chi alvo 1987
TRANSCRIPT
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IONIC HYDRATION MODEL FOR UNI-UNIVALENT
ELECTROLYTES. CALCULATION OF THE MEAN
HYDRATION NUMBER FROM EXPERIMENTAL SOLUTE
ACTIVITY COEFFICIENT AND WATER ACTIVITY DATA
M.R. GENNERODECHIALVO~II~ A.C. CHIALVO
Programa de Electroquimica Aplicada e lngenieria Electroquimica (PRELINE), Facultad de Ingenieria
Quimica (U.N.L.), Santiago de1 Ester0 2829, 3000 Santa Fe, Argentina
(Received 16 July 1985; in revisedform 14 July 1986)
Abstract A new thermodynamic treatment is proposed to evaluate the mean hydration number and to
correlate the experimental mean molal activity coefficients. The model is interpreted in terms of consecutive
hydration equilibria. The results are comparable to those given in the literature.
“A,“,
i
_i
m
l?Z*
n1
n:
n2
Y
Y’
PSRS
IDSRS
NOMENCLATURE
distance of closest approach
solvent activity (PSRS, scale concentration:
molar fraction)
mean hydration number
mean ionic hydration number
hydration degree of the cation
hydration degree of the anion
molality (referred to the total solvent)
molality (referred to the free solvent)
number of moles of free solvent
number of moles of total solvent
number of moles of solute (electrolyte as a
whole)
number of moles of the i-hydrated cation
number of moles of the j-hydrated anion
anion
cation
ionic specific constants
chemical potential of the solvent
chemical potential of the solute
chemical potential of the i-hydrated cation
chemical potential of the j-hydrated anion
standard state chemical potential of the solute
(IDSRS, scale concentration: molality)
activity coefficient (IDSRS, scale concentra-
tion: molar fraction)
activity coefficient (IDSRS, scale concentra-
tion: molality)
mean ionic activity coefficient (scale concentra-
tion: molality)
mean ionic activity coefficient (scale concentra-
tion: molar fraction)
Perfect Solution Reference State
Ideal Solution Reference State
INTRODUCTION
Different ionic hydration models[l-61 have been
proposed to quantitatively explain discrepancies ob-
served between the experimental activity coefficients of
aqueous electrolyte solutions and the theoretical ones
calculated using
the
Debye-Huckel equation. More
recent theoretical[7-91 and experimental[ 10-121
studies have shown that water structure surrounding
an ion in solution cannot be described by simple
models such as those included in the previously
mentioned literature. In this respect, an important step
forward was made by Stokes and Robinson[13] by
extending their primitive model and taking into ac-
count the occurrence of consecutive equilibrium steps
in the cation hydration process. Critical comments
were made since anion hydration was neglected in their
model and there is enough experimental evidence on
both ions being hydrated[1 12]. Besides, similar
models for the anion hydration in gas phase had
already been developed[ 14, 151.
This paper describes a new approach to ionic
hydration, taking into account the ion-solvent interac-
tion of both species.
On
this basis, the mean hydration
number of uni-univalent electrolytes is evaluated.
FUNDAMENTALS
The behavior of water molecules around an ion in
solution may be described by the changes produced in
their spatial distribution. In the closest region around
simple ionic species, the solvent exhibits a more
ordered structure. This order decreases with an in-
creasing distance to the ion, the mobility of water
molecules consequently increasing. A dynamic equilib-
rium between both free and bound solvent molecules is
thus established. This description ignores the presence
of other species originated by ion-ion interactions such
as ion-pairing, dissociation equilibria, etc Con-
sequently, this model based on a mechanism of
consecutive hydration equilibria is restricted to the
case of simple, uni-univalent, unassociated electroiytes
having low to moderate concentrations.
The fundamental hypotheses in the proposed model
are:
(a) The hydration process is developed through a
series of consecutive equilibria:
A- +jH20=[A(H10)j]- 0
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M.
R. GENNERO DE CHIALVO AND A. C. CHI LVO
The assumed extreme value for the hydration of
each ion is S(m) = 55.51/2 m, which corresponds to the
maximum availability of solvent molecules for each
ion.
The total free energy of the solution may be written
as
SW
S(m)
G = PI + c
n(iMj)+
c
WA4
(3)
j-0 i=o
and also in the form
By considering (1) and (2)
di) = di = 0) +jh
(5)
p i) = p i =
O)+ip,
(6)
and the following mass balances:
S(m) SW
n1 =
n, +
1 in(j)+C
in(i)
(7)
j=O i=o
S(m) S(m)
n2 = C
n(j) = C
n(i).
(8)
j=O i=o
From Equations (3)-(8)
p2 = p( j = 0) + ~(i = 0).
(9)
By developing the chemical potentials in Equation
obtained:
r;,,
1
m*(j =
O)m* i = 0)
y’+ j = 0,
i = 0)
= m:x(j = O)x(i = 0)’
14)
The resolution of Equation (14) requires first to
evaluate the concentration of non-hydrated ions.
Using the hydration equilibrium constants of the
reaction (l), Equation (5) and developing the solvent
chemical potential in the perfect solution reference
state (PSRS-scale concentration: molar fraction), it
follows:
m*(j)=r.(~~o)i”_‘(j=O) (15)
with the following boundary condition:
S(m)
jzom*(i) = m: =
m2
(1 - 0.018 mzh)
(16)
where the mean hydration number, which is the sum of
the mean ionic hydration numbers of the anion and the
cation, is defined as:
s(m)
s(m)
1
jm* (j) C im*(i )
h=h,+h,=j=“mr +‘=Om
(17)
By combining Equations (15) and (16)
m*(j = 0) =
m2
.(18)
(9) in the ideal dilute solution reference state
(IDSRS-scale concentration: molality) and taking into
account that the molal concentration of a hydrated
species is
m* (j) = m(j)/(l
-0.018
m2 h),
the following
can be obtained:
/1;’ + RTln
(mz y:,,)’ = p”’ (j = 0) + p” (i = 0)
+ RTln {m* j = O)m*(i = O)[y’ , (j = 0, i = O)]‘ }.
(10)
By reordering Equation (10) and taking limit values,
the equation of the solute chemical potential becomes:
= p”’ (j = 0) + p’O’
i = 0) + RT
In x( j = 0)
x(i = 0)
(11)
where
lim m*(j)
~ = x(j)
(12)
m2-10 m2
lim m*(i)
__ = x i).
(13)
mJ-~ m2
Equations (12) and (13) give the ratio between the
species having a given hydration degree (j or i) and the
total of species having different hydration degrees, at
infinite dilution for the anion and the cation,
respectively.
From Equations (lo)-(13), a basic relation between
the exnerimental and the mean ionic activity coefficient
corresponding to the non-hydrated ionic species is
A similar equation can be obtained for the cation.
(b) At infinite dilution, the relation between the
species having a hydration degree j and all hydrated
species has been defined as x(j). It is assumed that the
amount of the anhydrous species [x( j = 0)] as well as
that of highly hydrated species [x(j B hA)] are negli-
gible. Hence, the functional dependence between x(j)
and the hydration degree is approached through a
simple distribution law given by:
x(j) = (x(j =
O)+k,j’)K,j =
K j)x j = 0). 19)
By taking into account the boundary condition:
Xx(j) = 1, and considering x(j = 0) 1,
k,
can be
eliminated, the x(j) expression becoming:
j2K;j
x(j) = _,
(20)
For the cation, a similar equation is obtained. The
hydration constant can also be written as
K(j)
= x(j)/x(j = 0).
Therefore, by combining Equations
(18) and (20), and assuming that m(j = 0) 4 I:
m(j >
l),
it follows that:
m*(j = 0) =
I
Hi =
0)
(21)
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Ionic
hydration model for uni-univalent electrolytes
333
where activity coefficients remain to be evaluated.
(c) Finally, it should be noted that, apart from the
coulombic effect, there is a contribution to deviation
from the ideal behavior
due to the increase in the ionic
species size with an increasing hydration degree.
Conway and Verrall[16] analyzed this effect by split-
ting the activity coefficient into two contributions (y
= y,y.,), where ye is the activity coefficient calculated
using the Debye-Hucket equation and y,, is the non-
electrostatic contribution in this case given by:
(-+55.5’J
*(j)
Y”,(i 2 1) =
i m2
m2
/
.m*(i) I
55.51
exp(j--1)
’ 2
-_
h
m2
>
~
*A
1
x
m2
m*(j) 55.51
(22)
1
--_+++_.
m2
Jm
jJ
yne j=O)= 1.
(23)
Taking into account Equation (23), and considering
that the electrostatic effect is the same for both species,
it follows
y(j)ly(j = 0) = r.,(j).
(24)
Expressions similar to Equations (19b(24) are ob-
tained for the cations.
The molality activity coefficient y’* (j = 0,
i = 0),
on
the other hand, is calculated using the molar fraction
activity coefficient y + (j = 0,
i = 0)
which is obtained
from the Debye-Huckel expression, using the follow-
ing conversion equation[l]
y,~ ci = o
,
i = o) = Y f (j = 0, i = O)(L -O.Olgm )
[l-O.O18m,(h-v)]
.
(25)
By introducing Equations (21) and (25) in Equation
(14), the final expression relating experimental data
with the parameters of the proposed model is:
The other expressions needed for numerically
evaluating Equation (26) are obtained replacing
Equation (21) in Equations (17) and (22).
RESULTS
The calculation of mean hydration numbers was
made using experimental data on ionic activity coef-
ficients and water activity at 25”C, the concentration
ranging from 0.1 to l.Om[17]. The correlation was
carried out, at each concentration, by applying
Equation (26)along with Equations (17), (21), (24) and
the DebyeHuckel expression. In the latter, the molar
concentration was employed in ionic strength calcu-
lations, the distance of closest approach being taken as
adjustable parameter.
The following electrolytes were analyzed: NaCl,
NaBr, LiCl, LiBr. KC1 and KBr. Tables 1-6 show the
results obtained by computer calculations, the vari-
ation of the mean hydration number being given as a
function of electrolyte composition. Similarly, the
experimental activity coefficients are compared with
those correlated using Equation (26), such comparison
Table 1. Sodium chloride
m
h
Yexo &al
0 1
2.521 0.778 0.775
0.2
2.518 0.735 0.730
0.3
2.515 0.710 0.105
0.4 2.512 0.693 0.689
0.5 2.509 0.681 0.678
0.6
2.506 0.673 0.67 1
0.7 2.503 0.667 0.666
0.8
2.499 0.662 0.662
0.9 2.496 0.659 0.660
1.0 2.493
0.657
0.659
1.2 2.486 0.654 0.659
1.4
2.419
0.655
0.662
1.6 2.471 0.657 0.665
1.8 2.464 0.662 0.671
2.0 2.456 0.668
0.677
2.5 2.435 0.688
0.696
3.0 2.412 0.714
0.722
3.5
2.387 0.746 0.755
4.0 2.360 0.783
0.190
4.5 2.330 0.826
0.830
5.0 2.296 0.874
0.873
5.5 2.258 0.928 0.924
6.0 2.218 0.986 0.976
Table 2. Lithium chloride
m h
ysxp
Yt
0 1 4.941 0.790
0.799
0.2 4.907 0.757 0.755
0.3
4.872
0.744 0.741
0.4 4.836
0.740 0.735
0.5 4.798
0.739
0.735
0.6 4.174
0.743
0.737
0.7 4.745
0.748
0.742
0.8 4.702
0.755
0.749
0.9 4.670 0.764 0.757
1.0 4.628
0.774 0.766
1.2 4.554 0.196
0.788
1.4 4.488 0.823
0.813
1.6
4.410
0.853 0.841
1.8
4.317 0.885
0.878
2.0
4.260
0.921 0.904
2.5
4.063
1.026 0.991
3.0
3.770
1.156
1.098
3.5
3.611
1.317
1.223
4.0 3.435
1.510
1.356
4.5
3.304
1.741
1.524
5.0 3.203
2.020
1.712
5.5
3.105
2.340
2.129
6.0 3.024
2.720
2.223
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M. R. GENNERO
DE CHIALVO AND
A. C.CHIALVO
Table 3. Potassium chloride
Table 5. Lithium bromide
m
h
0.1
2.078
0.2 2.078
0.3 2.077
0.4 2.077
0.5 2.076
Z: : 2.076.075
0.8 2.075
0.9 2.074
1.0 2.074
1.2 2.073
1.4 2.072
1.6 2.070
1.8 2.069
2.0 2.068
2.5 2.066
3.0 2.064
3.5 2.061
4.0 2.059
4.5 2.056
Yex, &II
_
0.770 0.762
0.718 0.711
0.688 0.681
0.666 0.661
0.649 0.646
0.637.626 0.635.626
0.618 0.619
0.610 0.614
0.604 0.610
0.593 0.604
0.586 0.601
0.580 0.599
0.576 0.599
0.573 0.600
0.569 0.607
0.569 0.614
0.572 0.633
0.577 0.650
0.583 0.670
Table 4. Sodium bromide
m
h
7-P
Y cal
0.1 5.123 0.796 0.794
0.2 5.092 0.766 0.764
0.3 5.061 0.756 0.752
0.4 5.030 0.752 0.748
0.5 4.997 0.753 0.750
0.6 4.964 0.758 0.754
0.7 4.929 0.767 0.761
0.8 4.893 0.777 0.770
0.9 4.855 0.789 0.780
1.0 4.817 0.803 0.792
1.2 4.736 0.837 0.818
1.4 4.623 0.874 0.858
1.6 4.526 0.917 0.892
1.8 4.423 0.964 0.930
2.0 4.325 1.015 0.970
2.5 4.110 1.161 1.084
3.0 3.915 1.341 1.218
3.5 3.705 1.584 1.377
4.0 3.535 1.897 1.561
4.5 3.470 2.280 1.785
5.0 3.245 2.740 2.011
5.5 3.140 3.270 2.286
6.0 3.070 3.920 2.609
m
h
Ysxp Y i
0.1 2.712 0.782 0.779
0.2 2.707 0.741 0.741
0.3 2.702 0.719 0.718
0.4 2.697 0.704 0.705
0.5 2.692 0.697 0.696
0.6 2.687 0.692 0.691
0.7 2.681 0.689 0.688
0.8 2.676 0.687 0.686
0.9 2.670 0.687 0.685
1.0 2.665 0.687 0.686
1.2 2.655 0.692 0.689
1.4 2.640 0.699 0.695
1.6 2.637 0.706 0.702
1.8 2.619 0.718 0.711
2.0 2.606 0.731 0.721
2.5 2.578 0.768 0.751
3.0 2.545 0.812 0.787
3.5 2.504 0.865 0.828
4.0 2.464 0.929 0.875
Table 6. Potassium bromide
being made up to concentrations appreciably greater
than the one. used in fitting calculations.
The specific constants calculated for each ion,
having an inverse dependence on the mean ionic
hydration number, are:
KLi: 2.15
K,,: 9.0
K,: 140.0
K,,: 80.0
K,,: 17.0
and the values for the a” constant obtained for each
salt are:
a ,: 4.16 C s,: 4.47
(I ~~,: 3.89 a ,,: 4.28
a : 3.22
agB :
3.40.
m h
YeXp YL
0.1 2.270 0.772 0.766
0.2 2.268 0.722 0.718
0.3 2.266 0.693 0.689
0.4 2.264 0.673 0.670
0.5 2.262 0.657 0.656
E 0.258.260 0.636.646 0.648.641
0.8 2.257 0.629 0.635
0.9 2.254 0.622 0.630
1.0 2.252 0.617 0.626
1.2 2.248 0.608 0.623
1.4 2.244 0.602 0.623
1.6 2.240 0.598 0.622
1.8 2.237 0.595 0.622
2.0 2.233 0.593 0.627
2.5 2.224 0.593 0.639
3.0 2.213 0.595 0.655
3.5 2.204 0.600 0.674
4.0 2.196 0.608 0.694
4.5 2.189 0.6 16 0.718
5.0 2.182 0.626 0.749
DISCUSSION
The expression herein dealt with takes into account
different contributions to the experimental activity
coefficient of simple, unassociated, uni-univalent elec-
trolytes for concentrations ranging from 0.1 to 1.0 m.
The influence of ion-ion interaction appreciably de-
creases as concentration increases. The Debye-Huckel
approach including well-known mechano-statistical
deficiencies, was used to evaluate the coulombic contri-
bution; its use seems adequate since, in the concentra-
tion range considered, the contribution of long-range
interactions is lower than the one related to the
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Ionic hydration model for u univalent electrolytes
335
hydration process.
The ionic hydration phenomenon,
on the other hand, originates a decrease of the free
water content in the solution, thus increasing actual
concentration and appreciably affecting the activity
coefficient. Finally, deviations from ideality, produced
by species having different size, and stemming from
different ionic hydration degrees, are considered.
The hydration process occurs in successive equilib-
rium steps between the ions and the solvent, yet
following an evolution clearly different from that
proposed by Stokes and RobinsonC13J. The variation
of the hydrated species ratio at infinite dilution, x(j or
i).
as a function of the hydration degree (j or
i)
corresponds to a distribution of solvation states for-
mally similar to the one experimentally found by
Kebarle et al.[ 151 for the hydration of ions in the gas
phase, or to the distribution of coordination numbers
theoretically calculated by Chandrasekhar et ~I.[93 for
ions in solution.
The results obtained show an inverse dependence
between the mean ionic hydration number, II,-, and the
ionic radius for the
cations. In the case of the anions, h,
increases when the ionic radius increases. These facts
could be explained by considering the different-
ion-solvent interactions on the part of cations and
anions[
133.
On the other hand, a direct relation between the
distance of closest approach and the hydration degree
of the electrolyte should be noticed. This fact could be
explained by considering a0 as the sum of the anhydr-
ous ion radius and the average radius of the ionic
species with different hydration degrees, which form
the ionic cloud of the Debye-Huckel model. For
example, the results
obtained for the series
LiCl-NaCl-KC1 show that, despite the cation radius
increasing,
a” decreases as a consequence of the
marked reduction of the average radius of the hydrated
species produced by the reduction of the hydration
degree.
Finally, it is concluded that the mean ionic hydration
number, representing a weighted average of the dif-
ferent states occurring in solution. greatly depends on
concentration. This can be clearly appreciated in the
case of LiCl, where it was found to have a value close to
5 at infinite dilution, while in a 6 m solution its value
became 3.02. Similar situations are given in the remain-
ing systems analyzed in this work.
Another remarkable fact appeared when analyzing
the prediction capability of the equations developed in
this work: using the parameters correlated at concen-
trations up to 1 m, a good fitting of the experimental
data at molalities as high as 6 was observed.
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