## 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 controled by the law of mass action. In other words, a mineral AaBb dissolves/precipitates according to the reaction formula1

 (1) AaBb  =  aA + bB with equilibrium constant Ksp

defined by the law of mass action:

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

The curly brackets indicate ionic activities (valid for non-ideal solutions). The right hand side of Eq.(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 Eq.(2), Ksp is named the “solubility product” – more precisely: thermodynamic 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} ⇒ [a], [b]. This simplification leads to the so-called stoichiometric solublity product Ksp*. Therefore, we have to distinguish between:

 (4a) stoichiometric solubility product: Ksp* = [A]a[B]b (concentrations) (4b) thermodynamic 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 Ksp*).

Solubility product constants vary over several orders of magnitude. This favors the logarithmic notation (similar to the definition of pH):

 (5) pKsp = - log10 Ksp

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

Classification rule (maybe the simplest one):

 insoluble: Ksp ≤ 1 ⇔ pKsp ≥ 0 soluble: Ksp > 1 ⇔ pKsp < 0

Solubility S

The solubility is the maximum amount of solute (mineral, salt) that can be dissolved in 1 Liter water at equilibrium. Since we are searching for concentrations (instead activities), we will employ Eq.(4a). Together with the mass-balance condition in Eq.(1) there are two equations to start with:

 (6) mass balance: [A]/a = [B]/b   bzw.   [B] = (b/a)[A] (7) stoichiometric solubility product: Ksp* = [A]a[B]b

Inserting Eq.(6) into Eq.(7) yields Ksp* = [A]a+b (b/a)a+b and, after rearranging,

 (8) $[A] \ = \ \sqrt[a+b]{\dfrac{K^{*}_{sp}}{(b/a)^b}}$

Dividing Eq.(8) by a, we obtain the (molar) solubility

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

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

The law of mass action in Eq.(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. This non-equilibrium state is described by the ion activity product (IAP). It has the same form as the equilibrium constant Ksp, but involves the actual activities, {A}actual and {B}actual:

 (11) real solution: IAP = {A}aactual {B}bactual

The ratio between IAP and Ksp enters the definition of the saturation index:

 (12) saturation index: SI = log10 (IAP / Ksp)

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 → saturated (in equilibrium) SI < 0 IAP < Ksp → undersaturated SI > 0 IAP > Ksp → supersaturated

Examples

The following example calculations are provided:

Remarks & Footnotes

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