Mineral Solubility and Saturation Index

Equilibrium between Solid Phases and Aqueous Solution

In chemical thermodynamics, the dissolution or precipitation of solid phases (minerals, salts) is controlled by the law of mass action. In other words, a mineral AaBb dissolves/precipitates according to the reaction formula12

 (1) AaBb  =  a A + b B with equilibrium constant Ksp

defined by the law of mass action:3

 (2) $$K_{sp} \,=\, \dfrac{\{A\}^{a}\{B\}^{b}}{\{A_{a}B_{b}\}} \,=\, \{A\}^{a}\{B\}^{b}$$

The right hand side of 2 follows from the fact that the activity of a pure phase is, by convention, equal to 1:

 (3) pure solid phase: {AaBb} = 1

Inspired by the simple product form of 2, Ksp is named the “solubility product” — more precisely: thdyn solubility product.

Thermodynamic vs. Stoichiometric Solubility Product

When the solubility of the mineral is low, only a small amount of ions will be dissolved (resulting in a water of low ionic strength). In this case, the activities of the ions can be replaced by the concentrations: {A}, {B} $$\Longrightarrow$$ [A], [B]. This simplification leads to the so-called stoichiometric solubility product Ksp*. Therefore, we have to distinguish between:

 (4a) stoichiometric solubility product: Ksp* = [A]a [B]b (concentrations) (4b) thdyn solubility product: Ksp = {A}a {B}b (activities)

For a given solid phase (mineral, salt) both quantities can differ considerably — especially in the case of highly soluble salts. Thus, when comparing literature data it is advisable to check which type of solubility product, Ksp or Ksp*, is presented.

Of both quantities, Ksp plays the fundamental role, because it refers to ideal as well as to non-ideal solutions. Hydrochemistry models and software, including aqion, rely on Ksp data (rather than on Ksp*).

Solubility products Ksp vary over several orders of magnitude. This favors the notation based on the decadic logarithm (similar to the definition of pH):

 (5) pKsp = – lg Ksp

pKsp values for about 200 minerals and salts are listed here.

Classification rule (maybe the simplest one):

 insoluble: Ksp ≤ 1 $$\Longleftrightarrow$$ pKsp ≥ 0 soluble: Ksp > 1 $$\Longleftrightarrow$$ pKsp < 0

Molar Solubility S

Don’t confuse “solubility” with the “solubility product constant” Ksp or Ksp* introduced above. The molar solubility [mol/L] is the maximum amount of solute (mineral, salt) that can be dissolved in 1 Liter water at equilibrium.

In “congruent dissolution” the solid substance dissolves without changing the stoichiometry. That is, the composition of the solid and the composition of the solute remain the same, the ion ratio A:B = a:b is fixed. This statement is manifest in the definition of the molar solubility:

 (6) $$S\ \equiv \ \dfrac{[A]}{a} = \ \dfrac{[B]}{b}$$

Inserting S into 4a yields:

 (7) Ksp*  =  [A]a [B]b  =  (S·a)a (S·b)b  =  Sa+b aa bb

After rearranging, we obtain the relationship between the molar solubility S and the stoichiometric solubility product Ksp*:

 (8) molar solubility: $$S \ = \ \dfrac{[A]}{a} = \ \dfrac{[B]}{b} \ = \ \sqrt[a+b\,]{\dfrac{K^{*}_{sp}}{a^a\,b^b}}$$

The smaller the Ksp value, the less soluble the solid.

The two-ion formula can easily be generalized for three or more ions (using the same arguments). For the substance “AaBbCc” we get:

 (9) $$S \ = \ \dfrac{[A]}{a} = \ \dfrac{[B]}{b} \ = \ \dfrac{[C]}{c} \ = \ \sqrt[a+b+c\,]{\dfrac{K^{*}_{sp}}{a^a\,b^b\,c^c}}$$

Note.  Equations (8) and (9) are approximations for several reasons: (i) the existence of other ions in the solution is neglected, (ii) complex formation is neglected, (iii) activity corrections are neglected.

Ion Activity Product (IAP) and Saturation Index (SI)

The law of mass action in 2 determines the activities at the state of equilibrium, {A}eq and {B}eq:

 (10) chem equilibrium: Ksp = {A}aeq {B}beq

However, a real solution may not be in the state of equilibrium. The non-equilibrium state is described by the ion activity product (IAP).4 It has the same math form as the equilibrium constant Ksp, but involves the actual activities, {A}actual and {B}actual:

 (11) non-equilibrium: IAP = {A}aactual {B}bactual

The decadic logarithm of the ratio of IAP to Ksp defines of the saturation index:

 (12) saturation index: SI $$\,=\, \lg \left( \dfrac{\textrm{IAP}}{K_{sp}} \right)$$

The saturation index is a useful quantity to determine whether the water is saturated, undersaturated, or supersaturated with respect to the given mineral:

 SI = 0 IAP = Ksp $$\longrightarrow$$ saturated  (in chem equilibrium) SI < 0 IAP < Ksp $$\longrightarrow$$ undersaturated SI > 0 IAP > Ksp $$\longrightarrow$$ supersaturated

Examples

The following example calculations are provided:

Remarks & Footnotes

1. This equation is a shorthand for:  AaBb(s)  =  a A(aq) + b B(aq).

2. In order to keep the notation straight, electrical charges are omitted in the equations.

3. The curly brackets indicate activities (as “effective concentrations” for non-ideal solutions).

4. In chemical thdyn, IAP is also known as the reaction coefficient Q and Ksp as the equilibrium constant K. Chemical equilibrium is established if Q=K.