In all  previous lectures, we have clung unquestioningly to the concept of thermodynamic equilibrium as the basis for our view of the world (or, at least, our view of geochemical speciation). There are, of course, countless examples from our everyday observations of the world around us that things are not always in thermodynamic equilibrium. Our very existence as organisms is perhaps the most profound one; thermodynamics dictates that we should not exist (as protoplasm), but rather should all be CO₂ and water by now. In natural groundwaters, there are many examples of non-equilibrium, particularly among redox species (i.e., many redox couples are not in equilibrium with one another, hence there is no real system pE) and among mineral phases.

                         (Eq.-1)

Letís assume weíre starting out with an arbitrary amounts of each of the species involved ñ A, B, C, and D. The system will proceed toward a state of thermodynamic equilibrium described by a mass action expression and an equilibrium constant, K:

       (Eq.-2)

Obviously, Eq.-2 does not just apply to individual atoms or molecules, but to an scales, this occurs as a result of diffusion, which implies a certain time dependency, particularly if one or more of the constituents is in short supply.

 Activation energy and temperature. Sometimes, even when two potentially reacting atoms or molecules readily come into contact with one another, there is an energy "hump" that must be overcome in order for the reaction to occur, despite the fact that the reaction is

     (Eq.-3)

 Available surface area. For heterogeneous reactions, it is intuitive that the rate of reaction will depend on the amount of surface area available for mass transfer between the aqueous phase and a mineral surface.

 Biological effects. For certain types of reactions, notably redox reactions, the reaction process, although favorable, involves the breaking of covalent bonds that may be quite strong. As a result, otherwise favorable redox reactions may proceed sluggishly if at all. However, many types of bacteria have evolved which extract energy from redox reactions. They catalyze the reactions using enzymes that aid in breaking the bonds. Consequently, reaction times are substantially increased.

Given all of this, while we have an understanding of some of the factors that determine reaction rates, we lack a single unifying model that can predict reaction rates for arbitrary reactions in typical groundwater geochemical settings (although the development of various kinetic theories is an active area of research). Thus, as a practicality, we are left with empirical rate laws extracted from the analysis of experimental data. Letís have a brief look at some of the more commonly encountered differential rate laws:

                     (Eq.-4)

Higher-order rate law expressions are also possible. For example, a second-order rate law relationship, expressed as,

                   (Eq.-8)

is often suitable for systems in which a collision between two molecules forms a rate-limiting step in the sequence of a chemical reaction. A solution to Eq.-8 equation can be shown to be:

                           (Eq.-9)

                   (Eq.-10)

The slower of the two reactions in the sequence (A ý  B or B ý  C) will be the rate-limiting step in the overall transformation of A into C.

Closed form expressions for concentrations of each member of a reaction sequence have been developed. While we will not repeat the details, or the actual solutions for that matter, here in this lecture,

Now suppose that we consider the reaction sequence given by Eq.-10 and assume that the reaction kinetics are described by first-order rate laws. Letís start out with a fixed quantity of A (say [A] =1) and no B or C. Letís assume in one case that the reaction rate for A ý  B is 0.014^-1 and B ý  C is 0.069_-1 (the units of time are unimportant for this illustration). Species C is assumed to be stable (i.e., it doesnít transform into anything else). The concentration histories of these three species under this scenario are shown on Figure 1. In this case, because A transforms more slowly than B, itís transformation is the rate-limiting step in forming C The concentration of B never becomes very large as it is formed slowly but is transformed quickly.

Figure 1. Concentrations of species in the A ý  B ý  C reaction sequence over time, with the first reaction (A ý  B) as the rate-determining (i.e., "slow") step.

What would happen if we swapped the reaction rates, so that B ý  C became the rate-limiting step? The result is shown on Figure 2, where, in contrast to the first case, there is a significant accumulation of B in the system early in the simulation.

Figure 2. Concentrations of species in the A ý  B ý  C reaction sequence over time, with the second reaction (B ý  C) as the rate-determining (i.e., "slow") step.

Now, letís take this example one step further. Suppose we take a look at a continuous point source model of solute concentration in a homogeneous aquifer of uniform thickness that extends infinitely in all directions, with uniform groundwater flow. This model takes the form:

(Eq.-12)

The details of this model and its terms are unimportant for the illustration. What is important is that we can include  the transformation sequence that weíve used for the reaction series A ý  B ý  C into Eq.-12 as part of the convolution integral (details omitted). This allows us to simulate, in a very idealized sort of way, the concentration distribution of A, B, and C as a function of space and time, assuming a prescribed influx of A at x =0, y = 0. Postulating a set of self-consistent values for the parameters in Eq.-12, and assuming the same rate-limiting relationships discussed above, the profiles of the concentration distributions along the plume centerline are shown on Figures 3 and 4.

Simulation results indicate the plume is much more enriched in B under the second scenario the first, as expected. In both scenarios, peak concentrations of the transformation products (B and C) occur downstream of the source.

Figure 3. Concentrations of species in the A ý  B ý  C reaction sequence with distance from the source of A at x = 0 at t = 100, with the first reaction (A ý  B) as the rate-determining (i.e., "slow") step.

 of A at x = 0 at t = 100, with the second reaction (B ý  C) as the rate-determining (i.e., "slow") step.

In the real world, the rate of transformation of a particular species will depend not only, in some cases, on the concentration of the species itself but also on the concentrations of other species in the system. A well-known example among contaminant hydrologists is the rate at which benzene, a constituent of gasoline, transforms through a coupled redox reaction (via microbial mediation) into CO₂. A number of electron acceptors are usually available in most groundwater environments that could, thermodynamically, oxidize benzene to CO₂. However, the rates by which benzene is oxidized have been shown infield and laboratory studies to vary significantly, depending on which electron acceptor is in fact being utilized as an oxidizing agent. As a general rule, transformation rates are the fastest when dissolved O₂ is the oxidizing agent, probably because the energy gain for the microorganisms is the greatest. Transformation rates under anaerobic conditions (i.e., in the absence of dissolved O₂) are much lower.

Consider, as an example, a situation where benzene is continuously released into an aquifer, perhaps as a result of a leaking underground gasoline storage tank. Now, letís look at the simplest kinetic model, one in which benzene transforms through a first-order rate law with a constant transformation coefficient of 0.003 day^-1 (a value typical of those reported under aerobic conditions). Using this rate constant, and a numerical solute transport model using reasonable aquifer properties and benzene leaching rates, a plausible benzene plume after 20 years of leaching might look something like that depicted on Figure 5.

Figure 7. Concentrations of dissolved O₂ (in mg/L, or parts-per-million) after 20 years. The dark region, indicating anaerobic conditions where the concentration of dissolved O₂ ~ 0, results from, and limits, the biodegradation of benzene.

Letís generalize a bit for a moment and look beyond groundwater chemistry per se. Suppose you are designing procedures to enable large telescopes or space exploration spacecraft to use to collect data that would be indicative of life on other planets. Given what we know about the biogeochemical environment here on Earth, what sort of things might you look for that would be indicative of life (Hint: think about gases you could measure in a planetís atmosphere using spectrographic techniques).

ANSWER

LECTURE 1: Commonly Used Geochemical Speciation Models and Solution Techniques.

You may e-mail me questions and comments.

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

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