Ion Exchange Equilibria

A binary ion exchange reaction between ion A (charge zA) and ion B (charge zB) may be written as:

Selectivity coefficient (selectivity quotient)

where Cs are the concentrations of the ions in the exchanger and Cs those in solution. Various concentration units are commonly used (molar, molal, equivalent fraction, etc.). When zA O zB, the numerical value of kA/B depends on the choice of the concentration units. The selectivity coefficient (kA/B) usually changes as a function of exchanger composition (kA/B _ f (CA); see Figure 1) and also as a function of the total concentration (or ionic strength) of the external solution, especially in concentrated solutions.

Corrected selectivity coefficient By making the activity (nonideality) correction for the solution phase, the so-called corrected selectivity coefficient is obtained:

k _ C A aB _ C A CB yB _k y^ [3] kA/B ^Za _ZB _ z^-ZA/^ZB TTZB kA/B TZB [3]

where superscript bars refer to the ions in a solid ion exchanger. Various equilibrium quantities are used to measure and estimate the efficiency of the ion exchanger for a given separation task. The most common of these include the following.

where aA, aB are the activities of ions A, B in the solution. This quantity is independent of the total concentration of the external solution by definition and thus reflects the pure exchanger-ion interactions contributing to the selectivity. In dilute solutions,

Figure 1 Selectivity coefficient A*Na/H (see eqn [2]) for Na #/H # exchange is sulfonated polystyrene/divinylbenzene (DVB) resins as a function Na # equivalent fraction in resin at different degrees of cross-linking (nominal DVB content). Circles, DVB 5.5%; squares, DVB 15%; triangles, DVB 25%. (Data from Helfferich FG, 1995.)

Figure 1 Selectivity coefficient A*Na/H (see eqn [2]) for Na #/H # exchange is sulfonated polystyrene/divinylbenzene (DVB) resins as a function Na # equivalent fraction in resin at different degrees of cross-linking (nominal DVB content). Circles, DVB 5.5%; squares, DVB 15%; triangles, DVB 25%. (Data from Helfferich FG, 1995.)

k a/b + &A/B- However coefficient usually composition (CA).

the corrected selectivity varies with the exchanger tion than B (CA«CB, CA«CB), kA/B and CB are essentially constant (CB + Q, the ion exchange capacity) and:

Thermodynamic equilibrium constant The thermodynamic equilibrium constant KA/B:

/rZB/rZA

Km~ aZa,Zb a b a a c ZbcZa yZ^ ?ZA C ZACZB y ZB yZB

X ABCZA yZZA X ZACAB yZB

ZB ZB

can be obtained by integrating the corrected selectivity coefficient as a function of exchanger composition. According to Argesinger et al. and Hogfeldt et al.:

where EA is the equivalent fraction of A in the exchanger (Ea = ZaCa/(zaCa + ZbCb)) and KH is the corrected selectivity coefficient written with mole fractions (X) as concentration units for the ions in the exchanger:

Under these circumstances, kd depends only on the concentration of ion B and, on a logarithmic scale, the slope of kd equals — zA/zB, the ratio of cation charges.

Experimentally determined graphs of log kd vs log CB (eqn [9]) are frequently used in research to study sorption mechanisms, the charges of the exchanging species and in the estimation of exchanger performance, e.g. in water purification (estimation of processing capacity) and in ion chromatography (estimation of retention volume). Great care should be taken, however, in the interpretation of the data and in making sure that the assumptions leading to eqn [9] are valid. Because of the widespread use of distribution coefficients in ion exchange, it is useful to emphasize this point by taking a binary uni-univalent exchange (zA = zB = 1) as an example here. For this equilibrium, eqn [9] can be further manipulated to give:

Gaines and Thomas supplemented the abstract thermodynamic treatment to include the contributions of salt imbibition and water activity changes, which need to be considered when ions are exchanged in concentrated solutions.

Distribution coefficient (distribution constant, distribution ratio) Various distribution constants and coefficients are used to measure the ion exchange equilibria. In general, the distribution coefficient kd of ion A is defined as a concentration ratio in the exchanger and solution:

Za Zb

Cs/Na

This quantity is only a constant under special conditions. In general, kd depends on the ionic composition of the exchanger and the solution. For a binary exchange, one obtains from eqns [2] and [7] that:

Under the special condition that A is present in the solution and in exchanger at much lower concentra-

This equation now shows that, in fact, the condition for linear dependence of kd on Ca (log kd = log (kA/BQ) — log Cb) is that CA«CB/kA/B. Thus, even if CA«CB, the dependence may not be linear if the selectivity coefficient is very large. This feature of kd is shown as calculated examples in Figure 2. It can be seen that, if the selectivity coefficient is low, kd falls linearly with the concentration of the macro-ion B, on a logarithmic scale with a slope of — 1, as eqn [9] implies. However, when the selectivity increases, the kd starts to level off at lower concentrations of B and ultimately becomes independent of CB when the selectivity is very high. In the studies of highly selective exchangers (zeolites and some other inorganic materials, chelating resins) such independence of kd on the macro-ion concentration is often observed and every now and then the incorrect conclusion is made that the uptake of trace ions is not ion exchange but some sort of surface adsorption reaction. Figure 2 also shows an interesting feature of the link between kd and selectivity: in dilute solutions the kds tend to a common value, which is determined by the ratio i

Figure 2 Calculated values (eqn [10]) for the distribution coefficient kd for trace ion A (CA = 10 6 mol L 1) in binary univalent A #/B # exchange as a function of macro-ion concentration Cb at different values of the selectivity coefficient Aa/b. Dotted line, Aa/b = 5 000 000; dashed line, Aa/b _ 50 000; continuous line, Aa/b — 500. Ion exchange capacity of the exchanger 2.0 mmol g 1.

Figure 2 Calculated values (eqn [10]) for the distribution coefficient kd for trace ion A (CA = 10 6 mol L 1) in binary univalent A #/B # exchange as a function of macro-ion concentration Cb at different values of the selectivity coefficient Aa/b. Dotted line, Aa/b = 5 000 000; dashed line, Aa/b _ 50 000; continuous line, Aa/b — 500. Ion exchange capacity of the exchanger 2.0 mmol g 1.

of Q/Ca. The value of selectivity thus becomes unimportant in dilute solutions.

In general, eqn [9] is valid for several parallel trace ion exchange reactions (A, C, D ...) in the presence of one common macro-ion B, since the ions that are present at trace level will have a negligible effect on other trace ion equilibria.

Separation factor Separation factor is usually used in ion exchange chromatography to estimate the separability of two trace ions. Considering the separation of two trace ions A and C using macro ions B as an eluent, one obtains for the separation factor (aA/C):

b in Figure 3). If the isotherm lies on the diagonal of the presentation (EA = EA), the exchanger has no preference for either ion A or B (curve c) and a bending of the isotherm towards the EA-axis indicates that the exchanger is nonselective. The magnitude of the selectivity coefficient cannot be always deduced from the isotherm because when zA O zB, the shape of the isotherm depends strongly on the total ion concentration (CT) in the solution. This behaviour arises from the difference between the cation charges, which can be clearly seen if the equation for the selectivity coefficient is expressed in terms of equivalent fractions and rearranged:

1 Za

In the case that zA O zC, the separation factor increases as the concentration of B is decreased.

Ion exchange isotherm An ion exchange isotherm is a function that represents the ionic composition of the exchanger (EA) as a function of the ionic composition of the solution (EA), or vice versa, at constant temperature (Figure 3). Traditionally, the selectivity of the exchanger is estimated from the isotherm. If the isotherm is concave towards the axis representing the ion concentration in the exchanger, the ion exchanger is considered to be selective for that ion (curves a and

At a given point EA on the isotherm (the left-hand side of eqn [12] constant), the ratio EA/EB must decrease as CT is decreased when zA > zB. Thus, the relative concentration of ion A must decrease with decreasing CT. This feature, the increased preference of an ion exchanger for the ion having a higher charge with the dilution of the solution, is called electroselec-tivity. It should be also noted that the ion exchanger may prefer ion A strongly even though the value of the selectivity coefficient is equal to or less than unity (see calculated examples in Figure 3).

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