J. Biomedical Science and Engineering, 2010, 3, 484-487
doi:10.4236/jbise.2010.35067 Published Online May 2010 (http://www.SciRP.org/journal/jbise/
JBiSE
).
Published Online May 2010 in SciRes. http://www.scirp.org/journal/jbise
The determination of acidity of the dilute solutions of weak
multibasic organic acids
Elene Kvaratskhelia, Ramaz Kvaratskhelia
R. Agladze Institute of Inorganic Chemistry and Electrochemistry, Tbilisi, Georgia.
Received 12 February 2010; revised 1 March 2010, accepted 5 March 2010.
ABSTRACT
The new theoretical method for the accurate deter-
mination of acidity of dilute solutions of weak multi-
basic organic acids (which are widely used in medi-
cine, pharmacology, various branches of industry
and participate in important biological processes in
living organisms) is suggested. The concepts of the
contributions of the separate dissociation steps to the
[H+] value, xm, are used for an analysis of complex
equilibria of the processes of dissociation of these
acids. The cases of weak dibasic and tribasic organic
acids with the “overlapping” dissociation equilibria
and a general case of weak multibasic acids, HnA, are
considered. From the conditions of equality of the
concentrations of various ionized and non-ionized
forms in the dilute solutions of weak multibasic or-
ganic acids the areas of dominance of these forms in
connection with the corresponding xm values are
formulated.
Keywords: Dibasic Acids; Tribasic Acids; Hydrogen
Ions Concentration; Equations
1. INTRODUCTION
Weak multibasic organic acids are widely used in medi-
cine, pharmacology, chemical, food and cosmetic indus-
tries. Some of these acids participate in a series of im-
portant biological processes occurring in living organ-
isms (for example, in the Krebs cycle). The majority of
drugs are weak acids and/or bases. Their biopharmaceu-
tical properties are directly connected with the dissocia-
tion constants and degrees of these compounds, conse-
quently, with acidity of their solutions. The latter is the
very factor which affects in physiological systems the
rate at which the compound is able to diffuse across
membranes and various obstacles, determines the ac-
id-base homeostasis and enzyme kinetics in the cell and
in the body. It is possible to say that an acidity of the
weak multibasic organic acids determines, as a rule, all
their useful (or harmful) properties.
Many weak multibasic organic acids have compara-
tively close values of the dissociation constants for the
various steps; this fact causes their simultaneous partici-
pation in the determining the hydrogen ion concentration
in solutions of these acids (i.e., “overlapping” dissocia-
tion equilibria). In this paper a new theoretical method
for determination of acidity of the dilute (0.0001-0.1
mol·d m–3) solutions of such acids is suggested.
2. RESULTS AND DISCUSSION
2.1. Dibasic Acids
Dibasic acids form the most numerous group of weak
multibasic organic acids with the “overlapping” equilib-
ria effect. In dilute aqueous solutions the primary and
secondary steps of dissociation are
  HAHAH2 (1)
  2
AHHA (2)
In our previous communications [1,2] we have used
the concepts of the contributions to the total hydrogen
ion concentration, [H+], being assigned to the primary
and secondary dissociation steps, x1 and x2, such that x1
+ x2 = [H+]. The corresponding mass-action equations
for both steps dissociation constants are
11 F
xc
xx
F
xc
xxH
K
1
2
2
2
1
1
21
1
)]([







(3)
22 F
xx
)xx(x
F
xx
x]H[
K
21
212
21
2
2





(4)
where K1 and K2 are the thermodynamic dissociation
constants, c is the total (analytical) concentration of acid,
F1 and F2 are the quotients of the activity coefficients:
AH
HAH
f
ff
F
2
1

 (5)
-
HA
AH
f
ff
F
2
2 (6)

E. Kvaratskhelia et al. / J. Biomedical Science and Engineering 3 (2010) 484-487 485
The values of the activity coefficients may be appro-
ximated by the Debye-Huckel equation:
IBa
IAz
f
i
i
i

1
log
2
10 (7)
where ai is the cation-anion distance of closest approach,
A and B are constants depending on the properties of
water at given temperature, zi is the charge of ion. The
ionic strength is given by . The activity co-
efficient of undissociated acid is assumed to be unity.
212xxI 
According to the (3) and (4) the x1 and x2 values (and
then their sum – the [H+] value) can be calculated suc-
cessively by an iterative solution of two quadratic equa-
tions:



























1
1
2
2
2
1
1
1
1
14
2
1
F
cK
x
F
K
F
K
x (8)



























2
12
2
1
2
2
1
2
2
2
4
2
1
F
xK
x
F
K
x
F
K
x (9)
We suggest also the empirical equation for the fast
approximate determination of the pH values of dilute
(0.0001-0.01 mol·dm–3) solutions of weak dibasic (and
tribasic with the low K3 values) organic acids:
cpKpKpH lg)14.0185.1(8.0489.1 11  (10)
The maximum value of the relative error for this equa-
tion for a series of weak dibasic and tribasic organic ac-
ids with the pK1 values in the interval: 2.5-5 does not
exceed 5% (the relative error is the ratio of the differ-
ence between the approximate pH value and corre-
sponding accurate value, divided by the approximate pH
value, and converted to percent).
2.2. Tribasic Acids
In case of weak tribasic organic acids with the “over-
lapping” dissociation equilibria, the mass-action equa-
tions may be expressed as follows:
1
1
21321
1
1
21
1
))(()]([ F
xc
xxxxx
F
xc
xxΗ
Κ






(11)
2
21
32321
2
21
32
2
))(()]([F
xx
xxxxx
F
xx
xxΗ
Κ






(12)
31233
33
23 23
[]Ηx(xxx)x
Κ3
F
F
xx xx



(13)
where
AH
AHH
f
ff
F
3
2
1

 (14)



AH
HAH
f
ff
F
2
2
2 (15)



2
3
3
HA
AH
f
ff
F (16)
and 321 32 xxxI



The x1, x2 and x3 values (and then their sum – the
[H+] value) can be calculated successively by an itera-
tive solution of three quadratic equations:


































1
1
32
2
2
3
1
1
3
1
1
12
4
2
1
F
cK
xxxx
F
K
x
F
K
x (17)


































2
12
31
2
2
1
2
2
1
2
2
23
4
2
1
F
xK
xxxx
F
K
x
F
K
x (18)



























3
23
2
21
3
3
21
3
3
3
4
2
1
F
xK
xx
F
K
xx
F
K
x (19)
2.3. Acids with the Higher Basicity
It is necessary at first to consider the general case of
the weak multibasic organic acid HnA with the “over-
lapping” dissociation equilibria. For this case we may
write the equations connecting the values of x1, x2,
x3, … xn−1, xn with the concentrations of various anions:
x1 = [Hn−1A–] + [Hn−2A2−] + [Hn−3A3−] + … + [HA(n−1)−]
+ [An−] (20)
x2 = [Hn−2A2−] + [Hn−3A3−] + … + [HA(n−1)− ] + [An−] (21)
x3 = [Hn−3A3−] + … + [HA(n−1)− ] + [An−] (22)
xn−1 = [HA(n−1)−] + [An−] (23)
xn = [An−] (24)
In a general form for the m dissociation step we may
write:
xm = [Hn−mAm−] + xm+1 (25)
The total hydrogen ion concentration may be ex-
pressed as follows:




 n
m
n
m
mmn x m
11
m]Η[]H[
 (26)
The mass-action equation for the m dissociation step
may be expressed by the following equations:
m
mm
n
m
mmm
m
mm
mm
mF
xx
xxx
F
xx
xxΗ
Κ











1
1
1
1
1
)(
)]([ (27)
where

486 E. Kvaratskhelia et al. / J. Biomedical Science and Engineering 3 (2010) 484-487
--m
A
mn
H
-m
A
mn
HH
mf
ff
F
)1(
)1( 


 (28)
The equation for an ionic strength may be written as
follows:


n
1m
m
mxI (29)
In case of weak organic acids with the high (more
than tribasic) basicity the conversion of (27) to the
forms of (8-9, 17-19) leads to the very complicated
expressions. That is why we suggest to solve the com-
plicated problem of determining of acidity of the
high-basic acids solutions by more simple method. Let
us consider this method for the most difficult case of
hexabasic mellitic (benzenehexacarboxylic) acid. First,
assume that this acid can be treated as a tribasic acid
(taking into account that the main contribution to the
[H+] value is made by first three dissociation steps).
Then the x1, x2 and x3 values are determined succes-
sively by an iterative solution of (17-19), where the
values of F1, F2 and F3 were assumed to be unity. The
obtained x1, x2 and x3 values are then used for the de-
termination of the initial estimate of [H+]. Then, with
the aid of [H+] value and the iterative solution of the
following equations (which are obtained from (27) for
the corresponding dissociation steps):
44
5434
4][
][
FHK
xFHxK
x



 (30)
55
6545
5][
][
FHK
xFHxK
x



 (31)
66
56
6][ FHK
xK
x

 (32)
the initial values of x4, x5 and x6 are determined (assum-
ing that F4, F5 and F6 values to be unity). These values
are used for a correction to the [H+] value:


6
1
]H[
m
m
x
and then obtaining the final (for this stage) x4, x5 and x6
values. Then with the aid of the following equations
(where F4, F5 and F6 values are assumed to be unity):
11
211
1][
][
FHK
xFHcK
x



 (33)
22
3212
2][
][
FHK
xFHxK
x



 (34)
33
4323
3][
][
FHK
xFHxK
x



 (35)
improved x1, x2 and x3 values are obtained. At the fol-
lowing stage, with the aid of the obtained six xm values,
the ionic strength I is calculated with the aid of (29).
The values of the activity coefficients of H+ and all
anions are approximated by the Debye-Huckel (7).
With the aid of the activity coefficient values and (28)
the F1, F2, F3, F4, F5 and F6 values are calculated. Us-
ing these values in (30) to (35), corrected values of all
six xm values are obtained. With the aid of the latters,
the final [H+] value is determined.
2.4. The Use of the Xm Concept for a Determination
of the Concentrations of the Ionized and
Non-Ionized Forms and their Distribution in
the Dilute Solutions of Weak Multibasic Acids
The determination of the xm values gives us also the op-
portunity to calculate the important dissociation pa-
rameters: the concentrations of all ionized and non-ion-
ized forms of weak multibasic organic acids in their di-
lute solutions. For this goal in general case of HnA acid
can be used (25) and (26). Taking into account also the
following equation:
1
][ xcAHn

(36)
We can formulate the conditions of an equality of the
concentrations of ionized and non-ionized forms:


n
m
mn xxcAHH
2
1
2]:[][ (37)
211 2]:[][ xxcAHAH nn

 (38)
321
2
2]:[][ xxxcAHAH nn 

 (39)
11
]:[][ 

 mmn
m
mnxxxcAHAH (40)
nn
nxxcAHA  
1
]:[][ (41)
Taking into account these conditions, we may formu-
late the areas of dominance of various ionized and non-
ionized forms of acid:


 n
m
mn xxcAHH
2
1
2]:[][ (42)
(and vice versa)
211 2]:[][ xxcAHAH nn

 (43)
(and vice versa)
3212 ]:[][ xxxcAHAH n
2
n

 (44)
(and vice versa)
11
]:[][ 

 mmn
m
mnxxxcAHAH (45)
(and vice versa)
nn
nxxcAHA  
1
]:[][ (46)
(and vice versa)

E. Kvaratskhelia et al. / J. Biomedical Science and Engineering 3 (2010) 484-487 487
3. CONCLUSIONS
Many weak multibasic organic acids have the compara-
tively close values of the dissociation constants of the
different steps. This fact causes the participation of all
steps in formation the hydrogen ions total concentration
in the solutions of such acids. We suggest the new theo-
retical method for a determination of acidity of these
solutions using the concepts of the contributions of the
separate dissociation steps to the [H+] value, xm. The
equations for the accurate calculation of the [H+] values
in cases of dibasic and tribasic acids and also in general
case of weak multibasic acid, HnA, are suggested. The
comparatively simple method for a determination of
acidity of the dilute solutions of high-basic (more than
tribasic) acids is also described. With the aid of the for-
mulated by us conditions of an equality of the concentra-
tions of various ionized and non-ionized forms in the
dilute solutions of weak multibasic organic acids the
areas of dominance of these forms in connection with
the corresponding xm values are formulated.
REFERENCES
[1] Kvaratskhelia, E. and Kvaratskhelia, R. (2007) The
degrees of dissociation of weak multibasic organic acids.
Joural of Solution Chemistry, 36(6), 787-792.
[2] Kvaratskhelia, R. and Kvaratskhelia, E. (2000) Voltam-
metry of dicarboxylic acids on solid electrodes in aque-
ous and mixed media. Russian Journal of Electro-
chemisry, 36(3), 330-333.
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