Activities and Mass-Action Equations

EQUATIONS FOR SPECIATION AND FORWARD MODELING

Activities and Mass-Action Equations

In this section the activities of aqueous, exchange, and surface species are defined and the mass-action relations for each species are presented. Equations are derived from the mass-action expression for the number of moles of each species in the chemical system in terms of the master variables. These equations are then differentiated with respect to the master variables. Later, these equations for the number of moles of a species and the partial derivatives will be substituted into the constituent mole-balance, charge-balance, and phase-equilibria functions.

Mass-Action and Activity-Coefficient Equations for Aqueous Species

PHREEQC allows speciation or equilibration with respect to a single aqueous phase. However, multiple aqueous phases may be defined in the course of a run and an aqueous phase may be defined as a mixture of one or more aqueous phases (see MIX keyword in data input section). The dissolved species in the aqueous phase are assumed to be in thermodynamic equilibrium, except in initial solution calculations, when equilibrium may be restricted to obtain only among the species of each element valence state. The unknowns for each aqueous species are the activity, a_i, activity coefficient,

, molality, m_i, and number of moles in solution, n_i, of each aqueous species, i. The following relationships apply to all aqueous species (except aqueous electrons and water itself):

and

, where

is the mass of water in the aqueous phase.

PHREEQC rewrites all chemical equations in terms of master species. There is one master aqueous species associated with each element (for example, Ca⁺² for calcium) or element valence state (for example, Fe⁺³ for ferric iron) plus the activity of the hydrogen ion, the activity of the aqueous electron, and the activity of water. For PHREEQC, the identity of each aqueous master species is defined with SOLUTION_MASTER_SPECIES keyword data block. (See Description of Data Input.) The numerical method reduces the number of unknowns to be a minimum number of master unknowns, and iteratively refines the values of these master unknowns until a solution to the set of algebraic equations is found. The master unknowns for aqueous solutions are the natural log of the activities of master species, the natural log of the activity of water, , the ionic strength, , and the mass of solvent water in an aqueous solution, W_aq.

Equilibrium among aqueous species in an ion-association model requires that all mass-action equations for aqueous species are satisfied. For example, the association reaction for the aqueous species is . The log K for this reaction at 25^oC is 2.3, which results in the following mass-action equation:

(1) .

In general, mass-action equations can be written as follows:

(2) ,

where c_m,i is the stoichiometric coefficient of master species m in species i. The values of c_m,i may be positive or negative. For PHREEQC, terms on the right-hand side of an association reaction are assigned negative coefficients and terms on the left-hand side are assigned positive coefficients. K_i is an equilibrium constant that is dependent on temperature, and m ranges over all master species. The same formalism applies to master species, where the mass-action equation is simply .

For aqueous species the equation, derived from the mass-action expression, for the total number of moles of species i is

(3) .

The Newton-Raphson method uses the total derivative of the number of moles with respect to the master unknowns. The total derivative is

(4) .

Activity coefficients of aqueous species are defined with the following equations:

(5) ,

which is referred to as the Davies equation, or

(6) ,

which is referred to as either the extended Debye-Hückel equation, if b_i is zero, or the WATEQ Debye-Hückel equation (see Truesdell and Jones, 1974), if b_i is not equal to zero. A and B are constants dependent only on temperature, is the ion-size parameter in the extended Debye-Hückel equation, and b_i are ion-specific parameters fitted from mean-salt activity-coefficient data in the WATEQ Debye-Hückel equation, and z_i is the ionic charge of aqueous species i. Unless otherwise specified in the database file or the input data set, the Davies equation is used for charged species. For uncharged species, the first term of the activity coefficient equation is zero, and unless otherwise specified b_i is assumed to be 0.1 for all uncharged species.

The partial derivatives of these activity coefficient equations with respect to ionic strength are

(7) ,

for the Davies equation and

(8) ,

for the extended or WATEQ Debye-Hückel equation.

For data input to PHREEQC, the chemical equation for the mole-balance and mass-action expression, the log K and its temperature dependence, and the activity coefficient parameters for each aqueous species are defined through the SOLUTION_SPECIES keyword data block. Master species for elements and element valence states are defined with the SOLUTION_MASTER_SPECIES keyword data block. Composition of a solution is defined with the SOLUTION keyword data block. (See Description of Data Input.)

Mass-Action Equations for Exchange Species

Ion-exchange equilibria are included in the model through additional, heterogeneous mass-action equations. PHREEQC allows multiple exchangers, termed an "exchange assemblage", to exist in equilibrium with the aqueous phase. The approach uses mass-action expressions based on half-reactions between aqueous species and a fictive unoccupied exchange site (Appelo and Postma, 1993) for each exchanger. This unoccupied exchange site is the master species for the exchanger and the log of its activity is an additional master unknown. Its identity is defined with EXCHANGE_MASTER_SPECIES keyword data block. (See Description of Data Input.) However, the master species is not included in the mole-balance equation for the exchanger, forcing its physical concentration to be zero. Its activity is also physically meaningless, but is such that all of the exchange sites are filled by other exchange species.

The unknowns for exchange calculations are the activity, , which is defined to be the equivalent fraction in PHREEQC, and the number of moles, , of each exchange species, , of exchanger e. The equivalent fraction is the number of moles of sites occupied by an exchange species divided by the total number of exchange sites. The activity of an exchange species is defined as follows: , where is the number of equivalents of exchanger, e, occupied by the exchange species, and is the total number of exchange sites for the exchanger, in equivalents. Note that is the total number of equivalents of the exchanger in the system which is not necessarily equal to the number of equivalents per kilogram of water (eq/kg H₂O), because the mass of water in the system may be more or less than 1 kg.

Equilibrium among aqueous and exchange species requires that all mass-action equations for the exchange species are satisfied. The association reaction for the exchange species is , where is the exchange master species for the default database. The use of equivalent fractions for activities and this form for the chemical reaction is known as the Gaines-Thomas convention (Gaines and Thomas, 1953) and is the convention used in the default database for PHREEQC. [It is also possible to use the Gapon convention in PHREEQC, which uses equivalent fraction, but writes the exchange reaction as . See Appelo and Postma (1993) for more discussion.] The log K for calcium exchange in the default database file is 0.8, which results in the following mass-action equation:

(9) .

In general, mass-action equations can be written as follows:

(10) ,

where m varies over all master species, including exchange master species, is the stoichiometric coefficient of master species, m, in the association half reaction for exchange species i_e. The values of may be positive or negative. For PHREEQC, terms on the right-hand side of an association reaction are assigned negative coefficients and terms on the left-hand side are assigned positive coefficients. , is a half-reaction selectivity constant.

For an exchange species, the equation for the total number of moles of species i_e is

(11) .

The natural log of the activity of the master species of the exchanger is an additional master unknown in the numerical method. The total derivative of the number of moles of species i_e with respect to the master unknowns is

(12) .

For data input to PHREEQC, the chemical equation for the mole-balance and mass-action expression and the log K and its temperature dependence for each exchange species are defined through the EXCHANGE_SPECIES keyword data block. Exchange master species are defined with the EXCHANGE_MASTER_SPECIES keyword data block. Number of exchange sites and exchanger composition are defined with the EXCHANGE keyword data block. (See Description of Data Input.)

Mass-Action Equations for Surface Species

Surface-complexation processes are included in the model through additional, heterogeneous mass-action equations, and charge-potential relations. PHREEQC allows multiple surface complexers, termed a "surface assemblage", to exist in equilibrium with the aqueous phase. Two formulations of the mass-action equations for surface species are available in PHREEQC: (1) including an electrostatic potential term and (2) excluding any potential term. The two principle differences between the formulation of exchange reactions and surface reactions are that exchange reactions are formulated as half reactions, which causes the master species not to appear in any mole-balance equations, and the exchange species are expected to be neutral. Surface reactions are not half-reactions, so the master species is a physically real species and appears in mole-balance equations, and surface species may be anionic, cationic, or neutral. If the Dzombak and Morel (1990) model, which includes an electrostatic effects, is used, additional equations and mass-action terms are included because of surface charge and surface electrostatic potential.

The basic theory for surface-complexation reactions including electrostatic potentials is presented in Dzombak and Morel (1990). The theory assumes that the number of active sites, T_s (equivalents, eq), the specific area, A_s (meters squared per gram, m²/g), and the mass, S_s (g), of the surface are known. The activity of a surface species is assumed to be equal to its molality (moles of surface species per kilogram of water, even though surface species are conceptually in the solid phase). The two additional master unknowns are (1) the quantity, , where F is the Faraday constant, is the potential at surface s, R is the gas constant, and T is temperature in Kelvin and (2) the natural log of the activity of the master surface species. The identity of the master surface species is defined with SURFACE_MASTER_SPECIES keyword data block. (See Description of Data Input.) Note that the quantity is defined with a 2 in the denominator of the term on the right hand side. This is a different master unknown than that used in Dzombak and Morel (1990), but produces the same results as their model because all equations are written to be consistent with this master unknown.

If "HfoOH" is used to represent a neutral surface-complexation site ("Hfo", Hydrous ferric oxide, is used in the default database files), the association reaction for the formation of a negatively charged site (it is an association reaction in the sense that the defined species is on the right hand side of the equation) can be written as follows:

(13) ,

and the mass-action expression including the electrostatic potential term is

(14) ,

where is the intrinsic equilibrium constant for the reaction, is a factor that accounts for the work involved in moving a charged species (H⁺) away from a charged surface. In general, the equation for surface species i_s is

(15) ,

where i_s is the i^th surface species for surface s, m varies over all master species, including surface master species, is the stoichiometric coefficient of master species, m, in the association half reaction for surface species i_s. The values of may be positive or negative. For PHREEQC, terms on the right-hand side of an association reaction are assigned negative coefficients and terms on the left-hand side are assigned positive coefficients. , is the intrinsic equilibrium constant, and is the net change in surface charge due to the formation of the surface species.

For a surface species, the equation for the total number of moles of species i_s is

(16) .

The total derivative of the number of moles of species i_s with respect to the master unknowns is

(17) .

The second formulation of mass-action equations for surface species excludes the electrostatic potential term in the mass-action expression (-no_edl identifier in the SURFACE keyword data block). The equation for the number of moles of a surface species is the same as equation 16, except the factor involving does not appear. Likewise, the total derivative of the number of moles is the same as equation 17, except the final term is absent.

For data input to PHREEQC, the chemical equation for the mole-balance and mass-action expression and the log K and its temperature dependence of surface species are defined through the SURFACE_SPECIES keyword data block. Surface master species or types of surface sites are defined with the SURFACE_MASTER_SPECIES keyword data block. The number of sites, the composition of the surface, the specific surface area, and the mass of the surface are defined with the SURFACE keyword data block. (See Description of Data Input.)

Mass-Action and Activity-Coefficient Equations for Aqueous Species
Mass-Action Equations for Exchange Species
Mass-Action Equations for Surface Species

User's Guide to PHREEQC - 07 MAY 96

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