Solubility Products and Solubility Concepts

Abstract

The chapter refers to a general concept of solubility product Ksp of sparingly soluble hydroxides and different salts and calculation of solubility of some hydroxides, oxides, and different salts in aqueous media. A (criticized) conventional approach, based on stoichiometry of a reaction notation and the solubility product of a precipitate, is compared with the unconventional/correct approach based on charge and concentration balances and a detailed physicochemical knowledge on the system considered, and calculations realized according to generalized approach to electrolytic systems (GATES) principles. An indisputable advantage of the latter approach is proved in simulation of static or dynamic, two-phase nonredox or redox systems.

1. Introduction

The problem of solubility of various chemical compounds occupies a prominent place in the scientific literature. This stems from the fact that among various properties determining the use of these compounds, the solubility is of the paramount importance. Among others, this issue has been the subject of intense activities initiated in 1979 by the Solubility Data Commission V.8 of the IUPAC Analytical Chemistry Division established and headed by S. Kertes [1], who conceived the IUPAC-NIST Solubility Data Series (SDS) project [2, 3]. Within 1979–2009, the series of 87 volumes, concerning the solubility of gases, liquids, and solids in liquids or 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 of Aqueous Solubility Data [5].

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

The distinguishing feature of a chemical compound sparingly soluble in a particular medium is the solubility product K_sp value. In practice, the known K_sp values are referred only to aqueous media. One should note, however, that the expression for the solubility product and then the K_sp 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 K_sp’s values referred to a particular precipitate. Moreover, as will be stated below, the expression for K_sp must not necessarily contain ionic species. On the other hand, factual or seeming lack of K_sp’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 formation of ionic species: H₂L⁺¹ in acidic and L⁻¹ in alkaline media.

The K_sp 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 redox systems.

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 an organized manner—not just imitative, but focused on heuristics. This viewpoint is in accordance with constructivist teaching, based on the belief that learning occurs, as learners are actively involved in a process of meaning and knowledge construction, as opposed to passively receiving information [19].

2. Definitions and formulation of solubility products

The K_sp value refers to a two-phase system where the equilibrium solid phase is a sparingly soluble precipitate, whose K_sp 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 K_sp (usually represented by solubility constant pK_sp = −logK_sp value) are known for stoichiometric precipitates of A_aB_b or A_aB_bC_c type, related to dissociation reactions:

$K_{\text{sp}}={\left[\text{A}\right]}^{\text{a}}{\left[\text{B}\right]}^{\text{b}}\text{for}{A}_a{B}_b=\text{aA}+\text{bB},\text{or}$E1

$K_{\text{sp}}={\left[\text{A}\right]}^{\text{a}}{\left[\text{B}\right]}^{\text{b}}{\left[\text{C}\right]}^{\text{c}}\text{for}{A}_a{B}_b{C}_c=\text{aA}+\text{bB}+\text{cC}$E2

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 A_aB_bC_c are known as ternary salts [24], e.g., struvite, dolomite, and hydroxyapatite Ca₅(PO₄)₃OH.

The solubility products for precipitates of A_aB_b 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⁻¹, S⁻², 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

$HgS={\text{Hg}}^{+2}+{\text{S}}^{–2}(K_{\text{sp}}=\left[{\text{Hg}}^{+2}\right][{\text{S}}^{-2}])$E3

although polar covalent bond exists between its constituent atoms [26]. Very low solubility product value (pK_sp = 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^–2] calculated at [Hg⁺²] = [S^–2] = 1/N_A exceeds K_sp, 1/N_A² > K_sp (N_A – Avogadro’s number); the concentration 1/N_A = 1.66∙10^–23 mol/L corresponds to 1 ion in 1 L of the solution. The scheme of dissociation into elemental species [14]

$HgS=Hg+S (K_{\text{sp1}}=\text{[}Hg\text{]}[S])$E4

is far more favored from thermodynamic viewpoint; nonetheless, the solubility product (K_sp) for HgS is commonly formulated on the basis of reaction (3). We obtain pK_sp1 = pK_sp – 2A(E₀₁−E₀₂), where E₀₁ = 0.850 V for Hg⁺² + 2e^–1 = Hg, E₀₂ = –0.48 V for S + 2e^–1 = S^–2, 1/A = RT/F⋅ln10, A = 16.92 for 298 K; then pK_sp1 = 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

$Hg{I}_2=Hg+{\text{I}}_2(K_{\text{sp1}}=[Hg][{\text{I}}_2])$E5

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

$Hg{I}_2={\text{Hg}}^{+2}+2{\text{I}}^{-1}(K_{\text{sp}}=[{\text{Hg}}^{+2}]{[{\text{I}}^{-1}]}^2,p{K}_{\text{sp}}=28.55)$E6

we obtain pK_sp1 = pK_sp – 2A(E₀₁–E₀₃), where

$E_{01}=0.850\text{V}{\text{for Hg}}^{+2}+2{\text{e}}^{-1}=Hg,E_{03}=0.621{\text{V for I}}_2+2{\text{e}}^{-1}=2{\text{I}}^{-1};\text{then}p{K}_{\text{sp1}}=\mathrm{20.80.}$

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

${\text{HgCl}}_4^{-2}+{\text{H}}_2\text{S}=HgS+4{\text{Cl}}^{–1}+2{\text{H}}^{+1}$E7

Eq. (7) can be applied to formulate the related solubility product, K_sp2, for HgS. To be online with customary requirements put on the solubility product formulation, Eq. (7) should be rewritten into the form

$HgS+4{\text{Cl}}^{–1}+2{\text{H}}^{+1}={\text{HgCl}}_4^{-2}+{\text{H}}_2\text{S}$E7a

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

$K_{\text{sp2}}=\frac{{\text{[HgCl}}_4^{-2}{\text{][H}}_{\text{2}}\text{S]}}{{{\text{[Cl}}^{-\text{1}}\text{]}}^{\text{4}}{{\text{[H}}^{+\text{1}}\text{]}}^{\text{2}}},(p{K}_{sp2}=17.33)$E8

where [HgCl₄^–2] = 10^15.07[Hg⁺²][Cl^–1]⁴, [H₂S] = 10^20.0[H⁺¹]²[S^–2], K_sp (Eq. (3)).

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

$MgN{H}_{4}P{O}_4=\text{M}{\text{g}}^{+2}+\text{N}{\text{H}}_4^{+1}+\text{P}{\text{O}}_4^{-3}({\text{K}}_{\text{sp}}=[{\text{Mg}}^{+2}][{\text{NH}}_4^{+1}][{\text{PO}}_4^{-3}])$E9

$MgN{H}_{4}P{O}_4=\text{M}{\text{g}}^{+2}+\text{N}{\text{H}}_3+\text{H}\text{P}{\text{O}}_4^{-2}({\text{K}}_{\text{s}\text{p}1}=[\text{M}{\text{g}}^{+2}][\text{N}{\text{H}}_3][\text{H}\text{P}{\text{O}}_4^{-2}]={\text{K}}_{\text{s}\text{p}}{\text{K}}_{1\text{N}}/{\text{K}}_{3\text{P}})$E10

$\begin{array}{c}MgN{H}_{4}P{O}_4+{\text{H}}_2\text{O}=\text{M}\text{g}\text{O}{\text{H}}^{+1}+\text{N}{\text{H}}_3+{\text{H}}_2\text{P}{\text{O}}_4^{-1}({\text{K}}_{\text{s}\text{p}2}=[\text{M}\text{g}\text{O}{\text{H}}^{+1}][\text{N}{\text{H}}_3][{\text{H}}_2\text{P}{\text{O}}_4^{-1}]\hfill \\ ={\text{K}}_{\text{s}\text{p}}{\text{K}}_1^{\text{OH}}{\text{K}}_{1\text{N}}{\text{K}}_{\text{W}}/({\text{K}}_{2\text{P}}{\text{K}}_{3\text{P}})\hfill \end{array}$E11

where K_1N = [H⁺¹][NH₃]/[NH₄⁺¹], K_2P = [H⁺¹][HPO₄^–2]/[H₂PO₄^–1], K_3P = [H⁺¹][PO₄^–3]/[HPO₄^–2], [MgOH⁺¹] = K₁^OH[Mg⁺²][OH^–1], K_W = [H⁺¹][OH^–1].

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]

$Hg+S=HgS$E8422

reverse to Eq. (4). Some precipitates can be optionally considered as the species of A_aB_b or A_aB_bC_c type. For example, the solubility product for MgHPO₄ can be written as K_sp = [Mg⁺²][HPO₄^–2] or K_sp1 = [Mg⁺²][H⁺¹][PO₄^–3] = K_spK_3P.

The ferrocyanide ion Fe(CN)₆^–4 (with evaluated stability constant K₆ ca. 10³⁷) can be considered as practically undissociated, i.e., Fe(CN)₆^–4 is kinetically inert [28], and then it does not give Fe⁺² and CN^–1 ions. The solubility product of Zn₂Fe(CN)₆ is K_sp = [Zn⁺²]²[Fe(CN)₆^–4]. Therefore, consideration of Zn₂Fe(CN)₆ as a ternary salt with K_sp1 = [Zn⁺²]²[Fe²⁺][CN^–1]⁶ = K_sp/K₆ is not acceptable.

In principle, the solubility product values are formulated for stoichiometric compounds, and specified as such in the related tables. However, some precipitates obtained in laboratory have nonstoichiometric composition, e.g., dolomite Ca_1+xMg_1-x(CO₃)₂ [22, 23], Fe_xS [29]. In particular, Fe_xS can be rewritten as Fe⁺²_pFe⁺³_qS; from the relations: 2p + 3q − 2 = 0 and p + q = x, we get q/p = 2(1 − x)/(3x − 2).

In this context, some remark needs a formulation of K_sp for some hydroxyoxides (e.g., FeOOH) and oxides (e.g., Ag₂O). The related solubility products are formulated after completion of the corresponding reactions with water, e.g., FeOOH + H₂O = Fe(OH)₃, Fe₂O₃∙xH₂O + (3 − x)H₂O = 2Fe(OH)₃ ⇒ Fe(OH)₃ = Fe⁺³ + 3OH^–1 ⇒ K_sp = [Fe⁺³][OH^–1]³; Ag₂O + H₂O = 2AgOH ⇒ AgOH = Ag⁺¹ + OH^–1 ⇒ K_sp = [Ag⁺¹][OH^–1], see it in the context with gcd(a,b) = 1.

The solubility product can be involved not only with dissociation reaction. For example, the dissolution reaction Ca(OH)₂ + 2H⁺¹ = Ca⁺² + 2H₂O [30], characterized by K_sp1 = [Ca⁺²]/[H⁺¹]², is involved with K_sp = [Ca⁺²][OH^–1]² in the relation K_sp1 = K_sp/K_w². In Ref. [31], the solubility product is associated with formation (not dissociation) of a precipitate.

3. Solubility product(s) for MnO₂

The scheme presented above cannot be extended to all oxides. For example, one cannot recommend the formulation of this sequence for MnO₂, i.e., MnO₂ + 2H₂O = Mn(OH)₄ ⇒ Mn(OH)₄ = Mn⁺⁴ + 4OH^–1 ⇒ K_sp0 = [Mn⁺⁴][OH^–1]⁴; Mn⁺⁴ ions do not exist in aqueous media, and MnO₂ is the sole Mn(+4) species present in such systems. In effect, K_sp0 for MnO₂ is not known in the literature, compare with Ref. [32]. However, the K_sp for MnO₂ can be formally calculated according to an unconventional approach, based on the disproportionation reaction

$5Mn{O}_2+4{\text{H}}^{+1}=2{\text{MnO}}_4^{-1}+3{\text{Mn}}^{+2}+{\text{H}}_2\text{O}$E12

reverse to the symproportionation reaction 2MnO₄⁻¹ + 3Mn⁺² + H₂O = 5MnO₂ + 4H⁺¹. The K_sp = K_sp1 value can be found there on the basis of E₀₁ and E₀₂ values [33], specified for reactions:

${\text{MnO}}_4^{-1}+4{\text{H}}^{+1}+3{\text{e}}^{-1}=Mn{O}_2+2{\text{H}}_2\text{O(}{E}_{01}=1.692\text{V)}$E13

$Mn{O}_2+4{\text{H}}^{+1}+2{\text{e}}^{-1}={\text{Mn}}^{+2}+2{\text{H}}_2\text{O(}{E}_{02}=1.228\text{V)}$E14

Eqs. (13) and (14) are characterized by the equilibrium constants:

$K_{\text{e1}}=\frac{{\text{[MnO}}_{\text{2}}{\text{][H}}_{\text{2}}{\text{O]}}^{\text{2}}}{{\text{[MnO}}_4^{-1}{\text{][H}}^{+\text{1}}{\text{]}}^{\text{4}}{{\text{[e}}^{-\text{1}}\text{]}}^{\text{3}}},K_{\text{e2}}=\frac{{\text{[Mn}}^{+\text{2}}{\text{][H}}_{\text{2}}{\text{O]}}^{\text{2}}}{\text{[}Mn{O}_2{\text{][H}}^{+\text{1}}{\text{]}}^{\text{4}}{{\text{[e}}^{-\text{1}}\text{]}}^{\text{2}}}$E15

defined on the basis of mass action law (MAL) [14], where logK_e1 = 3⋅A⋅E₀₁, logK_e2 = 2⋅A⋅E₀₂, A = 16.92. From Eqs. (13) and (14), we get

$\begin{array}{c}2Mn{O}_2+4{\text{H}}_2\text{O}+3Mn{O}_2+12{\text{H}}^{+1}+6{\text{e}}^{-\text{1}}=2{\text{MnO}}_4^{-1}+8{\text{H}}^{+1}\\ +6{\text{e}}^{-1}+3{\text{Mn}}^{+2}+6{\text{H}}_2\text{O}\end{array}$E16

Assuming [MnO₂] = 1 and [H₂O] = 1 on the stage of the K_sp1 formulation for reaction (16), equivalent to reaction (12), we have

$K_{\text{sp1}}=\frac{{{\text{[MnO}}_4^{-1}\text{]}}^{\text{2}}{{\text{[Mn}}^{+\text{2}}\text{]}}^{\text{3}}}{{{\text{[H}}^{+\text{1}}\text{]}}^{\text{4}}}$E17

and then

$K_{\text{sp1}}={\text{(}{K}_{\text{e2}}\text{)}}^{\text{3}}\cdot {\text{(}{K}_{\text{e1}}\text{)}}^{-\text{2}}$E18

$p{K}_{\text{sp1}}={\text{3logK}}_{\text{e2}}-{\text{2logK}}_{\text{e1}}=\text{6}A\text{(}{E}_{\text{01}}-E_{\text{02}}\text{)}=\text{6}\cdot \text{16}\text{.92}\cdot \text{(1}\text{.692}-\text{1}\text{.228)}=\text{47}\text{.11}$E19

The solubility products with MnO₂ involved can be formulated on the basis of other reactions. For example, addition of

${\text{Mn}}^{+2}={\text{Mn}}^{+3}+{\text{e}}^{-1}$E20

to Eq. (14) gives

$Mn{O}_2+4{\text{H}}^{+1}+2{\text{e}}^{-1}+{\text{Mn}}^{+2}={\text{Mn}}^{+2}+2{\text{H}}_2\text{O}+{\text{Mn}}^{+3}+{\text{e}}^{-1}$E21

Multiplication of Eq. (21) by 3, and then addition to Eq. (13a)

$Mn{O}_2+2{\text{H}}_2\text{O}={\text{MnO}}_4^{-1}+4{\text{H}}^{+1}+3{\text{e}}^{-1}$E8423

(reverse to Eq. (13)) gives the equation

$\begin{array}{c}3Mn{O}_2+12{\text{H}}^{+1}+6{\text{e}}^{-1}+3{\text{Mn}}^{+2}+Mn{O}_2+2{\text{H}}_2\text{O}\hfill \\ =3{\text{Mn}}^{+2}+6{\text{H}}_2\text{O}+3{\text{Mn}}^{+3}+3{\text{e}}^{-1}+{\text{MnO}}_4^{-1}+4{\text{H}}^{+1}+3{\text{e}}^{-1}\hfill \end{array}$E22

and its equivalent form, obtained after simplifications,

$4Mn{O}_2+8{\text{H}}^{+1}=3{\text{Mn}}^{+3}+{\text{MnO}}_4^{-1}+4{\text{H}}_2\text{O}$E22a

Eq. (22) and then Eq. (22a) is characterized by the solubility product

$K_{\text{sp2}}=\frac{{\text{[MnO}}_4^{-1}{\text{][Mn}}^{+\text{3}}{\text{]}}^{\text{3}}}{{{\text{[H}}^{+\text{1}}\text{]}}^{\text{8}}}={\text{(}{K}_{\text{e2}}\text{)}}^{\text{3}}\cdot {\text{(}{K}_{\text{e3}}\text{)}}^{-\text{3}}\cdot {\text{(}{K}_{\text{e1}}\text{)}}^{-\text{1}}$E23

where

$K_{\text{e3}}=\frac{{\text{[Mn}}^{+\text{2}}\text{]}}{{\text{[Mn}}^{+\text{3}}\text{][}{e}^{-\text{1}}\text{]}}$E24

for Mn⁺³ + e⁻¹ = Mn⁺² (E₀₃ = 1.509 V) (reverse to Eq. (20)), logK_e3 = A⋅E₀₃. Then

${\mathit{pK}}_{\text{sp2}}=3A\bullet (E_{01}-2E_{02}\mathrm{+}{E}_{03})+37.82$E25

Formulation of K_spi for other combinations of redox and/or nonredox reactions is also possible. This way, some derivative solubility products are obtained. The choice between the “output” and derivative solubility product values is a matter of choice. Nevertheless, one can choose the K_sp3 value related to the simplest expression for the solubility product K_sp3 = [Mn⁺²][MnO₄⁻²] involved with reaction 2MnO₂ = Mn⁺² + MnO₄⁻².

As results from calculations, the low K_spi (i = 1,2,3) values obtained from the calculations should be crossed, even in acidified solution with the related manganese species presented in Figure 1. In the real conditions of analysis, at C_a = 1.0 mol/L, the system is homogeneous during the titration, also after crossing the equivalence point, at Φ = Φ_eq > 0.2; this indicates that the corresponding manganese species form a metastable system [34], unable for the symproportionation reactions.

Figure 1.

The log[X_i] versus Φ relationships for different manganese species X_i, plotted for titration of V₀ = 100 mL solution of FeSO₄ (C₀ = 0.01 mol/L) + H₂SO₄ (C_a = 1.0 mol/L) with V mL of C = 0.02 mol/L KMnO₄; Φ = C·V/(C₀·V₀). The species X_i are indicated at the corresponding lines.

4. Calculation of solubility

In this section, we compare two options applied to the subject in question. The first/criticized option, met commonly in different textbooks, is based on the stoichiometric considerations, resulting from dissociation of a precipitate, characterized by the solubility product K_sp value, and considered a priori as an equilibrium solid phase in the system in question; the solubility value obtained this way will be denoted by s^* [mol/L]. The second option, considered as a correct resolution of the problem, is based on full physicochemical knowledge of the system, not limited only to K_sp value (as in the option 1); the solubility value thus obtained is denoted as s [mol/L]. The second option fulfills all requirements expressed in GATES and involved with basic laws of conservation in the systems considered. Within this option, we check, among others, whether the precipitate is really the equilibrium solid phase. The results (s^*, s) obtained according to both options (1 and 2) are compared for the systems of different degree of complexity. The unquestionable advantages of GATES will be stressed this way.

4.1. Formulation of the solubility s^*

The solubility s^* will be calculated for a pure precipitate of: (1^o) A_aB_b or (2^o) A_aB_bC_c type, when introduced into pure water. Assuming [A] = a∙s^* and [B] = b∙s^*, from Eq. (1), we have

${\text{s}}^{\ast }={\left(\frac{K_{\text{sp}}}{{\text{a}}^{\text{a}}\cdot {\text{b}}^{\text{b}}}\right)}^{\text{1/(a}+\text{b)}}$E26

and assuming [A] = a∙s^*, [B] = b∙s^*, [C] = c∙s^*, from Eq. (2), we have

${\text{s}}^{\ast }={\left(\frac{K_{\text{sp}}}{{\text{a}}^{\text{a}}\cdot {\text{b}}^{\text{b}}\cdot {\text{c}}^{\text{c}}}\right)}^{\text{1/(a}+\text{b}+\text{c)}}$E27

As a rule, the formulas (26) and (27) are invalid for different reasons, indicated in this chapter. This invalidity results, among others, from inclusion of the simplest/minor species in Eq. (26) or (27) and omission of hydroxo-complexes + other soluble complexes formed by A, and proto-complexes + other soluble complexes, formed by B. In other words, not only the species entering the expression for the related solubility product are present in the solution considered. Then the concentrations: [A], [B] or [A], [B], and [C] are usually minor species relative to the other species included in the respective balances, considered from the viewpoint of GATES [8].

4.2. Dissolution of hydroxides

We refer first to the simplest two-phase systems, with insoluble hydroxides as the solid phases. In all instances, s^* denotes the solubility obtained from stoichiometric considerations, whereas s relates to the solubility calculated on the basis of full/attainable physicochemical knowledge related to the system in question where, except the solubility product (K_sp), other physicochemical data are also involved.

Applying formula (26) to hydroxides (B = OH⁻¹): Ca(OH)₂ (pK_sp1 = 5.03) and Fe(OH)₃ (pK_sp2 = 38.6), we have [35]

$Ca{(OH)}_2={\text{Ca}}^{+2}+2{\text{OH}}^{-1}(K_{\text{sp1}}=[{\text{Ca}}^{+2}]{[{\text{OH}}^{-1}]}^2,{\text{s}}^{*}={(K_{\text{sp1}}/4)}^{1/3}=0.0133\text{mol}/\text{L})$E28

$Fe{(OH)}_3={\text{Fe}}^{+3}+3{\text{OH}}^{-1}(K_{\text{sp2}}=[{\text{Fe}}^{+3}]{[{\text{OH}}^{-1}]}^3,{\text{s}}^{*}={(K_{\text{sp2}}/27)}^{1/4}=0.98\times {10}^{–10}\text{mol}/\text{L})$E29

respectively. However, Ca⁺² and Fe⁺³ form the related hydroxo-complexes: [CaOH⁺¹] = 10^1.3·[Ca⁺²][OH⁻¹] and: [FeOH⁺²] = 10^11.0·[Fe⁺³][OH⁻¹], [Fe(OH)₂⁺¹] = 10^21.7·[Fe⁺³][OH⁻¹]²; [Fe₂(OH)₂⁺⁴] = 10^25.1·[Fe⁺³]²[OH⁻¹]² [31]. The corrected expression for the solubility of Ca(OH)₂ is as follows

$\text{s}=[{\text{Ca}}^{+2}]+[{\text{CaOH}}^{+1}]$E30

Inserting [Ca⁺²] = K_sp1/[OH⁻¹]² and [OH⁻¹] = K_W/[H⁺¹], [H⁺¹] = 10^−pH (pK_W = 14.0 for ionic product of water, K_W) into the charge balance

$2[{\text{Ca}}^{+2}]+[{\text{CaOH}}^{+1}]+[{\text{H}}^{+1}]–[{\text{OH}}^{-1}]=0$E31

we get, by turns,

$\begin{array}{c}2\cdot {10}^{-5.03}/{[{\text{OH}}^{-1}]}^2+{10}^{1.3}\cdot {10}^{-5.03}/[{\text{OH}}^{-1}][{\text{H}}^{+1}]-[{\text{OH}}^{-1}]=0\\ \Rightarrow 2\cdot {10}^{-5.03+28-2\text{pH}}+{10}^{1.3}\cdot {10}^{-5.03+14-\text{pH}}+{10}^{-\text{pH}}–{10}^{\text{pH}-14}=0\end{array}$E9000

$\text{y(pH)}=2\cdot {10}^{22.97-2\text{pH}}+{10}^{10,17-\text{pH}}+{10}^{-\text{pH}}–{10}^{\text{pH}-14}=0$E32

where $\text{pH}=-{\text{log[H}}^{+\text{1}}\text{]}$. Applying the zeroing procedure to Eq. (30), we get pH₀ = 12.453 (Table 1), where: [Ca⁺²] = 0.0116, [CaOH⁺¹] = 0.00656, s = 0.0182 mol/L (Eq. (28)). As we see, [CaOH⁺¹] is comparable with [Ca⁺²], and there are none reasons to omit [CaOH⁺¹] in Eq. (28).

pH	y(pH)	[OH^−1]	[Ca^+2]	[CaOH^+1]
12.451	0.000377	0.02825	0.01169	0.006592
12.452	0.000193	0.02831	0.01164	0.006577
12.453	8.30E-06	0.02838	0.01159	0.006561
12.454	−0.000176	0.02844	0.01153	0.006546
12.455	−0.000359	0.02851	0.01148	0.006531

Table 1.

Zeroing the function (30) for the system with Ca(OH)₂ precipitate introduced into pure water (copy of a fragment of display).

The alkaline reaction in the system with Ca(OH)₂ results immediately from Eq. (29): [OH⁻¹] – [H⁺¹] = $2[{\text{Ca}}^{+2}]+[{\text{CaOH}}^{+1}]>0.$

Analogously, for the system with Fe(OH)₃, we have the charge balance

$3[{\text{Fe}}^{+3}]+2[{\text{FeOH}}^{+2}]+[{\text{Fe(OH)}}_2^{+1}]+4[{\text{Fe}}_2{(\text{OH})}_2^{+4}]+[{\text{H}}^{+1}]–[{\text{OH}}^{-1}]=0$E33

and then

$\text{y(pH)}=3\cdot {10}^{3.4-3\text{pH}}+2\cdot {10}^{0.4-2\text{pH}}+{10}^{-2.9-\text{pH}}+4\cdot {10}^{3.9-4\text{pH}}+{10}^{-\text{pH}}–{10}^{\text{pH}-14}=0$E34

Eq. (32) zeroes at pH₀ = 7.0003 (Table 2), where the value

pH	y(pH)	[Fe^+3]	[FeOH^+2]	[Fe(OH)2^+1]	[Fe2(OH)2^+4]
7.0001	7.99E-11	2.510E-18	2.511E-14	1.259E-10	7.936E-25
7.0002	3.38E-11	2.508E-18	2.510E-14	1.258E-10	7.929E-25
7.0003	−1.23E-11	2.507E-18	2.508E-14	1.258E-10	7.921E-25
7.0004	−5.84E-11	2.505E-18	2.507E-14	1.258E-10	7.914E-25
7.0005	−1.04E-10	2.503E-18	2.506E-14	1.257E-10	7.907E-25

Table 2.

Zeroing the function (32) for the system with Fe(OH)₃ precipitate introduced into pure water (copy of a fragment of display).

$\text{s}=[{\text{Fe}}^{+3}]+[{\text{FeOH}}^{+2}]+[{\text{Fe(OH)}}_2^{+1}]+2[{\text{Fe}}_2{(\text{OH})}_2^{+4}]$E35

is close to s ≅ [Fe(OH)₂⁺¹] = 10^–9.9. Alkaline reaction for this system, i.e., [OH⁻¹] > [H⁺¹], results immediately from Eq. (30), and pH₀ = 7.0003 (>7).

At pH = 7, Fe(OH)₂⁺¹ (not Fe⁺³) is the predominating species in the system, [Fe(OH)₂⁺¹]/[Fe⁺³] = 10^21.7–14 = 5·10⁷, i.e., the equality/assumption s^* = [Fe⁺³] is extremely invalid. Moreover, the value [OH⁻¹] = 3·s^* = 2.94·10^–10 = 10^–9.532, i.e., pH = 4.468; this pH-value is contradictory with the inequality [OH⁻¹] > [H⁺¹] resulting from Eq. (31). Similarly, extremely invalid result was obtained in Ref. [36], where the strong hydroxo-complexes were totally omitted, and weak chloride complexes of Fe⁺³ ions were included into considerations.

Taking only the main dissociating species formed in the solution saturated with respect to Fe(OH)₃, we check whether the reaction Fe(OH)₃ = Fe(OH)₂⁺¹ + OH⁻¹ with K_sp1 = [Fe(OH)₂⁺¹][OH⁻¹] = 10^21.7·10^–38.6 = 10^–16.9 can be used for calculation of solubility $\text{s'}={\text{(}{K}_{\text{sp1}}\text{)}}^{\text{1/2}}$ for Fe(OH)₃; the answer is also negative. Simply, the main part of OH⁻¹ ions originates here from dissociation of water, where the precipitate has been introduced, and then Fe(OH)₂⁺¹ and OH⁻¹ differ significantly. As we see, the diversity in K_sp value related to a precipitate depends on its dissociation reaction notation, which disqualifies the calculation of s^* based solely on the K_sp value. This fact was not stressed in the literature issued hitherto.

Concluding, the application of the option 1, based on the stoichiometry of the reaction (29), leads not only to completely inadmissible results for s⁺, but also to a conflict with one of the fundamental rules of conservation obligatory in electrolytic systems, namely the law of charge conservation.

Similarly, critical/disqualifying remarks can be related to the series of formulas considered in the chapter [37], e.g., K_sp = 27(s^*)⁴ for precipitates of A₃B and AB₃ type, and K_sp = 108(s^*)⁵ for A₂B₃ and A₃B₂. For Ca₅(PO₄)₃OH, the formula K_sp = 84375(s^*)⁹ (!) was applied [38].

As a third example let us take a system, where an excess of Zn(OH)₂ precipitate is introduced into pure water. It is usually stated that Zn(OH)₂ dissociates according to the reaction

$Zn{(OH)}_2={\text{Zn}}^{+2}+2{\text{OH}}^{-1}$E36

applied to formulate the expression for the solubility product

$K_{\text{sp3}}=[{\text{Zn}}^{+2}]{[{\text{OH}}^{-1}]}^2(p{K}_{\text{sp3}}=15.0)$E37

The soluble hydroxo-complexes Zn(OH)_i⁺²⁻ⁱ (i=1,…,4), with the stability constants, K_i^OH, expressed by the values logK_i^OH = 4.4, 11.3, 13.14, 14.66, are also formed in the system in question. The charge balance (ChB) has the form

$2[{\text{Zn}}^{+2}]+[{\text{ZnOH}}^{+1}]–[{\text{Zn(OH)}}_3^{-1}]-2[{\text{Zn(OH)}}_4^{-2}{\text{]+[H}}^{+1}]-[{\text{OH}}^{-1}]=0$E38

i.e., 2·10⁻¹⁵/[OH⁻¹]² + 10^4.4·10⁻¹⁵/[OH⁻¹] – 10^13.14·10⁻¹⁵∙[OH⁻¹] – 2·10^14.66·10⁻¹⁵∙[OH⁻¹]² = 0

$\text{y(pH)}=2\cdot {10}^{13-2\text{pH}}+{10}^{3.4-\text{pH}}–{10}^{-15.86+pH}–2\cdot {10}^{-28.34+2\text{pH}}+{10}^{-\text{pH}}–{10}^{\text{pH}-14}=0$E39

The function (39) zeroes at pH₀ = 9.121 (see Table 3). The basic reaction of this system is not immediately stated from Eq. (38) (there are positive and negative terms in expression for [OH⁻¹] − [H⁺¹]). The solubility s value

pH	[OH^−1]	[Zn^+2]	[ZnOH^+1]	[Zn(OH)2]	[Zn(OH)3^−1]	[Zn(OH)4^−2]	y(pH)	s [mol/L]
9.118	1.3122E-05	5.8076E-06	1.9143E-06	0.0002	1.8113E-07	7.8705E-11	2.2702E-07	0.00020743
9.119	1.3152E-05	5.7810E-06	1.9099E-06	0.0002	1.8155E-07	7.9068E-11	1.3858E-07	0.00020740
9.120	1.3183E-05	5.7544E-06	1.9055E-06	0.0002	1.8197E-07	7.9433E-11	5.0322E-08	0.00020737
9.121	1.3213E-05	5.7280E-06	1.9011E-06	0.0002	1.8239E-07	7.9800E-11	−3.7750E-08	0.00020734
9.122	1.3243E-05	5.7016E-06	1.8967E-06	0.0002	1.8281E-07	8.0168E-11	−1.2564E-07	0.00020731
9.123	1.3274E-05	5.6755E-06	1.8923E-06	0.0002	1.8323E-07	8.0538E-11	−2.1335E-07	0.00020728

Table 3.

Zeroing the function (39) for the system with Zn(OH)₂ precipitate introduced into water; pK_W = 14.

$\text{s}=[{\text{Zn}}^{+2}]+[{\text{ZnOH}}^{+1}]+[{\text{Zn(OH)}}_2]+[{\text{Zn(OH)}}_3^{-1}]+[{\text{Zn(OH)}}_4^{-2}]=2.07·{10}^{-4}$E8433

calculated at this point is different from s^* = (K_so3/4)^1/3 = 6.3⋅10⁻⁶, and [OH⁻¹]/[Zn⁺²] ≠ 2; such incompatibilities contradict application of this formula.

4.3. Dissolution of MeL₂-type salts

Let us refer now to dissolution of precipitates MeL₂ formed by cations Me⁺² and anions L⁻¹ of a strong acid HL, as presented in Table 4. When an excess of MeL₂ is introduced into pure water, the concentration balances and charge balance in two-phase system thus formed are as follows:

Me^+2	MeOH^+1	Me(OH)2	Me(OH)3^−1	L^−1	MeL^+1	MeL2	MeL3^−1	MeL4^−2	MeL2
	logK1^OH	logK2^OH	logK3^OH		logK1	logK2	logK3	logK4	pKsp
Hg^+2	10.3	21.7	21.2	I^−1	12.87	23.82	27.60	29.83	28.54
Pb^+2	6.9	10.8	13.3	I^−1	1.26	2.80	3.42	3.92	8.98
				Cl^−1	1.62	2.44	2.04	1.0	4.79

Table 4.

logK_i^OH and logK_i values for the stability constants K_i and K_j of soluble complexes Me(OH)_i^+2-i and MeL_j^+2-j and pK_sp values for the precipitates MeL₂; [MeL_i^+2-i] = K_i[Me⁺²][L⁻¹]ⁱ, K_sp = [Me⁺²][L⁻¹]².

$\text{[}Me{L}_2\text{]}+{\text{[Me}}^{+\text{2}}\text{]}+{\displaystyle {\displaystyle \sum _{i=\text{1}}^I}{\text{[Me(OH)}}_i^{+\text{2}-i}\text{]}}+{\displaystyle {\displaystyle \sum _{j=\text{1}}^J}{\text{[MeL}}_j^{+\text{2}-j}\text{]}}=C_{\text{Me}}$E40

$\text{2[}Me{L}_2\text{]}+{\text{[L}}^{-1}\text{]}+{\displaystyle {\displaystyle \sum _{j=\text{1}}^J}j{\text{[MeL}}_j^{+\text{2}-j}\text{]}}=C_{\text{L}}$E41

${\text{[H}}^{+\text{1}}\text{]}-{\text{[OH}}^{-\text{1}}\text{]}+{\text{2[Me}}^{+\text{2}}\text{]}+{\displaystyle {\displaystyle \sum _{i=\text{1}}^I}\text{(2}-i{\text{)[Me(OH)}}_i^{+\text{2}-i}\text{]}}+{\displaystyle {\displaystyle \sum _{j=\text{1}}^J}\text{(2}-j{\text{)[MeL}}_j^{+\text{2}-j}\text{]}}-{\text{[L}}^{-\text{1}}\text{]}=\text{0}$E42

where [MeL₂] denotes the concentration of the precipitate MeL₂. At C_L = 2C_Me, we have

${\text{2[Me}}^{+\text{2}}\text{]}+\text{2}{\displaystyle {\displaystyle \sum _{i=\text{1}}^I}{\text{[Me(OH)}}_i^{+\text{2}-i}\text{]}}+{\displaystyle {\displaystyle \sum _{j=\text{1}}^J}\text{(2}-j{\text{)[MeL}}_j^{+\text{2}-j}\text{]}}={\text{[L}}^{-\text{1}}\text{]}$E43

From Eqs. (40) and (41)

$\alpha ={\text{[H}}^{+\text{1}}\text{]}-{\text{[OH}}^{-\text{1}}\text{]}={\displaystyle {\displaystyle \sum _{i=\text{1}}^I}i{\text{[Me(OH)}}_i^{+\text{2}-i}\text{]}}$E44

i.e., reaction of the solution is acidic, [H⁺¹] > [OH⁻¹]. Applying the relations for the equilibrium constants:

[Me⁺²][L⁻¹]² = K_sp, [Me(OH)_i⁺²⁻ⁱ] = K_i^OH[Me⁺²][OH⁻¹]ⁱ (i = 1,…, I), [MeL_j^+2−j] = K_j[Me⁺²][L⁻¹]^j (j = 1,…, J)

from Eqs. (43) and (44) we have

${\text{2[Me}}^{+\text{2}}{\text{]}}^{\text{3/2}}\cdot \text{(1}+\text{(1}+{\displaystyle {\displaystyle \sum _{i=\text{1}}^I}{x}_i}\text{)}+K_{\text{sp}}{}^{\text{1/2}}\cdot {\text{[Me}}^{+\text{2}}\text{]}\cdot {\displaystyle {\displaystyle \sum _{j=\text{1}}^J}\text{(2}-j\text{)}{K}_j{\text{[L}}^{-\text{1}}]}-K_{\text{sp}}{}^{\text{1/2}}=\text{0}$E45

where

$\begin{array}{c}[{\text{Me}}^{+2}]=\frac{\alpha }{{\displaystyle {\displaystyle \sum _{i=1}^I}i\cdot x_i}};\alpha ={\text{[H}}^{+\text{1}}\text{]}-{\text{[OH}}^{\text{-1}}\text{]}={\text{10}}^{-\text{pH}}-{\text{10}}^{\text{pH}-p{K}_{\text{W}}}{\text{;[L}}^{-\text{1}}\text{]}\hfill \\ ={\left(\frac{K_{\text{sp}}}{{\text{[Me}}^{+2}\text{]}}\right)}^{\text{1/2}};x_i=K_i^{\text{OH}}\cdot {\text{(}{K}_{\text{W}}{\text{/[H}}^{+\text{1}}\text{])}}^i\hfill \end{array}$E8000

In particular, for I = 3, J = 4 (Table 4), we have

$\text{2}\cdot \left(\text{1}+{\displaystyle {\displaystyle \sum _{i=\text{1}}^{\text{3}}}{\text{x}}_i}\right)\cdot \frac{{\text{[Me]}}^{\text{2}}}{K_{\text{sp}}{}^{\text{1/2}}}+K_{\text{1}}\cdot {\text{[Me]}}^{\text{3/2}}-(K_{\text{3}}\cdot K_{\text{sp}}+1)\cdot {\text{[Me]}}^{\text{1/2}}-\text{2}\cdot K_{\text{4}}\cdot K_{\text{sp}}{}^{\text{3/2}}=\text{0}$E46

Applying the zeroing procedure to Eq. (46) gives the pH = pH₀ of the solution at equilibrium. At this pH₀ value, we calculate the concentrations of all species and solubility of this precipitate recalculated on s_Me and s_L. When zeroing Eq. (46), we calculate pH = pH₀ of the solution in equilibrium with the related precipitate. The solubilities are as follows:

$s=s_{\text{Me}}={\text{[Me}}^{+\text{2}}\text{]}+{\displaystyle {\displaystyle \sum _{i=\text{1}}^I}{\text{[Me(OH)}}_i{}^{+\text{2}-i}\text{]}}+{\displaystyle {\displaystyle \sum _{j=\text{1}}^J}{\text{[MeL}}_j{}^{+\text{2}-j}\text{]}}$E47

$s=s_{\text{L}}={\text{[L}}^{-\text{1}}\text{]}+{\displaystyle {\displaystyle \sum _{j=\text{1}}^{\text{4}}}j\left[{\text{MeL}}_j{}^{+\text{2}-j}\right]}$E48

The calculations of s_Me and s_L for the precipitates specified in Table 4 can be realized with use of Excel spreadsheet, according to zeroing procedure, as suggested above (Table 1).

pH	[Pb^+2]	[PbOH^+1]	[Pb(OH)2]	[Pb(OH)3^-1]	[PbCl^+1]	[PbCl2]	[PbCl3^−1]	[PbCl4^−2]	[Cl^−1]	y
4.5343	0.010749606	2.92208E-05	7.94315E-11	8.59592E-18	0.017405892	0.004466836	6.90723E-05	2.44685E-07	0.038842191	0.000138249
4.5344	0.010744657	2.92141E-05	7.94315E-11	8.5979E-18	0.017401884	0.004466836	6.90882E-05	2.44798E-07	0.038851136	7.7139E-05
4.5345	0.01073971	2.92074E-05	7.94315E-11	8.59988E-18	0.017397878	0.004466836	6.91041E-05	2.44911E-07	0.038860083	1.60945E-05
4.5346	0.010734765	2.92007E-05	7.94315E-11	8.60186E-18	0.017393872	0.004466836	6.912E-05	2.45023E-07	0.038869032	-4.48848E-05
4.5347	0.010729823	2.91939E-05	7.94315E-11	8.60384E-18	0.017389867	0.004466836	6.91359E-05	2.45136E-07	0.038877983	-0.000105799

Table 5.

Fragment of display for PbCl₂.

For PbI₂: pH₀ = 5.1502, s_Pb = 6.5276∙10⁻⁴, s_I = 1.3051∙10⁻³, see Table 6. The difference between s_I and 2s_Pb = 1.3055∙10⁻³ results from rounding the pH₀-value.

pH	[Pb^+2]	[PbOH^+1]	[Pb(OH)2]	[Pb(OH)3^−1]	[PbI^+1]	[PbI2]	[PbI3^−1]	[PbI4^−2]	[I^−1]	y
5.15	0.000630817	7.07789E-06	7.94152E-11	3.54735E-17	1.47894E-05	6.60693E-07	3.54853E-09	1.44576E-11	0.001288393	0.000138249
5.1501	0.000630527	7.07626E-06	7.94152E-11	3.54816E-17	1.4786E-05	6.60693E-07	3.54935E-09	1.44643E-11	0.001288689	7.7139E-05
5.1502	0.000630236	7.07463E-06	7.94152E-11	3.54898E-17	1.47826E-05	6.60693E-07	3.55016E-09	1.44709E-11	0.001288986	1.60945E-05
5.1503	0.000629946	7.073E-06	7.94152E-11	3.5498E-17	1.47792E-05	6.60693E-07	3.55098E-09	1.44776E-11	0.001289283	-4.48848E-05
5.1504	0.000629656	7.07137E-06	7.94152E-11	3.55061E-17	1.47758E-05	6.60693E-07	3.5518E-09	1.44843E-11	0.00128958	-0.000105799

Table 6.

Fragment of display for PbI₂.

For HgI₂: pH₀ = 6.7769, s_Hg = 1.91217∙10⁻⁵, s_I = 3.82435∙10⁻⁵, see Table 7. The difference between s_I and 2s_Hg = 3.82434∙10⁻⁵ results from rounding the pH-value. The concentration [HgI₂] = K₂K_sp = 1.90546∙10⁻⁵ is close to the s_Hg value. For comparison, 4(s^*)³ = K_sp ⟹ s^* = 1.93∙10⁻¹⁰, i.e., s^*/s ≈ 10⁻⁵.

pH	[Hg^+2]	[HgOH^+1]	[Hg(OH)2]	[Hg(OH)3^−1]	[HgI^+1]	[HgI2]	[HgI3^−1]	[HgI4^−2]	[I^−1]	y
6.7767	2.99681E-15	3.57569E-12	5.37106E-08	1.01569E-15	2.17936E-09	1.90546E-05	1.12634E-08	1.87646E-13	9.81003E-08	1.35932E-10
6.7768	2.99398E-15	3.57313E-12	5.36844E-08	1.01543E-15	2.17833E-09	1.90546E-05	1.12688E-08	1.87824E-13	9.81467E-08	7.72021E-11
6.7769	2.99114E-15	3.57056E-12	5.36583E-08	1.01517E-15	2.1773E-09	1.90546E-05	1.12741E-08	1.88002E-13	9.81932E-08	1.8567E-11
6.777	2.98831E-15	3.568E-12	5.36322E-08	1.0149E-15	2.17627E-09	1.90546E-05	1.12794E-08	1.88181E-13	9.82398E-08	-3.99731E-11
6.7771	2.98548E-15	3.56544E-12	5.3606E-08	1.01464E-15	2.17524E-09	1.90546E-05	1.12848E-08	1.88359E-13	9.82863E-08	-9.84182E-11

Table 7.

Fragment of display for HgI₂.

4.4. Dissolution of CaCO₃ in the presence of CO₂

The portions 0.1 g of calcite CaCO₃ (M = 100.0869 g/mol, d = 2.711 g/cm³) are inserted into 100 mL of: pure water (task A) or aqueous solutions of CO₂ specified in the tasks: B1, B2, B3, and equilibrated. Denoting the starting (t = 0) concentrations [mol/L]: C^o for CaCO₃ and ${\text{C}}_{{\text{CO}}_{\text{2}}}$ for CO₂ in the related systems, on the basis of equilibrium data collected in Table 8:

1. (A) we calculate pH = pH₀₁ and solubility s = s(pH₀₁) of CaCO₃ at equilibrium in the system;

2. (B1) we calculate pH = pH₀₂ and solubility s = s(pH₀₂) of CaCO₃ in the system, where ${\text{C}}_{{\text{CO}}_{\text{2}}}$ refers to saturated (at 25 ^oC) solution of CO₂, where 1.45 g CO₂ dissolves in 1 L of water [39].

3. (B2) we calculate minimal ${\text{C}}_{{\text{CO}}_{\text{2}}}$ in the starting solution needed for complete dissolution of CaCO₃ in the system and the related pH = pH₀₃ value, where s = s(pH₀₃) = C^o;

4. (B3) we plot the logs_Ca versus V, pH versus V and logs_Ca versus pH relationships for the system obtained after addition of V mL of a strong base MOH (C_b = 0.1) into V₀ = 100 mL of the system with CaCO₃ presented in (B1). The quasistatic course of the titration is assumed.

No.	Reaction	Expression for the equilibrium constant	Equilibrium data
1	CaCO3 = Ca^+2 + CO3^−2	[Ca^+2][CO3^−2] = Ksp	pKsp = 8.48
2	Ca^+2 + OH^−1 = CaOH^+1	[CaOH^+1] = K10[Ca^+2][OH^−1]	logK10 = 1.3
3	H2CO3 = H^+1 + HCO3^−1	[H^+1][HCO3^−1] = K1[H2CO3]	pK1 = 6.38
4	HCO3^−1 = H^+1 + CO3^−2	[H^+1][CO3^−2] = K2[HCO3^-1]	pK2 = 10.33
5	Ca^+2 + HCO3^−1 = CaHCO3^+1	[CaHCO3^+1] = K11[Ca^+2][HCO3^−1]	logK11 = 1.11
6	Ca^+2 + CO3^−2 = CaCO3	[CaCO3] = K12[Ca^+2][CO3^−2]	logK12 = 3.22
7	Ca(OH)2 = Ca^+2 + 2OH^−1	[Ca^+2][OH^−1]^2 = Ksp1	pKsp1 = 5.03
8	H2O = H^+1 + OH^−1	[H^+1][OH^−1] = KW	pKW = 14.0

Table 8.

Equilibrium data.

The volume 0.1/2.711 = 0.037 cm³ of introduced CaCO₃ is negligible when compared with V₀ at the start (t = 0) of the dissolution. Starting concentration of CaCO₃ in the systems: A, B1, B2, B3 is C^o = (0.1/100)/0.1 = 10⁻² mol/L. At t > 0, concentration of CaCO₃ is c^o mol/L. The balances are as follows:

${\text{C}}^{\text{o}}={\text{c}}^{\text{o}}+[{\text{Ca}}^{+2}]+[{\text{CaOH}}^{+1}]+[{\text{CaHCO}}_3^{+1}]+[{\text{CaCO}}_3](\text{for A},\text{B}1,\text{B}2,\text{B}3)$E49

${\text{C}}^{\text{o}}={\text{c}}^{\text{o}}+[{\text{CaHCO}}_3^{+1}]+[{\text{CaCO}}_3]+[{\text{H}}_2{\text{CO}}_3]+[{\text{HCO}}_3^{-1}]+{\text{CO}}_3^{-2}]\text{ (for A)}$E50

$\begin{array}{c}{\text{C}}^{\text{o}}+{\text{C}}_{\text{CO}2}={\text{c}}^{\text{o}}+\left[{\text{CaHCO}}_3^{+1}\right]+\left[{\text{CaCO}}_3\right]+\left[{\text{H}}_2{\text{CO}}_3\right]+\left[{\text{HCO}}_3^{-1}\right]\\ +\left[{\text{CO}}_3^{-2}\right]\left(\text{for}\text{B}1,\text{B}2,\text{B}3\right)\end{array}$E51

$\begin{array}{c}\left[{\text{H}}^{+1}\right]-\left[{\text{OH}}^{-1}\right]+2\left[{\text{Ca}}^{+2}\right]+\left[{\text{CaOH}}^{+1}\right]+\left[{\text{CaHCO}}_3^{+1}\right]-\left[{\text{HCO}}_3^{-1}\right]\\ -2\left[{\text{CO}}_3^{-2}\right]=0\left(\text{for A},\text{B}1,\text{B}2\right)\end{array}$E52

$\begin{array}{c}\left[{\text{H}}^{+1}\right]-\left[{\text{OH}}^{-1}\right]+\left[{\text{M}}^{+1}\right]+2\left[{\text{Ca}}^{+2}\right]+\left[{\text{CaOH}}^{+1}\right]+\left[{\text{CaHCO}}_3^{+1}\right]\\ -\left[{\text{HCO}}_3^{-1}\right]-2\left[{\text{CO}}_3^{-2}\right]=0\left(\text{for B}3\right)\end{array}$E52a

where [M⁺¹] = C_bV/(V₀+V).

• For (A)

  From Eqs. (49) and (50), we have

  $\left[{\text{Ca}}^{+2}\right]+\left[{\text{CaOH}}^{+1}\right]=\left[{\text{H}}_2{\text{CO}}_3\right]+\left[{\text{HCO}}_3^{-1}\right]+\left[{\text{CO}}_3^{-2}\right]$E53

  Considering the solution saturated with respect to CaCO₃ and denoting: f₁ = 10^16.71−2pH + 10^10.33−pH + 1, f₂ = 1 + 10^pH−12.7, from Eq. (53) and Table 1, we have the relations:

  $\begin{array}{l}\left[{\text{Ca}}^{+2}\right]f_2=\left[{\text{CO}}_3^{-2}\right]\cdot f_1\Rightarrow \left[{\text{Ca}}^{+2}\right]={10}^{-4.24}\cdot {\left(f_1/f_2\right)}^{0.5};\left[{\text{CO}}_3^{-2}\right]={10}^{-4.24}\cdot {\left(f_2/f_1\right)}^{0.5};\hfill \\ \left[{\text{CaOH}}^{+1}\right]={10}^{\text{pH}-16.94}\cdot {\left(f_1/f_2\right)}^{0.5};\hfill \\ \begin{array}{l}\left[{\text{CaCO}}_3\right]={10}^{-5.26};\left[{\text{CaHCO}}_3^{+1}\right]={10}^{2.96-\text{pH}};\left[{\text{HCO}}_3^{-1}\right]={10}^{6.09-\text{pH}}\cdot {\left({\text{f}}_2/{\text{f}}_1\right)}^{0.5};\left[{\text{H}}^{+1}\right]\hfill \end{array}\hfill \\ ={10}^{-\text{pH}};\left[{\text{OH}}^{-1}\right]={10}^{\text{pH}-14}.\hfill \end{array}$E8300

  Inserting them into the charge balance (52), rewritten into the form

  $\begin{array}{c}z=z\left(\text{pH}\right)={10}^{-\text{pH}}-{10}^{\text{pH}-14}+2\cdot {10}^{-4.24}\cdot {\left(f_1/f_2\right)}^{0.5}+{10}^{\text{pH}-16.94}\cdot {\left(f_1/f_2\right)}^{0.5}\\ +{10}^{2.96-\text{pH}}–{10}^{6.09-\text{pH}}\cdot {\left(f_2/f_1\right)}^{0.5}-2\cdot {10}^{-4.24}\cdot {\left(f_2/f_1\right)}^{0.5}\end{array}$E54

  and applying the zeroing procedure to the function (54), we find pH₀₁ = 9.904, at z = z(pH₀₁) = 0. The solubility s = s(pH) of CaCO₃, resulting from Eq. (49), is

  $\text{s}=\left[{\text{Ca}}^{+2}\right]+\left[{\text{CaOH}}^{+1}\right]+\left[{\text{CaHCO}}_3^{+1}\right]+\left[{\text{CaCO}}_3\right]$E55
  $={10}^{-4.24}\cdot {\left(f_1/f_2\right)}^{0.5}+{10}^{\text{pH}-16.94}\cdot {\left(f_1/f_2\right)}^{0.5}+{10}^{2.96-\text{pH}}+{10}^{-5.26}$E55a

  We have s = s(pH = pH₀₁) = 1.159⋅10⁻⁴ mol/L.

• For (B1)

  Subtraction of Eq. (49) from Eq. (51) gives

  $\begin{array}{c}\left[{\text{H}}_2{\text{CO}}_3\right]+\left[{\text{HCO}}_3^{-1}\right]+\left[{\text{CO}}_3^{-2}\right]-\left(\right[{\text{Ca}}^{+2}]+{[\text{CaOH}}^{+1}\left]\right)={\text{C}}_{\text{CO}2}\hfill \\ \Rightarrow \left[{\text{CO}}_3^{-2}\right]\cdot f_1–\left[C{a}^{+2}\right]\cdot f_2-{\text{C}}_{\text{CO}2}=0\Rightarrow {\left[{\text{Ca}}^{+2}\right]}^2\cdot f_2+{\text{ C}}_{\text{CO}2}\cdot \left[{\text{Ca}}^{+2}\right]–K_{\text{sp}}\cdot f_1=0\hfill \end{array}$E8400

  In this case,

  ${\text{[Ca}}^{+\text{2}}\text{]}=\frac{\sqrt{{{\text{(C}}_{{\text{CO}}_{\text{2}}})}^{\text{2}}+\text{4}\cdot K_{\text{sp}}\cdot f_{\text{1}}\cdot f_{\text{2}}}-{\text{C}}_{{\text{CO}}_{\text{2}}}}{\text{2}\cdot f_{\text{2}}}$E56

  where ${\text{C}}_{{\text{CO}}_{\text{2}}}$ = 1.45/44 = 0.0329 mol/L. Eq. (55) has the form

  $\text{s}=\left[{\text{Ca}}^{+2}\right]\cdot f_2+{10}^{2.96-\text{pH}}+{10}^{-5.26}$E57

  and the charge balance is transformed into the zeroing function

  $\begin{array}{c}z=z\left(\text{pH}\right)={10}^{-\text{pH}}-{10}^{\text{pH}-14}+\left[{\text{Ca}}^{+2}\right]\cdot (2+{10}^{\text{pH}-12.7})+{10}^{2.96-\text{pH}}\\ -\left[{\text{CO}}_3^{-2}\right]\cdot \left({10}^{10.33-\text{pH}}+2\right)\end{array}$E58

  where [CO₃⁻²] = 10^-8.48/[Ca⁺²], and [Ca⁺²] is given by Eq. (56). Eq. (58) zeroes at pH = pH₀₂ = 6.031. Then from Eq. (57) we calculate s = s(pH₀₂) = 6.393∙10⁻³ mol/L, at pH = pH₀₂ = 6.031.

• For (B2)

  At pH = pH₀₃, where c^o = 0, i.e., s = C^o, the solution (a monophase system) is saturated toward CaCO₃, i.e., the relation [Ca⁺²][CO₃⁻²] = K_sp is still valid. Applying Eqs. (56) and (57), we find pH values zeroing Eq. (58) at different, preassumed ${\text{C}}_{{\text{CO}}_{\text{2}}}$ values. Applying these pH-values in Eq. (57), we calculate the related s = s(pH, ${\text{C}}_{{\text{CO}}_{\text{2}}}$) values (Eq. (57), Table 9). Graphically, ${\text{C}}_{{\text{CO}}_{\text{2}}}$ = 0.100 is found at pH₀₃ = 5.683, as the abscissa of the point of intersection of the lines: s = s(pH) and s = C^o = 0.01. Table 9 shows other, preassumed s = C^o values.

• For (B3)

  We apply again the formulas used in (B1) and (B2), and the charge balance (Eq. (52a)), which is transformed there into the function

  $\begin{array}{c}z=z\left(\text{pH},\text{V}\right)={10}^{-\text{pH}}-{10}^{\text{pH}-14}+{\text{C}}_{\text{b}}V/(V_0+V)+\left[{\text{Ca}}^{+2}\right]\cdot (2+{10}^{\text{pH}-12.7})\\ +{10}^{2.96-\text{pH}}-\left[{\text{CO}}_3^{-2}\right]\cdot \left({10}^{10.33-\text{pH}}+2\right)\end{array}$E59

  applied for zeroing purposes, at different V values. The data thus obtained are presented graphically in Figures 2a–c. The data presented in the dynamic solubility diagram (Figure 2b), illustrating the solubility changes affected by pH changes (Figure 2a) resulting from addition of a base, MOH; Figure 2c shows a synthesis of these changes. Solubility product of Ca(OH)₂ is not crossed in this system.

${\text{C}}_{{\text{CO}}_{\text{2}}}$	0.090	0.091	0.092	0.093	0.094	0.095	0.096	0.097	0.098	0.099	0.100	0.101	0.102
pH	5.716	5.712	5.709	5.706	5.702	5.699	5.696	5.693	5.690	5.687	5.683	5.680	5.577
s	9.58E-3	9.64E-3	9.67E-3	9.70E-3	9.77E-3	9.80E-3	9.84E-3	9.87E-3	9.91E-3	9.94E-3	10.01E-3	10.06E-3	10.10E-3

Table 9.

The set of points used for searching the ${\text{C}}_{{\text{CO}}_{\text{2}}}$ value at s = C^o = 0.01; at this point, we have pH₀₃ = 5.683.

Figure 2.

Graphical presentation of the data considered in (b3): (a) pH versus V, (b) log s_Ca versus V, (c) log s_Ca versus pH relationships.

5. Nonequilibrium solid phases in aqueous media

Some solids when introduced into aqueous media (e.g., pure water) may appear to be nonequilibrium phases in these media.

5.1. Silver dichromate (Ag₂Cr₂O₇)

The equilibrium data related to the system, where Ag₂Cr₂O₇ is introduced into pure water, were taken from Refs. [33, 40, 41], and presented in Table 10. A large discrepancy between pK_sp2 values (6.7 and 10) in the cited literature is taken here into account. We prove that Ag₂Cr₂O₇ changes into Ag₂CrO₄.

Reaction	Equilibrium data
H2O = H^+1 + OH^-1	pKw = 14.0
${\text{H}}_2{\text{CrO}}_4={\text{H}}^{+}+{\text{HCrO}}_4^{-1}$	pK1 = 0.8
${\text{HCrO}}_4^{-1}{\text{= H}}^{\text{+}}+{\text{CrO}}_4^{-2}$	pK2 = 6.5
${\text{HCr}}_2{\text{O}}_7^{-1}={\text{H}}^{+1}+{\text{Cr}}_2{\text{O}}_7^{-2}$	logK3 = 0.07
$2{\text{HCrO}}_4^{-1}={\text{Cr}}_2{\text{O}}_7^{-2}+{\text{H}}_2\text{O}$	logK4 = 1.52
Ag^+1 + OH^−1 = AgOH	logK1^OH = 2.3
${\text{Ag}}^{+1}+2{\text{OH}}^{-1}=\text{Ag}{(\text{OH})}_2^{-1}$	logK2^OH = 3.6
${\text{Ag}}^{+1}+3{\text{OH}}^{-1}=\text{Ag}{(\text{OH})}_3^{-2}$	logK3^OH = 4.8
$A{g}_{2}Cr{O}_4=2{\text{Ag}}^{+1}+{\text{CrO}}_4^{-2}$	pKsp1 = 11.9
$A{g}_{2}C{r}_2{O}_7=2{\text{Ag}}^{+1}+{\text{Cr}}_2{\text{O}}_7^{-2}$	pKsp2 = 6.7
$AgOH={\text{Ag}}^{\text{+1}}+{\text{OH}}^{-1}$	pKsp3 = 7.84

Table 10.

Physicochemical equilibrium data relevant to the Ag₂Cr₂O₇ + H₂O system (pK = −logK), at “room” temperatures.

On the dissociation step, each dissolving molecule of Ag₂Cr₂O₇ gives two ions Ag⁺¹ and 1 ion Cr₂O₇⁻², where two atoms of Cr are involved; in the contact with water, these ions are hydrolyzed, to varying degrees. In the initial step of the dissolution, before the saturation of the solution with respect to an equilibrium solid phase (not specified at this moment), we can write the concentration balances

$2\left[A{g}_{2}C{r}_2{O}_7\right]+\left[{\text{Ag}}^{\text{+1}}\right]\text{+}\left[\text{AgOH}\right]+\left[{\text{Ag(OH)}}_2^{-1}\right]+\left[{\text{Ag(OH)}}_3^{-2}\right]=2{\text{C}}_0$E60

$2\left[A{g}_{2}C{r}_2{O}_7\right]+\left[{\text{H}}_2{\text{CrO}}_4\right]\text{+}\left[{\text{HCrO}}_4^{-1}\right]+\left[{\text{CrO}}_4^{-2}\right]+2\left[{\text{HCr}}_2{\text{O}}_7^{-1}\right]+2\left[{\text{Cr}}_2{\text{O}}_7^{-2}\right]=2{\text{C}}_0$E61

where 2C₀ is the total concentration of the solid phase in the system, at the moment (t = 0) of introducing this phase into water, [Ag₂Cr₂O₇] is the concentration of this phase at a given moment of the intermediary step. As previously, we assume that addition of the solid phase (here: Ag₂Cr₂O₇) does not change the volume of the system in a significant degree, and that Ag₂Cr₂O₇ is added in a due excess, securing the formation of a solid (that is not specified at this moment), as an equilibrium solid phase. The balances in Eqs. (60) and (61) are completed by the charge balance

$\begin{array}{c}\hfill \left[{\text{H}}^{+1}\right]-\left[{\text{OH}}^{-1}\right]+\left[{\text{Ag}}^{+1}\right]-\left[\text{Ag}{\left(\text{OH}\right)}_2^{-1}\right]-2\left[\text{Ag}{\left(\text{OH}\right)}_3^{-2}\right]-\left[{\text{HCrO}}_4^{-1}\right]\\ \hfill -2\left[{\text{CrO}}_4^{-2}\right]-\left[{\text{HCr}}_2{\text{O}}_7^{-1}\right]-2\left[{\text{Cr}}_2{\text{O}}_7^{-2}\right]=0\end{array}$E62

used, as previously, to formulation of the zeroing function, y = y(pH), and the set of relations for equilibrium data specified in Table 10. From these relations, we get

$\begin{array}{c}\hfill {\text{[H}}_{\text{2}}{\text{CrO}}_{\text{4}}\text{]}={\text{10}}^{\text{7}\text{.3}-\text{2pH}}\cdot {\text{[CrO}}_{\text{4}}{}^{-2}\text{];}{\text{[HCrO}}_{\text{4}}{}^{-1}\text{]}={\text{10}}^{\text{6}\text{.5}-\text{pH}}\cdot {\text{[CrO}}_{\text{4}}{}^{-\text{2}}\text{]};\\ \hfill {\text{[HCr}}_{\text{2}}{\text{O}}_{\text{7}}{}^{-1}\text{]}={\text{10}}^{\text{14}\text{.59}-\text{3pH}}\cdot {{\text{[CrO}}_{\text{4}}{}^{-\text{2}}\text{]}}^{\text{2}};\end{array}$E63

${\text{[Cr}}_{\text{2}}{\text{O}}_{\text{7}}{}^{-\text{2}}\text{]}={\text{10}}^{\text{14}\text{.52}-\text{2pH}}\cdot {{\text{[CrO}}_{\text{4}}{}^{-\text{2}}\text{]}}^{\text{2}}$E63a

Denoting by 2c₀ (< 2C₀) the total concentration of dissolved Ag and Cr species formed, in a transition stage, from Ag₂Cr₂O₇, we can write

$\left[{\text{Ag}}^{+1}\right]+\left[\text{AgOH}\right]+\left[\text{Ag}{\left(\text{OH}\right)}_2^{-1}\right]+\left[\text{Ag}{\left(\text{OH}\right)}_3^{-2}\right]=2{\text{c}}_0$E64

$\left[{\text{H}}_2{\text{CrO}}_4\right]+\left[{\text{HCrO}}_4^{-1}\right]+\left[{\text{CrO}}_4^{-2}\right]+2\left[{\text{HCr}}_2{\text{O}}_7^{-1}\right]+2\left[{\text{Cr}}_2{\text{O}}_7^{-2}\right]=2{\text{c}}_0$E65

From Table 10 and formulas (63)–(65) we get the relations:

$(\text{a})\left[{\text{Ag}}^{+1}\right]=2{\text{c}}_0/g_0;2g_2{\left[{\text{CrO}}_4^{-2}\right]}^2+g_1\left[{\text{CrO}}_4^{-1}\right]–2{\text{c}}_0=0\Rightarrow (b)\left[{\text{CrO}}_4^{-2}\right]=\frac{{\text{(}{g}_1^2+\text{16}\cdot {\text{c}}_{\text{0}}{g}_{\text{2}})}^{\text{0}\text{.5}}-g_{\text{1}}}{\text{4}\cdot g_{\text{2}}}$E66

where g₀ = 1 + 10^pH−11.7 + 10^2pH−24.4 + 10^3pH−37.2; g₁ = 10^7.3−2pH + 10^6.5−pH + 1; g₂ = 10^14.59−3pH + 10^14.52−2pH. Applying them in Eq. (62), we get the zeroing function

$\text{y}=\text{y}\left(\text{pH}\right)={10}^{-\text{pH}}–{10}^{\text{pH}}{}^{-14}+g_3\cdot \left[{\text{Ag}}^{+1}\right]–g_4\cdot \left[{\text{CrO}}_4^{-2}\right]-g_5{\cdot \left[{\text{CrO}}_4^{-2}\right]}^2$E67

where g₃ = 1 – 10^2pH−24.4 – 2∙10^3pH−37.2; g₄ = 10^6.5−pH + 2; g₅ = 10^14.59−3pH + 2∙10^14.52−2pH, and [Ag⁺¹] and [CrO₄⁻²] are defied above, as functions of pH.

The calculation procedure, realizable with use of Excel spreadsheet, is as follows. We assume a sequence of growing numerical values for 2c₀. At particular 2c₀ values, we calculate pH = pH(2c₀) value zeroing the function (67), and then calculate the values of the products: q₁ = [Ag⁺¹]²[CrO₄⁻²]/K_sp1 and q₂ = [Ag⁺¹]²[Cr₂O₇⁻²]/K_sp2, where: [Ag⁺¹], [CrO₄⁻²], and [Cr₂O₇⁻²] are presented above (Eqs. (66a), (66b) and (63a), resp.), pK_sp1 = 11.9, pK_sp2 = 6.7. As results from Figure 3, where logq₁ and logq₂ are plotted as functions of 2c₀; logq₁ = 0 ⇔ q₁ = 1 ⇔ [Ag⁺¹]²[CrO₄⁻²] = K_sp1 at lower 2c₀ value, whereas logq₂ < 0 ⇔ q₂ < 1 ⇔ [Ag⁺¹]²[Cr₂O₇⁻²] < K_sp2, both for pK₂ = 6.7 and 10, cited in the literature. The x₁=1 value is attained at 2c₀ = 3.5∙10⁻⁴ ⟹ c₀ = 1.75∙10⁻⁴; then Ag₂CrO₄ precipitates as the new solid phase, i.e., total depletion of Ag₂Cr₂O₇ occurs. It means that Ag₂Cr₂O₇ is not the equilibrium solid phase in this system. This fact was confirmed experimentally, as stated in [42], i.e., Ag₂Cr₂O₇ is transformed into Ag₂CrO₄ upon boiling with H₂O; at higher temperatures, this transformation proceeds more effectively. Concluding, the formula s^* = (K_sp2/4)^1/3 applied for K_sp2 = [Ag⁺¹]²[Cr₂O₇⁻²] is not “the best answer,” as stated in Ref. [43].

Figure 3.

The convergence of logq₁ and logq₂ to 0 value; K_sp1 is attained at lower 2c₀ value.

The system involved with Ag₂CrO₄ was also considered in context with the Mohr’s method of Cl⁻¹ determination [44–46]. As were stated there, the systematic error in Cl⁻¹ determining according to this method, expressed by the difference between the equivalence (eq) volume (V_eq = C₀V₀/C) and the volume V_end corresponding to the end point where the K_sp1 for Ag₂CrO₄ is crossed, equals to

$V_{\text{eq}}-V_{\text{end}}=\frac{K_{\text{sp}}}{\text{C}}\cdot {\left(\frac{{\text{C}}_{\text{01}}{\text{V}}_{\text{0}}}{K_{\text{sp1}}}\right)}^{\text{0}\text{.5}}\cdot {\text{(}{V}_{\text{0}}+V_{\text{end}}\text{)}}^{\text{0}\text{.5}}-\frac{\text{1}}{\text{C}}\cdot {\left(\frac{K_{\text{sp1}}}{{\text{C}}_{\text{01}}{\text{V}}_{\text{0}}}\right)}^{\text{0}\text{.5}}\cdot {\text{(}{V}_{\text{0}}+V_{\text{end}}\text{)}}^{\text{1}\text{.5}}$E8435

where K_sp = [Ag⁺¹][Cl⁻¹] (pK_sp = 9.75), V₀ is the volume of titrant with NaCl (C₀) + K₂CrO₄ (C₀₁) titrated with AgNO₃ (C) solution; V_end = V_eq at C₀₁ = (1 + V_end/V₀)∙K_sp1/K_sp.

All calculations presented above were realized using Excel spreadsheets. For more complex nonequilibrium two-phase systems, the use of iterative computer programs, e.g., ones offered by MATLAB [8, 47], is required. This way, the quasistatic course of the relevant processes under isothermal conditions can be tested [48].

5.2. Dissolution of struvite

The fact that NH₃ evolves from the system obtained after leaving pure struvite pr1 in contact with pure water, e.g., on the stage of washing this precipitate, has already been known at the end of nineteenth century [49]. It was noted that the system obtained after mixing magnesium, ammonium, and phosphate salts at the molar ratio 1:1:1 gives a system containing an excess of ammonium species remaining in the solution and the precipitate that “was not struvite, but was probably composed of magnesium phosphates” [50]. This effect can be explained by the reaction [20]

$3MgN{H}_{4}P{O}_4=M{g}_3{\left(P{O}_4\right)}_2+{\text{HPO}}_4^{-2}+{\text{NH}}_3+2{\text{NH}}_4^{+1}$E68

Such inferences were formulated on the basis of X-ray diffraction analysis, the crystallographic structure of the solid phase thus obtained. It was also stated that the precipitation of struvite requires a significant excess of ammonium species, e.g., Mg:N:P = 1:1.6:1. Struvite (pr1) is the equilibrium solid phase only at a due excess of one or two of the precipitating reagents. This remark is important in context with gravimetric analysis of magnesium as pyrophosphate. Nonetheless, also in recent times, the solubility of struvite is calculated from the approximate formula s^* = (K_sp1)^1/3 based on an assumption that it is the equilibrium solid phase in such a system.

Struvite is not the equilibrium solid phase also when introduced into aqueous solution of CO₂ (${\text{C}}_{{\text{CO}}_{\text{2}}}$, mol/L), modified (or not) by free strong acid HB (C_a, mol/L) or strong base MOH (C_b, mol/L).

The case of struvite requires more detailed comments. The reaction (68) was proved theoretically [20], on the basis of simulated calculations performed by iterative computer programs, with use of all attainable physicochemical knowledge about the system in question. For this purpose, the fractions

$\begin{array}{cc}{q}_1& =\left[{\text{Mg}}^{+2}\right]\left[{\text{NH}}_4^{+1}\right]\left[{\text{PO}}_4^{-3}\right]/K_{\text{sp}1},q_2={\left[{\text{Mg}}^{+2}\right]}^3{\left[{\text{PO}}_4^{-3}\right]}^2/K_{\text{sp}2},q_3\\ & =\left[{\text{Mg}}^{+2}\right]\left[{\text{HPO}}_4^{-2}\right]/K_{\text{sp}3},q_4\left[{\text{Mg}}^{+2}\right]{\left[{\text{OH}}^{-1}\right]}^2/K_{\text{sp}4}\end{array}$E69

were calculated for: pr1 = MgNH₄PO₄ (pK_sp1 = 12.6), pr2 = Mg₃(PO₄)₂ (pK_sp2 = 24.38), pr3 = MgHPO₄ (pK_sp3 = 5.5), pr4 = Mg(OH)₂ (pK_sp4 = 10.74) and are presented in Figure 4, at an initial concentration of pr1, equal C⁰ = [pr1]_t=0 = 10⁻³ mol/L (pC⁰ = (ppr1)_t=0 = 3); ppr1 = −log[pr1]. As we see, the precipitation of pr2 (Eq. (68)) starts at ppr1 = 3.088; other solubility products are not crossed. The changes in concentrations of some species, resulting from dissolution of pr1, are indicated in Figure 5, where s is defined by equation [20]

Figure 4.

Plots of logq_i versus ppr1 = −log[pr1] relationships, at (ppr1)_t=0 = 3; i = 1,2,3,4 refer to pr1, pr2, pr3 and pr4, respectively.

Figure 5.

The speciation curves for indicated species resulting from dissolution of pr1 at (ppr1)_t=0 = 3.

$\begin{array}{c}\hfill \text{s}={\text{s}}_{\text{Mg}}=\left[{\text{Mg}}^{+2}\right]+\left[{\text{MgOH}}^{+1}\right]+\left[{\text{MgH}}_2{\text{PO}}_4^{+1}\right]+\left[\text{M}\text{g}\text{H}\text{P}{\text{O}}_4\right]+\left[{\text{MgPO}}_4^{-1}\right]\hfill \\ \hfill +\left[{\text{MgNH}}_3^{+2}\right]+\left[\text{Mg}{\left({\text{NH}}_3\right)}_2^{+2}\right]+\left[\text{Mg}{\left({\text{NH}}_3\right)}_3^{+2}\right]\hfill \end{array}$E70

involving all soluble magnesium species are identical in its form, irrespective of the equilibrium solid phase(s) present in this system. Moreover, it is stated that pH in the solution equals ca. 9–9.5 (Figure 6); this pH can be affected by the presence of CO₂ from air. Under such conditions, NH₄⁺¹ and NH₃ occur there at comparable concentrations [NH₄⁺¹] ≈ [NH₃], but [HPO₄⁻²]/[PO₄⁻³] = 10^12.36−pH ≈ 10³. This way, the scheme (10) would be more advantageous, provided that struvite is the equilibrium solid phase; but it is not the case, see Eq. (68). The reaction (68) occurs also in the presence of CO₂ in water where struvite was introduced.

Figure 6.

The pH versus log[pr2] relationship; pr2 = Mg₃(PO₄)₂, at [ppr1]_t=0 = 3. The numbers at the corresponding lines indicate ${\text{pCO}}_{\text{2}}=-{\text{logC}}_{{\text{CO}}_{\text{2}}}$ values; ${\text{pCO}}_{\text{2}}=\infty$ ⇔ ${\text{C}}_{{\text{CO}}_{\text{2}}}$= 0.

After introducing struvite pr1 (at pC₀ = [ppr1]_t=0 = 2) into alkaline (C_b = 10⁻² mol/L KOH, pC_b = 2) solution of CO₂ (pCO₂ = 4), the dissolution is more complicated and proceeds in three steps, see Figure 7.

Figure 7.

The speciation curves for indicated species ${\text{X}}_{\text{i}}{}^{{\text{z}}_{\text{i}}}$, resulting from dissolution of pr1 = MgNH₄PO₄, at (pC₀, pCO₂, pC_b) = (2, 4, 2); s′ is defined by Eq. (71).

In step 1, pr4 precipitates first, pr1 + 2OH⁻¹ = pr4 + NH₃ + HPO₄⁻², nearly from the very start of pr1 dissolution, up to ppr1 = 2.151, where K_sp2 is attained. Within step 2, the solution is saturated toward pr2 and pr4. In this step, the reaction expressed by the notation 2pr1 + pr4 = pr2 + 2NH₃ + 2H₂O occurs up to total depletion of pr4 (at ppr1 = 2.896). In this step, the reaction 3pr1 + 2OH⁻¹ = pr2 + 3NH₃ + HPO₄⁻¹ + 2H₂O occurs up to total depletion of pr1, i.e., the solubility product K_sp1 for pr1 is not crossed. The curve s′ (Figure 7) is related to the function

${\text{s}}^{\prime }=\text{s}+[{\text{MgHCO}}_3^{+1}]+[{\text{MgCO}}_3]$E71

where s is expressed by Eq. (70).

6. Solubility of nickel dimethylglyoximate

The precipitate of nickel dimethylglyoximate, NiL₂, has soluble counterpart with the same formula, i.e., NiL₂, in aqueous media. If NiL₂ is in equilibrium with the solution, concentration of the soluble complex NiL₂ assumes constant value: [NiL₂] = K₂∙[Ni²⁺][L⁻]² = K₂∙K_sp, where K₂ = 10^17.24, K_sp = [Ni²⁺][L⁻]² = 10^−23.66 [14, 17, 18], and then [NiL₂] = 10^−6.42 (i.e., log[NiL₂] = −6.42). The concentration [NiL₂] is the constant, limiting component in expression for solubility s = s_Ni of nickel dimethylglyoximate, NiL₂. Moreover, it is a predominant component in expression for s in alkaline media, see Figure 8. This pH range involves pH of ammonia buffer solutions, where NiL₂ is precipitated from NiSO₄ solution during the gravimetric analysis of nickel; the expression for solubility

Figure 8.

Solubility curves for nickel dimethylglyoximate NiL₂ in (a) ammonia, (b) acetate+ammonia, and (c) citrate+acetate+ ammonia media at total concentrations [mol/L]: C_Ni = 0.001, C_L = 0.003, C_N = 0.5, C_Ac = 0.3, C_Cit = 0.1 [14].

$\text{s}={\text{s}}_{\text{Ni}}={\text{[Ni}}^{+\text{2}}\text{]}+{\text{[NiOH}}^{+1}\text{]}+{\text{[NiSO}}_{\text{4}}\text{]}+{\displaystyle {\displaystyle \sum _{\text{i}=\text{1}}^{\text{6}}}{\text{[Ni(NH}}_{\text{3}}{\text{)}}_{\text{i}}{}^{+2}\text{]}+{\text{[NiL}}_{\text{2}}\text{]}}$E72

The effect of other, e.g., citrate (Cit) and acetate (Ac) species as complexing agents can also be considered for calculation purposes, see the lines b and c in Figure 8. The presence of citrate does not affect significantly the solubility of NiL₂ in ammonia buffer media, i.e., at pH ≈ 9, where s_Ni ≅ [NiL₂].

Calculations of s = s_Ni were made at C_Ni = 0.001 mol/L and C_L = 0.003 mol/L HL, i.e., at the excessive HL concentration equal C_L – 2C_Ni = 0.001 mol/L. Solubility of HL in water, equal 0.063 g HL/100 mL H₂O (25^oC) [51], corresponds to concentration 0.63/116.12 = 0.0054 mol/L of the saturated HL solution, 0.003 < 0.0054. Applying higher C_L values needs the HL solution in ethanol, where HL is fairly soluble. However, the aqueous-ethanolic medium is thus formed, where equilibrium constants are unknown. To avoid it, lower C_Ni and C_L values were applied in calculations. The equilibrium data were taken from Ref. [31].

The soluble complex having the formula identical to the formula of the precipitate occurs also in other, two-phase systems. In some pH range, concentration of this soluble form is the dominant component of the expression for the solubility s. As stated above, such a case occurs for NiL₂. Then one can assume the approximation

$\text{s}=K_2{K}_{\text{sp}}$E73

Similar relationship exists also for other precipitates. By differentiation of Eq. (73) with respect to temperature T at p = const, and application of van’t Hoff’s isobar equation for K₂ and K_sp, we obtain

$\frac{\text{1}}{\text{s}}\cdot {\left(\frac{\partial s}{\partial T}\right)}_p=\frac{1}{R{T}^2}\cdot \left(\text{Δ}{G}_1^o+\text{Δ}{G}_2^o\right)$E74

where

$\begin{array}{ccc}\text{Δ}{G}_1^o=R{T}^2\cdot {\left(\frac{\partial \ln K_{sp}}{\partial T}\right)}_p& \text{and}& \text{Δ}{G}_2^o=R{T}^2\cdot {\left(\frac{\partial \ln K_2}{\partial T}\right)}_p\end{array}$E8436

Because, as a rule,

$\begin{array}{ccc}{\left(\frac{\partial K_{sp}}{\partial T}\right)}_p>0& \text{and}& {\left(\frac{\partial K_2}{\partial T}\right)}_p<0\end{array}$E8437

then $\text{Δ}{G}_{\text{1}}^{\text{o}}>\text{0}$ and $\text{Δ}{G}_{\text{2}}^{\text{o}}<\text{0}$, and Eq. (74) can be rewritten into the form

$\frac{\text{1}}{\text{s}}\cdot {\left(\frac{\partial s}{\partial T}\right)}_p=\frac{1}{R{T}^2}\cdot (|\text{Δ}{G}_1^o|-|\text{Δ}{G}_2^o|)$E75

If $|\text{Δ}{G}_{\text{1}}^{\text{o}}|\approx |\text{Δ}{G}_{\text{2}}^{\text{o}}|$ within the temperature range (T₀, T), the value of s is approximately constant. Let T₀ denote the room temperature (at which,as a rule—all the equilibrium constants are determined) and T ≠ T₀ is the temperature at which the precipitate is filtered and washed. In this case, the solubility s and then theoretical accuracy of gravimetric analysis does not change with temperature.

7. Calculation of solubility in dynamic redox systems

7.1. Preliminary information

The redox system presented in this section is resolvable according to generalized approach to redox systems (GATES), formulated by Michałowski (1992) [8]. According to GATES principles, the algebraic balancing of any electrolytic system is based on the rules of conservation of particular elements/cores Y_g (g = 1,…, G), and on charge balance (ChB), expressing the rule of electroneutrality of this system; the terms element and core are then distinguished. The core is a cluster of elements with defined composition (expressed by its chemical formula) and external charge that remains unchanged during the chemical process considered, e.g., titration. For ordering purposes, we assume: Y₁ = H, Y₂ = O,…. For modeling purposes, the closed systems, composed of condensed phases separated from its environment by diathermal (freely permeable by heat) walls, are considered; it enables the heat exchange between the system and its environment. Any chemical process, such as titration, is carried out under isothermal conditions, in a quasistatic manner; constant temperature is one of the conditions securing constancy of equilibrium constants values. An exchange of the matter (H₂O, CO₂, O₂,…) between the system and its environment is thus forbidden, for modeling purposes. The elemental/core balance F(Y_g) for the g-th element/core (Y_g) (g = 1,…, G) is expressed by an equation interrelating the numbers of Y_g-atoms or cores in components of the system with the numbers of Y_g-atoms/cores in the species of the system thus formed; we have F(H) for Y₁ = H, F(O) for Y₂ = O, etc.

The key role in redox systems is due to generalized electron balance (GEB) concept, discovered by Michałowski as the Approach I (1992) and Approach II (2006) to GEB; both approaches are equivalent:

$\text{Therefore},\text{Approach}\text{II}\text{to GEB}\leftrightarrow \text{Approach}\text{I}\text{to}\text{GEB}$E76

GEB is fully compatible with charge balance (ChB) and concentration balances F(Y_g), formulated for different elements and cores. The primary form of GEB, pr-GEB, obtained according to Approach II to GEB is the linear combination

$\text{pr-GEB}=\text{2}\cdot F(\text{O})–F(\text{H})$E77

Both approaches (I and II) to GEB were widely discussed in the literature [7–12, 14, 15, 17, 18, 34, 52–74], and in three other chapters in textbooks [75–79] issued in 2017 within InTech. The GEB is perceived as a law of nature [9, 10, 17, 67, 71, 73, 74], as the hidden connection of physicochemical laws, as a breakthrough in the theory of electrolytic redox systems. The GATES refers to mono- and polyphase, redox, and nonredox, equilibrium and metastable [20, 21–23, 78, 79] static and dynamic systems, in aqueous, nonaqueous, and mixed-solvent media [69, 72], and in liquid-liquid extraction systems [53]. Summarizing, Approach II to GEB needs none prior information on oxidation numbers of all elements in components forming a redox system and in the species in the system thus formed. The Approach I to GEB, considered as the “short” version of GEB, is useful if all the oxidation numbers are known beforehand; such a case is obligatory in the system considered below. The terms “oxidant” and “reductant” are not used within both approaches. In redox systems, 2∙F(O) – F(H) is linearly independent on CHB and F(Y_g) (g ≥ 3,…, G); in nonredox systems, 2∙F(O) – F(H) is dependent on those balances. This property distinguishes redox and nonredox systems of any degree of complexity. Within GATES, and GATES/GEB in particular, the terms: “stoichiometry,” “oxidation number,” “oxidant,” “reductant,” “equivalent mass” are considered as redundant, old-fashioned terms. The term “mass action law” (MAL) was also replaced by the equilibrium law (EL), fully compatible with the GATES principles. Within GATES, the law of charge conservation and law of conservation of all elements of the system tested have adequate importance/significance.

A detailed consideration of complex electrolytic systems requires a collection and an arrangement of qualitative (particular species) and quantitative data; the latter ones are expressed by interrelations between concentrations of the species. The interrelations consist of material balances and a complete set of expressions for equilibrium constants. Our further considerations will be referred to a titration, as a most common example of dynamic systems. The redox and nonredox systems, of any degree of complexity, can be resolved in analogous manner, without any simplifications done, with the possibility to apply all (prior, preselected) physicochemical knowledge involved in equilibrium constants related to a system in question. This way, one can simulate (imitate) the analytical prescription to any process that may be realized under isothermal conditions, in mono- and two-phase systems, with liquid-liquid extraction systems included.

7.2. Solubility of CuI in a dynamic redox system

The system considered in this section is related to iodometric, indirect analysis of an acidified (H₂SO₄) solution of CuSO₄ [14, 64]. It is a very interesting system, both from analytical and physicochemical viewpoints. Because the standard potential E₀ = 0.621 V for (I₂, I⁻¹) exceeds E₀ = 0.153 V for (Cu⁺², Cu⁺¹), one could expect (at a first sight) the oxidation of Cu⁺¹ by I₂. However, such a reaction does not occur, due to the formation of sparingly soluble CuI precipitate (pK_sp = 11.96).

This method consists of four steps. In the preparatory step (step 1), an excess of H₂SO₄ is neutralized with NH₃ (step 1) until a blue color appears, which is derived from Cu(NH₃)_i⁺² complexes. Then the excess of CH₃COOH is added (step 2), to attain a pH ca. 3.6. After subsequent introduction of an excess of KI solution (step 3), the mixture with CuI precipitate and dissolved iodine formed in the reactions: 2Cu⁺² + 4I⁻¹ = 2CuI + I₂, 2Cu⁺² + 5I⁻¹ = 2CuI + I₃⁻¹ is titrated with Na₂S₂O₃ solution (step 4), until the reduction of iodine: I₂ + 2S₂O₃⁻² = 2I⁻¹ + S₄O₆⁻², I₃⁻¹ + 2S₂O₃⁻² = 3I⁻¹ + S₄O₆⁻² is completed; the reactions proceed quantitatively in mildly acidic solutions (acetate buffer), where the thiosulfate species are in a metastable state. In strongly acidic media, thiosulfuric acid disproportionates according to the scheme H₂S₂O₃ = H₂SO₃ + S [80].

7.3. Formulation of the system

We assume that V mL of C mol/L Na₂S₂O₃ solution is added into the mixture obtained after successive addition of: V_N mL of NH₃ (C₁) (step 1), V_Ac mL of CH₃COOH (C₂) (step 2), V_KI mL of KI (C₃) (step 3), and V mL of Na₂S₂O₃ (C) (step 4) into V₀ mL of titrand D composed of CuSO₄ (C₀) + H₂SO₄ (C₀₁). To follow the changes occurring in particular steps of this analysis, we assume that the corresponding reagents in particular steps are added according to the titrimetric mode, and the assumption of the volumes additivity is valid.

In this system, three electron-active elements are involved: Cu (atomic number Z_Cu = 29), I (Z_I = 53), S (Z_S = 16). Note that sulfur in the core SO₄⁻² is not involved here in electron-transfer equilibria between S₂O₃⁻² and S₄O₆⁻²; then the concentration balance for sulfate species can be considered separately.

The balances written according to Approach I to GEB, in terms of molar concentrations, are as follows:

• Generalized electron balance (GEB)

  $\begin{array}{l}\begin{array}{l}({\text{Z}}_{\text{Cu}}–2)([{\text{Cu}}^{+2}]+[{\text{CuOH}}^{+1}]+[\text{Cu}{(\text{OH})}_2]+[\text{Cu}{(\text{OH})}_3^{-1}]+[\text{Cu}{(\text{OH})}_4^{-2}]+[{\text{CuNH}}_3^{+2}]\hfill \end{array}\hfill \\ \begin{array}{c}+[\text{Cu}{({\text{NH}}_3)}_2^{+2}]+[\text{Cu}{({\text{NH}}_3)}_3^{+2}]+[\text{Cu}{({\text{NH}}_3)}_4^{+2}]+[{\text{CuCH}}_3{\text{COO}}^{+1}]+[\text{Cu}{({\text{CH}}_3\text{COO})}_2])\end{array}\hfill \\ +({\text{Z}}_{\text{Cu}}-2+{\text{Z}}_{\text{I}}-5)[{\text{CuIO}}_3^{+1}]\begin{array}{c}+({\text{Z}}_{\text{Cu}}-1)([{\text{Cu}}^{+1}]+[{\text{CuNH}}_3^{+1}]+[\text{Cu}{({\text{NH}}_3)}_2^{+1}])+({\text{Z}}_{\text{Cu}}\end{array}\hfill \\ +{\text{Z}}_{\text{I}})[Cu{I}_{(\text{s})}]+({\text{Z}}_{\text{Cu}}+2{\text{Z}}_{\text{I}}+1)[{\text{CuI}}_2^{-1}]\begin{array}{c}+({\text{Z}}_{\text{I}}+1)[{\text{I}}^{-1}]+(3{\text{Z}}_{\text{I}}+1)[{\text{I}}_3^{-1}]+2{\text{Z}}_{\text{I}}([{\text{I}}_2]\end{array}\hfill \\ +\text{a}\cdot [I_{2(\text{s})}])+({\text{Z}}_{\text{I}}-1)([{\text{HIO]+[IO}}^{-1}\text{]})+({\text{Z}}_{\text{I}}-5)([{\text{HIO}}_3]+[{\text{IO}}_3^{-1}])\begin{array}{c}+({\text{Z}}_{\text{I}}-7)([{\text{H}}_5{\text{IO}}_6]\end{array}\hfill \\ +[{\text{H}}_4{\text{IO}}_6^{-1}]+[{\text{H}}_3{\text{IO}}_6^{-2}])+2\cdot ({\text{Z}}_{\text{S}}-2)([{\text{H}}_2{\text{S}}_2{\text{O}}_3]+[{\text{HS}}_2{\text{O}}_3^{-1}])+[{\text{S}}_2{\text{O}}_3^{-2}])\begin{array}{c}+4\cdot ({\text{Z}}_{\text{S}}–2.5)\end{array}\hfill \\ [{\text{S}}_4{\text{O}}_6^{-2}]+({\text{Z}}_{\text{Cu}}–1+2\cdot ({\text{Z}}_{\text{S}}–2))[{\text{CuS}}_2{\text{O}}_3^{-1}]+({\text{Z}}_{\text{Cu}}–1+4\cdot ({\text{Z}}_{\text{S}}–2))[\text{Cu}{({\text{S}}_2{\text{O}}_3)}_2^{-3}]\hfill \\ \begin{array}{c}+({\text{Z}}_{\text{Cu}}–1+6\cdot ({\text{Z}}_{\text{S}}–2))[\text{Cu}{\left({\text{S}}_2{\text{O}}_3\right)}_3^{-5}]-(({\text{Z}}_{\text{Cu}}-2){\text{C}}_0{\text{V}}_0+({\text{Z}}_{\text{I}}+1){\text{C}}_3{\text{V}}_{\text{KI}}+2\cdot ({\text{Z}}_{\text{S}}-2)\end{array}\hfill \\ \begin{array}{c}\text{CV})/{\text{(V}}_0{\text{+V}}_{\text{N}}{\text{+V}}_{\text{Ac}}{\text{+V}}_{\text{KI}}\text{+V)}=0\end{array}\hfill \end{array}$E78

• CHB

  $\begin{array}{l}\begin{array}{l}[{\text{H}}^{\text{+1}}]–[{\text{OH}}^{-1}\text{]}+[{\text{K}}^{+1}{\text{]+[Na}}^{+1}]-[{\text{HSO}}_{\text{4}}{}^{-1}]-2[{\text{SO}}_4{}^{-2}]+[{\text{NH}}_4{}^{+1}]-[{\text{CH}}_3{\text{COO}}^{-1}]\hfill \end{array}\hfill \\ +2[{\text{Cu}}^{+2}]\begin{array}{c}{\text{+[CuOH}}^{+1}]-[\text{Cu}{(\text{OH})}_3{}^{-1}]–2[\text{Cu}{(\text{OH})}_{\text{4}}{}^{-\text{2}}]+2[{\text{Cu(NH}}_{\text{3}}{)}^{+2}]\end{array}\hfill \\ +2[\text{Cu}{({\text{NH}}_{\text{3}})}_{\text{3}}{}^{+2}]+2[\text{Cu}{({\text{NH}}_{\text{3}})}_3{}^{+2}]\begin{array}{c}+2\text{Cu}{({\text{NH}}_3)}_4{}^{+2}]+[{\text{CuCH}}_3{\text{COO}}^{+1}]-[{\text{I}}^{-1}{\text{]-[I}}_3{}^{-1}\text{]}\end{array}\hfill \\ –[{\text{IO}}^{-1}]-[{\text{IO}}_3{}^{-1}]-[{\text{H}}_4{\text{IO}}_6{}^{-1}]-2[{\text{H}}_3{\text{IO}}_6{}^{-2}]\begin{array}{c}+[{\text{Cu}}^{+1}]+[{\text{CuNH}}_3{}^{+1}]+[\text{Cu}{({\text{NH}}_3)}_2{}^{+1}]\end{array}\hfill \\ -[{\text{CuI}}_2{}^{-1}]+[{\text{CuIO}}_3{}^{+1}]-[{\text{HS}}_2{\text{O}}_3{}^{-1}]-2[{\text{S}}_2{\text{O}}_3{}^{-2}]\begin{array}{c}-2[{\text{S}}_4{\text{O}}_6{}^{-2}]-[{\text{CuS}}_2{\text{O}}_3{}^{-1}]\end{array}\hfill \\ -3[\text{Cu}{({\text{S}}_2{\text{O}}_3)}_2{}^{-3}]-5[\text{Cu}{({\text{S}}_2{\text{O}}_3)}_3{}^{-5}]=0\hfill \end{array}$E79

• F(Cu)

  $\begin{array}{l}[{\text{Cu}}^{+2}]+[{\text{CuOH}}^{+1}]+[\text{Cu}(\text{OH}{)}_2]+[\text{Cu}(\text{OH}{)}_3{}^{-1}]+[\text{Cu}(\text{OH}{)}_4{}^{-2}]+[{\text{CuSO}}_4]+[{\text{CuNH}}_3{}^{+2}]\\ +[\text{Cu}({\text{NH}}_3{)}_2{}^{+2}]+[\text{Cu}({\text{NH}}_3{)}_3{}^{+2}]+[\text{Cu}({\text{NH}}_3{)}_4{}^{+2}]+[{\text{CuCH}}_{\text{3}}{\text{COO}}^{+1}]+[\text{Cu}({\text{CH}}_3\text{COO}{)}_2]\\ +[{\text{Cu}}^{\text{+1}}]+[{\text{CuNH}}_3{}^{+1}]+[\text{Cu}{\left({\text{NH}}_3\right)}_2{}^{+1}]+[{\text{CuI}}_2{}^{-1}]+[Cu{I}_{(\text{s})}]+[{\text{CuIO}}_3{}^{+1}]+[{\text{CuS}}_2{\text{O}}_3{}^{-1}]\\ +[\text{Cu}({\text{S}}_2{\text{O}}_3{)}_2{}^{-3}]+[\text{Cu}({\text{S}}_2{\text{O}}_3{)}_3{}^{-5}]–{\text{C}}_0{\text{V}}_0{\text{/(V}}_0+{\text{V}}_{\text{N}}+{\text{V}}_{\text{Ac}}+{\text{V}}_{\text{KI}}+\text{V})=0\end{array}$E80

• F(SO₄)

  $[{\text{HSO}}_4{}^{-1}]+[{\text{SO}}_4{}^{-2}]+[{\text{CuSO}}_4]–{\text{C}}_{01}{V}_0/({\mathrm{V}}_0+{\mathrm{V}}_{\text{N}}+{\mathrm{V}}_{\text{Ac}}+{\mathrm{V}}_{\text{KI}}+\mathrm{V})=0$E81

• F(NH₃)

  $\begin{array}{c}\hfill [{\text{NH}}_4{}^{+1}]+[{\text{NH}}_3]+[{\text{CuNH}}_3{}^{+2}]+2[\text{Cu}{({\text{NH}}_3)}_2{}^{+2}]+3[\text{Cu}{({\text{NH}}_3)}_3{}^{+2}]+4[\text{Cu}{\left({\text{NH}}_3\right)}_4{}^{+2}]\\ \hfill +[{\text{CuNH}}_3{}^{+1}]+2[\text{Cu}{({\text{NH}}_3)}_2{}^{+1}]–{\text{C}}_1{V}_{\text{N}}/({\text{V}}_0+{\mathrm{V}}_{\text{N}}+{\mathrm{V}}_{\text{Ac}}+{\mathrm{V}}_{\text{KI}}+\mathrm{V})=0\end{array}$E82

• F(CH₃COO)

  $\begin{array}{c}\hfill [{\text{CH}}_3\text{COOH}]+[{\text{CH}}_3{\text{COO}}^{-1}]+[{\text{CuCH}}_3{\text{COO}}^{+1}]+2[\text{Cu}{({\text{CH}}_3\text{COO})}_2]\\ \hfill -{\text{C}}_2{\text{V}}_{\text{Ac}}/({\mathrm{V}}_0+{\mathrm{V}}_{\text{N}}+{\mathrm{V}}_{\text{Ac}}+{\mathrm{V}}_{\text{KI}}+\mathrm{V})=0\end{array}$E83

• F(K)

  $[{\text{K}}^{+1}]={\text{C}}_3{V}_{\text{KI}}/({\mathrm{V}}_0+V_{\text{N}}+{\mathrm{V}}_{\text{Ac}}+{\mathrm{V}}_{\text{KI}}+\mathrm{V})$E84

• F(I)

  $\begin{array}{l}[{\text{I}}^{-1}]+3[{\text{I}}_3{}^{-1}]+2([{\text{I}}_2]+[I_2{}_{(\text{s})}])+[\text{HIO}]+[{\text{IO}}^{-1}]+[{\text{HIO}}_3]+[{\text{IO}}_3{}^{-1}]+[{\text{H}}_5{\text{IO}}_6]+[{\text{H}}_4{\text{IO}}_6{}^{-1}]\\ +[{\text{H}}_3{\text{IO}}_6{}^{-2}]+2[{\text{CuI}}_2{}^{-1}]+[Cu{I}_{(\text{s})}]+[{\text{CuIO}}_3{}^{+1}]–{\text{C}}_3{\text{V}}_{\text{KI}}/(V_0+V_{\text{N}}+V_{\text{Ac}}+V_{\text{KI}}+V)=0\end{array}$E85

• F(S)

  $\begin{array}{l}2[{\text{H}}_2{\text{S}}_2{\text{O}}_3]+2[{\text{HS}}_2{\text{O}}_3{}^{-1}]+2[{\text{S}}_2{\text{O}}_3{}^{-2}]+4[{\text{S}}_4{\text{O}}_6{}^{-2}]+2[{\text{CuS}}_2{\text{O}}_3{}^{-1}]+4[\text{Cu}({\text{S}}_2{\text{O}}_3{)}_2{}^{-3}]\\ +6[\text{Cu}({\text{S}}_2{\text{O}}_3{)}_3{}^{-5}]–2\text{CV/}(V_0+V_{\text{N}}+V_{\text{Ac}}+V_{\text{KI}}+V)=0\Rightarrow [{\text{H}}_2{\text{S}}_2{\text{O}}_3]+[{\text{HS}}_2{\text{O}}_3{}^{-1}]+[{\text{S}}_2{\text{O}}_3{}^{-2}]\\ +2[{\text{S}}_4{\text{O}}_6{}^{-2}\left]+\right[Cu{S}_2{O}_3{}^{-1}\left]+2\right[Cu{\left(S_2{O}_3\right)}_2{}^{-3}]+3[\text{Cu}({\text{S}}_2{\text{O}}_3{)}_3{}^{-5}]\\ –\text{CV/}({\text{V}}_0+{\text{V}}_{\text{N}}+{\text{V}}_{\text{Ac}}+{\text{V}}_{\text{KI}}+\text{V})=0\end{array}$E86

• F(Na)

  $[{\text{Na}}^{+1}]=2\text{CV/}(V_0+V_{\text{N}}+V_{\text{Ac}}+V_{\text{KI}}+V)$E87

The GEB is presented here in terms of the Approach I to GEB, based on the “card game” principle, with Cu (Eq. (80)), I (Eq. (85)) as S (Eq. (86)) as “players,” and H, O, S (Eq. (81)), C (from Eq. (83)), N (from Eq. (82)), K, Na as “fans.” There are together 47 species involved in 2 + 6 = 8, Eqs. (78)–(83), (85), (86) and two equalities; [K⁺¹] (Eq. (84)) and [Na⁺¹] (Eq. (87)) are not involved in expressions for equilibrium constants, and then are perceived as numbers (not variables), at a particular V-value. Concentrations of the species in the equations are interrelated in 35 independent equilibrium constants:

$\begin{array}{l}[{\text{H}}^{+1}]={10}^{-\text{pH}},[{\text{OH}}^{-1}]={10}^{\text{pH}}{}^{-14}(p{K}_{\text{W}}=14),[{\text{CuOH}}^{+1}]={10}^{7.0}\cdot [{\text{Cu}}^{+2}][{\text{OH}}^{-1}],[\text{Cu}(\text{OH}{)}_2]={10}^{13.68}\cdot [{\text{Cu}}^{+2}][{\text{OH}}^{-1}{]}^2,\\ [\text{Cu}(\text{OH}{)}_3{}^{-1}]={10}^{17.0}\cdot [{\text{Cu}}^{+2}][{\text{OH}}^{-1}{]}^3,[\text{Cu}(\text{OH}{)}_4^{-2}]={10}^{18.5}\cdot [{\text{Cu}}^{+2}][{\text{OH}}^{-1}{]}^4,[{\text{CuNH}}_3^{+2}]={10}^{3.39}\cdot [{\text{Cu}}^{+2}][{\text{NH}}_3],\\ [\text{Cu}({\text{NH}}_3{)}_2^{+2}]={10}^{7.33}\cdot [{\text{Cu}}^{+2}][{\text{NH}}_3{]}^2,[\text{Cu}({\text{NH}}_3{)}_3^{2+}]={10}^{10.06}\cdot [{\text{Cu}}^{+2}][{\text{NH}}_3{]}^3,[\text{Cu}({\text{NH}}_3{)}_4^{+2}]={10}^{12.03}\cdot [{\text{Cu}}^{+2}][{\text{NH}}_3{]}^4,\\ [{\text{CuSO}}_4]={10}^{2.36}\cdot [{\text{Cu}}^{+2}][{\text{SO}}_4^{-2}],[{\text{NH}}_4^{+1}]={10}^{9.35}\cdot [{\text{H}}^{+1}][{\text{NH}}_3],[{\text{HSO}}_4^{-1}]={10}^{1.8}\cdot [{\text{H}}^{+1}][{\text{SO}}_4^{-2}],\\ [{\text{CH}}_3\text{COOH}]={10}^{4.65}\cdot [{\text{H}}^{+1}][{\text{CH}}_3{\text{COO}}^{-1}],[{\text{Cu}}^{+1}][{\text{I}}^{-1}]={10}^{-11.96}(\text{solubitlity}\text{product}\text{for}CuI),\\ [{\text{CuI}}_2^{-1}]={10}^{8.85}\cdot [{\text{Cu}}^{+1}][{\text{I}}^{-1}{]}^2,[{\text{CuIO}}_3^{+1}]={10}^{0.82}\cdot [{\text{Cu}}^{+2}][{\text{IO}}_3^{-1}],[{\text{CuCH}}_3{\text{COO}}^{+1}]={10}^{2.24}\cdot [{\text{Cu}}^{+2}][{\text{CH}}_3{\text{COO}}^{-1}],\\ [\text{Cu}({\text{CH}}_3\text{COO}{)}_2]={10}^{3.30}\cdot [{\text{Cu}}^{+2}][{\text{CH}}_3{\text{COO}}^{-1}{]}^2,[{\text{HS}}_2{\text{O}}_3^{-1}]={10}^{1.72}\cdot [{\text{H}}^{+1}][{\text{S}}_2{\text{O}}_3^{-2}],[{\text{H}}_2{\text{S}}_2{\text{O}}_3]={10}^{2.32}\cdot [{\text{H}}^{+1}{]}^2[{\text{S}}_2{\text{O}}_3^{-2}],\\ [{\text{CuS}}_2{\text{O}}_3^{-1}]={10}^{3.30}\cdot [{\text{Cu}}^{+1}][{\text{S}}_2{\text{O}}_3^{-2}],[\text{Cu}({\text{S}}_2{\text{O}}_3{)}_2^{-3}]={10}^{12.2}\cdot [{\text{Cu}}^{+1}][{\text{S}}_2{\text{O}}_3^{-2}{]}^2,[\text{Cu}({\text{S}}_2{\text{O}}_3{)}_3^{-5}]={10}^{13.8}\cdot [{\text{Cu}}^{+1}][{\text{S}}_2{\text{O}}_3^{-2}{]}^3\end{array}$E8420

Applying A = 16.92 [16], we have

$\begin{array}{l}\begin{array}{l}[{\text{Cu}}^{+2}]=[{\text{Cu}}^{+1}]\cdot {10}^{\text{A}(\text{E}-0.153)};[{\text{I}}_2]={[{\text{I}}^{-1}]}^2\cdot {10}^{2\text{A}(\text{E}–0.621)},s=1.33\cdot {10}^{-3}\text{mol/L}(\text{solubility}\text{of}{I}_2{}_{(\text{s})}),\hfill \end{array}\hfill \\ \begin{array}{l}[{\text{I}}_3^{-1}]={[{\text{I}}^{-1}]}^3\cdot {10}^{2\text{A}(\text{E}–0.545)},[{\text{IO}}^{-1}]=[{\text{I}}^{-}]\cdot {10}^{2\text{A}(\text{E}–0.49)+2\text{pH}–28},[\text{HIO}]={10}^{10.6}\cdot [{\text{H}}^{+1}][{\text{IO}}^{-1}],[{\text{IO}}_3^{-1}]\hfill \end{array}\hfill \\ =[{\text{I}}^{-1}]\cdot {10}^{6\text{A}(\text{E}–1.08)+6\text{pH}},\hfill \\ \begin{array}{l}[{\text{HIO}}_3]={10}^{0.79}\cdot [{\text{H}}^{+1}][{\text{IO}}_3^{-1}],[{\text{H}}_5{\text{IO}}_6]=[{\text{I}}^{-1}]\cdot {10}^{8\text{A}(\text{E}–1.24)+7\text{pH}},[{\text{H}}_4{\text{IO}}_6^{-1}]\hfill \end{array}\hfill \\ =[{\text{H}}_5{\text{IO}}_6]\cdot {10}^{-3.3+\text{pH}},[{\text{H}}_3{\text{IO}}_6^{-2}]=[{\text{I}}^{-1}]\cdot {10}^{8\text{A}(\text{E}–0.37)+9\text{pH}–126}.\hfill \end{array}$E8438

In the calculations made in this system according to the computer programs attached to Ref. [64], it was assumed that V₀ = 100, C₀ = 0.01, C₀₁ = 0.01, C₁ = 0.25, C₂ = 0.75, C₃ = 2.0, C₄ = C = 0.1; V_N = 20, V_Ac = 40, V_K = 20. At each stage, the variable V is considered as a volume of the solution added, consecutively: NH₃, CH₃COOH, KI, and Na₂S₂O₃, although the true/factual titrant in this method is the Na₂S₂O₃ solution, added in stage 4.

The solubility s [mol/L] of CuI in this system (Figures 8a and b) is put in context with the speciation diagrams presented in Figure 9. This precipitate appears in the initial part of titration with KI (C₃) solution (Figure 8a) and further it accompanies the titration, also in stage 4 (Figure 8b). Within stage 3, at V ≥ C₀V₀/C₃, we have

Figure 9.

The speciation plots for indicated Cu-species within the successive stages. The V-values on the abscissas correspond to successive addition of V mL of: 0.25 mol/L NH₃ (stage 1); 0.75 mol/L CH₃COOH (stage 2); 2.0 mol/L KI (stage 3); and 0.1 mol/L Na₂S₂O₃ (stage 4). For more details see text.

$\begin{array}{l}s=s_3=[{\text{Cu}}^{+2}\text{]}+{\displaystyle {\displaystyle \sum _{i=1}^4}[\text{Cu}{(\text{OH})}_i^{+2-i}]}+{\displaystyle {\displaystyle \sum _{i=1}^4}[\text{Cu}{({\text{NH}}_3)}_i^{+2}]+[{\text{CuSO}}_4]}+[{\text{CuIO}}_3^{+1}]\hfill \\ +{\displaystyle {\displaystyle \sum _{i=1}^2}[\text{Cu}{({\text{CH}}_3\text{COO})}_i^{+2-i}]}+[{\text{Cu}}^{+1}\text{]}+[{\text{CuI}}_2{}^{-1}\text{]}+{\displaystyle {\displaystyle \sum _{i=1}^2}[\text{Cu}{({\text{NH}}_3)}_i^{+1}]}\hfill \end{array}$E88

and in stage 4

$s=s_4=s_3+{\displaystyle {\displaystyle \sum _{i=1}^3}[{\text{Cu(S}}_2{\text{O}}_3{\text{)}}_i^{+1-2i}]}$E89

The small concentration of Cu⁺¹ (Figure 9, stage 3) occurs at a relatively high total concentration of Cu⁺² species, determining the potential ca. 0.53–0.58 V, [Cu⁺²]/[Cu⁺¹] = 10^{A(E – 0.153)}, see Figure 10a. Therefore, the concentration of Cu⁺² species determine a relatively high solubility s in the initial part of stage 3. The decrease in the s value in further parts of stage 3 is continued in stage 4, at V < V_eq = C₀V₀/C = 0.01∙100/0.1 = 10 mL. Next, a growth in the solubility s₄ at V > V_eq is involved with formation of thiosulfate complexes, mainly CuS₂O₃⁻¹ (Figure 9, stage 4). The species I₃⁻¹ and I₂ are consumed during the titration in stage 4 (Figure 9d). A sharp drop of E value at V_eq = 10 mL (Figure 10b) corresponds to the fraction titrated Φ_eq = 1.

Figure 10.

Plots of E versus V for (a) stage 3 and (b) stage 4.

The course of the E versus V relationship within the stage 3 is worth mentioning (Figure 10a). The corresponding curve initially decreases and reaches a “sharp” minimum at the point corresponding to crossing the solubility product for CuI. Precipitation of CuI starts after addition of 0.795 mL of 2.0 mol/L KI (Figure 11a). Subsequently, the curve in Figure 10a increases, reaches a maximum and then decreases. At a due excess of the KI (C₃) added on the stage 3 (V_K = 20 mL), solid iodine (I_2(s), of solubility 0.00133 mol/L at 25^oC) is not precipitated.

Figure 11.

Solubility s of CuI within stage 3 (a) and stage 4 (b).

8. Final comments

The solubility and dissolution of sparingly soluble salts in aqueous media are among the main educational topics realized within general chemistry and analytical chemistry courses. The principles of solubility calculations were formulated at a time when knowledge of the two-phase electrolytic systems was still rudimentary. However, the earlier arrangements persisted in subsequent generations [81], and little has changed in the meantime [82]. About 20 years ago, Hawkes put in the title of his article [83] a dramatic question, corresponding to his statement presented therein that “the simple algorithms in introductory texts usually produce dramatic and often catastrophic errors”; it is hard not to agree with this opinion.

In the meantime, Meites et al. [84] stated that “It would be better to confine illustrations of the solubility product principle to 1:1 salts, like silver bromide (…), in which the (…) calculations will yield results close enough to the truth.” The unwarranted simplifications cause confusion in teaching of chemistry. Students will trust us enough to believe that a calculation we have taught must be generally useful.

The theory of electrolytic systems, perceived as the main problem in the physicochemical studies for many decades, is now put on the side. It can be argued that the gaining of quantitative chemical knowledge in the education process is essentially based on the stoichiometry and proportions.

Overview of the literature indicates that the problems of dissolution and solubility calculation are not usually resolved in a proper manner; positive (and sole) exceptions are the studies and practice made by the authors of this chapter. Other authors, e.g., [13, 85], rely on the simplified schemes (ready-to-use formulas), which usually lead to erroneous results, expressed by dissolution denoted as s^* [mol/L]; the values for s^* are based on stoichiometric reaction notations and expressions for the solubility product values, specified by Eqs. (1) and (2). The calculation of s^* contradicts the common sense principle; this was clearly stated in the example with Fe(OH)₃ precipitate. Equation (27) was applied to struvite [50] and dolomite [86], although these precipitates are nonequilibrium solid phases when introduced into pure water, as were proved in Refs. [20–23]. The fact of the struvite instability was known at the end of nineteenth century [49]; nevertheless, the formula s^* = (K_sp)^1/3 for struvite may be still encountered in almost all textbooks and learning materials; this problem was raised in Ref. [15]. In this chapter, we identified typical errors involved with s^* calculations, and indicated the proper manner of resolution of the problem in question.

The calculations of solubility s^*, based on stoichiometric notation and Eq. (3), contradict the calculations of s, based on the matter and charge preservation. In calculations of s, all the species formed by defined element are involved, not only the species from the related reaction notation. A simple zeroing method, based on charge balance equation, can be applied for the calculation of pH = pH₀ value, and then for calculation of concentrations for all species involved in expression for solubility value.

The solubility of a precipitate and the pH-interval where it exists as an equilibrium-solid phase in two-phase system can be accurately determined from calculations based on charge and concentration balances, and complete set of equilibrium constant values referred to the system in question.

In the calculations performed here we assumed a priori that the K_sp values in the relevant tables were obtained in a manner worthy of the recognition, i.e., these values are true. However, one should be aware that the equilibrium constants collected in the relevant tables come from the period of time covering many decades; it results from an overview of dates of references contained in some textbooks [31, 85] relating to the equilibrium constants. In the early literature were generally presented the results obtained in the simplest manner, based on K_sp calculation from the experimentally determined s* value, where all soluble species formed in solution by these ions were included on account of simple cations and anions forming the expression for K_sp. In many instances, the K_sp^* values should be then perceived as conditional equilibrium constants [87]. Moreover, the differences between the equilibrium constants obtained under different physicochemical conditions in the solution tested were credited on account of activity coefficients, as an antidote to any discrepancies between theory and experiment.

First dissociation constants for acids were published in 1889. Most of the stability constants of metal complexes were determined after the announcement 1941 of Bjerrum’s works, see Ref. [88], about ammine-complexes of metals, and research studies on metal complexes were carried out intermittently in the twentieth century [89]. The studies of complexes formed by simple ions started only from the 1940s; these studies were related both to mono- and two-phase systems. It should also be noted that the first mathematical models used for determination of equilibrium constants were adapted to the current computing capabilities. Critical comments in this regard can be found, among others, in the Beck [90] monograph; the variation between the values obtained by different authors for some equilibrium constants was startling, and reaching 20 orders of magnitude. It should be noted, however, that the determination of a set of stability constants of complexes as parameters of a set of suitable algebraic equations requires complex mathematical models, solvable only with use of an iterative computer program [91–93].

The difficulties associated with the resolution of electrolytic systems and two-phase systems, in particular, can be perceived today in the context of calculations using (1^o) spreadsheets (2^o) iterative calculation methods. In (1^o), a calculation is made by the zeroing method applied to the function with one variable; both options are presented in this chapter.

The expression for solubility products, as well as the expression of other equilibrium constants, is formulated on the basis of mass action law (MAL). It should be noted, however, that the underlying mathematical formalism contained in MAL does not inspire trust, to put it mildly. For this purpose, the equilibrium law (EL) based on the Gibbs function [94] and the Lagrange multipliers method [95–97] with laws of charge and elements conservation was suggested lately by Michałowski.

From semantic viewpoint, the term “solubility product” is not adequate, e.g., in relation to Eq. (8). Moreover, K_sp is not necessarily the product of ion concentrations, as indicated in formulas (4), (5), and (11). In some (numerous) instances of sparingly soluble species, e.g., sulfur, solid iodine, 8-hydroxyquinoline, dimethylglyoxime, the term solubility product is not applied. In some instances, e.g., for MnO₂, this term is doubtful.

One of the main purposes of the present chapter is to familiarize GEB within GATES as GATES/GEB to a wider community of analysts engaged in electrolytic systems, also in aspect of solubility problems.

In this context, owing to large advantages and versatile capabilities offered by GATES/GEB, it deserves a due attention and promotion. The GATES is perceived as a step toward reductionism [19, 71] of chemistry in the area of electrolytic systems and the GEB is considered as a general law of nature; it provides the real proof of the world harmony, harmony of nature.