Ind. Eng. Chem. Res. 1999, 38, 251-259
251
Comparison between Heterogeneous and Homogeneous MASS
Action Models in the Prediction of Ternary Ion Exchange Equilibria
Jose L. Valverde,* Antonio de Lucas, and Juan F. Rodrı́guez
Chemical Engineering Department, Faculty of Chemistry, University of Castilla-La Mancha,
Campus Universitario s/n, 13004, Ciudad Real, Spain
In this paper two models for the prediction of multicomponent ion exchange equilibria were
compared: one, a heterogeneous model, based on the mass action law in which ideal behavior
for both the solution and the solid phase and the heterogeneity of ion exchange sites has been
assumed, and the other a homogeneous model, based on the same mass action law in which
nonideal behavior for both the solution and the solid phase has been taken into account.
Considering counterions with different valences, the optimum value of the fraction of functional
groups was obtained from binary systems data. It was shown that the energy distribution was
practically symmetric. Likewise, a procedure for decreasing the number of adjustable parameters, based on the determination of the minimum linearly independent equilibria considering
each equilibrium as a heterogeneous reaction, was developed. The adjustable parameters
(namely, the equilibrium constants forced to satisfy the triangle rule) were used in the prediction
of the ternary equilibrium data. In all cases the prediction of the heterogeneous model was in
good agreement with experimental data. However, although the homogeneous model was
somewhat superior to the heterogeneous model in the prediction of ternary systems, the
comparison between both models did not allow us to distinguish statistically between them.
Introduction
Prediction of multicomponent ion exchange equilibrium is needed for the design of exchangers that operate
over a wide range of conditions. A theoretical model
that allows the prediction of equilibrium behavior of
multicomponent systems would therefore be extremely
useful. A test of an ion exchange model is that it be
able to predict accurately the composition of one phase
that is in equilibrium with the other phase of known
composition. The typical approach is to develop thermodynamic models which, based on binary equilibrium
data, can predict multicomponent equilibria. The models proposed in the literature can be divided in two main
groups: those describing the ion exchange in terms of
the law of mass action and those regarding the ion
exchange as a phase equilibrium.
In early works, which belong to the first group,1,2 the
ideality of the ion exchange equilibria was assumed with
a constant separation factor and the activity coefficients
of all components both in the solution and in the solidphase equal to unity. This model assumes that the
presence of other counterions does not affect the equilibrium exchange between two particular ions, which
implies constant selectivities coefficients. However, the
behavior of systems where the selectivity coefficients
change with the resin composition cannot be described
on the basis of these models.3 To solve this problem it
was necessary to introduce models that consider nonideal or real systems. Smith and Woodburn4 successfully applied the Wilson equation to correlate the
activity coefficients in the solid phase with the composition. To calculate the binary interaction parameter in
the Wilson equation, these authors developed a computational method which consists of obtaining the set
* Author to whom correspondence should be addressed.
E-mail: [email protected].
of values of the thermodynamic equilibrium constant
and the two binary interaction constants in the Wilson
equation which best fit the experimental equilibrium
data. If the activity coefficients of the salts in the solution are known, the activity coefficient of each ion in
the solid phase and the equilibrium constant can then
be estimated from experimental data. Recently Vamos
and Haas5 used the Margules equation to model the dependency of the resin phase activity coefficients with
the resin composition. However, statistical analysis of
data fits indicates that although the Wilson and Margules equations were able to model adequately the resin
phase nonidealities, the Wilson model provided the
superior data fits. Shallcross et al.,6 on the basis of the
model by Smith and Woodburn, proposed the use of the
Pitzer equation7 to describe single ion activity coefficients in the liquid phase and the Wilson model for the
activity coefficients in the solid phase. In previous
works, de Lucas et al.8-10 developed a similar model
which was used for the prediction of the ternary and
quaternary equilibrium data. In this case, the activity
coefficients in the liquid phase were calculated by using
the Meissner and Kusik method.11 Likewise, a method
to reduce the number of parameters was developed by
Allen et al.12 and Allen and Addison13 and later used
by de Lucas et al.14 by forcing the Wilson parameters
to satisfy some constraints based on empirical relationships.
In the models of the second group, ion exchange is
treated as an adsorption process. In this sense, the first
attempts to explain and to predict ion exchange equilibria were based on the phenomena of adsorption and
ion exchange, using the Freundlich and Langmuir
isotherms for the characterization of equilibrium. More
recently, Novosad and Myers15 and Myers and Byington16 suggested that the ion exchange should be treated
in a similar way as the phenomenon of adsorption.
10.1021/ie980312z CCC: $18.00 © 1999 American Chemical Society
Published on Web 11/21/1998
252 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999
These authors defined a thermodynamic variable,
namely, the excess surface, which can be experimentally
determined solely from measurements in the bulk liquid
phase, thereby eliminating the ambiguity between adsorbed and exchanged ions.
In a previous paper10 different models for characterizing the ion exchange equilibrium in binary mixtures
were compared. Three models were used: the Langmuir model, that proposed by Novosad and Myers15 and
Myers and Byington,16 and one based on the law of mass
action. It was observed that the model based on the
law of mass action, which takes into account the effects
of temperature, the total ionic concentration of the
external solution, and the activity coefficients of the two
phases in addition to the equilibrium constant, was the
model which best represented the experimental binary
equilibrium isotherms in all of the systems studied.
Likewise, predictions of multicomponent ion exchange
equilibria with the law of the mass action are more
accurate than those made with the other models.17
Recently, Melis et al.18 developed a new model based
on the mass action law. This new model differs from
all models of the first group discussed above in that it
assumes that both the fluid and the solid-phase behave
ideally, while it acknowledges the heterogeneity of ion
exchange sites. In other words, in this approach they
assumed that the effect of mixture nonidealities is
smaller than that due to the resin heterogeneity and
can be neglected. This corresponds to introducing the
idea of Myers and Byington16 accounting for the heterogeneous nature of the solid in the framework of the
equilibrium models based on the law of mass action.
These authors introduced the heterogeneity of ion
exchange functional groups by assuming a given distribution for the standard free-energy change of the ion
exchange process. Assuming the simplest situation in
order to diminish the number of adjustable parameter,
they consider that the resin could be characterized by
using only two types of equally abundant functional
groups. Accordingly, the total resin capacity is given
by the sum of the capacity of the resin with respect to
the functional groups of each type. Representing p1 and
p2 as the fraction of functional groups of type 1 and 2,
respectively, it is clear that
p1 + p2 ) 1
stants forced to satisfy the triangle rule) were used in
the prediction of the ternary equilibrium data. The
results obtained with this model were compared to those
of the homogeneous one based on the law of mass action
described in previous papers in order to verify which
model is more accurate to predict multicomponent ion
exchange equilibria.
Mathematical Formulation of the
Heterogeneous MASS Action Model
The following mathematical formulation is based on
the previous one by Melis et al.18
Binary Systems. For a given binary system
β+
β+
R+
βAR+
s + RBr T βAr + RBs
where AR+ and Bβ+ are the two exchanging ions and r
and s indicate the resin and the solution phases,
respectively. The thermodynamic equilibrium constant
for such a reaction is given by
K(T) )
)β(aBβ+
)R
(aAR+
r
s
(aAR+
)β(aBβ+
)R
s
r
(3)
in terms of the activities of the involved ionic species.
This quantity is defined by the thermodynamic relationship
∆G°(T) ) -RT ln K(T)
(4)
where ∆G° is the standard free-energy change associated with the ion exchange process. By assuming that
the ion exchange equilibrium is ideal and neglecting the
effects due to the swelling and hydration of the resin,
the equilibrium constant can be written as
K(T) )
)β(CBβ+
)R
(qAR+
r
s
(CAR+
)β(qBβ+
)R
r
s
(5)
where qi and Ci are the concentrations in the resin and
in the solution phase, respectively. Introducing the
ionic fraction of the generic ith ionic species
(1)
The values of the parameters p1 and p2 should be, for
a given resin, the same for all binary and multicomponent systems. The reliability of the proposed model in
describing binary uptake data as well as in predicting
the behavior of multicomponent systems was tested by
comparison with experimental data available in the
literature.
The aim of this paper is to compare between a
heterogeneous model and a homogeneous mass action
model in the prediction of ion exchange equilibria. On
the basis of the binary and ternary experimental data
of Ca2+, Mg2+, K+, Na+, and H+ obtained in previous
works,8,10,17 in this paper we discuss the influence of the
fraction of functional groups (p1 and p2) on describing
experimental binary data where counterions with different valences are involved. Likewise, a procedure for
decreasing the number of adjustable parameters, based
on the determination of the minimum linearly independent equilibria when considering each equilibrium as a
heterogeneous reaction, was developed. In this way, the
adjustable parameters (namely, the equilibrium con-
(2)
νiCi
Xi )
)
νiCi
N
Nc
∑ νkCk
(6)
k)1
Yi )
νiqi
)
Nc
∑ νkqk
νiqi
q0
(7)
k)1
where Ci represents the concentration of ith ionic
speciesin solution and νi its charge. Nc is the total
number of counterion species. N is the total ionic
concentration in the solution phase or the solution
normality, qi is the solute concentration in the solid
phase and q0 is the total capacity of the resin. Upon
substituting in eq 5 the equilibrium constant becomes
K(T) )
()
YAβ R+XBR β+ q0
Xβ R+YR β+ N
A
B
β-R
(8)
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 253
If we assumed that the resin is characterized by two
type of functional groups, the total resin capacity is
given by
q0 ) q0,1 + q0,2
(9)
where q0,1 and q0,2 represent the capacity of the resin
with respect to the functional groups of type 1 and 2,
respectively.
The fractions of functional groups of type 1 and 2, p1
and p2, are defined by
pi )
q0,i
q0
i ) 1, 2
(10)
Table 1. Matrix of Stoichiometric Coefficients
no. of
binary
system
1
H+
s
2
Na+
s
3
K+
s
4
Mg2+
s
5
Ca2+
s
6
H+
r
7
Na+
r
8
K+
r
9
Mg2+
r
10
Ca2+
r
1
2
3
4
5
6
7
8
9
10
+1
+1
+2
+2
0
0
0
0
0
0
-1
0
0
0
+1
+2
+2
0
0
0
0
-1
0
0
-1
0
0
+2
+2
0
0
0
-1
0
0
-1
0
-1
0
+1
0
0
0
-1
0
0
-1
0
-1
-1
-1
-1
-2
-2
0
0
0
0
0
0
+1
0
0
0
-1
-2
-2
0
0
0
0
+1
0
0
+1
0
0
-2
-2
0
0
0
+1
0
0
+1
0
+1
0
-1
0
0
0
+1
0
0
+1
0
+1
+1
These two (Nc - 1) equations along with the stoichiometric condition
being
Nc
2
pi ) 1
∑
i)1
(11)
The ionic fraction of the generic ith ionic species on
the jth functional group is given by
Yj,i )
νiqj,i
q0,j
( )
β
R
Yj,A
R+XBβ+ q0,j
Xβ R+YR β+ N
A
j,B
β-R
(12)
j ) 1, 2
(13)
where
(
Kj(T) ) exp -
)
∆G°j(T)
RT
(14)
The overall resin composition is obtained by considering for each component the quantities present on each
type of functional group:
Nf)2
YAR+ )
∑
j)1
pjYj,AR+
(15)
Nf)2
YBβ+ )
∑
j)1
pjYj,Bβ+
On the other hand, the following stoichiometric condition has to be satisfied:
Yj,AR+ + Yj,Bβ+ ) 1
(16)
Multicomponent Systems. Let us consider a multicomponent system containing Nc counterions. Selecting a reference one, say the Ncth ion, we must consider
the (Nc - 1) equilibrium relations between this and each
one of the other counterions. Thus
( )
(Yj,iνi+)νNc(XNcνNc+)νi q0,j
Kj,i(T) )
(X ν +)νNc(Y ν +)νi N
ii
j,Nc Nc
for j ) 1, 2 and i ) 1, 2, ..., (Nc - 1).
ν
νNc-νi
(17)
j ) 1, 2
(18)
and the following equation
Nf)2
Yiνi )
j ) 1, 2
and the equilibrium constant of the exchange process
occurring on the same functional groups is
Kj(T) )
Yj,i i ) 1
∑
i)1
∑
j)1
pjYj,iνi
(19)
lead to the composition of phase adsorbed on sites of
each type and the overall resin composition. In the last
equation, Nf is the number of types of functional groups.
Results and Discussion
In the following, the heterogeneous mass action model
will be referred to as HMAM and the homogeneous mass
action law described in previous works8-10 will be
referred to as LAM.
Binary Systems. The equilibrium experimental
data of the 10 binary systems combining Ca2+, Mg2+,
K+, Na+, and H+ on the same resin (Amberlite IR-120)
at different total ionic concentrations (0.1, 0.3, and 0.5
N) and three temperatures (283, 303, and 323 K) have
been presented in previous works.10,17 The physical
properties of Amberlite IR-120 also has been reported
in these works. A total exchange capacity of 3.130
equiv/L of resin has been assumed. The electrolytes
used were CaCl2‚2H2O, MgCl2‚6H2O, KCl, NaCl, and
HClaq.
According to Melis et al.,18 the energy distribution of
the active sites of the resin should not change with the
type of ion; therefore, the value of p1 has to be equal for
all the systems, whether binary or multicomponent systems. A meaningful value of the fraction of functional
groups p1 and, hence, p2, can be obtained by a nonlinear
regression procedure. This procedure, used in earlier
works,19,20 is based on the Marquardt’s algorithm.21 In
this way, for the 606 experimental data points corresponding to the 10 binary systems, 61 parameters must
be considered: 60 equilibrium constants (two for each
binary system at each of the three work temperatures)
and the fraction of functional group 1, p1.
To diminish the number of adjustable parameters, the
following procedure based on an earlier paper by Schuber and Hofmann22 has been carried out. The procedure
consists of finding a set of linearly independent key
equilibria from a given equilibrium pattern.
The 10 binary systems here considered can be written
as follows:
254 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999
+
+
+
Na+
s + Hr T Nar + Hs
(A)
+
+
+
K+
s + Hr T Kr + Hs
(B)
+
2+
+
Mg2+
s + 2Hr T Mgr + 2Hs
(C)
+
2+
+
Ca2+
s + 2Hr T Car + 2Hs
(D)
+
+
+
K+
s + Nar T Kr + Nas
(E)
Mg2+
s
+
2Na+
r
T
Mg2+
r
+
2Na+
s
(F)
+
2+
+
Ca2+
s + 2Nar T Car + 2Nas
(G)
+
+
2+
Mg2+
s + 2Kr T Mgr + 2Ks
(H)
+
2+
+
Ca2+
s + 2Kr T Car + 2Ks
(I)
2+
2+
2+
Ca2+
s + Mgr T Car + Mgs
(J)
In general
βRARRis+ + RRBRβir+ T βRARRir+ + RRBRβis+
(20)
where R ) 1, 10.
+
In this way, the following species occur: H+
s , Nas ,
+
2+
2+
+
+
+
2+
2+
Ks , Mgs , Cas , Hr , Nar , Kr , Mgr , and Car . In
Table 1 is shown the matrix of stoichoimetric coefficients. The stoichiometric coefficients are positive for
the products, negative for the reactants, and zero for
the inert components. By applying the Gaussian algorithm to the matrix of stoichiometric coefficients yields
the matrix
(
+1
0
0
0
0
0
0
0
0
0
-1
+1
0
0
0
0
0
0
0
0
0
-1
+2
0
0
0
0
0
0
0
0
0
-1
+1
0
0
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
0
0
+1
-1
0
0
0
0
0
0
0
0
0
+1
-2
0
0
0
0
0
0
0
0
0
+1
-1
0
0
0
0
0
0
0
0
0
+1
0
0
0
0
0
0
)
(21)
In other words, four key equilibria were found,
namely systems A-D. In this way, the following
relationships between the standard Gibbs free energy
corresponding to functional groups of type 1 and 2 may
be established:
(∆G°j(T))5 + (-∆G°j(T))2 - (-∆G°j(T))1
(-∆G°j(T))6 ) (-∆G°j(T))3 - 2(-∆G°j(T))1
(-∆G°j(T))7 ) (-∆G°j(T))4 - 2(-∆G°j(T))1
(-∆G°j(T))8 ) (-∆G°j(T))3 - 2(-∆G°j(T))2
(22)
(-∆G°j(T))9 ) (-∆G°j(T))4 - 2(-∆G°j(T))2
(-∆G°j(T))10 ) (-∆G°j(T))4 - (-∆G°j(T))3
where j ) 1, 2 and T ) 283, 303, 323 K.
Those relationships are equivalent to assume the
triangle rule for the equilibrium constants. This ther-
modynamic constraint allows us to diminish the number
of parameters from 61 to 25 and ensures the thermodynamic consistency of the equilibrium parameters thus
obtained.
As mentioned above, all of data (606) corresponding
to the 10 binary systems were simultaneously fitted.
According to eq 20, the objective function to be minimized is given by the following equation:
606
χ2 )
{(YA,exp - YA,th)k2 + (YB,exp - YB,th)k2}
∑
k)1
(23)
The results obtained for the free energy change on
each functional group are shown in Table 2. The
fraction of functional groups of type 1, p1, is equal to
0.477. This value is close to 0.5, as assumed by Melis
et al.18 In fact, the confidence interval of p1 evaluated
in the same way as in a previous work20 is 0.313 < p1
(t0.477) < 0.640 with a significance of 0.05.
In Figure 1 is shown the influence of p1 on χ2. As
seen, the trend observed corroborates the previous
statistical analysis. This result would indicate that the
energy distribution is practically symmetric.
In Figures 2-4 are compared the HMAM and LAM
models. It can be seen that the accuracy of the HMAM
model in reproducing experimental data is comparable
to that of LAM. Likewise, by means of an example, in
Figures 5 and 6 are shown the ionic fraction of different
species in the resin phase on functional groups of type
1 and 2. The capability of HMAM in describing the
experimental data where counterions with different
valences are involved can be also verified (Figure 3). It
clearly appears that the effect of the solution normality,
which increases the equilibrium uptake as the solution
becomes more diluted, is properly accounted for by the
model. However, it can be observed that the experimental data is just better reproduced with the LAM
model than with the HMAM model. From the mathematical viewpoint, this result is expected. In the LAM
model, each binary data at a given temperature were
interpreted by fitting three parameters, namely, the
equilibrium constant and the two binary interaction
parameters of the Wilson model. As previously reported
by Shallcross et al.6 and de Lucas et al.,8 the regression
procedure does not imply that the triangle rule for the
equilibrium constants has to be completely obeyed. In
fact, the deviations of this rule are, in some cases, higher
than 15% in the temperature range investigated (Table
3). According to Shallcross et al.,6 these differences
could be also attributed to the experimental error.
On the contrary, the HMAM model here used is forced
in the fitting procedure of all binary data to satisfy the
triangle rule for the equilibrium constants. Furthermore, the experimental error, in addition to the assumption that considers the behavior of the aqueous
solution to be ideal, could have a clear negative effect
on the quality of the fit. In this way the slight
differences in the prediction of experimental data by
both models could be justified.
The heterogeneity of the resin may be explained
attending to different factors. Polymeric supports, such
as ion exchange resins, are semirigid because their
structure is influenced by molecules present in the bulk
phase. In most cases swelling provides accessibility to
new active groups. Therefore, one may distinguish
between two types of accessibility, one related to the
morphology in the inert bulk environment (noninterac-
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 255
Table 2. Standard Gibbs Free Energy Change Corresponding to Functional Groups of Types 1 and 2a
T (K)
283
303
323
a
(J/mol)
HNa
HK
HMg
HCa
NaK
NaMg
NaCa
KMg
KCa
MgCa
∆G1°
∆G2°
∆G1°
∆G2°
∆G1°
∆G2°
-2597.8
-81.0
-1825.8
-136.4
-1674.6
-849.8
-5130.0
370.5
-4107.0
316.8
-3450.4
-278.2
-6814.8
1219.2
-3489.9
-1750.3
-2224.3
-4172.2
-6465.5
-4081.2
-6756.9
-5165.9
-7446.2
-5849.0
-2532.2
451.5
-2281.2
453.2
-1775.7
571.6
-1619.1
1381.1
161.7
-1477.6
1124.9
-2472.5
-1269.8
-3919.2
-3105.3
-4893.2
-4096.9
-4149.3
3445.3
478.1
4724.2
-2383.9
4676.4
-3615.7
3794.6
-4822.2
1457.2
-5799.5
-545.5
-5292.5
349.3
-5300.3
-3267.0
-3415.6
-5221.9
-1676.8
+
+
+
In this table, for example, HNa corresponds to the binary equilibrium Na+
s + Hr T Nar + Hs .
Figure 1. Influence of p1 on χ2.
Figure 3. Comparison between calculated results of both HMAM
and LAM models and experimental values of the ionic fraction in
the resin phase as a function of the ionic fraction in the solution
phase for the ion exchange equilibrium Ca2+/Na+ at 283 K.
Figure 2. Comparison between calculated results of both HMAM
and LAM models and experimental values of the ionic fraction in
the resin phase as a function of the ionic fraction in the solution
phase for the ion exchange equilibrium K+/H+ at 323 K.
tion accessibility) and another reflecting the influence
of the solvent and adsorbed molecules (interaction
accessibility). According to Buttersack,23 the corresponding fraction of accessible functional groups cannot
be considered as a statistical distribution of groups since
the degree of cross-linking should be lower in the surface
region, which should in turn result in higher degrees of
surface sulfonation. On the other hand, the interaction
accessibility appears to be related to the ratio of styrene
to DVB. These types of accessibility could explain the
observed heterogeneous distribution of ion exchange
centers.
Figure 4. Comparison between calculated results of both HMAM
and LAM models and experimental values of the ionic fraction in
the resin phase as a function of the ionic fraction in the solution
phase for the ion exchange equilibrium Ca2+/Mg2+ at 283 K.
On the other hand, if the surface were not heterogeneous, one would expect the relative separation to be
approximately constant. Nonidealities in the bulk and
surface phase would produce only minor deviations from
this rule. But if the surface contains high- and lowenergy sites, then the separation factor for the preferentially adsorbed species should decrease with its ionic
256 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999
using Marquardt’s algorithm21 in a similar way as done
in previous works.19,20,24
In Figures 7 and 8 are compared the capability of both
models in predicting ternary experimental equilibrium
data. It can be seen that the accuracy of the two models
is approximately the same. Deviations higher than 10%
are mainly related to the equivalent ionic fractions with
values lower than 0.2. According to de Lucas et al.,8
since the experimental resin composition is determined
by material balance, higher deviations can be expected
at low concentration values, due to experimental data
scatter. To establish a better comparison between both
models, a given ternary system constituted by the ions
A, B, and C was defined by the following objective
function:
Ε)
Figure 5. Ionic fraction of K+ of the functional groups 1 and 2
evaluated by the HMAM model for the ion exchange equilibrium
K+/H+ at 323 K.
1
NE
NE
{(YA,exp - YA,th)k2 + (YB,exp - YB,th)k2 +
∑
k)1
(YC,exp - YC,th)k2} (24)
In Table 4 are shown the results reached with the
HMAM and LAM models. It can be proven that the
accuracy of the two models is similar. However, in the
same way as in the binary systems where counterions
with different valences are involved, the prediction of
ternary systems with the LAM model is just better than
that with the HMAM model. A perusal of those tables
indicates that for the full K+-Na+-H+ ternary data set
the objective function for the HMAM model is, in value,
lower than that of the LAM model. In the case of the
Ca2+-Na+-H+ and Ca2+-Mg2+-H+ ternary systems,
the ratio of objective functions ΕHMAM/ΕLAM is higher
than 2 in practically all cases.
The quality of the prediction achieved with the
HMAM and LAM models also can be determined by
using the F-test in a similar way as Allen et al.12 If it
is assumed that eq 24 is a measurement of the variance,
the following F-statistic could be defined:
F0 )
Ca2+
Figure 6. Ionic fraction of
of the functional groups 1 and 2
evaluated by the HMAM model for the ion exchange equilibrium
Ca2+/K+ at 323 K and 0.3 N total concentration.
Table 3. Triangle Rule for Equilibrium Constants (LAM
Model)
system
283 K
303 K
323 K
KNaCaK2HNaKCaH
KKCaK2HKKCaH
KNaKKHNaKKH
KKCaK2NaKKCaNa
KHMgKMgNaKCaH
1.10
1.19
0.97
1.02
0.88
1.05
1.11
0.99
1.04
0.93
0.89
1.12
0.85
0.91
1.02
fraction and with the total solution normality. Figure
3 clearly shows this behavior.
Ternary Systems. The equilibrium experimental
data of the ternary system combining Ca2+, Mg2+, K+,
Na+, and H+ on the same resin (Amberlite IR-120) at
different total ionic concentrations (0.1, 0.3, and 0.5 N)
and three temperatures (283, 303, and 323 K) have been
reported in previous works.8,17 The resin composition
of the ternary systems has been predicted from the
parameters obtained in the above section by solving the
set of nonlinear equations constituted by eqs 17-19
ΕHMAM
ΕLAM
ΕLAM
F0 )
ΕHMAM
if ΕHMAM > ΕLAM
(25)
if ΕHMAM e ΕLAM
In Table 4 is also shown the random variable F(ν1,ν2),
where the degree of freedom, ν1 and ν2, are in this case
equal to NΕ. The results of these computations show
that there is not a statistically significant reduction in
the quality of the prediction with both models. The
systems for which there is a significant lessening (at
0.25 level) are those in which counterions with different
valences are presents. In any case, the differences in
the objective functions for the two models are not
statiscally significant. In other words, the only conclusion to be drawn in comparing the two models by their
minimized objective function is that the model which
provides the objective functions smaller in value is
superior to the other for a given data set.
Summarizing, although the LAM model is just superior to the HMAM model in the prediction of the ternary
systems, the comparison between both models does not
allow us to distinguish statistically between them. In
all cases, however, the HMAM model predicts values
to be in good agreement with experimental data. These
results clearly demonstrate that the potential of the
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 257
Figure 7. Comparison between calculated and experimental
values of the ionic fraction in the resin phase for the ion exchange
equilibrium K+/Na+/H+ at 283, 303, and 323 K: (a) K+, (b) Na+,
and (c) H+.
Figure 8. Comparison between calculated and experimental
values of the ionic fraction in the resin phase for the ion exchange
equilibrium Ca2+/Mg2+/H+ at 283, 303,p and 323 K: (a) Ca2+, (b)
Mg2+ and (c) H+.
258 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999
Table 4. Comparison between HMAM and LAM Models in the Prediction of Ternary Systems
system
T (K)
normality
NE
ΕLAM
ΕHMAM
F0
F(NE,NE) at 0.05 level
significance
K-Na-H
K-Na-H
K-Na-H
K-Na-H
K-Na-H
K-Na-H
K-Na-H
K-Na-H
K-Na-H
Ca-Na-H
Ca-Na-H
Ca-Na-H
Ca-Na-H
Ca-Na-H
Ca-Na-H
Ca-Na-H
Ca-Na-H
Ca-K-H
Ca-K-Na
Ca-Mg-H
Ca-Mg-H
Ca-Mg-H
Ca-Mg-H
Ca-Mg-H
283
283
283
303
303
303
323
323
323
283
283
283
303
303
323
323
323
303
303
283
283
303
323
323
0.1
0.3
0.5
0.1
0.3
0.5
0.1
0.3
0.5
0.1
0.3
0.5
0.3
0.5
0.1
0.3
0.5
0.3
0.3
0.1
0.5
0.3
0.1
0.5
6
8
6
9
5
5
7
7
8
5
5
5
5
5
5
5
5
10
10
5
5
9
5
5
0.0100
0.0124
0.0114
0.0056
0.0077
0.0147
0.0061
0.0095
0.0062
0.0039
0.0065
0.0116
0.0057
0.0072
0.0022
0.0040
0.0040
0.0072
0.0331
0.0066
0.0028
0.0062
0.0017
0.0063
0.0075
0.0088
0.0089
0.0034
0.0051
0.0121
0.0037
0.0070
0.0038
0.0085
0.0184
0.0222
0.0280
0.0302
0.0069
0.0127
0.0171
0.0072
0.0213
0.0188
0.0077
0.0158
0.0055
0.0023
1.333
1.409
1.281
1.647
1.510
1.215
1.649
1.357
1.632
2.179
2.831
1.914
4.912
4.194
3.136
3.175
4.275
1.000
1.554
2.848
2.750
2.548
3.235
2.739
4.284
3.438
4.284
3.179
5.050
5.050
3.787
3.787
3.438
5.050
5.050
5.050
5.050
5.050
5.050
5.050
5.050
2.978
2.978
5.050
5.050
3.179
5.050
5.050
not significant
not significant
not significant
significant at 0.25 level
not significant
not significant
not significant
not significant
not significant
significant at 0.25 level
significant at 0.25 level
significant at 0.25 level
significant at 0.1 level
significant at 0.1 level
significant at 0.25 level
significant at 0.25 level
significant at 0.1 level
not significant
not significant
significant at 0.25 level
significant at 0.25 level
significant at 0.1 level
significant at 0.25 level
significant at 0.25 level
HMAM model in predicting the equilibrium ion exchange data for binary and ternary systems is similar
to that of LAM model. On the other hand, the HMAM
model has a clear advantage with respect to LAM model.
The procedure for decreasing the number of adjustable
parameters, based on the determination of the minimum linearly independent equilibria considering each
equilibrium as a heterogeneous reactions, allows us to
predict the equilibrium constants for the two functional
groups of a new binary equilibrium system. As abovementioned, this prediction is, from the thermodynamic
point of view, more consistent than that carried out with
the LAM model. Consider, for example, the system A/D
different from the 10 here studied, where A is Ca2+,
Mg2+, K+, Na+, or H+. The resulting equilibrium
constants for the two functional groups of the binary
system constituted by D and another ion different from
A, could be predicted by applying the triangle rule.
On the other hand, the programming effort for the
prediction of the equilibrium experimental data of the
ternary system is similar in both models. In this
situation, either model could be equally used for the
prediction of multicomponent ion exchange equilibria.
However, as concluded in the previous section, the
programming effort for the HMAM model in the evaluation of the standard Gibbs free energy corresponding
to functional groups of type 1 and 2 from binary systems
is always greater than that for the LAM model with no
restrictions. The computational effort would moreover
increase if the nonideality of the aqueous phase was
considered.
Conclusions
In this paper heterogeneous and homogeneous models
(HMAM and LAM, respectively) based on the mass
action law have been compared. The influence of the
distribution of functional groups on the prediction of the
binary equilibria, where counterions with different
valences are involved, has been studied. The best fit is
obtained when the energy distribution is practically
symmetric. Likewise, a procedure for decreasing the
number of adjustable parameters, based on determination of the minimum linearly independent equilibria
considering each equilibrium as a heterogeneous reaction, has been developed. The adjustable parameters
(namely, the equilibrium constants forced to satisfy the
triangle rule) have been used in the prediction of the
ternary equilibrium data. The quality of the prediction
achieved with the heterogeneous and homogeneous
models has been determined by using the F-test. In all
cases the heterogeneous model predicted values to be
in good agreement with experimental data. However,
in the same way as in the binary systems, where
counterions with different valences are involved, the
prediction of ternary ones with the homogeneous model
is slightly better than that with the heterogeneous
model. Although the homogeneous model is somewhat
superior to the heterogeneous model in the prediction
of ternary systems, the comparison between both models
does not in general distinguish statistically between
them. This result clearly demonstrates the potential
of the heterogeneous model in predicting the equilibrium ion exchange data for binary and ternary systems.
Finally, if the programming effort is considered, either
model could be equally used to predict the multicomponent ion exchange equilibria.
Nomenclature
ai ) activity of ith ion
Ci ) concentration of ith ion in the solution phase, mol/L
F ) F-test value
F0 ) function defined by eq 25
∆G° ) standard Gibbs free energy change, J/mol
(∆Gj°) ) standard Gibbs free energy change corresponding
to functional groups of type j
K ) thermodynamic equilibrium constant
Kj ) thermodynamic equilibrium constant for functional
groups of type j
Kj,i ) thermodynamic equilibrium constant between ion i
and reference counterion for functional groups of type j
N ) total concentration or normality of the solution phase
Nc ) number of counterions in the system
Nf ) number of types of functional groups
NΕ ) number of experiments carried out with a given
ternary system
pj ) fraction of functional groups of type j
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 259
qi ) concentration of ith ion in the resin phase, mol/L
qj,i ) concentration of ith ion in the resin phase on
functional groups of type j, mol/L
q0 ) total ion exchange capacity, mol/L
q0,j ) ion exchange capacity of functional groups of type j,
mol/L
R ) ideal gas constant
T ) temperature
Xi ) ionic fraction of ith ion in the solution phase
Yi ) ionic fraction of ith ion in the resin phase
Yj,i ) ionic fraction of ith ion in the resin phase on
functional groups of type j
Greek Letters
R, β, ν ) stoichiometric coefficients or ion charge
ν ) stoichiometric coefficients or ion charge, degrees of
freedom
Ε ) function defined by eq 24
χ ) objective function defined by eq 23
Subscripts
exp ) experimental
r ) resin phase
s ) solution phase
th ) predicted
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Received for review May 20, 1998
Revised manuscript received September 29, 1998
Accepted October 2, 1998
IE980312Z