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*Inorganics*
**2017**,
*5*(3),
42;
doi:10.3390/inorganics5030042

Article

Silver(I) Extraction with Benzo-18-Crown-6 Ether from Water into 1,2-Dichloroethane: Analyses on Ionic Strength of the Phases and their Equilibrium Potentials

^{1}

Graduate School of Science, Chiba University, Chiba 263-8522, Japan

^{2}

Department of Chemistry, Faculty of Science, Chiba 263-8522, Japan

*

Correspondence: Tel.: +81-43-290-2786

Academic Editor:
Duncan H. Gregory

Received: 19 May 2017 / Accepted: 25 June 2017 / Published: 30 June 2017

## Abstract

**:**

Extraction constants (K

_{ex}& K_{ex±}) for the extraction of silver picrate (AgPic) by benzo-18-crown-6 ether (B18C6) into 1,2-dichloroethane (DCE) were determined at 298 K and various ionic strength (I)-values of a water phase with or without excess HNO_{3}. Here the symbols, K_{ex}and K_{ex±}, were defined as [AgLPic]_{DCE}/P and [AgL^{+}]_{DCE}[Pic^{−}]_{DCE}/P with P = [Ag^{+}][L]_{DCE}[Pic^{−}] and L = B18C6, respectively; [ ]_{DCE}refers to the concentration of the corresponding species in the DCE phase at equilibrium. Simultaneously, K_{D,Pic}(= [Pic^{−}]_{DCE}/[Pic^{−}]) and K_{1,DCE}(= K_{ex}/K_{ex±}) values for given I and I_{DCE}values were determined, where the symbol I_{DCE}shows I of the DCE phase. Also, equilibrium potential differences (Δφ_{eq}) based on the Pic^{−}transfer at the water/DCE interface were obtained from the analysis of the K_{D,Pic}[= K_{D,Pic}^{S}exp{−(F/RT) Δφ_{eq}}] values; the symbol K_{D,Pic}^{S}shows K_{D,Pic}at Δφ_{eq}= 0 V. On the basis of these results, I dependences of logK_{ex}and logK_{ex±}and I_{DCE}ones of logK_{1,DCE}and logK_{ex±}were examined. Extraction experiments of AgClO_{4}and AgNO_{3}by B18C6 into DCE were done for comparison. The logK_{ex±}-versus-Δφ_{eq}plot for the above Ag(I) extraction systems with Pic^{−}, ClO_{4}^{−}, and NO_{3}^{−}gave a good positive correlation.Keywords:

extraction constants; conditional distribution constants of ions; equilibrium potential difference between water and organic phases; ionic strength; silver salts; benzo-18-crown-6 ether; 1,2-dichloroethane## 1. Introduction

It is well known that crown compounds (L) extract alkali and alkaline-earth metal ions (M

^{z}^{+}, z = 1, 2) from water (w) into various diluents [1,2,3,4]. In many extraction experiments, extraction constants for L have been determined so far [1,2,3,4,5,6,7,8]. For example, the two representative constants, K_{ex}and K_{ex±}, for the extraction of a univalent metal salt (M^{I}A) by L have been defined as [MLA]_{org}/P [2,3] and [ML^{+}]_{org}[A^{−}]_{org}/P [1,4] with P = [M^{+}][L]_{org}[A^{−}], respectively. Generally, the K_{ex}value is effective for the evaluation of an extraction-ability and -selectivity of L against M^{+}into low-polar diluents, while the K_{ex±}value is for those of L into high-polar ones. Here, the subscript ″org″ denotes an organic phase and A^{−}does a univalent pairing anion. For the latter K_{ex±}, its thermodynamic equilibrium constants have been reported [4]. For the former K_{ex}, its thermodynamic treatment seems to be few. The authors were not able to find out the study with respect to a dependence of logK_{ex}on the ionic strength (I) of the w phase.Presences of equilibrium potential differences (Δφ

_{eq}) between aqueous and diluent solutions have been recently reported for the extraction of some M^{+}or M^{2+}with L [5,6,7]. This symbol Δφ_{eq}was defined as ∑{inner potential (φ ) of ionic species in the w phase} − ∑{φ of those in the org or diluent phase} [8], according to the definition [9,10], φ_{w}− φ_{org}, of an interfacial equilibrium potential-difference in the electrochemistry at liquid/liquid interfaces. In the above studies, an approximation method [5,6,7,8] for the Δφ_{eq}determination, namely the method with a use of a ″conditional″ distribution constant (K_{D,A}) of A^{−}into the org phase, has been described in comparison with its more-precise method with solving higher-degree equations [6]. Here the ″conditional″ is due to the fact that the K_{D,A}values change depending on the Δφ_{eq}ones, even at fixed pressure and temperature. However, it is still not clarified whether the Δφ_{eq}values determined by the K_{D,A}values equal those coming from the distribution of M^{+}into the org phases or not.In the present paper, we determined the K

_{ex}, K_{ex±}, and K_{D,A}values [6,8] at 298 K by the extraction experiments into 1,2-dichloroethane (DCE) with silver picrate (AgPic) and benzo-18-crown-6 ether (B18C6), in order to elucidate mainly the above two subjects for I and Δφ_{eq}. The same experiments were performed under the condition of the presence of excess HNO_{3}in the w phases. Then, an ion-pair formation constant (K_{1,DCE}/mol^{−1}·dm^{3}) for Ag(B18C6)^{+}Pic^{−}in the DCE phase, DCE saturated with water, and the Δφ_{eq}values were calculated from the relations, K_{1,DCE}= K_{ex}/K_{ex±}[6,8] and Δφ_{eq}= −(2.303RT/F){logK_{D,A}− log(K_{D,A}standardized at Δφ_{eq}= 0 V)} [11], respectively. Here, R, T, and F are usual meanings. On the basis of these data, the dependences of logK_{ex}and logK_{ex±}on the I values and those of logK_{1,DCE}and logK_{ex±}on the I values (I_{DCE}) of the DCE phases were examined. Moreover, a relation between the Δφ_{eq}values determined by the K_{D,A}ones and the conditional distribution constants (K_{D,Ag}) of Ag^{+}into the DCE phases was discussed indirectly. For comparison, the K_{ex±}and K_{ex}values were experimentally determined at 298 K for the AgClO_{4}- and AgNO_{3}-B18C6 extraction into DCE. As basic data, the K_{D,Ag}^{S}value was determined in terms of a simple Ag^{+}Pic^{−}extraction experiment into DCE. The symbol K_{D,Ag}^{S}denotes the distribution constant of Ag^{+}into the DCE phase standardized at Δφ_{eq}= 0 V, that is, the standard distribution one.## 2. Results

#### 2.1. Determination of LogK_{D,Ag}^{S}

According to our previous paper [12], the K
with
and K

_{D,M}^{S}value has been obtained from a plot of D_{A}′ versus [A^{−}] based on the equation
D

_{A}′ = [A]_{t,org}/[A^{−}] = K_{ex,MA}[A^{−}] + K_{D, ±}
K

_{D, ±}^{2}= K_{D,M}^{S}·K_{D,A}^{S}= K_{D,M}·K_{D,A}_{ex,MA}= K_{MA,org}(K_{D, ±})^{2}(= [MA]_{org}/[M^{+}][A^{−}]), where K_{MA,org}is [MA]_{org}/[M^{+}]_{org}[A^{−}]_{org}and [A]_{t,org}denotes a total concentration, [MA]_{org}+ [A^{−}]_{org}, of A(−I) in the org phase. A regression analysis of the plot (see Figure 1) yields a straight line with a slope of K_{ex,AgPic}and an intercept of K_{D, ±}. We call this K_{D, ±}a mean distribution constant.Using the K

_{D,Pic}^{S}value (= 10^{−1.011}[12]), we immediately can obtain the K_{D,Ag}^{S}one from Equation (2). The thus-determined values were logK_{D, ±}= −3.74 ± 0.04, logK_{ex,AgPic}= −1.49 ± 0.05, and logK_{AgPic,DCE}= 5.992 ± 0.008 and then the logK_{D,Ag}^{S}value became −6.47 ± 0.04 from the logarithmic form of Equation (2). This K_{D,Ag}^{S}value was used only for the K_{AgL,DCE}calculation (see Table 1 and Table 2).#### 2.2. Composition Determination of Complex Species Extracted into DCE

Compositions of species extracted into DCE have been determined by a plot of 2logD or log(D/[A

^{−}]) versus log[L]_{org}[2,7,13]. When the slope of both plots is in unity, it independently gives the compositions of ML^{+}with A^{−}or MLA as the extracted major species. Among their plots, experimental slopes of all the log(D/[Pic^{−}])-versus-log[B18C6]_{org}plots were less than unity, suggesting the dissociation of Ag(B18C6)Pic in the DCE phases [13]. On the other hand, the 2logD-versus-log[L]_{org}plots were in the slope ranges of 1.01–1.11 for the AgPic extraction with L = B18C6, in the slopes of 1.09 for the AgClO_{4}one and of 1.08 for the AgNO_{3}one (Figure 2). These results indicate the AgB18C6^{+}extraction into DCE with A^{−}= Pic^{−}, ClO_{4}^{−}, or NO_{3}^{−}. That is, the extraction systems were accompanied with the dissociation process, Ag(B18C6)A_{DCE}$\rightleftharpoons $ AgB18C6^{+}_{DCE}+ A^{−}_{DCE}(see Appendix).#### 2.3. Determination of Various Equilibrium Constants for the Extraction Systems

For the determination of K
Rearranging this equation, we immediately obtain
under the assumption of [MLA]

_{D,A}, K_{ex±}, and K_{ex}, the following extraction-constant parameter (K_{ex}^{mix}/mol^{−2}·dm^{6}) has been employed [7,8,11].
logK

_{ex}^{mix}= log{([MLA]_{org}+ [ML^{+}]_{org}+ [M^{+}]_{org})/P}
logK

_{ex}^{mix}≈ log{K_{ex}+ (K_{D,A}/[M^{+}][L]_{org})}
≈ log{K

_{ex}+ (K_{ex±}/P)^{1/2}}_{org}+ [ML^{+}]_{org}>> [M^{+}]_{org}. So we can determine the K_{D,A}and K_{ex±}values from the plots of logK_{ex}^{mix}versus −log([M^{+}][L]_{org}) {see Equation (3a)} and −(1/2)logP {see Equation (3b)}, respectively, together with the K_{ex}values. Figure 3 and Figure 4 show examples for such plots. Additionally, the ion-pair formation constant, K_{1,org}, for MLA in the org phase was calculated from K_{1,org}= K_{ex}/K_{ex±}for a given I_{org}on average.Similarly, a complex formation constant (K

_{AgB18C6,DCE}/mol^{−1}·dm^{3}) for AgB18C6^{+}in the DCE phase was estimated from the thermodynamic relation of K_{AgB18C6,DCE}= K_{ex±}/(K_{D,Ag}^{S}·K_{D,Pic}^{S}). Table 1 and Table 2 list the logarithmic K_{D,A}, K_{ex±}, K_{ex}, K_{1,DCE}, and K_{AgB18C6,DCE}values thus-obtained, together with the average I and I_{DCE}values.#### 2.4. Determination of Equilibrium Potential Differences between the Water and DCE Phases

The logK
with
where the symbols, K

_{D,A}values were obtained from the plots based on Equation (3a). Next, the equilibrium potential differences Δφ_{eq}can be evaluated from using the following equation [11]:
Δφ

_{eq}= −(2.303RT/F)(logK_{D,A}− logK_{D,A}^{S})
logK

_{D,A}^{S}= (F/2.303RT)Δφ_{A}°′_{D,A}^{S}and Δφ_{A}°′, are called the standard distribution constant of A^{−}into the org phase, namely K_{D,A}at Δφ_{eq}= 0 V, and a standard formal potential for the A^{−}transfer at the w/diluent interface, respectively; see Section 2.1 for the K_{D,Pic}^{S}value. The thus-evaluated values are listed in Table 1 and Table 2.## 3. Discussion

#### 3.1. I Dependences of LogK_{ex} and LogK_{ex±}

The dependences of logK

_{ex}and logK_{ex±}on I are considered below. On the basis of their definitions, the K_{ex}value can be dependent on [M^{+}] and [A**], while, in addition to these concentrations, the K**^{−}_{ex±}value can be on [ML^{+}]_{org}and [A**]**^{−}_{org}. Therefore, K_{ex}is mainly a function of I, while K_{ex±}is a function with the two parameters, I and I_{org}; the relations [14] of [M^{+}] = a_{M}/y_{+}(I) and [A^{−}]_{org}= a_{A,org}/y_{−,org}(I_{org}) hold as examples (see below for the symbols a and y). The I dependence of logK_{ex±}is of an approximate.Figure 5 shows the logK
where the symbols, a

_{ex}-versus-I plot for the AgPic-B18C6 extraction system with DCE; in the plot, the average value of I was employed as I (x-axis) of each system. Using the extended Debye-Hückel (DH) equation [14], the extraction constant (K_{ex}^{0}) at I → 0 mol·dm^{−3}is expressed as
logK

_{ex}^{0}= log([MLA]_{org}/a_{M}[L]_{org}a_{A}) = logK_{ex}− log(y_{+}y_{−})
$$=\mathrm{log}{K}_{\mathrm{ex}}+2A\sqrt{I}/(1+B\xe5\sqrt{I})$$

_{j}and y, denote the activity of species j (= M^{+}, A^{−}) and its activity coefficient with I, respectively [14]. Rearranging Equation (5), the following equation was obtained immediately:
$$\mathrm{log}{K}_{\mathrm{ex}}=\mathrm{log}{{K}_{\mathrm{ex}}}^{0}-2A\sqrt{I}/(1+B\xe5\sqrt{I})$$

The regression analysis of the plot in Figure 5 based on this equation yielded the regression line with log(K

_{ex}^{0}/mol^{−2}·dm^{6}) = 5.28 ± 0.25 and Bå = 44 ± 661 mol^{−1/2}·dm^{3/2}Å at r = 0.024 and N (number of data) = 10, the fixed A value (= 0.5114) in pure water, and 298 K. Considering the error of the experimental Bå value, it is difficult to discuss the Bå or å value in this result. When the three parameters, logK_{ex}^{0}, A, and Bå, had been used for the regression analysis, it gave the results of logK_{ex}^{0}> 0, A < 0, and Bå < 0. Consequently, we gave up such an analysis.Similarly, the extraction constant (K
where the subscript “ML” means the complex ion ML
where K

_{ex±}^{0}) at I → 0 is expressed as
logK

_{ex±}^{0}= log(a_{ML,org}a_{A,org}/a_{M}[L]_{org}a_{A})
= logK

_{ex±}+ log(y_{ML,org}y_{−,org}) − log(y_{+}y_{−})^{+}. Rearranging this equation, we can immediately obtain
$$\mathrm{log}{K}_{\mathrm{ex}\pm}=\mathrm{log}{{K}_{\mathrm{ex}\pm}}^{0}\prime -2A\sqrt{I}/(1+B\xe5\sqrt{I}),$$

_{ex±}^{0}′ denotes K_{ex±}^{0}/(y_{ML,org}y_{−,org}) (= [ML^{+}]_{org}[A^{−}]_{org}/a_{M}[L]_{org}a_{A}). Unfortunately, the analysis of the plot based on Equation (6a) did not yield the suitable result which satisfies the condition of Bå > 0.On the other hand, using the Davies equation without Bå instead of the extended DH equation [14], logK

_{ex±}^{0}′ = 0.60 ± 0.11 and A = 2.05 ± 0.38 mol^{−1/2}·dm^{3/2}were obtained (Figure 6). The Davies equation is logy = −A${z}^{2}\{\sqrt{I}$/(1 + $\sqrt{I}$) − 0.3I} [14], where z shows a formal charge of ionic species with a sign (refer to the Introduction).The analysis of the logK

_{ex}-versus-I plot by the Davies equation yielded logK_{ex}^{0}= 5.29 ± 0.11 with A = 0.08 ± 0.40 mol^{−1/2}·dm^{3/2}. Within the calculation error of ±0.3, this logK_{ex}^{0}value was in accord with 5.3 determined by the DH equation (see above in this section).#### 3.2. I_{DCE} Dependence of LogK_{ex±}

Applying the DH limiting law [14] for the system and rearranging Equation (6) at org = DCE, we can easily obtain
Hence, a plot of logK

$$\mathrm{log}{K}_{\mathrm{ex}\pm}\approx \mathrm{log}{{K}_{\mathrm{ex}\pm}}^{0}\prime \prime +2{A}_{\mathrm{DCE}}\sqrt{{I}_{\mathrm{DCE}}}.$$

_{ex±}versus I_{DCE}yields logK_{ex±}^{0}′′ and A_{DCE}values immediately. Here, K_{ex±}^{0}′′ is defined as a_{ML,DCE}a_{A,DCE}/([M^{+}][L]_{DCE}[A^{−}]) (=y_{+}y_{−}K_{ex±}^{0}). Figure 7 shows its plot for the AgPic-18C6 extraction systems with DCE.Also the average values of I

_{DCE}were used for the plot (see Section 3.1) and the y_{+}y_{−}value in K_{ex±}^{0}′′ was estimated on average (N = 10) to be 0.76 ± 0.14. This product was calculated from the ion size parameters, a(Ag^{+}) = 2.5 and a(Pic^{−}) = 7 Å, in water [15]. A plot analysis gave logK_{ex±}^{0}′′ = −0.45 ± 0.24 and A_{DCE}= 116 ± 46 mol^{−1/2}·dm^{3/2}at r = 0.665. Accordingly, introducing y_{+}y_{−}in logK_{ex±}^{0}′′ = logy_{+}y_{−}+ logK_{ex±}^{0}, the logK_{ex±}^{0}value became –0.36 ± 0.24. The experimental A_{DCE}value was much larger than its theoretical one (= 10.6 mol^{−1/2}·dm^{3/2}) for a pure DCE at 298 K. This difference between these A_{DCE}values may be due to simple errors caused by the narrow experimental I_{DCE}-range of (0.097–2.2) × 10^{−5}mol·dm^{−3}or to the condition where the diluent DCE was saturated with water.#### 3.3. I_{DCE} Dependences of LogK_{1,DCE}

The thermodynamic ion-pair formation constant (K
Rearranging this equation at org = DCE and ML
A plot of logK

_{1,org}^{0}) at I_{org}→ 0 is described as
logK

_{1,org}^{0}= log([MLA]_{org}/a_{ML,org}a_{A,org}) = logK_{1,org}− log(y_{ML,org}y_{−,org})^{+}= AgB18C6^{+}can give the following equation:
$$\mathrm{log}{K}_{1,\mathrm{DCE}}=\mathrm{log}{{K}_{1,\mathrm{DCE}}}^{0}+\mathrm{log}({y}_{\mathrm{AgB}18\mathrm{C}6,\mathrm{DCE}}{y}_{-,\mathrm{DCE}})\approx \mathrm{log}{{K}_{1,\mathrm{DCE}}}^{0}-2{A}_{\mathrm{DCE}}\sqrt{{I}_{\mathrm{DCE}}}$$

_{1,DCE}versus I_{DCE}is shown in Figure 8. The plot analysis yielded the regression line with logK_{1,DCE}^{0}= 5.89 ± 0.19 and A_{DCE}= 152 ± 37 mol^{−1/2}·dm^{3/2}at r = 0.821 and N = 10. This A_{DCE}value overlaps with the value (= 116) determined above (see Section 3.2), within the calculation error (= 46) and much larger than the theoretical one too. The authors cannot clearly explain the larger experimental A_{DCE}values, as similar to Section 3.2.The logarithmic value, logK

_{1,DCE}^{av}, of simple average-K_{1,DCE}one was 5.36 ± 0.42 in the I_{DCE}range of (0.097-2.2) × 10^{−5}mol·dm^{−3}at N = 10 and was smaller than the logK_{1,DCE}^{0}value (= 5.9 at I → 0). Although the experimental I_{DCE}values were adequately small (I_{DCE}<< 0.001), the magnitude of K_{1,DCE}decreased with an increase in I_{DCE}. Also, the logK_{1,DCE}^{av}value was smaller than the logK_{AgPic,DCE}one (= 6.0, see Section 2.1). From the logK_{ex±}^{0}′ value (= 0.6) in Section 3.1 and the logK_{ex±}^{0}one (= −0.36) in 3.2, we obtained log(y_{AgB18C6,DCE}y^{−}_{,DCE}) (= logK_{ex±}^{0}− logK_{ex±}^{0}′) = −0.96 ± 0.27. Hence, the logK_{1,DCE}value was estimated to be 4.93 {= logK_{1,DCE}^{0}+ log(y_{AgB18C6,DCE}y_{−,DCE}) = 5.89−0.96}, being somewhat smaller than the logK_{1,DCE}^{av}value (= 5.4). These facts indicate that the logK_{1,DCE}^{av}value is not properly reflective of the logK_{1,DCE}one in Equation (7).On the other hand, the log(K

_{AgB18C6,DCE}/mol^{−}^{1}·dm^{3}) values were calculated from the relation logK_{ML,org}= logK_{ex±}− logK_{D,M}^{S}⋅K_{D,A}^{S}for a given I_{DCE}. Here, we assumed that, considering the smaller I_{DCE}values, the ratio, y_{Ml,DCE}/y_{+,DCE}, of the activity coefficients in the thermodynamic complex-formation constant, K_{ML,DCE}^{0}, equals unity. Accordingly, the approximation that an average value among the K_{AgB18C6,DCE}ones equals the K_{AgB18C6,DCE}^{0}value becomes valid. Consequently, as its logarithmic value, 7.77 ± 0.25 was obtained on average (N = 12) at 298 K.#### 3.4. A Trend between LogK_{D,Pic} and Log(I_{DCE}/I)

From a plot of logK

_{D,Pic}versus log(I_{DCE}/I), we obtained a theoretical line of logK_{D,Pic}= log(I_{DCE}/I) − (0.09 ± 0.12) at r = 0.398 (Figure 9) under the condition of the fixed slope of unity, except for the points in the I range of 0.024-0.26 mol·dm^{−}^{3}. This trend suggests that the K_{D,Pic}values are basically proportional to the ratios of (I_{DCE}/I), as reported before [16]. The deviation of the five points in Figure 9 can come from an effect of the higher I values in (I_{DCE}/I). An answer to this deviation is explained as follows.For the case of only MA (or mixture of MX with small excess organic acid, HA) in w phase with L, we can propose the following equation for I and I
with the charge balance equation of [M
in the org phase. The assumption that [AgB18C6

_{org}.
I = (1/2)([M

^{+}] + [ML^{+}] + [A^{−}]) = [A^{−}]^{+}] + [ML^{+}] = [A^{−}]. Similarly, the authors can obtain
I

_{org}= (1/2)([M^{+}]_{org}+ [ML^{+}]_{org}+ [A^{−}]_{org}) ≈ [A^{−}]_{org}^{+}]_{DCE}is much larger than [Ag^{+}]_{DCE}, namely [AgB18C6^{+}]_{DCE}≈ [Pic^{−}]_{DCE}, was employed for the experiments. From Equations (8) and (8a), one can see easily the experimental relation that (I_{DCE}/I) is basically proportional to K_{D,Pic}.On the other hand, for the present case of mixture of MX with the small excess HA and excess HX (strong acid) in the w phase with L, the corresponding equations are
and
with the assumptions that [M
and then its ratio becomes smaller than the K

I = (1/2)([M

^{+}] + [H^{+}] + [ML^{+}] + [HL^{+}] + [A^{−}] + [X^{−}]) ≈ [A^{−}] + [X^{−}]
I

_{org}= (1/2)([M^{+}]_{org}+ [H^{+}]_{org}+ [ML^{+}]_{org}+ [HL^{+}]_{org}+ [A^{−}]_{org}+ [X^{−}]_{org}) ≈ [A^{−}]_{org}^{+}] + [ML^{+}] + [H^{+}] ≈ [A^{−}] + [X^{−}] and [M^{+}]_{org}+ [H^{+}]_{org}+ [ML^{+}]_{org}+ [HL^{+}]_{org}≈ [A^{−}]_{org}(>> [X^{−}]_{org}), respectively. Therefore, Equations (9) and (9a) give the experimental relation of
(I

_{DCE}/I) ≈ [Pic^{−}]_{DCE}/([Pic^{−}] + [NO_{3}^{−}]) = K_{D,Pic}/(1 + [NO_{3}^{−}]/[Pic^{−}])_{D,Pic}value in the case of [NO_{3}^{−}] >> [Pic^{−}]. When [NO_{3}^{−}] nearly equals [Pic^{−}] in Equation (9b), the log(I_{DCE}/I) value deviates from the logK_{D,Pic}one by +0.3: that is, (I_{DCE}/I) ≈ K_{D,Pic}/2. However, such a deviation is comparable to experimental errors. Thus, Equation (9b) explains well the deviation of the five points from the regression line in Figure 9. Obviously, the deviation becomes larger, when an excess of HX, such as HNO_{3}, was added in the w phase and X^{−}less distribute into the org phase than A^{−}does (for example, see the K_{D,Pic}^{S}& K_{D,NO3}^{S}values in Table 3).#### 3.5. Δφ_{eq} Dependences of LogK_{M/ML} and LogK_{ex±}

The logK
at 298 K with
Here, the term of log(K

_{M/ML}, defined as log([ML^{+}]_{org}/[M^{+}][L]_{org}) [8], can be resolved as follows and calculated from logK_{ex±}− logK_{D,A}.
logK

_{M/ML}= logK_{D,M}+ logK_{ML,org}= 16.90Δφ_{eq}+ logK_{D,M}^{S}·K_{ML,org}
logK

_{D,M}= (F/2.303RT)Δφ_{eq}+ logK_{D,M}^{S}_{D,M}^{S}⋅K_{ML,org}) has to be a constant, because these two equilibrium constants are independent of Δφ_{eq}; logK_{ML,org}= −(F/2.303RT) (standard formal potential of the ML^{+}formation in the org phase) [8]. The magnitude of the K_{M/ML}(or K_{M/ML}^{S}) value shows an incorporation-ability into the org phase of L against M^{+}. Then, the plot of logK_{M/ML}versus Δφ_{eq}based on Equation (10) can yield a straight line with the slope of 17 V^{−1}and the intercept of logK_{D,M}^{S}⋅K_{ML,org}.Figure 10 shows its plot, of which the experimental regression line was logK

_{M/ML}= (16.8 ± 6.1)Δφ_{eq,Pic}+ (1.30 ± 0.50) at r = 0.775 and N = 7. Here, the three data (the squares in Figure 10) were neglected from the calculation of the line, because their I values were much larger than the values of the extraction systems without the presence of excess HNO_{3}in the w phase (see Table 1 and Table 2). The Δφ_{eq,Pic}values show the Δφ_{eq}ones obtained from the experimental logK_{D,Pic}values. The predictable intercept value was calculated to be 1.30 (= logK_{D,Ag}^{S}+ logK_{AgB18C6,DCE}= −6.47 + 7.77) ± 0.25, being in accord with the experimental value. Similarly, the slope value was in good agreement with its theoretical one (= 17). These facts indicate that the Δφ_{eq,Pic}values essentially correspond to the Δφ_{eq}ones in logK_{D,Ag}{see Equation (10a)}. In other words, the relation of logK_{Ag/AgB18C6}= (slope) × Δφ_{eq,Pic}+ logK_{D,Ag}^{S}·K_{AgB18C6,DCE}is satisfied.A plot of logK

_{ex±}versus Δφ_{eq}for the A^{−}= Pic^{−}, ClO_{4}^{−}, and NO_{3}^{−}systems also gave a regression line with a slope of 19.3 ± 2.0 V^{−1}and an intercept of −1.23 ± 0.18 at r = 0.974 and N = 7 in the narrow I range (see Table 1 for their basic data). This slope was very close to the theoretical value (= 17). As similar to the previous results [11], these results indicate that the plot satisfies the relation of logK_{ex±}= 16.90Δφ_{eq}+ logK_{D,A}+ logK_{D,Ag}^{S}·K_{AgB18C6,DCE}(= 16.90Δφ_{eq}+ logK_{D,A}+ 1.30). The (logK_{D,A}+ 1.3) term corresponds to the intercept [11] within ±0.3 at least.#### 3.6. Evaluation of LogK_{ex±} Based on LogK_{D,A}^{S}

Using the logK
by using Equations (10a) and (4). It is difficult to accurately determine the K

_{M/ML}^{S}and logK_{D,A}^{S}values, we can immediately evaluate the logK_{ex±}value as follows. A thermodynamic cycle gives
logK

_{ex±}= logK_{M/ML}^{S}+ logK_{D,A}^{S}= logK_{M/ML}+ logK_{D,A}_{D,M}^{S}{or K_{D,M}(Δφ_{eq}): the function of Δφ_{eq}, see Equation (10a)} and K_{ML,org}values. On the other hand, it is comparatively easy to determine the K_{M/ML}^{S}value. So, if the logK_{M/ML}^{S}{or logK_{M/ML}(Δφ_{eq})} value is determined for given ML^{+}and diluent, then the logK_{ex±}values can be calculated from Equation (11) with the logK_{D,A}^{S}{or logK_{D,A}(Δφ_{eq})} ones. In this study, we determined the logK_{Ag/AgL}^{S}value to be 1.30 from the data (see Table 1) of the AgPic-B18C6 extraction systems with DCE. Calculated logK_{ex±}values for some A^{−}are listed in Table 3, together with several experimental values.The determination of the K

_{ex±}values will be experimentally difficult for the Cl^{−}to I^{−}extraction systems, because of the precipitation [18] of their silver salts. The same is true of the SCN^{−}extraction system, because of its low solubility product (= 1.0 × 10^{−12}mol^{2}·dm^{−6}[18]) against Ag^{+}in water. Also, AgN_{3}, which is a white insoluble crystal, is explosive [19]. Therefore, the experimental K_{ex±}values were determined at 298 K for the ClO_{4}^{−}and NO_{3}^{−}extraction systems (see Table 1). Considering the differences (0.03–0.26) between the logK_{D,A}^{S}values at a fixed A^{−}in Table 3 and the standard deviation (= 0.3) of the logK_{Ag/AgB18C6}^{S}value, these experimentally-obtained logK_{ex±}values are very close to the values evaluated here.Similar results were obtained for the NaA-B18C6 extraction into DCE. Their logK

_{ex±}values evaluated from logK_{Na/NaB18C6}^{S}= 0.53, which were calculated here, were −2.8 for A^{−}= MnO_{4}^{−}and −2.3 for ClO_{4}^{−}. Their experimental logK_{ex±}values were −2.23 [13] at I = 0.0077 mol·dm^{−3}for MnO_{4}^{−}and −3.65 ± 0.07 at 0.074 for ClO_{4}^{−}of which the value was re-calculated from the data reported before [8]. These differences, ≤ ∣1.4∣, between the evaluated and experimental values were larger than those for the AgA-B18C6 extraction systems.The above fact indicates that the logK

_{ex±}values can be evaluated from a sum of the logK_{Ag/AgB18C6}^{S}and logK_{D,A}^{S}(or logK_{Ag/AgB18C6}& logK_{D,A}) ones. Namely, the order, A^{−}= NO_{3}^{−}< ClO_{4}^{−}<< Pic^{−}, in logK_{ex±}for a given extraction system at fixed AgB18C6^{+}and DCE is fundamentally predicted from that of logK_{D,A}^{S}(see Table 3). Thus, for the systems where the extraction experiments are difficult, the present procedure becomes easy to evaluate the logK_{ex±}values. Also, the experimental intercepts (≈ logK_{ex±}[13]) of the straight lines in Figure 2 support this order: the intercepts were −4.05 for A^{−}= NO_{3}^{−}, −0.92 for ClO_{4}^{−}, and 2.20 for Pic^{−}.## 4. Materials and Methods

#### 4.1. Materials

An aqueous solution of a commercial Ba(OH)

_{2}·8H_{2}O (≥98%, Wako Pure Chemical Industries, Tokyo, Japan) and a solution with 2 equivalents of HPic·nH_{2}O (≥99.5%, Wako Pure Chemical Industries, Tokyo, Japan) were mixed, that of Ag_{2}SO_{4}(≥99.5%, Kanto Chemicals Co. Ltd., Tokyo, Japan) was added in the mixture, and immediately BaSO_{4}precipitated. After standing the mixture overnight, the thus-obtained yellow solution with the precipitate was filtered and then its filtrate was evaporated by a rotary evaporator (type RE1-N, Iwaki, Japan) in order to deposit a fine yellow crystal. The crystal obtained was filtered and dried in vacuo for 21 h. Found: 97.29% for Ag by the AAS measurements at 328.1 nm; 101% for Pic^{−}by spectrophotometric measurements at 355.0 nm. An amount of water in the AgPic crystal was not detected by a Karl-Fischer titration. This crystal was employed only for the AgPic extraction experiments without B18C6.Concentrations of the aqueous solution with AgNO

_{3}(≥99.8%, Kanto Chemicals Co. Ltd., Tokyo, Japan) and that with AgClO_{4}(97%, Aldrich, Missouri, MO, USA) were determined by a precipitation titration with NaCl (99.98% ± 0.01%, Wako: standard reagent for volumetric analysis, Wako Pure Chemical Industries, Tokyo, Japan). A commercial DCE (guaranteed-pure reagent, Kanto Chemicals Co. Ltd., Tokyo, Japan) was treated with the same procedure as that described previously [13,16]. All other chemicals used in this study were of guaranteed-pure reagent grade.#### 4.2. Extraction Experiments

Procedures for both the AgPic extraction experiments and the AgA extraction ones with B18C6 into DCE were essentially the same as those [8,20] reported before. The latter experiments were performed by using mixtures of AgNO

_{3}with HPic in the w phases. The total concentration range of Ag(I) was 0.00041 to 0.043 mol·dm^{−3}for the AgPic extraction and the ranges of AgNO_{3}, HPic, and B18C6 were (1.5 or 5.0) × 10^{−4}, (3.3 or 3.4) × 10^{−3}, and (0.4–7.5) × 10^{−4}mol·dm^{−3}, respectively, for the Ag(I) extraction with B18C6. The extraction of AgB18C6^{+}with NO_{3}^{−}or Ag(B18C6)NO_{3}was negligible, compared with that of AgB18C6^{+}with Pic^{−}or Ag(B18C6)Pic (see Figure 2 or the logK_{ex±}and logK_{ex}values in Table 1). After the extraction operations, the w phases were in the pH ranges of 2.68–2.74 at the system of I = 0.0025 mol·dm^{−3}and 2.70–3.37 at that of 0.0031 (see Table 1).For the AgPic extraction by B18C6 into DCE in the presence of “excess HNO

_{3}“ in the w phases, the total concentrations of AgNO_{3}and HPic were fixed at 1.5 × 10^{−4}and 0.0033 or 0.0034 mol·dm^{−3}, respectively. Under this condition, the total concentration, [HNO_{3}]_{t}, of HNO_{3}in the w phase was changed in the range of 0.025 to 0.25 mol·dm^{−3}. After the extraction operations, the w phases were in the pH ranges of 1.64 and 1.65 at [HNO_{3}]_{t}= 0.025 mol·dm^{−3}, 1.34–1.38 at 0.050, 1.06 at 0.10, and 0.62–0.68 at 0.25.## 5. Conclusions

The I or I

_{DCE}dependences of the logK_{ex±}and logK_{ex}values for the present extraction systems were clarified experimentally. Their experimental equations were logK_{ex}= 5.3 – 2×0.51$\sqrt{I}/(1+44\sqrt{I}$) and logK_{ex±}= 0.6 − 2×2.0$\left\{\sqrt{I}/\left(1+\sqrt{I}\right)-0.3I\right\}$ = −0.5 + 2×116$\sqrt{{I}_{\mathrm{DCE}}}$. However, the magnitudes of these changes in logK_{ex±}or logK_{ex}with I or I_{DCE}were insignificant in practical separation. It was also suggested that the style of M(I) employed in the extraction experiments with L, either the simple salt MPic or the mixture of MX with HPic and excess HX, largely control whether the logK_{D,Pic}values are dependent on the log(I_{DCE}/I ) ones or not. The logK_{ex±}-versus-Δφ_{eq}plot for the Pic^{−}, ClO_{4}^{−}, and NO_{3}^{−}systems yielded the good positive correlation. On the basis of the plot of logK_{Ag/AgB18C6}versus Δφ_{eq,Pic}, it was indirectly proved that the Δφ_{eq}values obtained from the K_{D,Pic}ones is common to those from the K_{D,Ag}values. Moreover, the logK_{ex±}values were predicted from the sum of the logK_{D,A}^{S}and logK_{Ag/AgB18C6}^{S}ones for given MB18C6^{+}and DCE at least. At the same time, the order in K_{ex±}reflected that in K_{D,A}^{S}.## Author Contributions

Yoshihiro Kudo and Satoshi Ikeda conceived and designed the experiments; Saya Morioka, Shuntaro Tomokata, and Satoshi Ikeda performed the experiments; Yoshihiro Kudo and Satoshi Ikeda analyzed the data; Yoshihiro Kudo wrote the paper.

## Conflicts of Interest

The authors declare no conflict of interest.

## Appendix A

The extraction of AgX by L into the org phase was analyzed at X
were considered to be
and

^{−}= ClO_{4}^{−}or NO_{3}^{−}, L = B18C6, and org = DCE as follows. The component equilibria constituting the overall extraction equilibrium,
Ag

^{+}+ L_{org}+ X^{−}⇌ AgLX_{org}or AgL^{+}_{org}+ X^{−}_{org},
L ⇌ L

_{org}(the symbol of the equilibrium constant: K_{D,L}),
Ag

^{+}+ L + X^{−}⇌ AgL^{+}+ X^{−}(K_{AgL}),
Ag

^{+}_{org}+ L_{org}⇌ AgL^{+}_{org}(K_{AgL,org}),
AgL

^{+}_{org}+ X^{−}_{org}⇌ AgLX_{org}(K_{1,org})From the reactions (A2)–(A4), the following equilibria, Ag
and
Rearranging these three equations for [Ag]
and

^{+}$\rightleftharpoons $ Ag^{+}_{org}(K_{D,Ag}), X^{−}$\rightleftharpoons $ X^{−}_{org}(K_{D,X}), and AgL^{+}$\rightleftharpoons $ AgL^{+}_{org}(K_{D,AgL}), necessarily appear in the component equilibria. Apparently, these three distribution constants are the conditional distribution ones (see the Introduction). The mass balance equations based on the processes (A1)–(A4) became
[Ag]

_{t}= [Ag^{+}] + [AgL^{+}] + [Ag^{+}]_{org}+ [AgL^{+}]_{org}+ [AgLX]_{org},
[L]

_{t}= [L] + [AgL^{+}] + [L]_{org}+ [AgL^{+}]_{org}+ [AgLX]_{org},
[X]

_{t}= [X^{−}] + [X^{−}]_{org}+ [AgLX]_{org}
(= [X

^{−}] + [Ag^{+}]_{org}+ [AgL^{+}]_{org}+ [AgLX]_{org}, based on a charge balance equation)._{t}, [L]_{t}, and [X]_{t}, we can obtain easily
$$\left[{\mathrm{Ag}}^{+}\right]=\frac{{\left[\mathrm{Ag}\right]}_{\mathrm{t}}-Ab}{1+{K}_{\mathrm{AgL}}\left[\mathrm{L}\right]}$$

$${\left[\mathrm{L}\right]}_{\mathrm{org}}\approx \frac{{K}_{\mathrm{D},\mathrm{L}}\left({\left[\mathrm{L}\right]}_{\mathrm{t}}-Ab\right)}{1+{K}_{\mathrm{D},\mathrm{L}}+{K}_{\mathrm{AgL}}\left[{\mathrm{Ag}}^{+}\right]}$$

[X

^{−}] (= I) = [X]_{t}– AbHere, K
After the above handlings, we determined the K

_{AgL}, Ab, and K_{D,L}denote the complex formation constant (mol^{−1}·dm^{3}) for AgL^{+}in water, the analytical concentration (= [Ag^{+}]_{org}+ [AgL^{+}]_{org}+ [AgLX]_{org}) of Ag(I) in the org phase, and the distribution constant of L into the org phase, respectively. Especially, the Ab values were directly determined in terms of AAS measurements. Also, the I_{org}value can be expressed as ([Ag^{+}]_{org}+ [AgL^{+}]_{org}+ [X^{−}]_{org})/2 = [X^{−}]_{org}. In Equation (A6), the Ab term was approximately expressed as [AgL^{+}]_{org}+ [AgLX]_{org}(>> [Ag^{+}]_{org}). Using Equations (A5) and (A6), we determined the [Ag^{+}] and [L]_{org}values by the successive approximation computation [2,13,21] and then obtained K_{ex}^{mix}values from the relation of
K

_{ex}^{mix}= Ab/([Ag^{+}][L]_{org}[X^{−}])_{D,A}and K_{ex±}values, together with the K_{ex}value(s), by the plots (see Section 2.3) reported before [7,8,11,13]. The above extraction model can be essentially applied for systems with another diluent, M(I), or L.## References

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**Figure 1.**A plot of D

_{Pic}′ vs. [Pic

^{−}] for the AgPic extraction into 1,2-dichloroethane (DCE). The straight line was D

_{Pic}′ = (1.81 × 10

^{−4}){1 + (1.81 × 10

^{−4})(9.81 × 10

^{5})[Pic

^{−}]} at r (correlation coefficient) = 0.996.

**Figure 2.**Plots of 2logD vs. log[L]

_{DCE}for the AgPic (circle: without excess HNO

_{3}; diamond: 0.05 mol·dm

^{−3}HNO

_{3}in the w phase), AgClO

_{4}(triangle), and AgNO

_{3}(square) extraction with B18C6 (= L) into DCE.

**Figure 3.**A plot of logK

_{ex}

^{mix}vs. −log([Ag

^{+}][L]

_{DCE}) for the AgClO

_{4}extraction with B18C6 (= L) into DCE. The regression line was logK

_{ex}

^{mix}= log{9.7 × 10

^{2}+ (7.4 × 10

^{−3})/([Ag

^{+}][L]

_{DCE})} at r = 0.891.

**Figure 4.**A plot of logK

_{ex}

^{mix}vs. −(1/2)logP for the AgClO

_{4}extraction with B18C6 (= L) into DCE. The regression line was logK

_{ex}

^{mix}= log(2.5 × 10

^{2}+ $\sqrt{{0.057}_{5}/P}$) at r = 0.999.

**Figure 5.**A plot of logK

_{ex}vs. I for the AgPic extraction with B18C6 into DCE. See the text for the regression line (broken line). Error bars were added for only the present experimental values in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. There are some cases where the bars were smaller than the points, such as circle and square. Especially, all the bars were smaller than the size of the circles in Figure 5.

**Figure 6.**A plot of logK

_{ex±}vs. I for the AgPic extraction with B18C6 into DCE. The regression line (broken line) based on the Davies equation was logK

_{ex±}= 0.596 – 2 × 2.05$\left\{\sqrt{I}/\left(1+\sqrt{I}\right)-0.3I\right\}$ at r = 0.884 and N = 10.

**Figure 7.**A plot of logK

_{ex±}vs. I

_{DCE}for the AgPic extraction with B18C6 into DCE. See the text for the regression line (broken line).

**Figure 8.**A plot of logK

_{1,DCE}vs. I

_{DCE}for the AgPic extraction with B18C6 into DCE. The regression line was logK

_{1,DCE}= 5.89 − 2×(1.52 × 10

^{2})$\sqrt{{I}_{\mathrm{DCE}}}$ (broken line) at r = 0.821 and N = 10.

**Figure 9.**A plot of logK

_{D,Pic}vs. log(I

_{DCE}/I ) for the AgPic extraction with B18C6 into DCE. The broken line shows a theoretical one for the slope fixed at unity: see the text. The plots (square) were of the extraction with the excess addition of HNO

_{3}in the w phases.

**Figure 10.**A plot of logK

_{Ag/AgL}vs. Δφ

_{eq}for the AgPic extraction with B18C6 (= L) into DCE. See the text for the regression line (broken line).

**Table 1.**Fundamental data for the extraction with AgPic, AgClO

_{4}, or AgNO

_{3}(= AgA) and B18C6 (= L) into DCE at 298 K.

A^{−} (I ^{1}/10^{−3}) | logK_{ex±} | logK_{D,A}[Δφ _{eq}/V] | logK_{ex} | logK_{1,DCE}(I _{DCE}^{1}/10^{−5}) | logK_{AgL,DCE} |
---|---|---|---|---|---|

Pic^{−} (2.5) | 0.33 ± 0.03 | −2.40 ± 0.03 [0.082] | 5.31 ± 0.03 | 4.98 ± 0.05 (0.93) | 7.81 |

(2.7)^{2} | 0.51 | −2.60 [0.094] | 5.336 | 4.82 (1.1) | 7.99 ^{3} |

(2.8)^{2} | 0.25 | −2.33 [0.078] | 5.17 | 4.92 (0.40) | 7.73 ^{3} |

(3.1) | 0.40 ± 0.09 | −2.07 ± 0.05 [0.063] | 4.93 ± 0.05 | 4.53 ± 0.10 (2.2) | 7.88 |

(3.6) ^{2} | 0.17 | −2.70 [0.10] | 5.55 | 5.38 (0.64) | 7.65 ^{3} |

ClO_{4}^{−} (2.8) | −1.24 ± 0.02 | −2.13 ± 0.11 [−0.032] | 2.99 ± 0.11 | 4.23 ± 0.11 (2.1) | 8.07 |

NO_{3}^{−} (3.1) | −4.40 ± 0.07 | −3.47 ± 0.04 [−0.14] | 1.04 ± 0.03 | 5.44 ± 0.08 (0.11) | 7.98 |

^{1}Unit: mol·dm

^{−3};

^{2}Values in this line were cited from Ref. [6];

^{3}Values re-calculated from the relation logK

_{AgL,DCE}= logK

_{ex±}− logK

_{D,Ag}

^{S}·K

_{D,Pic}

^{S}= logK

_{ex±}+ 7.481, since the K

_{D,Ag}

^{S}value was re-determined in this study.

**Table 2.**Fundamental data for the extraction with AgPic and B18C6 (= L) into DCE in the presence of excess HNO

_{3}in the water phases at 298 K.

I/10^{−2} mol·dm^{−3} | logK_{ex±} | logK_{D,Pic}[Δφ _{eq}/V] | logK_{ex} | logK_{1,DCE}(I _{DCE} ^{1}/10^{−6}) | logK_{AgL,DCE} |
---|---|---|---|---|---|

2.4 | 0.38 ± 0.10 | −2.33 ± 0.05 [0.078] | 5.35 ± 0.04 | 4.97 ± 0.11 (3.0) | 7.87 |

5.0 | 0.00 ± 0.06 | −2.30 ± 0.03 [0.076] | 5.41 ± 0.03 | 5.41 ± 0.07 (2.0) | 7.49 |

9.7 ^{2} | −0.13 | −1.68 [0.040] | 5.07 | 5.20 (5.5) | 7.35 ^{3} |

11 | −0.23 ± 0.06 | −2.26 ± 0.02 [0.074] | 5.45 ± 0.02 | 5.68 ± 0.06 (0.97) | 7.26 |

26 | −0.78 ± 0.06 | −1.88 ± 0.03 [0.051] | 5.11 ± 0.02 | 5.89 ± 0.07 (1.3) | 6.70 |

**Table 3.**Log K

_{ex±}values evaluated from the logK

_{D,A}

^{S}and logK

_{Ag/AgL}

^{S}values at B18C6 (= L), DCE, and 298 K.

A^{−} | logK_{D,A}^{S 1} | logK_{ex±} | |
---|---|---|---|

Evaluated ^{2} | Experimental | ||

Cl^{−} | −7.99, −8.135 ^{3} | −6.69 | - ^{4} |

Br^{−} | −6.57, −6.74 ^{3} | −5.27 | - ^{4} |

N_{3}^{−} | −6.42 | −5.12 | - ^{4} |

NO_{3}^{−} | −5.91, −5.94 ^{3} | −4.61, −4.64 ^{3} | −4.40 |

I^{−} | −4.50, −4.62 ^{3} | −3.20 | - ^{4} |

SCN^{−} | −4.21, −4.47 ^{3} | −2.91 | - ^{4} |

MnO_{4}^{−} | −3.33 | −2.03 | - ^{5} |

ClO_{4}^{−} | −2.84, −3.00 ^{3} | −1.5_{4}, −1.70 ^{3} | −1.24 |

Pic^{− 6} | −1.01 _{1} | - ^{7} | 0.35 ± 0.12 ^{8}−0.36 ± 0.24 ^{9} |

^{1}Ref. [12];

^{2}Values calculated from logK

_{ex±}= 1.30 + logK

_{D,A}

^{S}{see Equation (11)}. They have the error of 0.3 at least because of the standard deviation of logK

_{Ag/AgL}

^{S};

^{3}Ref. [17];

^{4}Not determinable, probably because of precipitation etc. See the text;

^{5}Not determined here;

^{6}The K

_{Ag/AgL}

^{S}value was determined, based on the data of the extraction experiments. See Table 1;

^{7}Not evaluated;

^{8}Average value in the I range of (2.5–3.6) × 10

^{−3}mol·dm

^{−3}. See Table 1;

^{9}Value at I

_{DCE}→ 0. See the text.

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