Hydration thermodynamics and hydrodynamic (Stokes) radius of the lanthanide ions

Many biochemical and physiological properties depend on the size of ions and the thermodynamic quantities of ion hydration. The diffusion coefficient (D) of lanthanide (III) ions (Ln+3) in solution assumed (1.558-1.618 ×10−9 m2 s−1) by Einstein–Smoluchowski relation. The association constant (KA) of Ln+3 ions was calculated (210.3-215.3 dm3 mole-1) using the Shedlovsky method, and the hydrodynamic radius calculated (1.515-1.569 ×10−10 m) by the Stokes-Einstein equation. The thermodynamic parameters (ΔGo, ΔSo) also calculated by used suitable relations, while ΔHo, values are obtained from the literature. ΔGo, for ion hydration, has negative values in the range (13.25-13.30 KJ/mole), and a negative ΔSo, results have been shown in the limit (11.016-12.506 KJ/ K. mole).


Introduction
Lanthanides are critical components used in various recent technologies, from batteries of electric cars, wind turbines magnets, security inks, optical glasses, hard disk drives, and lasers [1,2]. In medicine and nanotechnology the lanthanide has important development in human health like sensing fluorescent tags, drug delivery, diagnosis cancer, treatment of burn wounds, and bone tissue disease [3][4][5][6][7]. Lanthanides (in full body examine studies) are found in trace amount in numerous organs like bones, kidney, spleen, and liver. In eyes the amount of the lanthanides found to be different in an extensive range. Studies in 21 biochemistry signed to the greatest significant of lanthanide concentration accrued in many organs vary with different diseases steps progress [8].
The significant biochemical properties of lanthanide series (Ln +3 ) ( Fig. 1) are due to their 4f electrons (Ln +3 = [Xe] 4f 1-14 5d 1 6s 2 ) [1]. In body tissues, electrolytes (ions) are important in a number of functions like neuron activation, body osmotic fluids pressure and muscle contractions. The ability function of lanthanides for numerous biological properties in many biomolecules (ex. in proteins and bone) is replace calcium, without essentially replacing for it role [9,10]. The toxicity of Ln +3 ions can be measured by its amount of aberration from pertinent reference essential ion such as Ca +2 [8]. Ln +3 salts in water as part of the solution chemistry have varied inferences in many fields such as element tracing, materials processing and medicine [11].
Many bio-chemical and physiological properties, like effectiveness of a membrane separation process, depend on the size of ions [12]. Ion-water associations in aqueous environment can be termed as M z+ (H 2 O) n where (n) is the molecules of water interacting with the ion M z+ by hydrogen bonds (HB) to build a hydration (solvation) shells (Fig. 2), so the effect size is the hydrated radius [13,14], and this supplies the next independent effects to the thermodynamic quantities of ion hydration.

Fig. 2: A-The solvation shells of M 3+ ion (e.g. M 3+ of group 3) in water solution [15], B-Hydrogen bonds between water molecules.
The present work reports conductance measurements (Electro-Analytical Measurements), hydrodynamic (Stokes) radius, association constant and, thermodynamics properties for ion hydration of lanthanide ions in water solution at temperature, 298.15 K.
Calculation procedure and results 1. Conductivity, mobility and diffusion coefficient of lanthanide ions in water solution at T=298.15 K. The solvation process and ionic hydration of ions play a very significant role in a solution chemistry [16]. The conductance transport property is important because it provides benefit information about ion association and ion-solvent hydration [17]. At infinite dilution conductance and diffusion properties can be accumulated to velocity, because, the interionic forces which repel transfer of ions disappear in a solution. Ion hydration is the greatest interactions, which is generally used in the chemistry of solutions. [18]. Limiting conductance of Ln +3 ions in water is obtained from literature [20]. The calculation procedure of molar conductivity, λ, mobility, µ, and diffusion coefficient, D, for lanthanide ions in water at temperature 298.15 K are based on references [21][22][23]. The results are listed in Table 1 and as shown in Figs. 3 and 4.  The lanthanide series ions are basically found in the (+3) oxidation state with a number of water molecules surrounding the ion by hydrogen bonds in their aqua complexes in water solution. This affects their transport properties like conductance. From   [24] is the main observed geometry in solution [8]. Ln 3+ ions ionic radius decreases through the ions series because the (4f) electrons have slight effect on bonding (lanthanide contraction). Ionic radius for Ln 3+ series is in the range of (1.03-0.86 ×10 -10 m (C.N.6), 1.16-0.98×10 -10 m (C.N.8) and 1.22-1.03 ×10 -10 m (C.N.9)) for La 3+ to Lu 3+ respectively, Pr 3+ ion is of the same radius as Ca 2+ (1.12 for C.N.8).
In aqueous solutions, HB interactions describe the arrangement of ions with water molecules association [13]. Hydrodynamic (Stokes) radius of a solute is the radius of hard sphere that diffuses at the same ion or solute speed in water solution. The hydrodynamic radius (Stokes radius) for Ln 3+ diffusing ions can be obtained via Stokes-Einstein equation (Eq. (3)) [26].
The results of equation 3 and the ionic radius of Ln 3+ in different coordination numbers [27,28] are listed in Table 2, and demonstrated in Fig.6.

Fig. 6: The hydrodynamic and ionic radiuses (r ×10 -10 m) for lanthanide series in different coordination numbers.
The 4f electrons have little effect on lanthanide series ions bonding (the lanthanide contraction), consequently, the values from Table 2 and Fig.6, shows that: The ionic radius for lanthanide series at different coordination numbers is very closer in values (with a difference 0.17×10 -10 m, 0.18×10 -10 m, 0.19×10 -10 m, in lanthanide series at C.N. 6, 8 and 9 respectability) and decreased as the atomic number increases in series, while the hydrodynamic radii in opposite side rearrangement with narrow differences in value (less than 0.1×10 -10 m, through the series). The hydrodynamic radii for Gd 3+ -Lu 3+ are larger than that for La 3+ -Eu 3+ ions. This is relevant to the fact that a smaller ion interacts with larger molecules of water as it transfers via a solution. The hydrodynamic radius depends on the ionic size and the charge of the central ion. Water molecules interacted more with Gd 3+ -Lu 3+ , which have charge densities (Z/r) than with La 3+ -Eu 3+ of less charge densities. This indicates that Gd 3+ -Lu 3+ ions hold more hydration shell from water molecules. 3. Thermodynamic functions of lanthanide ions in water at 298.15 K: Ion hydration in solution gives benefit quantities of ion-water association like diffusion coefficients and thermodynamic properties [29]. The chemo-physical properties of HB transmit the intermolecular modifications as a consequence to hydrogen bonds formation in solution. So HB affects the dynamic and thermodynamic properties of solvents [30]. The conductance values of ion hydration at infinitive dilution allow the determination of association and thermodynamic constants.
The attraction between ions and water molecules in ion hydration is measured by association constant (K A ). To calculate the association constant Shedlovsky method is used (Eqs. (4, 5 and 6)) [30]. (7) : Limiting molar conductivity (S m 2 mol -1 ), C: solution concentration (mol dm -3 ), : molar conductivity (S m 2 mol -1 ), A, and B are constants that depend on water solvent viscosity, dielectric constant, temperature, and ion charge. Gibbs free energy ΔG o , ("a measure of usable energy in the system") via the association interaction is resulted from Eq.(8) [30].
(8) R (J K −1 mol −1 ), constant of gas. Gibbs-Helmholtz relation (Eq. 9) [30] has been applied to evaluate ΔS o , the change of entropy ("the state function that gives a measure of disorder or randomness"). The Ln +3 ion hydration enthalpies, ΔH o , ("the association heat change when one mole of a compound results from elements") are obtained from literature [29].
(9) The data obtained from Eqs. (4, 7, 8 and 9) are listed in Table 3. The results from hydrodynamic radii indicates that the number of water molecules associated with lanthanide series ions are very near (because the charge densities (Z/r) depends on the ionic size (very closer) and the charge of the central ion (equal to +3)), So this affected the thermodynamics association interaction parameters in near values as a results with little differences.
From Table 3 the results show that: The Gibbs free energy ΔG o , for ion hydration has negative values because the association is favored rather than the dissociation in solution. A negative ΔS o , the change of entropy results in a decrease of disorder of ions in water solution, because of increase in the solvation of Ln +3 ions. This can be accumulated to decrease in the freedom degree through forming ion hydration. The hydration enthalpies ΔH o , are of negative values in solution, and this indicates that the interaction between ions with a negative values exothermic processes.

Conclusion
Molar conductivities of Ln +3 ions in water solution have been reported at T = 298.15 K according to the limiting molar conductivities. Shedlovsky method was used to calculate the association constant. Stokes-Einstein relation was applied to calculate the hydrodynamic radius.
The 4f electrons have very little effect on lanthanide series ions bonding, so the ionic radius of lanthanide series at different coordination numbers were of very close values (with a difference 0.17×10 -10 m, 0.18×10 -10 m, 0.19×10 -10 m, in lanthanide series at C.N. 6, 8 and 9 respectively) and decreased as the atomic number increases through the series. The hydrodynamic radius of Ln +3 ions increased as the ionic radius decreases because of the increase in charge densities through the lanthanides series, with little difference in values (less than 0.1×10 -10 m), via the series. The hydro-dynamic radii of Gd 3+ -Lu 3+ are larger than that of La 3+ -Eu 3+ ions. This is relevant to the fact that a small ion interacts with larger molecules of water as it transfers via a solution. The results from hydrodynamic radii indicated that the number of water molecules associated with lanthanide series ions is very near (because the charge densities (Z/r) depends on the ionic size (very closer) and the charge of the central ion (equal to +3)), so this effected on thermodynamics association interaction parameters in near values as a results with little differences. The Ln +3 ions hydration enthalpy, ΔH o , was obtained from literature. Gibbs free energy ΔG o and the change of entropy, ΔS o are resulted from the appropriate equations. ΔG o , for ion hydration has negative values because the association is favored rather than the dissociation in solution. A negative ΔS o , results indicate a decrease in disorder of ions in water solution, because of increase in the solvation of Ln +3 ions. ΔH o values were negative in solution, and this refers to exothermic association process.