Journal of Analytical Sciences, Methods and Instrumentation, 2012, 2, 1-4
http://dx.doi.org/10.4236/jasmi.2012.21001 Published Online March 2012 (http://www.SciRP.org/journal/jasmi)
1
Formulation of the System of Isohydric Solutions
Tadeusz Michałowski1*, Agustin G. Asuero2
1Faculty of Engineering and Chemical Technology, Technical University of Cracow, Kraków, Poland; 2Department of Analytical
Chemistry, The University of Seville, Seville, Spain.
Received November 6th, 2011; revised November 30th, 2011; accepted December 22nd, 2011
ABSTRACT
The isohydricity (pH constancy) as the property referred to mixtures of acids or bases, is illustrated on a simple example
of the solutions: HL (C0, mol/L) and HL (C mol/L) mixed according to titrimetric mode (pH titration). A new derivation
of the Michałowski formula 1
2
010
p
K
CCC  expressing this property is presented, and its applicability for determi-
nation of pK1 = – logK1 value is indicated. The principle of the isohydric method of pK1 determination is also outlined.
Keywords: Isohydric Solutions; pH Titration
1. Introduction
The term “isohydric” refers to solutions of the same hy-
drogen-ion concentration. According to Arrhenius’ state-
ment [1], expressed in more contemporary terms, “if two
solutions of the same pH are mixed, pH of the mixture is
unchanged, regardless the composition of the solutions”.
This statement is not valid, however, when referred on
any pair of electrolytic systems.
To prove it, let us take, for example, the pair of solu-
tions: C1 = 10–2.5 ≈ 0.003 mol/L HCN (pK1 = 9.2) and C2 = 1
mol/L AgNO3 ( = 2.3 for Ag+ + OH– = AgOH)
OH
1
log K
[2]. From the approximate formulae: 11 H CK



and OH
21
H W CK K


 we get pH = 5.85, for both
solutions. However, as were stated in [3], silver ions
when added into HCN solution act as a strong acid, gen-
erating protons mainly in the complexation reaction Ag+
+ 2HCN = 2 + 2H+, and pH of the mixture
drops abruptly. So, the isohydricity property is limited to
the systems where only acid-base equilibria are involved.
This property was formulated first by Michałowski [4]
for different pairs of acid-base systems, then generalized
on more complex mixtures of acid-base systems, and
extended on mixtures containing basal salts and binary-
solvent systems [5]. Moreover, the isohydricity concept
was the basis for a very sensitive method of determina-
tion of dissociation constants values [4,5].

AgCN 
The present article intends to familiarize the readers
with this interesting concept [6] that is in some relevance
with buffering action [7,8] and pH-static titration [9,10]
principles. Isohydricity concept is also referred to acid-
base homeostasis in living organisms [11].
2. Principle of Isohydricity
It is usually stated that an addition of a strong acid HB (C
mol/L) into a weak acid HL (C0 mol/L) decreases pH
value of the resulting solution and, consequently, shifts
(retracts) the HL dissociation, according to Le Chatel-
ier’s principle. As results from Figure 1, the decrease in
pH value (dpH/dV < 0) and then retracting the dissocia-
tion of HL occurs at higher C values, whereas the dilu-
tion effect, expressed by dpH/dV > 0, predominates at
lower C-values. The related effect depends, however, on
the HL strength, expressed by its dissociation constant K1
value.
Figure 1. The effect of addition of V mL of C = 10–pC mol/L
strong acid HB into V0 = 10 mL of C0 = 0.1 mol/L weak acid
(pK1 = 4.0). The titration curves V = V0·(α – C0/z)/(C – α),
where α = 10–pH – 10pH–14 , z = + 1 are plotted for
different pC = – logC values indicated at the corresponding
curves; pC = 1, 2, 3, ∞.
1pH
10pK 
*Corresponding author.

Formulation of the System of Isohydric Solutions
2
1
[H][L]
[HL]
K


 (1)
and on the relative concentrations (C0, C) of both acids,
i.e. HL and HB. Under special conditions, expressed by
the set of (C0, C, pK1) values [4], pH = const (i.e.,
dpH/dV = 0) when mixing the solutions in different pro-
portions; it is just the subject of the present note.
3. Formulation of the Isohydricity Concept
The simplest system of isohydric solutions is composed
of a strong monoprotic acid HB and a weak monoprotic
acid HL, characterized by pK1 = – logK1 value, where K1
is expressed by Equation (1). In order to derive the rela-
tion for the isohydricity concept, let us consider the titra-
tion of V0 mL of C0 mol/L HL with V mL of C mol/L HB.
From charge and concentration balances
HOH BL

  
 

(2)
0
BCV
VV


  (3)

00
0
HL LCV
VV

 
  (4)
we get

00
00
HOH1 CV
CV n
VV VV

  
 

 (5)
where
[HL]
[HL] L
n


(6)
i.e.

1
1
L
1HL LH
K
n
K




 
  
  (7)
Mixing the solutions can be made according to titri-
metric mode, in quasistatic manner, under isothermal
conditions; it enables some changes in equilibrium con-
stants, affected by thermal effects, to be avoided. As will
be seen later, the ionic strength (I) of the related mixture
is also secured; it acts in favour of constancy of K1 and
ionic product of water, KW = [H+][OH–] during the titra-
tion in the system of isohydric solutions. This way, the
terms: [H+] – [OH–] = [H+] – KW/[H+] and 1 – n (Equa-
tion (7)) in Equation (5) are constant at any stage of titra-
tion in the isohydric system. In particular, at the start for
the titration, i.e. for V = 0, from (5) we have

0
1
H
OHn C

 
 (8)
Comparing the right sides of (5) and (8), we get, by
turns:
 
00
0
00
0
11
1
CV CV nn
VV VV
C
nC
C

 


(9)
HOHC


  
  (10)
Then we get, by turns,
0
1
H
C
KC

1


  
 

(11)
1
0
1
OH 1
W
KC
KC



 
 


(12)
1/2
0
12
4
111
2
W
CK C
KCC

 


 

 

(13)
Assuming that 2
4
W
KC1
, from (13) we get the
formula [4]
1
2
010
p
K
CCC  (14)
Identical formula is obtained for reverse titration,
where V0 mL of C mol/L HB is titrated with V mL of C0
mol/L HL. It means that the isohydricity condition is
fulfilled for the set (C0, C, pK1) where the relationship
(14) is valid, independently on the volume of titrant
added. The related curves expressed by Equation (14) are
plotted for different pK1 in Figure 2 within (pC, pC0)
coordinates. The curves appear nonlinearity for lower
pK1 values and are linear, with slope 2, for pK1 greater
than ca. 6. This regularity can be stated from the Equa-
tion (14) transformed by turns:



11
1
2
0
01
101 10
2log110
pK pK
pC pK
CC C
pCpC pK


 
  (15)
pC
0
Figure 2. The plots of pC0 = –logC0 vs. pC = –logC relation-
ships obtained on the basis of Equation (14), for different
pK1 values indicated at the corresponding lines.

Formulation of the System of Isohydric Solutions 3
and valid for pC significantly smaller than pK1.
It can also be noticed that ionic strength (I) of the so-
lution remains constant during the titration, i.e. it is in-
dependent on the volume V of the titrant added. Namely,
from (2), (10)-(12) we get

 
–––
101 0
I0.5 HOHBL
0.51 1
W
K
CCKKCCC

   
 
 
(16)
It is the unique property in titrimetric analyses, ex-
ploited in the new method of pK1 determination, suggested
in [4,5]. In the light of the Debye-Hückel theory, the con-
stancy in ionic strength (I) is, apart from constancy in
temperature T and dielectric permeability  , one of the
properties securing constancy in K1 and KW values. Refer-
ring again to Figure 1, from Equation (14) we calculate

11
0
0.5 104 1011
pK pK
C

 

C.
For C0 = 0.1, pK1 = 4.0 we have pC = 2.507.
4. An Isohydric Method of Acidity Constant
Determination
The isohydricity property can be perceived as a valuable
tool applicable for determination [4,5] of acidity constant
(pK1 = – logK1) for a weak acid HL, see Equation 1. For
this purpose, a series of pairs of solutions (HB (C), HL
()) is prepared, where C and are interrelated in
the formula
*
0i
C*
0i
C
1
*2
010 i
pK
i
CCC    (17)
where 1i (i = 1,···, n) are the numbers chosen from
the vicinity of the true pK1 value for HL. Within each
pair (HB (C), HL ()), the pH titrations HB (C, V) 
HL (C, V0), are made (see Figure 3). In [4,5], at V0 = 3
mL, n = 5 titrations were made up to V = 4 mL. The re-
sults of titrations are approximated by straight lines pH =
ai + bi·V (see Figure 4), where bi is the slope of the re-
lated line. The plots of the slopes vs. 1i values are
usually arranged along the straight or hyperbolic line
(Figure 5). The point of intersection with 1
pK
*
0i
*
0i
C
pK
pK

line,
corresponding to the slope b = 0 (compare with Figure 3)
provides an evaluation of the true pK1 value, 1 pK

=
pK1. Additional, n + 1 th titration HB (C, V)  HL
(C, V0), made at 0, 1
*
0, 1n1
2* 10
p
K
nprovides a con-
firmation of this pK1 value, if bn+1 value obtained in this
titration is relatively small.
CCC
 
5. Final Comments
The new derivation of the formula (14) expressing the
isohydricity condition for the simplest case of a mixture
composed of strong monoprotic (HB, C mol/L) and weak
monoprotic (HL, C0 mol/L), when mixed according to
titrimetric mode, is presented. The roles of titrand and
Figure 3. The simulated pH vs. V relationships plotted for
the titration HB (C, V)  HL (, V0), at pK1 = 2.87, V0 = 3,
C = 0.01, and

0i
C

0i
C calculated from Equation (17), at dif-
ferent (indicated)

1i
pK values.
Figure 4. Exemplary approximation of experimental (e)
points {(Vj, pHj) | j = 1,···,N} by straight line (m) [4].
Figure 5. Linear (lin) and hyperbolic (hyp) approximation
of experimental (exp, ▲) points in (
1
p
K, b) co-ordinates [4].

Formulation of the System of Isohydric Solutions
4
titrant in such systems can be reversed. The formulae for
more complicated systems are given in [4]. The iso-
hydricity property can be formulated for the systems
where only acid-base equilibria occur Equation (14),
relates analytical concentrations of strong acid HB and
weak acid HL. Such an interesting property is not di-
rectly relevant to buffering action. Nevertheless, it is
on-line with a general property desired from buffering
systems.
The systems of isohydric solutions have a unique fea-
ture, not stated in other acid-base systems. It is the con-
stancy of ionic strength (I), not caused by presence of a
basal electrolyte. The conjunction of properties: pH =
const, I = const, together with constancy of temperature
(T = const) provided a useful tool for a sensitive method
of determination of pK1 values for weak acids, as indi-
cated and applied in [4,5]. This method is illustrated with
some examples taken from [4].
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