Application of Molar Volumes Obtained From a Solvent Extraction Method for Estimation of Sizes of Metal Complexes With Crown Ethers in Phases

Molar volumes (Vj) of ion-pair complexes (MLAz = j; z = 1, 2) and simple complex ions (CdL ) without a pairing anion A obtained at 298 K from the plots of the regular solution theory and the geometrical relation Vj = (4π/3)Rj 3 were used for an estimation of their apparent radii (Rj) in liquid phases. Here, the cases for M z+ = Li-Cs, Cd, and Pb and L = 18-crown-6 ether (18C6) and benzo-18C6 (B18C6) with A = Cl-I and picrate ion (Pic) were examined. The thus-estimated Rj values were related to h(ML ), λmax, and (r+ + r−) values, where these symbols refer to the number of water molecules attached to the univalent complex ions, ML, extracted into water-saturated nitrobenzene, absorption maxima of Pic with ML in water-saturated chloroform, and the sum in an effective ionic radius, r+ or r−, between the central M and the pairing A, respectively. The data points of j = Li(18C6)Pic, Na(18C6)Pic, M(B18C6)Pic with M = Na-Rb(I), Cd18C6, Cd(18C6)Cl2, Cd(18C6)Br2, and Cd(18C6)I2 showed larger deviations from the line of Rj = (r+ + r−). While j = M(18C6)Pic with M = K-Cs(I) and M(18C6)Pic2 with Cd(II) and Pb(II) were on the line. It was demonstrated that essentially the former groups are present as water-separated and -shared ion pairs in the phases, while the latter ones as contact pairs.


Introduction
Crown compounds (L) extract alkali and alkaline-earth metal ions (M z+ ; z = 1, 2) from water into various diluents (Yajima et al., 2000;Takeda, 2002;Levitskaia et al., 2007;Makrlí k et al., 2008;Kudo et al., 1990;Kudo et al., 2017a;Kudo et al., 2017b;Zakurdaeva et al., 2018), such as nitrobenzene (NB) and chloroform (CF).Particularly, the extraction behaviors of alkali metal picrates (LiPic-CsPic) have been reported so far for L = 18-crown-6 ether (18C6) and its benzo-derivatives, benzo-18C6 (B18C6) and dibenzo-18C6 (DB18C6) (Takeda, 2002;Yajima et al., 2000).In course of such studies, Takeda and coworkers have analyzed their extraction systems by applying the regular solution theory (RST) for the systems and then determined the solubility parameters and molar volumes (V j /cm 3 mol −1 ) of L (= j) and its ion pair complexes, ML + Pic − (Takeda, 2002).On the other hand, our groups have systematically studied the Cd(II) and Pb(II) extraction with 18C6 (Kudo et al., 2018;Kudo & Tomokata, 2018;Kudo et al., 2014;Kudo et al., 2015;Kudo et al., 2013a;Kudo et al., 2013b) or B18C6 (Kudo et al., 2017a) into various diluents and applied the same RST analyses for their systems (Kudo et al., 2018;Kudo & Tomokata, 2018;Kudo et al., 2014;Kudo et al., 2015;Kudo et al., 2013a).Here the analyses are based on the plots of log K ex,ip vs. log K D,L ; the symbols K ex,ip and K D,L denote an ion-pair extraction constant of complex ions, ML 2+ , with A − into an organic (org) phase and a distribution constant of the free L into the org one, respectively.In such studies, we developed a procedure which estimates apparent radii (R j ) of the complex ion CdL 2+ at L = 18C6 and those of its ion pair complexes CdLA 2 (A − = Cl − -I − , Pic − ) in phases from the experimental V j values (Kudo et al., 2018).Consequently, it has been clarified that the R j values depend on the Cd-A bond lengths in the crystals, such as Cd(18C6)A 2 with A(−I) = Cl-I (Kudo et al., 2018).
In the present work, we tried to extend such a procedure to the MPic extraction systems with 18C6 and B18C6 and made discussion for the R j in the Cd(II) extraction systems (Kudo et al., 2018) more quantitative.In particular, it was demonstrated that the ion pair complexes, M(18C6)Pic z with M = K-Cs at z = 1 and with M = Cd and Pb at z = 2, were present as contact ion pairs.

Basic Procedure for an R j Estimation
According to Takeda's study (Takeda, 2002) in which the RST has been applied for extraction systems with MA and L, the relation of log K ex,ip with log K D,L has been derived as where V j , f(C), and K 1 denote the molar volume of j = MLA and L, a function with cohesive energy densities (C/J cm −3 ), and an ion-pair formation constant (mol −1 dm 3 ) for MLA in water (phase), respectively.Using the relation of log K ex,ip = log K ex + log K D,L − log K ML (Takeda & Gotō, 1979;Takeda & Katō, 1979), one can easily obtain the K ex,ip values from the extraction experiments and the K D,L values from the reference (Takeda, 2002).Here, the symbols, K ex and K ML , denote an overall extraction constant (= [MLA] org /[M + ][L] org [A − ]) and a complex formation constant (= [ML + ]/[M + ][L]) for ML + in water, respectively.So if the V L values are determined with some methods (Takeda, 2002;Bernal, 2001), then we can immediately obtain the V MLA ones from the slopes of the log K ex,ip -vs.-logK D,L plots, called the RST plots (Takeda, 2002;Yajima et al., 2000).Table 1 summarizes the thus-obtained V j values reported previously in the reference (Takeda, 2002) for the alkali metal picrate (MPic) extraction systems with L = 18C6 and B18C6.Shannon, 1976; Katsuta et al., 2007.b Molar volumes determined by the RST plots.See Ref. Takeda, 2002.c Standard errors.d See the text and Ref. Iwachido et al., 1980.e See the text and Ref. Kudo et al., 1990.f Not determined or reported.
After the V j determination with the plots, we calculated the R j values from the following procedure (Kudo et al., 2018).First, we thought the shape of j = MLPic or ML + to be approximately a globe with the radius R j and then the V j values in a cm 3 mol −1 unit divided by (6.0221  10 23 mol −1 )(10 −24 cm 3 Å −3 ) (= 0.60221 cm 3 Å −3 mol −1 ) were converted to those in a Å 3 molecule −1 unit.Lastly, we calculated these values from the geometrical relation, V j = (4π/3)R j 3 , and these converted V j values.The results are listed in Table 1, together with the h(ML + ) and λ max values reported before.Here the symbols h(ML + ) and λ max refer to the number of water molecules attached to ML + extracted into water-saturated NB (Iwachido et al, 1980) and absorption maxima (nm) of species with Pic − in water-saturated CF (Kudo et al., 1990), respectively.
The same is essentially true of the MLA 2 extraction systems.As similar to Eq. ( 1), the following equation (Kudo & Tomokata, 2018) for the MA 2 extraction with L was applied.
where the symbols K ex,ip and β 2 are defined as , respectively.Also, the function of g(C) corresponds to that of f(C) in Eq. (1).

For Correlations of the Parameters h(ML + ) and λ max With Effective Ionic Radii r +
In order to examine meanings of these parameters, we plotted h(ML + ) for the NB system and λ max −1 for the CF system against r + −1 for M = Li-Cs(I) and L = 18C6 (see Table 1 for their data).Both the plots showed positive correlation: their correlation coefficients were 0.98 8 for the former and 0.959 for the latter (Figs. 1 & 2, respectively).These findings indicate that the both parameters are essentially proportional to the coulombic forces.The h(M18C6 + ) values directly reflect the strength of the ion-dipole interaction between M + and H 2 O, namely the strength of the M + -OH 2 bond.The h(NaL + ) values were in the order L = benzo-15-crown-5 ether {B15C5; h(NaL + ) = 0.8} < 15C5 (0.9) < B18C6 (1.1) < 18C6 (1.2) (Iwachido et al., 1980).This order was the same as that of the logarithms of the average K 1,NB values (Kudo et al., 2017a), where K 1,NB was defined as [NaLPic] NB /[NaL + ] NB [Pic − ] NB .Therefore, we assumed that the h(ML + ) values reflect the comparable number of water molecules attached to the ML + Pic − extracted into the NB phase.Similarly, the λ max −1 values reflect a change of the electronic state on Pic − due to the ion-ion interaction and are indirect parameters for the interaction between M + and Pic − .Consequently, the both parameters mainly depend on the strength of the electrostatic interaction between M + in/on 18C6 and H 2 O or Pic − , not scarcely that between -M18C6 + ‖ and H 2 O or Pic − , in the NB or CF phase.That is, Pic − and H 2 O do not recognize the size of the M18C6 + , but mainly do that of M + .Seeing from another point of view, h(ML + ) is the direct parameter for the interaction between M + and H 2 O (schematically LM + -OH 2 ), but becomes the indirect one in the present application for MLPic.On the other hand, λ max −1 is the indirect parameter for the interaction with M + and Pic − (LM + -Pic − ), but becomes the direct one for its application.
On the basis of the above results, we can see that, in MLPic, the smaller the sizes of the central M + , the larger the amounts of H 2 O which interacts with its M + become.Also, the smaller their sizes, the larger the energetic change of the electronic state on Pic − become.It is highly important that the two parameters fundamentally express the interaction with the central M + in/on L.

2.3
For the Plots of R j vs. r + , (r + + r − ), h(ML + ), and λ max −1 for the MLPic System These plots were of all V-shaped curves as shown in Fig. 3.They were the plots with the inflection points at M = K(I).
Especially, the plot (Fig. 3) of R j vs. r + is important, because the line produced for M(I) = K-Cs has a positive slope ( +1.7): the R MLPic order was M(I) = Li > Na > K < Rb < Cs at L = 18C6 with an increase in r + .This fact that their R j values are proportional to the r + ones in the M = K-Cs(I) range suggests the direct interaction between the M + in/on L and Pic − {see the (r + + r − ) values in Table 1 for comparison}.On the other hand, the same plot showed us an abnormality of the R j values at j = Li(18C6)Pic and Na(18C6)Pic.This suggests an indirect interaction between M + in 18C6 and Pic − , for example the interaction between them via hydrated H 2 O.
In contrast with the R j vs. r + plot in Fig. 3, the R j values were proportional to the h(M18C6 + ) in the M range of Li-K(I): the R M(18C6)Pic order was M(I) = Cs > Rb > K < Na < Li with an increase in h(M18C6 + ).This order for Li-K(I) shows the hydration of these M(18C6)Pic {and accordingly reflects the relation of h(M18C6 + )  a/r + + b as shown in Fig. 1}.The same was also observed in the R j vs. λ max −1 plot.
From the above results, we can see that Li(18C6) + Pic − and Na(18C6) + Pic − in the phases have a hydrated structure of M + in 18C6, a wrapped one of M + by the flexible 18C6, or do the mixture of their structures, since the ionic sizes of Li + and Na + (see Table 1) are smaller than the cavity size {= 1.34-1.43Å (Takeda, 1991)} of 18C6.That is, these complexes are suggested to be a water-separated, L-separated, or water-shared ion pairs.The above structures were self-consistently supported by Fig. 4. The plots of j = K(18C6)Pic to Cs(18C6)Pic were on the hypothetical straight line of R j = (r + + r A ) at A = Pic(−I) within the calculation errors, while those of j = Li(18C6)Pic and Na(18C6)Pic deviated largely from its line.Here the symbol r A is equivalent with r − in Table 1.These results indicate that the three formers are contact ion pairs.Also, these structures are not conflict with the fact that the K 1 0 (K) value {= [KLPic]/[KL + ][Pic − ] = 738 mol −1 dm 3 at I  0 mol dm −3 (Kudo, 2013)} is larger than the K 1 0 (Li) and K 1 0 (Na) ones {= 52 & 62 at I  0 (Kudo, 2013)} at L = 18C6.Here K 1 {see Eq. ( 1)} expresses the equilibrium in water, ML(OH 2 ) x + + Pic - ML(OH 2 ) y Pic + (x − y)H 2 O at x ≥ y.In such a reaction, x > 0 should be predicted for M = Li(I) and Na(I), while x > y  0 for K(I).For the structure of such M18C6(OH 2 ) x + in water, it has been proposed with an X-ray diffraction study that one molecule of water interacts with Na18C6 + and two ones of water do with K18C6 + and Cs18C6 + (Ozutsumi et al., 1989).The same is true of the MPic-B18C6 extraction system, although there were less data, compared with those of the 18C6 system (see Table 1).Additionally, a size effect of the benzo group seems to appear in the difference (= 0.5-0.9Å) between the R j and (r + + r − ) values.Also, taking the hydrophobicity of the benzo group, the less-flexibility of the B18C6 skeleton, compared with the 18C6 one (Kudo et al., 2017a), and the finding of the h(MB18C6 + ) vs. r + plot (see Fig. 1) into account, we can imagine that the K(B18C6)Pic and Rb(B18C6)Pic are the contact ion pairs.The same analysis can be effective for the MPic-DB18C6 extraction system (Takeda, 2002;Yajima et al., 2000); see Ebata & Inokuchi 2012 in reference to the structure of MDB18C6 + in the gas phase.(Kudo & Tomokata, 2018;Kudo et al., 2013a;Kudo et al., 2018).These three values agreed with each other within calculation errors (= ±0.4-0.6 Å) listed in Table 2.This larger deviation may be caused by defects of conditions applied for the pseudo-plots.However, we cannot discuss this matter in more detail.Shannon, 1976;  Katsuta et al., 2007.b Molar volumes determined by the RST & pseudo-RST plots.See Refs.Kudo & Tomokata, 2018;  Kudo et al., 2013a; Kudo et al., 2018; Kudo et al., 2014.c Standard errors.d Not determined.e The d Cd-A values by X-ray diffraction measurements of crystals.See Refs.Paige & Richardson, 1983; Hazell, 1988; Kennedy, 1990.f Value for the I(−I) system.See Ref. Kudo et al., 2018.g Value for the Br(−I) system.See Ref. Kudo et al., 2013a.h Value for the Cl(−I) system.See Ref. Kudo & Tomokata, 2018.i r Cd-OH2 value of [Cd(18C6)(OH 2 ) 2 ] 2+ by DFT study.See Ref. Harrinton et  al., 2009.j r Cd-OH2 value of [Cd(OH 2 ) 6 ] 2+ in water by X-ray diffraction measurements.See Ref. Ohtaki et al., 1974.k  r Pb-ONO2 value of [Pb(NO 3 ) 2 (18C6)] 0 in crystal by X-ray diffraction measurements.See Ref. Rogers & Bond, 1992.As similar to Fig. 4, Fig. 5 shows the plot of R j vs. (r + + r-).As can be seen from this figure, the R j values for j = Cd(18C6)Pic 2 and Pb(18C6)Pic 2 were comparably on the straight line of R j = (r + + r-).Comparing these results with those for the M(18C6)Pic system, we can easily suppose that these two complexes with M(II) are of contact ion pairs in the phases.Since the (r + + r-) values were larger than the Cd-A bond lengths (d Cd-A ) in their crystals, Cl-Cd18C6-Cl, Br-Cd18C6-Br, and I-Cd18C6-I (Table 2), we thought here the former values to become a measure of their species -in solutions‖ for the comparison.In other words, this means that the line of R j = (r + + r − ) in Fig. 5 expresses an ionic bond character of the Cd-A bond appeared in the solutions (also refer to Fig. 4).Comparing R j with (r + + r − ) at j = Cd(B18C6)Pic 2 in Table 2, this ion pair seems to be simply the contact form.On the other hand, the other ion pairs and the complex ion can take the hydrated forms, such as A-Cd(18C6)(OH 2 ) x -A and Cd(18C6)(OH 2 ) x 2+ , in the phases, respectively.Such ion pairs are speculated to be the water-separated or water-shared form, because the donor numbers (DN) of the representative diluents, such as benzene (DN = 0.1), 1,2-dichloroethane (0), and NB (4.4), used in the extraction experiments are much smaller than that (33) of water (Izutsu, 2002;Gutmann, 1978).It is interesting that Cd18C6 2+ shows the higher water affinity, as similar to Ca(18C6)(OH 2 ) 4.7 2+ (Iwachido et al., 1980;Kudo et al., 2018).

Conclusion
It was proved that the R j values are estimated from the V j ones for the water/diluent systems determined with the RST and pseudo-RST plots and then reflect the sizes of MLA z with z = 1 and 2 and CdL 2+ in the corresponding phases.
Although extraction experiments are needed for the about ten diluents, the proposed procedure can be convenient one, compared with some spectroscopic methods, such as X-ray diffraction analysis of solution samples.Unfortunately, it is now difficult to establish a liquid phase in which the objective species are present.However, its procedure can have the higher cost performance in such solution-chemical studies.

Figure 3 .
Figure 3. Plots of R j vs. r + for the MLPic systems with M = Li-Cs(I) and L = 18C6 (circle) and B18C6 (square)

Figure 4 .
Figure 4. Plot of R j vs. (r + + r − ) for the M(18C6)Pic systems with M = Li, Na (square), and K-Cs (circle).The broken line is of R j = (r + + r A ) with A = Pic(−I)The same is true of the MPic-B18C6 extraction system, although there were less data, compared with those of the 18C6 system (see Table1).Additionally, a size effect of the benzo group seems to appear in the difference (= 0.5-0.9Å) between the R j and (r + + r − ) values.Also, taking the hydrophobicity of the benzo group, the less-flexibility of the B18C6 skeleton, compared with the 18C6 one(Kudo et al., 2017a), and the finding of the h(MB18C6 + ) vs. r + plot (see Fig.1) into account, we can imagine that the K(B18C6)Pic and Rb(B18C6)Pic are the contact ion pairs.The same analysis can be effective for the MPic-DB18C6 extraction system(Takeda, 2002;Yajima et al., 2000); see Ebata & Inokuchi 2012 in reference to the structure of MDB18C6 + in the gas phase.
Plots of R j vs. (r + + r-) for the MLA 2 System Table 2 lists the V j and R j values for the MLA 2 extraction systems with M = Cd(II) and Pb(II), L = 18C6 and B18C6, A = Cl-I(−I) and Pic(−I), and without A(−I) (Kudo et al., 2018), together with the (r + + r − ) values.The V Cd18C6 values were obtained from pseudo-RST plots for the Cl(−I), Br(−I), and I(−I) extraction systems

Table 1 .
Fundamental data for the MLPic extraction systems at 298 K