Lecture No. 11. More Handy Computational Tools

Let’s stop for a moment, look at what we’ve covered so far, and get a peek at where the course is going from here.

Course Overview

Up until now we have mostly looked at general principles of environmental geochemistry. We have covered:

• authigenic minerals that form in surface and ground water systems;
• sulfate and sulfide chemistry in natural systems;
• redox equations and how to balance them;
• redox chemistry and electrochemistry as they relate to environmental systems;
• fundamentals of chemical measurement of environmental samples, sample collection, and the procedural definition of dissolved vs. suspended loads;
• contaminant phases;
• chemical reaction types;
• sediment movement in streams;
• movement of contaminants (and other substances) in sediment;
• methods for sediment separation (filtration, etc.);
• distribution coefficients for the partitioning of contaminants between solution and sediments;
• "total recoverable" measurements;
• calculation of dissolved, suspended, and/or total load per day using stream discharge measurements;
• grain size controls on transport by sediment;
• E_h-pH diagrams, their uses, and their limitations;
• Berner’s classification of geochemical environments and what characterizes each type of environment;
• phase diagrams other than E_h-pH, such as log c-pe.

Where do we go from here?

We are going to cover a few more computational tools: log ratio and log c-pH diagrams, their uses, and their limitations. Then we will shift gears and begin covering

• case studies, e.g., Butte, the Milltown Reservoir, and other cases of environmental contamination;
• types of contaminated environments, such as acid drainage regions and pit lakes;
• special cases of contaminant chemistry such as methylation of metals and metalloids.

We will also pick up a few more tools as we go along — but the P-chem-intensive and analytical chemistry-intensive part of the course will soon be behind us.

A Word about the Midterm Exam

There is, as promised, a simple E_h-pH diagram on the exam. There is at least one pH-controlled equilibrium. Now let’s revisit the aluminum diagram that was constructed in class. We had total aluminum activity at 1 M. We took the solubility product of Al(OH)₃, plugged in 1 M Al, and calculated the activity of OH^– that would make the ion activity product equal to the solubility product. At that OH^– activity, the first crystals of Al(OH)₃ would form.

But then what? As Al(OH)₃ formed, the Al³⁺ activity would fall, and Al(OH)₃ would stop precipitating. If 99+% of the aluminum is still in solution, what is the dominant species? I would say that the boundary ought to be at the pH where exactly half the aluminum is still in solution. In the case of our 1 M aluminum system, {Al³⁺} would be 0.5 M. Aluminum activity in solution is at 0.5 M when exactly half the Al is precipitated. It generally makes a small difference in pH, but I think it is important. That is why I have stipulated this approach on the exam.

Sediment Overview

Sediments and their influences were covered on several different days in large, difficult-to-digest chunks. Here is an overview of what it all means.

1. A lot of the metals in water reside in the sediments rather than in solution.
2. The way these metals are bound to the sediments varies from metal to metal and from sediment to sediment, and it changes with changes in pH and E_h.
3. Metals can reside on sediments in 3 ways: within the mineral lattice, bound to mineral surfaces as a coating, and bound to organic matter.
4. These 3 solute-binding mechanisms are expanded to define 5 categories of sediment-bound metals: exchangeable, bound to carbonates, bound to oxyhydroxides, bound to organics, and included within mineral lattices. An analytical scheme has been worked out to identify and separate these contaminant categories.
5. Even though bedload holds a significant portion of contaminant phases and some metals actually favor larger sediment particles, bedload accounts for only a minor part of contaminant transport. This is because bedload travels very slowly. Sand often moves only a few hundred meters a year. Gravel may move a few meters a year. On the other hand, suspended sediment moves at the same speed as the stream — kilometers per hour. (Reason #1 why suspended load is usually measured and bedload is disregarded.)
6. Suspended load transport is fairly easy to measure, while bedload transport is extremely hard to measure. (Reason #2 why bedload is often disregarded.)
7. The fine sediment, which tends to be in suspension, has a vastly larger surface area per unit mass than coarser sediment. Therefore, all other things being equal, the major part of all surface binding (ion exchange, adsorption, etc.) will happen on fine sediments.
8. All other things, however, are not always equal.
9. Sediment transport can fairly easily be measured using total recoverable metal as an approximation. It is especially useful at measuring the portion of a metal that can be hazardous (or beneficial, in the case of a nutrient) to biota.
10. Horowitz’s hypothesis (assuming essentially all metal transport and all metal sorption are on the fine particles) does not hold water.
11. Even so, we usually determine transport rates based on the fines, which travel in suspension, for the reasons given above.
12. We still must keep in mind that coarse sediments are the preferred substrates for some metals; that coatings on these sediments are very important adsorption and ion-exchange sites; and that coarse sediments often serve as major sources and sinks for solutes.

Activity Ratio Diagrams

Here is another very useful tool. There is one of these in the Elder paper, page 11. Once we go through this exercise you will understand it better.

In this type of diagram we plot all activities in terms of the logarithms of their ratios to one specific activity, vs. pe at fixed pH, or vs. pH at fixed pe. For example, we might plot {H₂S}, {HS^–}, and {SO₄^2–} in terms of their ratios to the activity of elemental sulfur. We refer to these plots as log [{i}/{S}]-pe plots. These are most useful when there are simple relationships for these ratios that involve only {S} and one other species activity without extraneous variables.

We will construct a diagram for sulfur at pH 10. For sulfate, we can use the equilibrium constant expression from Equation 4:

The log of the sulfate ratio has a slope of 6/pe and an intercept of 43.8 when pe=0.

Now for the {HS^–} ratio:

This line has a slope of –2/pe and an intercept of –12.2.

If we plot these two lines, we find that S is forbidden to exist any time either {SO₄^2–} or {HS^–} is larger, i.e., any time either log [{i}/{S}] is greater than zero. (See overhead.) Sulfur is forbidden by sulfate at all pe values above –7.3. It is forbidden by HS^– at all pe values smaller than –6.1.

We can do the same thing for pH 4. Now HS^– is not an important species, but H₂S is.

4.8 = log {H₂S} – log {S} + 2 pH + 2 pe

This log ratio has a slope of –2/pe and an intercept of –3.2.

Meanwhile the equation for the sulfate ratio is changed because the pH is lower:

When we plot the ratios at pH 4, we find we now have a region in which S is not forbidden – the pe range from –1.6 to +0.7. The question of whether S is permitted depends on whether {S} is > 1. If we specify that {S_T} is 10^–3 M, then we know that S can exist only if log [{i}/{S} is less than –3 – in the pe range from –0.1 to +0.2.

The same log ratio can be plotted against pH for equilibria that depend on pH but not pe. Consider the log [{i}/{Cu²⁺}] diagram from Elder’s paper. This is a plot for soluble and solid copper species in the presence of carbonate. Since {Cu²⁺} is fixed at 10^–2 M, any solid phase for which the log of the ratio exceeds 2 is permitted. On the other hand, any solute phase for which the log ratio exceeds zero will be the predominant one – unless there are more than one, in which case the one with the highest log ratio predominates.

(Note that Elder does not plot anything with a log ratio below zero.)

Examining the plot shows how these relationships interact. Note that from ~ pH 3.5 to ~ pH 7, malachite is the predominant solid phase. From ~ pH 4.5 to pH 7, malachite can exist (log ratio > 2). Above pH 7, tenorite predominates. Azurite is never predominant in the presence of an aqueous solution of Cu²⁺. (In fact, azurite is unstable in the presence of water; it hydrates to form malachite. Many 19^th-Century painters who painted in the West used azurite as their blue pigment. After a few decades in humid Eastern art galleries, the paintings acquired beautiful green skies.)

In solution, we see that the dissolved copper carbonate complex takes over from Cu²⁺ around pH 6. At higher pH, various other soluble copper complexes exist. We can read all this at a glance once we know the meaning of log [{i}/{Cu²⁺}].

pH-log c Diagrams

We have looked at diagrams in E_h-pH (or pe-pH) space with log c fixed, and we have looked at diagrams in pe-log c space at fixed pH, as well as concentration ratios vs. pe and pH.

Now we will address the remaining combination in pH-log c space at constant pe. These diagrams are generally handy when acid-base equilibria are the dominant ones and E_h (or pe) either is not important or can be fixed.

We will construct a pH-log c diagram for a fairly simple system and one for a more complex system. However, we will first review a little acid-base chemistry.

Consider a monoprotic acid (one with just one ionizable proton). Its dissociation equilibrium is

HA ó H⁺ + A^–

Its dissociation constant is given by the equilibrium expression:

There is a shorthand expression called ionization fraction. In brief, a ₀ is the fraction of the acid that is unionized, and a ₁ is the fraction ionized. If C_T is the total solution concentration of the acid, then

[HA] = a ₀ C_T and [A^–] = a ₁ C_T

Since

and {HA} = {H⁺} {A^–}/K_a,

and C_T = [HA] + [A^–],

and we assume that activity, {i}, is approximately equal to concentration, [i],

By similar reasoning,

There are tables listing a ₀ and a ₁ at various pH values for a number of acids. This is really no big deal for a monoprotic acid, but for acids with several ionizable hydrogens (such as citric, phosphoric, or arsenic acids) the higher ionization fractions, a ₂, a ₃, … get to be more laborious to calculate.

Now we will put together a pH-log c diagram for ammonia. We only need to deal with two equilibria and a third that is derived from the other two. The two equilibria are

NH₄⁺ ó NH₃ + H⁺

and

H₂O ó H⁺ + OH^–

The derived equilibrium is

NH₃ + H₂O ó NH₄⁺ + OH^–

The equilibrium expressions are listed below:

Now let’s put together a log c-pH diagram for ammonia. We need one other initial assumption: total ammonia concentration. We will set C_T = 1« 10^–3.

From the water equilibrium, we know that

K_w = [H⁺] [OH^–]

so

log K_w = –14 = log [H⁺] + log [OH^–] = log [OH^–] – pH

Therefore the equations for log [H⁺] and log [OH^–] are

log [H⁺] = – pH

and

log [OH^–] = pH – 14

From our initial conditions,

C_T = [NH₃] + [NH₄⁺]

For the ammonia species,

log K_a = –9.3 = log [NH₃] + log [H⁺] – log [NH₄⁺]

–9.3 = log [NH₃] – pH – log [NH₄⁺]

Therefore

log [NH₄⁺] = log [NH₃] – pH + 9.3

and

log [NH₃] = log [NH₄⁺] + pH – 9.3

So log [NH₄⁺] has a slope of –1 vs. pH, and log [NH₃] has a slope of +1 vs. pH.

When [NH₃] = [NH₄⁺], pH = 9.3.

We can plot the system point at pH 9.3 and log c = –3 – log 2 = – 3.301. The log [H⁺] and log [OH^–] intersect at pH 7 and log c = –7. Above the system point, C_T is limiting for NH₃ and NH₄⁺; we draw smooth lines connecting their lines with the log C_T line at –3. We now have a completed log c-pH diagram for ammonia at 10^–3 M.

Next: We will plot a pH-log c diagram for carbonic acid, then move on to the Milltown Reservoir case study.

To previous lecture: Lecture No 10. Some Handy Computational Tools

To next lecture: Lecture No 12. Carbonic acid. Milltown Case Study.

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