II. The Thermodynamics of Ion-Exchange Equilibria
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KEITH W.T. GOULDING
218
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
activities.
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
NS [Bv+IU
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 =
NvB[Au
N r A [Bv
+
+
1”
1”
where
f
A
= -[A1
NA
(7)
Gaines and Thomas (1953) also defined an adsorbed-ion activity coefficient,
g, but using equivalent fractions, E. Thus
THERMODYNAMICS AND POTASSIUM EXCHANGE
219
(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
PA[BV+lU
(9)
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
220
KEITH W.T.GOULDING
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)
+
L
In K, dE,
In the Vanselow convention, this becomes
In K
=
I:
InK,dE,
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).
THERMODYNAMICS AND POTASSIUM EXCHANGE
22 1
where mA and mg are molarities. It thus represents a selectivity coefficient
uncorrected for activities in solution.
B. STANDARD
STATES
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),
vp[A-(soil),]
+ up[B"+] = up[B-(soil),] + vp[AU+1
(15)
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
(16)
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
Implications
Reference
Activity = mole fraction when the Activity = molarity as concentration Can calculatef,Kv, etc., but all depend on Argersinger er al. (1950)
latter = 1
+o
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
+0
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
+0
for cations when mole fraction = 0.5
equilibrium
THERMODYNAMICS AND POTASSIUM EXCHANGE
223
C. IONICSTRENGTH
AND HYSTERESIS
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.
+
D. STANDARD FREEENERGIES,ENTHALPIES,AND ENTROPIES
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
KEITH W. T. GOLJLDING
224
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
(17)
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
equation
AG"
=
AH" - TAP
(18)
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)
R
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.
E. ADSORBED-ION
ACTIVITY
COEFFICIENTS
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,
THERMODYNAMICS AND POTASSIUM EXCHANGE
225
and
For Ca2+ + K + exchange, these become
In g,
= EK(ln K,- 1) -
In K, dEK
and
2 In g, = (l-EK) (1-ln K,)
+
I
1
In K , dE,
(23)
EK
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.
KEITH W . T . GOULDING
226
.4-
O
u)
4.0-B
L)
c
0
0.2
0.4
0.6
0.8
1.0
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).
F. EXCESSFIJNCITONS
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
equations:
UVACE= vE,RT In fA
+ u(l -E,)RT
ASE = [AHE - ACE]/T
In fB
(24)
(26)
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-
227
THERMODYNAMICS AND POTASSIUM EXCHANGE
morillonite by Laudelout et al. (1968b) and K -Ca2
on soils by Talibudeen (1971, 1972).
+
+
and K -AP
+
+
exchange
G . INCOMPLETE
EXCHANGE
AND MIXED
EXCHANGERS
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
separately.
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.
H. TERNARY
EXCHANGE
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
228
KEITH W.T.GOULDING
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.
+
+
+
+
+
+
+
+
+
111. CALORIMETRY IN ION-EXCHANGE STUDIES
A. HISTORY
AND TECHNIQUES
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