Speciation Calculations

This chapter focuses on one of the most common questions asked about natural chemical systems: what are the concentrations or activities of the different species present in a system at complete chemical equilibrium? We might be concerned, for example, with oxygen or sulfur fugacities, with the activities of complex ions, or activity ratios of reduced and oxidized species of the same component. In practice, these calculations range from trivially simple to enormously complex, depending on the number of species (and components) in the system. We will follow roughly this order—from trivial to complex—and outline some of the most common approaches used in performing speciation calculations. This simplest procedure is probably used most often, and works best with systems containing relatively few chemical species. As a general rule of thumb, you might try this if there are fewer than 10 species, but move on to another more sophisticated method for more complicated systems. As an example, we will solve for the equilibrium concentrations of all species in an acetic acid + water solution of a given concentration, m. Specifically, we might be interested in the pH of a 0.1 m HAc solution, but in calculating this we will also get the activities of all other species, whether we need them or not. This is one of the simplest examples imaginable, but the method works exactly the same way with more complicated systems. An excellent reference on this general approach is Butler (1964, Chap. 3). There are six steps to follow:… 1. Write all species of relevance or interest. Count the number of unknown species. You will need this many equations. 2. Write all known equilibrium constant equations. 3. If there are charged species, write a charge balance equation. 4. Write all known mass balance equations. 5. You should now have the same number of equations as unknown species. Reduce these by algebraic substitution to one (or two) equations that can be solved for the unknown concentrations. At first, assume all activity coefficients are 1.0.

Redox Systems

Up until this point we have dealt with familiar intensive variables such as temperature, pressure, density, and molar thermodynamic properties (molar entropies, free energies, and so on). There exists another, equally important intensive variable that we have used implicitly, but have not yet discussed in sufficient detail—the oxidation state of a system. This involves concepts and applications so useful to Earth scientists that we devote a complete chapter to this single variable. Except for nuclear processes, most chemical behavior is determined by electron distributions and interactions. From this point of view, the oxidation state of an atom is among the most fundamental of all its properties. Most elements can exist in multiple valences with each state usually displaying quite different behavior from the others. As an example, consider the element sulfur.

Gaseous Solutions

The procedures described in Chapter 15 are well suited to solid and liquid solutions and could also be applied to gases, but in fact, other approaches are generally used. The main reason for this is partly historical; much work was done early in the history of physical chemistry on the behavior of gases, and these methods have continued to evolve to the present day. We have also just seen that the Margules equations become very unwieldy with multi-component systems. Because true gases are completely miscible, natural gases often contain many different components, so the Margules approach is not very suitable. Unfortunately, the most successful alternative methods described in this section are also quite unwieldy; however, they do not become much more complicated for multi-component gases than they are for the pure gases themselves, and this is a definite advantage. We have seen that with real, non-ideal gases, all the thermodynamic properties are described if we know the T, P, and the fugacity coefficient. For gaseous solutions, the fugacity coefficient for each component generally depends on the concentrations and types of other gaseous species in the same mixture. All gases, whether pure or multi-component, should approach ideality at higher T and lower P; conversely, non-ideality is most pronounced in dense, low-temperature gases where intermolecular forces are strongest. The challenge here is to find an equation of state that can adequately cover this range of conditions for gases of many different constituents. In the following discussion we first briefly outline some of the equations of state used to describe pure gases. We will introduce these from the molecular point of view since this helps understand the physical basis (and limitations) of each model. Each of these equations of state can then be applied to mixtures of gases using a set of rules which we describe at the end of this section.

Solid Solutions

Real solutions of practical interest to Earth scientists do not behave ideally, although some do come fairly close. The problem of course lies in the stringent and unrealistic physical models we have prescribed for ideal solutions. The molecules of a gas do interact with each other, molecular forces within mixed component liquids really are non-uniform, and the different ions substituting for each other in solids are never exactly alike. So why bother defining an ideal solution in the first place if real systems do not behave that way? In fact, the ideal solution is a very useful artifice. It is something simple against which the behaviour of real solutions can be measured and compared. Our most fundamental definition of an ideal solution was With this as our reference, we can define a non-ideal solution as one for which the activity coefficient of each component i differs from unity The activity coefficient is the single quantity that expresses all deviations from non-ideality for each component of a solution. As we shall see, parameters other than the activity coefficient itself are frequently used to describe non-ideal behavior, but these could, if we wished, be related back to (15.1). Note that we say ⋎i, ≠1.0 in general; there are times when ⋎i = 1.0 for specific conditions (one set of T, P, Xi, etc.) even in highly non-ideal systems. This is just coincidental and certainly does not mean that the system is ideal at that particular point—the activity coefficient would have to be unity under all possible conditions for that to be the case.

Ideal Solutions

The chemical constituents of a solution can be varied — added, subtracted and interchanged or substituted for each other — within limits ranging from complete (e.g., gases) to highly restricted (trace components in quartz). Adding or subtracting chemical constituents to or from a phase involves changes in energy, which will be discussed in the following sections. For example, if two components A and B are mixed together, the Gibbs energy of a solution of the two mixed must be less than the sum of the Gibbs energies of the two separately for the spontaneous reaction to take place. That is, if we mix nA moles of component A and nA moles of component B, their combined total G is (nAGA + nBGB) where GA and GB are the molar free energies of A and B. If G(A,B) is the total free energy of the resulting solution, then necessarily if the solution took place spontaneously. Alternatively, dividing through by nA + nB, where XA and XB are the mole fractions. Thus if A is albite and B is anorthite, then (A,B) is plagioclase, and we say that the plagioclase solid solution is more stable than a "mechanical mixture" of grains of albite and anorthite. On the other hand if A is diopside and B is anorthite, little or no mutual solution takes place because in this case so that no spontaneous solution reaction takes place. The term "mechanical mixture" in this context nicely conveys the idea of quantities of mineral grains mixed together and not reacting, but does not work quite so well if A and B are other things such as water and halite, or water and alcohol. Nevertheless, the term is traditionally used no matter what the nature of the solution constituents, and no harm is done as long as we remember that "mechanical mixture" means that the constituents considered do not react with each other, whatever their physical nature.

Thermodynamic Properties of Simple Systems

We have now introduced several thermodynamic parameters that are useful in dealing with energy transfers (U, H, G, etc.). We wish now to see how these quantities are measured and where to find values for them. In later chapters we will see how they are used in detail. However, we have an immediate problem in that we cannot measure the energy parameters U, H, G and A, as discussed in Chapter 4. Because we do not know the absolute values of either the total or molar version of these variables, we are forced to deal only with their changes in processes or reactions of interest to us. But we obviously cannot tabulate these changes for every reaction of potential interest; there are too many. We must tabulate some sort of energy term for each pure substance so that the changes in any reaction between them can be calculated. In the example in §5.7 of water at — 2°C changing to ice at — 2°C, we said that AG was negative. How can we know this without carrying out a research program on the thermodynamic properties of ice and supercooled water? We begin by explaining how this is done. The problem created by not having absolute energy values is handled very conveniently by determining and tabulating, for every pure compound, the difference between the (absolute) G or H of the compound itself and the sum of the (absolute) G or H values of its constituent elements. In other words, AG or AH is determined for the reaction in which the compound is formed from its elements (in their stable states). These differences can be determined experimentally in spite of not knowing the absolute values involved.

Thermodynamic Terms

Thermodynamics is the science that deals with energy differences and transfers between systems, and with systematizing and predicting what transfers will take place. Such fundamental topics naturally find application in all branches of science, and have been of interest since the earliest beginnings of science. In general, since we are dealing with energy transfers between systems, most of what follows has to do with what the entities (equilibrium states) are from which and to which energy is being transferred, and the boundaries or walls through which or by which the transfer is effected. It is in these considerations that we first see the differences between natural systems (reality) and our models of these systems. System refers to any part of the universe we care to choose, whether the contents of a crucible, a cubic centimeter in the middle of a cooling magma, or the solar system. Depending on the nature of the discussion, it must be more or less clearly defined and separated (in fact or in thought) from the rest of the universe, which then becomes known as the system's surroundings. At the outset, we will effect an enormous simplification by considering only systems that are unaffected by electrical, magnetic, or gravitational fields, and in which particles are sufficiently large that surface effects can be neglected. Each of these topics can be incorporated into the basic thermodynamic network to be developed, but it is a nuisance to carry them all along from the beginning, and a great deal can be done without considering them at all. More exactly, a great deal can be done if we choose to consider systems where these fields and surfaces play a minor role. Clearly we would not get very far if we tried to understand the solar system without considering gravitational fields. Chemical and geochemical problems however commonly do not need to have these factors included in order to be understood. In science, when a problem or a phenomenon such as the solar system or the boiling of water is said to be understood, what is usually meant is that we have a model of the phenomenon which is satisfactory at some level, and about which virtually all scientists agree.

Mathematical Background

Thermodynamics, like other sciences, has a theoretical side, expressed in mathematical language, and a practical side, in which experiments are performed to produce the physical data required and interpreted by the theoretical side. The mathematical side of thermodynamics is simple and elegant and is easily derived from first principles. This might lead to the conclusion that thermodynamics is a simple subject, one that can be easily absorbed early in one's education before going on to more challenging and interesting topics. This is true, if by learning thermodynamics one means learning to manipulate its equations and variables and showing their interrelationships. But for most students the subject is actually far from simple, and for professors it is a considerable challenge to present the necessary material intelligibly. The equations and the variables are somehow related to the real world of beakers and solutions, fuels and engines, rocks and minerals, and it is this interface that provides most of the difficulties. What do variables such as entropy and free energy really mean, and what physical processes do the equations describe? The difficulty in understanding and using thermodynamics is conceptual, not mathematical. We will attempt to explain the relationship between the mathematical and the physical sides of thermodynamics, but it is advisable first to review the mathematics involved and subsequently to define the terms used in thermodynamics. The mathematics required for thermodynamics consists for the most part of nothing more complex than differential and integral calculus. However, several aspects of the subject can be presented in various ways that are either more or less mathematically based, and the "best" way for various individuals depends on their mathematical background. The more mathematical treatments are elegant, concise, and satisfying to some people, and too abstract and divorced from reality for others. In this book we attempt to steer a middle-of-the-road course. We review in the first part of this chapter those aspects of mathematics that are absolutely essential to an understanding of thermodynamics. The chapter closes with mathematical topics that, although not essential, do help in understanding certain aspects of thermodynamics.

The Equilibrium Constant

Consider a chemical reaction, involving any number of reactants and products and any number of phases, which may be written where Mi represents the chemical formulae of the reaction constituents and Vi represents the stoichiometric coefficients, negative for reactants and positive for products. An example would be where, if MI is SiO2(s), M2 is H2O, and M3 is H4SiO4(ag), and v1 = —I, i/2 = —2, and V3 = 1 , the reaction is Now let's recall (from Chapter 3) what we mean by an equation such as (13.3). If there are no constraints placed on the system containing M1, M2, and M3 other than T and P (or T and V; U and V; S and P; etc.) then M1, M2, and M3 react until they reach an equilibrium state characterized by a minimum in the appropriate energy potential as indicated by expressions like dGT,p = 0. A corollary of this equilibrium relationship, to be fully developed in the next chapter, is that the sums of the chemical potentials of the reactants and products must be equal. In the example, this would be or or in general terms, No notation is necessary for the phases involved because μ must be the same in every phase in the system. However, if more than the minimum two constraints apply to the system, then any equilibrium state achieved will be in our terms a metastable state, (14.25) does not apply, and the difference in chemical potential between products and reactants is not zero. In our example, a solution might be supersaturated with H4SIO4 but prevented from precipitating quartz by a nucleation constraint, so that μ H4SiO4 — μ SiO2 ~ 2 μ H2o > 0.

Applications to Simple Systems

Thus far we have developed just about all the thermodynamic concepts required by Earth scientists with the exception of those needed to deal with solutions. Since all naturally occurring substances are solutions of one kind or another (although some can usefully be treated as pure substances), this is quite an important limitation, and we will proceed to discuss the treatment of solutions in Chapter 10. However, a great deal can be done with the thermodynamics of pure systems, and in this chapter we discuss a couple of applications of the concepts so far developed which are of particular interest to Earth scientists—the thermal effects associated with adiabatic volume changes, and the T-P phase diagrams of pure minerals. All systems experience a change in volume in response to changes in pressure. We have discussed this mostly from the point of view of the work accomplished by isobaric volume changes in Chapter 4, but it is even more informative to consider the temperature changes accompanying volume changes. The best way to do this is to consider only cases uncomplicated by heat entering or leaving the system, i.e., adiabatic processes. Such processes, although yet another "hypothetical limiting case," serve as useful end-members in considering actual processes in real systems. The most familiar everyday example is the hand-held bicycle pump, which most cyclists at least know gets quite warm during pumping (compressing air). This process, while not strictly adiabatic (bicycle pumps are not well insulated) is sufficient to show that volume changes can be associated with temperature changes, and it is not difficult to see in this case why—a great deal of energy in the form of work is being added to the gas, and some of it is being used to warm the gas. It seems reasonable to suppose, too, that by reversing the process—suddenly expanding the gas—it would experience a temperature decrease. This much may seem intuitively reasonable, perhaps even obvious. What is not so obvious is the fact, first investigated by Joule and Thompson in 1853, that some substances do not warm but cool during compression, and that in fact all substances have a range of conditions where they warm on expansion and another where they cool on expansion. When the expansions are at constant enthalpy, these two ranges are separated by the Joule-Thompson inversion curve.