# 7.1 Thermodynamics and Mineral Saturation Indices

Thermodynamics is the study of energy transformations and evolved from the need to understand and improve the efficiency of machines during the 19^{th} century. Some important thermodynamic concepts such as specific heat originated in the 18^{th} century, but the observations and the theory that integrated the work of several famous scientists were made in the 19^{th} century.

All substances were found to have a characteristic heat content, *H*, that could be measured in several different ways such as heat capacity, heat of oxidation, and heat of dissolution. However, when petroleum fuel is combusted in an engine, not all the known heat content is transformed into mechanical or electrical energy. A substantial amount of that energy is irreversibly lost and unavailable for useful work. For example, instead of completely turning the gasoline combustion into mechanical energy to power an automobile, the engine also heats up, the metal parts expand, vibrations occur throughout the vehicle, and that is lost energy. Some of the metal in the engine is irreversibly oxidized, that is lost energy involving a chemical reaction. That dissipation of energy is known as entropy, *S*, and when multiplied by the absolute temperature, *T*, has the same units as the heat content. Hence the useful or available work from some process is the difference between the total and the dissipated heat or *H* – *TS* which is a function called the Gibbs free energy function, *G*. Energy is also involved during any chemical reaction and when the free energy is considered per mole of reaction, *∂G/∂n*, it is known as the chemical potential, *µ*. The chemical potential is the energy available for dissolution/precipitation of minerals, redox reactions, sorption reactions, ionic and molecular diffusion. It tells us whether a chemical reaction is possible or not. Knowing whether a mineral dissolution or precipitation reaction is possible for a given set of physico-chemical conditions is useful when interpreting mineral reactions in groundwaters.

Because of the relationships between the variables: pressure, *P*; temperature, *T*; volume, *V*; entropy, *S*; and free energy, *G*; the chemical potential can be expressed using Equation 1.

µ = µ^{o} + RT lnX |
(1) |

where:

µ^{o} |
= | chemical potential of the substance in a defined standard state usually referenced to 25 ^{o}C and 1 bar pressure for ideal conditions (joules mol^{-1}) |

R |
= | molal gas constant (8.3144 joules mol^{–}^{1} K^{–}^{1}) |

X |
= | mole fraction (dimensionless) |

Non-ideal conditions are accounted for by a coefficient called the activity coefficient, *λ* (dimensionless), and the product *λX* is the activity (dimensionless). For aqueous solutions we normally use the expression *γm*, where *m* is the molality and *γ* is the activity coefficient when molal concentrations are used. Hence, the activity is expressed as shown in Equation 2.

a = γm |
(2) |

The chemical potential is expressed as Equation 3.

µ = µ^{o} + RT lna |
(3) |

For practical applications, we just need to know the activity coefficient because the molality is measured, and the standard state is a matter of careful definition. There are several options for calculating the activity coefficient depending on the concentration range and available data. Because most groundwaters of interest are relatively dilute, speciation computations are not very sensitive to the theoretical model chosen for the activity coefficient.

Consider the fluorite dissolution reaction shown in Equation 4.

CaF^{2} → Ca^{2+} + 2F^{−} |
(4) |

The solubility product constant is expressed by Equation 5.

[latex]\displaystyle K_{sp}=\frac{a_{Ca^{2+}}\ a_{F^{-}}^{2}}{a_{CaF_{2}}}=10^{-10.6}[/latex] | (5) |

This equilibrium is known as the law of mass action. If an acid mine water is saturated with respect to fluorite and some lime (CaO) is added to it, more fluorite will precipitate to return the solution to equilibrium. The reaction in Equation 4 is driven to the left to achieve equilibrium. For dilute solutions, the activity of water can be taken as unity. When fluorite is pure and in its most crystalline state, it can also be taken as unity leading to Equation 6.

[latex]\displaystyle K_{sp}=a_{Ca^{2+}}\ a_{F^{-}}^{2}[/latex] | (6) |

Equation 6 is known as the ion-activity product regardless of whether the solution is at equilibrium or not. At equilibrium solubility and 25 °C and 1 bar pressure, the ion-activity product is a constant value and equal to the solubility product constant regardless of the concentrations of Ca^{2+} and F^{–}. Given a water analysis with concentrations of all the major dissolved constituents, the activities of the ions can be calculated, then the ion-activity product can be calculated for any chosen mineral formula for which there is a solubility product constant. Several computer programs exist that perform this calculation. Solubility product constants (or their equivalent free energies) are thermodynamic properties compiled in databases of computer codes and in books and scientific papers. By comparing the mineral ion-activity product of a water composition to the solubility product constant, the extent to which a water has reached solubility equilibrium can be tested. For the fluorite example, the degree of saturation or saturation ratio, *Ω*, is calculated using Equation 7.

[latex]\displaystyle \mathit{\Omega }=\frac{\left ( a_{Ca^{2+}}\ a_{F^{-}}^{2} \right )_{sample}}{\left ( a_{Ca^{2+}}\ a_{F^{-}}^{2} \right )_{equilibrium}}=\frac{\left ( a_{Ca^{2+}}\ a_{F^{-}}^{2} \right )_{sample}}{K_{sp}}[/latex] | (7) |

When the ratio is unity, the water is at solubility equilibrium. If the ratio is less than one, the water is undersaturated and the mineral, if present, should dissolve. Because of the large range of values encountered in natural waters, the log *Ω* or *saturation* *index* (*SI*) is used for practical applications. Hence,

*SI*< 0, indicates undersaturation, so the mineral should dissolve if present; and,*SI*> 0, indicates supersaturation, so the mineral should precipitate.

Note that these are thermodynamic calculations, and they state what is possible, not necessarily what occurs. Given a negative *SI*, the mineral should dissolve. However, a mineral may be so slow to dissolve, such as quartz, that it might not be an important source of dissolved silica in a natural water. Similarly, given a positive *SI*, a mineral may not precipitate because of inhibitory factors such as calcite in the presence of dissolved magnesium. Seawater is supersaturated with respect to calcite, but pure calcite does not precipitate because of the inhibitory effect of high Mg concentrations. Aragonite, a mineral less stable than calcite, and some high-Mg calcite precipitate instead. Additional information on the rates of mineral dissolution and precipitation, the subject of kinetics, is necessary to determine how fast (and how likely) a given mineral is to dissolve or precipitate in a natural water.

The concept of *SI* is used routinely to interpret water-rock interactions and it is a key to the interpretation of how groundwaters gain high concentrations of F.