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 A_{a}B_{b} dissolves/precipitates according to the reaction formula^{1}^{2}
(1)  A_{a}B_{b} = a A + b B  with equilibrium constant K_{sp} 
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:  {A_{a}B_{b}} = 1 
Inspired by the simple product form of 2, K_{sp} 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 socalled stoichiometric solubility product K_{sp}^{*}. Therefore, we have to distinguish between:
(4a)  stoichiometric solubility product:  K_{sp}^{*} = [A]^{a} [B]^{b}  (concentrations) 
(4b)  thdyn solubility product:  K_{sp} = {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, K_{sp} or K_{sp}^{*}, is presented.
Of both quantities, K_{sp} plays the fundamental role, because it refers to ideal as well as to nonideal solutions. Hydrochemistry models and software, including aqion, rely on K_{sp} data (rather than on K_{sp}^{*}).
Solubility products K_{sp} vary over several orders of magnitude. This favors the notation based on the decadic logarithm (similar to the definition of pH):
(5)  pK_{sp} = – lg K_{sp} 
pK_{sp} values for about 200 minerals and salts are listed here.
Classification rule (maybe the simplest one):
insoluble:  K_{sp} ≤ 1  \(\Longleftrightarrow\)  pK_{sp} ≥ 0 
soluble:  K_{sp} > 1  \(\Longleftrightarrow\)  pK_{sp} < 0 
Molar Solubility S
Don’t confuse “solubility” with the “solubility product constant” K_{sp} or K_{sp}^{*} 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)  K_{sp}^{*} = [A]^{a} [B]^{b} = (S·a)^{a} (S·b)^{b} = S^{a+b} a^{a} b^{b} 
After rearranging, we obtain the relationship between the molar solubility S and the stoichiometric solubility product K_{sp}^{*}:
(8)  molar solubility:  \(S \ = \ \dfrac{[A]}{a} = \ \dfrac{[B]}{b} \ = \ \sqrt[a+b\,]{\dfrac{K^{*}_{sp}}{a^a\,b^b}}\) 
The smaller the K_{sp} value, the less soluble the solid.
The twoion formula can easily be generalized for three or more ions (using the same arguments). For the substance “A_{a}B_{b}C_{c}” 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:  K_{sp} = {A}^{a}_{eq} {B}^{b}_{eq} 
However, a real solution may not be in the state of equilibrium. The nonequilibrium state is described by the ion activity product (IAP).^{4} It has the same math form as the equilibrium constant K_{sp}, but involves the actual activities, {A}_{actual} and {B}_{actual}:
(11)  nonequilibrium:  IAP = {A}^{a}_{actual} {B}^{b}_{actual} 
The decadic logarithm of the ratio of IAP to K_{sp} 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 = K_{sp}  \(\longrightarrow\)  saturated (in chem equilibrium) 
SI < 0  IAP < K_{sp}  \(\longrightarrow\)  undersaturated 
SI > 0  IAP > K_{sp}  \(\longrightarrow\)  supersaturated 
Examples
The following example calculations are provided:
 dissolution of calcite
 dissolution of gypsum
Remarks & Footnotes

This equation is a shorthand for: A_{a}B_{b}(s) = a A(aq) + b B(aq). ↩

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

The curly brackets indicate activities (as “effective concentrations” for nonideal solutions). ↩

In chemical thdyn, IAP is also known as the reaction coefficient Q and K_{sp} as the equilibrium constant K. Chemical equilibrium is established if Q=K. ↩