Lattice Energies from Hydration Enthalpies: Some acid-base and Structural Considerations

In the present work, using reference values for the hydration enthalpies for a series of mono, di, tri and tetra cations, as well as reference values for the lattice energies of a series of nono, di, tri and tetrahalides, it is shown that reliable lattice energies for such halides can be calculated by UPOT = (ΔHhyd+ + ΔHhyd-), by UPOT = (ΔHhyd+ + 2ΔHhyd-), by UPOT = (ΔHhyd+ + 3ΔHhyd-) or by UPOT = (ΔHhyd+ + 4ΔHhyd-) for mono, di, tri and tetrahalides, respectively. Linearized improved versions of such simply equations, parametrized in order to take into account factors such as dilution and entropic contributions, were also obtained. Lattice energies for a series of halides and other salts are calculated by using the obtained empirical equations, providing results in very good agreement with literature reference values. Furthermore, a series of empirical equations were derived, relating several acid-base parameters with lattice energy. It is shown that the cation and anion volumes (obtained by X-ray data), are very closely related with the cation and anion absolute hardness, that is, are very closely relates with the frontier (HOMO and LUMO) orbitals energies.


I. INTRODUCTION
Lattice energy is a prominent parameter in chemistry, since it could be related with a series of properties of a given compound, such as s olubility, melting point, etc. (Dasent, 1982). Furthermore, hydration enthalpy is one of the fundamental quantities for the thermodynamics of aqueous systems.
Most recently, we have been developed an empirical equation to calculate the lattice energies for metal monohalides from average orbital electronegativities (de Farias, 2017). In the present work, are derives empirical equation that allows the calculation of lattice energies for +1, +2, +3 and +4 salts (specially halides) based only on hydration enthalpies.

II.
METHODOLOGY, RESULTS AND DISCUSSION The up to date hydration enthalpies for group 1 monocations and group 17 monoanions (Housecroft, 2017) as well as the lattice energies (UPOT) to the respective halides (Glasser, 2000;Mu, 2000) are summarized in Table 1. As can be verified, the sum of cations and anion hydrations enthalpies are in very good agreement with the lattice energies for the respective metal halides. Taking into account the uncertainties that there are in both, UPOT and ΔHhyd values, such agreement is really quite good. Hence, the following equation can be derived: where ΔHhyd + and ΔHhydare the hydration enthalpies of the respective cation and anion. When lattice energy is plotted as a function of the sum of the respective cation and anion hydration enthalpies, the curve shown in Figure 1 Such phenomena (UPOT = ΔHhyd + + ΔHhyd -) can be explained if we take into account that in the solid state (where cations are surrounded by anions and anions by cations, e.g. in a 6:6 environment, as in NaCl), or in solution (where both, cations and anions are surrounded by the solvent molecules), both, cations and anions are "looking for" (thermodynamic) stability.
In these systems, stability means to interact with positive of negative species in order to equalize their electronic chemical potentials (Parr, 1978), and such stability is achieved by exothermic interactions, with the total amount of energy required by the cation (or by the anion) been the same, no matter if the interactions occurs with other anions (or cations) in the solid state or, as in a aqueous solutions, with the negative (or positive) poles of the solvent molecules. The same procedures were repeated to group 2 halides, and the respective data are summarized in Table  2. The experimental hydration enthalpies for group 2 dications are those provided by (Smith, 1977). The agreement between reference and lattice energies calculated by using the equation: are very good, as verified in Table 2 data, and Figure 2.

Fig. 2: Lattice energies for group 2 halides, as function of the sum of the hydration enthalpies to the respective cations and (x 2) the hydration enthalpies to the anions.
When lattice energy is plotted as a function of the sum of the respective cation and (plus 2) anion hydration enthalpies, the curve shown in Figure 2 (r = 0.9775) is obtained, from which the following empirical equation is derived: The same procedures were repeated to group some halides, and the respective data are summarized in Table 3. The experimental hydration enthalpies for trications are those provided by (Smith, 1977). In Tables  1-3, the UPOT values taken as references are those previously reported (Glasser, 2000;Mu, 2000).
The agreement between reference and lattice enthalpies calculated by using the equation: is very good, as verified in Table 3 data. When lattice energy is plotted as a function of the sum of the respective cation and (x 3) anion hydration enthalpies, the curve shown in Figure 3 (r = 0.9515) is obtained, from which the following empirical equation is derived: The same procedures were repeated to some +4 cations halides, and the respective data are summarized in Table 4. The experimental hydration enthalpies fo r tetracations are those provided by (Smith, 1977).
The agreement between reference and lattice enthalpies calculated by using the equation: is very good, as verified in Table 4 data. When lattice energy is plotted as a function of the sum of the respective cation and (x 4) anion hydration enthalpies, the curve shown in Figure 4 ( Of course, Eq. (2), (4), (6) and (8) are improved versions of Eq. (1), (3), (5) and (7), and are parametrized in order to take into account factors such as dilution and entropic contributions (Persson, 2010;Hünenberger, 2011).
In order to verify the reliability and general application of Eq.(1), (3) and (5), they were employed to calculate the lattice energies for a series of salts. Despite the fact that the equations were obtained based on data for halides, they were also applied to salts with another kind of anions. The employed auxiliary data and the obtained results are summarized in Table 5. Of course, is possible to apply the values calculated by Eq.(1), (3) and (5) in Eq.
(2), (4) and (6) and obtain a new set of calculated values. In Table 5, the experimental hydration enthalpies for cations are those provided by (Smith, 1977). Except for F -, Cl -, Brand I -, for which were used the values provided by Housecroft (Housecroft, 2017), the hydration enthalpies for anions are those provided by (Smith, 1977).
As can be verified from Table 5 data, Eq. (1) works very well for CuF and AgF. However, as the anion hardness decreases, the agreement between calculated and reference values turns bad. This is a surprisingly result, since Cu + and Ag + are soft acids, and Fis hard base. For example, when applying average orbital electronegativities to calculated lattice energies (de Farias, 2017), it was verified (in agreement with HSAB theory) that the worst results were obtained, exactly, to CuF and AgF.
So, it is possible to suppose that Eq.(1) works better for compounds for which a zero or minor CFSE is computed (a natural conclusion, since it was obtained by using experimental data for group 1 halides).
The spectrochemical series for the halides is F -> Cl -> Br -> I - (Pfennig, 2015)., and all halides anions are weaker field ligands than water. Since, considering only the halides, Fis the ligand with the strongest field, this is the explanation why to exchange four water molecules by four Fions in the coordination sphere of Cu(I) leads to a very good lattice energy calculated by using Eq. (1), whereas the results turns progressively bad for Cl -, Brand I -.
It is also necessary to consider that, despite the fact that Li + is a hard acid and that Cu + is a soft acid, four coordinated Li + (Mähler, 2012) and four coordinated Cu + (Shannon, 1976) have the same radius: 60 pm. Hence, like in Kapustinskii equation (Kapustinskii, 1956), eq.(1) is closely related with the cation radius.
Furthermore, the number of water molecules in the coordination sphere increases form Li + to Cs + (Persson, 2010;Mähler, 2012), and then, whereas Li + is also four coordinated (like Cu + ), Na + and K + , for example, have six and eight water molecules in their coordination sphere (Persson, 2010;Mähler, 2012). Then, the entropic contribution is more prominent for Cu + than to Cu 2+ halides, if the lattice energies are calculated by using hydration enthalpy data.
Based on the results obtained to Ag + halides (Table 5) can be concluded that Eq.(1) provides underestimated lattice energy values for compounds with a high degree of covalence, and that such disagreement (between calculated and reference values) increases as the degree of covalence increases. Since Ag + is a soft acid, the degree of covalence increases from F -, Cl -(hard bases) to Br -(borderline base) and I -(soft base). For Au(I) halides the obtained results are really not good. However, Is necessary to remember that for gold, (Z = 79), relativistic contributions matters (Leszczynski, 2010), and that gold is the element with the (proportionally) higher relativistic contraction/effects.
Multiplying the lattice energy values calculated using Eq. (1), by γ, "corrected" lattice energy values are calculated for gold, and are shown between parenthesis in Table 5. Is worth noting that, considering the relativistic corrected values, the agreement between calculated and reference values increases from Clto I -, in agreement with the fact the Au + is a soft acid and Clis a hard base, Bra borderline base and Ia soft base.
A relativistic correction is also necessary for thallium halides. For Th, Z= 81, and γ = 1.240.
As can be verified from In Eq.(11), (+) and (-) superscripts were included to differentiate between cation and anion volumes.
It is noteworthy that have been shown that (Tissander, 1998) absolute hydration enthalpy values can be calculated from a set of cluster-ion solvation data, without the use of extra thermodynamic assumptions. Hence, could be concluded that the empirical equations obtained in the present work (Eq. 15, for example), can also be related with the previously derived hydration enthalpy equations, based on cluster-pair-based approximation.