Al+LwTIcA
CHIMICA
ACTA
Analytica
Chimica Acta 313 (1995) 139-143
Letter to the Editor
Comment on “effect of physico-chemical heterogeneity
complexants’ ’ by Buffle et al.
of natural
Ivica Rtii6
Center
for Marine Research Zagreb, “Ruaer Bo&wiC” Institute, Zagreb, Croatia
Received 30 September
1994; revised 30 March 1995; accepted 12 April 1995
Sir: A series of papers has been published by
Buffle and co-workers
[l-3]
on the subject of
physico-chemical
heterogeneity of natural complexants where they introduce inconsistent
concept of
complexation
in natural aquatic environment.
They
divide complexing sites of many natural complexants
into major and minor site types. Major sites according to their concept, amounting to up to 90% of the
total sites present, should be considered as chemically homogeneous,
producing s-shaped jumps in
logarithmic
plots of experimental
titration curves
(carboxylate, phenolate and Fe-OH groups are mentioned as examples), which are characterized
by
relatively low binding strengths. Minor sites, according to this concept, comprise a relatively small fraction of the total sites, should involve a large variety
of site types with different binding strengths. Consequently, their titration curves are supposed to show a
straightline devoid of a large number of small jumps
(N, S, S-H and Mn-OH functional groups are mentioned as examples). Two types of minor sites are
conceptually
recognised, the so called background
and dominant sites. However, there is no experimental evidence for this concept. It is known that heterogeneity of binding is not a characteristic of some
special type of functional groups, and it can be a
result of pure physical heterogeneity
of chemically
almost homogeneous surfaces. So-called major sites
generally do not produce titration curves presented in
Fig. la from Ref. [3]. In most cases they produce a
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single s shaped gradual jump where characteristics of
individual complexes are melted in an overall titration curve. Very often there is no clear limit between
major and minor sites on the titration curve because
major sites also exhibit significant degree of heterogeneities themselves. Therefore, statistics of complexation can explain the behaviour of both major
and minor sites with similar distribution functions
and in practice these two groups of sites cannot be
clearly separated. In addition, there is no experimental evidence for the existence of the so-called dominant minor sites. On the contrary, relatively smooth
titration curves are generally reported, and continuous model of binding strength distribution
can in
most cases describe well the experimental data. The
concept of heterogeneity in the form of many small
jumps on the titration curve, merging into a line
(whose slope, if far from unity, can be a measure of
the heterogeneity of binding) is not a new one. This
concept is well known indeed from the theory of gas
adsorption at heterogeneous surfaces [4,5]. Buffle et
al. [3] claim “The purpose of this paper is to explain
the chemical reasons for background sites and how
they are related to the overall buffering action of
these sites”. However, there is no solid evidence for
this statement in their paper. The fact is that physical
adsorption of gases exhibits the same effects of
heterogeneity as those reported for surface complexation interaction. From this it is clear that the shape
of the titration curve is a result of heterogeneity of
I. Ru.?iE/Analytica Chimica Acta 313 (1995) 139-143
140
binding without respect to the nature or mechanism
of binding itself. Changes of pH or other chemical
parameters can be used to study the nature of this
interaction. The linear relationship used by Buffle et
al. for the titration curve in the logarithmic plot is
equivalent to the classical Freundlich isotherm:
@=A.(K,#
(1)
where K, is the ground level binding strength (complexing stability constant or adsorption equilibrium
constant for local Langmuirian
isotherm),
m the
heterogeneity parameter, C solute concentration,
A a
constant and 0 the surface coverage. This fact,
which was first suggested by Zeldowitsch [6] a long
time ago, was confirmed rigorously also by Halsey
and Taylor [7] and Sips [8], using the power distribution function for binding strength K or the Boltzmann distribution function for the binding energy E:
F=(K/K,)-m
=exp(-mE/RT)
(2)
However, Buffle et al. completely ignored more
general theoretical results which can explain the
experimental
results which deviate from the linear
relationship of the Freundlich type in the logarithmic
plot for the titration curve. Glueckauf [9] explained
that the Freundlich isotherm fails to predict complete
monolayer formation equivalent to the total complex-
ing capacity because of the unnatural limits of integration used to derive the total surface coverage:
0, = jmt3( E,K,C)
0
. dF
(3)
Here O(E,K,C)
is the so-called local adsorption
isotherm (corresponding
to a model for the adsorption at the homogeneous surfaces). Instead, he introduced the upper limit to the distribution
function
F I 1, which corresponds to the lower limit of the
energy of binding E 2 0, and this enables to predict
well the complexation near the equivalency point. In
addition, he also introduced the upper limit to the
energy of binding E s E, corresponding to the lower
limit to the distribution function F 2 q, which enables one to predict the appearance of the Henry
region at the extremely
low surface coverages.
Therefore the correct model for complexation
or
submonolayer adsorption should be described by:
0, =
/‘@(E,K,C)
. dF
(4)
4
The complete analytical solution of
for a Langmuir type of local isotherm
mann distribution of adsorption energies
by Rudnitskii and Alekseev [lo] in terms
this integral
and Boltzwas derived
of hyperge-
0
-1
-6
-a
-7
-6
-5
-4
-3
-2
-1
0
1
2
log (K,Cl
Fig. I. ‘&e rigorous solution from Eq. 5 for m = l/2
and different
q values. (1) q = 0, (2) q = 0.001, (3) q = 0.01 and (4) 9 = 0.1.
I. RuiiC/Analytica
ometric functions
RuZC [ll]:
and using
o=
l;-
(F(l,m;m+
-q*F(l,m;m
numerical
solution
by
l/K&)
+ l;-ql’“/K,C))/(l-q)
This general rigorous solution is described in Fig. 1
for m = l/2 and different q values. Without the
upper limit for the adsorption energy (E + w> this
general analytical solution [12] becomes simpler:
@=F(l,m;m+l;-l/K,C)=7r*m
rnr ) + (K,C
since they used the following
E:
power series instead of
&E’
(5)
. ( K,C) “/sin(
141
Chimica Acta 313 (1995) 139-143
. m/( 1 - m))
In our recent paper [ll] we demonstrated
that
rigorous solution of Eq. 4 can do the same and in
addition it can predict the Henry behaviour at extremely low surface coverages.
In order to predict the formation of a monolayer,
i.e., total complexing capacity, several authors reported semi-empirical
relations. Sips [8,15] himself
suggested the use of the so-called generalized Langmuir:
or=
*F( 1,l - m;2 - m; - K,C)
(8)
(~,c)~/(l/~
and generalized
+ (~,c)“)
Freundlich
(9
isotherms:
(6)
0,
The first part of the expression on the right side of
Eq. 6 is the well known Freundlich isotherm reported
earlier by Sips [81. The second part enables prediction of the complete monolayer
formation.
From
such a rigorous solution one can recognise that Freundlich behaviour can be observed only inside a
certain concentration range depending on the q values. In the case of relatively large q values the
heterogeneity of adsorption or complexation can be
completely lost [9], and Langmuirian,
i.e., Henry
behaviour can only be detected from experiments in
the entire concentration
region. Therefore, Buffle et
al. [l] are wrong in claiming that a few dominant
minor sites “are responsible for the small deviations
from linearity observed on these titration curves”.
Deviations from the linearity of titration curves in
the logarithmic plot (i.e., from the Freundlich behaviour) experimentally
observed can be in most
cases explained be the rigorous solution of Eq. 4.
Deviations from the linearity below the Freundlich
region have been interpreted by Dubinin [12] and
Dubinin
and Astakhov
[13] via semi-empirical
isotherms:
log 0,
= a - b . (log( KC))’
(7)
(i = 2 in the case of the Dubinin-Radushkewich
isotherm). Cerofolini et al. [14] claimed erroneously
that a combination of Boltzmann and quasi-Gaussian
distribution
functions is necessary to explain both
Freundlich
and Dubinin-Radushkewich
isotherms
= (K,C/(
1 + K,C))”
(10)
while van Riemsdijk et al. [16] reported a multicomponent version of the following isotherm:
0,
= K&/(1+
(K,C)m)l’m
(11)
Isotherms 9 and 11 correspond to quasi-Gaussian
adsorption energy distributions, and the van Riemsdijk isotherm predicts the appearance of a Henry
region at low surface coverages. Recently, we compared these isotherms in the region of concentrations
near the equivalence point to the data on the titration
of Tjeukemeer lake water with Cu(I1) reported by
Verweij et al. [17]. The concentration range available
for titration of four orders of magnitude was not
large enough to reach beyond the Freundlich region
(the parameter of heterogeneity
m = l/2 was obtained). Better agreement was obtained with the rigorous solution of Eq. 4 than with Eqs. 9, 10 and 11
[Ml. The shape of the rigorous solution of Eq. 4 and
Eq. 10 both resemble the shape of the local Langmuir isotherm at very high coverages. However, the
semi-empirical
generalized Freundlich isotherm cannot predict the constant
A, because in order to
describe well the formation of the monolayer
it
should be set to unity. Therefore, semi-empirical
isotherms, especially those based on quasi-Gaussian
distribution of adsorption energies, are not suitable
for the interpretation
of titration curves. If experimental titration curves can be obtained in the region
near the equivalency point, then K, and the com-
142
I. RuiiC/Andytica
Chimica Acta 313 (1995) 139-143
plexing capacity could be extracted from the data as
parameters of complexation. The complexing capacity can be determined
from the van den BergRuiiC-Lee
plot in the same way as was recently
proposed for a discrete model of binding strengths
[18]. Without the knowledge of complexing capacity
the complete interpretation
of the titration curve
would be extremely difficult. The K, value can then
also be determined from the inverse mean of individual stabilities, which for a continuous
model of
binding strengths should read:
l/K;
= (l/K,)
. /lexp(
4
= lim ,&(
@/C(l -
-E/RT)
0))
*dF
(12)
In most cases 4 < 1, and for this purpose can be
set to zero. For the Boltzmann distribution function
of adsorption energies K; = K, . (1 + l/m> [Ml. If,
in addition, the titration window is wide enough,
then the heterogeneity
parameter m could also be
determined from the slope of the titration curve in
the logarithmic plot at lower coverages. Without the
tn parameter, only the overall quantity K; can be
extracted from the experimental data or the approximative value for K, could be obtained as the weakest individual stability constant using the procedure
described for a discrete model of binding strengths
[18]. In cases where the region near the equivalence
point cannot be reached by titration experiments, and
if the deviation from the straight line is observed at
higher concentrations
applicable, then rigorous solution of Eq. 4 could be used to determine both K,
and complexing capacity. When only the Freundlich
region is available at higher concentrations
applicable, then these two basic parameters for submonolayer adsorption or complexation cannot be obtained
except if the reliable value for the complexing capacity could be obtained in any other way. Because the
complete rigorous solution of Eq. 4 is known, the
DEF method introduced first by Gamble et al. [19211 and later extended by Buffle et al. [1,3] is
unnecessarily
complex, and at the end producing
only an overall quantity K,* which would be certainly influenced by the range of validity of Freundlich behaviour, or which is even worse by available titration window. In conclusion, we recommend
for the interpretation of titration data the following
procedure:
(1) Determination
of the complexing capacity or
estimation of its value in any possible way, because
without this quantity the basic parameters of the
complexation cannot be obtained. The corrected version for determination of this quantity from the van
den Berg-RuiiC-Lee
plot [18] could be used for this
purpose if experimental data are available near the
equivalence point.
(2) K,, the ground level stability constant, can be
obtained from the inverse mean of individual stability constants KI*. Such an overall value can be
obtained together with the complexing capacity from
the van den Berg-Ruiic-Lee
plot or by fitting the
experimental
and theoretical curves in the plot of
log(O/C(l
- 0)) vs. log C [18].
(3) If the Freundlich behaviour can be observed at
least in a part of the titration curve, then the heterogeneity parameter m can be extracted from the experimental data.
(4) If the deviation from the Freundlich to Henry
region can be observed at the lowest coverages, then
the maximum energy of adsorption E can be determined.
A set of these four parameters is enough to fit the
entire region of the titration curve. If the complexing
capacity is not known it could be replaced with a
certain measure of the amount of binding material
and only a constant equivalent to K,* could be
obtained as an operationally defined parameter from
the data inside the Freundlich region. It has been
recognized from the titration experiments with different trace metals in natural waters that all sites are
not available for complexation with every metal and
therefore the complexing capacity is an important
parameter for the interpretation
of complexation in
aquatic systems (contrary to the claim of Buffle et al.
[31).
Acknowledgements
The author is grateful to Dr. Dieter Britz for his
support and kind interest in the part of this work
done at the Chemistry Department of the University
of Aarhus, Denmark. This work was also supported
by the National Science Foundation of Croatia.
I. RuZ/Analytica
Chimica Acta 313 (1995) 139-143
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