Chapter 3: Chemical Properties and Principles

Chemical Properties and Principles

The chemical and biochemical constituents in groundwater determine its usefulness for industry, agriculture, and the home. Dissolved constituents in the water provide clues on its geologic history, its influence on the soil or rock masses through which it has passed, the presence of hidden ore deposits, and its mode of origin within the hydrologic cycle. Chemical processes in the groundwater zone can influence the strength of geologic materials, and in situations where they are not recognized, can cause failure of artificial slopes, dams, mining excavations, and other features of importance to man. It is becoming increasingly common for industrial, agricultural, and domestic wastes to be stored or disposed on or beneath the land surface. This can be a safe or hazardous practice, the consequences of which depend greatly on the chemical and microbiological processes in the groundwater zone. In the study of landscape evolution the assumption is commonly made that the physical processes of mechanical erosion, thermal expansion and contraction, frost action, and slope movements are the dominant influences, but on closer examination it is often found that chemical processes in the groundwater zone are the controlling influences.

The purpose of this chapter is to describe the geochemical properties and principles that control the behavior of dissolved constituents in the groundwater environment. A more comprehensive coverage of the study and interpretation of the chemical characteristics of natural water is provided by Hem (1970) and by Stumm and Morgan (1970). Most of the geochemical principles described in this chapter are based on equilibrium concepts. Examples described in Chapter 7 indicate that many hydrochemical processes in the groundwater zone proceed slowly toward chemical equilibrium and some rarely achieve equilibrium. At times, the reader may doubt the usefulness of equilibrium approaches. Equilibrium concepts or models have great value, however, because of their capability for establishing boundary conditions on chemical processes. Differences between observed hydrochemical conditions and computed equilibrium conditions can provide insight into the behavior of the system and at a minimum can provide a quantitative framework within which appropriate questions can be posed.

3.1 Groundwater and Its Chemical Constituents

Water and Electrolytes

Water is formed by the union of two hydrogen atoms with one oxygen atom. The oxygen atom is bonded to the hydrogen atoms unsymmetrically, with a bond angle of 105°. This unsymmetrical arrangement gives rise to an unbalanced electrical charge that imparts a polar characteristic to the molecule. Water in the liquid state, although given the formula H2O or HOH, is composed of molecular groups with the HOH molecules in each group held together by hydrogen bonding. Each group or molecular cluster is estimated to have an average of 130 molecules at 0°C, 90 molecules at 20°C, and 60 molecules at 72°C (Choppin, 1965). H180O90 is an approximate formula for the cluster at 20°C.

Water is unusual in that the density of the solid phase, ice, is substantially lower than the density of the liquid phase, water. In the liquid phase the maximum density is achieved at 4°C. With further cooling below this temperature there is a significant density decrease.

All chemical elements have two or more isotopes. In this book, however, we will be concerned only with the isotopes that provide useful hydrological or geochemical information. The formula H2O is a gross simplification from the structural viewpoint and is also a simplification from the atomic viewpoint. Natural water can be a mixture of the six nuclides listed in Table 3.1. The atomic nature of the hydrogen isotopes is illustrated in Figure 3.1. Eighteen combinations of H—O—H are possible using these nuclides. 2H216O, 1H218O, 3H217O are some examples of the molecules that comprise water, which in its most common form is 1H216O. Of the six isotopes of hydrogen and oxygen in Table 3.1, five are stable R and one, 3H, known as tritium, is radioactive, with a half-life of 12.3 years.

Table 3.1 Natural Isotopes of Hydrogen, Oxygen, and Radioactive Carbon and Their Relative Abundance in Water of the Hydrologic Cycle

Isotope Relative abundance
1H proteum 99.984 Stable
2H deuterium 0.016 Stable
3H tritium 0 – 10–15 Radioactive
half-life 12.3 years
16O oxygen 99.76 Stable
17O oxygen 0.04 Stable
18O oxygen 0.20 Stable
14C carbon <0.001 Radioactive
half-life 5730 years

Figure 3.1 Isotropes of hydrogen.

Pure water contains hydrogen and oxygen in ionic form as well as in the combined molecular form. The ions are formed by the dissociation of water,

\ce{H_2O <=> H + HO-} (3.1)

where the plus and minus signs indicate the charge on the ionic species. Hydrogen can occur in vastly different forms, as illustrated in Figure 3.2. Although the ionic form of hydrogen in water is usually expressed in chemical equations as H+, it is normally in the form H3O+, which denotes a hydrogen core surrounded by oxygen with four electron-cloud pairs. In discussions of groundwater mineral interactions, a process known as proton transfer denotes the transfer of an H+ between components or phases.

Figure 3.2 Four forms of hydrogen drawn to relative scale. (a) The hydrogen atom, a proton with one electron. (b) The hydrogen molecule, two separated protons in a cloud of two electrons. (c) The hydrogen core, or H+, a proton. (d) The hydronium ion, oxygen with four electron cloud pairs, three of which are protonated in H3O+.

Water is a solvent for many salts and some types of organic matter. Water is effective in dissolving salts because it has a very high dielectric constant and because its molecules tend to combine with ions to form hydrated ions. The thermal agitation of ions in many materials is great enough to overcome the relatively weak charge attraction that exists when surrounded by water, thus allowing large numbers of ions to dissociate into aqueous solution. Stability of the ions in the aqueous solution is promoted by the formation of hydrated ions. Each positively charged ion, known as a cation, attracts the negative ends of the polar water molecules and binds several molecules in a relatively stable arrangement. The number of water molecules attached to a cation is determined by the size of the cation. For example, the small cation Be2+ forms the hydrated ion Be(H2O) 2+. Larger ions, such as Mg2+ or Al3+, have hydrated forms such as Mg(H2O)62+ and Al(H2O)63+. Negatively charged species, known as anions, exhibit a much weaker tendency or hydration. In this case the anions attract the positive ends of the polar water molecules. The sizes of the ions in their hydrated form are important with respect to many processes that occur in the groundwater environment.

As a result of chemical and biochemical interactions between groundwater and the geological materials through which it flows, and to a lesser extent because of contributions from the atmosphere and surface-water bodies, groundwater contains a wide variety of dissolved inorganic chemical constituents in various concentrations. The concentration of total dissolved solids (TDS) in groundwater is determined by weighing the solid residue obtained by evaporating a measured volume of filtered sample to dryness. The solid residue almost invariably consists of inorganic constituents and very small amounts of organic matter. The TDS concentrations in groundwater vary over many orders of magnitude. A simple but widely used scheme for categorizing groundwater based on TDS is presented in Table 3.2. To put the concentration ranges in perspective, it may be useful to note that water containing more than 2000–3000 mg/\ell TDS is generally too salty to drink. The TDS of seawater is approximately 35,000 mg/\ell.

Table 3.2 Simple Groundwater Classification Based on Total Dissolved Solids

Category Total dissolved solids
(mg/\ell or g/m3)
Fresh water 0 – 1000
Brackish water 1000 – 10,000
Saline water 10,000 – 100,000
Brine water More than 100,000

Groundwater can be viewed as an electrolyte solution because nearly all its major and minor dissolved constituents are present in ionic form. A general indication of the total dissolved ionic constituents can be obtained by determining the capability of the water to conduct an applied electrical current. This property is usually reported as electrical conductance and is expressed in terms of the conductance of a cube of water 1 cm2 on a side. It is the reciprocal of electrical resistance and has units known as siemens (S) or microsiemens (\mu S) in the SI system. In the past these units have been known as millimhos and micromhos. The values are the same; only the designations have changed. The conductance of groundwater ranges from several tens of microsiemens for water nearly as nonsaline as rainwater to hundreds of thousands of microsiemens for brines in deep sedimentary basins.

A classification of the inorganic species that occur in groundwater is shown in Table 3.3. The Concentration categories are only a general guide. In some ground-waters, the concentration ranges are exceeded. The major constituents in Table 3.3 occur mainly in ionic form and are commonly referred to as the major ions (Na+, Mg2+, Ca2+, Cl, \ce{HCO^-_3}, SO42–). The total concentration of these six major ions normally comprises more than 90% of the total dissolved solids in the water, regardless of whether the water is dilute or has salinity greater than seawater.

Table 3.3 Classification of Dissolved Inorganic Constituents in Groundwater

Major constituents (greater than 5 mg/\ell)
Bicarbonate Silicon
Calcium Sodium
Chloride Sulfate
Magnesium Carbonic acid
Minor constituents (0.01 – 10.0 mg/\ell)
Boron Nitrate
Carbonate Potassium
Fluoride Strontium
Trace constituents (less than 0.1 mg/\ell)
Aluminum Molybdenum
Antimony Nickel
Arsenic Niobium
Barium Phosphate
Beryllium Platinum
Bismuth Radium
Bromide Rubidium
Cadmium Ruthenium
Cerium Scandium
Cesium Selenium
Chromium Silver
Cobalt Thallium
Copper Thorium
Gallium Tin
Germanium Titanium
Gold Tungsten
Indium Uranium
Iodide Vanadium
Lanthanum Ytterbium
Lead Yttrium
Lithium Zinc
Manganese Zirconium
SOURCE: Davis and De Wiest, 1966.

The concentrations of the major, minor, and trace inorganic constituents in groundwater are controlled by the availability of the elements in the soil and rock through which the water has passed, by geochemical constraints such as solubility and adsorption, by the rates (kinetics) of geochemical processes, and by the sequence in which the water has come into contact with the various minerals occurring in the geologic materials along the flow paths. It is becoming increasingly common for the concentrations of the dissolved inorganic constituents to be influenced by man’s activities. In some cases contributions from man-made sources can cause some of the elements listed as minor or trace constituents in Table 3.3 to occur as contaminants at concentration levels that are orders of magnitude above the normal ranges indicated in this table.

Organic Constituents

Organic compounds are those that have carbon and usually hydrogen and oxygen as the main elemental components in their structural framework. By definition, carbon is the key element. The species H2CO3, CO2, \ce{HCO^-_3}, and CO32, which are important constituents in all groundwater, however, are not classified as organic compounds.

Dissolved organic matter is ubiquitous in natural groundwater, although the concentrations are generally low compared to the inorganic constituents. Little is known about the chemical nature of organic matter in groundwater. Investigations of soil water suggest that most dissolved organic matter in subsurface flow systems is fulvic and humic acid. These terms refer to particular types of organic compounds that persist in subsurface waters because they are resistant to degradation by microorganisms. The molecular weights of these compounds range from a few thousand to many thousand grams. Carbon is commonly about half of the formula weight. Although little is known about the origin and composition of organic matter in groundwater, analyses of the total concentrations of dissolved organic carbon (DOC) are becoming a common part of groundwater investigations. Concentrations in the range 0.1–10 mg/\ell are most common, but in some areas values are as high as several tens of milligrams per liter.

Dissolved Gases

The most abundant dissolved gases in groundwater are N2, O2, CO2, CH4 (methane), H2S, and N2O. The first three make up the earth’s atmosphere and it is, therefore, not surprising that they occur in subsurface water. CH4, H2S, and N2O can often exist in groundwater in significant concentrations because they are the product of biogeochemical processes that occur in nonaerated subsurface zones. As will be shown later in this chapter and in Chapter 7, the concentrations of these gases can serve as indicators of geochemical conditions in groundwater.

Dissolved gases can have a significant influence on the subsurface hydro-chemical environment. They can limit the usefulness of groundwater and, in some cases, can even cause major problems or even hazards. For example, because of its odor, H2S at concentrations greater than about 1 mg/\ell renders water unfit for human consumption. CH4 bubbling out of solution can accumulate in wells or buildings and cause explosion hazards. Gases coming out of solution can form bubbles in wells, screens, or pumps, causing a reduction in well productivity or efficiency. Radon 222 (222Rn), which is a common constituent of groundwater because it is a decay product of radioactive uranium and thorium, which are common in rock or soil, can accumulate to undesirable concentrations in unventilated basements. Decay products of radon 222 can be hazardous to human health.

Other species of dissolved gases, which occur in groundwater in minute amounts, can provide information on water sources, ages, or other factors of hydrologic or geochemical interest. Noteworthy in this regard are Ar, He, Kr, Ne, and Xe, for which uses in groundwater studies have been described by Sugisaki (1959, 1961) and Mazor (1972).

Concentration Units

To have a meaningful discussion of the chemical aspects of groundwater, the relative amounts of solute (the dissolved inorganic or organic constituents) and the solvent (the water) must be specified. This is accomplished by means of concentrations units. Various types of concentration units are in use.

Molality is defined as the number of moles of solute dissolved in a 1-kg mass of solution. This is an SI unit with the symbol mol/kg. The derived SI symbol for this quantity is mB, where B denotes the solute. A is normally used to designate the solvent. One mole of a compound is the equivalent of one molecular weight.

Molarity is the number of moles of solute in 1 m3 of solution. The SI unit for molarity is designated as mol/m3. It is useful to note that 1 mol/m3, equals 1 mmol/\ell. Moles per liter, with the symbol mol/\ell, is a permitted unit for molarity in the SI system and is commonly used in groundwater studies.

Mass concentration is the mass of solute dissolved in a specified unit volume of solution. The SI unit for this quantity is kilograms per cubic meter, with the symbol kg/m3. Grams per liter (g/\ell) is a permitted SI unit. The most common mass concentration unit reported in the groundwater literature is milligrams per liter (mg/\ell). Since 1 mg/\ell equals 1 g/m3, there is no difference in the magnitude of this unit (mg/\ell) and the permitted SI concentration unit (g/m3).

There are many other non-Sl concentration units that commonly appear in the groundwater literature. Equivalents per liter (ep\ell) is the number of moles of solute, multiplied by the valence of the solute species, in 1 liter of solution:

\text{ep}\ell = \frac{\text{moles of solute} \times \text{valence}}{\text{liter of solution}}

Equivalents per million (epm) is the number of moles of solute multiplied by the valence of the solute species, in 106 g of solution, or this can be stated as the number of milligram equivalents of solute per kilogram of solution:

\text{epm} = \frac{\text{moles of solute} \times \text{valence}}{10^6 \text{g of solution}}

Parts per million (ppm) is the number of grams of solute per million grams of solution

\text{ppm} = \frac{\text{grams of solute}}{10^6 \text{g of solution}}

For nonsaline waters, 1 ppm equals 1 g/m3 or 1 mg/\ell. Mole fraction (XB) is the ratio of the number of moles of a given solute species to the total number of moles of all components in the solution. If nB is moles of solute, nA is moles of solvent, and nC, nD, . . . denote moles of other solutes, the mole fraction of solute B is

X_B = \frac{n_B}{n_\text{A} + n_\text{B} + n_\text{C} +n_\text{D} + ...}

or XB for aqueous solutions can be expressed as

X_B = \frac{m_B}{55.5 + \sum m_{\text{B, C, D}}...}

where m denotes molality.

In procedures of chemical analysis, quantities are most conveniently obtained by use of volumetric glassware. Concentrations are therefore usually expressed in the laboratory in terms of solute mass in a given volume of water. Most chemical laboratories report analytical results in milligrams per liter or, in SI units, as kilograms per cubic meter. When the results of chemical analyses are used in a geochemical context, it is usually necessary to use data expressed in molality or molarity, since elements combine to form compounds according to relations between moles rather than mass or weight amounts. To convert between molarity and kilograms per cubic meter or milligrams per liter, the following relation is used:

\text{molarity} = \frac{\text{milligrams per liter of grams per cubic meter}}{1000 \times \text{formula weight}}

If the water does not have large concentrations of total dissolved solids and if the temperature is close to 4°C, 1 \ell of solution weighs 1 kg, in which case molality and molarity are equivalent and 1 mg/\ell = 1 ppm. For most practical purposes, water with less than about 10,000 mg/\ell total dissolved solids and at temperatures below about 100°C can be considered to have a density close enough to 1 kg/\ell for the unit equivalents above to be used. If the water has higher salinity or temperature, density corrections should be used when converting between units with mass and volume denominators.

3.2 Chemical Equilibrium

The Law of Mass Action

One of the most useful relations in the analysis of chemical processes in groundwater is the law of mass action. It has been known for more than a century that the driving force of a chemical reaction is related to the concentrations of the constituents that are reacting and the concentrations of the products of the reaction. Consider the constituents B and C reacting to produce the products D and E,

b\text{B} + c\text{C} \ce{<=>} d\text{D} + e\text{E} (3.2)

where b, c, d, and e are the number of moles of the chemical constituents B, C, D, E, respectively.

The law of mass action expresses the relation between the reactants and the products when the reaction is at equilibrium,

K = \frac{[D]^d[E]^e}{[B]^b[C]^c} (3.3)

where K is a coefficient known as the thermodynamic equilibrium constant or the stability constant. The brackets specify that the concentration of the constituent is the thermodynamically effective concentration, usually referred to as the activity. Equation (3.3) indicates that for any initial condition, the reaction expressed in Eq. (3.2) will proceed until the reactants and products attain their equilibrium activities. Depending on the initial activities, the reaction may have to proceed to the left or to the right to achieve this equilibrium condition.

The law of mass action contains no parameters that express the rate at which the reaction proceeds, and therefore tells us nothing about the kinetics of the chemical process. It is strictly an equilibrium statement. For example, consider the reaction that occurs when groundwater flows through a limestone aquifer composed of the mineral calcite (CaCO3). The reaction that describes the thermodynamic equilibrium of calcite

\ce{CaCO3 <=> Ca^{2+} + CO3^{2-}} (3.4)

This reaction will proceed to the right (mineral dissolution) or to the left (mineral precipitation) until the mass-action equilibrium is achieved. It may take years or even thousands of years for equilibrium to be achieved. After a disturbance in the system, such as an addition of reactants or removal of products, the system will continue to proceed toward the equilibrium condition. If the temperature or pressure changes, the system will proceed toward a new equilibrium because the magnitude of K changes. If disturbances are frequent compared to the reaction rate, equilibrium will never be achieved. As we will see in Chapter 7, some chemical interactions between groundwater and its host materials never do attain equilibrium.

Activity Coefficients

In the law of mass action, solute concentrations are expressed as activities. Activity and molality are related by

a_i = m_i \gamma_i (3.5)

where ai is the activity of solute species i, mi the molality, and \gamma_i the activity coefficient. \gamma carries the dimensions of reciprocal molality (kg/mol), and ai is therefore dimensionless. Except for waters with extremely high salt concentrations, \gamma_i is less than 1 for ionic species. In the previous section, activity was referred to as the thermodynamically effective concentration, because it is conceptually convenient to consider it to be that portion of mi that actually participates in the reaction. The activity coefficient is therefore just an adjustment factor that can be used to convert concentrations into the form suitable for use in most thermodynamically based equations.

The activity coefficient of a given solute is the same in all solutions of the same ionic strength. Ionic strength is defined by the relation

I = \frac{1}{2} \sum m_i z_i ^2 (3.6)

where mi is the molality of species i, and zi is the valence, or charge, that the ion carries. For groundwater, in which the six common major ions are the only ionic constituents that exist in significant concentration,

I = \ce{\frac{1}{2}[(Na+) + 4(Mg^{2+}) +4(Ca{2+}) + (HCO^-_3) + (Cl-) + 4(SO^{2-}_4)]} (3.7)

where the quantities in parentheses are molalities. To obtain values for \gamma_i the graphical relations of \gamma versus [shown in Figure 3.3 can be used for the common inorganic constituents, or at dilute concentrations a relation known as the Debye-Hückel equation can be used (Appendix IV). At ionic strengths below about 0.1, the activity coefficients for many of the less common ions can be estimated from the Kielland table, which is also included in Appendix IV. For a discussion of the theoretical basis for activity coefficient relations, the reader is referred to Babcock (1963). Comparison between experimental and calculated values of activity coefficients are made by Guenther (1968).

Equilibrium and Free Energy

From a thermodynamic viewpoint the equilibrium state is a state of maximum stability toward which a closed physicochemical system proceeds by irreversible processes (Stumm and Morgan, 1970). The concepts of stability and instability for a simple mechanical system serve as an illustrative step toward development of the thermodynamic concept of equilibrium. Similar examples have been used by Guggenheim (1949) and others. Consider three different “equilibrium” positions of a rectangular box on a horizontal surface [Figure 3.4(a)]. Position 3 is the most stable position that the box can achieve. In this position the gravitational potential energy is at a minimum, and if the position is slightly disturbed, it will return to the condition of stable equilibrium.

Figure 3.3 Activity coefficient versus ionic strength relations for common ionic constituents in groundwater.

In position 1, the box is also in an equilibrium position to which it will return if only slightly disturbed. But in this position the potential energy is not a minimum, so it is referred to as a condition of metastable equilibrium. If the box in position 2 is disturbed only slightly, the box will move to a new position. Position 2 is therefore a condition of unstable equilibrium.

An analogy between the mechanical system and the thermodynamic system is illustrated in Figure 3.4(b). Following the development by Stumm and Morgan (1970), a hypothetical, generalized energy or entropy profile is shown as a function of the state of the system. The conditions of stable, metastable, and unstable equilibrium are represented by troughs and peaks on the energy or entropy function. If the chemical system exists in closed conditions under constant temperature and pressure, its response to change can be described in terms of a particular energy function known as the Gibbs free energy, named after Willard Gibbs, the founder of classical thermodynamics. This direction of possible change in response to change in a composition variable is that accompanied by a decrease in Gibbs free energy. State C is the most stable state because it has an absolute minimum Gibbs free energy under closed-system conditions at constant temperature and pressure. State A is stable with respect to infinitesimally near states of the system, but is unstable with respect to a finite change toward state C. Natural processes proceed toward equilibrium states and never away from them. Therefore, thermodynamic equilibrium is found in metastable and stable conditions of equilibrium but not in unstable equilibrium.

Figure 3.4 Concepts in mechanical and chemical equilibrium. (a) Metastable, unstable, and stable equilibrium in a mechanical system. (b) Metastability, instability, and stability for different energetic states of a thermodynamic system (after Stumm and Morgan, 1970).

The driving force in a chemical reaction is commonly represented by the Gibbs free energy of reaction denoted as \Delta G_r. For systems at constant temperature and pressure, \Delta G_r represents the change in internal energy per unit mass and is a measure of the reaction’s capability to do nonmechanical work. Since in this text our objective in the use of thermodynamic data focuses on determining the directions in which reactions will proceed and on obtaining numerical values for equilibrium constants, there is little need to directly consider the thermodynamic components that make up \Delta G_r. For a development of the theory of chemical thermodynamics, the reader is referred to the text by Denbigh (1966), and the comprehensive discussion of the thermodynamics of soil systems by Babcock (1963).

The condition of chemical equilibrium can be defined as

\sum free-energy: products – \sum free-energy: reactants = 0 (3.8)

The next step in this development is to relate free-energy changes of reactions to their equilibrium constants. To do this a convenient free-energy accounting system is needed. The standard free energy of formation, \Delta G^0_f, is defined as the free energy of the reaction to produce 1 mol of a substance from the stable elements under conditions that are specified as standard-state conditions. The standard free energy of elements in their most stable pure chemical state is assigned a value of zero by convention. Similarly, it is convenient to take as zero the \Delta G^0_f of hydrogen ion. For example, carbon as graphite and oxygen as O2 have \Delta G^0_f values of zero, but 1 mol of gaseous carbon dioxide has a \Delta G^0_f value of –386.41 kJ (–92.31 kcal), which is the energy released when CO2 forms from the stable elements in their standard state. The standard state of pure water is defined as unity at the temperature and pressure of the reaction, and for solutes the standard state is a unimolal concentration in a hypothetical condition where the activity coefficient is unity, or, in other words, in a condition where the activity equals the molality. For gas, the standard state is pure (ideal) gas at 1 bar total pressure at the temperature of the reaction. This system of arbitrarily defined standard states may at first seem unnecessarily complex, but in practice it leads to a tidy consistent system of bookkeeping. A more detailed discussion of standard states is provided by Berner (1971).

The standard free-energy change of reaction, \Delta G^0_r, is the sum of the free energies of formation of the products in their standard”states minus the free energies of formation of the reactants in their standard states:

\Delta G_r^0 = \sum \Delta G_f^0 \hspace{1mm} \text{products} - \sum \Delta G_f^0 \hspace{1mm} \text{reactants} (3.9)

For the general reaction in Eq. (3.2), the change in free energy of the reaction is related to the standard free-energy change and to the activities of each of the reactants and products, measured at the same temperature, by the expression

\Delta G_r = \Delta G_r^0 + RT \hspace{1mm}\text{ln} \frac{[D]^d[E]^e}{[B]^b[C]^c} (3.10)

where R is the universal gas constant and T is temperature in degrees Kelvin. At 25°C, R = 8.314 J/K • mol or 0.001987 kcal/K • mol. Conversion of temperatures on the Celsius scale to those on the Kelvin scale is made through the relation K = °C + 273.15. For a chemical reaction to proceed spontaneously as written, \Delta G_r, must be less than zero, or, in other words, there must be a net decrease in free energy. If \Delta G_r > O, the reaction can only proceed from right to left. If \Delta G_r = O, the reaction will not proceed in either direction, in which case the equilibrium condition has been achieved. In accordance with our definition of the standard state for solutes (unimolal conditions where \gamma = 1, \Delta G^0_r = \Delta G_r in the standard state because [D]d[E]e/[B]b[C]c = 1, and hence the natural logarithm of this term is zero. Substitution of the equilibrium constant relation [Eq. (3.3)] into Eq. (3.10) yields, for equilibrium conditions,

\Delta G_r^0 = - RT \hspace{1mm}\text{ln} K (3.11)

For standard-state conditions, the equilibrium constant can be obtained from free-energy data by means of the relations

\log K = -0.175 \Delta G_r^0 (\text{for} \hspace{1mm} \Delta G_r^0 \hspace{1mm} \text{in kJ/mol})

\log K = -0.733 \Delta G_r^0 (\text{for} \hspace{1mm} \Delta G_r^0 \hspace{1mm} \text{in kcal/mol})

where \Delta G^0_r can be obtained from Eq. (3.9) using \Delta G^0_f data. Values for \Delta G^0_f at 25°C and 1 bar have been tabulated for thousands of minerals, gases, and aqueous species that occur in geologic systems (Rossini et al., 1952; Sillen and Martell, 1964, 1971). Less comprehensive tables that are convenient for student use are included in the texts by Garrels and Christ (1965), Krauskopf (1967), and Berner (1971).

Compared to the abundance of \Delta G^0_f data for conditions of 25°C and 1 bar total pressure, there is a paucity of data for other temperatures and pressures. Pressure has only a slight effect on \Delta G^0_f values and consequently has little influence on the equilibrium constant. For practical purposes, the variation in K over the fluid pressures normally encountered in the upper few hundred meters of the earth’s crust are negligible. Changes of several degrees, however, can cause significant changes in the equilibrium constant. To obtain estimates of K values at other temperatures, an expression known as the van’t Hoff relation, named after a Dutch physical chemist who made important contributions in the late 1800s’ and early 1900’s to the understanding of solution behavior, can be used:

\log K_T = \log K_{T^*} - \frac{\Delta H_{T^*}}{2.3 R}\left( \frac{1}{T} - \frac{1}{T^*} \right) (3.13)

where T* is the reference temperature, usually 298.15 K (25°C), T the temperature of the solution, and \Delta H_T, the enthalpy. Enthalpy data for many of the minerals, gases, and dissolved species of interest are tabulated in the tables referred to above. Since the van’t Hoff equation considers only two temperatures and assumes a linear relationship between them, it yields only approximate values. The best approach is to develop specific interpolation relations from free-energy data over a wide range of temperatures, if such data are available.

To illustrate the use of \Delta G^0_f data to obtain equilibrium constants, consider the calcite dissolution reaction as expressed in Eq. (3.4). \Delta G^0_f values for pure CaCO3, Ca2+, and CO32 are –1129.10, –553.04, and –528.10 kJ, respectively, at 25°C and 1 bar. The standard free energy of the reaction is therefore

\Delta G^0_r = (-553.04 - 528.10) - (-1129.10)

From Eq. (3.12) we obtain, for 25°C and 1 bar,

\log K_{\text{calcite}} = -8.40 \hspace{0.5cm} \text{or} \hspace{0.5cm} K_c = 10^{-8.40}

Dissolved Gases

When water is exposed to a gas phase, an equilibrium is established between the gas and the liquid through the exchange of molecules across the liquid-gas interface. If the gaseous phase is a mixture of more than one gas, an equilibrium will be established for each gas. The pressure that each gas in the mixture exerts is its partial pressure, which is defined as the pressure that the specific component of the gas would exert if it occupied the same volume alone. Dalton’s law of partial pressures states that in a mixture of gases, the total pressure equals the sum of the partial pressures. The partial pressure of a vapor is also referred to as the vapor pressure.

Groundwater contains dissolved gases as a result of (1) exposure to the earth’s atmosphere prior to infiltration into the subsurface environment, (2) contact with soil gases during infiltration through the unsaturated zone, or (3) gas production below the water table by chemical or biochemical reactions involving the groundwater, minerals, organic matter, and bacterial activity.

Probably the most important of the dissolved gases in groundwater is CO2. Two reactions that describe the interaction between gaseous CO2 and its dissolved species are,

\ce{CO2 (g) + H2O <=> CO2(aq) + H2O} (3.14)

\ce{CO2(g) + H2O <=> H2CO3(aq)} (3.15)

where the suffixes (g) and (aq) denote gaseous and dissolved species, respectively. The ratio of CO2 (aq)/H2CO3 is much greater than unity in aqueous solutions; however, it is customary to denote all dissolved CO2 in water as H2CO3 (carbonic acid). This usage results in no loss of generality as long as consistency is maintained elsewhere in the treatment of this dissolved molecular species. These matters are discussed in detail by Kern (1960).

The partial pressure of a dissolved gas is the partial pressure with which the dissolved gas would be in equilibrium if the solution were in contact with a gaseous phase. It is common practice to refer to the partial pressure of a solute such as H2CO3 or dissolved O2 even though the water may be isolated from the gas phase. For example, we can refer to the partial pressure of dissolved CO2 in groundwater even though the water is isolated from the earth’s atmosphere and from the gases in the open pore spaces above the water table.

In dilute solutions the partial pressure of a solute, expressed in bars (1 bar = 105 N/m2), is proportional to its molality. This is a statement of Henry’s law. It is applicable to gases that are not very soluble, such as CO2, O2, N2, CH4, and H2S. From application of the law of mass action to Eq. (3.15),

K_\ce{CO2} = \ce{\frac{H2CO3}{[H2O][CO2(g)]}} (3.16)

Because the activity of H2O is unity except for very saline solutions and because the partial pressure of CO2 in bars is equal to its molality, Eq. (3.16) can be expressed as

K_\ce{CO2} = \frac{\ce{H2CO3}}{\gamma_\ce{CO2} \cdot P_\ce{CO2}} (3.17)

where \gamma_{\ce{CO2}} is the activity coefficient for dissolved CO2 and P_{\ce{CO2}}, is the partial pressure in bars. With this expression the partial pressure of CO2 that would exist at equilibrium with a solution of specified H2CO3 activity can be computed. The activity coefficients for uncharged solute species such as dissolved gases (CO2, O2, H2S, N2, etc.) are greater than unity. The solubility of these gases in water therefore decreases with increasing ionic strength. This effect is known as the salting-out effect.

In addition to its dependence on ionic strength, the activity coefficient can be influenced by the type of electrolyte present in the water. For example, at a given ionic strength, CO2 is less soluble (i.e., has a larger activity coefficient) in a NaCl solution than in a KCl solution. Most geochemical problems of interest in groundwater hydrology involve solutions at ionic strengths of less than 0.1 or 0.2. It is common practice, therefore, for the activity coefficient of the dissolved gas to be approximated as unity. Consequently, under these conditions, Eq. (3.17) reduces to the relation

K_\ce{CO2} = \frac{\ce{H2CO3}}{P_\ce{CO2}} (3.18)

3.3 Association and Dissociation of Dissolved Species

The Electroneutrality Condition

Before proceeding with a discussion of the processes and consequences of the chemical interactions between groundwater and the geologic materials through which it flows, the behavior of dissolved constituents in the liquid phase without interactions with solid phases will be considered.

A fundamental condition of electrolyte solutions is that on a macroscopic scale, rather than the molecular scale, a condition of electroneutrality exists. The sum of the positive ionic charges equals the sum of the negative ionic charges, or

\sum zm_c = \sum zm_a (3.19)

where z is the ionic valence, mc the molality of cation species, and ma the molality of anion species. This is known as the electroneutrality equation, or the charge-balance equation, and it is used in nearly all calculations involving equilibrium interactions between water and geologic materials.

An indication of the accuracy of water analysis data can be obtained using the charge-balance equation. For example, if a water sample is analyzed for the major constituents listed in Table 3.3, and if the concentration values are substituted into Eq. (3.19) as

\ce{(Na+) + 2(Mg^{2+}) + 2(Ca^{2+}) = (Cl-) + (HCO^-_3) + 2(SO4^{2-})} (3.20)

The quantities obtained on the left- and right-hand sides of the equation should be approximately equal. Silicon is not included in this relation because it occurs in a neutral rather than in a charged form. If significant deviation from equality occurs, there must be (1) analytical errors in the concentration determinations or (2) ionic species at significant concentration levels that were not included in the analysis. It is common practice to express the deviation from equality in the form

E = \frac{\sum zm_c - \sum zm_a}{\sum zm_c + \sum zm_a} 100 (3.21)

where E is the charge-balance error expressed in percent and the other terms are as defined above.

Water analysis laboratories normally consider a charge-balance error of less than about 5% to be acceptable, although for some types of groundwater many laboratories consistently achieve results with errors that are much smaller than this. It should be kept in mind that an acceptable charge-balance error may occur in situations where large errors in the individual ion analyses balance one another. Appraisal of the charge-balance error therefore cannot be used as the only means of detecting analytical errors.

For the purpose of computing the charge-balance error, the results of chemical analyses are sometimes expressed as millequivalents per liter. When these units are used, the valence terms are omitted from Eq. (3.20).

Dissociation and Activity of Water

In the liquid state water undergoes the equilibrium dissociation,

\ce{H2O <=> H+ + OH-} (3.22)

which, from the law of mass action, can be expressed as

K_w = \ce{\frac{[H]^+[OH^-]}{[H2O]}} (3.23)

where brackets denote activities. It will be recalled that the activity of pure water is defined as unity at standard-state conditions. The reference condition of 25°C at 1 bar will be used. Since water vapor at low or moderate pressures behaves as an ideal gas, the activity of water in aqueous solution can be expressed as

\ce{H2O} = \frac{P_\ce{H2O}}{P^*_\ce{H2O}} (3.24)

where P^*_{\ce{H2O}} is the partial pressure of the vapor for pure water and P_{\ce{H2O}}, is the partial pressure of the vapor for the aqueous solution. At 25°C and 1 bar, the activity of water in a solution of NaCl at a concentration similar to that in seawater, which is approximately 3%, is 0.98, and in a 20% NaCl solution is 0.84. Thus, except for highly concentrated waters such as brines, the activity of water can, for practical purposes, be taken as unity. In this case

K_w = \ce{[H+][OH^-]} (3.25)

Values of Kw at temperatures between 0 and 50°C are listed in Table 3.4. Because the effect of fluid pressure is very slight, this expression is also acceptable for pressures as high as about 100 bars. At 1000 bars and 25°C, the activity of water is 2.062 (Garrels and Christ, 1965).

Table 3.4 Equilibrium Constants for Dissociation of Water, 0–60ºC

t (ºC) Kw × 10–14
0 0.1139
5 0.1846
10 0.2920
15 0.4505
20 0.6809
25 1.008
30 1.469
35 2.089
40 2.919
45 4.018
50 5.474
55 7.297
60 9.614
SOURCE: Garrels and Christ, 1965.

Since pH is defined as the negative logarithm of the hydrogen-ion activity, water at 25°C and pH 7 has equal H+ and OH activities ([H+] = [OH] = 1.00 × 10–7). At lower temperatures the equality of H+ and OH activities occurs at higher pH values and vice versa for higher temperatures. For example, at 0°C the equality occurs at a pH 7.53 and at 50°C at pH 6.63.

Polyprotic Acids

The most important acid in natural groundwater and in many contaminated groundwaters is carbonic acid (H2CO3), which forms when carbon dioxide (CO2) combines with water [Eq. (3.15)]. Carbonic acid can dissociate in more than one step by transferring hydrogen ions (protons) through the reactions

\ce{H2CO2 <=> H+ + HCO^-_3} (3.26)

\ce{HCO^-_3 <=> H+ + CO3^{2-}} (3.27)

Because hydrogen ions are commonly referred to as protons by chemists and because more than one hydrogen-ion dissociation is involved, carbonic acid is known as a polyprotic acid. Another polyprotic acid that occurs in groundwater, although in much smaller concentrations than carbonic acid, is phosphoric acid, which dissociates in three steps:

\ce{H3PO4 <=> H2PO^-_4 + H+} (3.28)

\ce{H2PO^-_4 <=> HPO4^{3-} + H+} (3.29)

\ce{HPO4^{2-} <=> PO4^{3-} + H+} (3.30)

Since the dissociation equations for the polyprotic acids all involve H+, it is possible to calculate the fraction of the acid in its molecular form or in any one of its anionic forms as a function of pH. For example, for carbonic acid, dissociation constants for Eqs. (3.26) and (3.27) can be expressed according to the law of mass action as

K_\ce{H2CO3} = \ce{\frac{[H+][HCO^-_3]}{[H2CO3]}} (3.31)

K_\ce{HCO^-_3} = \ce{\frac{[H+][CO3^{2-}]}{[HCO^-_3]}} (3.32)

A mass-balance expression for the carbon in the acid and its dissociated anionic species, expressed in molality, is

\text{DIC} = \ce{(H2CO3) + (HCO^-_3) + (CO3^{2-})} (3.33)

where DIC is the concentration of total dissolved inorganic carbon in these species. If we select an arbitrary value of 1 for DIC, and reexpress Eq. (3.33) in terms of pH, \ce{HCO^-_3}, K_{\ce{H2CO3}}, and K_{\ce{HCO^-_3}}, and then in terms of pH, CO32, and the dissociation constants, equations for the relative concentration of H2CO3, \ce{HCO^-_3}, and CO32as a function of pH are obtained. They are expressed graphically in Figure 3.5(a).

Figure 3.5 Distribution of major species of (a) dissolved inorganic carbon and (b) inorganic phosphorus in water at 25°C.

At lower pH, H2CO3 is the dominant species and at high pH, CO32 is the dominant species. Over most of the normal pH range of groundwater (6–9), \ce{HCO^-_3} is the dominant carbonate species. This is why \ce{HCO^-_3}, rather than CO32 or H2CO3, is listed in Table 3.3 as one of the major dissolved inorganic constituents in groundwater. Following a similar analysis, the relative concentrations of the dissolved phosphate species shown in Figure 3.5(b) are obtained. In the normal pH range of groundwater, \ce{H2PO^-_4} and HPO42 are the dominant species.

Ion Complexes

Chemical analyses of dissolved constituents in groundwater indicate the total concentrations of the constituents, but not the form in which the constituents occur in the water. Some constituents are present almost entirely in the simple ionic form. For instance, chlorine is present as the chloride ion, Cl. Calcium and magnesium, however, are present in the free ionic form, Ca2+ and Mg2+, in inorganic ion associations such as the uncharged (zero-valence) species, \ce{CaSO^{\circ}_4}, \ce{CaCO^{\circ}_3}, \ce{MgSO^{\circ}_4}, and \ce{MgCO^{\circ}_3}, and the charged associations, \ce{CaHCO^{+}_3} and \ce{MgHCO^{+}_3}. These charged and uncharged associations are known as complexes or in some cases, as ion pairs. The complexes form because of the forces of electrical attraction between the ions of opposite charge. Some inorganic species such as aluminum occur in dissolved form as Al3+, as the positively charged complex or ion pair, [Al(OH)]2+, and as complexes with covalent bonds such as [Al2(OH)2]4+, [Al6(OH15)]3+, and [Al(OH)4]. The total dissolved concentration of an inorganic species Ci can be expressed as

C_i = \sum C_{\text{free ion}} + \sum C_{\text{inorganic complexes}} + \sum C_{\text{organic complexes}} (3.34)

The occurrence of ion complexes can be treated using the law of mass action. For example, the formation of \ce{CaSO^{\circ}_4} can be expressed as

\ce{Ca^{2+} + SO4^{2-} <=> CaSO^o_4} (3.35)

with the equilibrium relation

K_{\ce{CaSO4^o}} = \ce{\frac{[Ca^{2+}][SO4^{2-}]}{[CaSO^o_4]}} (3.36)

where K_{\ce{CaSO^{\circ}_4}} is the thermodynamic equilibrium constant, sometimes referred to as the dissociation constant, and the terms in brackets are activities. Concentration values for the free ions are related to activities through the ionic strength versus activity coefficient relations described in Section 3.2. The activity coefficient for the neutral complex, \ce{CaSO^{\circ}_4}, is taken as unity. Values for K_{\ce{CaSO^{\circ}_4}} and equilibrium constants for other inorganic pairs and complexes can be computed using Eq. (3.12).

Table 3.5 shows the results of a chemical analysis of groundwater expressed in both milligrams per liter (or grams per cubic meter) and molality. The concentrations of free ions and inorganic ion complexes were calculated from the total analytical concentrations in the manner described below. In this sample (Table 3.5), the only complexes that occur in appreciable concentrations are those of sulfate; 18%, of the total sulfate is complexed. When groundwater has large sulfate concentrations, sulfate complexes are normally quite important. The procedure by which the concentrations of the free-ion complexes in Table 3.5 were calculated is described by Garrels and Christ (1965) and Truesdell and Jones (1974).

Table 3.5 Chemical Analysis of Groundwater Expressed as Analytical Results and as Computed Dissolved Species

  Analytical results from laboratory Computed dissolved species
Dissolved constituent mg/\ell or g/m3 Molality × 10–3 Free-ion
(molality × 10–3)
ion pairs*
(molality × 10–3)
ion pairs†
(molality × 10–3)
ion pairs‡
(molality × 10–3)
Ca 136 3.40 2.61 0.69 0.09 0.007
Mg 63 2.59 2.00 0.47 0.12 0.004
Na 325 14.13 14.0 0.07 0.06 0.001
K 9.0 0.23 0.23 0.003 <0.0001 <0.0001
Cl 40 1.0 1.0      
SO4 640 6.67 5.43      
HCO3 651 10.67 10.4      
CO3 0.12 0.020 0.0086      
DIC 147.5 12.29 CO32–, DIC calculated from HCO3 and pH data
Temp. = 10°C, pH = 7.20, partial pressure CO2 (calculated) = 3.04 × 10–2 bar
\sum cations (analytical) = 26.39 meq/\ell \hspace{1cm} \sum anions (analytical) = 25.17
\sum cations (computed) = 23.64 meq/\ell \hspace{1cm} \sum anions (computed) = 22.44
Error in cation-anion total (charge-balance error) (analytical) = 2.9%
Error in cation-anion totals (charge-balance error) (computed) = 2.7%

*\text{SO}_4^{2-} complexes = CaSO^{\text{o}}_4, MgSO^{\text{o}}_4, NaSO^{\text{o}}_4, Na_2, SO^{\text{o}}_4, KSO^-_4.
\ce{HCO^-_3} complexes = CaHCO^-_3, MgHCO^-_3, NaHCO^{\text{o}}_3.
\text{CO}_3^{2-} complexes = CaCO^{\text{o}}_3, MgCO^{\text{o}}_3, NaCO^-_3.

Inorganic constituents in groundwater can also form dissolved complexes with organic compounds such as fulvic and humic acids. In natural groundwater, which rarely has dissolved organic carbon at concentrations of more than 10 mg/\ell, complexing of major ions with the dissolved organic matter is probably insignificant. In contaminated groundwaters, however, the movement of hazardous inorganic compounds as organic complexes can be very important.

Calculation of Dissolved Species

Depending on the methods of analysis used in the laboratory, results of analysis. of inorganic carbon may be expressed as total dissolved inorganic carbon (DIC) or as \ce{HCO^-_3}. Each of these types of data can be used, in conjunction with pH values to compute the concentrations of H2CO3, CO32, \ce{HCO^-_3} or DIC, and the partial pressure of CO2. Equations (3.18), (3.20), (3.31), (3.32), and (3.33) serve as a basis for the calculations. If the water is nonsaline, the activity of H2O and the activity coefficients for CO2 and H2CO3 are taken as unity. It must be kept in mind that Eqs. (3.18), (3.31), and (3.32) are expressed in activities, whereas Eqs. (3.20) and (3.33) require molalities. If in the chemical analysis of the water \ce{HCO^-_3} concentration and pH are determined, Eq. (3.31) can be used, along with Eq. (3.5) for conversion between concentrations and activities, to obtain the activity and concentration of H2CO3. Substitution of the H2CO3 activity in Eq. (3.18) yields the partial pressure of CO2 in bars. The activity of CO32 can be computed from Eq. (3.32) and then converted to concentration by Eq. (3.5). Substitution of the concentration values in Eq. (3.33) then yields the concentration of DIC. The accuracy of the calculated result is strongly dependent on the accuracy of pH measurement. To obtain reliable pH data, it is necessary to make the pH measurements in the field. This is discussed further in Section 3.9. In the following illustration of the method for calculating free-ion and complex concentrations, it will be assumed that only the cation-sulfate complexes occur in significant concentrations. The equilibrium relations of interest are, therefore,

K_{\ce{CaSO4^o}} = \ce{\frac{[Ca^{2+}][SO4^{2-}]}{[CaSO^o_4]}} (3.37)

K_{\ce{MgSO4^o}} = \ce{\frac{[Mg^{2+}][SO4^{2-}]}{[MgSO^o_4]}} (3.38)

K_{\ce{NaSO4^-}} = \ce{\frac{[Na^{2+}][SO4^{2-}]}{[NaSO^-_4]}} (3.39)

From the conservation of mass principle, we can write

\ce{Ca(total) = (Ca^{2+}) + (CaSO^o_4)} (3.40)

\ce{Mg(total) = (Mg^{2+}) + (MgSO^o_4)} (3.41)

\ce{Na(total) = (Na+) + (NaSO^-_4)} (3.42)

\ce{SO4(total) = (SO4^{2+}) + (CaSO^o_4) + (MgSO^o_4) + (NaSO^-_4)} (3.43)

The concentrations of Ca(total), Mg(total), Na(total), and SO4(total) are those obtained from the laboratory analysis. We therefore have seven equations and seven unknowns (Na+, Mg2+, Ca2+, SO42, \ce{NaSO^-_4}, \ce{MgSO^{\circ}_4}, and \ce{CaSO^{\circ}_4}). The equations can be solved manually using the method of successive approximations described by Garrels and Christ (1965). Conversion between activities and molalities is accomplished using the ionic strength versus activity coefficient relations indicated in the discussions of Eqs. (3.5) and (3.6). In many cases the ionic strength calculated from the total concentration values has acceptable accuracy. In saline solutions, however, the ionic strength should be adjusted for the effect of complexes.

The process of computing the concentrations of free ions and complexes can be quite tedious and time-consuming, particularly when the sulfate, bicarbonate, and carbonate complexes are all included in the calculations. In recent years it has become common for the computations to be done by digital computer. There are several well-documented and widely used computer programs available for this purpose. Two of the most readily available programs are those by Truesdell and Jones (1974), which were used to obtain the results listed in Table 3.5, and Kharaka and Barnes (1973). Processing of chemical data on groundwater using programs of this type is becoming a relatively standard procedure in situations where one wishes to interpret chemical analyses in a geochemical framework.

3.4 Effects of Concentration Gradients

Diffusion in solutions is the process whereby ionic or molecular constituents move under the influence of their kinetic activity in the direction of their concentration gradient. Diffusion occurs in the absence of any bulk hydraulic movement of the solution. If the solution is flowing, diffusion is a mechanism, along with mechanical dispersion, that causes mixing of ionic or molecular constituents. Diffusion ceases only when concentration gradients become nonexistent. The process of diffusion is often referred to as self-diffusion, molecular diffusion, or ionic diffusion.

The mass of diffusing substance passing through a given cross section per unit time is proportional to the concentration gradient. This is known as Fick’s first law. It can be expressed as

F = -D\frac{dC}{dx} (3.44)

where F, which is the mass flux, is the mass of solute per unit area per unit time [M/L2T]; D is the diffusion coefficient [L2/T]; C is the solute concentration [M/L3]; and dC/dx is the concentration gradient, which is a negative quantity in the direction of diffusion. The diffusion coefficients for electrolytes in aqueous solutions are well known. The major ions in groundwater (Na+, K+, Mg2+, Ca2+, Cl, \ce{HCO^-_3}, SO42–) have diffusion coefficients in the range 1 × 10–9 to 2 × 10–9 m2/s at 25°C (Robinson and Stokes, 1965). The coefficients are temperature-dependent. At 5°C, for example, the coefficients are about 50%, smaller. The effect of ionic strength is very small.

In porous media the apparent diffusion coefficients for these ions are much smaller than in water because the ions follow longer paths of diffusion caused by the presence of the particles in the solid matrix and because of adsorption on the solids. The apparent diffusion coefficient for nonadsorbed species in porous media, D*, is represented by the relation

D^* = \omega D (3.45)

where \omega, which is less than 1, is an empirical coefficient that takes into account the effect of the solid phase of the porous medium on the diffusion. In laboratory studies of diffusion of nonadsorbed ions in porous geologic materials, \omega values between about 0.5 and 0.01 are commonly observed.

From Fick’s first law and the equation of continuity, it is possible to derive a differential equation that relates the concentration of a diffusing substance to space and time. In one dimension, this expression, known as Fick’s second law, is

\frac{\partial C}{\partial t} = D^* \frac{\partial^2C}{\partial x^2} (3.46)

To obtain an indication of the rates at which solutes can diffuse in porous geological materials, we will consider a hypothetical situation where two strata containing different solute concentrations are in contact. It will be assumed that the strata are saturated with water and that the hydraulic gradients in these strata are negligible. At some initial time, one of the strata has solute species i at concentration C0. In the other bed the initial concentration of C is small enough to be approximated as zero. Because of the concentration gradient across the interface, the solute will diffuse from the higher concentration layer to the lower concentration layer. It will also be assumed that the solute concentration in the higher concentration layer remains constant with time, as would be the case if the solute concentration were maintained at an equilibrium by mineral dissolution. Values of C in the x direction over time t can be calculated from the relation (Crank, 1956)

C_i(x,t) = C_0 \text{erfc}(x/2\sqrt{D^*t}) (3.47)

where erfc is the complementary error function (Appendix V). Assuming a value of 5 × 10–10 m2/s for D*, the solute concentration profile at specified time intervals can be computed. For instance, if we choose a relative concentration C/C0 of 0.1 and a distance x of 10 m, Eq. (3.47) indicates that the diffusion time would be approximately 500 years. It is evident, therefore, that diffusion is a relatively slow process. In zones of active groundwater flow its effects are usually masked by the effects of the bulk water movement. In low-permeability deposits such as clay or shale, in which the groundwater velocities are small, diffusion over periods of geologic time can, however, have a strong influence on the spatial distribution of dissolved constituents. This is discussed further in Sections 7.8 and 9.2.

Laboratory investigations have shown that compacted clays can act as semipermeable membranes (Hanshaw, 1962). Semipermeable membranes restrict the passage of ions while allowing relatively unrestricted passage of neutral species. If the pore waters in strata on either side of a compacted clay layer have different ionic concentrations, the concentration of the water in these strata must also be different. Because water molecules as uncharged species can move through semi-permeable clay membranes, it follows that under conditions of negligible hydraulic gradients across the membrane, movement from the higher water-concentration zone (lower salinity zone) to the lower water-concentration zone (higher salinity Zone) would occur by diffusion. If the higher salinity zone is a closed system. movement of water into the zone by diffusion across the clay will cause the fluid pressure in it to rise. If the lower-salinity zone is a closed system, its fluid pressure will decline. This process of development of a pressure differential across the clay is known as osmosis. The equilibrium osmotic pressure across the clay is the pressure differential that would exist when the effect of water diffusion is balanced by the pressure differential. When this occurs, migration of water across the clay ceases. In laboratory experiments the osmotic pressure across a semipermeable membrane separating solutions of different concentrations is measured by applying a pressure differential just sufficient to prevent water diffusion. In sedimentary basins osmosis may cause significant pressure differentials across clayey strata even if the equilibrium osmotic pressure differential is not achieved.

Many equations have been used to express the relation between the osmotic pressure differential and the difference in solution concentration across semi-permeable membranes. One of these, which can be derived from thermodynamic arguments (Babcock, 1963), is

P_0 = \frac{RT}{V_{\ce{H2O}}} \text{ln} \left( \ce{\frac{[H2O]^I}{[H2O]^{II}}} \right) (3.48)

where P0 is the hydrostatic pressure differential caused by osmosis, R is the gas constant (0.0821 liter • bar/K • mol), T is degrees Kelvin, \bar{V}_{\ce{H2O}}, is the molal volume of pure water, (0.018 \ell/mol at 25°C) and [H2O]I and [H2O]II are the activities of water in the more saline solution and the less saline solution, respectively. Values for the activity of water in various salt solutions are listed in Robinson and Stokes (1965). Using Eq. (3.48), it can be shown that salinity differences that are not uncommon in groundwater of sedimentary basins can cause large osmotic pressures, provided of course that there is a compacted, unfractured clay or shale separating the salinity zones. For example, consider two sandstone aquifers, I and II, separated by a layer of compacted clay. If the water in both of the aquifers has high NaCl concentrations, one with 6% NaCl and the other with 12% NaCl, the H2O activity ratio will be 0.95, which upon substitution in Eq. (3.48) yields an osmotic pressure difference between the two aquifers of 68 bars. This is the equivalent of 694 m of hydrostatic head (expressed in terms of pure water). This would indeed be a striking head differential in any sedimentary basin. For large osmotic pressure differentials to actually occur, however, it is necessary for the hydro-stratigraphic conditions to be such that osmotic pressure develops much more quickly than the pressure that is dissipated by fluid flow from the high-pressure zone and by flow into the low-pressure zone.

3.5 Mineral Dissolution and Solubility

Solubility and the Equilibrium Constant

When water comes into contact with minerals, dissolution of the minerals begins and continues until equilibrium concentrations are attained in the water or until all the minerals are consumed. The solubility of a mineral is defined as the mass of the mineral that will dissolve in a unit volume of solution under specified conditions. The solubilities of minerals that are encountered by groundwater as it moves along its flow paths vary over many orders of magnitude. Thus, depending on the minerals that the water has come into contact with during its flow history, groundwater may be only slightly higher in dissolved solids than rainwater, or it many become many times more salty than seawater.

Table 3.6 indicates the solubilities of several sedimentary minerals in pure water at 25°C and 1 bar total pressure. This table also lists the dissolution reactions for these minerals and the equilibrium constants for the reactions at 25°C and 1 bar. The solubility of carbonate minerals is dependent on the partial pressure of CO2. The solubilities of calcite and dolomite at two partial pressures (103 bar and 101 bar) are listed in Table 3.6 as an indication of the range of values that are relevant for natural groundwater.

Table 3.6 Dissociation Reactions, Equilibrium Constants, and Solubilities of Some Minerals That Dissolve Congruently in Water at 25°C and 1 Bar Total Pressure

Mineral Dissociation reaction Equilibrium constant,
Solubility at pH 7 (mg/\ell or g/m3)
Gibbsite Al2O3 • 2H2O + H2O = 2Al3+ + 6OH 10–34 0.001
Quartz SiO2 + 2H2O = Si(OH)4 10–3.7 12
Hydroxylapatite Ca5OH(PO4)3 = 5Ca2+ + 3PO43– +OH 10–55.6 30
Amorphous silica SiO2 + 2H2O = Si(OH)4 10–2.7 120
Fluorite CaF2 = Ca2+ + 2F 10–9.8 160
Dolomite CaMg(CO3)2 = Ca2+ + Mg2+ + 2CO32– 10–17.0 90,* 480†
Calcite CaCO3 = Ca2+ + CO32– 10–8.4 100,* 500†
Gypsum CaSO4 • 2H2O = Ca2+ + SO42– + 2H2O 10–4.5 2100
Sylvite KCl = K+ + Cl 10+0.9 264,000
Epsomite MgSO4 • 7H2O = Mg2+ + SO42– + 7H2O –– 267,000
Mirabillite Na2SO4 • 10H2O = 2Na+ + SO42– + 10H2O 10–1.6 280,000
Halite NaCl = Na+ + Cl 10+1.6 360,000
*Partial pressure of CO2 = 10–3 bar.
†Partial pressure of CO2 = 10–1 bar.
SOURCE: Solubility data from Seidell, 1958.

Comparison of the mineral solubilities and equilibrium constants indicates that the relative magnitudes of the equilibrium constant are a poor indication of the relative solubilities of the minerals because in the equilibrium relations, the activities of the ions or molecules are raised to the power of the number of moles in the balanced dissociation expression. For example, the solubility of calcite in pure water at PCO2 = 101 bar is 500 mg/\ell, and the solubility of dolomite under the same conditions is nearly the same (480 mg/\ell), but the equilibrium constants differ by eight orders of magnitude because the term [CO32] is raised to the second power in the Kdol expression. Another example is hydroxylapatite, which has a solubility of 30 mg/\ell at pH 7 and yet has an equilibrium constant of 1055.6, a value that might give the erroneous impression that this mineral has no significant solubility.

All the minerals listed in Table 3.6 normally dissolve congruently. This statement means that the products of the mineral dissolution reaction are all dissolved species. Many minerals that affect the chemical evolution of groundwater dissolve incongruently; that is, one or more of the dissolution products occur as minerals or as amorphous solid substances. Most aluminum silicate minerals dissolve incongruently. The feldspar, albite, is a good example

{\underbrace{\ce{NaAlSi3O8}}_\text{albite} + \ce{H2CO3} + \frac{9}{2}\ce{H2O <=>}

\ce{Na+ + HCO^-_3 + 2H4SiO4} + \underbrace{\frac{1}{2} \ce{Al2Si2O5(OH)4}}_\text{kaolinite} \text{(s)}

In this reaction albite dissolves under the leaching action of carbonic acid (H2CO3) to produce dissolved products and the clay mineral kaolinite. This is a common reaction in groundwater zones in granitic terrain. From the law of mass action

K_{\text{alb-kaol}} = \ce{\frac{[Na+][HCO^-_3][H4SiO4]^2}{[H2CO3]}} (3.50)

where the equilibrium constant K depends on temperature and pressure. If the partial pressure of CO2 is specified, it is evident from Eqs. (3.18), (3.31), and (3.32) that [H2CO3] and [\ce{HCO^-_3}] are also specified. The solubility of albite and other cation aluminosilicates increases with increasing partial pressure of CO2.

Effect of Ionic Strength

Comparison of the solubilities of minerals in pure water versus water with a high salt content indicates that the salinity increases the solubilities. This is known as the ionic strength effect because the increased solubility is caused by decreases in activity coefficients as a result of increased ionic strength. For example, the expression for the equilibrium constant for gypsum can be

K_{\text{gyp}} = [\gamma_{\ce{Ca^{2+}}} \cdot \gamma_{\ce{SO4^{2-}}}] [\ce{(Ca^{2+})(SO4^{2-})}] (3.51)

where \gamma is the activity coefficient and the species in parentheses are expressed in molality. Figure 3.3 indicates that as ionic strength increases, \gamma_{\ce{Ca^{2+}}} and the \gamma_{\ce{SO4^{2-}}} value decrease. To compensate, in Eq. (3.51), the concentrations of Ca2+ and SO42 must increase. This results in greater solubility of the mineral under the specified conditions of temperature and pressure. This effect is illustrated in Figure 3.6, which shows that the solubility of gypsum more than triples as a result of the ionic strength effect. Other examples, described in Chapter 7, indicate that the ionic strength effect can play an important role in the chemical evolution of natural and contaminated groundwater.

Figure 3.6 Solubility of gypsum in aqueous solutions of different NaCl concentrations, 25°C, and 1 bar (after Shternina, 1960).

The Carbonate System

It is estimated that over 99% of the earth’s carbon exists in carbonate minerals, the most important of which are calcite, CaCO3, and dolomite, CaMg(CO3)2. In nearly all sedimentary terrain and in many areas of metamorphic and igneous rocks, groundwater is in contact with carbonate minerals during at least part of its flow history. The ability of the groundwater zone to minimize adverse effects of many types of pollutants can be dependent on interactions that involve water and carbonate minerals. Interpretation of carbon 14 age dates of groundwater requires an understanding of how the water has interacted with these minerals.

At equilibrium, the reactions between water and the carbonate minerals calcite and dolomite can be expressed as

K_{\text{cal}} = \ce{[Ca^{2+}][CO3^{2-}]} (3.52)

K_{\text{dol}} = \ce{[Ca^{2+}][Mg^{2+}][CO3^{2-}]^2} (3.53)

where the equilibrium constants depend on temperature and pressure.

If the minerals dissolve in water that has an abundant supply of CO2(g) at a constant partial pressure, the concentration of dissolved CO2 (expressed as carbonic acid, H2CO3) remains constant, as indicated by Eq. (3.18). It is instructive to represent the calcite dissolution process as

\ce{CaCO3 + H2CO3 -> Ca^{2+} + 2HCO^-_3} (3.54)

which indicates that the dissolution is accompanied by consumption of carbonic acid. The higher the P_{\ce{CO2}}, the greater is the amount of H2CO3, available for consumption, and hence the reaction proceeds farther to the right to achieve equilibrium.

An aqueous system in which the dissolved CO2 is constant because of relatively unobstructed interaction with an abundant gaseous environment of constant P_{\ce{CO2}}, such as the earth’s atmosphere, is commonly referred to in the context of mineral dissolution as an open system. If the H2CO3 consumed by mineral-water reactions is not replenished from a gaseous reservoir, the system is denoted as a closed system.

Substitutions of Eqs. (3.18), (3.31), and (3.32) in Eq. (3.52) and rearranging yields

\ce{[H+]} = \left\{ \frac{K_{\ce{H2CO3}} \cdot K_{\ce{HCO^-_3}} \cdot K_{\ce{CO2}}}{K_{\text{cal}}} \cdot P_{\ce{CO2}} [\ce{Ca^{2+}}] \right\}^{1/2} (3.55)

The bracketed terms are activities and P_{\ce{CO2}}, is expressed in bars. Values for the equilibrium constants in the range 0–30°C are listed in Table 3.7. At 25°C, Eq. (3.55) simplifies

\ce{[H+]} = 10^{-4.9} \{[\ce{Ca^{2+}}]P_{\ce{CO2}} \}^{1/2} (3.56)

Table 3.7 Equilibrium Constants for Calcite, Dolomite, and Major Aqueous Carbonate Species in Pure Water, 0–30°C, and 1 Bar Total Pressure

\text{p}K_{\text{CO}_2}^* \text{p}K_{\text{H}_2\text{CO}_3}^* \text{p}K_{\text{HCO}_3^-} \text{p}K_{\text{cal}} \text{p}K_{\text{dol}}
0 1.12 6.58 10.62 8.340 16.56
5 1.20 6.52 10.56 8.345 16.63
10 1.27 6.47 10.49 8.355 16.71
15 1.34 6.42 10.43 8.370 16.79
20 1.41 6.38 10.38 8.385 16.89
25 1.47 6.35 10.33 8.400 17.0
30 1.67 6.33 10.29 8.51 17.9
*pK = –log K.
SOURCES: Garrels and Christ, 1965; Langmuir, 1971.

Table 3.8 Rules for Assigning Oxidation States and Some Examples

Rules for assigning oxidation states:

  1. The oxidation state of free elements, whether in atomic or molecular form, is zero.
  2. The oxidation state of an element in simple ionic form is equal to the charge on the ion.
  3. The oxidation state of oxygen in oxygen compounds is –2. The only exceptions are O2, O3 (see rule 1), OF2 (where it is +2), and peroxides such as H2O2 and other compounds with –0–0– bonds,
    where it is –1.
  4. The oxidation state of hydrogen is +1 except in H2 and in compounds where H is combined with a less electronegative element.
  5. The sum of oxidation states is zero for molecules, and for ion pairs or complexes it is equal to the formal charge on the species.


Carbon compounds Sulfur compounds Nitrogen compounds Iron compounds
Substance C state Substance S state Substance N State Substance Fe state
\ce{HCO^-_3} +IV S 0 N2 0 Fe 0
CH2O 0 FeS2 –I NH4 +III FeCO3 +II
C6H12O6 0 FeS –II NO^-_2 +V Fe2O3 +III
CH4 –IV SO32– +IV NO^-_3 –III Fe(OH)3 +III
SOURCE: Gymer, 1973; Stumm and Morgan, 1970.

To calculate the solubility of calcite under the specified CO2 partial pressure, another equation is required. At this stage in this type of mineral-water equilibrium problem, it is appropriate to make use of the electroneutrality equation. For the case of calcite dissolution in pure water, the electroneutrality expression is

\ce{2(Ca^{2+}) + (H+) = (HCO^-_3) + 2(CO3^{2-}) + (OH-)} (3.57)

The terms in this equation are expressed in molality. For the P_{\ce{CO2}}, range of interest in groundwater studies, the (H+) and (OH) terms are negligible compared to the other terms in this equation. Equations (3.56) and (3.57) can be combined and with substitution of Eqs. (3.18), (3.31), and (3.32) can be expressed as a polynomial in terms of two of the variables and the activity coefficients. For a specific iterative solutions by computer can be obtained. Manual solutions can also be obtained with little difficulty using the method of successive approximations outlined by Garrels and Christ (1965) and Guenther (1975). As a first approximation a convenient approach is to assume that (\ce{HCO^-_3}) is large compared to (CO32). Figure 3.5(a) indicates that this assumption is valid for solutions with pH values less than about 9, which includes nearly all natural waters, provided that concentrations of cations that complex CO32 are low. Equation (3.57) therefore reduces to

\ce{(Ca^{2+}) = \frac{HCO^-_3}{2}} (3.58)

After calcite has dissolved to equilibrium at a specified P_{\ce{CO2}}, the dissolved species in the water can now be obtained through the following steps: (1) assign an arbitrary value of [H+] to Eq. (3.55) and then calculate a value for [Ca2+]; (2) estimate an ionic strength value using the [Ca2+] obtained from step (1) and an \ce{HCO^-_3} value obtained from Eq. (3.58); (3) obtain an estimate for \gamma_{ce{Ca^{2+}}} and \gamma_\ce{HCO^-_3} from Figure 3.3 and then calculate (Ca2+) from the relation (Ca2+) = [Ca2+]/\gamma_{ce{Ca^{2+}}}; (4) using the specified P_{\ce{CO2}}, and the [H+] chosen in step (1), calculate [\ce{HCO^-_3}] from Eqs. (3.18) and (3.31); (5) convert [\ce{HCO^-_3}] to \ce{(HCO^-_3)} through the activity coefficient relation; and (6) compare the (Ca2+) value obtained from step (1) with the calculated value of \ce{(HCO^-_3)}/2 from step (5). If the two computed values are equal or nearly so, Eq. (3.57) has been satisfied and a solution to the problem has been obtained. If they are unequal, the sequence of computational steps must be repeated using a new selection for [H+]. In these types of problems an acceptable solution can usually be obtained after two or three iterations. The results of these types of calculations for equilibrium calcite dissolution in pure water under various fixed P_{\ce{CO2}}, and temperature conditions are illustrated in Figure 3.7, which indicates that the solubility is strongly dependent on the P_{\ce{CO2}}, and that the equilibrium values of [H+] or pH vary strongly with P_{\ce{CO2}}. The calculation procedure did not include ion pairs such as \ce{CaCO^{\circ}_3} and \ce{CaHCO^+_3}, which occur in small concentrations in dilute aqueous solutions saturated with CaCO3. From Figure 3.7 it is apparent that the Ca2+ and \ce{HCO^-_3} concentration lines are parallel (just 0.30 unit apart). This indicates that the reaction in Eq. (3.54) accurately represents the dissolution process under the range of P_{\ce{CO2}}, conditions that are characteristic of the groundwater environment, where P_{\ce{CO2}}, is almost invariably greater than 10–4 bar.

Figure 3.7 Dissolved species in water in equilibrium with calcite as a function of PCO2, open-system dissolution (after Guenther, 1975).

This explains why \ce{HCO^-_3} rather than CO32 is the dominant ionic species of dissolved inorganic carbon in groundwater.

If water becomes charged with CO2, which may occur because of contact with the atmosphere of the earth or with the soil-zone atmosphere, and then comes into contact with calcite or dolomite in a zone isolated from the gaseous CO2 source, such as the groundwater zone, dissolution will occur but the concentration of dissolved species at equilibrium will be different. In this process of closed-system dissolution, carbonic acid is consumed and not replenished from outside the system as dissolution proceeds. For this condition Eq. (3.18) indicates that the P_{\ce{CO2}}, must also decline as the reaction proceeds toward equilibrium.

The carbonate minerals are less soluble under closed-system conditions, and have higher equilibrium pH values. In the closed-system case, the dissolved inorganic carbon is derived from the dissolved CO2 present as dissolution begins and from the calcite and/or dolomite which dissolves. In the open-system case, CO2 continues to enter the solution from the atmosphere as dissolution proceeds. In this case the total dissolved inorganic carbon consists of carbon from the initial CO2 and from the replenished CO2 and also from the minerals. As indicated in Chapter 7, these differences can be crucial to the interpretation of the chemical evolution of groundwater in carbonate terrain and in the evaluation of carbon 14 age dates.

The Common-Ion Effect

In some situations the addition of ions by dissolution of one mineral can influence the solubility of another mineral to a greater degree than the effect exerted by the change in activity coefficients. If an electrolyte that does not contain Ca2+ or CO32 is added to an aqueous solution saturated with calcite, the solubility of calcite will increase because of the ionic strength effect. However, if an electrolyte is added which Contains either Ca2+ or CO32, calcite will eventually precipitate because the product [Ca2+][CO32] must adjust to attain a value equal to the equilibrium constant Kcal. This process is known as the common-ion effect.

Water moving through a groundwater zone that contains sufficient Ca2+ and CO32 for their activity product to equal Kcal may encounter strata that contain gypsum. Dissolution of gypsum,

\ce{CaSO4} \cdot 2\ce{H2O -> Ca^{2+} + SO4^{2-} + H2O} (3.59)

causes the ionic strength to increase and the concentration of Ca2+ to rise. Expressed in terms of molality and activity coefficients, the equilibrium expression for calcite is

K_{\text{cal}} = \gamma_{\ce{Ca^{2+}}} \cdot \gamma_{\ce{CO3^{2-}}} \cdot \ce{(Ca^{2+})(CO3^{2-})}

Gypsum dissolution causes the activity coefficient product \gamma_{\ce{Ca^{2+}}} \cdot \gamma_{\ce{CO3^{2-}}} to decrease. But because of the contribution of (Ca2+) from dissolved gypsum, the product (Ca2+)(CO32) increases by a much greater amount. Therefore, for the solution to remain in equilibrium Vsllh respect to calcite, precipitation of calcite must occur.

The solubilities of calcite and gypsum in water at various NaCl concentrations are shown in Figure 3.8. For a given NaCl content, the presence of each mineral, calcite or gypsum, causes a decrease in the solubility of the other. Because of the ionic strength effect, both minerals increase in solubility at higher NaCl concentrations.

Disequilibrium and the Saturation Index

Considering Eq. (3.2) in a condition of disequilibrium, the relation between the reactants and the products can be expressed as

Q = \frac{[B]^b[C]^c}{[D]^d[E]^e} (3.60)

where Q is the reaction quotient and the other parameters are as expressed in Eq. (3.3).

Figure 3.8 Solubility of gypsum and calcite in water with various concentrations of dissolved NaCl, 25°C, PCO2 = 1 bar (after Shternina and Frolova, 1945).

The following ratio is a useful comparison between the status of a mineral dissolution-precipitation reaction at a particular point in time or space and the thermodynamic equilibrium condition:

S_i = \frac{Q}{K_{\text{eq}}} (3.61)

where Si is called the saturation index. For calcite in contact with groundwater (see Section 3.2), the saturation index is

S_i = \frac{\ce{[Ca^{2+}][CO3^{2-}]}}{K_{\text{cal}}} (3.62)

The ion activities in the numerator can be obtained from analysis of groundwater samples and Eq. (3.32) and the equilibrium constant Kcal, can be obtained from the free-energy data, or directly from equilibrium-constant tabulations, such as Table 3.7.

If Si > 1, the water contains an excess of the ionic constituents. The reaction [Eq. (3.4)] must therefore proceed to the left, which requires that mineral precipitation occur. If Si < l, the reaction proceeds to the right as the mineral dissolves. If Si = 1, the reaction is at equilibrium, which means that it is saturated with respect to the mineral in question. With the saturation index relation, it is possible for specified mineral-water reactions to compare the status of actual water samples to computed equilibrium conditions. For the saturation index to be of interest, the mineral need not actually be present in the groundwater zone. Knowledge of the mineralogical composition is necessary, however, if one desires to obtain a detailed understanding of the geochemical behavior and controls on the water.

Some authors express the saturation index in logarithmic form, in which case an index value of zero denotes the equilibrium condition. The saturation index is in some publications denoted as the disequilibrium index because in some situations groundwater is more generally in disequilibrium than in equilibrium with respect to common minerals.

3.6 Oxidation and Reduction Processes

Oxidation States and Redox Reactions

Many reactions that occur in the groundwater environment involve the transfer of electrons between dissolved. gaseous, or solid constituents. As a result of the electron transfer there are changes in the oxidation states of the reactants and the products. The oxidation state, sometimes referred to as the oxidation number, represents a hypothetical charge that an atom would have if the ion or molecule were to dissociate. The oxidation states that can be achieved by the most important multioxidation state elements that occur in groundwater are listed in Table 3.8, which also contains some rules that can be used to deduce the oxidation state from the formula of a substance. Sometimes there are uncertainties in the assignment of electron loss or electron gain to a particular atom, especially when the reactions involve covalent bonds. In this book Roman numerals are used to represent oxidation states and Arabic numbers represent actual valence.

In oxidation-reduction reactions, which will be referred to as redox reactions, there are no free electrons. Every oxidation is accompanied by a reduction and vice versa, so that an electron balance is always maintained. By definition, oxidation is the loss of electrons and reduction is the gain in electrons. This is illustrated by expressing the redox reaction for the oxidation of iron:

\ce{O2 + 4Fe^{2+} + 4H+ = 4Fe^{3+} + 2H2O} (3.63)

For every redox system half-reactions in the following form can be written:

\text{oxidized state} + ne = \text{reduction state} (3.64)

The redox reaction for iron can therefore be expressed in half-reactions,

\ce{O2 + 4H+} +4e = \ce{2H2O} \hspace{1cm} \text{(reduction)} (3.65)

\ce{4Fe^{2+} = 4Fe^{3+}} + 4e \hspace{1cm} \text{(oxidation)} (3.66)

In the reduction half-reaction the oxidation state of oxygen goes from zero (oxygen as O2) to –II (oxygen in H2O). There is, therefore, a release of four electrons because 2 mol of H2O forms from 1 mol of O2 and 4 mol of H+. In the oxidation half-reaction, 4 mol of Fe(+II) goes to 4 mol of Fe (+III), with a gain of four electrons. The complete redox reaction [Eq. (3.63)] expresses the net effect of the electron transfer and therefore contains no free electrons. When writing half-reactions, care must be taken to ensure that the electrons on each side of the equation are balanced. These reactions need not involve oxygen or hydrogen, although most redox reactions that occur in the groundwater zone do involve one or both of these elements. The concept of oxidation and reduction in terms of changes in oxidation states is illustrated in Figure 3.9.

A list of half-reactions that represent most of the redox processes occurring in groundwater is presented in Table 3.9.

Figure 3.9 Oxidation and reduction in relation to oxidation states.

Table 3.9 Redox Half-Reactions for Many Constituents That Occur in the Groundwater Environment

(1) \frac{1}{4}\ce{O2 + H+} + e = \frac{1}{2} \ce{H2O} (18) \frac{1}{8}\ce{SO4^{2-}} + \frac{9}{8}\ce{H+} + e = \frac{1}{8}\ce{HS^-} + \frac{1}{2}\ce{H2O}
(2) \ce{H+} + e = \frac{1}{2}\ce{H2O} (19) \frac{1}{2}\ce{S(s) + H+} + e = \frac{1}{2}\ce{H2S(g)}
(3) \ce{H2O} + e = \frac{1}{2}\ce{H2(g) + OH-} (20) \ce{Fe^{3+}} + e = \ce{Fe^{2+}}
(4) \frac{1}{5}\ce{NO^-_3} + \frac{6}{5}\ce{H+} + e = \frac{1}{10}\ce{N2(g)} + \frac{3}{5}\ce{H2O} (21) \ce{Fe(OH)3(s) + HCO^-_3 + 2H+} + e = \ce{Fe(CO3)(s) + 3H2O}
(5) \frac{1}{2}\ce{NO^-_3 + H+} + e = \frac{1}{2}\ce{NO^-_2} + \frac{1}{2}\ce{H2O} (22) \ce{Fe(OH)3(s) + 3H+} + e = \ce{Fe^{2+} + 3H2O}
(6) \frac{1}{8}\ce{NO^-_3} + \frac{5}{4}\ce{H+} + e = \frac{1}{8}\ce{NH^+_4} + \frac{3}{8}\ce{H2O} (23) \ce{Fe(OH)3(s) + H+} + e = \ce{Fe(OH)2(s) + H2O}
(7) \frac{1}{6}\ce{NO^-_2} + \frac{5}{4}\ce{H+} + e = \frac{1}{6}\ce{NH^+_4} + \frac{1}{3}\ce{H2O} (24) \frac{1}{2}\ce{FeS2(s) + 2H+} + e = \frac{1}{2}\ce{Fe^{2+} + H2S(g)}
(8) \frac{1}{4}\ce{NO^-_3} + \frac{5}{4}\ce{H+} + e = \ce{N2O(g)} + \frac{3}{8}\ce{H2O} (25) \frac{1}{2}\ce{FeS2(s) + S(s)} + e = \frac{1}{2}\ce{FeS2(s)}
(9) \frac{1}{2}\ce{NO^-_2} + \frac{3}{2}\ce{H+} + e = \frac{1}{4}\ce{N2O(g)} + \frac{3}{4}\ce{H2O} (26) \frac{1}{16}\ce{Fe^{2+}} + \frac{1}{8}\ce{SO4^{2-}} + e = \frac{1}{16}\ce{FeS2(s)} + \frac{1}{2}\ce{H2O}
(10) \frac{1}{6}\ce{N2(g)} + + \frac{4}{5}\ce{H+} + e = \frac{1}{3}\ce{NH^+_4} (27) \frac{1}{14}\ce{Fe(OH)2(s)} + \frac{1}{7}\ce{SO4^{2-}} + \frac{9}{7}\ce{H+} + e = \frac{1}{14}\ce{FeS2(s)} + \frac{5}{7}\ce{H2O}
(11) \frac{1}{4}\ce{CH2O} + \ce{H+} + e = \frac{1}{4}\ce{CH2O} +  \frac{1}{4}\ce{H2O} (28) \frac{1}{14}\ce{Fe(CO)3(s)} + \frac{1}{7}\ce{SO4^{2-}} + \frac{17}{14}\ce{H+} + e = \frac{1}{14}\ce{FeS2(s)} + \frac{4}{7}\ce{H2O} + \frac{1}{14}\ce{HCO^-_3}
(12) \frac{1}{4}\ce{CO2(g)} + \ce{H+} + e = \frac{1}{4}\ce{CH2O} +  \frac{1}{4}\ce{H2O} (29) \frac{1}{2}\ce{MnO2(s)} + \frac{1}{2}\ce{HCO^-_3} + \frac{3}{2}\ce{H+} + e = \frac{1}{2}\ce{MnCO3(s)} + \frac{3}{8}\ce{H2O}
(13) \frac{1}{2}\ce{CH2O} + \ce{H+} + e = \frac{1}{2}\ce{CH3OH} (30) \ce{Mn^{2+}} + 2e = \text{Mn(s)}
(14) \frac{1}{4}\ce{CO2(g) + H+} + e = \frac{1}{8}\ce{CH4(g)} \frac{1}{4}\ce{H2O} (31) \frac{1}{2}\ce{MnCO3(s)} + \frac{1}{2}\ce{H+} + e = \frac{1}{2}\text{Mn(s)} + \frac{1}{2}\ce{HCO^-_3}
(15) \frac{1}{2}\ce{CH3OH + H+} + e = \frac{1}{2}\ce{CH4(g)} + \frac{1}{2}\ce{H2O} (32) \text{MnOOH(s)} + \ce{HCO^-_3 + 2H+} + e = \ce{MnCO3 + 2H2O}
(16) \frac{1}{6}\ce{SO4^{2-}} + \frac{4}{3}\ce{H+} + e = \frac{2}{3}\ce{H2O} (33) \ce{MnO2 + H+} + e = \text{MnOOH}
(17) \frac{1}{8}\ce{SO4^{2-}} + \frac{5}{4}\ce{H+} + e = \frac{1}{8}\ce{H2S(g)} + \frac{1}{2}\ce{H2O}    

Consumption of Oxygen and Organic Matter

Unpolluted rivers and lakes generally have oxidizing conditions because of mixing with oxygen from the earth‘s atmosphere. The tendency in groundwater systems, however, is toward oxygen depletion and reducing conditions. Because the water that circulates through the groundwater zone is generally isolated from the earth’s atmosphere, oxygen that is consumed by hydrochemical and biochemical reactions is not replenished. In order for reduction of inorganic constituents to occur, some other constituents must be oxidized. The oxidized compound is generally organic matter. The reactions are catalyzed by bacteria or isolated enzymes that derive energy by facilitating the process of electron transfer. In the present discussion we will assume that reactions proceed in an appropriate thermodynamic direction, without clarification of the associated biochemical processes. To illustrate the process of organic-matter oxidation, a simple carbohydrate, CH2O, is used as the electron donor, even though numerous other organic compounds, such as polysaccharides, saccharides, fatty acids, amino acids, and phenols, may be the actual organic compound involved in the redox process

\ce{\frac{1}{4}CH2O + \frac{1}{4}H2O = \frac{1}{4}CO2(g) + H+} + e (3.67)

To obtain full reactions for redox processes, the half-reaction for the oxidation of organic matter, represented by Eq. (3.67), can be combined with many of the half-reactions for reduction of inorganic compounds given in Table 3.9. Combinations of Eq. (3.67) and reaction (1) in Table 3.9 yields the redox relation

\ce{O2(g) + CH2O = CO2(g) + H2O} (3.68)

which represents the process of organic-matter oxidation in the presence of bacteria and free molecular oxygen. This redox process is the main source of dissolved CO2. CO2 combines with H2O to produce H2CO3 [Eq. (3.l5)], which is an acid of considerable strength when viewed in a geochemical context.

Because the solubility of O2 in water is low (9 mg/\ell at 25°C and 11 mg/\ell at 5°C), and because O2 replenishment in subsurface environments is limited, oxidation of only a small amount of organic matter can result in consumption of all the dissolved O2. For example, from the mass conservation relations inherent in Eq. (3.68), oxidation of only 8.4 mg/\ell (0.28 mmol/\ell of CH2O would consume 9 mg/\ell (0.28 mmol/\ell) of O2. This would result in the water having no dissolved O2. Water that infiltrates through the soil zone is normally in contact with soil organic matter. O2 consumption and CO2 production is therefore a widespread process in the very shallow part of the subsurface environment.

Table 3.10 lists some redox reactions in which dissolved oxygen is consumed. In all these reactions, H+ ions are produced. In many groundwater systems the H+ ions are consumed by reactions with minerals. The pH therefore does not decrease appreciably. In some systems, however, minerals that react in this manner are not present, in which case the oxidation processes cause the water to become acidic.

Table 3.10 Some Inorganic Oxidation Processes That Consume Dissolved Oxygen in Groundwater

Process Reaction*  
Sulfide oxidation \ce{O2} + \frac{1}{2}\ce{HS-} = \frac{1}{2}\ce{SO4^{2-}} + \frac{1}{2}\ce{H+} (1)
Iron oxidation \frac{1}{4}\ce{O2} + \ce{Fe^{2+}} + \ce{H+} = \ce{Fe^{3+}} + \frac{1}{2}\ce{H+} (2)
Nitrification \ce{O2} + \frac{1}{2}\ce{NH^+_4} = \frac{1}{2}\ce{NO^-_3} + \ce{H+} + \frac{1}{2}\ce{H+} (3)
Manganese oxidation \ce{O2} + \ce{Mn^{2+} + 2H2O = 2MnO2(s) + 4H+} (4)
Iron sulfide oxidation† \frac{15}{4}\ce{O2} + \ce{FeS2(s)} + \frac{7}{2}\ce{H2O} = \ce{Fe(OH)3(s) + 2SO4^{2-} + 4H+} (5)
*(s), solid.
†Expressed as a combined reaction.

When all the dissolved O2 in groundwater is consumed, oxidation of organic matter can still occur, but the oxidizing agents (i.e., constituents that undergo reduction) are \ce{NO^-_3}, MnO2, Fe(OH)3, SO42, and others, as indicated in Table 3.11. As these oxidizing agents are consumed, the groundwater environment becomes more and more reduced. If the processes proceed far enough, the environment may become so strongly reducing that organic compounds may undergo anaerobic degradation. An equation for this process, which represents the conversion of organic matter to methane and carbon dioxide, is shown by reaction (5) in Table 3.11. The sequence of redox processes represented by reactions (1) to (5) in Table 3.11 proceed from aerobic oxidation through to methane fermentation provided that (1) organic matter in a consumable form continues to be available in the water, (2) the bacteria that mediate the reactions have sufficient nutrients to sustain their existence, and (3) the temperature variations are not large enough to disrupt the biochemical processes. In many groundwater systems one or more of these factors is limiting, so the groundwater does not proceed through all the redox stages. The evolution of groundwater through various stages of oxidation and reduction is described in more detail in Chapter 7.

Table 3.11 Some Redox Processes That Consume Organic Matter and Reduce Inorganic Compounds in Groundwater

Process Equation*  
Denitrification† \ce{CH2O} + \frac{4}{5}\ce{NO^-_3} = \frac{2}{5}\ce{N2(g) + HCO^-_3} + \frac{1}{5}\ce{H+} + \frac{2}{5}\ce{H2O} (1)
Manganese(IV) reduction \ce{CH2O + 2MnO2(s) + 3H+ = 2Mn^{2+} + HCO^-_3 + 2H2O} (2)
Iron(III) reduction \ce{CH2O + 4Fe(OH)3(s) + 7H+ = 4Fe^{2+} + HCO^-_3 + 10H2O} (3)
Sulfate reduction‡ \ce{CH2O} + \frac{1}{2}\ce{SO4^{2-}} = \frac{1}{2}\ce{HS- + HCO^-_3} + \frac{1}{2}\ce{H+} (4)
Methane fermentation \ce{CH2O} + \frac{1}{2}\ce{H2O} = \frac{1}{2}\ce{CH4} + \frac{1}{2}\ce{HCO^-_3} + \frac{1}{2}\ce{H+} (5)
*(g), gaseous or dissolved form; (s), solid.
†CH2O represents organic matter; other organic compounds can also be oxidized.
‡H2S exists as a dissolved species in the water: HS + H+ = H2S. H2S is the dominant species at pH < 7.

Equilibrium Redox Conditions

Aqueous solutions do not contain free electrons, but it is nevertheless convenient to express redox processes as half-reactions and then manipulate the half-reactions as if they occur as separate processes. Within this framework a parameter known as the pE is used to describe the relative electron activity. By definition,

\text{p}E = -\log[e] (3.69)

pE, which is a dimensionless quantity, is analogous to the pH expression for proton (hydrogen-ion) activity. The pE of a solution is a measure of the oxidizing or reducing tendency of the solution. In parallel to the convention of arbitrarily assigning \Delta G^{\circ} = 0 for the hydration of H+ (i.e. K_{\ce{H+}} = 0 for the reaction H+ + H2O = H3O+) the free-energy change for the reduction of H+ to H2(g) [H+ + e = \frac{1}{2}H2(g)] is zero. pE and pH are functions of the free energy involved in the transfer of 1 mol of protons or electrons, respectively.

For the general half-reaction

b\text{B} + c\text{C} + n\text{e} = d\text{B} + e\text{E} (3.70)

the law of mass action can be written as

K = \frac{[\text{D}]^d[\text{E}]^e}{[\text{e}]^n[\text{B}]^b[\text{C}]^c} (3.71)

For example, consider the oxidation of Fe(II) to Fe(III) by free oxygen:

\frac{1}{2} \ce{O2 + 2H+} + 2e = + \ce{H2O} \hspace{1cm} \text{(reduction)} (3.72)

\ce{2Fe^{2+} = 2Fe^{3+}} + 2e \hspace{1cm} \text{(oxidation)} (3.73)

\overline{\ce{\frac{1}{2}O2 + 2Fe^{2+} + 2H+ = 2Fe^{3+} + H2O} \hspace{1cm} \text{(redox reaction)}} (3.74)

In this book the equilibrium constants for half-reactions are always expressed in the reduction form. The oxidized forms and electrons are written on the left and the reduced products on the right. This is known as the Stockholm or IUPAC (International Union of Physical and Analytical Chemistry) convention. Expressing the half-reactions [Eq. (3.72) and (3.73)] in terms of equilibrium constants [Eq. (3.7l)] for conditions at 25°C and 1 bar yields

K = \frac{1}{P^{1/2}_{\ce{O2}}[\ce{H+}]^2[e]^2} = 10^{41.55} (3.75)

K = \frac{\ce{Fe^{2+}}}{[\ce{Fe^{3+}}][e]} = 10^{12.53} (3.76)

The numerical values for the equilibrium constants were computed from Eq. (3.12) using Gibbs’ free-energy data for 25°C and 1 bar. To obtain expressions for redox conditions expressed as pE, Eqs. (3.75) and (3.76) can be rearranged to yield

\text{p}E = 20.78 + \frac{1}{4} \log (P_{\ce{O2}}) - \text{pH} (3.77)

\text{p}E = 12.53 + \log \left(\ce{\frac{[Fe^{3+}]}{[Fe^{2+}]}}\right) (3.78)

If the redox reaction [Eq. (3.74)] is at equilibrium, and if the concentrations of iron, the P_{\ce{O2}}, and the pH are known, the pE obtained from both these relations is the same. Even though there may be many dissolved species in the solution involved in reactions with electron and hydrogen ion transfer, at equilibrium there is only one pE condition, just as there is only one pH condition. In groundwater systems there is an interdependency of pH and pE. Nearly all the reactions listed in Table 3.9 involve both electron and proton transfers. If equilibrium is assumed, the reactions that include pH can be written as pE expressions. Graphical representations of pH–pE relations are described below.

Although the discussion above was based entirely on the assumption that the redox processes are at equilibrium, in field situations the concentrations of oxidizable and reducible species may be far from those predicted using equilibrium models. Many redox reactions proceed at a slow rate and many are irreversible. It is possible, therefore, to have several different redox levels existing in the same locale. There is also the possibility that the bacteria required to catalyze many of the redox reactions exist in microenvironments in the porous media that are not representative of the overall macroenvironment in which the bulk flow of groundwater occurs. Equilibrium considerations can, however, greatly aid in our efforts to understand in a general way the redox conditions observed in subsurface waters. Stumm and Morgan (1970), in their comprehensive text on aquatic chemistry, state: “In all circumstances equilibrium calculations provide boundary conditions towards which the systems must be proceeding. Moreover, partial equilibria (those involving some but not all redox couples) are approximated frequently, even though total equilibrium is not reached. . . . Valuable insight is gained even when differences are observed between computations and observations. The lack of equilibrium and the need for additional information or more sophisticated theory are then made clear” (p. 300).

The redox condition for equilibrium processes can be expressed in terms of pE (dimensionless), Eh (volts), or \Delta G (joules or calories). Although in recent years pE has become a commonly used parameter in redox studies, Eh has been used in many investigations, particularly prior to the 1970’s. Eh is commonly referred to as the redox potential and is defined as the energy gained in the transfer of 1 mol of electrons from an oxidant to H2. The h in Eh indicates that the potential is on the hydrogen scale and E symbolizes the electromotive force. pE and Eh are related by

\text{p}E = \frac{nF}{2.3RT}Eh (3.79)

where F is the faraday constant (9.65 × 104 C • mol1), R the gas constant, T the absolute temperature, and n the number of electrons in the half-reaction. For reactions at 25°C in which the half-reactions are expressed in terms of transfer of a single electron, Eq. (3.79) becomes

\text{p}E = 16.9 Eh (3.80)

Eh is defined by a relation known as the Nernst equation,

Eh\text{(volts)} = Eh^\text{o} + \frac{2.3 RT}{nF} \log \left( \frac{\text{[oxidant]}}{\text{[reductant]}} \right) (3.81)

where Eh° is a standard or reference condition at which all substances involved are at unit activity and n is the number of transferred electrons. This is a thermodynamic convenience. Unit activities could only exist in solutions of infinite dilution; this condition is therefore only hypothetical. The equation relating Eh° directly to the equilibrium constant is

Eh^\text{o} = \frac{RT}{nF} \text{ln} \hspace{1mm} K (3.82)

In the study of aqueous systems the same objectives can be met using either pE or Eh to represent redox conditions. pE is often the preferred parameter because its formulation follows so simply from half-cell representations of redox reactions in combination with the law of mass action. Facility in making computations interchangeably between pE and Eh is desirable because tabulations of thermodynamic data for redox reactions are commonly expressed as Eh° values and because in some aqueous systems a convenient method of obtaining an indication of the redox conditions involves measurements of electrode potentials as voltage.

Microbiological Factors

Microorganisms catalyze nearly all the important redox reactions that occur in groundwater. This means that although the reactions are spontaneous thermodynamically, they require the catalyzing effect of microorganisms in order to proceed at a significant rate. Although it is not customary for microorganisms to be regarded as important components of the groundwater environment, their influence cannot be dismissed if we wish to understand the causes and effects of redox processes.

The microorganisms that are most important in redox processes in the groundwater zone are bacteria. Other types of microorganisms, such as algae, fungi, yeasts, and protozoans, can be important in other aqueous environments. Bacteria generally range in size from about 0.5 to 3 \mu m. They are small compared to the pore sizes in most nonindurated geological materials and are large in relation to the size of hydrated inorganic ions and molecules. The catalytic capability of bacteria is produced by the activity of enzymes that normally occur within the bacteria. Enzymes are protein substances formed by living organisms that have the power to increase the rate of redox reactions by decreasing the activation energies of the reactions. They accomplish this by strongly interacting with complex molecules representing molecular structures halfway between the reactant and the product (Pauling and Pauling, 1975). The local molecular environment of many enzyme reactions is very different from the bulk environment of the aqueous-system.

Bacteria and their enzymes are involved in redox processes in order to acquire energy for synthesis of new cells and maintenance of old cells. An important step in the process of bacterial cell growth is the construction of molecules forming an energy-storage substance known as adenosine triphosphate (ATP). After its formation, molecules of this high-energy material can by hydrolyzed through a sequence of energy-releasing reactions that provide for synthesis of new cell material. The growth of bacteria is therefore directly related to the number of moles of ATP formed from the available nutrients. Some of the energy obtained from the redox reactions is maintenance energy required by bacterial cells for such things as mobility, to prevent an undesirable flow of solutes either into or out of the cell, or for resynthesis of proteins that are constantly degrading (McCarty, 1965).

For bacteria to be able to make use of an energy yield from a redox reaction, a minimum free-energy change of approximately 60 kJ/mol between the reactants and the products is required (Delwiche, 1967). The main source of energy for bacteria in the groundwater zone is the oxidation of organic matter.

Bacteria that can thrive only in the presence of dissolved oxygen are known as aerobic bacteria. Anaerobic bacteria require an absence of dissolved oxygen. Facultative bacteria can thrive with or without oxygen. The lower limit of dissolved O2 for the existence of most aerobic bacteria is considered to be about 0.05 mg/\ell, but some aerobic species can persist at lower levels. Since most methods commonly used for measuring dissolved O2 have a lower detection limit of about 0.1 mg/\ell, it is possible that aerobic bacteria can mediate redox reactions in situations that might appear to be anaerobic based on the lack of detectable oxygen.

Bacteria of different varieties can withstand fluid pressures of many hundreds of bars, pH conditions from 1 to 10, temperatures from near 0 to greater than 75°C, and salinities much higher than that of seawater. They can migrate through porous geological materials and in unfavorable environments can evolve into resistant bodies that may be activated at a later time (Oppenheimer, 1963). In spite of these apparent characteristics of hardiness, there are many groundwater environments in which organic matter is not being oxidized at an appreciable rate. As a result, the redox conditions have not declined to low levels even though hundreds or thousands of years or more have been available for the reactions to proceed. If the redox reactions that require bacterial catalysis are not occurring at significant rates, a lack of one or more of the essential nutrients for bacterial growth is likely the cause. There are various types of nutrients. Some are required for incorporation into the cellular mass of the bacteria. Carbon, nitrogen, sulfur, and phosphorous compounds and many metals are in this category. Other nutrients are substances that function as electron donors or energy sources, such as water, ammonia, glucose, and H2S, and substances that function as electron acceptors, such as oxygen, nitrate, and sulfate. Macronutrients are those substances that are required in large amounts as direct building blocks in cell construction. Micronutrients are required in amounts so small as to be difficult to detect. The macronutrient requirements of many bacteria are similar or identical. The micronutrient requirements are more likely to differ from species to species (Brock, 1966).

Although bacteria play an important role in the geochemical environment of groundwater, the study of bacteria at depths below the soil zone is in its infancy. The next decade or two should yield interesting developments in this area of research.

pE–pH Diagrams

Graphs that show the equilibrium occurrence of ions or minerals as domains relative to pE (or Eh) and pH are known as pE–pH or Eh–pH diagrams. During the 1950’s diagrams of this type were developed by M. J. N. Pourbaix and coworkers at the Belgian Center for Study of Corrosion as a practical tool in applied chemistry. The results of this work are summarized by Pourbaix et al. (1963). Following the methods developed by the Belgian group, R. M. Garrels and coworkers pioneered applications in the analysis of geological systems. The use of pE–pH diagrams has become widespread in geology, limnology, oceanography, and pedology. In groundwater quality investigations, considerable emphasis is now being placed on developing an understanding of the processes that control the occurrence and mobility of minor and trace elements. pE–pH diagrams are an important aid in this endeavor. The following discussion of these diagrams is only a brief introduction. The redox condition will be represented by pE rather than Eh, but this is just a matter of convenience. Comprehensive treatments of the subject are presented in the texts by Garrels and Christ (1965), Stumm and Morgan (1970), and Guenther (1975). A concise outline of methods for construction of Eh–pH diagrams is provided by Cloke (1966).

Since we are interested in the equilibrium occurrence (i.e., stability) of dissolved species and minerals in aqueous environments, an appropriate first step in the consideration of pE–pH relations is to determine conditions under which H2O is stable. From the redox half-reactions

\ce{O2(g) + 4H+ + 4}e = 2 \ce{H2O} (3.83)

\ce{2H+} + 2e = \ce{H2(g)} (3.84)

we obtain for conditions at 25°C,

\text{p}E = 20.8 - \text{pH} + \frac{1}{4} \log P_{\ce{O2}} (3.85)

\text{p}E = -\text{pH} - \frac{1}{2} \log P_{\ce{H2}} (3.86)

These relations plot as straight lines (1 and 2) on the pE–pH diagram shown in Figure 3.10(a).

As an example explanation of the stability domains of ions and minerals, the Fe–H2O system represented in Figure 3.10 will be considered. In groundwater, iron in solution is normally present mainly as Fe2+ and Fe3+. These are the only species that are accounted for in our analysis. In a more detailed treatment, complexes such as Fe(OH)2+, \ce{Fe(OH)^+_2}, and \ce{HFeO^-_2} would be included. The solid compounds that can occur in the Fe–H2O system are listed in Table 3.12. A series of reduction reactions involving a solid material (iron compound) and H+ and e as reactants and a more reduced solid compound and water as products can written for the compounds in this table. For example,

\ce{Fe(OH)3 + H+} + e = \ce{Fe(OH)2 + H2O} (3.87)

Expressing this reaction in mass-action form, with activities of water and the solid phases taken as unity (for reasons indicated in Section 3.2), yields

K = \frac{1}{[\text{H}][e]} (3.88)

Figure 3.10 pE–pH diagrams, 25°C and 1 bar. (a) Stability field for water: (b) construction lines for the Fe–H2O system (see the text for equations representing number-designated lines): (c) completed diagram showing stability fields for major dissolved species and solid phases.

Table 3.12 Oxides and Hydroxides in the Fe–H2O System

Oxidation state Solid substances
0 Fe
II and III Fe3O4
III Fe2O3,

and in logarithmic form,

\log K - \text{pH} - \text{p}E = 0 (3.89)

The equilibrium constant in this equation can be obtained using Eq. (3.12) and tables of values of Gibbs’ free energy of formation (\Delta G^0_f), as indicated in Section 3.2. Equation (3.89) is represented as a line on a pE–pH diagram as Shown in Figure 3.10(b) (line 3). In the pE–pH domain above this line, Fe(OH)3 is stable; below the line it is reduced to Fe(OH)2. These are known as the stability fields for these two solid compounds of iron. Lines representing the many other reduction equations obtained by reacting the solids in Table 3.12 with H+ and e to form more reduced compounds and H2O can be constructed on the pE–pH diagram. However, these lines are located outside the stability field for H2O [i.e., above and below lines (1) and (2)], consequently are of no interest in groundwater studies.

In most studies of natural waters, interest is focused on the dissolved species as well as on the mineral phases. Therefore, information on the equilibrium concentrations of dissolved species is commonly included on pE–pH diagrams. For illustration, the Fe–H2O system will be considered further. The oxidation state of iron in Fe(OH)3 is +III. The dissociation of moderately crystalline Fe(OH)3 in water is

\ce{Fe(OH)3 + 3H+ = Fe^{3+} + 3H2O} \hspace{1cm} \Delta G_r^0 = -1.84 \text{kJ} (3.90)

The law of mass action yields

K_{\ce{Fe(OH)3}} = \ce{\frac{[Fe^{3+}]}{[H+]^3}} (3.91)

From Eq. (3.12), a value of +0.32 is obtained for log K. The mass-action relation can be expressed as

\log \ce{[Fe^{3+}] = 0.32 - 3 \hspace{1mm} \text{pH}} (3.92)

which plots as a vertical line on the pE–pH diagram. If the pH is specified, the line obtained from this expression represents the equilibrium activity of Fe3+ that will exist in an aqueous solution in contact with the solid phase, Fe(OH)3. Equation (3.92) indicates that Fe3+ activity increases at lower pH values. In the construction of pE–pH diagrams, a common procedure is to choose a pH condition at which the activity of the dissolved species is at a level considered to be negligible, The choice of this level depends on the nature of the problem. For illustration purposes, two lines are shown on Figure 3.10(b) [lines (4) and (5), which represent Fe3+ activities of 105 and 106]. Although in theory these lines represent activities, and therefore are dimensionless, they can be valid as representing molality, because in low-salinity solutions activity coefficients are nearly equal to unity.

Under lower pE conditions, Fe2+ is the important species of dissolved iron. The reaction of interest is

\ce{Fe(OH)2 + 3H+ = Fe^{2+} + 2H2O} \hspace{1cm} \Delta G_r^0 = +26.33 \text{kJ} (3.93)

From the mass-action relation, the following expression is derived:

\log \ce{[Fe^{2+}] = 10.23 - 2 pH} (3.94)

For [Fe2+] values of 105 and 106, this equation is represented in Figure 3.10 by lines (6) and (7), respectively. The lines have been superimposed only on the part of the diagram in which Fe(OH)2 is the stable solid phase. But Fe2+ also exists at some equilibrium activity in the part of the diagram in which Fe(OH)3 is the stable solid phase. Fe2+ and Fe(OH)3 are related by the redox half-reaction

\ce{Fe(OH)3 + 3H+} + e = \ce{Fe^{2+} + 3H2O} \hspace{1cm} \Delta G_r^0 = -76.26 \text{kJ} (3.95)

From the mass-action relation

\log \ce{[Fe^{2+}]} = \log K_{\ce{Fe(OH)3}} - 3\text{pH} - \text{p}E (3.96)

where log KFe(OH)3 = 13.30. On Figure 3.10(b) this expression is represented as lines (8) and (9) for [Fe2+] values of 10–5 and 10–6, respectively.

Figure 3.10(c) is a “cleaned-up” version of the pE–pH diagram. It illustrates the general form in which pE–pH diagrams are normally presented in the literature. It is important to keep in mind that the boundary lines between solid phases and dissolved species are based on specified activity values, and that the validity of all lines as thermodynamically defined equilibrium conditions is dependent on the reliability of the free-energy data used in construction of the diagram. In the example above, there is considerable uncertainty in the position of some of the boundaries because the solid phase, Fe(OH)3, is a substance of variable crystallinity which has different \Delta G^0_f values depending on its crystallinity.

In Chapter 9, pE–pH diagrams are used in the consideration of other dissolved constituents in groundwater. Although some pE–pH diagrams appear complex, their construction can be accomplished by procedures not much more elaborate than those described above.

3.7 Ion Exchange and Adsorption


Porous geological materials that are composed of an appreciable percentage of colloidal-sized particles have the capability to exchange ionic constituents adsorbed on the particle surfaces. Colloidal particles have diameters in the range 10–3–10–6 mm. They are large compared to the size of small molecules, but are sufficiently small so that interfacial forces are significant in controlling their behavior. Most clay minerals are of colloidal size. The geochemical weathering products of rocks are often inorganic, amorphous (uncrystallized or poorly crystallized) colloids in a persistent metastable state. These colloidal weathering products may occur as coatings on the surfaces of much larger particles. Even a deposit that appears to be composed of clean sand or gravel can have a significant colloid content.

Ion-exchange processes are almost exclusively limited to colloidal particles because these particles have a large electrical charge relative to their surface areas. The surface charge is a result of (1) imperfections or ionic substitutions within the crystal lattice or (2) chemical dissociation reactions at the particle surface. Ionic substitutions cause a net positive or negative charge on the crystal lattice. This charge imbalance is compensated for by a surface accumulation of ions of opposite charge, known as counterions. The counterions comprise an adsorbed layer of changeable composition. Ions in this layer can be exchanged for other ions providing that the electrical charge imbalance in the crystal lattice continues to be balanced off.

In geologic materials the colloids that characteristically exhibit surface charge caused primarily by ionic substitution are clay minerals. The common clay minerals can be subdivided into five groups: the kaolinite group, the montmorillonite group (often referred to as the smectite group), the illite group, the chlorite group, and the vermiculite group. Each group may include a few or many compositional and structural varieties with separate mineral names. The five groups, however, are all layer-type aluminosilicates. The structure and composition of these groups is described in detail in the monographs on clay mineralogy by Grim (1968) and on ion exchange by van Olphen (1963).

Silica, which is the most common oxide in the earth’s crust and one of the simpler oxides, is characterized by electrically charged surfaces. The surfaces contain ions that are not fully coordinated and hence have unbalanced charge. In a vacuum the net charge is extremely small. On exposure to water, the charged sites are converted to surface hydroxide groups that control the charge on the mineral surface. Surface charge is developed because of the dissociation of the adsorbed OH groups on the particular surface. To neutralize this charge, an adsorbed layer of cations and anions forms in a zone adjacent to the hydroxylated layer. Parks (1967) states that hydroxylated surface conditions should be expected on all oxide materials that have had a chance to come to equilibrium with the aqueous environment. Depending on whether the hydroxyl-group dissociation is predominantly acidic or basic, the net charge on the hydroxylated layer can be negative or positive, Surface charge may also be produced by adsorption of charged ionic complexes.

The nature of the surface charge is a function of pH. At low pH a positively charged surface prevails; at a high pH a negatively charged surface develops. At some intermediate pH, the charge will be zero, a condition known as the zero point of charge (pHzpc). The tendency for adsorption of cations or anions therefore depends on the pH of the solution.

Cation Exchange Capacity

The cation exchange capacity (CEC) of a colloidal material is defined by van Olphen (1963) as the excess of counter ions in the zone adjacent to the charged surface or layer which can be exchanged for other cations. The cation exchange capacity of geological materials is normally expressed as the number of milliequivalents of cations that can be exchanged in a sample with a dry mass of 100 g. The standard test for determining the CEC of these materials involves (1) adjustment of the pore water pH to 7.0, (2) saturation of the exchange sites with \ce{NH^+_4} by mixing the soil sample with a solution of ammonium acetate, (3) removal of the absorbed \ce{NH^+_4} by leaching with a strong solution of NaCl (Na+ replaces \ce{NH^+_4} on the exchange sites), and (4) determination of the \ce{NH^+_4} content of the leaching solution after equilibrium has been attained. CEC values obtained from standard laboratory tests are a measure of the exchange capacity under the specified conditions of the test. For minerals that owe their exchange capacity to chemical dissociation reactions on their surfaces, the actual exchange capacity can be strongly dependent on pH.

The concept of cation exchange capacity and its relation to clay minerals and isomorphous substitution is illustrated by the following example adapted from van Olphen (1963). Consider a montmorillonitic clay in which 0.67 mol of Mg occurs in isomorphous substitution for Al in the alumina octahedra of the crystal lattice. The unit cell formula for the montmorillonite crystal lattice can be expressed as


where Ex denotes exchangeable cations. It will be assumed that the exchangeable cations are entirely Na+. From the atomic weights of the elements, the formula weight of this montmorillonite is 734. Hence, from Avogadro’s number, 734 g of this clay contains 6.02 × 10–23 unit cells. The unit cell is the smallest structural unit from which clay particles are assembled. Typical unit cell dimensions for montmorillonite determined from X-ray diffraction analyses are 5.15 Å and 8.9 Å (angstroms) in the plane of the octahedral-tetrahedral sheets. The spacing between sheets varies from 9 to 15 Å depending on the nature of the adsorbed cations and water molecules. The total surface area of 1 g of clay is

\frac{1}{734} \times 6.02 \times 10^{23} \times 2 \times 5.15 \times 8.9\buildrel _\circ \over {\mathrm{A}}^2\text{/g = 750 m}^2/\text{g}

To balance the negative charge caused by Mg substitution, 0.67 mol of monovalent cations, in this case Na+, is required per 734 g of clay. Expressed in the units normally used, the cation exchange capacity is therefore

\text{CEC} = \frac{0.67}{734} \times 10^3 \times 100 = 91.5 \hspace{1mm} \text{meq/100g}

which is equivalent to 0.915 × 6.02 × 1020 monovalent cations per gram.

Since the number of cations that are required to balance the surface charge per unit mass of clay and the surface area per unit mass of clay are now known, the surface area available for each monovalent exchangeable cation can be calculated:

\frac{750 \times 10^{20}}{0.915 \times 6.02 \times 10^{20}} = 132 \buildrel _\circ \over {\mathrm{A}}^2\text{/ion}

The hydrated radius of Na+ is estimated to be in the range 5.6-7.9 Å, which corresponds to areas of 98.5–196.1 Å2. Comparison of these areas to the surface area available per monovalent cation indicates that little more than a monolayer of adsorbed cations is required to balance the surface charge caused by isomorphous substitution.

A similar calculation for kaolinite indicates that for this clay the surface area is 1075 ml2/g (Wayman, 1967). The cation exchange capacity for kaolinite is typically in the range 1–10 meq/100 g, and therefore a monolayer of adsorbed cations would satisfy the charge-balance requirements.

Mass-Action Equations

Following the methodology used in consideration of many of the other topics covered in the chapter, we will develop quantitative relations for cation exchange processes by applying the law of mass action. To proceed on this basis it is assumed that the exchange system consists of two discrete phases, the solution phase and the exchange phase. The exchange phase consists of all or part of the porous medium. The process of ion exchange is then represented simply as an exchange of ions between these two phases,

a\text{A} + b\text{B}(ad) = a\text{A} + b\text{B} (3.97)

where A and B are the exchangeable ions, a and b are the number of moles, and the suffix (ad) represents an adsorbed ion. The absence of this suffix denotes an ion in solution. From the law of mass action,

K_{\text{A-B}} = \frac{[\text{A}_{\text{(ad)}}]^a\text{[B]}^b}{[\text{A}]^a [\text{B}_{\text{(ad)}}]^b} (3.98)

where the quantities in brackets represent activities. For the exchange between Na+ and Ca2+, which is very important in many natural groundwater systems, the exchange equation is

\ce{2Na+ + Ca(ad) = Ca^{2+} + 2Na(ad)} (3.99)

K_{\text{Na-Ca}} = \frac{\ce{[Ca^{2+}][Na_{(ad)}]^2}}{\ce{[Na+]^2[Ca_{(ad)}]}} (3.100)

The activity ratio of ions in solution can be expressed in terms of molality and activity coefficients as

\frac{\text{[B]}^b}{\text{[A]}^a} = \frac{\gamma_{\text{B}}^{b}(\text{B})^b}{\gamma_{\text{A}}^{a}(\text{A})^a} (3.101)

where activity coefficient values (\gamma_A, \gamma_B) can be obtained in the usual manner (Section 3.2). For Eq. (3.98) to be useful it is necessary to obtain values for the activities of the ions adsorbed on the exchange phase. Vanselow (1932) proposed that the activities of the adsorbed ions be set equal to their mole fractions (Section 3.2 includes a definition of this quantity). The mole fractions of A and B are

N_{\text{A}} \frac{\text{(A)}}{\text{(A) + (B)}} \hspace{1cm} \text{and} \hspace{1cm} N_{\text{B}} \frac{\text{(B)}}{\text{(A) + (B)}}

where (A) and (B), expressed in moles, are adsorbed constituents. The equilibrium expression becomes

\bar{K}_{\text{(A-B)}} = \frac{\gamma_{\text{B}}^{\text{b}}(\text{B})^b}{\gamma_{\text{A}}^{\text{a}}(\text{A})^a} \frac{N_{\text{A(ad)}}^{\text{a}}}{N_{\text{B(ad)}}^{\text{b}}} (3.102)

Vanselow and others have found experimentally that for some exchange systems involving electrolytes and clays, \bar{K} is a constant. Consequently, \bar{K} has become known as the selectivity coefficient. In cases where it is not a constant, it is more appropriately called a selectivity function (Babcock, 1963). In many investigations the activity coefficient terms in Eq. (3.101) are not included. Babcock and Schulz (1963) have shown, however, that the activity coefficient effect can be particularly important in the case of monovalent-divalent cation exchange.

Argersinger et al. (1950) extended Vanselow’s theory to more fully account for the effects of the adsorbed ions. Activity coefficients for adsorbed ions were introduced in a form analogous to solute activity coefficients.

\gamma_{\text{A(ad)}} = \frac{[\text{A}_{\text{(ad)}}]}{N_{\text{A(ad)}}} \hspace{1cm} \text{and} \hspace{1cm} \gamma_{\text{B(ad)}} = \frac{[\text{B}_{\text{(ad)}}]}{N_{\text{B(ad)}}} (3.103)

The mass-action equilibrium constant, KA–B is therefore related to the selectivity function by

K_{\text{A-B}} = \frac{\gamma_{\text{A(ad)}}^{\text{a}}} {\gamma_{\text{A(ad)}}^b} \bar{K}_{\text{(A-B)}} (3.104)

Although in theory this equation should provide a valid method for predicting the effects of ion exchange on cation concentrations in groundwater, with the notable exceptions of the investigations by Jensen and Babcoek (1973) and El-Prince and Babcock (1975), cation exchange studies generally do not include determination of K and \gamma(ad), values. Information on selectivity coefficients is much more common in the literature. For the Mg2+–Ca2+ exchange pair, Jensen and Babcock and others have observed that the selectivity coefficient is constant over large ranges of ratios of (Mg2+)(ad)/(Ca2+)(ad) and ionic strength. \bar{K}_{\ce{Mg-Ca}} values are typically in the range 0.6–0.9. This indicates that Ca2+ is adsorbed preferentially to Mg2+.

Interest in cation exchange processes in the groundwater zone commonly focuses on the question of what will happen to the cation concentrations in groundwater as water moves into a zone in which there is significant cation exchange capacity. Strata that can alter the chemistry of groundwater by cation exchange may possess other important geochemical properties. For simplicity these are excluded from this discussion. When groundwater of a particular composition moves into a cation exchange zone, the cation concentrations will adjust to a condition of exchange equilibrium. The equilibrium cation concentrations depend on initial conditions, such as: (1) cation concentrations of the water entering the pore space in which the exchange occurs and (2) the mole fractions of adsorbed cations on the pore surfaces immediately prior to entry of the new pore water. As each new volume of water moves through the pore space, a new equilibrium is established in response to the new set of initial conditions. Continual movement of groundwater through the cation exchange zone can be accompanied by a gradually changing pore chemistry, even though exchange equilibrium in the pore water is maintained at all times. This condition of changing equilibrium is particularly characteristic of cation exchange processes in the groundwater zone, and is also associated with other hydrochemieal processes where hydrodynamic flow causes continual pore water replacement as rapid mineral-water reactions occur.

The following example illustrates how exchange reactions can influence groundwater chemistry. Consider the reaction

\ce{(Mg^{2+}) + (Ca^{2+})_{ad} <=> (Ca^{2+}) + (Mg^{2+})_{ad}} (3.105)

which leads to

\bar{K}_{\text{Mg-Ca}} = \frac{\gamma_{\text{Ca}}(\ce{Ce}^{2+})}{\gamma_{\text{Mg}}(\ce{Mg}^{2+})} \frac{N_{\text{Mg}}}{N_{\text{Ca}}} (3.106)

where \bar{K}_{\ce{Mg-Ca}} is the selectivity coefficient, \gamma denotes activity coefficient, (Ca2+) and (Mg2+) are molalities, and NMg, and NCa, are the mole fractions of adsorbed Mg2+ and Ca2+. At low and moderate ionic strengths, the activity coefficients of Ca2+ and Mg2+ are similar (Figure 3.3), and Eq. (3.106) can be simplified to

\bar{K}_{\text{Mg-Ca}} = \frac{(\ce{Ce}^{2+})}{(\ce{Mg}^{2+})} \frac{N_{\text{Mg}}}{N_{\text{Ca}}} (3.107)

In this example, exchange occurs when groundwater of low ionic strength with Mg2+ and Ca2+ molalities of 1 × 10–3 flows through a clayey stratum with a cation exchange capacity of 100 meq/100 g. Concentrations of other cations in the water are insignificant. It is assumed that prior to entry of the groundwater into the clay stratum, the exchange positions on the clay are shared equally by Mg2+ and Ca2+. The initial adsorption condition is therefore NMg = NCa. To compute the equilibrium cation concentrations, information on the porosity or bulk dry mass density of the clay is required. It is assumed that the porosity is 0.33 and that the mass density of the solids is 2.65 g/cm3. A reasonable estimate for the bulk dry mass density is therefore 1.75 g/cm3. It is convenient in this context to express the cation concentrations in solution as moles per liter, which at low concentrations is the same as molality. Since the porosity is 0.33, expressed as a fraction, each liter of water in the clayey stratum is in contact with 2 × 103 cm3 of solids that have a mass 5.3 × 103 g. Because the CEC is 1 meq/g and because 1 mol of Ca2+ or Mg2+ = 2 equivalents, 5.3 × 103 g of clay will have a total of 5.3 equivalents, which equals 1.33 mol of adsorbed Mg2+ and 1.33 mol of Ca2+. It is assumed that the groundwater flows into the water-saturated clay and totally displaces the original pore water. The Ca2+ and Mg2+ concentrations in the groundwater as it enters the clayey stratum can now be calculated. A \bar{K}_{\ce{Mg-Ca}} value of 0.6 will be used, and it will be assumed that pore-water displacement occurs instantaneously with negligible hydrodynamic dispersion. Because the initial conditions are specified as NMg = NCa, a liter of water is in contact with clay that has 1.33 mol of Mg2+ and 1.33 mol of Ca2+ on the exchange sites. Compared to the concentrations of Mg2+ and Ca2+ in the groundwater, the adsorbed layer on the clay particles is a large reservoir of exchangeable cations.

Substitution of the initial values into the right-hand side of Eq. (3.107) yields a value for the reaction quotient [Eq. (3.60)]:

Q_{\text{Mg-Ca}} = \frac{1 \times 10^{-3}}{1 \times 10^{-3}} \times \frac{0.5}{0.5} = 1

For the reaction to proceed to equilibrium with respect to the new pore water, Q_{\ce{Mg-Ca}} must decrease to a value of 0.6 to attain the condition of Q = K. This occurs by adsorption of Ca2+ and release of Mg2+ to the solution. The equilibrium is achieved when (Ca2+) = 0.743 × 10–3, (Mg2+) = 1.257 × 10–3, NCa = 0.500743, and NMg = 0.499257. The ratio of adsorbed cations is not changed significantly, but the (Mg2+)/(Ca2+) ratio for the dissolved species has increased from 1 to 1.7. If the groundwater continues to flow through the clayey stratum, the equilibrium cation concentrations will remain as indicated above until a sufficient number of pore volumes pass through to cause the ratio of adsorbed cations to gradually change. Eventually, the NMg/NCa, ratio decreases to a value of 0.6, at which time the clay will no longer be capable of changing the Mg2+ and Ca2+ concentrations of the incoming groundwater. If the chemistry of the input water changes, the steady-state equilibrium will not be achieved.

This example illustrates the dynamic nature of cation exchange equilibria. Because exchange reactions between cations and clays are normally fast, the cation concentrations in groundwater can be expected to be in exchange equilibrium, but many thousands or millions of pore volumes may have to pass through the porous medium before the ratio of adsorbed cations completely adjusts to the input water. Depending on the geochemical and hydrologic conditions, time periods of millions of years may be necessary for this to occur.

Exchange involving cations of the same valence is characterized by preference for one of the ions if the selectivity coefficient is greater or less than unity. The normal order of preference for some monovalent and divalent cations for most clays is

Affinity for adsorption

Cs+ > Rb+ > K+ > Na+ > Li+

Stronger \ce{->} weaker

Ba2+ > Sr2+ > Ca2+ > Mg2+

The divalent ions normally have stronger adsorption affinity than the monovalent ions, although this depends to some extent on the nature of the exchanger and the concentration of the solutions (Wiklander, 1964). Both affinity sequences proceed in the direction of increasing hydrated ionic radii, with strongest adsorption for the smaller hydrated ions and weakest adsorption for the largest ions. It must be kept in mind, however, that the direction in which a cation exchange reaction proceeds also depends on the ratio of the adsorbed mole fractions at the initial condition and on the concentration ratio of the two ions in solution. For example, if we consider the Mg–Ca exchange condition used in the equilibrium calculations presented above but alter the initial condition of adsorbed ions to NMg = 0.375 and NCa = 0.625, there would be no change in the Mg2+ and Ca2+ concentrations as the groundwater passes through the clay. If the initial adsorbed ion conditions were such that the NMg/NCa ratio was less than 0.6, the exchange reaction would proceed in the reverse direction [to the right in Eq. (3.105)], thereby causing the ratio (Mg2+)/(Ca2+) to decrease. This indicates that to determine the direction in which an ion exchange reaction will proceed, more information than the simple adsorption affinity series presented above is required.

The most important cation exchange reactions in groundwater systems are those involving monovalent and divalent cations such as Na+–Ca2+, Na+–Mg2+, K+–Ca2+, and K+–Mg2+. For these reactions,

\ce{2A+ + B(ad) = B^{2+} + 2A(ad)} (3.108)

\bar{K}_{\text{A-B}} = \frac{\text{[B]}^{2+}N_{\text{A}}^2}{[\text{A}^+]^2 N_{\text{B}}} (3.109)

The Na+–Ca2+ exchange reaction is of special importance when it occurs in montmorillonitic clays (smectite) because it can cause large changes in permeability. Clays of the montmorillonite group can expand and contract in response to changes in the composition of the adsorbed cation between the clay platelets. The hydrated radii of Na+ and Ca2+ are such that two hydrated Na+ require more space than one Ca2+. Hence, replacement of Ca2+ by Na+ on the exchange sites causes an increase in the dimension of the crystal lattice. This results in a decrease in permeability. This can cause a degradation in the agricultural productivity of soils.

3.8 Environmental Isotopes

Since the early 1950’s naturally occurring isotopes that exist in water in the hydrologic cycle have been used in investigations of groundwater and surface water systems. Of primary importance in these studies are tritium (3H) and carbon 14 (14C), which are radioactive, and oxygen 18 (18O) and deuterium (2H), which are nonradioactive (Table 3.1). The latter are known as stable isotopes. Tritium and deuterium are often represented as T and D, respectively. 3H and 14C are used as a guide to the age of groundwater. 18O and 2H serve mainly as indicators of groundwater source areas and as evaporation indicators in surface-water bodies.

In this text these four isotopes are the only environmental isotopes for which hydrogeologic applications are described. For discussions of the theory and hydrologic or hydrochemical use of other naturally occurring isotopes, such as carbon 13, nitrogen 15, and sulfur 34, the reader is referred to Back and Hanshaw (1965), Kreitler and Jones (1975), and Wigley (1975). There are many situations where isotopic data can provide valuable hydrologic information that could not otherwise be obtained. Sophisticated techniques for the measurement of the above- mentioned isotopes in water have been available for several decades, during which time the use of these isotopes in groundwater studies has gradually increased.


Prior to the advent of large aboveground thermonuclear tests in 1953, 14C in the global atmosphere was derived entirely from the natural process of nitrogen transmutation caused by bombardment of cosmic rays. This 14C production has been estimated to be about 2.5 atoms/s • cm2 (Lal and Suess, 1968). Oxidation to CO2 occurs quickly, followed by mixing with the atmospheric CO2 reservoir. The steady-state concentration of 14C in the atmosphere is about one 14C atom in 1012 atoms of ordinary carbon (12C). Studies of the 14C content of tree rings indicate that this concentration of 14C has varied only slightly during the past 7000 years. Other evidence suggests that there have been no major shifts in the atmospheric 14C concentrations during the past several tens of thousands of years.

The law of radioactive decay describes the rate at which the activity of 14C and all other radioactive substances decreases with time. This is expressed

A = A_02^{-t/T} (3.110)

where A0 is the radioactivity level at some initial time, A the level of radioactivity after time t, and T the half-life of the isotope. This law, in conjunction with measurements of the 14C content of groundwater, can be used as a guide to groundwater age. In this context the term age refers to the period of time that has elapsed since the water moved deep enough into the groundwater zone to be isolated from the earth’s atmosphere.

Use of 14C for dating of groundwater was first proposed by Münnich (1957), following the development of techniques for 14C dating of solid carbonaceous materials pioneered by the Nobel laureate W. F. Libby in 1950. When water moves below the water table and becomes isolated from the earth’s CO2 reservoir, radioactive decay causes the 14C content in the dissolved carbon to gradually decline. The expression for radioactive decay [Eq. (3.110)] can be rearranged, and upon substitution of T = 5730 years

t = -8720 \hspace{1mm}\text{ln} \left( \frac{A}{A_0} \right) (3.111)

where A0 is the specific activity (disintegrations per unit time per unit mass of sample) of carbon 14 in the earth’s atmosphere, A the activity per unit mass of sample, and t the decay age of the carbon. In groundwater investigations 14C determinations are made on samples of inorganic carbon that are extracted from samples of groundwater that generally range in volume from 20 to 200 \ell. The mass of carbon needed for accurate analysis by normal methods is about 3 g. The 14C values obtained in this manner are a measure of the 14C content of the CO2(aq), H2CO3, CO32, and \ce{HCO^-_3} in the water at the time of sampling. 14C may also be present in dissolved organic carbon such as fulvic and humic acids, but this 14C source is small and is normally not included in studies of groundwater age.

The specific activity of 14C in carbon that was in equilibrium with the atmosphere of the earth prior to atmospheric testing of thermonuclear devices is approximately 10 disintegrations per minute per gram (dpm/g). Modern equipment for 14C measurement can detect 14C activity levels as low as approximately 0.02 dpm/g. Substitution of these specific activities in Eq. (3.111) yields a maximum age of 50,000 years. It must be emphasized that this is an apparent age of the dissolved inorganic carbon. To gain some useful hydrologic information from this type of data, it is necessary to determine the source of the inorganic carbon. Calcite or dolomite occur in many groundwater environments. Carbon that enters the groundwater by dissolution of these minerals can cause dilution of the 14C content of the total inorganic carbon in the water. This is the case because in most groundwater systems the calcite and dolomite are much older than 50,000 years. Their carbon is therefore devoid of significant amounts of 14C and is often referred to as “dead” carbon. To obtain 14C estimates of the actual groundwater age it is necessary to determine the extent to which this dead carbon has reduced the relative 14C content of the groundwater. An indication of how this can be done is described in Chapter 7.


The occurrence of tritium in waters of the hydrological cycle arises from both natural and man-made sources. In a manner similar to 14C production, 3H is produced naturally in the earth’s atmosphere by interaction of cosmic-ray-produced neutrons with nitrogen. Lal and Peters (1962) estimated that the atmospheric production rate is 0.25 atoms/s • cm2. In 1951, Van Grosse and coworkers discovered that 3H occurred naturally in precipitation. Two years later large quantities of man-made tritium entered the hydrological cycle as a result of large-scale atmospheric testing of thermonuclear bombs. Unfortunately, few measurements of natural tritium in precipitation were made before atmospheric contamination occurred. It has been estimated that prior to initiation of atmospheric testing in 1952, the natural tritium content of precipitation was in the range of about 5–20 tritium units (Payne, 1972). A tritium unit is the equivalent of 1 tritium atom in 1018 atoms of hydrogen. Since the half-life of 3H is 12.3 years, groundwater that was recharged prior to 1953 is therefore expected to have 3H concentrations below about 2–4 TU. The first major source of man-made 3H entered the atmosphere during the initial tests of large thermonuclear devices in 1952. These tests were followed by additional tests in 1954, 1958, 1961, and 1962 before the moratorium on atmospheric testing agreed upon by the United States and the USSR.

Since the onset of thermonuclear testing, the tritium content in precipitation has been monitored at numerous locations in the northern hemisphere and at a smaller but significant number of locations in the southern hemisphere. Considering the data separately by hemispheres, there is a strong parallelism in 3H concentration with time, although absolute values vary from place to place (Payne, 1972). In the southern hemisphere, 3H values are much lower because of the higher ratio of oceanic area to land area. The longest continuous record of 3H concentrations in precipitation is from Ottawa, Canada, where sampling was begun in 1952. The 3H versus time record for this location is shown in Figure 3.11. The trends displayed in this graph are representative of the 3H trends recorded elsewhere in the northern hemisphere. Tritium data obtained by the International Atomic Energy Agency (IAEA) from a global sampling network enable the estimation of 3H versus time trends for areas in which there are no sampling stations or only short-term records. At a given latitude the concentrations of tritium in precipitation at sampling stations near the coast are lower than those inland because of dilution from oceanic water vapor, which is low in tritium.

Measurements of tritium concentrations can be a valuable aid in many types of groundwater investigations. If a sample of groundwater from a location in the northern hemisphere contains tritium at concentration levels of hundreds or thousands of TU, it is evident that the water, or at least a large fraction of the water, originally entered the groundwater zone sometime after 1953. If the water has less than 5–10 TU, it must have originated prior to 1953. Using routine methods for measurement of low-level tritium in water samples, concentrations as low as about 5–10 TU can be detected.

Figure 3.11 Variations of tritium in precipitation (mean monthly concentrations, TU) at Ottawa, Canada.

Using special methods for concentrating 3H from water samples, values as low as about 0.1 TU can be measured. If samples contain no detectable 3H in routine measurements, it is usually reasonable to conclude that significant amounts of post-1953 water are not present. Post-1953 water is often referred to as modern water or bomb tritium water.

Tritium data from detailed sampling patterns can sometimes be used to distinguish different age zones within the modern-water part of groundwater flow systems. For this type of tritium use, the stratigraphic setting should be simple so that complex flow patterns do not hinder the identification of tritium trends. In situations where the 3H concentrations of two adjacent flow zones are well defined, tritium data can be useful to distinguish zones of mixing. The usefulness of tritium in groundwater studies is enhanced by the fact that it is not significantly affected by reactions other than radioactive decay.

Oxygen and Deuterium

With the advent of the mass spectrometer, it became possible in the early 1950’s to make rapid accurate measurements of isotope ratios. Of special interest to hydrologists are the ratios of the main isotopes that comprise the water molecule, 18O/16O and 2H/1H. The isotope ratios are expressed in delta units (\partial) as per mille (parts per thousand or ‰) differences relative to an arbitrary standard known as standard mean ocean water (SMOW):

\newcommand*\rfrac[2]{{}^{#1}\!/_{#2}}\partial\rfrac{0}{00} = [(R - R_{\text{standard}} / R_{\text{standard}})] \times 1000 (3.112)

where R and Rstandard are the isotope ratios. 2H/1H or 18O/16O, of the sample and the standard, respectively. The accuracy of measurement is usually better than ±0.2‰ and at ±2‰ for \partial^{18}O and \partial^2H, respectively.

The various isotopic forms of water have slightly different vapor pressures and freezing points. These two properties give rise to differences in 18O and 2H concentrations in water in various parts of the hydrologic cycle. The process whereby the isotope content of a substance changes as a result of evaporation, condensation, freezing, melting, chemical reactions, or biological processes is known as isotropic fractionation. When water evaporates from the oceans, the water vapor produced is depleted in 18O and 2H relative to ocean water, by about 12–15‰ in 18O and 80–120‰ in 2H. When water vapor condenses, the rain or snow that forms has higher 18O and 2H concentrations than the remaining water vapor. As the water vapor moves farther inland as part of regional or continential atmospheric circulation systems, and as the process of condensation and precipitation is repeated many times, rain or snow becomes characterized by low concentrations of the heavy isotopes 18O and 2H. The 18O and 2H content of precipitation at a given locality at a particular time depends in a general way on the location within the continental land mass, and more specifically on the condensation-precipitation history of the atmospheric water vapor. Since both condensation and isotope fractionation are temperature-dependent, the isotopic composition of precipitation is also temperature-dependent. The combined efect of these factors is that (1) there are strong continental trends in the average annual isotopic composition of precipitation, (2) there is a strong seasonal variation in the time-averaged isotopic composition of precipitation at a given location, and (3) the isotopic composition of rain or snow during an individual precipitation event is very variable and unpredictable. In continental areas, rain values can vary between about 0 and –25‰ for 18O and 0 to –150‰ for 2H, even though the average annual values have little variation. Because of temperature changes in the zone of atmospheric condensation or isotopic depletion effects, large variations can even occur during individual rainfall events. Changes can also occur in the raindrop during its fall, especially at the beginning of a rainstorm and in arid or semiarid regions.

In deep subsurface zones where temperatures are above 50–100°C, the 18O and 2H content of groundwater can be significantly altered as a result of chemical interactions with the host rock. In shallower groundwater systems with normal temperatures, the concentrations of these isotopes are little, if at all, affected by chemical processes. In these flow regimes, 18O and 2H are nonreactive, naturally occurring tracers that have concentrations determined by the isotopic composition of the precipitation that falls on the ground surface and on the amount of evaporation that occurs before the water penetrates below the upper part of the soil zone. Once the water moves below the upper part of the soil zone, the 18O and 2H concentrations become a characteristic property of the subsurface water mass, which in many hydrogeologic settings enables the source areas and mixing patterns to be determined by sampling and analysis for these isotopes.

2H and 18O concentrations obtained from global precipitation surveys correlate according to the relation (Dansgaard, 1964)

\newcommand*\rfrac[2]{{}^{#1}\!/_{#2}}\partial^2 \text{H} \rfrac{0}{00} = 8\partial^{18}\text{O} \rfrac{0}{00} (3.113)

which is known as the meteoric water line. Linear correlations with coefficients only slightly different than this are obtained from studies of local precipitation. When water evaporates from soil- or surface-water bodies under natural conditions, it becomes enriched in 18O and 2H. The relative degree of enrichment is different than the enrichment that occurs during condensation. The ratio of \partial^{18}O/\partial^2H for precipitation that has partially evaporated is greater than the ratio for normal precipitation obtained from Eq. (3.113). The departure of 18O and 2H concentrations from the meteoric water line is a feature of the isotopes that can be used in a variety of hydrologic investigations, including studies of the influence of groundwater on the hydrologic balance of lakes and reservoirs and the effects of evaporation on infiltration.

3.9 Field Measurement of Index Parameters

Description of the laboratory techniques that are used in the chemical or isotopic analysis of water samples is beyond the scope of this text. For this type of information the reader is referred to Rainwater and Thatcher (1960) and US. Environmental Protection Agency (1974b). Our purpose here is to briefly describe methods by which several important index parameters are measured in field investigations. These parameters are specific electrical conductance, pH, redox potential, and dissolved oxygen. In groundwater studies each of these parameters can be measured in the field by immersing probes in samples of water or by lowering probes down wells or piezometers.

Electrical conductivity is the ability of a substance to conduct an electrical current. It has units of reciprocal ohm-meters, denoted in the SI system as siemens per meter (S/m). Electrical conductance is the conductivity of a body or mass of fluid of unit length and unit cross section at a specified temperature. In the groundwater literature electrical conductance has normally been reported as reciprocal milliohms or reciprocal microohms, known as millimhos and micromhos. In the SI system, 1 millimho is denoted as I millisiemen (mS) and 1 micromho as l microsiemen (\mu S).

Pure liquid water has a very low electrical conductance, less than a tenth of a microsiemen at 25°C (Hem, 1970). The presence of charged ionic species in solution makes the solution conductive. Since natural waters contain a variety of both ionic and uncharged species in various amounts and proportions, conductance determinations cannot be used to obtain accurate estimates of ion concentrations or total dissolved solids. As a general indication of total dissolved solids (TDS), however, specific conductance values are often useful in a practical manner. For conversion between conductance values and TDS, the following relation is used (Hem, 1970):

\text{TDS} = AC (3.114)

where C is the conductance in microsiemens or micromhos, TDS is expressed in g/m3 or mg/\ell, and A is a conversion factor. For most groundwater, A is between 0.55 and 0.75, depending on the ionic composition of the solution.

Measurements of electrical conductance can be made in the field simply by immersing a conductance cell in water samples or lowering it down wells and then recording the conductance on a galvanometer. Rugged equipment that is well suited for field use is available from numerous commercial sources. In groundwater studies conductance measurements are commonly made in the field so that variations in dissolved solids can be determined without the delay associated with transportation of samples to the laboratory. As distributions of groundwater conductance values are mapped in the field, sampling programs can be adjusted to take into account anomalies or trends that can be identified as the field work proceeds.

To avoid changes caused by escape of CO2 from the water, measurements of the pH of groundwater are normally made in the field immediately after sample collection. Carbon dioxide in groundwater normally occurs at a much higher partial pressure than in the earth’s atmosphere. When groundwater is exposed to the atmosphere, CO2 will escape and the pH will rise. The amount of pH rise for a given decrease in P_{\ce{CO2}} can be calculated using the methods described in Section 3.3. For field measurements of pH, portable pH meters and electrodes are generally used. Samples are usually brought to ground surface by pumping or by means of down-hole samplers rather than by lowering electrodes down wells. A detailed description of the theory and methods of pH measurement in water are presented by Langmuir (1970).

Dissolved oxygen is another important hydrochemical parameter that is commonly measured in the field by immersing a small probe in water samples or down wells. In a dissolved oxygen probe, oxygen gas molecules diffuse through a membrane into a measuring cell at a rate proportional to the partial pressure of oxygen in the water. Inside the sensor the oxygen reacts with an electrolyte and is reduced by an applied voltage. The current that is generated is directly proportional to the partial pressure of oxygen in the water outside the sensor (Back and Hanshaw, 1965). Rugged dissolved oxygen probes that connect to portable meters are commercially available. These probes can be lowered down wells or piezometers to obtain measurements that are representative of in situ conditions. Dissolved oxygen can also be measured in the field by a titration technique known as the Winkler method (U.S. Environmental Protection Agency, 1974b).

Dissolved oxygen probes of the type that are generally used have a detection limit of about 0.1 mg/\ell. High-precision probes can measure dissolved oxygen at levels as low as 0.01 mg/\ell. Even at dissolved oxygen contents near these detection limits, groundwater can have enough oxygen to provide considerable capability for oxidation of many types of reduced constituents. Eh or pE values can be computed from measured values of dissolved oxygen by means of Eq. (3.77). The concentration of dissolved oxygen is converted to P_{\ce{O2}}, using Henry’s law (P_{\ce{O2}} = O2 dissolved/K_{\ce{O2}}), where K_{\ce{O2}} at 25°C is 1.28 × 10–3 mol/bar). At pH 7, pE values obtained in this manner using dissolved oxygen values at the detection limits indicated above are 13.1 and 12.9, or expressed as Eh, 0.78 and 0.76 V, respectively. Figure 3.10 indicates that these values are near the upper limit of the pE–pH domain for water. If the water is saturated with dissolved oxygen (i.e., in equilibrium with oxygen in the earth’s atmosphere), the calculated pE is 13.6. For pE values calculated from dissolved oxygen concentrations to serve as a true indication of the redox condition of the water, dissolved oxygen must be the controlling oxidative species in the water with redox conditions at or near equilibrium. Measurements values of other dissolved multivalent constituents can also be used to obtain estimates of redox conditions of groundwater. Additional discussion of this topic is included in Chapter 7.

Another approach to obtaining estimates of the redox condition of groundwater is to measure electrical potential in the water using an electrode system that includes an inert metallic electrode (platinum is commonly used). Electrode systems known as Eh probes are commercially available. To record the electrical potential, they can be attached to the same meters used for pH. For these readings to have significance, the probes must be lowered into wells or piezometers or be placed in sample containers that prevent the invasion of air. For some groundwater zones the potentials measured in this manner are an indication of the redox conditions, but in many cases they are not. Detailed discussions of the theory and significance of the electrode approach to redox measurements are provided by Stumm and Morgan (1970) and Whitfield (1974).

Suggested Readings

BLACKBURN, T. R. 1969. Equilibrium, A Chemistry of Solutions. Holt, Rinehart and Winston, New York, pp. 93–111.

GARRELS, R. M., and C. L. CHRIST. 1965. Solutions, Minerals, and Equilibria. Harper & Row, New York, pp. 1–18, 50–71.

KRAUSKOPF, K. 1967. Introduction to Geochemistry. McGraw-Hill, New York, pp. 3–23, 29–54, 206–226, 237–255.

STUMM, W., and J. J. MORGAN. 1970. Aquatic Chemistry. Wiley-Interscience, New York, pp. 300–377.