General equation for determining the dissociation constants ofpolyprotic acids and bases from additive properties.
Part II. Applications
Issam Jano*, James E. Hardcastle
Department of Chemistry and Physics, Texas Woman's University, Denton, TX 76204, USA
Received 13 August 1998; received in revised form 25 January 1999; accepted 1 February 1999
Abstract
Dissociation constants of some polyprotic acids and bases are determined from spectrophotometric and reversed-phase high
performance liquid chromatographic data. A general equation relating additive properties of acids and bases to the pH of the
solution is used for this purpose. The method is tested by applying it to the calculation of overlapping pKas without the
prerequisite of measuring the limiting values of the property for the individual species that result from the dissociation of the
solute. The possibility of applying the same method to hyperpolarizability measurements is pointed out, and a procedure,
based on the general equation, for obtaining the activity coef®cients of the ionic species as a function of the ionic strength is
also suggested. # 1999 Elsevier Science B.V. All rights reserved.
Keywords: Dissociation constants; Polyprotic acids and bases; General method; Spectrophotometry; Reversed-phase high performance liquid
chromatography; Hyperpolarizability; Activity coef®cients
1. Introduction
In a previous paper [1], referred to as Part I, a
general equation allowing the calculation of the dis-
sociation constants of polyprotic acids and bases from
measurements of additive properties was derived. The
equation is
p � p0 �Pn
r�1 pr Ke�r� erx
1�Pnr�1 Ke�r� erx
; (1)
where p is the measured molar property and pr is the
limiting value of the property for the anion �HnÿrArÿ�.
Ke(r) is the effective dissociation constant of the anion,
and x is related to the pH of the solution
x � pH ln�10�:Eq. (1) allows the calculation of the dissociation
constants and the limiting values of the additive
property by a nonlinear iterative ®tting procedure.
The reader is referred to Part I for details. The purpose
of the present paper is to test the method and to show
the results of some applications from spectrophoto-
metry and reversed-phase high performance liquid
chromatography (RP-HPLC).
2. Spectrophotometry
Spectrophotometric measurements have been used
for the determination of the dissociation constants of
Analytica Chimica Acta 390 (1999) 267±274
*Corresponding author. Tel.: +1-940-898-2550; fax: +1-940-
898-2548.
0003-2670/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 0 0 3 - 2 6 7 0 ( 9 9 ) 0 0 1 7 1 - 3
acids and bases [2±6]. In dilute solutions Beer's law is
obeyed, and the measured molar absorptivity � of an
ionogenic solute is the sum of contributions from the
different entities that result from the dissociation of
the solute. Denoting the limiting molar absorptivities
of the individual solute species by �r, then the applica-
tion of Eq. (1) leads to the following relation:
� � �0 �Pn
r�1 �r Ke�r� erx
1�Pnr�1 Ke�r� erx
: (2)
We have used this relation to calculate the apparent
dissociation constants of some acids and bases. For the
sake of comparison, we selected compounds that have
been studied carefully by other authors using different
approaches. Albert and Serjeant [6] had reported
measurements and analyses for monoprotic and dipro-
tic acids and bases. Eq. (2) is ®tted to their data and the
results are summarized in the following section.
Fig. 1 shows the variation of the molar absorptivity
of acridine, C13H9N, as a function of the pH of the
solution at a constant ionic strength of 0.01. The
molarity of the solution is 0.0002 mol/l, and the
analytic wavelength is 403 nm. The experimental
points are marked as (x). The curve labeled (e) is
calculated by Eq. (2). The limiting molar absorptiv-
ities obtained by ®tting Eq. (2) to the experimental
data are �0�3.00�103 l/mol cm for the protonated
acridine (RNH�), and �1�1.04�103 l/mol cm for
the neutral molecule. The apparent dissociation con-
stant Kap was found equal to 2.265�10ÿ6. This cor-
responds to pKap�5.64�0.01. The limiting molar
absorptivities, calculated from the limiting absor-
bances reported by Albert and Serjeant [6], are
3.04�103 and 1.25�102 l/mol cm, respectively, and
the pKa (more precisely pKap) is 5.62�0.02. Albert
and Serjeant measured the limiting absorbances
assuming that at pH equal to 2, acridine exists com-
pletely in the protonated form, while at pH equal to
about 9.1 it exists in the neutral form. Our calculation
of the mole fractions (Eq. (9) below) showed that
acridine is indeed completely protonated at pH equal
to 2 or less, but the completely neutral form exists at
pH equal to or above 9.5. The X0 and X1 curves in
Fig. 1 show the variation of the mole fractions of the
protonated and neutral molecules. It is clear that at pH
less than 2, the molar absorptivity is due entirely to the
protonated acridine. In the pH range between 4 and 8,
the measured absorptivity has varying contributions
from both the protonated and neutral molecule. As the
pH increases, the measured molar absorptivity gradu-
ally approaches the limiting value of the neutral
molecule which is relatively low.
Benzidine (diaminobiphenyl) is also chosen as an
example of a compound having two overlapping pKas.
Albert and Serjeant [6] measured the molar absorp-
tivity of a 0.500 mol/l solution of this compound as a
function of the pH. The ionic strength was stable at
0.010, and the analytical wave length was 300 nm.
The authors used the lengthy and elaborate successive-
approximation approach for the determination of the
molar absorptivity of the monoprotonated benzidine,
and for the calculation of the pKas. They reported the
following results. �0�0.00 for the diprotonated solute,
R�NH�3 �2, �1�9.733�103 l/mol cm for the monopro-
tonated solute, RNH2NH�3 , and �2�1.644�104
l/mol cm for the neutral (nonprotonated) molecule,
R(NH2)2. The calculated average pKas were:
pKa1�3.43 and pKa2�4.65. On the other hand, when
the measured molar absorptivities vs. pH are ®tted into
Eq. (2) according to the method described in Part I, the
following results are obtained, �0�2.93�102,
�1�9.914�103, and �2�1.588�104 l/mol cm. The dis-
sociation constants obtained are Kap, 1�2.43�10ÿ4
and Kap, 2�2.35�10ÿ5. They correspond to
pKap, 1�3.61�0.01 and pKap, 2�4.63�0.01. The
Fig. 1. Molar absorptivity of acridine vs. pH. Wavelength, 403 nm.
(x) Experimental points, (e) fitted curve (Eq. (2)), (X0) mole
fraction of undissociated acridine, and (X1) mole fraction of
dissociated acridine (Eq. (9)).
268 I. Jano, J.E. Hardcastle / Analytica Chimica Acta 390 (1999) 267±274
differences between the values obtained by the suc-
cessive-approximation method and the values found
in this work are due to an inherent systematic bias of
the successive-approximation approach, as will be
shown shortly.
Fig. 2 shows the variation of � vs. pH of benzidine.
The points marked (x) are the experimentally mea-
sured ones, and the (e) curve is calculated using
Eq. (2). The (Xr) curves in Fig. 2 are the variations
of the respective mole fractions in terms of the pH of
the solution.
Fig. 2 shows clearly that in the pH range between
about 3 and 5, the diprotonated, monoprotonated, and
neutral species coexist, causing the pKas of benzidine
to overlap. It is also clear that the molar absorptivity at
pH between 1 and 2 is very small, but nevertheless is
not zero as was assumed by Albert and Serjeant. As
the pH increases, � increases and approaches the
limiting value, �2, of the neutral molecule which
becomes dominantly present in the solution.
To assess the accuracy of the successive-approx-
imation approach of Albert and Serjeant [6], we used
the parameters obtained by that approach to calculate
the � vs. pH curve using the exact Eq. (2). The curve
obtained in this manner is presented as curve (a) in
Fig. 3. The pH-scale is enlarged for clarity, and is
limited to the experimental pH range between 3 and 5
[6]. The points marked (x) are the experimental points,
and the line labeled (b) is calculated by the nonlinear
iterative method described in Part I. The systematic
error of the successive-approximation method is
obvious. Similar results are found with other com-
pounds.
As another example of overlapping pKas we con-
sidered the case of 3-aminobenzoic acid. This com-
pound has a carboxylic group (±COOH), and an NH2
group which exists as NH�3 in strong acidic solutions.
The experimental � vs. pH data [6] are ®tted into
Eq. (2). The iterative calculation lead to the following
result, �0�679 l/mol cm for the protonated molecule,
�1�275 l/mol cm for the ionized carboxylic group
(H�3 RCOOÿ), and �2�811 l/mol cm for the comple-
tely dissociated (H2NRCOOÿ) solute. The calculated
apparent dissociation constants are Kap, 1�7.04�10ÿ4
and Kap, 2�1.87�10ÿ5. They correspond to
pKap, 1�3.15�0.01 and pKap, 2�4.73�0.01, respec-
tively. Fig. 4 shows the variation of � vs. pH. The
experimental points are marked as (x), and the curve
(e) is calculated from Eq. (2). The Xr curves represent
the variations of the respective mole fractions. The �vs. pH curve has two sigmoid branches. One branch, in
the pH range below 4, corresponds to the ionization of
the carboxylic group, and the other branch at higher
pH values represents the dissociation of the protonated
Fig. 2. Molar absorptivity of benzidine vs. pH. Wavelength,
300 nm. (x) Experimental points, (e) fitted curve, and (X0), (X1),
(X2) mole fractions of undissociated, first dissociated and second
dissociated species, respectively.
Fig. 3. Molar absorptivity of benzidine vs. pH. (a) Successive-
approximation method ([6]), (x) experimental points, and (b) fitted
curve (Eq. (2)).
I. Jano, J.E. Hardcastle / Analytica Chimica Acta 390 (1999) 267±274 269
amine group, H�3 Nÿ. At pH�3.9 the mole fraction, X1,
of H�3 NRCOOÿ reaches a maximum value of 0.753
and becomes the main contributor to the absorbance.
The molar absorptivity of this entity at the analytic
wavelength ��280 nm is relatively small. For this
reason the � vs. pH curve goes through a minimum at
about pH�3.9. As the pH increases, the molar absorp-
tivity of 3-aminobenzoic acid increases and becomes
equal to the molar absorptivity of H2NRCOOÿ (8 1 1)
at about pH�9.0.
We also compared the successive-approximation
method with the method used in this work. For this
purpose, the parameters obtained by the ®rst method,
as reported by Albert and Serjeant [6], are utilized to
calculate the � vs. pH curve using Eq. (2). The para-
meters used are �0�681.3, �1�283.3, �2�812.1,
pKa1�3.075, and pKa2�4.798. Fig. 5 displays the
result. The dotted curve (b) is the one obtained with
the successive-approximation parameters. The line
labeled (a) is calculated according to the procedure
described in this work. The (x)-points represent the
experimental measurements. The systematic error of
the successive-approximation method is evident.
It is apparent from Figs. 3 and 5 that the successive-
approximation method has an inherent bias. It tends to
shift the calculated absorptivity towards lower pH as
compared with the experimental measurement. It also
has the disadvantage of requiring the premeasurement
of some of the limiting molar absorptivities. Besides,
it is not applicable for solutes having more than two
ionizable groups (n >2). The method presented in this
work, on the other hand, does not have any of these
limitations. The only requirement for its application is
that the analytic wavelength should be chosen such
that the different dissociated species have different
limiting molar absorptivities. To ®nd a suitable wave-
length, the absorption spectra of the acid or base at
different pH values should be examined, and the
proper wavelength is then selected [for more detail
see [6]].
3. Reversed-phase high performance liquidchromatography
It was recognized in 1970s that weak acids, bases,
and zwitterionic compounds could be separated by
reversed-phase high performance liquid chromatogra-
phy (RP-HPLC) using nonpolar stationary phases
[7±9]. It has been also demonstrated that RP-HPLC
can be used to determine the dissociation constants of
polyprotic acids and bases from the measurement of
Fig. 4. Molar absorptivity of 3-aminobenzoic acid vs. pH.
Wavelength, 280 nm. (x) Experimental points, (e) fitted curve,
and (X0), (X1), (X2) mole fractions of undissociated, first
dissociated and second dissociated species, respectively.
Fig. 5. Molar absorptivity of 3-aminobenzoic acid vs. pH. (x)
Experimental points, (a) fitted curve (Eq. (2)) and (b) successive-
approximation curve.
270 I. Jano, J.E. Hardcastle / Analytica Chimica Acta 390 (1999) 267±274
the retention factor of the solute as a function of the pH
of the mobile phase [10,11]. It was also shown that the
observed (measured) retention factor of the analyte is
the sum of weighted contributions from the species
that result from the dissociation of the analyte in the
mobile phase [11]. This implies that the observed
retention factor is an additive property and, conse-
quently, Eq. (1) can be applied. Denoting the mea-
sured retention factor of the acid or base as k, and the
limiting retention factors of the undissociated and
dissociated species as kr (r�0, 1, . . ., n), Eq. (1) is
then written as:
k � k0 �Pn
r�1 kr Ke�r� erx
1�Pnr�1 Ke�r� erx
: (3)
This equation has been derived on the basis of the
solvophobic theory of RP-HPLC [11,12], and was
used for the determination of the dissociation con-
stants. Some examples are presented here.
Hardcastle et al. [10] studied the chromatographic
behavior of some Leukotrienes which are usually
produced in very small quantities in the living cells.
In the following examples, the cases of Leukotriene
B4 (LTB4) having one carboxylic group, and Leuko-
triene E4 (LTE4) having three ionizable groups, are
considered. The formulas of these compounds are
shown in Fig. 6. The experimental details can be
found in [10].
The measured k vs. pH data for LTB4, obtained in
50% organic modi®er (acetonitrile) mobile phase, are
®tted into Eq. (3). The calculation produced the fol-
lowing results. The limiting retention factors are:
k0�3.86, k1�1.22 (k is dimensionless). The apparent
dissociation constant is Kap�9.124�10ÿ7 correspond-
ing to pKap�6.04�0.01. Fig. 7 displays the variation
of k as a function of the pH of the mobile phase. The
line labeled (k) is calculated with Eq. (3), and the
points designated as (x) represent the measured data.
The mole fractions of the undissociated and disso-
ciated solutes are represented by the (X0) and (X1)
graphs. The two mole-fraction curves intersect at
X�0.5 as expected. The pH at the intersection point
is equal to the pKap (6.04). This is generally true only
in the case of monoprotic acids and bases. In the case
of polyprotic acids and bases with overlapping pKas,
the pH at the intersection of two mole-fraction curves
does not necessarily equal a pKa of the acid. The proof
of this can be found elsewhere [11].
Fig. 7 reveals that at pH less than 3.5 LTB4 is
undissociated, and its retention factor is equal to the
limiting k0 (3.86). Above pH�8.5 LTB4 is completely
dissociated, and its observed retention factor is equal
to the limiting factor k1 (1.22).
Fig. 8 represents the case of LTE4. The experimen-
tal measurements are made in 55% organic-modi®er
mobile phase [10]. The parameters found by ®tting the
experimental data to Eq. (3) are k0�2.74, k1�0.97,
k2�0.40, k3�6.10, pKap, 1�3.09, pKap, 2�5.74, and
pKap, 3�9.44. The (k)-line in Fig. 8 is calculated,Fig. 6. (a) Leukotriene B4 (LTB4) and (b) Leukotriene E4 (LTE4).
Fig. 7. Chromatographic retention factor k of LTB4 vs. pH. (x)
Experimental points, (k) Fitted curve (Eq. (3)), and (X0), (X1) mole
fractions of undissociated and dissociated species, respectively.
I. Jano, J.E. Hardcastle / Analytica Chimica Acta 390 (1999) 267±274 271
and the points designated by (x) are the experimental
points. The variations of the mole fractions are repre-
sented by the Xr curves.
At low pH (acidic mobile phase) LTE4 is proto-
nated, and its retention factor is equal to the limiting
factor k0 (2.74). As the pH increases, the observed
retention factor k decreases, reaches a minimum, and
then increases to become equal to the limiting factor k3
(6.10). It can be seen from the variations of the mole
fractions of the different species that the retention of
the ®rst dissociated ion (having one ±COOÿ and
H�3 Nÿ group) is less than the retention of the proto-
nated acid. The lowest retention is that of the second
dissociated acid having two ionized carboxylic
groups. The completely dissociated acid has the high-
est retention (6.10). In other words, it is relatively the
least solvated in the mobile phase. The reason of this is
not clear. It could be due to some structural or steric
changes that partially block the ionized carboxylic
groups from being solvated in the mobile phase.
4. Hyperpolarizability
It was shown recently that dissociation constants of
weak monoprotic acids could be determined from
measurements of hyperpolarizabilities using hyper
Rayleigh scattering technique [13]. The hyperpolariz-
ability of the acid solution is related to the intensity of
the scattered second-harmonic light, I2!, according to
the following relation
B2 � I2!=�G I2!�; (4)
where B is the hyperpolarizability of the solution. I! is
the intensity of the incident light, and G a constant
depending on the scattering geometry and the wave-
length used in the measurement. G can be determined
by a separate calibration procedure. Eq. (4) helps
obtain the hyperpolarizability B of the solution by
measuring the intensities of the incident and scattered
lights. The hyperpolarizability has contributions from
both the solvent and solute molecules. If the solute is
an ionizable weak acid, HnA, then all the species in the
solution contribute to B, and we may write in general
B2 � A�S��2s � A
Xn
r�0
�HnÿrArÿ��2
r ; (5)
where A is Avogadro's number, and [S] the concen-
tration of the solvent (in mol/l). �s is the hyperpolar-
izability of the solvent molecule, and �r is the
hyperpolarizability of HnÿrArÿ. Since the concentra-
Fig. 8. Chromatographic retention factor k of LTE4 vs. pH. (x) Experimental points, (k) fitted curve, and (X0), (X1), (X2), (X3) mole fractions
of undissociated, first-, second-, and third-dissociated species, respectively.
272 I. Jano, J.E. Hardcastle / Analytica Chimica Acta 390 (1999) 267±274
tion of HnÿrArÿ may be expressed as a product of the
mole fraction X(r) and the molarity M of the acid,
Eq. (5) takes the following form:
B2 � A �S��2s � AM
Xn
r�0
X�r��2r : (6)
In dilute solutions, the concentration of the solvent
remains constant, and the hyperpolarizability, �s, can
be obtained from standardization experiment. The
hyperpolarizabilities of some common solvents are
reported [13]. Putting
g � �B2 ÿ A �S��2s �=A M; (7)
allows us to write Eq. (6) in the following form:
g �Xn
r�0
��r� �2r : (8)
The mole fraction X(r) is related to the pH (see Eq.
(12) of Part I):
X�r� � Ke�r� erx
1�Pnt�1 Ke�t� etx
: (9)
The substitution of X(r) into Eq. (8) yields
g � �20 �
Pnr�1 �
2r Ke�r� erx
1�Pnr�1 Ke�r� erx
: (10)
Since the quantity g is measurable, Eq. (10) can be
used to determine the dissociation constants of the
acid and the limiting hyperpolarizabilities of the acid
species.
The derivation of Eq. (10), as outlined above,
assumes that the acid species do not absorb at the
incident light frequency ! or the second-harmonic
frequency 2!. This is generally the case for dilute
solutions. However, in the case of signi®cant absorp-
tion, a correction on the measured scattered light
intensity can be made [13].
The pH of the acid solution can be changed by
changing the molarity of the acid. Therefore, the pH
and the hyperpolarizability can be measured at dif-
ferent acid molarities. Such measurements provide the
required data (i.e. hyperpolarizability vs. pH) for using
Eq. (10).
In the case of a monoprotic acid, the mole fractions
in Eq. (8) can be expressed in terms of the degree of
dissociation � of the acid, that is, X(HA)�1ÿ�, and
X(Aÿ)��. Therefore,
g � �1ÿ ���2HA � ��2
Aÿ : (11)
This equation can be written in a different form.
Substituting g from Eq. (7) and rearranging, the fol-
lowing relation may be obtained:
B2 � Nsol �2s � N0�1ÿ ���2
HA � N0 ��2Aÿ
where Nsol is the number density (molecule/ml) of the
solvent, and N0 is the number density of the acid
before dissociation. This equation was derived by Ray
and Das [13] and used for determining the dissociation
constants of several weak organic acids.
At present, to our knowledge, experimental data for
polyprotic acids an bases (n>2) are still lacking, there-
fore Eq. (10) will not be discussed further in this
report.
5. Determination of the activity coefficients ofindividual ions
In this section, the possibility of obtaining the
activity coef®cients of individual ions is discussed.
The effective dissociation constant Ke(r) of species
�HnÿrArÿ� is related to the activity-coef®cient ratio f(r)
(see Eq. 11 of Part I) as in the following equation:
log Ke�r� � log f �r� � log K�r�; (12)
where K(r) is the thermodynamic constant at zero
ionic strength where f(r)�1. This relation suggests
that the effective dissociation constants, Ke(r)s, of an
acid or base may be determined from the measure-
ments of an additive property at a set of different ionic
strengths. Log Ke(r) can then be extrapolated to zero
ionic strength to obtain the value of the thermody-
namic constant K(r):
log�Ke�r��I�0 � log K�r�: (13)
Eqs. (12) and (13), when combined, lead to the
following relation:
log�f �r��I � log�Ke�r��I ÿ log�Ke�r��I�0: (14)
This equation may be used to calculate the activity
coef®cient ratio f(r) (f(r)� 0/ r) at any ionic strength
I.
To obtain the activity coef®cients, r, of individual
species from the activity coef®cient ratios, f(r), the
I. Jano, J.E. Hardcastle / Analytica Chimica Acta 390 (1999) 267±274 273
activity coef®cient of one species must be known. It is
customary to consider the activity coef®cient of a
neutral molecule, such as undissociated carboxylic
acid, equal to 1. But in general this is not necessarily
true. The activity coef®cients are related to the non-
ideality of the solution which is due mainly to the
interactions among the component solute species [12].
Most neutral compounds of practical interest, includ-
ing zwitterions, have dipole moments, and their inter-
actions with other species in the solution may not be
negligible. This implies that polar acids and bases may
have activity coef®cients differing from unity. There-
fore, to obtain the activity coef®cient of the ionic
HnÿrArÿ from f(r), the activity coef®cient of the
neutral acid has to be calculated separately using a
proper model. Some semiempirical models may be
utilized for this purpose. For example, Pitzer had
developed a theory for treating the thermodynamic
properties of electrolyte solutions [14]. The theory
may be applied for estimating the activity coef®cient
of a neutral acid [see for example Eq. (22) of [15]].
This requires a proper adjustment of some empirical
parameters. On the other hand, the activity coef®cient
ratios, f(r)s, can be used to develop a scale of relative
activity coef®cients for ionogenic acids and bases.
Such a scale may prove to be convenient for the
determination of the thermodynamic properties of
the species that result from the dissociation of weak
polyprotic acids and bases.
It is also important to note here, that the theory
outlined in this paper considers the measured pH equal
to the thermodynamic pH as de®ned by pH�ÿlog
a(H�). In practice, this is only a (valid) approxima-
tion. Therefore, the dissociation constants and the
activity coef®cient ratios calculated from measure-
ments of additive properties vs. pH, as described
above, are meaningful only with reference to the
pH scale adapted in the measurements.
6. Conclusion
We have presented in Part I and Part II of this series
a general theory for the determination of the pKas of
polyprotic acids and bases from measurements of
additive properties in terms of the pH of the solution.
The computational procedure does not require any
approximation, and is applicable to different types of
additive properties. The theory also makes it possible
to trace the evolution of the mole fractions of the
entities in the solution as the pH changes. Thus, it
helps identify the major contributor to the measured
property at any pH value (Eq. (9)). This provides an
insight of the events that take place in the solution of
the analyte while the pH is changing. Applications to
spectrophotometry and reversed-phase high perfor-
mance liquid chromatography reveal the ef®ciency
of the procedure, especially for calculating overlap-
ping pKas. The possibility of applying the general
equation (Eq. (1)) to the hyperpolarizability measure-
ments is made apparent. The theory also suggests
clearly a method for obtaining relative activity coef®-
cients (rather than average coef®cients of positive
and negative ion-pairs) of the ionic species that result
from the dissociation of weak acids and bases. Such
relative activities may prove to be useful for studying
the thermodynamic properties of the mentioned
species.
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