Productos de solubilidad y conceptos de solubilidad

Tabla de contenidos

1. Introducción

El problema de la solubilidad de varios compuestos químicos ocupa un lugar destacado en la literatura científica. Esto se debe al hecho de que, entre varias propiedades que determinan el uso de estos compuestos, la solubilidad es de suma importancia. Entre otros, este tema ha sido objeto de intensas actividades iniciadas en 1979 por la Comisión de Datos de Solubilidad V.8 de la División de Química Analítica de la IUPAC establecida y dirigida por S. Kertes [ 1 ], quien concibió la IUPAC -NIST Solubility Data Series (SDS) proyecto [ 2 , 3 ]. Entre 1979 y 2009, se emitieron la serie de 87 volúmenes, en relación con la solubilidad de gases, líquidos y sólidos en líquidos o sólidos [ 3 ]; Uno de los volúmenes se refiere a la solubilidad de varios óxidos e hidróxidos [ 4 ]. Una extensa compilación de datos de solubilidad acuosa proporciona el Manual de datos de solubilidad acuosa [ 5 ].

Una observación . Los precipitados están marcados en negrita letras; Las especies / complejos solubles están marcados en letras normales.

La característica distintiva de un compuesto químico escasamente soluble en un medio particular es el valor del producto de solubilidad K _sp. En la práctica, los valores conocidos de K _sp se refieren solo a medios acuosos. Sin embargo, se debe tener en cuenta que la expresión para el producto de solubilidad y luego el valor K _sp de un precipitado dependen de la notación de una reacción en la que este precipitado está involucrado. De esto se deduce la aparente multiplicidad de los valores de K _sp referidos a un precipitado particular. Además, como se indicará a continuación, la expresión para K _sp no debe contener necesariamente especies iónicas. Por otro lado, se percibe una falta real o aparente de K _sp para algunos precipitados; este último tema se abordará aquí a MnO ₂, tomado como ejemplo.

Los productos de solubilidad se refieren a un gran grupo de sales e hidróxidos escasamente solubles y algunos óxidos, por ejemplo, Ag ₂ O [ 19459016], considerado en general como hidróxidos. Por cierto, otros óxidos, como MnO ₂, ZrO ₂, do No pertenecer a este grupo, en principio. Para ZrO ₂, las mediciones de solubilidad mostraron valores bastante bajos incluso en condiciones muy ácidas [ 6 ]. La solubilidad depende de la historia previa de estos óxidos, por ejemplo, el tostado previo elimina virtualmente la solubilidad de algunos óxidos. El yodo moderadamente soluble ( I ₂) se disuelve debido a la reducción u oxidación, o desproporción en medios alcalinos [ 7 – 12 ]; para I ₂, la solubilidad mínima en agua es un estado de referencia. Para 8-hidroxiquinolina, la solubilidad de la molécula neutra HL es un estado de referencia; aquí se produce un crecimiento en la solubilidad por la formación de especies iónicas: H ₂ L ⁺¹ en ácidos y L ⁻¹ en medios alcalinos.

El K _sp es el principal pero no el único parámetro utilizado para calcular la solubilidad de un precipitado. Las simplificaciones [ 13 ] practicadas a este respecto son inaceptables y conducen a resultados incorrectos / falsos, como se indica en [ 14 – 18 ]; Más constantes de equilibrio también están involucradas con los sistemas de dos fases. Estas objeciones, formuladas a la luz del enfoque generalizado de los sistemas electrolíticos (GATES) [ 8 ], donde s es la suma “ponderada” de las concentraciones de todas las especies solubles formadas por el precipitado, también se presentan en este capítulo, relacionado con los sistemas no redox y redox.

El cálculo de s proporciona una información de gran importancia, por ejemplo, desde el punto de vista de la gravimetría, donde el paso principal del análisis es la transformación cuantitativa de un analito adecuado en un precipitado escasamente soluble (sal, hidróxido). Aunque la precipitación y las operaciones analíticas adicionales generalmente se llevan a cabo a temperaturas mucho más altas que la temperatura ambiente, a las cuales se determinaron las constantes de equilibrio, los valores de s obtenidos a partir de los cálculos realizados sobre la base de datos de equilibrio relacionados con la temperatura ambiente son útiles en la elección de las condiciones óptimas a priori del análisis, asegurando la concentración mínima resumida de todas las formas solubles del analito, que permanecen en la solución, en equilibrio con el precipitado obtenido después de la adición de un exceso del precipitado agente; Este exceso se denomina relativo a la composición estequiométrica del precipitado. La capacidad de realizar cálculos apropiados, basados en todo el conocimiento fisicoquímico disponible, de acuerdo con las leyes básicas de conservación de la materia, profundiza nuestro conocimiento de los sistemas relevantes. Al mismo tiempo, produce la capacidad de adquirir conocimiento relevante de manera organizada, no solo imitativo, sino centrado en la heurística. Este punto de vista está de acuerdo con la enseñanza constructivista, basada en la creencia de que el aprendizaje ocurre, ya que los alumnos participan activamente en un proceso de construcción de significado y conocimiento, en lugar de recibir información pasivamente [ 19 ].

2. Definiciones y formulación de productos de solubilidad

El valor K _sp se refiere a un sistema de dos fases donde la fase sólida de equilibrio es un precipitado escasamente soluble, cuyo valor K _sp se mide / calcula de acuerdo con la expresión definida para el producto de solubilidad. Esta suposición significa que la solución con especies definidas está saturada contra este precipitado, a la temperatura y composición de la solución. Sin embargo, a menudo un precipitado, cuando se introduce en medios acuosos, no es la fase sólida de equilibrio, y luego este requisito fundamental no se cumple, como se indica en los ejemplos de los análisis fisicoquímicos de los sistemas con estruvita MgNH [19459018 ] 4 PO ₄ [ 20 , 21 ], dolomita CaMg (CO ₃ ) ₂ [ 22 , 23 ] y Ag ₂ Cr ₂ ] O ₇.

Los valores de los productos de solubilidad K _sp (generalmente representado por la constante de solubilidad pK _sp = −log [19459013 ] K _sp valor) son conocidos por precipitados estequiométricos de A _a B _b o A _a B _b ] C _c tipo, relacionado con reacciones de disociación:

K_{sp} = {[A [1945904] 9]]}^{a} {[B [19459052 ]]}^{b} para A_{a} B_{b} = [ 19459053]aA+bB,o

19459062] E1

K_{sp} [194590 = {[1945905 2] [A]}^{a} {[[ 19459053]B]}^{b} {[] C]}^{c} para {A A  ]}_{a} B_{b [ 19459051] C_{c} = aA] +bB+ [1945905 3]cC}

donde A y B o A, B y C son las especies que forman la precipitado relacionado; los cargos se omiten aquí, por simplicidad de notación. Los productos de solubilidad para precipitados más complejos son desconocidos en la literatura. Los precipitados A _a B _b C [ 19459018] c se conocen como sales ternarias [ 24 ], por ejemplo, estruvita, dolomita e hidroxiapatita Ca _{5 [ 19459016]} (PO ₄ ) ₃ OH .

Los productos de solubilidad para precipitados de A _a B _{b [ 19459019] tipo se encuentran con mayor frecuencia en la literatura. En estos casos, para A generalmente se colocan cationes simples de metales u oxidaciones [ 25 ]; por ejemplo, BiO ⁺¹ y UO ₂ ⁺² forman los precipitados: BiOCl y (UO [19459018 ] 2} ) ₂ (OH) ₂. Como B, se consideran aniones simples o más complejos, por ejemplo, Cl ⁻¹, S ⁻², PO ₄ ⁻³, Fe (CN) ₆ ⁻⁴, en AgCl, HgS, Zn ₃ (PO ₄ ) ₂ y Zn _{[19459015 ] 2} Fe (CN) ₆.

En diferentes libros de texto, los productos de solubilidad generalmente se formulan para reacciones de disociación, con iones como productos, también para HgS

[ 19459057] H g S = {Hg}^{[ 19459044] + 2} + S^{- [ 19459074] 2} (K_{sp} = [ 19459053][{Hg}^{+ 2] [1 9459055]}][S^{- 2 [ 19459050]}])

aunque existe un enlace covalente polar entre sus átomos constituyentes [ 26 ]. Valor de producto de muy baja solubilidad ( pK _sp = 52,4) para HgS realiza la disociación de acuerdo con el esquema presentado por la ecuación. ( 3 ) imposible, e incluso la formulación verbal del producto de solubilidad no es razonable. A saber, el producto iónico x = [Hg ⁺²] [S ^–2] calculado en [Hg ⁺²] = [ S ^–2] = 1 / N _A excede K _sp, 1 / N [ 19459014] _A ²> K _sp ( N _A – número de Avogadro) ; la concentración 1 / N _A = 1,66 ∙ 10 ^–23 mol / L corresponde a 1 ion en 1 L de la solución. El esquema de disociación en especies elementales [ 14 ]

H g [19459046 ] S = H g + S [ 19459047] (K_{sp1} =======][H g][S [ 19459047]])

es mucho más favorecido por la termodinámica punto de vista; no obstante, el producto de solubilidad ( K _sp) para HgS se formula comúnmente sobre la base de la reacción (3). Obtenemos pK _sp1 = pK _sp – 2 A ( E [19459018 ] 01 – E ₀₂), donde E ₀₁ = 0,850 V para Hg ⁺² + 2e ^–1 = Hg , E ₀₂ = –0,48 V para S + 2e ^–1 = S ^–2, 1 / A = RT / F⋅ln10, A = 16.92 para 298 K; entonces pK _sp1 = 7,4.

Las constantes de equilibrio generalmente se formulan para las notaciones de reacción más simples. Sin embargo, a este respecto, la ecuación. ( 4 ) es más simple que la ecuación. ( 3 ). Además, estamos “acostumbrados” a aplicar productos de solubilidad con iones (cationes y aniones) involucrados, pero esta costumbre puede ser fácilmente eliminada. Una observación similar puede referirse a la notación referida a la disociación elemental del precipitado de yoduro mercúrico

H g I_{2} = H [ 19459046] g + I_{2} (K_{sp1} = [H g g] [I_{2}])]

donde I ₂ denota una forma soluble de yodo en un sistema. Desde

H g I [ 19459058] 2 = {Hg}^{+++++]2} + 2 I^{-] 1} (K_{sp} =] [{Hg}^{+ 2}] {[[194590 53]I^{- 1}]}^{2 [ 19459075]}, p K_{sp} = [ 19459053]28.55)

obtenemos

19459013] pK _sp1 = pK _sp – 2 A ( E _{01 [ 19459019] – E ₀₃), donde}

E_{] 01} = [19459053 ]0.850V{para Hg}^{+ [19457474 [ ] 2}+2e^{- 1 [ 19459075]}=H g,E [ 19459047]03=0.621{V para I}_{2}+2e^{- 1}= [1945905 3]2I^{- 1};luegopK_{sp1}[ 19459052] =20.80.

Las especies en la expresión de productos de solubilidad no predominan en los sistemas químicos reales, por regla general. Sin embargo, la precipitación de HgS de una solución acidificada (HCl) de sal de mercurio con solución de H ₂ S puede presentarse en términos de especies predominantes; tenemos

{HgCl}_{4}^{- –  –  –  – 2} + H_{2} S = H g S + 4 {[19459044 ]}^{Cl} - 1 + 2]^{H + 1}

Eq. ( 7 ) se puede aplicar para formular el producto de solubilidad relacionado, K _sp2, para HgS . Para estar en línea con los requisitos habituales establecidos en la formulación del producto de solubilidad, ec. ( 7 ) debe reescribirse en la forma

H g S + 4 {Cl}^{- 1} + 2 H^{+ 1 [19459075 1 ]} = {HgCl}_{4}^{- [194574]] 2} + H_{2} S

E7a [1945 9033]

Aplicando la ley de acción de masas a la ec. ( 7a ), tenemos

K_{sp2} = {[HgCl}_{4}^{- –  –  –  – 2} {] [H}_{2} S] [ 19459050] {[Cl}^{- 1 [19455555 ]]}^{4} {{[H}^{+ 1} [194 59048]]}^{2}, (p [ 19459047]K_{s p 2}=[ 19459074] 17.33)

donde [HgCl ₄ [ ] –2 ] = 10 ^15.07 [Hg ⁺²] [Cl ^–1] ⁴, [H [19459018 ] 2 S] = 10 ^20.0 [H ⁺¹] ² [S ^–2], K _sp (ec. ( 3 )).

El producto de solubilidad para MgNH ₄ PO ₄ puede formularse en base a las reacciones:

M g N [19459058 ] H_{4} P O_{4} = M g^{+ 2} + N H_{4}^{+ 1 [ 19459050]} + P O_{4}^{- 3} (K_{sp} = [ 19459076] [{Mg}^{+ 2}] [ 19459053][{NH}_{4}^{+ 1 [ [ [ 19459050]}[{PO}_{4}^{----19459053]3}[194590 76]]

M g N H_{4} P O_{4}] = M g^{+ 2}] + N H_{3} + H P O_{4 [19459075 ]- 2} (K_{s s  s  s  ] p 1} = [M g [ 19459049]+ 2] [N {[ 19459048] H}_{3}_{] [H P O4- 2] = [19459045 ] K s p [1 9459049]}_{K_{1 N} / K_{3 P})}

E10

\begin{matrix} M g N H_{4} P O_{4} + H_{2} [1945904 8] O = M g O H^{+ 1} + N H_{3} + H_{2} P O_{4}^{- 1} (K_{s p 2} = [M g O H^{+ [1 9459053]1}] [N H_{3}] [H_{2} P O_{4}^{- 1}] \\ = K_{s p} K_{1}^{OH} K_{1 N} {[ 19459048] K}_{W} / (K_{2 P} K_{3 P}) \end{matrix}

E11

where K _1N = [H ⁺¹ ][NH ₃ ]/[NH ₄ ⁺¹ ], K _2P = [H ⁺¹ ][HPO ₄ ^–2 ]/[H ₂ PO ₄ ^–1 ], K _3P = [H ⁺¹ ][PO ₄ ^–3 ]/[HPO ₄ ^–2 ], [MgOH ⁺¹ ] = K ₁ ^OH [Mg ⁺² ][OH ^–1 ], K _W = [H ⁺¹ ][OH ^–1 ].

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 ]

reverse to Eq. ( 4 ). Some precipitates can be optionally considered as the species of A _a B _b or A _a B _b C _c type. For example, the solubility product for MgHPO ₄ can be written as K _sp = [Mg ⁺² ][HPO ₄ ^–2 ] or K _sp1 = [Mg ⁺² ][H ⁺¹ ][PO ₄ ^–3 ] = K _sp K _3P .

The ferrocyanide ion Fe(CN) ₆ ^–4 (with evaluated stability constant K ₆ ca. 10 ³⁷ ) 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 Zn ₂ Fe(CN) ₆ is K _sp = [Zn ⁺² ] ² [Fe(CN) ₆ ^–4 ]. 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 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+x Mg _1-x (CO ₃ ) ₂ [ 22 , 23 ], Fe _x S [ 29 ]. In particular, Fe _x S can be rewritten as Fe ⁺² _p Fe ⁺³ _q S ; from the relations: 2 p + 3 q − 2 = 0 and p + q = x , we get q / p = 2(1 − x )/(3 x − 2).

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 + H ₂ O = Fe(OH) ₃ , Fe ₂ O ₃ ∙xH ₂ O + (3 − x )H ₂ O = 2 Fe(OH) ₃ ⇒ Fe(OH) ₃ = Fe ⁺³ + 3OH ^–1 ⇒ K _sp = [Fe ⁺³ ][OH ^–1 ] ³ ; Ag ₂ O + H ₂ O = 2 AgOH ⇒ 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 + 2H ₂ O [ 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 _w ² . In Ref. [ 31 ], the solubility product is associated with formation (not dissociation) of a precipitate.

3. Solubility product(s) for MnO ₂

The scheme presented above cannot be extended to all oxides. For example, one cannot recommend the formulation of this sequence for MnO ₂ , i.e., MnO ₂ + 2H ₂ O = 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 MnO ₂ is the sole Mn(+4) species present in such systems. In effect, K _sp0 for MnO ₂ is not known in the literature, compare with Ref. [ 32 ]. However, the K _sp for MnO ₂ can be formally calculated according to an unconventional approach, based on the disproportionation reaction

5 M n O_{2} + 4 H^{+ 1} = 2 {MnO}_{4}^{- 1} + 3 {[19 459044]}^{Mn} + 2 + H_{2} O

E12

reverse to the symproportionation reaction 2MnO ₄ ⁻¹ + 3Mn ⁺² + H ₂ O = 5 MnO 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:

{MnO}_{4}^{- 1} + 4 H^{+ 1} + 3 e^{- 1} = M n [194590 58] O_{2} + 2 H_{2} O(E_{01} = 1.692 V)

E13

M n O_{2} + 4 H^{+ 1} + [19 459074] 2 e^{- 1} = {Mn}^{+ 2} + 2 H_{2} O(E_{02} = 1.228 V)

E14

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

K_{e1} = \frac{{[MnO}_{2} {][H}_{2} {O]}^{2}}{{[MnO}_{4}^{- 1} {][H}^{+ 1}] [194590 49]4 {{[e}^{- 1}]}^{3}}, K_{e2} = \frac{{[Mn}^{+ 2} {][H}_{2} {O]}^{2}}{[M n O_{2} {][H}^{+ 1}]^{4} {{[e}^{- 1}]}^{2}}

E15

defined on the basis of mass action law (MAL) [ 14 ], where log K _e1 = 3⋅ A ⋅ E ₀₁ , log K _{e2 [19459 019] = 2⋅ A ⋅ E ₀₂ , A = 16.92. From Eqs. ( 13 ) and ( 14 ), we get}

\begin{matrix} 2 M n O_{2} + 4 H_{2} O + 3 M n O_{2} + 12 H^{+ 1} + [194590 74] 6 e^{- 1} = 2 {MnO}_{4}^{- 1} + 8 H^{+ 1} \\ + 6 e^{- 1} + 3 {Mn}^{+ 2} + 6 H_{2} O \end{matrix}

E16

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

K_{sp1} = \frac{{{[MnO}_{4}^{- 1}]}^{2} {{[Mn}^{+ 2}]}^{3}}{{{[H}^{+ 1}]}^{4}}

E17

and then

K_{sp1} = {([194590 45] K e2)}^{3} \cdot {(K_{e1})}^{- 2}

E18

p K_{sp1} = {3logK}_{e2} - {2logK}_{e1 [194 59049]} = 6 A (E_{01} - E_{02}) = 6 \cdot 16 .92 \cdot (1 .692 - 1 .228) = 47 .11

E19

The solubility products with MnO ₂ involved can be formulated on the basis of o ther reactions. For example, addition of

to Eq. ( 14 ) gives

M n O_{2} + 4 H^{+ 1} + 2 e^{- 1} + {Mn}^{+ 2} = {Mn}^{+ 2} + [1 9459053]2H_{2}O+{Mn}^{+ 3}+e^{- 1}

E21

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

M n O_{2} + 2 H_{2} O = {MnO}_{4}^{- 1} + 4 H^{+ 1} + 3 e^{- 1} [194 59059]

E8423

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

\begin{matrix} 3 M n O_{2} + 12 H^{+ 1} + 6 e^{- 1} + 3 {Mn}^{+ 2} + M n O_{2} + 2 H_{2} O \\ = 3 {Mn}^{+ 2} + 6 H_{2} O + 3 {Mn}^{+ 3} + 3 e^{- 1} + {MnO}_{4}^{- 1} + 4 H^{+ 1} + 3 e^{- 1} \end{matrix}

E22

and its equivalent form, obtained after simplifications,

4 M n O [ 19459058] 2 + 8 H^{+ 1} = 3 {Mn}^{+ 3} + {MnO}_{4}^{- 1} + 4 H_{2} O

E22a

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

K_{sp2} = \frac{{[MnO}_{4}^{- 1} {][Mn}^{+ 3}]^{3}}{{{[H}^{+ 1}]}^{8}} = {(K_{e2})}^{3} \cdot {(K_{e3})}^{- 3} \cdot {(K_{e1})}^{- 1}

E23

where

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

E24

for Mn ⁺³ + e ⁻¹ = Mn ⁺² ( E ₀₃ = 1.509 V) (reverse to Eq. ( 20 )), log K _e3 = A ⋅ E ₀₃ . Then

{pK}_{sp2} = 3 A ∙ (E_{01} - 2 E_{02} + E_{03}) + 37.82

E25

Formulation of K _{sp i} for other combinations of redox and/or nonredox reactions is als o 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 ][MnO ₄ ⁻² ] involved with reaction 2 MnO ₂ = Mn ⁺² + MnO ₄ ⁻² .

As results from calculations, the low K _{sp i} ( 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[Xi] versus Φ relationships for different manganese species Xi, plotted for titration of V0 = 100 mL solution of FeSO4 (C0 = 0.01 mol/L) + H2SO4 (Ca = 1.0 mol/L) with V mL of C = 0.02 mol/L KMnO4; Φ = C·V/(C0·V0). The species Xi 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 _a B _b or (2 ^o ) A _a B _b C _c type, when introduced into pure water. Assuming [ A ] = a∙s ^* and [ B ] = b∙s ^* , from Eq. ( 1 ), we have

s^{*} = {(\frac{K_{sp}}{a^{a} \cdot b^{b}})}^{1/(a + b)}

E26

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

s^{*} = {(\frac{K_{sp}}{a^{a} \cdot b^{b} \cdot c^{c}})}^{1/(a + b + c)}

[1945906 2] 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 ]

C a {(O H)}_{2} = {Ca}^{+ 2} + 2 {OH}^{- 1} (K_{sp1} = [{Ca}^{+ 2}] {[{OH}^{- 1}]}^{2}, s^{*} = {({[1945 9046] K}_{sp1} / 4)}^{1 / 3} = 0.0133 mol / L)

E28

F e {(O H)}_{3} = {Fe}^{+ 3 [194590 75]} + 3 {OH}^{- 1} (K_{sp2} = [{Fe}^{+ 3}] {[{OH}^{- 1}]}^{3}, s^{*} = {[1 9459044]}^{(K_{sp2} / 27)} 1 / 4 = 0.98 \times 10^{- 10} mol / 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 ; [Fe ₂ (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

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 [{Ca}^{+ 2}] + [{CaOH}^{+ 1}] + [H^{+ 1}] - [{OH}^{- 1}] = 0

E31

we get, by turns,

\begin{matrix} 2 \cdot 10^{5.03} / {[{OH}^{- 1}]}^{2} + 10^{1.3} \cdot 10^{- 5.03} / [{OH}^{- 1}] [H^{[19459 052] + 1}] - [{OH}^{- 1}] = 0 \\ \Rightarrow 2 \cdot 10^{- 5.03 + 28 - 2 pH} + 10^{1.3} \cdot {10 [19459 075]}^{- 5.03 + 14 - pH} + 10^{- pH} - 10^{pH - 14} = 0 \end{matrix}

E9000

y(pH) = 2 \cdot 10^{[194 59074] 22.97 - 2 pH} + 10^{10, 17 - pH} + 10^{- pH} - 10^{pH - 14} = 0

E32

where $pH = - {log[H}^{+ 1}]$ . Applying the zeroing procedure to Eq. ( 30 ), we get pH ₀ = 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 [194592 91]	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 ] = $2 [{Ca}^{+ 2}] + [{CaOH}^{+ 1}] > 0.$

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

3 [Fe [1945 9050] + 3] + 2 [{FeOH}^{+ 2}] + [{Fe(OH)}_{2}^{+ 1}] + 4 [{Fe}_{2} {(OH)}_{2}^{+ 4 [194590 50]}] + [H^{+ 1}] - [{OH}^{- 1}] = 0

E33

and then

y(pH) = 3 \cdot 10^{3.4 - 3 pH} [194590 52] + 2 \cdot 10^{0.4 - 2 pH} + 10^{- 2.9 - pH} + 4 \cdot 10^{3.9 - 4 pH} + 10^{- pH} - {[1 9459044]}^{10} pH - 14 = 0

E34

Eq. ( 32 ) zeroes at pH ₀ = 7.0003 ( Table 2 ), where the value

pH	y(pH)	[Fe ⁺³ ]	[FeOH ⁺² ]	[Fe(OH) ₂ ⁺¹ ]	[Fe ₂ (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. [1 9459236]

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

s = [{Fe}^{+ 3}] + [{FeOH}^{+ 2}] + [{Fe(OH)}_{2}^{+ 1}] + 2 [{Fe}_{2} [1945 9102] (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 pH ₀ = 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 = 5·10 ⁷ , 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 $s’ = {(K_{sp1})}^{1/2 [19 459049]}$ 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 A ₃ B and AB ₃ type, and K _sp = 108(s ^* ) ⁵ for A ₂ B ₃ and A ₃ B ₂ . For Ca ₅ (PO ₄ ) ₃ OH , 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

applied to formulate the expression for the solubility product

K_{sp3} = [{Zn}^{+ 2}] {[{OH}^{- 1}]}^{2} (p K_{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 log K _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 [{Zn}^{+ 2}] + [{ZnOH}^{+ 1}] - [{Zn(OH)}_{3}^{- 1}] - 2 [{Zn(OH)}_{4}^{- [19 459053]2} {]+[H}^{+ 1}] - [{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

y(pH) = 2 \cdot 10^{13 - 2 pH} + 10^{3.4 - pH} - 10^{- 15.86 + p H} - 2 \cdot 10^{- 28.34 + 2 pH} + 10^{- pH} - 10^{pH - 14} = 0

E39

The function ( 39 ) zeroes at pH ₀ = 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	2.27 02E-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.

Ze roing the function ( 39 ) for the system with Zn(OH) ₂ precipitate introduced into water; pK _W = 14.

s = [{Zn}^{+ 2}] + [{ZnOH}^{+ 1}] + [{Zn(OH)}_{2}] + [{Zn(OH)}_{3}^{- 1 [19 459075]}] + [{Zn(OH)}_{4}^{- 2}] = 2.07 \cdot 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 MeL ₂ -type salts

Let us refer now to dissolution of precipitates MeL ₂ formed by cations Me ⁺² and anions L ⁻¹ 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:

Me ⁺²	MeOH ⁺¹	Me(OH) ₂	Me(OH) ₃ ⁻¹	L ⁻¹	MeL ⁺¹	MeL ₂	MeL ₃ ⁻¹	MeL ₄ ⁻²	MeL ₂
	log K ₁ ^OH	log K ₂ ^OH	log K ₃ ^OH		log K ₁	log K ₂	log K ₃	log K ₄	pK _sp
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.

log K _i ^{OH [19 459025] and log K _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 MeL ₂ ; [MeL _i ^+2- ⁱ ] = K _i [Me ⁺² ][L ⁻ ¹ ] ⁱ , K _s _p = [Me ⁺² ][L ⁻ ¹ ] ² .}

[M e L_{2}] + {[Me}^{+ 2}] + \sum_{i = 1}^{I} {[Me(OH)}_{i}^{+ 2 - i [1945 9050]}] + \sum_{j = 1}^{J} {[MeL}_{j}^{+ 2 - j}] = C_{Me}

E40

2[M e L_{2}] + {[L}^{- 1}] + \sum_{j = 1}^{J} j {[MeL}_{j}^{+ 2 - j}] = C_{L}

[19459 062] E41

{[H}^{+ 1}] - {[OH}^{- 1}] + {2[Me}^{+ 2}] + \sum_{i = 1}^{I} (2 - i {[1945 9044]}_{)[Me(OH)} i + 2 - i] + \sum_{j = 1}^{J} (2 - j {)[MeL}_{j}^{+ 2 - j}] - {[L}^{- [19459 048] 1}] = 0

E42

where [ MeL ₂ ] denotes the concentration of the precipitate MeL ₂ . At C _L = 2 C _Me , we have

{2[Me}^{+ 2}] + 2 \sum_{i = 1}^{I} {[Me(OH)}_{i}^{+ 2 - i}] + \sum_{[194590 44]}^{j} J (2 - j {)[MeL}_{j}^{+ 2 - j}] = {[L}^{- 1}]

E43

From Eqs. ( 40 ) and ( 41 )

α = {[H}^{+ 1}] - {[OH}^{- 1}] = \sum_{i = 1}^{I} i {[Me(OH)}_{i}^{+ 2 - [19459053 ]i}]

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

{2[Me}^{+ 2}]^{3/2} \cdot (1 + (1 + \sum_{i = 1}^{I} x_{i}) + K_{sp} {[19459391 ]}^{1/2} \cdot {[Me}^{+ 2}] \cdot \sum_{j = 1}^{J} (2 - j) K_{j} {[L}^{- 1}] - K_{sp [1945 9050]}^{1/2} = 0

E45

where

\begin{matrix} [{Me}^{+ 2}] = \frac{α}{\sum_{i = 1}^{I} i \cdot x_{i}}; α = {[H}^{+ 1}] - {[OH}^{-1}] = 10^{- pH} - 10^{pH - p K_{W}} {;[L}^{- 1 [194590 50]}] \\ = {(\frac{K_{sp}}{{[Me}^{+ 2}]})}^{1/2}; x_{i} = K_{i}^{OH} \cdot {(K_{W} {/[H}^{+ 1}])}^{i} \end{matrix}

E8000

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

2 \cdot [1945904 4] (1 + \sum_{i = 1}^{3} x_{i}) \cdot \frac{{[Me]}^{2}}{K_{sp}^{1/2}} + K_{1} \cdot {[19 459048] [Me]}^{3/2} - (K_{3} \cdot K_{sp} + 1) \cdot {[Me]}^{1/2} - 2 \cdot K_{4} \cdot K_{sp}^{3/2} = 0 [19 459050]

E46

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:

s = s_{Me} = {[Me}^{+ 2}] + \sum_{i = 1}^{I} {[Me(OH)}_{i}^{+ 2 - i}] [1945905 8] + \sum_{j = 1}^{J} {[MeL}_{j}^{+ 2 - j}]

E47

s = s_{L} = {[L}^{- 1} [1 9459048] ] + \sum_{j = 1}^{4} j [{MeL}_{j}^{+ 2 - j}]

E48

The calculations of s _Me and s _L for the precipitates specified in Table 4 can be realized with use of E xcel spreadsheet, according to zeroing procedure, as suggested above ( Table 1 ).

[19459292 ] 0.004466836

pH	[Pb ⁺² ]	[PbOH ⁺¹ ]	[Pb(OH) ₂ ]	[Pb(OH) ₃ ^-1 ]	[PbCl ⁺¹ ]	[PbCl ₂ ]	[PbCl ₃ ⁻¹ ]	[PbCl ₄ ⁻² ]	[Cl ⁻¹ ]	y
4.5343	0.010749606	2.92208E-05 [1 9459291]	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 [194592 91]	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	6.91359E-05	2.45136E-07	0.038877983	-0.000105799

Table 5.

Fragment of display for PbCl ₂ .

For PbI ₂ : pH ₀ = 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 pH ₀ -value.

pH	[Pb ⁺² ]	[PbOH ⁺¹ ]	[Pb(OH) ₂ ]	[Pb(OH) ₃ ⁻¹ ]	[PbI ⁺¹ ]	[PbI ₂ ]	[PbI ₃ ⁻¹ ]	[PbI ₄ ⁻² ]	[I ⁻¹ ]	y
5.15	0.000630817	7.07789E-06 [194592 91]	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	6.6 0693E-07	3.5518E-09	1.44843E-11	0.00128958	-0.000105799

Table 6.

Fragment of display for PbI ₂ .

For HgI ₂ : pH ₀ = 6.7769, s _Hg = 1.91217∙10 ⁻⁵ , s _I = 3.82435∙10 ⁻⁵ , see Table 7 . The difference between s _I and 2 s _Hg = 3.82434∙10 ⁻⁵ results from rounding the pH-value. The concentration [HgI ₂ ] = K ₂ K _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 .

[19 459292] 1.12848E-08

pH	[Hg ⁺² ]	[HgOH ⁺¹ ]	[Hg(OH) ₂ ]	[Hg(OH) ₃ ⁻¹ ]	[HgI ⁺¹ ]	[HgI ₂ ]	[HgI ₃ ⁻¹ ]	[HgI ₄ ⁻² ]	[I ⁻¹ ]	y
6.7767	2.99681E-15	3.57569E-12 [1945 9291]	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 [194592 91]	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	1.88359E-13	9.82863E-08	-9.84182E-11

Table 7.

Fragment of display for HgI ₂ .

4.4. Dissolution of CaCO ₃ in the presence of CO ₂

The portions 0.1 g of calcite CaCO ₃ ( M = 100.0869 g/mol, d = 2.711 g/cm ³ ) are inserted into 100 mL of: pure water (task A) or aqueous solutions of CO ₂ specified in the tasks: B1, B2, B3, and equilibrated. Denoting the starting ( t = 0) concentrations [mol/L]: C ^o for CaCO ₃ and $C_{{CO}_{2}}$ for CO ₂ in the related systems, on the basis of equilibrium data collected in Table 8 :

(A) we calculate pH = pH ₀₁ and solubility s = s(pH ₀₁ ) of CaCO ₃ at equilibrium in the system;
(B1) we calculate pH = pH ₀₂ and solubility s = s(pH ₀₂ ) of CaCO ₃ in the system, where $C_{{CO}_{2}}$ refers to saturated (at 25 ^o C) solution of CO ₂ , where 1.45 g CO ₂ dissolves in 1 L of water [ 39 ].
(B2) we calculate minimal $C_{{CO}_{2}}$ in the starting solution needed for complete dissolution of CaCO ₃ in the system and the related pH = pH ₀₃ value, where s = s(pH ₀₃ ) = C ^o ;
(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 V ₀ = 100 mL of the system with CaCO ₃ presented in (B1). The quasistatic course of the titration is assumed.

[19 459290] [H ⁺¹ ][CO ₃ ⁻² ] = K ₂ [HCO ₃ ^-1 ]

No.	Reaction	Expression for the equilibrium constant	Equilibrium data
1	CaCO ₃ = Ca ⁺² + CO ₃ ⁻²	[Ca ⁺² ][CO ₃ ⁻² ] = K _sp	pK _sp = 8.48
2	Ca ⁺² + OH ⁻¹ = CaOH ⁺¹ [194 59291]	[CaOH ⁺¹ ] = K ₁₀ [Ca ⁺² ][OH ⁻¹ ]	log K ₁₀ = 1.3
3	H ₂ CO ₃ = H ⁺¹ + HCO ₃ ⁻¹	[H ⁺¹ ][HCO ₃ ⁻¹ ] = K ₁ [H ₂ CO ₃ ]	pK ₁ = 6.38
4	HCO ₃ ⁻¹ = H ⁺¹ + CO ₃ ⁻²	pK ₂ = 10.33
5	Ca ⁺² + HCO ₃ ⁻¹ = CaHCO ₃ ⁺¹	[CaHCO ₃ ⁺¹ ] = K ₁₁ [Ca ⁺² ][HCO ₃ ⁻¹ ]	log K ₁₁ = 1.11
6	Ca ⁺² + CO ₃ ⁻² = CaCO _{3 [19459 019]}	[CaCO ₃ ] = K ₁₂ [Ca ⁺² ][CO ₃ ⁻² ]	log K ₁₂ = 3.22
7	Ca(OH) ₂ = Ca ⁺² + 2OH ⁻¹	[Ca ⁺² ][OH ⁻¹ ] ² = K _sp1	pK _sp1 = 5.03
8	H ₂ O = H ⁺¹ + OH ⁻¹	[H ⁺¹ ][OH ⁻¹ ] = K _W	pK _W = 14.0

The volume 0.1/2.711 = 0.037 cm ³ of introduced CaCO ₃ is negligible when compared with V ₀ at the start ( t = 0) of the dissolution. Starting concentration of CaCO ₃ in the systems: A, B1, B2, B3 is C ^o = (0.1/100)/0.1 = 10 ⁻² mol/L. At t > 0, concentration of CaCO ₃ is c ^o mol/L. The balances are as follows:

C^{o} = c^{o} + [{Ca}^{+ 2}] + [{CaOH}^{+ 1}] + [{CaHCO}_{3}^{+ 1}] + [ [ 19459053]{CaCO}_{3}](for A,B1,B2,B3)

E49

C^{o} = c^{o} + [{CaHCO}_{3}^{+ 1}] + [{CaCO}_{3}] + [H_{2} {CO}_{3}] + [{HCO}_{3}^{- 1}] + {CO}_{3}^{- 2}] (for A)

[1945 9062] E50

\begin{matrix} C^{o} + C_{CO 2} = c^{o} + [{CaHCO}_{3}^{+ 1}] + [{CaCO}_{3}] + [H_{2} {CO}_{3}] + [{HCO}_{3}^{- 1}] \\ + [{CO}_{3}^{- 2}] (for B 1, B 2, B 3) \end{matrix}

E51

\begin{matrix} [H^{+ 1}] - [{OH}^{- 1}] [1945 9053] + 2 [{Ca}^{+ 2}] + [{CaOH}^{+ 1}] + [{CaHCO}_{3}^{+ 1}] - [{HC O}_{3}^{- 1}] \\ - 2 [{CO}_{3}^{- 2}] = 0 (for A, B 1, B 2) \end{matrix}

[19 459061] E52

\begin{matrix} [H^{+ 1}] - [{OH}^{- 1}] + [M^{+ 1}] + 2 [{[19 459044]}^{Ca} + 2] + [{CaOH}^{+ 1}] + [{CaHCO}_{3}^{+ 1}] \end{matrix} - [{HCO}_{3}^{- 1}] - 2 [{CO}_{3}^{- 2}] = 0 (for B 3)

E52a

where [M ⁺¹ ] = C _b V /( V ₀ + V ).

For (A)

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

$[{Ca}^{+ 2}] + [{CaOH}^{+ 1}] = [H_{2} {CO}_{3}] + [ [194 59053]{HCO}_{3}^{- 1}] + [{CO}_{3}^{- 2}]$ E53

Considering the solution saturated with respect to CaCO ₃ and denoting: f ₁ = 10 ^16.71−2pH + 10 ^10.33−pH + 1, f ₂ = 1 + 10 ^pH−12.7 , from Eq. ( 53 ) and Table 1 , we have the relations:

$\begin{array}{l} [{Ca}^{+ 2}] f_{2} = [{CO}_{3}^{- 2}] \cdot f_{1} \Rightarrow [[1945905 4] Ca + 2] = 10^{- 4.24} \cdot {(f_{1} / f_{2})}^{0.5}; [{CO}_{3}^{- 2}] = 10^{- 4.24} \cdot {(f_{2} / f_{1})}^{0.5}; \\ [{CaOH}^{+ 1}] = 10^{pH - 16.94} \cdot {(f_{1} / f_{2})}^{0.5}; \\ \begin{array}{l} [{CaCO}_{3}] = 10^{- 5.26}; [{CaHCO}_{3}^{+ 1}] = 10^{2.96 - pH}; [{HCO}_{3}^{- 1}] = 10^{6.09 - pH} \cdot {(f_{2} / f_{1})}^{0.5}; [H^{+ 1}] \end{array} \\ = 10^{- pH}; [{OH}^{- 1}] = 10^{pH - 14} . \end{array}$ E8300

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

$\begin{matrix} z = z (pH) = 10^{- pH} - 10^{pH - 14} + 2 \cdot [194590 53]10^{- 4.24}\cdot{(f_{1} / f_{2})}^{0.5}+10^{pH - 16.94}\cdot{(f_{1} / f_{[194 59074] 2})}^{0.5} \\ + 10^{2.96 - pH} - 10^{6.09 - pH} \cdot {(f_{2} / f_{1})}^{0.5} - 2 \cdot 10^{- 4.24} \cdot {(f_{2} / f_{1})}^{0.5} \end{matrix}$ E54

and applying the zeroing procedure to the function ( 54 ), we find pH ₀₁ = 9.904, at z = z (pH ₀₁ ) = 0. The solubility s = s(pH) of CaCO ₃ , resulting from Eq. ( 49 ), is

$s = [{Ca}^{+ 2}] + [{CaOH}^{+ 1}] + [{CaHCO}_{3}^{+ 1}] + [1 9459052] [{CaCO}_{3}]$ E55

$= 10^{- 4.24} \cdot {(f_{1} / f_{2})}^{0.5} + 10^{[19459 048] pH - 16.94} \cdot {(f_{1} / f_{2})}^{0.5} + 10^{2.96 - pH} + 10^{- 5.26}$ E55a

We have s = s(pH = pH ₀₁ ) = 1.159⋅10 ⁻⁴ mol/L.
For (B1)

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

$\begin{matrix} [H_{2} {CO}_{3}] + [{HCO}_{3}^{- 1}] + [{CO}_{3}^{- 2}] [ 19459052] − ([{Ca}^{+ 2}] + {[CaOH}^{+ 1}]) = C_{CO 2} \\ \Rightarrow [{CO}_{3}^{- [19459 074] 2}] \cdot f_{1} - [C a^{+ 2}] \cdot f_{2} - C_{CO 2} = 0 \Rightarrow {[{Ca}^{+ 2 [1945905 0]}]}^{2} \cdot f_{2} + C_{CO 2} \cdot [{Ca}^{+ 2}] - K_{sp} \cdot f_{1} = 0 \end{matrix}$ E8400

In this case,

${[Ca}^{+ 2}] = \frac{\sqrt{{{(C}_{{CO}_{2}})}^{2} + 4 \cdot K_{sp} \cdot f_{1} \cdot [194 59045] f 2} - C_{{CO}_{2}}}{2 \cdot f_{2}}$ E56

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

$s = [{Ca}^{+ 2}] \cdot f_{2} + 10^{2.96 - pH} + 10^{- 5.26}$ E57

and the charge balanc e is transformed into the zeroing function

$\begin{matrix} z = z (pH) = 10^{- pH} - 10^{pH - 14} + [{Ca}^{+ 2}] \cdot (2 + 10^{pH - 12.7}) + 10^{2.96 - pH} \\ - [{CO}_{3}^{- 2}] \cdot ({10 [19 459075]}^{10.33 - pH} + 2) \end{matrix}$ E58

where [CO ₃ ⁻² ] = 10 ^-8.48 /[Ca ⁺² ], and [Ca ⁺² ] is given by Eq. ( 56 ). Eq. ( 58 ) zeroes at pH = pH ₀₂ = 6.031. Then from Eq. ( 57 ) we calculate s = s(pH ₀₂ ) = 6.393∙10 ⁻³ mol/L, at pH = pH ₀₂ = 6.031.
For (B2)

At pH = pH ₀₃ , where c ^o = 0, i.e., s = C ^o , the solution (a monophase system) is saturated toward CaCO ₃ , i.e., the relation [Ca ⁺² ][CO ₃ ⁻² ] = K _sp is still valid. Applying Eqs. ( 56 ) and ( 57 ), we find pH values zeroing Eq. ( 58 ) at different, preassumed $C_{{CO}_{2}}$ values. Applying these pH-values in Eq. ( 57 ), we calculate the related s = s(pH, $C_{{CO}_{2}}$ ) values (Eq. ( 57 ), Table 9 ). Graphically, $C_{{CO}_{2}}$ = 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

$\begin{matrix} z = z (pH, V) = 10^{- pH} - 10^{pH - 14} + {[19 459048] C}_{b} V / (V_{0} + V) + [{Ca}^{+ 2}] \cdot (2 + 10^{pH - 12.7}) \\ + {10 [1945 9075]}^{2.96 - pH} - [{CO}_{3}^{- 2}] \cdot (10^{10.33 - pH} + 2) \end{matrix}$ 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.

$C_{{CO}_{2}}$	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
[194590 15] 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

[1945 9295] Table 9.

The set of points used for searching the $C_{{CO}_{2}}$ value at s = C ^o = 0.01; at this point, we have pH ₀₃ = 5.683.

Figure 2.

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

Reaction	Equilibrium data
H ₂ O = H ⁺¹ + OH ^-1	pK _w = 14.0
$H_{2} {CrO}_{4} = H^{+} + {HCrO}_{4}^{- 1} [194590 59]$	pK ₁ = 0.8
${HCrO}_{4}^{- 1} {= H}^{+} + {CrO}_{4}^{- 2}$	pK ₂ = 6.5
${HCr}_{2} O_{[ 19459074] 7}^{- 1} = H^{+ 1} + {Cr}_{2} O_{7}^{- 2}$	log K ₃ = 0.07
$2 {HCrO}_{4}^{- 1} = {Cr}_{2} O_{7}^{- 2} + H_{2} O$	log K ₄ = 1.52
Ag ⁺¹ + OH ⁻¹ = AgOH	log K ₁ ^OH = 2.3
${Ag}^{+ 1} + 2 {OH}^{- 1} = Ag {(OH)}_{2}^{- 1}$	log K ₂ ^OH = 3.6
${Ag}^{+ 1} + 3 {OH}^{- 1} = [194590 48] Ag {(OH)}_{3}^{- 2}$	log K ₃ ^OH = 4.8
$A g_{2} C r O_{4} = 2 {Ag}^{+ 1} + {CrO}_{4}^{- 2}$	pK _sp1 = 11.9
$A g_{2} C r_{2} O_{7} = 2 {Ag}^{+ [194 59053]1} + {Cr}_{2} O_{7}^{- 2}$	pK _sp2 = 6.7
$A g O H = {Ag}^{+1} + {OH}^{- 1} [19459 059]$	pK _sp3 = 7.84

Table 10.

Physicochemical equilibrium data relevant to the Ag ₂ Cr ₂ O ₇ + H ₂ O system ( pK = −log K ), at “room” temperatures.

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 ⁻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 [A g_{2} C r_{2} O_{7}] + [{Ag}^{+ 1}] + [AgOH] + [{Ag(OH)}_{2}^{- 1}] + [{Ag(OH)}_{3}^{- 2}] = 2 C_{0}

E60

2 [A g_{2} C r_{2} O_{7}] + [H_{2} {CrO}_{4}] + [1 9459049][{HCrO}_{4}^{- 1}]+[{CrO}_{4}^{- 2}]+2[{HCr}_{2} O_{7}^{- 1}]+ [1945 9053]2[{Cr}_{2} O_{7}^{- 2}]=2C_{0}

E61

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 [19459 019] ] 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}

\begin{matrix} [H^{+ 1}] - [{OH}^{- 1}] + [{Ag}^{+ 1}] - [Ag {(OH)}_{2}^{- 1}] - 2 [Ag {(OH)}_{3}^{- 2}] - [{HCrO}_{4}^{[ 19459044]}] \\ - 2 [{CrO}_{4}^{- 2}] - [{HCr}_{2} O_{7}^{- 1}] - 2 [[1 9459044] {Cr}_{2} O_{7}^{- 2}] = 0 \end{matrix}

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{matrix} {[H}_{2} {CrO}_{4}] = 10^{7 .3 - 2pH} \cdot {[CrO}_{4}^{- 2}]; {[HCrO}_{4}^{- 1}] = 10^{6 .5 - pH} \cdot {[CrO}_{4}^{- 2}]; \\ {[HCr}_{2} O_{7}^{- 1}] = 10^{14 .59 - 3pH} \cdot {{[CrO}_{4}^{- 2}]}^{2}; \end{matrix}

E63

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

E63a

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

[{Ag}^{+ 1}] + [AgOH] [194 59053] + [Ag {(OH)}_{2}^{- 1}] + [Ag {(OH)}_{3}^{- 2}] = 2 c_{0}

E64

[H_{2} {CrO}_{4}] + [{HCrO}_{4}^{- 1}] + [{CrO}_{4}^{- 2}] [19459 052] + 2 [{HCr}_{2} O_{7}^{- 1}] + 2 [{Cr}_{2} O_{7}^{- 2}] = 2 c_{0}

E65

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

(a) [{Ag}^{+ 1}] = 2 c_{0} / g_{0}; 2 g_{2} {[{CrO}_{4}^{- 2}]}^{2} + g_{1} [{CrO}_{4}^{- 1}] - 2 c_{0} = [194 59053]0\Rightarrow(b)[{CrO}_{4}^{- 2}]=\frac{{(g_{1}^{2} + 16 \cdot c_{0} g_{2})}^{0 .5} - [1945 9045] g 1}{4 \cdot g_{2}}

E66

where g ₀ = 1 + 10 ^pH−11.7 + 10 ^2pH−24.4 + 10 ^3pH−37.2 ; g ₁ = 10 ^7.3−2pH + 10 ^6.5−pH + 1; g ₂ = 10 ^14.59−3pH + 10 ^14.52−2pH . Applying them in Eq. ( 62 ), we get the zeroing function

y = y (pH) = 10^{- pH} - 10^{pH}^{- 14} + g_{3} \cdot [{Ag}^{[19459 044] + 1}] - g_{4} \cdot [{CrO}_{4}^{- 2}] - g_{5} \cdot [{CrO}_{4}^{- 2}] [1945 9050] 2

E67

where g ₃ = 1 – 10 ^2pH−24.4 – 2∙10 ^3pH−37.2 ; g ₄ = 10 ^6.5−pH + 2; g ₅ = 10 ^14.59−3pH + 2∙10 ^14.52−2pH , and [Ag ⁺¹ ] and [CrO ₄ ⁻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 2c ₀ . At particular 2c ₀ values, we calculate pH = pH(2c ₀ ) value zeroing the function ( 67 ), and then calculate the values of the products: q ₁ = [Ag ⁺¹ ] ² [CrO ₄ ⁻² ]/K _sp1 and q ₂ = [Ag ⁺¹ ] ² [Cr ₂ O ₇ ⁻² ]/K _sp2 , where: [Ag ⁺¹ ], [CrO ₄ ⁻² ], and [Cr ₂ O ₇ ⁻² ] are presented above (Eqs. (66a), (66b) and (63a), resp.), pK _sp1 = 11.9, pK _sp2 = 6.7. As results from Figure 3 , where log q ₁ and log q ₂ are plotted as functions of 2c ₀ ; log q ₁ = 0 ⇔ q ₁ = 1 ⇔ [Ag ⁺¹ ] ² [CrO ₄ ⁻² ] = K _sp1 at lower 2c ₀ value, whereas log q ₂ < 0 ⇔ q ₂ < 1 ⇔ [Ag ⁺¹ ] ² [Cr ₂ O ₇ ⁻² ] < K _sp2 , both for pK ₂ = 6.7 and 10, cited in the literature. The x ₁ =1 value is attained at 2c ₀ = 3.5∙10 ⁻⁴ ⟹ c ₀ = 1.75∙10 ⁻⁴ ; then Ag ₂ CrO ₄ precipitates as the new solid phase, i.e., total depletion of Ag ₂ Cr ₂ O ₇ occurs. It means that Ag ₂ Cr ₂ O ₇ is not the equilibrium solid phase in this system. This fact was confirmed experimentally, as stated in [ 42 ], i.e., Ag ₂ Cr ₂ O ₇ is transformed into Ag ₂ CrO ₄ upon boiling with H ₂ O; at higher temperatures, this transformation proceeds more effectively. Concluding, the formula s ^* = ( K _sp2 /4) ^1/3 applied for K _sp2 = [Ag ⁺¹ ] ² [Cr ₂ O ₇ ⁻² ] is not “the best answer,” as stated in Ref. [ 43 ].

Figure 3.

The convergence of logq1 and logq2 to 0 value; Ksp1 is attained at lower 2c0 value.

The system involved with Ag ₂ CrO ₄ 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 = 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

V_{eq} - V_{end} = \frac{K_{sp}}{C} \cdot {(\frac{C_{01} V_{0}}{K_{sp1}})}^{0 .5} \cdot {(V_{0} + V_{end}) [19459 049]}^{0 .5} - \frac{1}{C} \cdot {(\frac{K_{sp1}}{C_{01} V_{0}})}^{0 .5} \cdot {(V_{0} + V [ 19459047]end)}^{1 .5}

E8435

where K _sp = [Ag ⁺¹ ][Cl ⁻¹ ] ( 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 ].

5.2. 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 ]

3 M g N H_{4} P O_{4} = M g_{3} (P O_{4} [19459050 ]) 2 + {HPO}_{4}^{- 2} + {NH}_{3} + 2 {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 CO ₂ ( $C_{{CO}_{2}}$ , 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{matrix} q_{1} & = [{Mg}^{+ 2}] [{NH}_{4}^{+ 1}] [{PO}_{4}^{- 3 [1945905 0]}] / K_{sp 1}, q_{2} = {[{Mg}^{+ 2}]}^{3} {[{PO}_{4}^{- 3}]}^{2 [19459055 ] / K_{sp 2}, q_{3}} \\ = [{Mg}^{+ 2}] [{HPO}_{4}^{- 2}] / K_{sp [1945907 4] 3}, q_{4} [{Mg}^{+ 2}] {[{OH}^{- 1}]}^{2} / K_{sp 4} \end{matrix}

E69

were calcu lated 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 ⁻³ mol/L (pC ⁰ = (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 logqi 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{matrix} s = s_{Mg} = [{Mg}^{+ 2}] + [{MgOH}^{+ 1}] + [{MgH}_{2} {PO}_{4}^{+ 1}] + [M g H P O_{4}] + [{MgPO}_{4}^{- 1}] [19459 050] \\ + [{MgNH}_{3}^{+ 2}] + [Mg {({NH}_{3})}_{2}^{+ 2}] + [Mg [194590 44] ({NH}_{3}) 3 + 2] \end{matrix}

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 CO ₂ from air. Under such conditions, NH ₄ ⁺¹ and NH ₃ occur there at comparable concentrations [NH ₄ ⁺¹ ] ≈ [NH ₃ ], but [HPO ₄ ⁻² ]/[PO ₄ ⁻³ ] = 10 ^12.36−pH ≈ 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.

Figure 6.

The pH versus log[pr2] relationship; pr2 = Mg3(PO4)2, at [ppr1]t=0 = 3. The numbers at the corresponding lines indicate pCO2=−logCCO2 values; pCO2=∞ ⇔ CCO2= 0.

After introducing struvite pr1 (at pC ₀ = [ppr1] _t=0 = 2) into alkaline ( C _b = 10 ⁻² 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 .

Figure 7.

The speciation curves for indicated species Xizi, resulting from dissolution of pr1 = MgNH4PO4, at (pC0, pCO2, pCb) = (2, 4, 2); s′ is defined by Eq. (71).

In step 1, pr4 precipitates first, pr1 + 2OH ⁻¹ = pr4 + NH ₃ + HPO ₄ ⁻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 + 2NH ₃ + 2H ₂ O occurs up to total depletion of pr4 (at ppr1 = 2.896). In this step, the reaction 3pr1 + 2OH ⁻¹ = pr2 + 3NH ₃ + HPO ₄ ⁻¹ + 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′ ( Figure 7 ) is related to the function

s^{'} = s + [{MgHCO}_{3}^{+ 1}] + [{MgCO}_{3}]

E71

where s is expressed by Eq. ( 70 ).

6. Solubility of nickel dimethylglyoximate

The precipitate of nickel dimethylglyoximate, NiL ₂ , has soluble counterpart with the same formula, i.e., NiL ₂ , in aqueous media. If NiL ₂ is in equilibrium with the solution, concentration of the soluble complex NiL ₂ assumes constant value: [NiL ₂ ] = K ₂ ∙[Ni ²⁺ ][L ⁻ ] ² = K ₂ ∙ K _sp , where K ₂ = 10 ^17.24 , K _sp = [Ni ²⁺ ][L ⁻ ] ² = 10 ^−23.66 [ 14 , 17 , 18 ], and then [NiL ₂ ] = 10 ^−6.42 (i.e., log[NiL ₂ ] = −6.42). The concentration [NiL ₂ ] is the constant, limiting component in expression for solubility s = s _Ni of nickel dimethylglyoximate, NiL ₂ . 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 NiL ₂ is precipitated from NiSO ₄ 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]: CNi = 0.001, CL = 0.003, CN = 0.5, CAc = 0.3, CCit = 0.1 [14].

s = s_{Ni} = {[Ni}^{+ 2}] + {[NiOH}^{+ 1}] + {[NiSO}_{4}] + \sum_{i = 1}^{6} {[Ni(NH}_{3})_{i}^{+ 2}] + {[NiL}_{2}]

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 NiL ₂ in ammonia buffer media, i.e., at pH ≈ 9, where s _Ni ≅ [NiL ₂ ].

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 – 2 C _Ni = 0.001 mol/L. Solubility of HL in water, equal 0.063 g HL/100 mL H ₂ O (25 ^o C) [ 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 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

\frac{1}{s} \cdot {(\frac{\partial s}{\partial T})}_{p} = \frac{1}{R T^{2}} \cdot (Δ G_{1}^{o} + Δ G_{2}^{o})

E74

where

\begin{matrix} Δ G_{1}^{o} = R T^{2} \cdot {(\frac{\partial \ln K_{s p}}{\partial T})}_{p} & and & Δ G_{2}^{o} = R T^{2} \cdot (\frac{\partial \ln K_{2}}{[19459 044]}) p \end{matrix}

E8436

Because, as a rule,

\begin{matrix} {(\frac{\partial K_{s p}}{\partial T})}_{p} > 0 & [ 19459048] and & {(\frac{\partial K_{2}}{\partial T})}_{p} < 0 \end{matrix}

E8437

then $Δ G_{1}^{o} > 0$ and $Δ {[ 19459046] G}_{2}^{o} < 0$ , and Eq. ( 74 ) can be rewritten into the form

\frac{1}{s} \cdot {(\frac{\partial s}{\partial T})}_{p} = \frac{1}{R T^{2}} \cdot (| Δ G_{1}^{o} | [1945905 2] − | Δ G_{2}^{o} |)

E75

If $| Δ G_{1}^{o} | \approx | Δ G_{2}^{o} |$ 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 ≠ 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.

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: 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 _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 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 \leftrightarrow Approach I to 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

pr-GEB = 2 \cdot F (O) - F (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 – [194597 08] 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 (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 ⁻¹ ) 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 ⁻¹ = 2 CuI + I ₂ , 2Cu ⁺² + 5I ⁻¹ = 2 CuI + I ₃ ⁻¹ is titrated with Na ₂ S ₂ O ₃ solution (step 4), until the reduction of iodine: I ₂ + 2S ₂ O ₃ ⁻² = 2I ⁻¹ + S ₄ O ₆ ⁻² , I ₃ ⁻¹ + 2S ₂ O ₃ ⁻² = 3I ⁻¹ + S [19 459018] 4 O ₆ ⁻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 H ₂ S ₂ O ₃ = H ₂ SO ₃ + S [ 80 ].

7.3. 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 ⁻² is not involved here in electron-transfer equilibria between S ₂ O ₃ ⁻² and S ₄ O ₆ ⁻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:

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} [H^{+ 1}] = 10^{- pH}, [{OH}^{- 1}] = 10^{pH}^{- 14} (p [19459047 ]K_{W}=14),[{CuOH}^{+ 1}]=10^{7.0}\cdot[{Cu}^{+ 2}][{OH}^{-1}],[Cu(OH)_{2 [1 9459051]] = 10^{13.68} \cdot [{Cu}^{+ 2}] [{OH}^{- 1}]^{2},} \\ [Cu (OH)_{3}^{- 1}] = 10^{17.0} \cdot [{Cu}^{+ 2}] [{OH}^{- 1}]^{3}, [Cu (OH)_{4}^{- 2}] = 10^{18.5} \cdot [ [19 459053]{Cu}^{+ 2}][{OH}^{- 1}]^{4},[{CuNH}_{3}^{+ 2}]=10^{3.39}\cdot[{Cu}^{+ 2}][{NH}_{3}], \\ [Cu ({NH}_{3})_{2}^{+ 2}] = 10^{7.33} \cdot [{Cu}^{+ 2}] [{NH}_{3}]^{2}, [Cu ({NH}_{3})_{3}^{2 +}] = 10^{10.06} \cdot [{Cu}^{+ 2}] [{NH}_{3}]^{3}, [Cu ({NH}_{3 [19459075 ]})_{4}^{+ 2}] = 10^{12.03} \cdot [{Cu}^{+ 2}] [{NH}_{3}]^{4}, \\ [{CuSO}_{4}] = 10^{2.36 [19459 050]} \cdot [{Cu}^{+ 2}] [{SO}_{4}^{- 2}], [{NH}_{4}^{+ 1}] = 10^{9.35} \cdot [H^{+ 1}] [[194 59045] NH 3], [{HSO}_{4}^{- 1}] = 10^{1.8} \cdot [H^{+1}] [{SO}_{4}^{- 2}], \\ [{CH}_{3} COOH [19459049 ]]=10^{4.65}\cdot[H^{+ 1}][{CH}_{3}{COO}^{- 1}],[{Cu}^{+ 1}][I^{- 1}]= [19459 053]10^{- 11.96}(solubitlityproductforC u I), \\ [{CuI}_{2}^{- 1}] = 10^{8.85} \cdot [{Cu}^{+ 1} [1945907 6] ] [I^{- 1}]^{2}, [{CuIO}_{3}^{+ 1}] = 10^{0.82} \cdot [{Cu}^{+ 2}] [{IO}_{3}^{- 1}] [194 59052] , [{CuCH}_{3} {COO}^{+ 1}] = 10^{2.24} \cdot [{Cu}^{+ 2}] [{CH}_{3} {COO}^{- 1}], \\ [Cu ({CH}_{3} COO)_{2}] = 10^{3.30} \cdot [{Cu}^{+ 2}] [{CH}_{3} {COO}^{- 1}]^{2}, [{HS}_{2} O_{[19459 074] 3}^{- 1}] = 10^{1.72} \cdot [H^{+ 1}] [S_{2} O_{3}^{- 2}], [H_{2} S_{2} O_{3 [194590 51]] = 10^{2.32} \cdot [H^{+ 1}]^{2} [S_{2} O_{3}^{- 2}],} \\ [{CuS}_{2} O_{3}^{- 1} \end{array}

E8420

Applying A = 16.92 [ 16 ], we have

[1 9459743] $\begin{array}{l} \begin{array}{l} [{Cu}^{+ 2}] = [{Cu}^{+ 1}] \cdot 10^{A (E - 0.153)}; [I_{2}] = {[I^{- 1}]}^{2} \cdot 10^{2 A (E - 0.621)}, s = 1.33 \cdot 10^{- 3} [1945 9048] mol/L (solubility of I_{2}_{(s)}), \end{array} \\ \begin{array}{l} [I_{3}^{- 1}] = {[I^{- [19459074 ] 1}]}^{3} \cdot 10^{2 A (E - 0.545)}, [{IO}^{- 1}] = [I^{-}] \cdot 10^{2 A (E [19 459052] – 0.49) + 2 pH - 28}, [HIO] = 10^{10.6} \cdot [H^{+ 1}] [{IO}^{- 1}], [{IO [19 459049]}_{3}^{- 1}] \end{array} \\ = [I^{- 1}] \cdot 10^{6 A (E - 1.08) + 6 pH}, \\ \begin{array}{l} [194 59477] \\ [{HIO}_{3}] = 10^{0.79} \cdot [H^{+ 1}] [{IO}_{3}^{- 1}], [H_{5} {IO}_{6}] [19459 053]=[I^{- 1}]\cdot10^{8 A (E - 1.24) + 7 pH},[H_{4}{IO}_{6}^{- 1}] \end{array} \end{array} [1945 9477] = [H_{5} {IO}_{6}] \cdot 10^{- 3.3 + pH}, [H_{3} {IO}_{6}^{- 2}] = [194590 76] [I^{- 1}] \cdot 10^{8 A (E - 0.37) + 9 pH - 126} .$ E8438

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

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} = [{Cu}^{+ 2}] + \sum_{i = 1}^{4} [Cu {(OH)}_{i}^{+ 2 - [1945905 3]i}] + \sum_{i = 1}^{4} [Cu {({NH}_{3})}_{i}^{+ 2}] + [{CuSO}_{4}] + [{[194 59048] CuIO}_{3}^{+ 1}] \\ + \sum_{i = 1}^{2} [Cu {({CH}_{3} COO)}_{i}^{+ 2 - i}] + [1945907 6] [{Cu}^{+ 1}] + [{CuI}_{2}^{- 1}] + \sum_{i = 1}^{2} [Cu {({NH}_{3})}_{i}^{[1 9459052] + 1}] \end{array}

E88

and in stage 4

s = s_{4} = s_{3} + \sum_{i = 1}^{3} [{Cu(S}_{2} O_{3})_{i}^{+ 1 - 2 i}]

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 = 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 ⁻¹ ( Figure 9 , stage 4). The species I ₃ ⁻¹ 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.

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 10 a 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.

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 = 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 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 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.

Post Views: 10

1. Introducción

2. Definiciones y formulación de productos de solubilidad

3. Solubility product(s) for MnO 2

Figure 1.

4. Calculation of solubility

4.1. Formulation of the solubility s *

4.2. Dissolution of hydroxides

Table 1.

Table 2. [1 9459236] Zeroing the function ( 32 ) for the system with Fe(OH) 3 precipitate introduced into pure water (copy of a fragment of display).

Table 3.

4.3. Dissolution of MeL 2 -type salts

Table 4.

Table 5.

Table 6.

Table 7.

4.4. Dissolution of CaCO 3 in the presence of CO 2

Figure 2.

5. Nonequilibrium solid phases in aqueous media

5.1. Silver dichromate (Ag 2 Cr 2 O 7 )

Table 10.

Figure 3.

5.2. Dissolution of struvite

Figure 4.

Figure 5.

Figure 6.

Figure 7.

6. Solubility of nickel dimethylglyoximate

Figure 8.

7. Calculation of solubility in dynamic redox systems

7.1. Preliminary information

7.2. Solubility of CuI in a dynamic redox system

7.3. Formulation of the system

Figure 9.

Figure 10.

Figure 11.

8. Final comments

3. Solubility product(s) for MnO ₂

4.1. Formulation of the solubility s ^*

Table 2. [1 9459236]

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

4.3. Dissolution of MeL ₂ -type salts

4.4. Dissolution of CaCO ₃ in the presence of CO ₂

5.1. Silver dichromate (Ag ₂ Cr ₂ O ₇ )