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II. The Thermodynamics of Ion-Exchange Equilibria

II. The Thermodynamics of Ion-Exchange Equilibria

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Many workers have suggested empirical relationships similar to Eq. (3) in an

attempt to define an equilibrium constant, because such a constant would be

valuable to soil science for predicting the state of the equilibrium at different

concentrations. Some of the better known exaniples are those of Ken (1928),

Vanselow (1932), and Gapon (1933); all were well reviewed by Bolt (1967) and

by Sposito (1981a). In a series of papers, Sposito (1977), Sposito and Mattigod

(1979), Oster and Sposito (1980), and El-Prince and Sposito (1981) have shown

that these empirical “constants” can be derived from thermodynamic principles.

However, in practical tests none of them have been found to be truly constant

over the whole of the exchange process, although some, such as Gapon’s constant, KG, have proved to be very useful in practice. They are thus better

described as equilibrium or selectiviry coeficients.

The true thermodynamic equilibrium constant is exactly what is required.

Unfortunately, it cannot be obtained directly because, although the activities of

ions in solution can be measured, those of adsorbed cations cannot. Nevertheless, the latter activities can be approximated by relating them to experimentally

measurable quantities, and as Sposito (1981a,b) shows, all, of the empirical

constants can be derived from Eq. (3) by choosing suitable expressions for the


The two most important forms of the selectivity coefficient as regards the

thermodynamics of K+ exchange are those of Vanselow (1932) and Gaines and

Thomas (1953). Vanselow (1932) approximated adsorbed-ion activities by mole

fractions, N, and wrote for the reaction in Eq. (1)

K, =

Nf; [Au+Iv


The Vanselow selectivity coefficient, K,, equals K only if the mixture is

ideal, that is, if activities = mole fractions (Guggenheim, 1967). Ca2+-Mg2+

exchange and K -Rb exchange most nearly approximate this. When the

mixture is not ideal, activities must be related to mole fractions by activity

coefficients, J and thus, in the Vanselow convention,



K =


N r A [Bv








= -[A1



Gaines and Thomas (1953) also defined an adsorbed-ion activity coefficient,

g, but using equivalent fractions, E. Thus



(The question of the use of different conventions to define adsorbed-ion activity

coefficients is discussed in full in Section II,E). Gaines and Thomas’ selectivity

coefficient, K,, is thus

K, =

Eij[A” +Iv



and, by the Gaines and Thomas convention,

The Gaines and Thomas thermodynamic treatment of ion exchange, from

which Eqs. (8)-(10) are taken, stimulated much research into ion exchange in

both “pure” clays and soils and other exchange materials such as resins and

zeolites (see Section IV,A). Therefore, when explaining the derivation of equations used, many workers have referred to the “Gaines and Thomas Method” or

the “Gaines and Thomas Treatment” (e.g., Hutcheon, 1966; Deist and Talibudeen, 1967b; Laudelout et al., 1968a; Talibudeen, 1981). However, others have

often referred to Argersinger et al. (1950) or to the “Argersinger Thermodynamic Approach” when making reference to the source of thermodynamicequations used (e.g., Jensen, 1973a).

Argersinger et al. (1950) and Hogfeldt and co-workers (Ekedahl et al., 1950;

Hogfeldt et al., 1950) first derived (independently) a set of general thermodynamic equations for ion exchange. They were based on the Vanselow (1932)

convention of mole fractions. Gaines and Thomas (1953), although referring to

Argersinger et al. (1950) and Ekedahl et al. (1950), made their own, thermodynamically more rigorous derivation of a set of equations based on equivalent fractions. For homovalent exchange (exchange between ions of the same

valency), mole and equivalent fractions are equal and so the two approaches give

the same results. For heterovalent exchange (exchange between ions of unequal

valency), mole and equivalent fractions are not equal and thus neither are most of

the thermodynamic parameters derived by the two methods. Any paper reporting

thermodynamic data must therefore be carefully examined for the convention

used, and of course direct comparison between data derived from the two conventions may not be possible (but see Section II,E).

As stated previously, K, or K, do not equal K unless the mixture is ideal.

However, K can be calculated from K, or K, by integrating over the whole

exchange [i.e., EB = 0 to 1, as shown by Gaines and Thomas (1953)]. For an

exchange as in &. (l), this gives a complex equation following Gaines and

Thomas’ convention



The last term of the equation represents the change in water activity (in effect

the change in water content of the soil) in going from an A-(soil) to a B-(soil).

This term has been found in practice to be negligible (Gaines and Thomas, 1955;

Laudelout and Thomas, 1965). The third term on the right-hand side of Eq. (11)

is made zero by the choice of suitable standard states (Section II,B), or by

assuming that g L = gff, which is not generally true. So there remains the

simplified form of Eq. (1 1) most often used

In K = (u-v)



In K, dE,

In the Vanselow convention, this becomes

In K




We now have basic equations for obtaining a selectivity coefficient, Eq.( 5 ) or

(9), and a thermodynamic equilibrium constant, Eq. (12) or (13), from experimental exchange equilibrium data. Other parameters are estimated as outlined in

Section II,D,E, and F.

Exchange isotherms are often presented in therodynamic analyses of exchange

data. These are plots of the equivalent fraction of an adsorbed cation against that

of the same cation in solution (Fig. 1). Their application is discussed in Section

IV,A. Also, sometimes an ‘‘uncorrected selectivity coefficient” is used, called

Khor KL (e.g., van Blade1 and Laudelout, 1967). This is given, again following

the Gaines and Thomas convention, by

Equivaknt fraction of K+

in solution


FIG.1. The exchange isotherm. A graph of the equivalent fraction of an adsorbed cation versus

its equivalent fraction in solution. This example: K+-CaZ+ exchange on soil showing hysteresis.

C a + K,0,K -+ Ca. After Deist and Talibudeen (1967a).


22 1

where mA and mg are molarities. It thus represents a selectivity coefficient

uncorrected for activities in solution.



To understand what these are and why they are important one must look at the

definition of ion activity and of equilibrium itself. The condition for chemical

equilibrium in any system is that the chemical potentials (p)of each component

of the system are equal throughout the system. Thus in a cation-exchange reaction such as that given in Eiq. (l),


+ up[B"+] = up[B-(soil),] + vp[AU+1


But p represents an intrinsic chemical property that cannot be identified with a

universal scale (such as temperature), nor accorded a reference value of zero in

the absence of the substance to which it refers. It is thus necessary to adopt a

conventional reference or standard state for the substance at which p is zero

(Sposito, 1981b; Talibudeen, 1981). The chemical potential in its standard state

is written as po,and it can be shown (see Sposito, 1981b, Chapter 2) that

p = po + R T l n a


where R is the gas constant, T is the absolute temperature, and a is the activity.

Thus the activity of an ion is a measure of the deviation of the chemical potential

of that ion from its value in the standard state, and the activity of an ion in its

standard state is 1.

Therefore, before thermodynamic quantities for exchange equilibria can be

calculated, standard states must be defined for each phase; their choice affects

greatly the value of such quantities and their physical interpretation. The various

standard states adopted for exchanger and solution phases were discussed in full

by Sposito (1981b). A list of the more important ones, and the practical results of

their use, is given in Table I. The only ones commonly used are those suggested

by Gaines and Thomas (1953), with a slight modification for practical reasons.

In practice, the standard state for adsorbed cations is taken as being a homoionic

exchanger in equilibrium with a solution of the saturating cation at constant ionic

strength. The experimental results can be obtained at several ionic strengths and

extrapolated back to zero, the standard state specified by Gaines and Thomas, as

suggested by van Blade1 and Laudelout (1967) (but see Section 11,C). However,

it appears that the values of activities in exchange reactions on soils and clays

depend very little on concentration (Jensen, 1973a; Jensen and Babcock, 1973);

this is a fortunate result, as such an extrapolation is rarely made in practice.

Table I

Some of the Standard States Used in Calculating the Thermodynamic Parameters of Cation-Exchange Equilibria



Standard states

Adsorbed phase

Solution phase



Activity = mole fraction when the Activity = molarity as concentration Can calculatef,Kv, etc., but all depend on Argersinger er al. (1950)

latter = 1


ionic strength

Homoionic exchanger in equilibrium Activity = molarity as concentration AG' expresses relative affiiity of exchanger Gaines and Thomas (1953)

with an infinitely dilute solution


for cations

of the ion

Activity = mole fraction when the Activity = molarity as concentration AGO expresses relative affiinity of exchanger Babcock (1963)

latter = 0.5. Components nor in


for cations when mole fraction = 0.5






van Blade1 and Laudelout (1967) found hysteresis of exchange isotherms

during heterovalent exchange reactions involving the selectively adsorbed NH,

ion (almost identical in size and hydration to K +). Hysteresis means that forward

and reverse exchange isotherms are not the same, as in Fig. 1. They also found a

large variation in the uncorrected selectivity coefficient, Kf, with ionic strength I

and suggested that both were caused by clay aggregation at finite ionic strength.

They reasoned that such aggregation would not occur at the standard state ionic

strength of zero. Therefore, to avoid the problem of hysteresis and the need to

calculate activity coefficients of ions in solution (y), they plotted log Kf against

(2J)f (finding this empirically to be a linear relationship) and extrapolated to (2Z)l

= 0 where, by definition, y = 1 and thus Kf = K,. This supported earlier

theoretical work by Laudelout and Thomas (1965), who had derived an equation

predicting a linear relationship between In K, and solution concentration at any

one cation ratio.

However, Laudelout et al. (1972) found a maximum change in In K , of only

9% in going from 0.01 M to 0.2 M ,showing that much of the variation in Kf is

corrected for by calculating activity coefficients in solution. In addition, although isotherms for heterovalent exchange do often exhibit hysteresis, selectivity for the ion of higher valency, as shown by the exchange isotherm or Kf,

increases continuously as ionic strength decreases. Thus, as the ionic strength

approaches zero, isotherms become rectangular (i.e., become increasingly close

to the x and y axes of the graph) and Kf tends to infinity (Barrer and Klinowski,

1974). Thus the log Kf versus ( U ) d relationship cannot have a finite linear slope

over a large range of I, and any extrapblation to (2J)f = 0 which gives a finite

value of log Kf is incorrect.

It would seem much more sensible in experimental work, therefore, to calculate y values and use K, at a known ionic strength to determine ion selectivity. It

is also worth noting that Barrer and Klinowski (1974) presented a method for

calculating exchange isotherms (and therefore K, values) at any solution concentration when an isotherm has been experimentally measured. Thus with modem computing methods little effort is required to measure cation selectivity in a

soil over a whole range of soil solution concentrations.



Many publications have examined cation selectivity during the exchange process by using selectivity coefficients and have drawn important conclusions from

them (e.g., with respect to potassium, see Bolt et al., 1963; van Schouwenberg

and Schuffelen, 1963; Marques, 1968). Often, however, the overall selectivity

or preference of the soil for one of a pair of cations is required, perhaps for



comparison with other cation pairs (Section IV,A) or of soils (Section IV,B).

This could be achieved through the thermodynamicequilibrium constant, which

integrates selectivity over the whole exchange process, although it is usually

expressed by the free energy function. The standard Gibbs free energy of exchange, AGO, is calculated from the experimentally determined thermodynamic

equilibrium constant, K, using the relationship

AGO = -RT In K


It is the difference in free energy between the two homoionic forms of the soil or

clay at the chosen standard state.

It has been stated that AGO defines the difference in the strength of binding

between the soil and the two cations (Drake, 1964; Deist and Talibudeen,

1%7a), but this is incorrect. The free energy term is the sum of ion binding

strength, expressed by the standard enthalpy of exchange, AH",and the degree of

order of the system, expressed by the standard entropy of exchange, Af'. The

relationship between these three standard functions is given by the familiar Gibbs






As well as being directly measurable by calorimetry (Section III), enthalpies

can be calculated from measurementsof the thermodynamicequilibriumconstant

at two temperatures, T, and T2,using the Van? Hoff equation

ln(K2/K,) =

-AH(11T2 - l/Tl)


The standard entropy of exchange is then calculated from AGO and AH'values

using Eq. (18). The physical interpretation of the three parameters is discussed

fully in Section IV.




Absorbed-ion activity coefficients are central to the development of a set of

thermodynamic equations describing cation exchange (Section 11,A). Although

they cannot be measured experimentally, they can be calculated from the measured selectivity coefficient (for derivations, see Gaines and Thomas, 1953;

Sposito, 1981b). For the general exchange reaction described in Eq.(l), and for

Coefficients (g) defined by equivalent fractions according to the Gaines and

Thomas conventions,

vln g,


E,[ln K, - (u-v)] - I S l n K , dE,




For Ca2+ + K + exchange, these become

In g,

= EK(ln K,- 1) -

In K, dEK


2 In g, = (l-EK) (1-ln K,)




In K , dE,



These equations have been used in the majority of papers where adsorbed-ion

activity coefficients have been calculated (e.g., Hutcheon, 1966; Deist and Talibudeen, 1967a,b; Goulding and Talibudeen, 1980). The equations forf values,

derived according to Vanselow ’s convention using mole fractions instead of

equivalent fractions, are of necessity slightly different (see Sposito, 198lb) and

have been used only by Jensen (1973a).

Activity coefficients, by definition, correct the equivalent or mole fraction

terms for departure from ideality (Section 11,A). They thus reflect the change in

the status, or fugacity, of the ion held at exchange sites, and thus the heterogeneity in the exchange process, as is shown experimentally in Section IV,B,C,

and D. Adsorbed-ion activity coefficients used in soil and clay studies have

almost always been calculated according to Gaines and Thomas’ (1953) procedure, but Sposito and Mattigod (1979) and Sposito (1981a,b) have questioned

this. They state that the Gaines and Thomas-type adsorbed-ion activity coefficients are not true thermodynamic activity coefficients because they are defined

by equivalent fractions [Eq.@))I rather than by mole fractions as in Vanselow’s

convention [Eq.(7)]. This problem has been discussed in detail by Goulding

(1983). Briefly, although the absolute values of the two types of coefficients are

not the same (except at Ei= 1, where gi=& = 1 by definition), plots of g j versus

Ei are very similar to those of f i versus Ei, as shown in Fig. 2, and result in

similar conclusions as to cation behavior during an exchange reaction. Also, as

will be shown later (Section IV,B,2), the girelate to heterogeneity as shown by

calorimetrically measured enthalpies of exchange and thus have a sensible physical interpretation. Sposito and Mattigod (1979) gave expressions relating gi and

fi, and also K , and K,.

In this article, the symbolfwill be used for adsorbed-ion activity coefficients,

and all values referred to have been calculated according to the Gaines and

Thomas convention.















Fractional K saturation

FIG.2. Adsorbed ion activity coefficients, calculated according to Vanselow’s v) and Gaines

and Thomas’ (g) conventions, as a function of fractional K + saturation for (A) Ca2+ + K+

exchange on Hanvell series soil, U.K. (Deist and Talibudeen, 1967a); (B) A13+ + K+ exchange on

Palm Garden Soil, Tea Research Institute, Sri Lanka (Talibudeen, 1972). From Goulding (1983).


Excess functions form the “ultimate” calculation from exchange equilibrium

data and have rarely been used in soil and clay studies. They account for the

properties of the exchange complex in terms of the activity coefficients of both

adsorbed ions and were first introduced in studies of ion exchange in zeolites

(Barrer et al., 1963). As was found for adsorbed-ion activity coefficients (Section II,E), they describe exchange heterogeneity qualitatively in soils and clays

(Goulding, 1980). Excess free energies (AGE), enthalpies (AHE), and entropies

(AYE) are calculated at chosen cation saturations (EB)from the following



+ u(l -E,)RT


In fB



A H E values can also be calculated from the temperature coefficient off, and fB

(Talibudeen, 1971).

Excess functions were used to describe NH,+-Sr2+ exchange on mont-



morillonite by Laudelout et al. (1968b) and K -Ca2

on soils by Talibudeen (1971, 1972).



and K -AP








Incomplete exchange implies that in an exchange such as that described in Eq.

(l), entering cations (B) cannot replace all the adsorbed cations (A). The reaction

is thus not completely reversible. There are three cases in which this can occur:

1. A time-dependent hysteresis occurs between forward and reverse isotherms

(i.e., a maximum in the equivalent fraction of the entering cation appears

to have been reached, but it increases with time)

2. A definite maximum content of B is reached which is less than complete

exchange and independent of temperature

3. A definite maximum is reached which varies with temperature.

The first case cannot be analyzed by equilibrium thermodynamics, but Barrer

et al. (1973) present a method for treating the second and third which will not be

examined in detail because it has not been used in soil or clay exchange work. It

involves separating exchange sites into those that can be occupied by A or B ions

and those that can only be occupied by A ions. Selectivity coefficients and

thermodynamic equilibrium constants are obtained for the two sets of sites


The ion-exchange complexes of soils are always mixed exchanger systems. As

Sposito (1981b) says, thermodynamic systems in soil may often be treated as if

they were homogeneous for the analysis of experimental data (and almost always

have been in ion-exchange work). But soils are truly polyfunctional ion exchangers and really should be treated as such. Sposito shows how this can be

achieved, based on work of Barrer and Klinowski (1979), again by splitting

exchange sites into classes, considering each separately, and then obtaining a

weighted geometric mean of the thermodynamic functions at the end. Such a

treatment is very complex and has not yet been used in practice, although Munns

(1976) separated K + adsorbed on volcanic ash soils into tightly and loosely

bound fractions by a similar procedure. However, modem computing methods

make the treatment of such mixed exchanger systems, and incomplete exchange,

perfectly feasible. Thus, although not yet of great importance in relation to K +

exchange, these methods may well prove more useful in the future.



Cation-exchange experiments in the laboratory can be restricted to binary

(two-cation) exchange. In the field, however, ion exchange is rarely binary,

although in many soils the real situation can be well approximated by considering



only the dominant cations (e.g., K+-Ca2+ in calcareous soils, K+-Na+ or

Ca2+-Na + in saline soils, and K -A13 in acid soils). As a move toward a

more realistic approximation of field conditions, attempts have been made to

develop a thermodynamic treatment of ternary (three-cation) exchange.

El-Prince and Babcock (1975) were the first to try this, basing their equations

on a model developed by Wilson (1964) for calculating activity coefficients for

three-component systems from mole fractions. It was thought then that all the

constants in the model could be calculated from binary exchange data. El-Prince

and Babcock (1975) calculated isotherms for Na+ -Rb+ -Cs exchange on

Chambers montmorillonite and for Na -K -Cs exchange on attapulgite.

These isotherms suggested that the qualitative selectivity rules that applied to

binary exchange also applied to ternary exchange, in that selectivity followed the

lyotropic series (Section IV,B,l). Wiedenfeld and Hossner (1978) used the same

equations for Ca2+-MgZ+-Na+ exchange in saline soils, and plotted threedimensional exchange isotherms. They found that the results were “in agreement with recognized properties of the cations,” in that Ca2+ and Mg2+ were

selectively adsorbed.

In neither of these reports were experimental data provided to test the model,

however. El-Prince et al. (1980) tested this “subregular model” of Wilson

(1964) against data for NH, -Ba2 -La3 exchange on a Nevada montmorillonite and found calculated results in “reasonably good agreement with experimental data.” The model has been questioned by Chu and Sposito (1981). They

calculated a set of general thermodynamic equations for ternary exchange and

showed with them that the subregular model was not solely dependent on binary

exchange data. One of the model constants required data from ternary exchange

for its calculation, although its value was often insignificant by comparison with

other terms. This perhaps explains the good agreement between calculated and

experimental results found by El-Prince et al. (1980). Unfortunately, Chu and

Sposito (1981) did not have enough experimental data for ternary exchange to

test their set of equations.













The enthalpy change of a chemical reaction expresses the gain or loss of heat

during the reaction. The reaction may be exothermic, in which case the change of

enthalpy is negative and heat is lost to the surroundings. Alternatively it may be

endothermic, in which case the enthalpy change is positive and heat is gained

from the surroundings. Very few reactions have an enthalpy change of zero. The

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