2. Definitions and formulation of solubility products

solids, were issued [3]; one of the volumes concerns the solubility of various oxides and hydroxides [4]. An extensive compilation of aqueous solubility data provides the Handbook

A remark. Precipitates are marked in bold letters; soluble species/complexes are marked in

The distinguishing feature of a chemical compound sparingly soluble in a particular medium is the solubility product Ksp value. In practice, the known Ksp values are referred only to aqueous media. One should note, however, that the expression for the solubility product and then the Ksp value of a precipitate depend on the notation of a reaction in which this precipitate is involved. From this it follows the apparent multiplicity of Ksp's values referred to a particular precipitate. Moreover, as will be stated below, the expression for Ksp must not necessarily contain ionic species. On the other hand, factual or seeming lack of Ksp's value for some precipitates is perceived; the latter issue be addressed here to MnO2, taken as an example.

Solubility products refer to a large group of sparingly soluble salts and hydroxides and some oxides, e.g., Ag2O, considered overall as hydroxides. Incidentally, other oxides, such as MnO2, ZrO2, do not belong to this group, in principle. For ZrO2, the solubility measurements showed quite low values even under a strongly acidic condition [6]. The solubility depends on the prior history of these oxides, e.g., prior roasting virtually eliminates the solubility of some oxides. Moderately soluble iodine (I2) dissolves due to reduction or oxidation, or disproportionation in alkaline media [7–12]; for I2, minimal solubility in water is a reference state. For 8-hydroxyquinoline, the solubility of the neutral molecule HL is a reference state; a growth in solubility is caused here by

The Ksp is the main but not the only parameter used for calculation of solubility s of a precipitate. The simplifications [13] practiced in this respect are unacceptable and lead to incorrect/false results, as stated in [14–18]; more equilibrium constants are also involved with two-phase systems. These objections, formulated in the light of the generalized approach to electrolytic systems (GATES) [8], where s is the "weighed" sum of concentrations of all soluble species formed by the precipitate, are presented also in this chapter, related to nonredox and

Calculation of s gives an information of great importance, e.g., from the viewpoint of gravimetry, where the primary step of the analysis is the quantitative transformation of a proper analyte into a sparingly soluble precipitate (salt, hydroxide). Although the precipitation and further analytical operations are usually carried out at temperatures far greater than the room temperature, at which the equilibrium constants were determined, the values of s obtained from the calculations made on the basis of equilibrium data related to room temperature are helpful in the choice of optimal a priori conditions of the analysis, ensuring the minimal, summary concentration of all soluble forms of the analyte, remaining in the solution, in equilibrium with the precipitate obtained after addition of an excess of the precipitating agent; this excess is referred to as relative to the stoichiometric composition of the precipitate. The ability to perform appropriate calculations, based on all available physicochemical knowledge, in accordance with the basic laws of matter conservation, deepens our knowledge of the relevant systems. At the same time, it produces the ability to acquire relevant knowledge in

the formation of ionic species: H2L+1 in acidic and L<sup>1</sup> in alkaline media.

of Aqueous Solubility Data [5].

94 Descriptive Inorganic Chemistry Researches of Metal Compounds

normal letters.

redox systems.

The Ksp value refers to a two-phase system where the equilibrium solid phase is a sparingly soluble precipitate, whose Ksp value is measured/calculated according to defined expression for the solubility product. This assumption means that the solution with defined species is saturated against this precipitate, at given temperature and composition of the solution. However, often a precipitate, when introduced into aqueous media, is not the equilibrium solid phase, and then this fundamental requirement is not complied, as indicated in examples of the physicochemical analyses of the systems with struvite MgNH4PO4 [20, 21], dolomite CaMg(CO3)2 [22, 23], and Ag2Cr2O7.

The values of solubility products Ksp (usually represented by solubility constant pKsp = �logKsp value) are known for stoichiometric precipitates of AaBb or AaBbCc type, related to dissociation reactions:

$$\mathbf{K\_{sp}} = [\mathbf{A}]^\mathbf{a}[\mathbf{B}]^\mathbf{b} \\ \text{for } \mathbf{A\_{a}} \\ \mathbf{B\_{b}} = \mathbf{a}\mathbf{A} + \mathbf{b} \\ \mathbf{B} \text{, or } \tag{1}$$

$$K\_{\rm sp} = \left[\text{A}\right]^{\rm a} \left[\text{B}\right]^{\rm b} \left[\text{C}\right]^{c} \text{for } \mathbf{A}\_{\mathbf{a}} \mathbf{B}\_{\mathbf{b}} \mathbf{C}\_{\mathbf{c}} = \mathbf{a} \mathbf{A} + \mathbf{b} \mathbf{B} + \text{c} \mathbf{C} \tag{2}$$

where A and B or A, B, and C are the species forming the related precipitate; charges are omitted here, for simplicity of notation. The solubility products for more complex precipitates are unknown in the literature. The precipitates AaBbCc are known as ternary salts [24], e.g., struvite, dolomite, and hydroxyapatite Ca5(PO4)3OH.

The solubility products for precipitates of AaBb type are most frequently met in the literature. In these cases, for A are usually put simple cations of metals, or oxycations [25]; e.g., BiO+1 and UO2 +2 form the precipitates: BiOCl and (UO2)2(OH)2. As B, simple or more complex anions are considered, e.g., Cl�<sup>1</sup> , S�<sup>2</sup> , PO4 �3 , Fe(CN)6 �4 , in AgCl, HgS, Zn3(PO4)2, and Zn2Fe(CN)6.

In different textbooks, the solubility products are usually formulated for dissociation reactions, with ions as products, also for HgS

$$\mathbf{H}\mathbf{g}\mathbf{S} = \mathbf{H}\mathbf{g}^{+2} + \mathbf{S}^{-2}(\mathbf{K}\_{\mathbf{\mathcal{P}}} = [\mathbf{H}\mathbf{g}^{+2}][\mathbf{S}^{-2}]) \tag{3}$$

although polar covalent bond exists between its constituent atoms [26]. Very low solubility product value (pKsp = 52.4) for HgS makes the dissociation according to the scheme presented by Eq. (3) impossible, and even verbal formulation of the solubility product is unreasonable. Namely, the ionic product x = [Hg+2][S–<sup>2</sup> ] calculated at [Hg+2] = [S–<sup>2</sup> ] = 1/N<sup>A</sup> exceeds Ksp, 1/N<sup>A</sup> 2 > Ksp (N<sup>A</sup> – Avogadro's number); the concentration 1/N<sup>A</sup> = 1.66∙10–<sup>23</sup> mol/L corresponds to 1 ion in 1 L of the solution. The scheme of dissociation into elemental species [14]

$$\mathbf{HgS} = \mathbf{Hg} + \mathbf{S} \ (K\_{\text{sp1}} = [\mathbf{Hg}][\mathbf{S}]) \tag{4}$$

is far more favored from thermodynamic viewpoint; nonetheless, the solubility product (Ksp) for HgS is commonly formulated on the basis of reaction (3). We obtain pKsp1 = pKsp – 2A (E01�E02), where <sup>E</sup><sup>01</sup> = 0.850 V for Hg+2 + 2e–<sup>1</sup> <sup>=</sup> Hg, <sup>E</sup><sup>02</sup> <sup>=</sup> –0.48 V for <sup>S</sup> + 2e–<sup>1</sup> = S–<sup>2</sup> , 1/A = RT/ F�ln10, A = 16.92 for 298 K; then pKsp1 = 7.4.

Equilibrium constants are usually formulated for the simplest reaction notations. However, in this respect, Eq. (4) is simpler than Eq. (3). Moreover, we are "accustomed" to apply solubility products with ions (cations and anions) involved, but this custom can easily be overthrown. A similar remark may concern the notation referred to elementary dissociation of mercuric iodide precipitate

$$\mathbf{H}\mathbf{g}\mathbf{I}\_2 = \mathbf{H}\mathbf{g} + \mathbf{I}\_2(K\_{\mathrm{sp1}} = [\mathbf{H}\mathbf{g}][\mathbf{I}\_2]) \tag{5}$$

where I2 denotes a soluble form of iodine in a system. From

$$\mathbf{H} \mathbf{g} \mathbf{I}\_2 = \mathbf{H} \mathbf{g}^{+2} + 2 \mathbf{I}^{-1} (\mathbf{K}\_{\text{sp}} = [\mathbf{H} \mathbf{g}^{+2}][\mathbf{I}^{-1}]^2, pK\_{\text{sp}} = 28.55) \tag{6}$$

we obtain pKsp1 = pKsp – 2A(E01–E03), where

$$E\_{01} = 0.850\,\text{V} \text{ for } \text{Hg}^{+2} + 2\text{e}^{-1} = \text{Hg}\,\text{ }E\_{03} = 0.621\,\text{V} \text{ for } \text{I}\_2 + 2\text{e}^{-1} = 2\text{I}^{-1}\text{; then }pK\_{\text{sp1}} = 20.80\,\text{J}$$

The species in the expression for solubility products do not predominate in real chemical systems, as a rule. However, the precipitation of HgS from acidified (HCl) solution of mercury salt with H2S solution can be presented in terms of predominating species; we have

$$\text{HgCl}\_4^{-2} + \text{H}\_2\text{S} = \text{HgS} + 4\text{Cl}^{-1} + 2\text{H}^{+1} \tag{7}$$

Eq. (7) can be applied to formulate the related solubility product, Ksp2, for HgS. To be online with customary requirements put on the solubility product formulation, Eq. (7) should be rewritten into the form

$$\text{HgS} + 4\text{Cl}^{-1} + 2\text{H}^{+1} = \text{HgCl}\_4^{-2} + \text{H}\_2\text{S} \tag{7a}$$

Applying the law of mass action to Eq. (7a), we have

$$K\_{sp2} = \frac{[\text{HgCl}\_4^{-2}][\text{H}\_2\text{S}]}{[\text{Cl}^{-1}]^4[\text{H}^{+1}]^2}, \quad (pK\_{sp2} = 17.33) \tag{8}$$

where [HgCl4 –2 ] = 1015.07[Hg+2][Cl–<sup>1</sup> ] 4 , [H2S] = 1020.0[H+1] 2 [S–<sup>2</sup> ], Ksp (Eq. (3)).

The solubility product for MgNH4PO4 can be formulated on the basis of reactions:

$$\text{MgNH}\_4\text{PO}\_4 = \text{Mg}^{+2} + \text{NH}\_4^{+1} + \text{PO}\_4^{-3} (\text{K}\_{\text{sp}} = [\text{Mg}^{+2}][\text{NH}\_4^{+1}][\text{PO}\_4^{-3}]) \tag{9}$$

$$\mathbf{M}\mathbf{g}\mathbf{N}\mathbf{H}\_4\mathbf{P}\mathbf{O}\_4 = \mathbf{M}\mathbf{g}^{+2} + \mathbf{N}\mathbf{H}\_3 + \mathbf{H}\mathbf{P}\mathbf{O}\_4^{-2} (\mathbf{K}\_{\mathbf{s}\mathbf{p}1} = [\mathbf{M}\mathbf{g}^{+2}][\mathbf{N}\mathbf{H}\_3][\mathbf{H}\mathbf{P}\mathbf{O}\_4^{-2}] = \mathbf{K}\_{\mathbf{s}\mathbf{p}}\mathbf{K}\_{\mathbf{l}\mathbf{N}}/\mathbf{K}\_{\mathbf{s}\mathbf{p}}\mathbf{} \tag{10}$$

$$\begin{aligned} \mathbf{MgNH\_4PO\_4} + \text{H\_2O} &= \text{MgOH}^{+1} + \text{NH}\_3 + \text{H\_2PO\_4}^{-1} (\text{K}\_{\text{sp2}} = [\text{MgOH}^{+1}][\text{NH}\_3][\text{H\_2PO\_4}^{-1}] \\ &= \text{K}\_{\text{sp}} \text{K}\_1^{\text{OH}} \text{K}\_{\text{IN}} \text{K}\_{\text{W}} / (\text{K}\_{\text{2P}} \text{K}\_{\text{3P}}) \end{aligned} \tag{11}$$

where K1N = [H+1][NH3]/[NH4 +1], K2P = [H+1][HPO4 –2 ]/[H2PO4 –1 ], K3P = [H+1][PO4 –3 ]/[HPO4 –2 ], [MgOH+1]=K1 OH[Mg+2][OH–<sup>1</sup> ], K<sup>W</sup> = [H+1][OH–<sup>1</sup> ].

Note that only uncharged (elemental) species are involved in Eqs. (4) and (5); H2S enters Eq. (8), and NH3 enters Eqs. (10) and (11). This is an extension of the definition/formulation commonly met in the literature, where only charged species were involved in expression for the solubility product. Note also that small/dispersed mercury drops are neutralized with powdered sulfur, according to thermodynamically favored reaction [27]
