2log KD,Pic + log KPb/PbL + log K1,Bz S (11) = 2log KD,Pic - 2(F/2.303RT)∆ϕeq + log KPb/PbL·K1,Bz

Here, the log KPb/PbL·K1,Bz term was in the range of 8.7 to 9.1 (see the data in Table2) and log S KD,Pic (=−8.208 [24] or −7.4473 [25]) equals log KD,Pic at ∆ϕeq = 0 V. Hence, we obtained to be −7.7 to Inorganics 2018, 6, 77 6 of 16

Inorganics 2018, 6, x FOR PEER REVIEW 6 of 16 S S −7.3 for the former KD,Pic value or the −6.2 to −5.8 for the latter one as the term of 2log KD,Pic + (see Tables 1 and 2). In addition, 2F/2.303RT becomes 33.80 V−1 at T = 298.15 K. Rearranging−1 Equation log KPb/PbL·K1,Bz (see Tables1 and2). In addition, 2 F/2.303RT becomes 33.80 V at T = 298.15 K. Rearranging(11), we can Equation immediately (11), wederive can immediately derive

log Kex± ≈ (−7.7 to −7.3) − 33.80ΔφEquation (11a) log Kex± ≈ (−7.7 to −7.3) − 33.80∆ϕEquation (11a) From the regression analysis of an experimental plot in Figure 4, the following line was obtained: logFrom Kex± = the(−5. regression3 ± 1.4) − (27. analysis3 ± 4.2)Δ ofφeq an at experimental|R| = 0.967, where plot the in Figure symbol4, theR denotes following a correlation line was obtained: coefficient. logThisKex ±regression= (−5.3 ± line1.4 )is− close(27. 3to± Equation4.2)∆ϕeq (11a)at |R which| = 0.967, was where estimated the symbolfrom theR experimentaldenotes a correlation K values. coefficient.This fact Thisindicates regression the presence line is close of dep, to Equation as similar (11a) to the which results was reported estimated previously from the experimental[7,8,12,16,21,22K ]. values. This fact indicates the presence of dep, as similar to the results reported previously [7,8,12,16,21,22].

4.5

) 4 6 dm -2

/mol 3.5 ± ex K

log ( 3

2.5 -0.36 -0.34 -0.32 -0.3 -0.28

Dep/V .

Figure 4. Plot of log Kex± versus dep for the Pb(II) extraction with Cd(NO3)2 and B18C6 into Bz. Figure 4. Plot of log Kex± versus dep for the Pb(II) extraction with Cd(NO3)2 and B18C6 into Bz. The The line corresponds to Equation (11a). line corresponds to Equation (11a).

2.6. I Dependences of log Kex and log Kex,ip 2.6. I Dependences of log Kex and log Kex,ip 0 2 The thermodynamic extraction constant of Kex is Kex = [PbLPic2]Bz/a [L]Bz(aPic) , where a The thermodynamic extraction constant of Kex is Kex0 = [PbLPic2]Bz/aPb[L]PbBz(aPic)2, where aPb andPb aPic 2+ − and aPic refer to activities of Pb and Pic in the water phase, respectively, and it was assumed that refer to activities of Pb2+ and Pic− in the water phase, respectively, and it was assumed that [PbLPic2]Bz 2+ − [PbLPic2]Bz is equal to the activity in the Bz phase, because2+ − PbL (Pic )2 is charge-less. The same is is equal to the activity in the Bz phase, because PbL (Pic )2 is charge-less. The same is true of [B18C6]Bz 0 true of [B18C6]Bz too. Taking the common logarithms of both sides0 of the Kex deﬁnition, we can obtain too. Taking the common logarithms of both sides of the Kex definition, we can obtain

0 0 22 log Klogex K=ex log = logKex K−ex log− log {y Pb{yPb(y(PicyPic))}} (12)(12)

2+ − with yPb = aPb/[Pb ] and2+ yPic = aPic/[Pic ]. Introducing− the extended Debye–Hückel (DH) equation with yPb = aPb/[Pb ] and yPic = aPic/[Pic ]. Introducing the extended Debye–Hückel (DH) [30,31] in Equation (12) and arranging it, the following equation was obtained: equation [30,31] in Equation (12) and arranging it, the following equation was obtained: 0 log Kex = log Kex − 6√A /(1 + Bå± √) (12a) log Kex = log Kex0 − 6A I/(1 + Ba± I (12a) Here, the DH equation was based on the mean activity√𝐼𝐼 coefficient√𝐼𝐼 and the symbol å± denote the ion-size parameter [30] in Å unit. Although the extended DH equation holds in the I range of ≤0.1 Here, the DH equation was based on the mean activity coefﬁcient and the symbol a± denote mol·dm−3 [30] as you know, we approximately employed it for the condition of I = 0.29 (see Table 1). the ion-size parameter [30] in Å unit. Although the extended DH equation holds in the I range Figure 5 shows curve-fittings of the plots for Equation (12a). of ≤0.1 mol·dm−3 [30] as you know, we approximately employed it for the condition of I = 0.29 (see Table1). Figure5 shows curve-ﬁttings of the plots for Equation (12a).

Inorganics 2018, 6, 77 7 of 16 Inorganics 2018, 6, x FOR PEER REVIEW 7 of 16

10 ) 6 dm -3 9.5 /mol ex K log (

9 0 0.1 0.2 0.3 0.4 0.5 0.6

I1/2/mol1/2 dm-3/2

1/2 Figure 5. Plot of log Kex versus I1/2 for the Pb(II) extraction with Cd(NO3)2 and B18C6 into Bz. The line Figure 5. Plot of log Kex versus I for the Pb(II) extraction with Cd(NO3)2 and B18C6 into Bz. The line is based on Equation (12a). is based on Equation (12a). √ √ ItsIts regression regression line was log Kex == (9.91 (9.91 ±± 0.03)0.03) − 6− ×6 (0.5114)× (0.5114)/(1 I+/(1 (3.4 +0 ± (3.4 0.406±) 0.4) at6) R =I )0.980, at R = where 0.980, −1/2 3/2 thewhere coefficient the coefﬁcient A wasA fixedwas ﬁxedto 0.5114 to 0.5114 mol−1/2 mol·dm3/2 ·[dm30] and[30 the] and å± value the a± invalue water in was water evaluated was evaluated to be √𝐼𝐼 √𝐼𝐼 10to beÅ (= 103.4 Å0/0.3291) (=3.40/0.3291) at 298 K. at 298 K. Similarly, the log Kex,ip valuesvalues were were analyzed. analyzed. Their Their constants constants were were expressed expressed as as

log Kex,ip = log Kex,ip0 − 6A √/(1 + Bå± ) √ (13) log Kex,ip = log Kex,ip0 − 6A I/(1 + Ba± I (13) with Kex,ip0 = [PbLPic2]Bz/aPbL(aPic√)𝐼𝐼2 = Kex,ip/y√PbL𝐼𝐼 (yPic)2. (13a) with K 0 = [PbLPic ] /a (a )2 = K /y (y )2 (13a) The regression analysis ex,ipof the plots yielded2 Bz logPbL KPicex,ip = (7.66ex,ip ± 0.03)PbL −Pic 6 × (0.5114) /(1 + (3.41 ± √ 0.51) ) at R = 0.975 and then the a± value was evaluated to be 10 Å. The accordance between Pb2+– The regression√ analysis of the plots yielded log Kex,ip = (7.66 ± 0.03) − 6 × (0.5114) I/(1 + − 2+ − √𝐼𝐼 2+ − Pic(3.4 distance± 0.5 ) andI) at PbB18C6R = 0.975 and–Pic then one thesuggestsa± value that was the evaluatedformer interaction to be 10 Å. between The accordance the Pb betweenand Pic 1√𝐼𝐼 1 ionsPb2+ –Picin water− distance saturated and PbB18C6with Bz is2+ equivalent–Pic− one suggests with the that latter the one former betwe interactionen PbB18C6 between2+ and thePic− Pb. 2+ and Pic− Itions is ininteresting water saturated that the with evaluated Bz is equivalent å± values with are the close latter to one the between sum (=11. PbB18C65 Å) 2+ofand the Picion−-.size 2+ − 0 parametersIt is interesting [32] between that Pb the (4.5 evaluated Å) anda Pic± values (7 Å) arefor closewater. to This the K sumex,ip value (=11.5 wasÅ) ofwell the in ion-size accord 7.66 −1 3 2+ − 0 0 0 9.91 withparameters that (=10 [32] betweenmol ·dm Pb) calcula(4.5ted Å) andfrom Pic the thermodynamic(7 Å) for water. ThiscycleK ofex,ip Kex,ipvalue ≈ Kex wasKD,L well/KPbL in (= accord10 × 0.943 3.19 7.66 −1 3 0 0 10with that/10 (=10). mol ·dm ) calculated from the thermodynamic cycle of Kex,ip ≈ Kex KD,L/KPbL (=109.91Considering× 100.943/10 that3.19 ).the Kex values are most precise ones of the some extraction constants determinedConsidering here (see that thetheirK exerrorsvalues in areTable most 1), precise the fair ones dependences of the some of extraction log Kex on constants I indicate determined a simple rolehere of (see Cd(NO their errors3)2 only in as Table the1 ),ionic the fairstrength dependences conditioner of log inK theex on presentI indicate extraction a simple systems. role of Cd(NO In other3)2 words,only as the the authors ionic strength were not conditioner be able to inclearly the present find out extraction positive or systems. negative In effects other words,of Cd(NO the3) authors2 on the presentwere not Pb(II) be able extraction to clearly with ﬁnd B18C6 out into positive Bz. or negative effects of Cd(NO3)2 on the present Pb(II) extraction with B18C6 into Bz. 2.7. IBz Dependences of log K1,Bz and log K2,Bz 2.7. IBz Dependences of log K1,Bz and log K2,Bz Using IBz and the DH limiting law [30], both log K1,Bz0 and log K2,Bz0 can be expressed as 0 0 Using IBz and the DH limiting law [30], both log K1,Bz and log K2,Bz can be expressed as log K1,Bz0 ≈ log K1,Bz – log yPbL,Bz = log K1,Bz + 4ABz (14) p K 0 ≈ K − y = K + A Bz I with yPbLPic,Bz ≈ yPic,Bz andlog 1,Bz log 1,Bz log PbL,Bz log 1,Bz 4 �Bz𝐼𝐼 Bz (14)

log K2,Bz0 = log K2,Bz – log (yPbLPic,Bz·yPic,Bz) = log K2,Bz + 2ABz . (15) with yPbLPic,Bz ≈ yPic,Bz and Rearranging Equations (14) and (15), we can obtain �𝐼𝐼Bzp log K2,Bz0 = log K2,Bz − log(yPbLPic,Bz·yPic,Bz) = log K2,Bz + 2ABz IBz. (15) log K1,Bz ≈ log K1,Bz0 − 4ABz (14a)

�𝐼𝐼Bz and log K2,Bz = log K2,Bz0 − 2ABz . (15a)

�𝐼𝐼Bz

Inorganics 2018, 6, 77 8 of 16

Rearranging Equations (14) and (15), we can obtain p log K1,Bz ≈ log K1,Bz0 − 4ABz IBz (14a)

Inorganics 2018, 6, x FOR PEER REVIEW p 8 of 16 and log K2,Bz = log K2,Bz0 − 2ABz IBz (15a) BasedBased on on Equations Equations (14a) (14a) and and (15a), (15a), we we prepared prepared Figure Figure6 from 6 from the the data data in Tablein Table2. 2.

) 7 3 dm -1 6 /mol

2,Bz 5 K or

1,Bz 4 K

log ( 3

2 0 0.0005 0.001 0.0015 0.002 0.0025 0.003

I 1/2/mol1/2 dm-3/2 Bz

1/2 FigureFigure 6. Plots6. Plots of logof logK1,Bz K1,Bz(circle) (circle) and and log logK2,Bz K2,Bz(square) (square) versus versusIBz IBz1/2for for the the Pb(II) Pb(II) extraction extraction with with Cd(NOCd(NO3)23)and2 and B18C6 B18C6 into into Bz. Bz. The The regression regression lines lines were were based based on on Equations Equations (14a) (14a) and and (15a). (15a). The The full full circlecircle was was omitted omitted from from the the regression regression analysis analysis of logof logK1,Bz K1,Bz. .

At the same time, these plots were analyzed by using the both equations. Their regression lines At the same time, these plots were analyzed√ by using the both equations. Their regression lines were log K1,Bz = (5.98± ± 0.49) −− 4 ×× (63 ± ±89) at R = 0.449 except for the point of [Cd]t/[Pb]t = 178 and were log K1,Bz = (5.98 0.49) 4 (63 89) √IBz at R = 0.449 except for the point of [Cd]t/[Pb]t = 178 1/2 andlog log K2,BzK2,Bz = (6.8= (6.81 ± 0.61 ±8)0.6 − 28 )×− (2822 × ±(282 195)± 195) atBz RIBz = at0.641.R = 0.641.These Theselines intersected lines intersected with each with other each otherat IBz 1/2 −3 1/2−3 −3/2 1/2 −3/2 �𝐼𝐼 at=IBz 2.7 × 10= 2. 7mol× 10·dmmol, yielding·dm log, yielding K1,BzBz = log log K2,BzK1,Bz = 5.3= log1. ThisK2,Bz fact= 5.3indicates1. This that, fact indicates in the lower that, IBz in range the −6 −3 −6 �𝐼𝐼 −3 lowerless IthanBz range 7.1 × less 10 than mol 7.·dm1 × 10, the molK2,Bz· dmvalues, theareK larger2,Bz values than are the larger K1,Bz ones. than theTheK 1,Bzlatterones. values The may latter be valuesestimated may beto estimatedactually tobe actuallythe smaller be the values smaller because values becauseof the ofapproximation the approximation [33] [for33] forthe theK1,Bz K1,Bzdeterminationdetermination (see (see Equation Equation (8) (8)).). Unlike Unlike the the case case of of thethe CdPicCdPic22–B18C6–B18C6 extractionextraction systemsystem [ 33[3],3], unfortunately,unfortunately, we we do do not not have have the the procedure procedure which which corrects corrects such such deviations deviations for for the the present present extraction extraction systems,systems, because because of of a lacka lack of of adequate adequate data data used used for for the the correction. correction. InIn addition, addition we, we tried tried curve-ﬁttings curve-fittings to to the the two two plots plots using using the the following following equations: equations:

log K1,Bz ≈ log K1,Bz0 − 4ABzp + b1IBz (14b) log K1,Bz ≈ log K1,Bz0 − 4ABz IBz + b1 IBz (14b) �𝐼𝐼Bz and log K2,Bz = log K2,Bz0 − 2ABzp + b2IBz. (15b) and log K2,Bz = log K2,Bz0 − 2ABz IBz + b2 IBz. (15b) √ Bz with the approximation of 1 >> (see Table 2). Here,�𝐼𝐼 the symbols b1 and b2 denote empirical with the approximation of 1 >> IBz (see Table2). 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 curve-fitting parameters [30,31] which� were𝐼𝐼Bz simply predicted in this study from the plot shapes Figure 6). The regression analyses of the plots at 298 K gave log K1,Bz = (6.16 ± 0.80) − 4 ±× (179 ± −247) × (see Figure√6). The regression analyses of the plots at 298 K gave log K1,Bz = (6.16 0.80) √4 5 5 5 + (2.±3 ± 2.7) I× 10 IBz at± R = 0.569× andI logR K2,Bz = (8.80 ± 0.9K8) − 2 × (1625± ± 603) − ×+ (7.6 ± ±3.3) × 10 IBz Bzat (179 247) Bz + (2.3 2.7) 10 Bz at = 0.569 and log 2,Bz = (8.80 0.98) 2 (1625 603) �Bz𝐼𝐼 + (7.0.914± (see3. ) ×Fig10ure5I 7).at 0.914Modifying (see Figure these7). equations Modifying like these the equationsDavies one like [30 the,31 Davies], their one2nd [and30,31 3rd], their terms 6 3 Bz √ 𝐼𝐼Bz √ 2 2 � 2 2ndbecame and 3rd −4 terms × (179 became ± 247−)(4 × (179 − (3.±2 247)(± 5.9) ×I Bz10−IBz(3.) 2and± 5. −9 )2× × 10(1625IBz) ± and 603−)(2 × (1625 − (2.±3 ±603)( 1.3) × I10Bz −IBz), −1/2 3/2 − (2.respectively.± 1. ) × 102 TheseI ), respectively. b1/4ABz and These b2/2AbBz/4 valuesA and of aboutb /2A 320values and 230 of about mol 320·dm and for 230 the mol Bz1/2 phases·dm3/2 are 3 3 Bz �𝐼𝐼Bz 1 Bz 2 Bz �𝐼𝐼Bz formuch the Bz larger phases than are 0.3 much [31 larger] for the than aqueous 0.3 [31] solution for the aqueous at 298 K. solution Equation at 298 (14b) K. Equationintersects (14b) Equation intersects (15b) around IBz1/2 = 3.3 × 10−3 mol1/2·dm−3/2, yielding log K1,Bz = log K2,Bz = 6.30, and then their two lines equal with each other within the experimental errors (see the plots in Figure 7).

Inorganics 2018, 6, 77 9 of 16

1/2 −3 1/2 −3/2 Equation (15b) around IBz = 3.3 × 10 mol ·dm , yielding log K1,Bz = log K2,Bz = 6.30, and then Inorganicstheir two 2018 lines, 6, equalx FOR PEER with REVIEW each other within the experimental errors (see the plots in Figure7). 9 of 16

9 ) 3 8 dm -1

/mol 7 2,Bz K

or 6 1,Bz K

log ( 5

4 0 0.0005 0.001 0.0015 0.002 0.0025 0.003

I 1/2/mol1/2 dm-3/2 Bz 1/2 Figure 7. Plots of log K1,Bz (circle) and log K2,Bz (square) versus IBz for the Pb(II) extraction with Figure 7. Plots of log K1,Bz (circle) and log K2,Bz (square) versus IBz1/2 for the Pb(II) extraction with Cd(NO3)2 and B18C6 into Bz. The regression lines were based on Equations (14b) and (15b). Cd(NO3)2 and B18C6 into Bz. The regression lines were based on Equations (14b) and (15b).

InIn comparisoncomparison of of Equations Equations (14a) (14a) and and (15a) (15a) with Equationswith Equations (14b) and (14b) (15b), and the (15b), point ofthe intersection point of 1/2 1/2 −3/2 intersectionchanged from changedIBz /mol from IBz·1/2dm/mol1/2=·dm 0.002−3/2 7= to0.002 0.0037 to3 ,0.003 while3, theirwhile corresponding their corresponding log K1,Bz log(=log K1,Bz K(=log2.Bz) Kvalue2.Bz) value changed changed from from 5.3 to 5.3 6.3. to 6.3. −5 −3 AtAt least least in in the the lower lower IIBzBz rangerange less less than than 1. 1.1 1× ×1010−5 molmol·dm·dm−3, the, K the2,Bz Kvalues2,Bz values are larger are larger than thanthe K the1,Bz ones.K1,Bz ones.From Fromthe results the results of the ofcalculation the calculation based basedon Equations on Equations (14a,b) (14a,b) and (15a and,b), (15a,b), the relation the relation of K1.Bz of< −5 −3 K2,Bz1.Bz holds< K2,Bz in theholds range in the less range than (0.7 less1– than1.1) × (0.7 10−51 –1.mol1)·dm× 10−3 (seemol above).·dm According(see above). to the According paper [8], to such the + apaper fact [suggests8], such aa fact structural suggests change a structural around change Pb(II) around in the Pb(II) reaction in the of reaction Pb(B18C6)Pic of Pb(B18C6)Pic+Bz + Pic−BzBz + − Pb(B18C6)PicPic Bz Pb(B18C6)Pic2,Bz, such as2,Bz Cd(18C6)Pic, such as Cd(18C6)Pic2,Bz of the 2,BzCd(ofII) theextraction Cd(II) extraction systems [ systems33]. Trends [33]. similar Trends to similar K1,Bz 2+ − − − ⇌

Table 3. ComparisonComparison between between Equations Equations (14a) (14a) and and (15a) (15a) and and Equations Equations (14b) (14b) and and (15b) (15b) in the in there- 1 productionre-production of the of the experimental experimental K1,BzK 1,Bzandand K2,BzK values2,Bz values 1 at 298at K. 298 K.

loglog KK1,Bz loglogK K2,Bz [Cd]t/[Pb]t 1,Bz 2,Bz [Cd]t/[Pb]t 2 2 2 2 EquationEquation (14a) (14a) |Dif.||Dif.|2 EquationEquation (14b) (14b) |Dif.||Dif.|2 Equation (15a) (15a) |Dif.||Dif.|2 Equation (15b) (15b) |Dif.||Dif.2| 0 5.81 0.2 6.16 0.2 6.44 0.6 6.98 0.1 0 5.81 0.2 6.16 0.2 6.44 0.6 6.98 0.1 1.061.06 5.55.588 0.20.2 5.65.61 1 0.20.2 5.95.91 1 0.10.1 5.55.54 4 0.20.2 26.626.6 5.55.566 0.10.1 5.65.61 1 0.10.1 5.85.87 7 0.30.3 5.55.50 0 0.10.1 88.5 5.66 0.4 5.62 0.4 6.08 0.6 5.87 0.4 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 0.0 178 5.28 0.8 5.96 0.1 5.24 0.4 5.62 0.0 1 2 See Table2 for these values. Absolute value for the difference between the experimental K1,Bz or K2,Bz value and 1 2 theirSee Table estimated 2 for one. these values. Absolute value for the difference between the experimental K1,Bz or K2,Bz value and their estimated one.

Inorganics 2018, 6, 77 10 of 16 Inorganics 2018, 6, x FOR PEER REVIEW 10 of 16

00 00 0 2.8. I Dependences of log Kex± , log K , and log K 2.8.Bz IBz Dependences of log Kex±0′, log KD,Pic0′, and log KPb/Pb/PbLPbL0 0 2 00 The thermodynamic equilibrium constant Kex± is equal to (y± ) Kex± , with y± = The thermodynamic equilibrium constant Kex±0 is equal to (y±,Bz)2Kex±0′,Bz, with y±,Bz = (yPbLPic,Bz·y,BzPic,Bz)1/2 1/2 00 + − 2 (y ·y ) and Kex± = [PbLPic ] [Pic ] /(a [L] (a ) ). Taking the common PbLPic,Bzand Kex±0′ Pic,Bz= [PbLPic+]Bz[Pic−]Bz/(aPb[L]Bz(aPic)2). TakingBz theBz commonPb Bz logarithmsPic of the both sides in this logarithmsequation and of the rearranging both sides it inwith this the equation DH limiting and rearranginglaw, we can iteasily with obtain the DH limiting law, we can easily obtain 0 2 2 0 p log0 =Kex± ′ =( log (Kex±/y(Pb(yPic) ))) = = log Kex± ++ 2ABz (16) log Kex±0 log Kex±/yPb yPic log Kex±0 2ABz IBz (16) Figure 8 shows the plot of log Kex±00 0′ versus IBz1/21/2 based on Equation� (16)𝐼𝐼Bz . The regression analysis Figure8 shows the plot of log Kex± versus IBz based on Equation (16).√ The regression analysis of this plot gave the equation of log K0ex±0 0′ = (3.11 ± 0.69) + 2 × (315 ± 196) at R = 0.680. From this of this plot gave the equation of log Kex± = (3.11 ± 0.69) + 2 × (315 ± 196) IBz at R = 0.680. From this 0 0 0 Kex,±0 value and the Kex0 one, we calculated log K2,Bz0 to be 6.80± ± 0.69, beingBz in good agreement with Kex,± value and the Kex one, we calculated log K2,Bz to be 6.80 0.69, being�𝐼𝐼 in good agreement with thatthat (=6.8) (=6.8) evaluated evaluated from from Equation Equation (15a). (15a).

5 ) 6

dm 4 -2 '/mol

0 3 ± ex K 2 log (

0 0 0.0005 0.001 0.0015 0.002 0.0025 0.003

I 1/2/mol1/2 dm-3/2 Bz

00 1/2 FigureFigure 8. 8.Plot Plot of logof logKex K±ex±0versus′ versusIBz IBz1/2 based onon EquationEquation (16) (16) for for the the Pb(II) Pb(II) extraction extraction with with Cd(NO Cd(NO3)23)2 00 2 andand B18C6 B18C6 into into Bz. Bz. The The symbol symbolK exK±ex±0′ waswas deﬁneddefined asas KKexex±±/y/PbyPb(yPic(y)Pic2. ) .

0 0 Similarly, the plot of log KD,Pic0 0′ versus I1/2Bz1/2 was performed in Figure 9, where KD,Pic0 0′ is defined Similarly, the plot of log KD,Pic versus IBz was performed in Figure9, where KD,Pic is deﬁned as [Pic− −]Bz/aPic. This plot is based on the equation as [Pic ]Bz/aPic. This plot is based on the equation log KD,Pic0′ = log (KD,Pic/yPic) = log KD,Pic0 + ABz p (17) log K 0 = log(K /y ) = log K 0 + A I (17) D,Pic0 D,Pic Pic D,Pic Bz Bz Bz The regression analysis yielded log KD,Pic0′ = (−3.00 ± 0.36) + (258 ± 210)�𝐼𝐼 at R = 0.578. 0 √ TheIn regressionaddition, the analysis analysis yielded was tried log Kby using0 = an (− equation3.0 ± 0.3 similar) + (258 to ±Equations210) I (14b)at R and= 0.578. (15b) with D,Pic 0 6 �𝐼𝐼Bz Bz 1 >> Its regression line was In√ addition, the analysis was tried by using an equation similar to Equations (14b) and (15b) with 1 >> IBzBzIts regression line was �𝐼𝐼 log KD,Pic0′ = (−4.25 ± 0.12) + (1949 ± 154) + (−4.76 ± 0.42) × 105IBz (17a) p 5 log K 0 = (−4.2 ± 0.1 ) + (1949 ± 154) I Bz+ (−4.7 ± 0.4 ) × 10 I (17a) at R = 0.995. Here,D,Pic the0 latter two5 terms2 are rearranged� into𝐼𝐼Bz (1949 ±6 154)(2 − (245Bz ± 29)IBz). This value, 245 mol−1/2·dm3/2, is comparable to the b1/4ABz (=~320) and b2/2ABz (=~230)√ values estimated �𝐼𝐼Bz atabove.R = 0.995. Here, the latter two terms are rearranged into (1949 ± 154)( IBz − (245 ± 29)IBz). −1/2 3/2 · 0 2+ This value,Lastly, 245 the mol log KPb/PbLdm′ (=log, is ([PbL comparable]Bz/aPb[L] toBz the)) valuesb1/4A Bzwere(=~320) analyzed. and bThis2/2A constantBz (=~230) was values related estimatedwith the above.log KPb/PbL0 (=log (aPbL,Bz/aPb[L]Bz)) value by the following equation: 00 2+ Lastly, the log KPb/PbL (=log ([PbL ]Bz/aPb[L]Bz)) values were analyzed. This constant was log0 KPb/PbL0′ = log (KPb/PbL/yPb) = log KPb/PbL0 + 4ABz . (18) related with the log KPb/PbL (=log (aPbL,Bz/aPb[L]Bz)) value by the following equation: The log KPb/PbL0′ values were plotted against the IBz1/2 ones. The regression�𝐼𝐼Bzp line based on Equation log K 00 = log(KPb/PbL/yPb) = log K 0 + 4ABz IBz (18) (18) was log KPb/PbL0′ = (3.546Pb/PbL ± 0.001) + 4 × (2.16 ± 0.16) at Pb/PbLR = 0.997, except for the two points of IBz = 4.4 × 10−7 and 1.7 × 10−6 mol·dm−3 (see Table 2). These two log KPb/PbL0′ values excluded from the �𝐼𝐼Bz regression analysis are included in the regression line within experimental errors. However, the ABz 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 ABz < 0.

Inorganics 2018, 6, 77 11 of 16

00 1/2 The log KPb/PbL values were plotted against the IBz ones. The√ regression line based on 00 Equation (18) was log KPb/PbL = (3.546 ± 0.001) + 4 × (2.16 ± 0.16) IBz at R = 0.997, except for −7 −6 −3 00 the two points of IBz = 4.4 × 10 and 1.7 × 10 mol·dm (see Table2). These two log KPb/PbL values excluded from the regression analysis are included in the regression line within experimental errors. However, the ABz value is much smaller than the others. In addition, the analysis was tried by using an Inorganics 2018, 6, x FOR PEER REVIEW 11 of 16 equation similar to Equations (14b) and (15b). However, its regression line showed the result of ABz < 0.

-2

-2.5 ' 0

D,Pic -3 K log

-3.5

-4 0 0.0005 0.001 0.0015 0.002 0.0025 0.003

I 1/2/mol1/2 dm-3/2 Bz

00 1/2 FigureFigure 9. 9.Plot Plot ofof log KD,Pic0′ versusversus IBz1/2IBz basedbased on Equation on Equation (17) fo (17)r the for Pb(II) the extraction Pb(II) extraction with Cd(NO with3)2 00 Cd(NOand B18C63)2 and into B18C6 Bz. The into symbol Bz. The K symbolD,Pic0′ wasK D,Picdefinedwas as K deﬁnedD,Pic/yPic as. TheKD,Pic error/y Picbars. The in the error figure bars a inre thethose ﬁgureof the are log those KD,Pic of values. the log KD,Pic values.

D,Pic0 0 Pb/PbL0 0 ex2±0 0 ByBy a combinationa combination with with log logKD,Pic K , the, the log logKPb/PbL K can can be be changed changed into into log logKex2 K± (=log (=log PbL,Bz Pic,Bz 22 Pb Bz Pic,Bz 2 2 Pb/PbL0 0 D,Pic0 0 ex2±0 0 (aPbL,Bz(a (a(Pic,Bza )) //aaPb[L][L]Bz(a(aPic,Bz) ))) ))= =log log KKPb/PbL + 2log+ 2log K KD,Pic. Thus,. Thus, the log the logK Kex2 value± value was wasestimated estimated to be 5 7 6 3 to−2.4 be − 2.4± 0.35 ± from0.37 from Equation Equation (17) (17)(the (the linear linear type) type) or or−4.9−4.9 ± 60.1± 0.1from3 from Equation Equation (17a (17a)) (the (the f (pf) (=p )a= +a bp+ + 2 2 ex±0 0 1 1,Bz00 8 bpcp+ cp typetype).). Using Using log log KKex± = 3.1= 3.1 obtained1 obtained from from Equation Equation (16) and loglogK K1,Bz == 5.95.98 fromfrom Equation Equation (14a) (14a) ex2±0 0 7 5 ex2±0 0 (the(the linear linear type), type), the the log logKex2 K± value value was was calculated calculated to be to− be2.8 −2.87 ± 0.8± 0.85. On. On the the other other hand, hand, the the log logKex2 K± 1 1 1,Bz0 0 6 valuevalue became became−3. −3.1 ± ±1. 1.1 in in the the calculation calculation with with log logK1,Bz K = = 6.1 6.16 from from Equation Equation (14b) (14b) (the (thef ( pf )(p =) a= +a bp+ bp+ + cp2cptype).2 type Except). Except for for−4.9 −4.9 from from Equation Equation (17a), (17 thea), valuesthe values calculated calculated from from the three the equationsthree equations agreed agreed with eachwith other each within other their within calculation their calculation errors. According errors. According to the thermodynamic to the thermodynamic cycle, the relation cycle, the of K relationex± = of Kex± = KPb/PbLK1,Bz2 (KD,Pic)2 holds. From this relation, we obtained0 log Kex±0 = 3.54 ± 0.72 (=log 0KPb/PbL0 + KPb/PbLK1,Bz(KD,Pic) holds. From this relation, we obtained log Kex± = 3.54 ± 0.72 (=log KPb/PbL + log 0 1,Bz0 0D,Pic0 K1,Bzlog K+ 2log + 2logKD,Pic K ). This). This value value is inis in agreement agreement with with that that (=3.1) (=3.1) calculated calculated from from Equation Equation (16) (16) within within thethe calculation calculation error error of of±0.7. ±0.7. In In addition, addition the, the same same calculation calculation was was performed performed with with the the values values obtained obtained 0 3 fromfrom the the polynomial polynomialEquations Equations (14b) (14b) and and (17a). (17Itsa). valueIts value was was 1.2 01.2± 0.8± 0.83, being, being much much smaller smaller than than 3.1. 3.1. TheseThese results results suggest suggest that that the the linear-type linear-type equation equation is the is more the more reliable reliable than the than polynomial-type the polynomia one,l-type from one, thefrom thermodynamic the thermodynamic points of points view. of view. Bz −1/2−1/2 3/23/2 FromFrom the the four four experimental experimentalABz/mol A /mol·dm·dm values values based based on Equationson Equations (14a), (14a), (15a), (15a), (16), and (16), (17), and except(17), forexcept the value for the obtained value fromobtained Equation from (18), Equation their average (18), their value average was estimated value was to be estimated 230. Consequently, to be 230. Bz −1/2 3/2 thisConsequenABz valuetly, for this Bz saturatedA value withfor Bz water saturated was about with water 2-times was larger about than 2-times that (=103.3 larger molthan that·dm (=103.3) −1/2 3/2 r calculatedmol ·dm for pure) calculated Bz with εforr = 2.275pure [Bz2] atwith 298.15 ε = K. 2.275 To agree [2] at with 298.15 this conclusion,K. To agree however, with this a reasonableconclusion, reasonhowever, will be a reasonable required for reason the omission will be of required the result for of the Equation omission (18). 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 (Kex,Pb/Kex,Cd) = log (D0,Pb/D0,Cd), (19)

log (Kex±,Pb/Kex±,Cd) = log (D+,Pb/D+,Cd), (20)

and log (KPb/PbL/KCd/CdL) = log (D2+,Pb/D2+,Cd) (21)

with

D0,Pb = [PbLPic2]Bz/[Pb2+] = Kex[L]Bz[Pic−]2, (22)

D+,Pb = [PbLPic+]Bz/[Pb2+] = Kex±[L]Bz[Pic−]/KD,Pic, (23)

Inorganics 2018, 6, 77 12 of 16

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 (Kex,Pb/Kex,Cd) = log (D0,Pb/D0,Cd), (19)

log (Kex±,Pb/Kex±,Cd) = log (D+,Pb/D+,Cd), (20)

and log (KPb/PbL/KCd/CdL) = log (D2+,Pb/D2+,Cd) (21) with 2+ − 2 D0,Pb = [PbLPic2]Bz/[Pb ] = Kex[L]Bz[Pic ] , (22) + 2+ − D+,Pb = [PbLPic ]Bz/[Pb ] = Kex±[L]Bz[Pic ]/KD,Pic, (23) 2+ 2+ and D2+,Pb = [PbL ]Bz/[Pb ] = KPb/PbL[L]Bz. (24)

Here, D0,Pb, D+,Pb, and D2+,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:

2+ 2+ log (D0,Pb/D0,Cd) = log ([PbLPic2]Bz/[CdLPic2]Bz) + log ([Cd ]/[Pb ]) (25)

2+ 2+ 2+ 2+ Assuming that [Cd ]/[Pb ] approximately equals [Cd ]t/[Pb ]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 (D0,Pb/D0,Cd) value was 7.09 in which the log Kex value (=9.448) was estimated at I = 0.095 [33] 2+ 2+ from the regression line of Figure5. From Equation (25) and [Cd ]t/[Pb ]t = 55.5 estimated from a correlation between [Cd]t/[Pb]t and I in Table1, the log ([PbLPic 2]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 D2+,Pb/D2+,Cd, if the Kex±,Cd and KCd/CdL values are determined about the CdPic2 extraction with B18C6 (=L) into Bz.

2.10. Relative Concentrations of the Three Species Extracted into Bz

+ 2+ We can immediately calculate relative concentrations of PbLPic2, PbLPic , and PbL in the Bz phases from the D0,Pb, D+,Pb, and D2+,Pb values, respectively [8,24,30]. For example, the percentage of the relative concentration of PbLPic2 can be obtained from 100D0,Pb/(D0,Pb + D+,Pb + D2+,Pb). In addition, the concentrations of PbLPic+ and PbL2+ were evaluated from similar equations. + 2+ The thus-calculated values were: 46% for PbLPic2, 10% for PbLPic , and 44% for PbL at −3 I = 0.0074 mol·dm (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. 2+ One can see easily that the distribution of PbLPic2 and PbL 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 0 2+ − of both PbLPic2 and PbL 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? Inorganics 2018, 6, 77 13 of 16

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 4 −1 −1 3 − with the calibration curve (ε356 = 1.45 × 10 cm ·mol ·dm [35]) for Pic . Using a calibration curve 3 −1 −1 3 (ε273 = 2.50 × 10 cm mol ·dm [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

−4 −3 −3 Aqueous solution containing 5.478 × 10 mol·dm Pb(NO3)2, 1.272 × 10 HPic, x Cd(NO3)2, 3 0.0198 HNO3, and y B18C6 was prepared and mixed with the equal volume (10 or 12 cm ) of Bz saturated with water in a stoppered glass tube of about 30 cm3. Here, x was fixed at 0 mol·dm−3, −4 −6 5.810 × 10 , 0.01459, 0.04847, or 0.09726 and, at a fixed x, y was changed in the ranges of 6.5 × 10 to 1.6 × 10−4 mol·dm−3 (see circle in Figure1), 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 −3 tubes, and some cubic centimeters of 0.1 mol·dm 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 2+ −1 3 2+ − + 2+ PbL [29](KCdL = 0.89 mol ·dm [18] was omitted; (2) Pb + Pic PbPic [21]; (3) Cd + − + + − − − Pic CdPic [1]; and (4) H + Pic HPic [33] in the water phase; (5) Pic Pic Bz; (6) HPic 2+ 2+ HPicBz [36]; (7) L LBz [2]; and (8) PbL PbL Bz between the water and Bz phases; and (9) 2+ − + + − PbL Bz + Pic Bz PbLPic Bz; and (10) PbLPic Bz + Pic Bz PbLPic2,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 Kex, Kex±, KPb/PbL, Kex,ip, and KD,Pic were determined at 298 K. 0 0 The same is also true of the K1,Bz and K2,Bz values at IBz → 0 for the simple Bz phases. It was Inorganics 2018, 6, 77 14 of 16

0 0 0 0 0 0 demonstrated that the thermodynamic relations, Kex ≈ Kex,ip KPbL/KD,L, K2,Bz = Kex /Kex± , Kex2± 0 0 2 0 0 0 0 0 0 0 2 = KPb/PbL (KD,Pic ) , Kex2± = Kex± /K1,Bz , and Kex± = KPb/PbL K1,Bz (KD,Pic ) , hold in the system. It seems that the linear equation is superior to the polynomial-type one for the IBz 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 Kex, Kex±, or K1,Bz values reported in different I or Iorg 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 Kex± 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 = 5 4 2.2 × 10 at B18C6 (=L) and satisfies a measure (=10 ) of the separation factor. The Kex,Pb/Kex,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.

Author Contributions: Y.K. and C.N. conceived and designed the experiments; T.N. performed the experiments; Y.K. and T.N. analyzed the data; C.N. and Y.K. contributed reagents/materials/analysis tools; and Y.K. wrote the paper. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

Kex Extraction constant for MLA2 + − Kex± Extraction constant for MLA with A I Ionic strength for the water phase − KD,A Conditional distribution constant of A into the org phase 2+ KM/ML Incorporative constant of M with L into the org phase Kex,ip Ion-pair extraction constant for MLA2 2+ − K1,org Ion-pair formation constant for ML with A in the org phase + D0, D+, D Distribution ratio for MLA2, that for MLA , that for mixture + Kex+ Extraction constant for MLA mix Kex Extraction-constant parameter Dep, ∆φeq Distribution equilibrium potential between the bulk water and org phases S − KD,Pic Standard distribution constant of Pic into the org phase 00 − ∆φPic Standard formal potential for the Pic transfer across the water/org interface + − K2,org Ion-pair formation constant for MLA with A in the org phase 2+ KD,PbL Conditional distribution constant of PbL into the org phase 2+ − Kex2± Extraction constant for ML with 2A KD,L Distribution constant of L into the org phase 2+ KPbL Complex formation constant of Pb with L in water 0 Kex Thermodynamic extraction constant of Kex å± Ion-size parameter, a mean value 0 Kex,ip Thermodynamic ion-pair extraction constant of Kex,ip 0 2+ − K1,Bz Thermodynamic ion-pair formation constant for ML with A in the Bz phase 0 + − K2,Bz Thermodynamic ion-pair formation constant for MLA with A in the Bz phase 0 Kex± Thermodynamic extraction constant of Kex± 0 2+ KPb/PbL Thermodynamic incorporative constant of Pb with L into the org phase 0 Kex2± Thermodynamic extraction constant of Kex2± 2+ D2+,Pb Distribution ratio for PbL Inorganics 2018, 6, 77 15 of 16

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