Mass action equations

By analogy with similar materials in which free elecU ons and electron holes are formed, NiO is called a p-type compound having vacant site Schottky defects, and ZnO is an n-type compound having interstitial Frenkel defects. The concentrations of these defects and their relation to the oxygen pressure in the suiTounding atmosphere can be calculated, for a dilute solution of defects by the application of a mass action equation. The two reactions shown above are represented by the equations... 

Assume that the total drug concentration [AT] is the sum of the free concentration [Afree] and the concentration bound to a site of adsorption [AD] (therefore, [Afiee] = [Ax] — [AD]). The mass action equation for adsorption is... 

If an aqueous ammonia solution which is 0.1M is employed, the concentration of NH4 ion as ammonium chloride or other ammonium salt necessary to prevent the precipitation of magnesium hydroxide can be readily calculated as follows. Substituting in the mass action equation ... 

Aqueous geochemists work daily with equations that describe the equilibrium points of chemical reactions among dissolved species, minerals, and gases. To study an individual reaction, a geochemist writes the familiar expression, known as the mass action equation, relating species activities to the reaction s equilibrium constant. In this chapter we carry this type of analysis a step farther by developing expressions that describe the conditions under which not just one but all of the possible reactions in a geochemical system are at equilibrium. 

How can we express the equilibrium state of such a system A direct approach would be to write each reaction that could occur among the system s species, minerals, and gases. To solve for the equilibrium state, we would determine a set of concentrations that simultaneously satisfy the mass action equation corresponding to each possible reaction. The concentrations would also have to add up, together with the mole numbers of any minerals in the system, to give the system s bulk composition. In other words, the concentrations would also need to satisfy a set of mass balance equations. 

At this point we can derive a set of governing equations that fully describes the equilibrium state of the geochemical system. To do this we will write the set of independent reactions that can occur among species, minerals, and gases in the system and set forth the mass action equation corresponding to each reaction. Then we will derive a mass balance equation for each chemical component in the system. Substituting the mass action equations into the mass balance equations gives a set... 

Each independent Reaction 3.22 in the system has an associated equilibrium constant Kj at the temperature of interest and, hence, a mass action equation of the... 

The final form of the governing equations is given by substituting the mass action equation (Eqn. 3.27) for each occurrence of ntj in the mass balance equations (Eqns. 3.28-3.31). The substituted equations are,... 

Like all formulations of the multicomponent equilibrium problem, these equations are nonlinear by nature because the unknown variables appear in product functions raised to the values of the reaction coefficients. (Nonlinearity also enters the problem because of variation in the activity coefficients.) Such nonlinearity, which is an unfortunate fact of life in equilibrium analysis, arises from the differing forms of the mass action equations, which are product functions, and the mass balance equations, which appear as summations. The equations, however, occur in a straightforward form that can be evaluated numerically, as discussed in Chapter 4. 

The governing equations are composed of two parts mass balance equations that require mass to be conserved, and mass action equations that prescribe chemical equilibrium among species and minerals. Water Aw, a set of species, 4/, the min-... 

We pose the problem for the remaining equations by specifying the total mole numbers Mw, Mi, and of the basis entries. Our task in this case is to solve the equations for the values of nw, mt, and - The solution is more difficult now because the unknown values appear raised to their reaction coefficients and multiplied by each other in the mass action Equation 4.7. In the next two sections we discuss how such nonlinear equations can be solved numerically. 

We attempt the calculation in the hope that error in estimating activity coefficients will not be so large as to render the results meaningless. In fact, the situation may be slightly better than might be feared because the activity coefficients appear in the numerator and denominator of the mass action equations, the error tends to cancel itself. 

To cast the model in general form, we begin with the basis shown in Equation 9.5 and write each sorption reaction in the form of Equation 9.7. The mass action equation corresponding to the reaction for each sorbed species Aq is... 

Substituting, the mass action equation (Eqn. 9.34) becomes, for the Gaines-Thomas convention,... 

To do so, we calculate the Jacobian matrix, which is composed of the partial derivatives of the residual functions with respect to the unknown variables. Differentiating the mass action equations for aqueous species Aj (Eqn. 4.2), we note that,... 

Here, vwq, vtq, v q, vmq, and vpq are coefficients in the reaction, written in terms of the basis B, for surface complex Aq. We have already shown (Eqn. 3.27) that the molality of each secondary species is given by a mass action equation ... 

In Reaction 10.11 the change Az in surface charge is —zq because the uncom-plexed sites Ap carry no charge. With this in mind, we can write a generalized mass action equation cast in the form of Equation 10.8,... 

The iteration step, however, is complicated by the need to account for the electrostatic state of the sorbing surface when setting values for mq. The surface potential T affects the sorption reactions, according to the mass action equation (Eqn. 10.13). In turn, according to Equation 10.5, the concentrations mq of the sorbed species control the surface charge and hence (by Eqn. 10.6) potential. Since the relationships are nonlinear, we must solve numerically (e.g., Westall, 1980) for a consistent set of values for the potential and species concentrations. 

To calculate a fixed activity path, the model maintains within the basis each species At whose activity at is to be held constant. For each such species, the corresponding mass balance equation (Eqn. 4.4) is reserved from the reduced basis, as described in Chapter 4, and the known value of a, is used in evaluating the mass action equation (Eqn. 4.7). Similarly, the model retains within the basis each gas Am whose fugacity is to be fixed. We reserve the corresponding mass balance equation (Eqn. 4.6) from the reduced basis and use the corresponding fugacity fm in evaluating the mass action equation. 

Despite the authority apparent in its name, no single rate law describes how quickly a mineral precipitates or dissolves. The mass action equation, which describes the equilibrium point of a mineral s dissolution reaction, is independent of reaction mechanism. A rate law, on the other hand, reflects our idea of how a reaction proceeds on a molecular scale. Rate laws, in fact, quantify the slowest or rate-limiting step in a hypothesized reaction mechanism. 

The situation is different for reactions of very hydrophilic ions, e.g. hydroxide and fluoride, because here overall rate constants increase with increasing concentration of the reactive anion even though the substrate is fully micellar bound (Bunton et al., 1979, 1980b, 1981a). The behavior is similar for equilibria involving OH" (Cipiciani et al., 1983a, 1985 Gan, 1985). In these systems the micellar surface does not appear to be saturated with counterions. The kinetic data can be treated on the assumption that the distribution between water and micelles of reactive anion, e.g. Y, follows a mass-action equation (9) (Bunton et al., 1981a). 

Analyses of the defect chemistry and thermodynamics of non-stoichiometric phases that are predominately ionic in nature (i.e. halides and oxides) are most often made using quasi-chemical reactions. The concentrations of the point defects are considered to be low, and defect-defect interactions as such are most often disregarded, although defect clusters often are incorporated. The resulting mass action equations give the relationship between the concentrations of point defects and partial pressure or chemical activity of the species involved in the defect reactions. 

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