Multispecies Reactive Transport---A Computer Mediated Distance Learning course by Walt W. McNab.

Updated:Nov. 29, 2001.
Copyright © 2000 by Walt W McNab, Concord, CA, U.S.A. 
All Rights Reserved.

Computer-Mediated Distance Learning

Course on

MULTISPECIES

REACTIVE TRANSPORT IN

GROUNDWATER

INSTRUCTOR:

WALT W McNAB

Senior Scientist
Exponent-Failure Analysis Associates
Oakland, California

TOPIC A: BASIC PROCESSES AND EQUATIONS

Lecture 5: Ion Exchange and Surface Complexation

Introduction

Previously, in Lectures 2 and 4, we have discussed homogeneous and heterogeneous chemical reactions and have distinguished between components dissolved in the aqueous phase and those that are included in (solid) mineral phases. In real groundwater systems, however, this simplified view of the world neglects the important "in-between" realm for geochemical components: sites on the surfaces of mineral phases. For charged aqueous components of chemical species, association, or complexation reactions with mineral surfaces arise because such surfaces may develop localized electrical charges and, hence, induce electrostatic attraction. For the reactive transport of contaminants, surface reactions may play an important role in determining the evolution of contaminant plumes, particularly in influencing the mobility of dissolved metals such as chromium, copper, and lead.

Reactions between aqueous components, or species, and mineral surfaces present a difficult topic in geochemistry. They are often described through the use of complex mathematical formulations. The details of the theory that underlies this description are far beyond the scope of this course. Because our interest is in the broad subject of reactive transport, which encompasses many processes and mechanisms, we will touch upon the topic of surface complexation in only an introductory manner. Nevertheless, we will bear in mind that many reactive transport models, particularly one that we will be using later in this course, feature mathematical tools that allow surface complexation effects to be considered alongside those of aqueous speciation, mineral precipitation/dissolution, and chemical transport.

Basic concepts in surface complexation

The development of electrical charges on mineral surfaces is a subject of great interest to soil scientists. From our perspective as contaminant hydrologists, we are most interested in are the following two types of surface charge effects:

(1) The negative charges that develop on the surfaces of clay minerals.

Clay minerals are referred to as sheet silicates because they consist of stacked sheets of silica-containing tetrahedral layers and aluminum-containing octahedral layers. Sheet silicates are generally categorized by the number of layers of each type in a representative molecular unit. In the 2:1 sheet silicates, the most common type and the one of most interest to reactive transport, contain two layers of Si-tetrahedral to each layer of Al-octahedra.

An idealized formula for such a mineral can be expressed as Al₄Si₈O₂₀(OH)₄. A mineral phase corresponding to this formula, known as pyrophyllite, really does exist in nature, but it is not very common. The reason for this is that different ions may substitute themselves into the tetrahedral sites in place of Si⁴⁺, while others may substitute themselves into some of the octahedral sites in place of Al³⁺. Bear in mind that clay minerals have a physical structure beyond just the stoichiometry shown by their chemical formulas. This means that the size of an ion can be as important, or more important, than the magnitude of its electric charge in determining how it can substituted by another ion at a tetrahedral or octahedral site. The result is that ions of different charge can substituted by Al and Si in the pyrophyllite structure; the implied change in overall electrical charge must be compensated for by the addition of yet other ions in the structure. The result is a general type of clay mineral known as a smectite clay that has a general chemical formula in the form

A close inspection of this formula suggests that the major ions that replace Al³⁺ in the octahedral

Metal oxyhydroxide mineral surfaces

Metal oxyhydroxides, notably common minerals associated with aluminum (e.g., Al(OH)₃, or gibbsite) and ferric iron (Fe(OH)₃, or ferric hydroxide, and FeO(OH), or goethite), like clay minerals, also develop surface charges. In the case of these minerals, the surface charges may be thought of as arising from the addition of protons, H⁺ (protonation), or the removal of protons (de-protonation) from mineral surfaces. Because the exchange of protons between the aqueous phase and mineral surfaces is controlled to a large extent by the solution pH, the magnitude, and even the sign, of the surface charge depends on the pH. This leads to the concept of point-of-zero-charge (PZC), which is the pH at which the surface charge of a metal oxyhydroxide mineral switches from positive (at low pH, where protons are abundant and protonation favorable) to negative (at high pH where protons are scarce and de-protonation favorable).

Given that clay minerals and metal oxyhydroxides have surface charges, what are the effects of those charges on the concentrations and/or activities of aqueous components or species? Soil physical chemists have developed three different conceptual models for the adsorption of charged species to mineral surfaces: inner-sphere adsorption, outer-sphere adsorption, and the formation of a diffuse layer (Figure 1). In inner-sphere adsorption, the charged ion attaches itself directly to the mineral surface, forming chemical bonds with atoms presented by the exposed structural surface. In outer-sphere adsorption, the charged ion is surrounded by water molecules.

Figure 1. Adsorption models based on soil physical chemistry.

Although the ion-water complex resides on the mineral surface by coulombic attraction, it does not form any real chemical bonds with the mineral surface because the interference posed by the water molecules. Finally, in the case of diffuse layer adsorption, the charged ion does not come into direct contact with the mineral surface but instead resides within a boundary layer on the mineral surface consisting of a "cloud" of like charged ions and water molecules.

Among the three adsorptive mechanisms, inner-sphere adsorption presents the strongest degree of attachment because of the chemical bonding mechanisms involved (both ionic and covalent in nature). Inner-sphere adsorption satisfies electrical charge imbalance locally on an atomic scale, whereas outer-sphere complexes are less locally focused, with diffuse layer ions even less so. In contrast to the outer-sphere complexes and ions held in the diffuse layer, bonding involved, ions adsorbed as inner-sphere complexes are not considered to be readily exchangeable.

Mathematical description

Simplified conceptual models describing the interactions of ions and mineral surfaces are all well and good, but how are we to make use of these concepts in a quantitative way in the context of our interest in reactive transport? There are several different approaches available to us. The choice of which one to use depends, of course, on the type of problem we are trying to solve:

Adsorption isotherm models

Adsorption isotherm is a mathematical expression that quantifies the partitioning of a component or species between the aqueous phase and the mineral surface. Under equilibrium conditions, three different expressions are commonly used. The selection of a particular isotherm depends on the nature of the component and the solid.

The linear isotherm, the simplest of these, merely assumes a proportional relationship between the mass of solute adsorbed to the surface and the concentration of the solute in the aqueous phase:

(Eq.-1)

where S denotes the mass of the solute adsorbed, per unit dry bulk mass of soil, and C is the aqueous concentration of the solute.

The Freundlich isotherm attempts to addresses the observed nonlinearity observed in certain cases in the ratio of adsorbed mass to aqueous concentration by introducing an empirical power law in the form:

(Eq.-2)

where S denotes the

While the Freunlich isotherm provides a more flexible means for fitting isotherms to experimental data, its empirical nature reduces its utility for understanding the mechanisms that shape adsorption curves (i.e., the S versus C relationship). The Langmuir isotherm, defined as,

(Eq.-3)

with b and K_d as empirical constants, provides a marginally more palatable alternative in that the Langmuir isotherm is developed based on the approach to saturation of a finite number of adsorption sites on a mineral surface. The development of this equation will not be repeated here, as it can be found in many soil science and surface chemistry textbooks.

Each of the adsorption isotherm models provides a convenient means for fitting quantitative descriptions of surface partitioning behavior to experimental data; their use is quite popular. In a subsequent lecture where we will discuss the coupling of chemical reaction equations with transport equations to form reactive transport models, we will see that adsorption isotherms can be used as a basis for extremely simple reactive transport models. In general, however, the empirical nature of these models renders them unsatisfying from a theoretical perspective. Moreover, difficulties in coupling adsorption isotherm relationships with charge balance and mass action relationships complicates the inclusion of these expressions in more general reactive transport models. Therefore, in large part, we will turn our attention elsewhere Ö

Electrostatic models

Electrostatic models are used to predict the distribution of ions near a charged surface based on potential theory. The approach is much more mechanism-oriented and microscopic in scope than that of the adsorption isotherm models, which is empirical and macroscopic. Several different types of electrostatic models exist, with various assumptions about the size of the adsorbing ions, their degree of interaction with one another, the nature of the adsorption sites (i.e., inner-sphere, outer-sphere, or diffuse layer), and so forth. All assume that the adsorbent surface is a uniform plane with some fixed charge density.

The diffuse double-layer model is perhaps the most familiar of the electrostatic models in use. It assumes that ions exist as point sources of charge that exist only within a diffuse cloud about the mineral surface (i.e., inner-sphere and outer-sphere adsorption is neglected). The models permits the calculation of the concentration of some charged species i as a function of distance, x, from the mineral surface according to:

(Eq.-4)

The electrical potential at x, f (x), can be calculated by solving the Poisson-Boltzmann differential equation (subject to a certain boundary condition):

(Eq.-5)

These equations may appear intimidating to the mathematically challenging. Donít worry; the diffuse necessarily have the relevant adsorption isotherm data available at our disposal.

Ion exchange models

Ion exchange models allow us to quantify the exchange of species or components between the aqueous phase and mineral surface in a manner resembling the familiar mass action expression. In this case, we simply assume that there are a number of exchange sites, X, associated with a soil and that the exchangeability between different cations can be described by an equilibrium constant. For example, for a reaction,

(Eq.-6)

a mass-action-like expression can be written as:

(Eq.-7)

The activities of the exchange sites are based on the mole fraction of the sites that are occupied by a given cation. Thus, if "q" refers to the number of exchange sites filled, then,

(Eq.-8)

and,

(Eq.-9)

Using the information provided in Eqs. 7-9, along with the number of exchange sites per unit volume or mass of soil, we can incorporate these equations directly into the charge and mass balance equations to obtain a complete model of the water-mineral system at equilibrium.

Questions

All of the preceding survey of information on surface complexation in aqueous systems may seem a bit daunting. Rather than focusing on the details, you should instead bear in mind that surface reactions are just one part of the whole picture in terms of all of the processes that play a role in reactive transport. Again, we leave the details of solving the equations up to the computer models; we will concern ourselves with developing conceptual models of practical problems and posing the problems to the model for solution. Weíll be considering specific effects of surface reactions on multispecies reactive transport in more detail in some of our examples in future lectures.

1. Cation Exchange Capacity. Suppose we have a clay mineral with the following composition: M_x(Si₈)Al_3.2Fe_0.2Mg_0.6O₂₀(OH)₄. Calculate its cation exchange capacity (in mol/kg) by determining the molecular weight of the mineral and the value of x. Note the following charges on the atoms in the molecule: Si(+4), Al(+3), Fe(+2), Mg(+2), O(-2), and H(+1).

2. Given the equation for a linear isotherm (Eq.-1), what is the ratio by which the concentration of a solute in solution is reduced by adsorption onto a solid phase? ( Hint: Take into account the value of K_d and the mass of solid present in a unit volume.)

ANSWERS

You are now ready to continue to LECTURE 6: Reaction Kinetics

You may e-mail me questions and comments.

Walt W. McNab
E-mail address: Walt McNab <WaltMcNab@prodigy.net>

ANSWERS

Question #1

If we construct a table of the atoms in a molecular unit of the clay mineral and the associated charges, we see that x must be given by the charge on the cation exchange element M as,

Species	Charge	Stoichiometry	Mol. wt. (gm/mol)
"M"	1	0.8	-
Si	4	8	28.09
Al	3	3.2	26.98
Fe	2	0.2	55.85
Mg	2	0.6	24.31
O	-2	20	15.9994
OH-	-1	4	17.0073

so that the total charge on the molecular unit is equal to zero. Ignoring the contribution of element M, we can calculate the molecular weight of the clay mineral by multiplying the stoichiometry of each species by its respective molecular weight, yielding 304.6 gm/mol. Since there are 0.8 mol of charge by exchangeable cations for each mol of the mineral, we can ,thus, calculate that the cation exchange capacity is about 2.6 mol/kg.

Question #2