Activity coefficient mass balance

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. 

This set of coupled differential equations can—as also expressed for the activated sludge model concept—be formulated in terms of a matrix. This matrix includes the relationships between the relevant components, processes, expressions, process rates and coefficients (Table 5.3). The mass balances shown in Equations (5.6) to (5.9) can be identified as columns in the matrix. 

HS, S, HCCU, CO3, RR NH, RR NCOO", H+, OH- and H2O. Hence there are twenty-three unknowns (m and Yj for all species except water plus x ). To solve for trie unknowns there are twenty-three independent equations Seven chemical equilibria, three mass balances, electroneutrality, the use of Equation (6) for the eleven activity coefficients and the phase equilibrium for xw. The problem is one of solving a system of nonlinear algebraic equations. Brown s method (21, 22) was used for this purpose. It is an efficient procedure, based on a partial pivoting technique, and is analogous to Gaussian elimination in linear systems of equations. 

Excluding activity coefficients, three relationships are required in addition to the nine thermodynamic equilibria in order to calculate concentrations of the 12 unknown species. These relationships are the mass balances for magnesium and chloride, and the electroneutrality equation. 

An important component of equilibrium calculations is the conversion between ion activities, which equilibrium constants refer to, and ion concentrations, which mass balance and electrical neutrality equations refer to. The conversion is made with activity coefficients defined by the relation ... 

Activity coefficients do not appear in the mass balance. The concentration of each species counts exactly the number of atoms of that species. 

Why do activity coefficients not appear in the charge and mass balance equations ... 

For infinitely dilute solutions activity coefficients approach unity so the activity and the concentration of an ion will be equal. For calculations involving more concentrated solutions corrections must be made using activity coefficients, especially when relating the calculated concentration of species to an imposed mass (mole) balance constraint. The activity coefficients can be calculated from a number of ion activity theories and the relevant equations for some of the commonly used ones are shown below. 

These parameters can be directly related back to the information contained in the EPM with n components (j) and m species (t). Application of the mass balance constraint equation requires that the concentration of each species must be known. Therefore, activity coefficients are computed if the ionic strength is already known from either the Davis or the extended Debye-Eluckel equation however, if ionic strength is unknown and has to be calculated, equation (5.134) can be converted to a general expression for the concentration of each species by substituting the expression for S to give... 

Table A.2 is model output for seawater freezing at 253.15 K. Beneath the title, the output includes temperature, ionic strength, density of the solution (p), osmotic coefficient amount of unfrozen water, amount of ice, and pressure on the system. Beneath this line are the solution and gaseous species in the system. The seven columns include species identification, initial concentration, final (equilibrium) concentration, activity coefficient, activity, moles in the solution phase, and mass balance. The mass balance column only contains those components for which a mass balance is maintained. The number of these components minus 1 is generally the number of independent components in the system (in this case, 8 — 1 = 7). The mass balances (col. 7) should equal the initial concentrations (col. 2). This mass balance comparison is a good check on the computational accuracy.

After writing mass balances, energy balances, and equilibrium relations, we need system property data to complete the formulation of the problem. Here, we divide the system property data into thermodynamic, transport, transfer, reaction properties, and economic data. Examples of thermodynamic properties are heat capacity, vapor pressure, and latent heat of vaporization. Transport properties include viscosity, thermal conductivity, and difiusivity. Corresponding to transport properties are the transfer coefficients, which are friction factor and heat and mass transfer coefficients. Chemical reaction properties are the reaction rate constant and activation energy. Finally, economic data are equipment costs, utility costs, inflation index, and other data, which were discussed in Chapter 2. 

Values of K, the thermodynamic association constants are given at 25°. The concentrations of ionic species in the solutions at any time can be determined from mass balance, electroneutrality, and the appropriate equilibrium constants as described previously (19, p, 85-92) by successive approximations for the ionic strength. The activity coefficients of Z-valent ionic species may be calculated from an extended form of the Debye-Huckel equation such as that proposed by Davies (20, p. 34-53). 

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... 

The flash problem is of course still subject to the mass balance condition in Eq. (35). The bubble point pressure (a = 0) is , and this is relatively easy to calculate because the activity coefficients t(x) are those at the feed composition X Ij ) and they are at worst very weak functions of pressure. The situation is different for the dew point pressure (a = 1), which is 1/. 

The TLM (Davis and Leckie, 1978) is the most complex model described in Figure 4. It is an example of an SCM. These models describe sorption within a framework similar to that used to describe reactions between metals and ligands in solutions (Kentef fll., 1988 Davis and Kent, 1990 Stumm, 1992). Reactions involving surface sites and solution species are postulated based on experimental data and theoretical principles. Mass balance, charge balance, and mass action laws are used to predict sorption as a function of solution chemistry. Different SCMs incorporate different assumptions about the nature of the solid - solution interface. These include the number of distinct surface planes where cations and anions can attach (double layer versus triple layer) and the relations between surface charge, electrical capacitance, and activity coefficients of surface species. 

With the help of operational stability constants valid for seawater (Table 6.7), a direct computation of the major inorganic species (without using individual activity coefficients) was carried out. The calculations consist essentially in solving 20 equations with 20 unknowns simultaneously. Twelve stability constants are available and nine mass balance relations (of the type... 

Depending on the nature of the class, the instructor may wish to spend more time with the basics, such as the mass balance concept, chemical equilibria, and simple transport scenarios more advanced material, such as transient well dynamics, superposition, temperature dependencies, activity coefficients, redox energetics, and Monod kinetics, can be skipped. Similarly, by omitting Chapter 4, an instructor can use the text for a water-only course. In the case of a more advanced class, the instructor is encouraged to expand on the material suggested additions include more rigorous derivation of the transport equations, discussions of chemical reaction mechanisms, introduction of quantitative models for atmospheric chemical transformations, use of computer software for more complex groundwater transport simulations, and inclusion of case studies and additional exercises. References are provided... 

Where t is time, z are the axial position in the column, qt is the concentration of solute i in the stationary phase in equilibrium with Cu the mobile phase concentration of solute /, u is the mobile phase velocity, Da is the apparent dispersion coefficient, and F is the phase ratio (Vs/Vm). The equation describes that the difference between the amounts of component / that enters a slice of the column and the amount of the same component that leaves it is equal to the amount accumulated in the slice. The fist two terms on the left-hand side of Eq. 10 are the accumulation terms in the mobile and stationary phase, respectively [109], The third term is the convective term and the term on the right-hand side of Eq. 10 is the diffusion term. For a multi component system there are as many mass balance equation, as there are active components in the system [13],... 

It turns out, however, that we will not need this or any other mass-balance equation to solve our rain pH problem. For rain, we know the pH is generally below about 6, so that mOH and mCO " are negligible relative to mH" and mHCOj. As a rule of thumb, any species whose concentration is less than 1% (two orders of magnitude) of another species concentration present is negligible in a charge-balance or mass-balance calculation. (How do we know these two species are negligible at this pH ) With this simplification, and ignoring activity coefficients in such a dilute water, we find expression (8.8) reduces to simply mH = mHCOj. 

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