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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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332

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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