Chemical potential solid-solution components

In the solid or liquid state the activity, a, is introduced to express the chemical potential of the components of a solution. It is defined by... 

Equilibrium between simple salts and aqueous solutions is often relatively easily demonstrated in the laboratory when the composition of the solid is invariant, such as occurs in the KCI-H2O system. However, when an additional component which coprecipitates is added to the system, the solid composition is no longer invariant. Very long times may be required to reach equilibrium when the reaction path requires shifts in the composition of both the solution and solid. Equilibrium is not established until the solid composition is homogeneous and the chemical potentials of all components between solid and aqueous phases are equivalent. As a result, equilibrium is rarely demonstrated with a solid solution series. 

The chemical potentials of the solid solution components can be eliminated from the liquidus equations given by Eqs. (8) and (9) using Eqs. (6) and... 

When a liquid solution and a solid solution are in equilibrium, both components exist in both phases. We again develop the pertinent equations for only one component, because the equations for the second component are identical except for the change of subscripts. The condition of equilibrium is the equality of the chemical potential of the component in the two phases, so... 

Thorstenson and Plummer (1977), in an elegant theoretical discussion (see section on The Fundamental Problems), discussed the equilibrium criteria applicable to a system composed of a two-component solid that is a member of a binary solid solution and an aqueous phase, depending on whether the solid reacts with fixed or variable composition. Because of kinetic restrictions, a solid may react with a fixed composition, even though it is a member of a continuous solid solution. Thorstenson and Plummer refer to equilibrium between such a solid and an aqueous phase as stoichiometric saturation. Because the solid reacts with fixed composition (reacts congruently), the chemical potentials of individual components cannot be equated between phases the solid reacts thermodynamically as a one-component phase. The variance of the system is reduced from two to one and, according to Thorstenson and Plummer, the only equilibrium constraint is IAP g. calcite = Keq(x>- where Keq(x) is the equilibrium constant for the solid, a function of... 

What we mean in this report by equilibrium and disequilibrium requires a brief discussion of definitions. Natural physicochemical systems contain gases, liquids and solids with interfaces forming the boundary between phases and with some solubility of the components from one phase in another depending on the chemical potential of each component. When equilibrium is reached by a heterogeneous system, the rate of transfer of any component between phases is equal in both directions across every interface. This definition demands that all solution reactions in the liquid phase be simultaneously in equilibrium with both gas and solid phases which make contact with that liquid. Homogeneous solution phase reactions, however, are commonly much faster than gas phase or solid phase reactions and faster than gas-liquid, gas-solid and... 

Let us consider an isothermal process in which 1 mol of pure solid component in equilibrium with its saturated solution transforms into a liquid state (melting). In equilibrium, the equality of the chemical potentials of this component in both the phases must be valid (e.g. for the component A)... 

The chemical potentials of each component in the solid solution may be written... 

Figure 7). Comparison of Figures 5 and 7 reveals that energy of an equilibrium mixture of the phases is minimized at the composition which gives maximum catalytic activity. It is apparent that at compositions where Af is a minimum, the difference in the chemical potentials of the components in the interfacial region and the equilibrium solid solutions is minimized. Thus, an interfacial region which is chemically similar to the saturated solid solutions appears optimum for maximum catalytic efficiency. 

If the different portions of a thermodynamic system are in true equilibrium the chemical potentials of the components of the system will be the same throughout, whatever the physical state. Thus the chemical potential of a vapor of a component is the same as that of the same component in the solid or liquid state, or of the component in solution. 

Now let us turn to a system of two components miscible in all proportions. The components could be gases, liquids or solids. The chemical potential of a component can conveniently be displayed using a graphical representation [1], The molar free energy of the solution G is defined as ... 

Figures 15.1b and c illustrate the two opposite cases where either enthalpy or entropy dominates the free energy function. The most difficult situation is that shown in Figure 15. Id, where entropy and enthalpy approximately balance each other. This produces a peculiar double-humped free energy curve as illustrated. Here the total free energy of the system is always lower than that of a mechanical mixture of the two pure end-members. However, this time there cannot be a complete solid solution from pure A to pure B as in the previous case. This is because we can draw a tangent (abed on the figure) that touches the free energy curve at two points (b and c). This means that two phases can coexist (having compositions and X ) in which the chemical potentials of each component are the same. That is, the chemical potential of A in both phases is given by the intercept a, and the chemical potential of B in both phases is given by the intercept at d.

Reaction rates at solid/solution interfaces are controlled by the area of the interface as well as by the chemical and physical conditions that occur there. Surface reactions are approximately confined to a two-dimensional region, so their rates are expressed in terms of how fast species are created per unit of surface area, and this means that the rates have imits of flux (/, mol/m sec). The flux notation (J) and terminology is used throughout this book. The environment at the solid/solution interface is a hybrid of the bulk solid and bulk solution, so models of the chemical and physical conditions controlling the reaction rates must account for this transitional character. Equilibrium thermodynamics provides a powerful starting point for constraining the surface conditions. At equilibrium the chemical potential of each component must be the same throughout the system, so the chemical potential of the components in the surface are equal to their chemical potentials in the solid and solution phases. At low temperatures, the slow rate of equilibration between the bulk solid and the surface may void this requirement for the solid but it should apply for the components in the bulk solution. Also, at equilibrium the principle of detailed balance requires that the rates of forward reactions in the interface must equal the rates of the reverse reactions. In addition, the forward and reverse reaction steps must be the same. Models of reaction rates at equilibrium are well constrained by these principles but as the system departs from equilibrium these requirements fall away and we must search for other principles to model interfacial reaction rates. 

If a binary system exhibits a solvus (a composition region in which two solid solutions coexist at equilibrium see Figures 17.12, 17.21 for examples), a convenient way of determining Wq values is to use the fact that the chemical potential of each component is the same in every pair of compositions at the same T (and P) on each side of the solvus. For example if we know that compositions and on a symmetrical solvus are at equilibrium, then from Equation (10.94) we write... 

We now consider chemical reactions in which one or more of the products or reactants is a solid or liquid. In the present chapter the discussion will be limited to cases where each of these solids or liquids is present in the system as a pure phase, i.e. when they do not take into solution appreciable amounts of the other components. Under these conditions the free energy of mixing, which has been shown to be an important part of the driving force of reaction, is limited to the gaseous phase. (The discussion of the case where there is an additional free energy of mixing in the condensed phases depends on a knowledge of the chemical potentials in solutions and will be deferred to Chapter 10.)... 

We will examine the methods for determining the activities, or the activity coefficients, of the components of a solid solution whose composition is known. Note in passing that this determination of the activity can quickly lead us to the chemical potential of that component, by knowledge of the chemical potential of that component in the reference state. 

A solution is a homogeneous mixture of two or more components (substances whose amounts can be independently varied). We ordinarily apply the name only to solid and liquid mixtures, although a gaseous mixture is also homogeneous. We begin with ideal solutions, which are defined to be solutions in which the chemical potential of each component is given for all compositions by the formula... 

For precise measurements, diere is a slight correction for the effect of the slightly different pressure on the chemical potentials of the solid or of the components of the solution. More important, corrections must be made for the non-ideality of the pure gas and of the gaseous mixture. With these corrections, equation (A2.1.60) can be verified within experimental error. 

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. 

In equilibrium the chemical potential must be equal in coexisting phases. The assumption is that the solid phase must consist of one component, water, whereas the liquid phase will be a mixture of water and salt. So the chemical potential for water in the solid phase fis is the chemical potential of the pure substance. However, in the liquid phase the water is diluted with the salt. Therefore the chemical potential of the water in liquid state must be corrected. X refers to the mole fraction of the solute, that is, salt or an organic substance. The equation is valid for small amounts of salt or additives in general ... 

Chemical potentials for the constituents of minerals are defined in a similar manner. All minerals contain substitutional impurities that affect their chemical properties. Impurities range from trace substitutions, as might be found in quartz, to widely varying fractions of the end-members of solid solutions series. Solid solutions of geologic significance include clay minerals, zeolites, and plagioclase feldspars, which are important components in most geochemical models. 

The difference between the chemical potential of a pure and diluted ideal gas is simply given in terms of the logarithm of the mole fraction of the gas component. As we will see in the following sections this relationship between the chemical potential and composition is also valid for ideal solid and liquid solutions. 

Equations (4.7) and (4.8) may be solved numerically or graphically. The latter approach is illustrated in Figure 4.2 by using the Gibbs energy curves for the liquid and solid solutions of the binary system Si-Ge as an example. The chemical potentials of the two components of the solutions are given by eqs. (3.79) and (3.80) as... 

Similar expressions are valid for the chemical potential of component B of the two phases. According to the equilibrium conditions given by eqs. (4.7) and (4.8), the solid and liquid solutions are in equilibrium when /j,s = and/tg =/Xgq, giving the two expressions... 

Stoichiometric saturation defines equilibrium between an aqueous solution and homogeneous multi-component solid of fixed composition (10). At stoichiometric saturation the composition of the solid remains fixed even though the mineral is part of a continuous compositional series. Since, in this case, the composition of the solid is invariant, the solid may be treated as a one-component phase and Equation 6 is the only equilibrium criteria applicable. Equations 1 and 2 no longer apply at stoichiometric saturation because, owing to kinetic restrictions, the solid and saturated solution compositions are not free to change in establishing an equivalence of individual component chemical potentials between solid and aqueous solution. The equilibrium constant, K(x), is defined identically for both equilibrium and stoichiometric saturation. 

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