Solubility of Rare Earth Chlorides in Ternary Water-Salt Systems in the Presence of a Fullerenol—C 60 (OH) 24 Nanoclusters at 25 ◦ C. Models of Nonelectrolyte Solubility in Electrolyte Solutions

: The solubility in triple water-salt systems containing NdCl 3 , PrCl 3 , YCl 3 , TbCl 3 chlorides, and water-soluble fullerenol C 60 (OH) 24 at 25 ◦ C was studied by isothermal saturation in ampoules. The analysis for the content of rare earth elements was carried out by atomic absorption spectroscopy, for the content of fullerenol—by electronic spectrophotometry. The solubility diagrams in all four ternary systems are simple eutonic, both consisting of two branches, corresponding to the crystallization of fullerenol crystal-hydrate and rare earth chloride crystal-hydrates, and containing one nonvariant point corresponding to the saturation of both solid phases. On the long branches of C 60 (OH) 24 *18H 2 O crystallization, a C 60 (OH) 24 decreases by more than 2 orders of magnitude compared to the solubility of fullerenol in pure water (salting-out effect). On very short branches of crystallization of NdCl 3 *6H 2 O, PrCl 3 *7H 2 O, YCl 3 *6H 2 O, and TbCl 3 *6H 2 O, the salting-in effect is clearly observed, and the solubility of all four chlorides increases markedly. The four diagrams cannot be correctly approximated by the simple one-term Sechenov equation (SE-1), and very accurately approximated by the three-term modiﬁed Sechenov equation (SEM-3). Both equations for the calculation of nonelectrolyte solubility in electrolyte solutions (SE-1 and SEM-3 models) are obtained, using Pitzer model of virial decomposition of excess Gibbs energy of electrolyte solution. It is shown that semi-empirical equations of SE-1 and SEM-3 models may be extended to the systems with crystallization of crystal-solvates.

The analysis for the content of rare earth elements was carried out by atomic abso tion spectroscopy (Perkin Elmer PinAAcle 500 atomic absorption spectrometer), the re tive error in the determination of rare earth elements is δREM 3+ ≈ 1 ÷ 3 rel. mass % at h concentrations of rare earth metals CREM 3+ ≈ 10 ÷ 700 g/dm 3 and δREM 3+ ≈ 3 ÷ 5 rel. mass % low concentrations of rare earth metals CREM 3+ ≈ 1 ÷ 10 g/dm 3 (here and after: REM = Nd, Y, Tb).
The analysis on the content of fullerenol were carried out by electron spectrophoto etry, according to the Bouguer-Lambert-Ber law (spectro-photometer UV 1280) by the tical density at the wavelength λ = 400 nm − D400 (see Figures  The absorption of REMCl3 in the near ultraviolet region at λ=400 nm can be neglec (see -REM solutions are practically absolutely transparent in the class violet region of the visible part of the spectrum. The relative error in the determination fullerenol was δC60(OH)24 ≈ 3-5 rel. mass % at high C60(OH)24 concentrations: CC60(OH)24 ≈ 0 5.6 g/dm 3 and δC60(OH)24 ≈ 5-10 rel. mass % at low C60(OH)24 concentrations: CC60(OH)24 ≈ 0 0.5 g/dm 3 .  The absorption of REMCl 3 in the near ultraviolet region at λ = 400 nm can be neglected (see -REM solutions are practically absolutely transparent in the classical violet region of the visible part of the spectrum. The relative error in the determination of fullerenol was δ C60(OH)24 ≈ 3-5 rel. mass % at high C 60 (OH) 24 concentrations: C C60(OH)24 ≈ 0.5 ÷ 5.6 g/dm 3 and δ C60(OH)24 ≈ 5-10 rel. mass % at low C 60 (OH) 24 concentrations: C C60(OH)24 ≈ 0.1 ÷ 0.5 g/dm 3 .           To determine the molality concentrations of components, it was also necessary able to translate volume concentrations of C(g/dm 3 ) into weight concentrations of C( %) or molality m(mole/kg H2O). To do this, it was necessary to determine the densi saturated ternary solutions. The determination was carried out using quartz pycnom with a working volume of about V~5 cm 3 , the standard liquid was distilled wate error in determining the density was ∆~0. 1 .% with temperature control acc ∆T~0.05 K.
Data on solubility in triple systems PrCl3-C60(OH)24-H2O, NdCl3-C60(OH)24-H2O C60(OH)24-H2O, and TbCl3-C60(OH)24-H2O at 25 °C are presented in Figures 7-10 a Table 1 in molalities of rare earth salts and fullerenol. Solubility diagrams in four ternary systems are simple eutonic, both consist o branches, corresponding to the crystallization of fullerenol crystal-hydrate and rare chloride crystal-hydrates, and contain one eutonic nonvariant point corresponding uration with both solid phases [12][13][14]. On the long branches of C60(OH)24*18H2O cr lization, a pronounced salting-out effect is observed-the solubility of C60(OH)24*1 decreases by almost 2 orders of magnitude compared to the solubility of fullerenol in water. On very short branches of crystallization of NdCl3*6H2O, PrCl3*7H2O, YCl3* and TbCl3*6H2O, the effect of salting-in is clearly observed, the solubility of cryst drates of four chlorides increases markedly. The long crystallization curves them C60(OH)24*18H2O in both cases have a rare concave-convex σ-id character with an tion point: To determine the molality concentrations of components, it was also necessary to be able to translate volume concentrations of C (g/dm 3 ) into weight concentrations of C (mass. %) or molality m (mole/kg H 2 O). To do this, it was necessary to determine the densities of saturated ternary solutions. The determination was carried out using quartz pycnometers with a working volume of about V ∼ 5 cm 3 , the standard liquid was distilled water. The error in determining the density was ∆ρ ∼ 0.1 rel.% with temperature control accuracy ∆T~0.05 K.
Data on solubility in triple systems PrCl 3 -C 60 (OH) 24 Table 1 in molalities of rare earth salts and fullerenol.
where: m i , C i , M i is the molality, concentration in mass. %, and the molecular weight of the i-th component of the solution.
Solubility diagrams in four ternary systems are simple eutonic, both consist of two branches, corresponding to the crystallization of fullerenol crystal-hydrate and rare-earth chloride crystal-hydrates, and contain one eutonic nonvariant point corresponding to saturation with both solid phases [12][13][14]. On the long branches of C 60 (OH) 24 *18H2O crystallization, a pronounced salting-out effect is observed-the solubility of C 60 (OH) 24 *18H2O decreases by almost 2 orders of magnitude compared to the solubility of fullerenol in pure water. On very short branches of crystallization of NdCl 3 *6H 2 O, PrCl 3 *7H 2 O, YCl 3 *6H 2 O, and TbCl 3 *6H 2 O, the effect of salting-in is clearly observed, the solubility of crystal-hydrates of four chlorides increases markedly. The long crystallization curves themselves C 60 (OH) 24 *18H 2 O in both cases have a rare concave-convex σ-id character with an inflection point: dm C60(OH)24 /dm REMCl3 = 0 and d 2 m C60(OH)24 /dm REMCl3

Sechenov Equation of the Solubility of Nonelectrolyte in Electrolyte Solutions
For

Modeling of C 60 (OH) 24 *18H 2 O Crystallization Branches in Ternary Systems
where: C 0 C60(OH)24 and C C60(OH)24 solubility of C 60 (OH) 24 in a binary system C 60 (OH) 24water and solubility of C 60 (OH) 24 in a ternary system-an aqueous solution of REMCl 3 (REM = Pr, Nd, Y, Tb), C REMCl3 is the concentration of REMCl 3 in a saturated solution, and A S is the constant of the Sechenov equation. The scale of concentrations-C is generally uncertain (it can be C V -g/dm 3 , molarity M-mole/dm 3 , molality m-mole/kg of solvent, and mole fraction X-a.u.). As is known, this equation often quite adequately describes the solubility of a nonelectrolyte or a weak electrolyte (usually gaseous) in solutions of a strong electrolyte in polar solvents (H 2 O, CH 3 OH, CH 3 CN . . . ) [9][10][11]. The results of the approximation are shown below in Figures 11-14 and in Table 2. The result of the approximation is generally not satisfactory enough, because, first, it is not accurate enough (see Figure the value of the standard deviations of the calculated values ln(m 0 C60(OH)24 /m C60(OH)24 ) from the experimentally obtained ones), and, secondly, the calculated curves do not at all convey the σ-id character of the experimental curves with inflection points (see Figures 11-14). The reasons for this discrepancy are as follows: The empirical Sechenov equation is written for the solubility branch of the unsolvated solid phase, and in our case the crystallization branch of the crystal-hydrate is described, i.e., the thermodynamic potential (solubility product-SP) of C 60 (OH) 24 *18H 2 O, which retains its value on the crystallization branch of this compound, has the form: where: a H2O is the activity of H 2 O in saturated solution, γ C60(OH)24 is the activity coefficient C 60 (OH) 24 in the molality scale with asymmetric normalization of redundant functions.
are formed from first-order associates, and so on. The aqueous solutions of C60( themselves are characterized by huge positive deviations from ideality, which often to the loss of diffusion stability by the solution [15][16][17]. It is unlikely that in saturated aqueous solutions PrCl3-C60(OH)24-H2O, Nd C60(OH)24-H2O, YCl3-C60(OH)24-H2O, TbCl3-C60(OH)24-H2O electrolytes NdCl3, YCl3, and TbCl3 can be considered as very strong, it is likely the formation pairs and larger ion associates in solutions.        Table). ln(m 0 C60(OH)24/mC60(OH)24) = A s mREMCl3 + B s mREMCl3 2   The C 60 (OH) 24 nanoclusters themselves form a complex hierarchical sequence of associates: First-order associates are formed from monomers, then second-order associates are formed from first-order associates, and so on. The aqueous solutions of C 60 (OH) 24 themselves are characterized by huge positive deviations from ideality, which often lead to the loss of diffusion stability by the solution [15,16].

Model Description of One-Parameter Sechenov Solubility Equation (SE-1)
Due to the fact that the Sechenov equation is historically considered to be empirical, we will give a derivation of this equation. Naturally, such a conclusion will be based on a semi-empirical model, in our case, the Pitzer model of strong electrolytes [17,18], and it is naturally impossible to call it strictly thermodynamic. The same conclusion is easily obtained using other semi-empirical models, such as the Bromley model [12] or the regular solutions model [14].
According to the physical sense, const = 0, because if: m NE → m 0 NE , ln(γ NE ) → 0 due to the conditions of asymmetric normalization for both E and NE.
Take this to describe the concentration dependence of the ln(γ NE ) simplified model of Pitzer [17] taking into account only the second virial coefficients (λ ij ) in the expansion of the excess Gibbs energy of solution (G ex ) on the numbers of moles of the cation and anion of the electrolyte-E (n c , n a ) and number of moles of NE (n NE ). We shall not take into account NE-NE interactions, as well as NE concentration in solution, are usually much smaller than the electrolyte, in our case, by 2 orders of magnitude. The physical sense of the second virial coefficients (λ ij ) is the following: It is reduced to RT energy of specific non-electrostatic interactions of 1 mole of i particles and 1 mole of j particles. Then: G ex /RT = n S f(I) + 1/n S [(λ c-c n c 2 ) + (λ a-a n a 2 ) + (λ c-a n c n a ) + (λ c-NE n c n NE ) + (λ a-NE n a n NE )] (9) where: n S is the moles number of the solvent in 1 kg, and I is the molar ionic strength of the solution: I = 1 2 [(n c /n S )z c 2 + n a /n S )z a 2 ] Function: characterizes the energy of non-specific electrostatic interactions, according to the Debye-Hückel theory, A ϕ is the Debye-Hückel constant, which depends on the temperature and permittivity of the solvent, and b = 1.2 [17] s, the non-variable Pitzer parameter. We assume, additionally, as in the Pitzer model, that λ ij (I) (if i,j are cation and anion) can exclusively be functions of I and do not depend on n NE . In addition, assume that λ c-NE and λ a-NE do not depend on I, because one of interacting particles (NE) has no charge. Then, given the fact that: where: So, the Sechenov equations in model SE-1 (8) and (4) are proved.

Model Description of Modified Three-Parameter Sechenov Solubility Equation (SEM-3)
To increase the accuracy of the approximation, we used a modified Sechenov solubility equation (SEM-3 where: A s , B s , D s are the empirical (fitting) constants of the model SEM-3. As authors know, such an approximation is original and at the same time forced due to the nontrivial convexconcave course of the solubility branches of the nonelectrolyte in the studied systems. The appearance of a quadratic and cubic term in equation (15) is quite justified, since, for example, the expression for the activity coefficient of electrolyte, in the most popular now semi-empirical model of K. Pitzer [17,18], includes variable parameters with weights m 2 and m 3 -these are the parameters B γ (I) and, C γ respectively in binary systems and θ, ψ in ternary systems. Let us consider the ternary system E(electrolyte)-NE(non-electrolyte)-S(solvent) and take into account virial coefficients of high order: here: µ ijk and ω ijk−NE are the third and fourth virial coefficients pf the decomposition of function G ex RT on the molar numbers of ions and NE. Pitzer [17] proposed to consider µ ijk independent on I, some of them (corresponds to the three ions with the same charge) are equal to zero µ iii = 0. Let us assume this, and additionally, that ω ijk−NE are also independent on I, and some of them are also equal to zero: ω iii−NE = 0.
So, one can easily find that: And after simplifying: So, the modified three-parameter Sechenov Equation (15) is proven and: The simulation results are shown in Figures 15-18 and in Table 2. As can be seen from the figures and the table: Firstly, the accuracy of the approximation has increased dramatically (the Parkinson's approximation accuracy criterion χ 2 /DoF is 3-8 times higher) and, secondly, the model begins to perfectly describe the nontrivial concave-convex σ-id course of the crystallization curve. The last three-parameter SEM-3 allows us to expand the scope of the model for describing the branches of solubility of non-electrolytes in electrolyte solutions to high concentrations of the latter (to a molalities range from 4 to 7 mol per kg of solvent).       Curiously, no two-parameter model is capable of such a description (see Table 2).

Conclusions
Solubility in triple systems С60(ОН)24-PrCl3-H2O, С60(ОН)24-С60(ОН)24-TbCl3-H2O, and С60(ОН)24-YCl3-H2O at 25 o C was studied. grams in four ternary systems are simple eutonic, consist of two branches, to the crystallization of fullerenol crystal-hydrate and rare earth chloride cr and contain one nonvariant point corresponding to saturation with both so the long branches of С60(ОН)24*18H2O crystallization, a pronounced salti observed-the solubility of С60(ОН)24 decreases by more than 2 orders of m pared to the solubility of fullerenol in pure water. On very short branche tion of PrCl3*7H2O, NdCl3*6H2O, YCl3*7H2O, and TbCl3*6H2O, the effect clearly observed, the solubility of all chlorides increases markedly. All diag accurately approximated by classical one-parameter Sechenov equation rately approximated by the three-parameter modified Sechenov equatio concentration range up to 4-7 molalities of electrolytes.

Conclusions
Solubility in triple systems C 60 (OH) 24 -PrCl 3 -H 2 O, C 60 (OH) 24 -NdCl 3 -H 2 O, C 60 (OH) 24 -TbCl 3 -H 2 O, and C 60 (OH) 24 -YCl 3 -H 2 O at 25 •C was studied. Solubility diagrams in four ternary systems are simple eutonic, consist of two branches, corresponding to the crystallization of fullerenol crystal-hydrate and rare earth chloride crystal-hydrates, and contain one nonvariant point corresponding to saturation with both solid phases. On the long branches of C 60 (OH) 24 *18H 2 O crystallization, a pronounced salting-out effect is observed-the solubility of C 60 (OH) 24 decreases by more than 2 orders of magnitude compared to the solubility of fullerenol in pure water. On very short branches of crystallization of PrCl 3 *7H 2 O, NdCl 3 *6H 2 O, YCl 3 *7H 2 O, and TbCl 3 *6H 2 O, the effect of salting-in is clearly observed, the solubility of all chlorides increases markedly. All diagrams are fairly accurately approximated by classical one-parameter Sechenov equation and very accurately approximated by the three-parameter modified Sechenov equation in the whole concentration range up to 4-7 molalities of electrolytes.