Received 3 July 2014; revised 3 August 2014; accepted 10 August 2014

ABSTRACT

A uniform procedure is suggested for calculation of the pH_t value(s) separating equilibrium solid phases in pH scale, at an excess of the precipitating agent. The pH_t value, related to pairs of precipitates formed from the species and , fulfils the relation, where F is a constant value involving pK_so’s for solubility products (’s) of these precipitates, and the equilibrium data, related to the species composing these precipitates.

Keywords:

Electrolytic Systems, Precipitates, pH-Intervals

1. Introduction

Some species are able to form different solid phases in aqueous media whose composition depends on pH-val- ue of these media. In particular, this was indicated for the systems obtained after introducing the ternary salts such as struvite [1] or dolomite [2] into pure water or aqueous solution of a strong base in presence/absence of CO₂, originating e.g. from air. Full physicochemical knowledge was involved in the algorithms used for calculations made according to iterative computer programs related to redox or non-redox, mono- or two-phase systems [3] - [8] .

^*Corresponding author.

This paper concerns calculations related to two-phase systems, and made with use of Excel spreadsheets. It refers to location of different equilibrium solid phases within defined pH-intervals [9] - [11] . The search of these pH-intervals is based on the simplified calculation procedure. The pH-values separating these intervals are named as transition points, and denoted as.

2. Formulation of the Transition Points

Let the precipitates and, characterized by solubility products:

(1)

(2)

be two equilibrium solid phases formed in an aqueous system involving and ions, together with the and species resulting from hydrolytic phenomena; other (possible) soluble complexes formed between the related species are omitted (not involved) in the related balances. The numbers: a, b, c, d, u, n, k and m in (1) and (2) satisfy the conditions of electro neutrality of the corresponding precipitates:

(3)

(4)

We assume that the Me-species are precipitated with an excess of the L-species; this excess is expressed by the molar concentration:

(5)

If the protonated species do not exist, then. Applying the stability constants of the proto-complexes, , we denote:

(6)

where

(7)

and. Assuming, and the equilibrium solid phases: (at) and (at), we state that at transitional value, the solubility products: and are fulfilled simultaneously, and then from (1) and (2) we get:

(8)

Applying in (8) the relations (3) and (4), we have and then, by turns,

(9)

where,. Similarly, when the relations: (2) and (10):

(10)

are valid simultaneously at, we have, by turns,

(11)

Note that is identical with at, , and then (see Equation (4)).

Equations (8) and (10) involve the term on the left side and defined numbers on the right side— irrespectively on the a, b, c, d, k and m values. The same regularity is fulfilled, after all, for different sets of parameters: a, b, c and k, in precipitates of type, where. From (6) we have

(12)

and then

(13)

In each case, is an increasing function of. This means, in particular, that larger values correspond to larger values. This circumstance is particularly important when arranging the equilibrium solid phases along the axis, when the number of possible solid phases is.

3. Transition Point for Carbonates

Many divalent cations form sparingly soluble carbonates MeCO₃ and hydroxides Me(OH)₂. In this case, we have:

(14)

(15)

(16)

(17)

(18)

The curve of vs. pH relationship is plotted in Figure 1 at. The numerical value of expression on the right side of Equation (18), related to defined Me⁺² ion, forms a straight line parallel to pH-axis (see Figure 1). The abscissa of the point of intersection of this line with the curve vs. relationship indicates the value, separating the -intervals for MeCO₃ and Me(OH)₂, as the equilibrium solid phases. For example, calculated for the pair (ZnCO₃, Zn(OH)₂) corresponds to (see Figure 1). The values found this way for different Me⁺² ions are collected in Table 1.

4. Transition Points for Lead Phosphates

For, we have, among others, three solid phases: PbHPO₄, Pb₃(PO₄)₂ and Pb(OH)₂, defined by the solubility products:

(19)

(20)

(21)

In this system, the physicochemical data related to another solid phases: Pb₅(PO₄)₃OH and Pb₄O(PO₄)₂ as precipitates are also cited in literature [12] [13] ; however, the solubility products for these species are formulated there in an unconventional manner. The unification of the solubility products to conventional notation will be the first, preparatory step for further considerations. The expressions for solubility products, formulated unconventionally, will be denoted as (asterisked, with the corresponding subscripts, specifying their stoichiometric composition). We have:

and then

The values:

(22)

(23)

refer to the reactions:, (see Appendix).

At, we assume (this assumption will be verified later) that the solubility products for PbHPO₄ and Pb₃(PO₄)₂ are fulfilled simultaneously. From Equations (19), (20) and (13) we get:

(24)

(25)

where (see Equation (7))

(26)

and, , , (on the basis of [9] , where, ,). The relation (24) agrees with Equation (9), for, , , ,. Similarly, when assuming that the solubility products for Pb₃(PO₄)₂ and Pb(OH)₂ are fulfilled simultaneously at, we get:

(27)

The complete set of values for, related to different pairs of precipitates specified in Equations (19)-(23), is presented in Table 2. Comparing the y-values in the first line of Table 2, we state that the lowest value corresponds to the pair (PbHPO₄, Pb₃(PO₄)₂); this means that Pb₃(PO₄)₂ follows PbHPO₄ on the -scale. Next, considering the -values in the second line of Table 2, we state that the lowest -value corresponds to the pair (Pb₃(PO₄)₂, Pb₅(PO₄)₃OH), i.e. Pb₅(PO₄)₃OH is the next precipitate on the pH-scale. Referring to the third line of Table 2, we state that the lower -value corresponds to the pair (Pb₅(PO₄)₃OH, Pb₄O(PO₄)₂), i.e. Pb₄O(PO₄)₂ is the next precipitate on the -scale. Finally, corresponds to the pair (Pb₄O(PO₄)₂, Pb(OH)₂). From the curve in Figure 1, we find the transition points as the abscissas for ; the values separating intervals of the equilibrium solid phases are specified in the lower part of the Table 2.

The curve of vs. pH relationship is plotted in Figure 2 at. In particular, the curve intersects the line at, separating the solid phases: PbHPO₄ and Pb₃(PO₄)₂ in -scale (see Table 2).

5. Crossing the pH Scale

In some cases, the precipitate of sparingly soluble salt is characterized by a relatively small solubility product value. Consequently, the value, separating the range of the salt and the corresponding hydroxide as the equilibrium solid phases, is significantly higher than the pH value, practically obtainable by addition of a strong base. In other instances, ions form soluble hydroxo-complexes up to, characterized by the stability constant value, with. When value of the solution is high―the hydroxide is not an equilibrium solid phase when, where is the total concentration of Me in the system, is defined by Equation (10).

As an example, let us take the precipitates: ZnS and Zn(OH)₂. Applying, (and for dissociation constants and of), we get. The as abscissa related to this -value is much higher than 14 (see Figure 3); what is more, it is much higher than values of a saturated strong base. Moreover, at high values, Zn(OH)₂ is transformed into soluble complexes, mainly .

Another example is the system with precipitates: CaC₂O₄ and Ca(OH)₂. Applying, (,for dissociation constants and of H₂C₂O₄), we get (see Figure 4); this value corresponds to, related to calculated value of 4.9 mol/L NaOH. The Ca(OH)₂ does not dissolve in an excess of strong base; Ca⁺² forms only one hydroxo-complex, CaOH⁺¹, and then Ca(OH)₂ is not dissolved in an excess of OH⁻¹ ions.

6. Final Comments

A simple, uniform method for determining the pH ranges of different precipitates as the equilibrium solid phases in aqueous systems with Me- and L-species is presented. The systems with two or more precipitates thus formed are discussed, together with the problem of ordering of appropriate precipitates along the pH scale. The above issues are applicable to the systems where soluble complexes of the and/or type are not formed or are relatively weak ones.

Solubility products can be defined in different ways. The lack of awareness of this fact can be a source of confusion, as results from examples taken from the literature. In particular, for the solubility product of PbHPO₄ we find the following values: 11.36 [14] , and ···23.80 [15] ―both are referred allegedly to the dissociation reaction. The third value, which we denote as, is significantly different from the previous ones; we can therefore assume that, in fact, it relates to dissociation reaction. Indeed, after introducing the dissociation constant concerning the reaction , we get, i.e., the value close to 11.36. The solubility product for Pb₅(PO₄)₃OH is also formulated improperly in [15] .

References

Appendix

As an example, let us consider the pair of precipitates defined by Equations (22) and (23). We have, by turns,