## Equilibrium Theories

Much theoretical work has been carried out to explain the nonideality of the ion exchange systems, e.g. for the calculation of ion exchange equilibria and to understand the factors that give rise to ion exchange selectivity. For ions in solution, sufficient theories exist to calculate the nonideality (activities) in the liquid phase. For the ions in the exchanger phase, no generally valid theories exist. For the calculation of ion exchange equilibria, it is always possible to measure the nonideality of the exchanger phase, e.g. the corrected selectivity coefficient kA/B as a function of exchanger ion composition at a given total solution concentration (CT) and then use the measured function for the calculation of equilibria at other Ct values from equation:

kA/B EBb " EAb yBBCT

E 2ZbZc cZaZcqZaZb ^ZaZc^ZaZb e ZaZce ZaZb c2ZbZc ^2ZbZc for the exchange reaction:

Parameter A can be determined from the measurements of the corrected selectivity coefficients of the binary equilibria i/j, k/i and k/j by curve fitting. Activity coefficients obtained by eqn [31] are then used in the binary equations of thermodynamic equilibrium constant (eqn [4]) for the calculation of ion exchange equilibria. This method has given accurate results even in four-component systems (e.g. Na/K/Ca/Mg in a strong acid cation resin). Several other related approaches have been developed.

The theories above are based on the measurements of nonideality and make no assumptions about the interactions that give rise to the selectivity. Thus, they do not allow the calculation of KA/B, k'A/B or j from the fundamental data or explain the changes of kA/B or j^. The first theory to explain the nonideality of the exchanger phase was developed by Kielland (the graphical presentations of Figure 1 are often called Kielland plots), who considered van der Waals-type interactions and showed that for the solid-phase activity coefficients:

In general, k'A/B and yA/yB are not known simultaneously, so iteration must be used to solve eqn [28]. The same approach can be extended to systems containing more than two counterions, e.g. for the ternary system the corrected selectivity coefficients kA/BC, kB/CA and kC/AB can be measured and used for the calculation of the equilibria. For instance, kA/BC is defined as:

2zbzca + zazcb + zazbc p 2zbzCa + zazcb # zazbc

kB/CA and kC/AB are defined accordingly. It is intrinsic to this method that it gives precise results provided that the selectivity coefficients are measured and described precisely within the exchanger composition range of interest. In practice this requires large number of measurements, which makes the method very laborious. Less effort is associated with ap-

Here, AR and BR denote the salt forms of the exchanger, R being the common anion. This theory predicts that the function kA/B = f (XA) is linear, which is in agreement with the observed behaviour in many cases. Quite often, the kA/B functions are not linear, but slightly, or even strongly, curved (see Figure 1). Some of these nonlinear functions can be explained by assuming that the exchanger has several types of exchange sites, each subsite having a characteristic selectivity coefficient k|. The measured overall kA/B decreases with XA as sites with higher selectivity are filled first. If these subsites behave ideally, sigmoidal curves are obtained for kA/B = f (XA). By assuming nonideal behaviour for the subsites, the kA/B functions exhibit a wide variety of different forms. This theory has been found to be consistent with the behaviour of several zeolite systems.

A related approach is to consider the different states that a given counterion may assume depending on the neighbouring counterions. In a polymer chain

(as in an organic ion exchange resin), considering the two nearest neighbours, each counterion can have three different energetic levels. As a consequence, in general k'A/B = f (XA) is a second-order polynomial function in XA, which is often the observed trend in organic resins (Figure 1). If two of the three energy levels are close to each other, the selectivity function is linear. Theories of this type are helpful in the calculation of ion exchange equilibria and in presenting the equilibria in a mathematical form, but they give no information about the magnitude of selectivity. In organic resins, various osmotic theories have been developed to estimate the relative magnitude of selectivity. The base in these theories is that:

where n is the osmotic pressure difference between the external solution and exchanger pore liquid and vA and vB are the partial molar volumes of A and B in the exchanger. The osmotic theory predicts the selectivity trend (I):

VIII. Na > K > Rb > Li > Cs IX. Na > K > Li > Rb > Cs

Most of these sequences have been observed in ion exchangers and they can be predicted from Eisenman theory, originally developed for selective glass electrodes. The theory considers cation exchange site and cation water (hydration) interaction energies. The free energy of exchange is obtained from:

where Fel is the coulombic interaction energy between cation and the anionic exchange site and Fhyd is the hydration energy of the cation. The coulombic interaction energy for a univalent cation can be calculated for widely separated sites from:

Li and for closely spaced sites:

observed in strong-acid cation resins, i.e. ions with smaller hydrated radius (smaller partial molar volume) are preferred, because replacing larger ions with smaller ones will reduce the swelling pressure. The same selectivity trend can also be predicted from purely electrostatic calculations. The dielectric theory:

where sZ and sS are the macro-permittivities of the exchanger and solution phases, respectively, predicts that in uni-univalent exchange, the selectivity decreases as the framework charge density increases for selective exchange (KA/B >1). This trend is commonly observed for zeolite ion exchange.

The selectivity sequence I for alkali metal ions, shown above, is common in organic resins have a low degree of cross-linking and in zeolites with low framework charge density. Other selectivity sequences appear as the degree of cross-linking or framework charge density increases:

VI. K > Na > Rb > Cs > Li where r + is the cation radius and r~ is the radius of the anionic exchange site. The anionic field strength decreases as r~ increases. Selectivity pattern I is exhibited by exchangers having a low field strength and cations are exchanged in the hydrated state with a preference for a smaller hydrated radius. As the field strength is increased, the less hydrated cations become desolvated and the selectivity patterns start to change. At high field strength, pattern XI is exhibited and cations are exchanged as bare cations.

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