Solubility Products and Solubility Concepts

The chapter refers to a general concept of solubility product K sp 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.


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 1979Within -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 MnO 2 , taken as an example.
Solubility products refer to a large group of sparingly soluble salts and hydroxides and some oxides, e.g., Ag 2 O, considered overall as hydroxides. Incidentally, other oxides, such as MnO 2 , ZrO 2 , do not belong to this group, in principle. For ZrO 2 , 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 (I 2 ) dissolves due to reduction or oxidation, or disproportionation in alkaline media [7][8][9][10][11][12]; for I 2 , 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: H 2 L +1 in acidic and L À1 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][15][16][17][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 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 (E 01 ÀE 02 ), where E 01 = 0.850 V for Hg +2 + 2e -1 = Hg, E 02 = -0.48 V for S + 2e -1 = S -2 , 1/A = RT/ FÁln10, 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 where I 2 denotes a soluble form of iodine in a system. From HgI 2 ¼ Hg þ2 þ 2I À1 ðK sp ¼ ½Hg þ2 ½I À1 2 , pK sp ¼ 28:55Þ ð 6Þ we obtain pK sp1 = pK sp -2A(E 01 -E 03 ), where E 01 ¼ 0:850 V for Hg þ2 þ 2e À1 ¼ Hg, E 03 ¼ 0:621 V for I 2 þ 2e À1 ¼ 2I À1 ; then pK sp1 ¼ 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 H 2 S solution can be presented in terms of predominating species; we have 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 Applying the law of mass action to Eq. (7a), we have The solubility product for MgNH 4 PO 4 can be formulated on the basis of reactions: Note that only uncharged (elemental) species are involved in Eqs. (4) and (5); H 2 S enters Eq. (8), and NH 3 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 ]. Therefore, consideration of Zn 2 Fe(CN) 6 as a ternary salt with K sp1 = [Zn +2 ] 2 [Fe 2+ ][CN -1 ] 6 = K sp / K 6 is not acceptable.
In this context, some remark needs a formulation of K sp for some hydroxyoxides (e.g., FeOOH) and oxides (e.g., Ag 2 O). The related solubility products are formulated after completion of the corresponding reactions with water, e.g., FeOOH The solubility product can be involved not only with dissociation reaction. For example, the dissolution reaction Ca(OH) 2 [32]. However, the K sp for MnO 2 can be formally calculated according to an unconventional approach, based on the disproportionation reaction reverse to the symproportionation reaction 2MnO 4 À1 + 3Mn +2 + H 2 O = 5MnO 2 + 4H +1 . The K sp = K sp1 value can be found there on the basis of E 01 and E 02 values [33], specified for reactions: Eqs. (13) and (14) are characterized by the equilibrium constants: defined on the basis of mass action law (MAL) [14], where logK e1 = 3ÁAÁE 01 , logK e2 = 2ÁAÁE 02 , A = 16.92. From Eqs. (13) and (14), we get Assuming [MnO 2 ] = 1 and [H 2 O] = 1 on the stage of the K sp1 formulation for reaction (16), equivalent to reaction (12), we have and then The solubility products with MnO 2 involved can be formulated on the basis of other reactions.
For example, addition of to Eq. (14) gives Multiplication of Eq. (21) by 3, and then addition to Eq. (13a) (reverse to Eq. (13)) gives the equation and its equivalent form, obtained after simplifications, Eq. (22) and then Eq. (22a) is characterized by the solubility product 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.

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.

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.
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.
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 applied to formulate the expression for the solubility product  Table 2. Zeroing the function (32) for the system with Fe(OH) 3 precipitate introduced into pure water (copy of a fragment of display).

Dissolution of MeL 2 -type salts
Let us refer now to dissolution of precipitates MeL 2 formed by cations Me +2 and anions L À1 of a strong acid HL, as presented in Table 4. When an excess of MeL 2 is introduced into pure water, the concentration balances and charge balance in two-phase system thus formed are as follows:   Table 3. Zeroing the function (39) for the system with Zn(OH) 2 precipitate introduced into water; pK W = 14.
From Eqs. (40) and (41) α i.e., reaction of the solution is acidic, Applying the relations for the equilibrium constants: from Eqs. (43) and (44) we have where In particular, for I = 3, J = 4 ( Applying the zeroing procedure to Eq. (46) gives the pH = pH 0 of the solution at equilibrium. At this pH 0 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 = pH 0 of the solution in equilibrium with the related precipitate. The solubilities are as follows: 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).
• For (B1) Subtraction of Eq. (49) from Eq. (51) gives In this case,   Table 9). Graphically, C CO2 = 0.100 is found at pH 03 = 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    (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.

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.

Silver dichromate (Ag 2 Cr 2 O 7 )
The equilibrium data related to the system, where Ag 2 Cr 2 O 7 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 Ag 2 Cr 2 O 7 changes into Ag 2 CrO 4 .
On the dissociation step, each dissolving molecule of Ag 2 Cr 2 O 7 gives two ions Ag +1 and 1 ion Cr 2 O 7

À2
, 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 2½Ag 2 Cr 2 O 7 þ ½Ag þ1 þ½AgOH þ ½AgðOHÞ À1 2 þ ½AgðOHÞ À2 where 2C 0 is the total concentration of the solid phase in the system, at the moment (t = 0) of introducing this phase into water, [Ag 2 Cr 2 O 7 ] 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: Ag 2 Cr 2 O 7 ) does not change the volume of the system in a significant degree, and that Ag 2 Cr 2 O 7 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 ½H þ1 À ½OH À1 þ ½Ag þ1 À ½AgðOHÞ À1 2 À 2½AgðOHÞ À2 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 Denoting by 2c 0 (< 2C 0 ) the total concentration of dissolved Ag and Cr species formed, in a transition stage, from Ag 2 Cr 2 O 7 , we can write pK sp2 = 6.7 AgOH ¼ Ag þ1 þ OH À1 pK sp3 = 7.84 Table 10. Physicochemical equilibrium data relevant to the Ag 2 Cr 2 O 7 + H 2 O system (pK = ÀlogK), at "room" temperatures.
The system involved with Ag 2 CrO 4 was also considered in context with the Mohr's method of Cl À1 determination [44][45][46]. As were stated there, the systematic error in Cl À1 determining according to this method, expressed by the difference between the equivalence (eq) volume (V eq = C 0 V 0 /C) and the volume V end corresponding to the end point where the K sp1 for Ag 2 CrO 4 is crossed, equals to ] (pK sp = 9.75), V 0 is the volume of titrant with NaCl (C 0 ) + K 2 CrO 4 (C 01 ) titrated with AgNO 3 (C) solution; V end = V eq at C 01 = (1 + V end /V 0 )•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].

Dissolution of struvite
The fact that NH 3 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] 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 CO 2 (C CO2 , 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 were calculated for: pr1 = MgNH 4 PO 4 (pK sp1 = 12.6), pr2 = Mg 3 (PO 4 ) 2 (pK sp2 = 24.38), pr3 = MgHPO 4 (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 C 0 = [pr1] t=0 = 10 À3 mol/L (pC 0 = (ppr1) t=0 = 3); ppr1 = Àlog [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] s 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) ] = 10 12.36ÀpH ≈ 10 3 . 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 CO 2 in water where struvite was introduced.   After introducing struvite pr1 (at pC 0 = [ppr1] t=0 = 2) into alkaline (C b = 10 À2 mol/L KOH, pC b = 2) solution of CO 2 (pCO 2 = 4), the dissolution is more complicated and proceeds in three steps, see Figure 7.

À2
, 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 + 2NH 3 + 2H 2 O occurs up to total depletion of pr4 (at ppr1 = 2.896). In this step, the reaction 3pr1 + 2OH À1 = pr2 + 3NH 3 + HPO 4 À1 + 2H 2 O occurs up to total depletion of pr1, i.e., the solubility product K sp1 for pr1 is not crossed. The curve s 0 (Figure 7) is related to the function where s is expressed by Eq. (70).  expression for s in alkaline media, see Figure 8. This pH range involves pH of ammonia buffer solutions, where NiL 2 is precipitated from NiSO 4 solution during the gravimetric analysis of nickel; the expression for solubility

Solubility of nickel dimethylglyoximate
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 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 NiL 2 . Then one can assume the approximation 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 K 2 and K sp , we obtain If jΔG o 1 j ≈ jΔG o 2 j within the temperature range (T 0 , T), the value of s is approximately constant. Let T 0 denote the room temperature (at which,as a rule-all the equilibrium constants are determined) and T 6 ¼ T 0 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.

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: Y 1 = H, Y 2 = 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 (H 2 O, CO 2 , O 2 ,…) 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 gatoms 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 Y 1 = H, F(O) for Y 2 = 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: Therefore, Approach II to GEB $ Approach I to GEB ð76Þ 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 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][76][77][78][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 mixedsolvent 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.

Solubility of CuI in a dynamic redox system
The system considered in this section is related to iodometric, indirect analysis of an acidified (H 2 SO 4 ) solution of CuSO 4 [14,64]. It is a very interesting system, both from analytical and physicochemical viewpoints. Because the standard potential E 0 = 0.621 V for (I 2 , I À1 ) exceeds E 0 = 0.153 V for (Cu +2 , Cu +1 ), one could expect (at a first sight) the oxidation of Cu +1 by I 2 . 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 H 2 SO 4 is neutralized with NH 3 (step 1) until a blue color appears, which is derived from Cu(NH 3 ) i +2 complexes. Then the excess of CH 3 COOH 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 À1 = 2CuI + I 2 , 2Cu +2 + 5I À1 = 2CuI + I 3

À1
is titrated with Na 2 S 2 O 3 solution (step 4), until the reduction of iodine:

Formulation of the system
We assume that V mL of C mol/L Na 2 S 2 O 3 solution is added into the mixture obtained after successive addition of: V N mL of NH 3 (C 1 ) (step 1), V Ac mL of CH 3 COOH (C 2 ) (step 2), V KI mL of KI (C 3 ) (step 3), and V mL of Na 2 S 2 O 3 (C) (step 4) into V 0 mL of titrand D composed of CuSO 4 (C 0 ) + H 2 SO 4 (C 01 ). 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 SO 4 À2 is not involved here in electron-transfer equilibria between S 2 O 3 À2 and S 4 O 6

À2
; 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) ½Cu þ2 þ ½CuOH þ1 þ ½CuðOHÞ 2 þ ½CuðOHÞ 3 À1 þ ½CuðOHÞ 4 À2 þ ½CuSO 4 þ ½CuNH 3 þ2 • F(CH 3 COO) • F(S) • F(Na) The GEB is presented here in terms of the Approach I to GEB, based on the "card game" principle, with Cu (Eq. (80)  In the calculations made in this system according to the computer programs attached to Ref. [64], it was assumed that V 0 = 100, C 0 = 0.01, C 01 = 0.01, C 1 = 0.25, C 2 = 0.75, C 3 = 2.0, C 4 = 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: NH 3 , CH 3 COOH, KI, and Na 2 S 2 O 3 , although the true/factual titrant in this method is the Na 2 S 2 O 3 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 (C 3 ) solution ( Figure 8a) and further it accompanies the titration, also in stage 4 ( Figure 8b). Within stage 3, at V ≥ C 0 V 0 /C 3 , we have and in stage 4 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 = C 0 V 0 /C = 0.01•100/0.1 = 10 mL. Next, a growth in the solubility s 4 at V > V eq is involved with formation of thiosulfate complexes, mainly CuS 2 O 3 À1 (Figure 9, stage 4). The species I 3 À1 and I 2 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.
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 (C 3 ) added on the stage 3 (V K = 20 mL), solid iodine (I 2(s) , of solubility 0.00133 mol/L at 25 o C) is not precipitated.

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 twophase 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][21][22][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 = pH 0 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 twophase 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][92][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][96][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 MnO 2 , 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.