4.1  Chemical Equilibria and Reaction Rates in Groundwater

F. Grant Ferris; Natalie Szponar; Brock A. Edwards

4.1 Chemical Equilibria and Reaction Rates in Groundwater

Descriptions of chemical processes in groundwater systems are usually formulated in terms of mass action and mass balance considerations, as well as reaction kinetics. The application of these basic aspects of physical chemistry and thermodynamics provides quantitative insight into the extent, direction, and rate of chemical reactions, including those involving microorganisms. A general reaction is shown in Equation 17.

aA + bB = cC + dD

(17)

The corresponding mass action equilibrium constant K is calculated as shown in Equation 18.

$\displaystyle K=\frac{\left\{C \right\}^{c}\left\{D \right\}^{d}}{\left\{A \right\}^{a}\left\{B \right\}^{b}}=e^{\frac{-\Delta G^{0}}{RT}}$ ; $\Delta G^{0}=-RT\textup{ln}K$

(18)

where:

{A}, {B}	=	activities of reactants (dimensionless)
{C}, {D}	=	activities of products (C,D) at equilibrium with stoichiometric coefficients given in lower case letters (dimensionless)
R	=	universal gas constant (ML²)/(T²°K mol)
T	=	temperature (°K)
ΔG⁰	=	standard Gibbs energy of reaction Δ (ML²)/(T² mol)

For a reversible reaction at equilibrium, the concentrations of reactants and products remain constant. This condition requires the rates of the forward (R_f) and reverse (R_r) reactions to be equal as shown in Equation 19 such that the equilibrium constant can be expressed as Equation 20 with corresponding forward (k_f) and reverse (k_r) rate coefficients.

R_f = k_f{A}^a{B}^b = R_r = k_r{C}^c{D}^d

(19)

$\displaystyle K=\frac{k_{f}}{k_{r}}=\frac{\left\{C \right\}^{c}\left\{D \right\}^{d}}{\left\{A \right\}^{a}\left\{B \right\}^{b}}$

(20)

The dependence of the rate coefficients on temperature and reaction activation energy is evident from the Arrhenius relationship (Equation 1).

A far more interesting condition, especially for groundwater systems, is when a reaction is not at equilibrium. Here, the Gibbs energy for a reaction (ΔG_r) to occur is described by Equation 21.

$\displaystyle \Delta G_{r}=\Delta G^{0}+RT\textup{ln}\frac{\left\{C \right\}^{c}\left\{D \right\}^{d}}{\left\{A \right\}^{a}\left\{B \right\}^{b}}$

(21)

For reactant and product activities observed away from equilibrium, defining the reaction quotient of products to reactants as Q, gives Equation 22.

$\displaystyle \Delta G_{r}=\Delta G^{0}+RT\textup{ln}Q=RT\textup{ln}\frac{Q}{K}$

(22)

From this expression, one finds that at equilibrium Q is the same as K, so ΔG_r= 0. If Q is less than K, ΔG_r is negative. This means the reaction is spontaneous. Conversely, ΔG_r= 0 is positive when Q is greater than K, indicating the reaction is not possible (unless energy is supplied from another spontaneous reaction). The capacity to use energy-yielding spontaneous reactions to drive energetically unfavorable reactions is a defining characteristic of biosynthetic processes in microbial metabolism.

Comparison of reaction quotients to equilibrium constants is widely applied to the study of mineral dissolution and precipitation reactions; however, subtle differences in terminology and interpretation exist. For dissolution reactions, the equilibrium constant is known as the solubility product constant (K_sp) and the reaction quotient is referred to as the ion activity product (IAP). The saturation index (SI) is defined by Equation 23.

$\displaystyle SI=\textup{log}_{10}\frac{IAP}{K_{sp}}$

(23)

When the IAP is equal to K_s_p, SI = 0 and the solution is said to be at equilibrium with respect to the mineral under consideration. Should IAP be greater than K_sp, the SI will be positive, which indicates the solution is oversaturated and mineral precipitation is possible. If the IAP is less than K_sp, the SI will be negative, which indicates the solution is undersaturated and mineral dissolution is possible.

License