## General Sorption Equilibrium Relationships for an Amphoteric Surface

The derivation of equilibrium relationships makes use of the following assumptions:

1. The functional groups are uniformly distributed across a plane surface.

2. Activity coefficients in the exchanger phase are assumed to be 1.

3. Any ion exchange develops as the replacement of one surface complex by a new one. As a consequence, an oxide valence is defined that is equal to the smallest common multiple of the valences of the counterions.

For the derivation of the equilibrium relationships a simple system with CP and Na + ions is considered. The protolytic reactions at the surface are considered as local equilibria that can be described by the mass action law. The formation of the two surface complexes can be described by the respective formation constants, K (see Table 1 for explanation of symbols used).

Table 1

Protonation

where c is the concentration of the species in parentheses in mol L_1.

Dissociation/neutralization

S-OH

Symbol

Ci)x

ST V

Unit

Definition m2 g"1 mol L"1 mol L"1

mol L"1

Fm"2

F m"2

Asm"2 V

Specific surface area Concentration of species i Concentration of species i in the Stern layer

Concentration of species iat position x in an electrical field Electric capacitance of capacitor formed by the layers of species iand j

Electric capacitance of capacitor formed by the surface and the layer of species i

Faraday constant

Equilibrium constant of surface reaction

Volume of liquid phase left-hand side expression of eqns [38] and [39]

Abbreviation, defined by eqn [22] Oxide loading with species i Maximum exchanger loading Generalized separation factor Gas constant Mass of sorbent

Abbreviation used to designate the surface

Temperature

Sign of charge

Dimensionless loading with species i

Probability of the presence of two adjacent ions, i at the surface Valence of species i Surface charge density of layer with ions i

Electrical potential at position x

The sorption of counterions leading to two further surface complexes is considered in an analogous way.

Sorption of anions

Sorption of cations

liquid phase by means of the Poisson-Boltzmann relationship:

Symbols with subscript's' designate concentrations in the respective Stern layer. These unknown quantities can be expressed in terms of the concentrations in the

Here c(i)x represents the concentration of species i at position x in an electric field, z(i) is the valence of species i, F is the Faraday constant, is the electrical potential at position x, R is the gas constant, and T is the temperature in Kelvin.

The uptake of anions develops as the simultaneous sorption of protons and chloride ions. Since we have a sequence of reactions, the equilibrium of this common sorption can be expressed by the product of the respective formation constants:

The uptake of cations, however, develops as the competitive sorption of protons and sodium ions. As a consequence, the equilibrium of the cation exchange is expressed by the ratio of the corresponding formation constants:

" c(S—OH2CI) Cl c(S—OH) • c(ClH) • c(H+ )

kh _c(S—OH) • c(Na + ) Na c(S—ONa) • c(H + )

of the surface with Cl" and Na + ions. After resolving the equations for the generalized separation factor one obtains:

The terms m(H, i) are given by the following equation:

qmax F

Multiplying by the volume of the liquid phase and dividing by the mass of sorbent, the concentrations of surface complexes are converted to exchanger loadings q(OH), q(Cl"), and q(Na + ) respectively; (q(OH)) denotes nonionized surface groups.

As a consequence, the first factor on the right-hand sides exclusively contains quantities that can be derived from experiments. Both expressions are designated as generalized separation factors, Q. After introducing dimensionless loadings with species i, y(i) " q(i)/qmax, where qmax is the maximum exchanger loading, the following expressions are obtained:

_q(OH) • c(Na + ) _y(OH) • c(Na + ) [19] QNa q(Na) • c(H+ ) y(Na) • c(H + ) [ ]

Similar relationships can be derived for any monovalent counterion.

The difference of electrical potentials in the exponential terms of eqns [16] and [17] needs consideration about the surface charge densities in the series of electric capacitors formed by the surface, Stern and diffuse layers. As has been shown in earlier publications, the unknown differences in the electrical potential can be expressed in terms of the loading where A0 is the specific surface area and C(S, i) denotes the capacitance of the capacitor formed by the surface and layer i. The derivation has been given in the literature.

For a system with an arbitrary number of monovalent counterions, the following relationship can be deduced from similar considerations:

i i = 2,3 2, n index of counterions j = 2,3 2 , i — 1 running index

Hydrogen ions are always taken as component '1'. The factors V(j) and V(i) are the signs of the charge of the ions having the values of — 1 for anions and + 1 for cations. The first summation comprises counter-ions from index '2' to 'i — 1' (closer to the surface than species'/'). The second summation considers the counterions with indices running from 'i' to 'n'. For a system with n counterions there are n— 1 equations [23]. By subsequent evaluation of data all constants log KH and m(H, i) can be derived from the multicom-ponent system.

During the exchange of one divalent counterion for two monovalent ions, one divalent ion replaces two adjacent monovalent species at the surface, where at least one divalent species of counterion is present, the sorbent has to be assumed to have divalent functional sites in the mathematical treatment (assumption 3). As a consequence, a surface complex consisting of two sites and two monovalent counterions is defined that is replaced by a surface complex which consists of two sites with one divalent counterion. For a system with H + , Na + , Cl~ and SO2~ the generalized

separation factors are then given by the following expressions:

c{S-0H2Cl, S-OH2Cl} c{S-0H, S-OH} • c(H + )2 • c(Cl)2

Here the subscript II refers to calculation under the assumption of divalent functional sites.

The probability of two adjacent monovalent counterions i at the surface is given by:

where £y( j) is the sum of dimensionless loadings of all monovalent counterions with valences + 1 and - 1.

By this means the above separation factors can be expressed as:

qhso4 =

y(S04) • {y(H)+y(Cl)+y(Na)} y(OH)2 • c(H + )2 • c(S02")

If divalent functional sites are assumed for systems with exclusively monovalent counterions, then the equilibrium parameters log K^ and mn(H, i) can be transformed to the parameters for monovalent functional sites according to:

systems with only monovalent counterions have to be converted corresponding to eqns [31] and [32].

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