Mass action equations equilibrium calculations

Place the final concentrations into the mass action equation and calculate the equilibrium constant ... 

The mass-action equations have been written in the same form as those given by Marynowski et al. (6) so that the equilibrium constants can be used directly. (Should more accurate data become available, the equilibrium yields calculated here will require revision.) The fourth equation, which applies to the heterogeneous equilibrium between carbon and nitrogen, is included for completeness but is unnecessary for the general solution. It can be shown that when the total pressure of the system is F, the partial pressure of cyanogen radicals is given by the equation ... 

Mass action equations. The first step in any calculation is to collate the mass action expressions that define the formation of the species. The way in which the formation constants can be used can be illustrated by considering a metal such as aluminium in an aqueous medium. Aluminium ions can undergo a number of hyrolysis reactions in water to form several hydroxy-metal complexes. The reactions can be written as the overall hydrolysis reactions and their associated equilibrium formation constants are shown below. 

It should be emphasized that an equilibrium can be established between a metastable phase(s) and its environment. Aragonite, for example, can be precipitated from seawater at 25°C, but it is unstable at Earth-surface T and P, and can persist metastably because of kinetic reasons. This statement is illustrated by the following calculation. We can use the free energies of formation of Table 6.1, and calculate the Gibbs free energy of reaction for the mass action equation representing aragonite-calcite equilibrium ... 

Methods for measurement of parameters used in SCM s have been described in the literature. Only a brief summary is presented here. Surface complexation model parameters that can be measured directly include, (1) the solid concentration, (2) surface site density, (3) surface area, and (4) equilibrium constants for the mass action equations describing all relevant adsorption reactions. The relation between surface charge and potential is calculated in geochemical equilibrium models. 

Numerical Methods and Data Structure. Both EQ3NR and EQ6 make extensive use of a combined method, using a "continued fraction" based "optimizer" algorithm, followed by the Newton-Raphson method, to make equilibrium calculations. The method uses a set of master or "basis" species to reduce the number of iteration variables. Mass action equations for the non-basis species are substituted into mass balance equations, each of which corresponds to a basis species. 

Since this equation has the same form as the mass action equation, the reaction is in equilibrium if Hi equals the reaction s equilibrium constant Kt. In this case, the fluid is saturated with respect to At. As a test of our calculations, for example, we would expect the fluid to be saturated with respect to any mineral in the equilibrium system. 

Of course, the ion exchange and surface complexation reactions can be easily included in this step of the model in a similar manner to the precipitation reactions. Likewise, the concentration of electron as a master species (pe value) is needed to calculate the concentration of the aqueous species involved in a redox couple. The mass action equation for each of these redox couples is combined to remove the concentration of the electron from the equations, as if a set of redox couple was in a redox equilibrium. 

Strategy The concentrations given are equilibrium concentrations. They have units of mol/L, so we can calculate the equilibrium constant K using the law of mass action [Equation (14.2)]. 

Secondary Ion Yields. The most successful calculations of secondary in yields are based on the local thermal equilibrium (LTE) model of Andersen and Hinthorne (1973), which assumes that a plasma in thermodynamic equilibrium is generated locally in the solid by ion bombardment. Assuming equilibrium, the law of mass action can be applied to find the ratio of ions, neutrals and electrons, and the Saha-Eggert equation is derived ... 

Fortunately, few of these variables are truly independent. Geochemists have developed a variety of numerical schemes to solve for equilibrium in multicomponent systems, each of which features a reduction in the number of independent variables carried through the calculation. The schemes are alike in that each solves sets of mass action and mass balance equations. They vary, however, in their choices of thermodynamic components and independent variables, and how effectively the number of independent variables has been reduced. 

The law of mass action, the laws of kinetics, and the laws ol distillation all operate simultaneously in a process of this type. Esterification can occur only when the concentrations of the acid and alcohol are in excess of equilibrium values otherwise, hydrolysis must occur The equations governing the rate of the reaction and the variation of the rale constant (as a function of such variables as temperature, catalyst strength, and proponion of reactants) describe Ihe kinetics of the liquid-phase reaction. The usual distillation laws must he modified, since must esterifications arc somewhat exothermic and reaction is occurring on each plate. Since these kinetic considerations are superimposed on distillation operations, each plate must be treated separately by successive calculations after Ihe extent of conversion has been determined. See also Distillation. 

The last equation, one of the most important physicochemical equations, expresses exactly the law of mass action, formulated for the first time by Guldberg and Waage in a less exact form. The equation enables the calculation of the equilibrium composition of a reaction mixture or determination of theoretically possible yields of chemical processes starting from the known value of the equilibrium constant K which can be determined by thermodynamic methods. 

The advantage of the application of the law of mass action is that the equilibrium exchange constant allows the calculation of other thermodynamic parameters, namely, the Gibbs free energy (Equation 1.84) ... 

WATEQ2 consists of a main program and 12 subroutines and is patterned similarly to WATEQF ( ). WATEQ2 (the main program) uses input data to set the bounds of all major arrays and calls most of the other procedures. INTABLE reads the thermodynamic data base and prints the thermodynamic data and other pertinent information, such as analytical expressions for effect of temperature on selected equilibrium constants. PREP reads the analytical data, converts concentrations to the required units, calculates temperature-dependent coefficients for the Debye-HKckel equation, and tests for charge balance of the input data. SET initializes values of individual species for the iterative mass action-mass balance calculations, and calculates the equilibrium constants as a function of the input temperature. MAJ EL calculates the activity coefficients and, on the first iteration only, does a partial speciation of the major anions, and performs mass action-mass balance calculations on Li, Cs, Rb, Ba, Sr and the major cations. TR EL performs these calculations on the minor cations, Mn, Cu, Zn, Cd, Pb, Ni, Ag, and As. SUMS performs the anion mass... 

Substituting the appropriate ideal expression for the activity of gaseous or dissolved species from Equation 14.8a or 14.8b leads to the forms of the mass action law and the equilibrium constant K already derived earlier in Section 14.3 for reactions in ideal gases or in ideal solutions. We write the mass action law for reactions involving pure solids and liquids and multiple phases by substituting unity for the activity of pure liquids or solids and the appropriate ideal expression for the activity of each gaseous or dissolved species into Equation 14.9. Once a proper reference state and concentration units have been identified for each reactant and product, we use tabulated free energies based on these reference states to calculate the equilibrium constant. 

Sections 15.4 and 15.5 outline methods for calculating equilibria involving weak acids, bases, and buffer solutions. There we assume that the amount of hydronium ion (or hydroxide ion) resulting from the ionization of water can be neglected in comparison with that produced by the ionization of dissolved acids or bases. In this section, we replace that approximation by a treatment of acid-base equilibria that is exact, within the limits of the mass-action law. This approach leads to somewhat more complicated equations, but it serves several purposes. It has great practical importance in cases in which the previous approximations no longer hold, such as very weak acids or bases or very dilute solutions. It includes as special cases the various aspects of acid-base equilibrium considered earlier. Finally, it provides a foundation for treating amphoteric equilibrium later in this section. 

The principle of electroneutrality in aqueous chemical systems states that the sum of the concentrations of all positively charged ions (expressed in equivalents) equals the sum of the concentrations of all negatively charged ions, so that the overall charge of the solution is zero. (If this were not true, we would be constantly bombarded with electrical shocks ) When an equation based on the principle of electroneutrality is combined with equations provided by conservation of mass, and by the mass action law, Eq. [1-12], the equilibrium chemical composition of a system can be calculated. 

The rates of production/consumption of ions have to be calculated in this step as independent phenomena from the previous step. According to the proposed hypotheses, these rates are high enough to consider that the local equilibrium is reached. In the present case, this means that concentration changes of the ions and species considered in this model can be calculated using the action mass equations of the involved chemical reactions. After the transport step, a set of preliminary values of concentrations, which are not at chemical equilibrium, is known for each volume element (C, ). The solution in each cell of the mass conservation equations, together... 

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