Mutual diffusion coefficients in electrolyte solutions

ELSEVIER

Journal of Molecular Liquids 73,74 (1997) 403411

Mutual diffusion coefficients in electrolyte solutions. 0.

Bernarda,

T. Cartailler”,

a Laboratoire 75252

d’Elcct,rochimie

Paris

Cedcx

b Department Rico,USA

URA

430, Universitb

Pierre

et Marie

Curie,

8 Rue Cuvier

05 FR.ANCE

of Physics,

P.O. Box 23343

University

of Puerto

Rico,

Rio Piedras,

Puerto

00931-3313

In this solutions

paper

the mutual

are calculakd

netted

Chain

level.

The

(HNC)

is found between

diffusion

coefficients

of LiCl,

NaCl

for a large range of concellt,rations a.nd Mean Spherical

theoretical

results

theory

are represented plateau

P. Turq”’ and L. Blum’

in terms

(MSA)

are compared

and experiment. of their

seen at high concentrations.

approximations

at the MC Millan

with experimental The

average

data.

mut,ual diffusion

ionic diameters.

A comparison

and KC1 in aqueous

(up to 1M) using the HyperGood

coefficients

Our theory

between

Mayer

agreement of the salts

explains

also the

HNC and MSA is made.

1. INTRODUCTION In previous Onsager-Fuoss modern port

work [I, 2, 3, 41 a Green’s hydrodynamic

equilibrium

coefficients

experiment

With

of clrct,rolyte

solutions

EPDF’s,

as other thermodynamic

theory

by a similar

based

It gives expressions

(HNC)

[ll,

theory

agreement

such as activity

with

approximation

12, 131 EPDF,

in a satisfactory

uses

for the trans-

[I] [2] [3] were obtained.

properties,

on the

[5] [6]. This theory

Using the mean spherical

electrolytes

equilibrium

quantities

response

and has led to very satisfactory

chain equation

and complex

were also described

linear

was developed

(EPDF).

lligh salt concentrations:

for simple

the same

function, relations

functions

lo] or t.he hypernetted

conductances coefficients

pair distribution

for fairly

(MSA)[7,8,9,

continuity

manner

the electric Self-diffusion (41.

coefficients,

as well

[14][15] were found to yield very good agreement

with

experiment. In this paper thermodynamic

we discuss

the case of mutual

and electrophoretic

a good test for the self-consistence justification

corrections

of the MSA-HNC

of this \rork is the abundant

tration

range.

mutual

diffusion

Moreover

new experimental

cocfhcients.

*to whom any corrrspo~&we

or chemical diffusion. This requires both [6]. Th e p resent theory will therefore be

The Taylor’s

numbrr

teclrniqltes tube

transport

of available

t.heory. The experimental data in the medium

made easier

tcclmique

should be addressed

0167-7322/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PI1 SO167-7322(97)00083-4

concen-

the determination

allows for fast and reasonably

of

404

precise mutual diffusion coefficients measurements, and increases theory of mutual diffusion, similar tc that of conductance.

the need for a modern

2. THEORY 2.1. Normal Modes In this paper wr cxtrnd our previous work [I, 2, 3,4] to the obtention of expressions for the concentration tlrpcndc~nce of the mutual dilTusion coefficients of ions in a continuum solvent. For each ionic spccirs i the hydrodynamic continuity rqua.tion can be written as:

2 +a.J;: = 0

(1)

where C; is the concentration of the i, J: its current density. J; is a function of the autodiffusion coeffirirnt at infinite dilution Dp, the charge Z;, the bulk concentration C,? and the activity corClicient of the i-th species [16]:

where (3) Here lig is Boltzmann’s

constant

and T is the absolnte

temperattIre

in degrees Kelvin.

or elecThe last term in the right hand side of eq.(2) includes the hydrodynamic trophoretic corrections due to all ions k. The cocficient nib which gives the equilibrium correlation of ions i and !i is calculated from the distribution functions hik(r): (4) where 7 is the pure solvent viscosity

and Pk the drnsity

in ions per litre.

The electric field l? verifies the Poisson equation:

-,

divE = where e is the elementary dielectric constant.

charge,

~0 the permitt.ivity

of vacuum

and & the solvent’s

As in our prcviolls work, we will assume small deviations from equilibrium: The equilibrium concrntrabion C,’ is indicated by t.hr superscript zero, and the deviations from equilibrium SCi are always small C; = Cf f SC,

(6)

405 Using the electronrutrality CCpZi

condition

for the equililxium

concentrations:

= 0

(7)

t then only the flur1.11ating trrms

in equation

(5) rema.in: (8)

Then equation

(1) 1.0 first order in 6Ci becomes:

(9) NOW we have for the mutual Di, = DpJij + piDf!!$X

coefficients

+ kBT& 3

Qij = Zje’&

D:3 = D,i -

0

(ZiDY +

the system of equations: + IceTp, C fiik?

Pj

1

k

kBTC Z/c&) k

2 Di,

(10)

3

Using the Normal-Riodcs method to solve the system mutual diffusion coefficient D, [17]: D

=

QnD;,

+

of equations

(9), we find for the

QnD;z

(11)

m &II

+

Q22

There are two ways to derive the mutual coefficient, Dzj 1) Using the electconeutrality the Helmholtz free energy

condition

and the symmetry

of the second derivative

of

(12) 2) Using the Gibbs-Dnhem

relations

we find: (13)

406

These two different approaches should be equiva.lmt for the exact pair correlation functions! but not necrssarily so for approximate tlleorirs. For this reason we have taken in this paper the first approach for the HNC calculalions and the second approach for the MSA treatment. We should remark that whilr the HNC is generally regarded as more accurate does not Iratl t,o explicit analytical expressions, while the MSA, which is simpler, and perhaps not. as acrllrat r, but leads to expliri( I;nmulas that. are reasonably accurate when the energy rollt v l’or t IIC thermodynamic quant itics is usctl.

2.2.

HNC approximation The non ideal trrms all,,,/apj

dlllri -

dPj

= -e;j(q =

Eq.(9) can hc rspressed

as follows: (14)

0)

where
(15) Here &j(q) is the Fourier transform of the direct correlation to ion-ion pair correlat,ion function by the Ornstein-Zernike

kjtr) = gijtr)

-

l = ct~(~)+

CPk k

function cij(r) which is related equation [18]:

(16)

JCik(T13)hkj(T32)diS

The second term of 111~right hand side of eq.(lT,) contains the mutual Coulombic interactions, which will modify the non ideal terms of fiik. However, because of the local electroneutrality condition is already included in 111~electric field equation eq.(8). The pair correla.tion function gij(r) and the dirrct correlation function “ii(r) are calculated using an appropriate closure relation. We choose the HNC closure [ll] [la] [13]: gij(T)

=

exp

( -g

+

hi,(r) -

where the solvent avera.ged ion-pair 2iZj-Z” Uij(r)

=

-

47r&eor

(17)

Cij(r))

R

+ P(r)

potentials

l/;j(r)

have the following form:

(18)

The first term represents the Coulombic interactions between ions i and j. The second term contains the short ranged part of the potential, which is appropriate solvent-averaged interaction. This allows LIS to calculate dZny;/apj which are used in the calculation of the mutual diffusion co&cient Eq.(12).

407

MSA approximation

2.3.

For the hard sphrre pressions

part of the compressibility,

we used the Percus-Yevick

(PY)

ex-

[19, 201:

apppy

(1 + 2iy = ~(1 - 0”

dp;

where C =

(19)

x(p+ + p_)o’/G. u

The MSA theory

yi&ls

sure or compressil)ility trostatic

contribllt

coefficient

is the ionic average

accurate

tliamcter.

thermodynamics

ro11t.r~ are notoriously

in the energy route only. The pres-

inaccurate.

ion is ol)t a.ined by differential

From the energy route the elec-

ion of the MSA cspression

of the osmotic

[9, lo]:

@PMSA

4ILBZ,2

----=

(20)

4( 1 + 2rcr)( 1 + rC7)2

dPi

where Ic is the invrrsc x2 = 4*LB

c

of the Debye’s

length,

given

by

&Z,2

LB is the Bjerrum’s

(21) length,

given by

e2

LB = &kBT r is a screening

paramrtrr

It is related 2r(l

+ ra)

of the MSA.

to K by the simple relation

= K

(23)

Let us now turn to t.he electrophoretic integration

of the function

$pj*

“Y

=

The

MSA

Bjerrum’s

fpA

corrections:

For the hard sphere

from the PY theory

contribution,

([19]) gives

(24)

1+2(_

electrostatic

contribution

to R;j

can bc expanded

in a power

series

LBztp,zj

(25)

37/JX(1 + ra)2

For the conductivity the electrophoretic

where the driving force of thr ion j is (ZjeE), term in the series.

contribution

of the conrrntrations

or we use higher

in the

The first order term is

would be the dominant gradient

hij deduced

1 - c/5 + <“/lo

length.

= --2

‘3

[9, IO]

ordrr

cancels

l?ither

term

implies that

since in eq. (13) the forces are proportional

and not the charges.

term in LB.

the electrophoretic

But for thr mutual diffusion eq.(25)

to the

we ignore this term altogether,

The next t,rrnm of these corrrlation

functions

can be

408

approximated expression

by hF5,“)*/2.

bySAl N 2

Z?Z,“L,‘,

- 2(;+

In order to conserve

erp( -2K(r

analytical

results,

we choose a simple

- o))

(26)

rz

r-0)”

We get to second ortlci

fl!?’ V

LiZ,Z/JjZ,’

N

39(1 + ray

where E,(x)

enp(Xk)&(Xa)

is the csponcnt,ial

integral defined

by:

(28) 3. RESULTS

AND

DISCUSSION

In this paper we studied electrolyte solutions of the monovalent salts: LiCl, NaCl and KCl. The mutual diffusion coefficients of these salts in solution are known for a large range of concentrations (i.e. from 10e3M to IM). The experimental data ([Zl]-1251);s compared to theorctiral calculations given by eq(l1) in the MSA and HNC. ._

I),,, (IO-” m* s-‘)

D, (10-9 1112s-1)

1.8 1.7 -

2,2 I 2

I

I

I

I

1.8

1.6 1.5 -

1.6

1.4 -

1.4

1.3 -\._.J.‘./ 1.2 **..._ ___” ....___..__._.__....... ...~~~~~~~--~ -------_________

1.2

,. I .’

.’

/.M’

/. /#’

1.1 1 0

0.2 fl

0.4 mol”‘/

0.6

0.8

1

0

I

I

I

I

0.2

0.4

0.6

0.8

1

dm3i2

Figure 1. Mutual diffusion coefficients D, as a function of the square root molarity: data, 0 HNC calculations 0 experimental and full line MSA calculat,ions.

Figure 2. Different contributions as a function of the square root molarity: full line all contributions, . . . all contributions except the second order electrophoretic term, -conlombic

contribution

sphere contributions trophoretic p111.s7:“)

terms

7:

without

7,F” and without and

hard elec-

. - .- 7; effect (7;”

without electrophoretic

terms.

409

First we use average ionic diameters to discuss the dependence with concentration of the mutual diffusion coeficients. In the last section a. comparison between MSA and HNC equations will be given. In our calculat.ions the autodifusion coefficient,s at infinite dilution are taken in the literature ([26]). VIC tliiTcrent ionic diameters RW adjusted in order to reproduce experimental data in a I;lrgr range of concentrations. Figure 1 presrnts rspcrimmtal mutual cliffusion coefficients for LiCl, NaCl and KC1 aqueous solutions and the calculated values given by MSA and HNC equations. The mutual diffusion cwfficicn!s bchaviour is similar for the three salts in aqueous solution: The mutual diffusion coefficients decreases with roncentra.tion a,t low concentrations and flattens out at higher concrnt,ra.tions. In all cases a good agrc-rmcnt is observed be(.wcen experimental values and theoretical ones. In table 1 we report the average diameters: ??

a(crys.)

??

u(A) is the average diameter

adjusted

to reproduce

conductivities

??

a(ri)

adjusted

to reproduce

activity

??

a(Dm) is the average diameter

Table 1 Different

is the crystallographic

is the average diameter

average diamrter

Diameter

tvne

o(crys.) g(A) 4%)

a(Dm)

diameter

adjusted

to reproduce

mutual

data [27].

coefficients

[14]

diffusion coefficients.

in .&. LiCl

NaCl

IiCl

2.49 3.1 4.1s 4.1s

3.14 2.95 3.76 3.35

2.78 3.1s 3:IS 3.25

All average diamctrrs are always bigger than tllc crystallographic diameters which is probably connected to ionic solvation. The comparison between u(crys.), u(h), u(a) and u(D,) do not allow us to extract a systematir law. There is no apparent systematics because the non ideal terms in the conductivity t.hcory ([l]), in the activity coefficients theory [14] and in the mutual diffusion coefficients model (this work) depend on different average diameters, whrre the individual ionic diamet,ers are weighted differently each time. The contributions to the non ideal in the mutua.1 diffusion are: ??

the activity

coefficient

??

the electrophorctic

effect dln~;/dCj

effect.

The activity cocficicnt 7; consists of two terms: the coulombic cont,ribution y,“.

the hard sphere contribution

-yysand

410

In figure 2 we show the effect of +yF (i.e. without ryS contributions terms) and the effect. of yi (i.e. without electrophoretic terms).

and electrophoretic

The coulombic part of the activity coefficient describes reasonably well the mutual diffusion coefficient, at low concentrations, but underestimates experimental values at higher concentrat,ions. Tllr hard sphere contribul.ions to the activity coefficient and the electrophoretic &Y-I. are small at low concentra(.ions, as expected. At higher conccnt,ra.l.ions, if we add only to ihe roulomhic pn.rt of activity coefficients the hard sphere contribution, the theoretical bchaviour overestimates the experimental data. To explain 1.11~p1atea.u of the experimental tla,ta at high concentrations we have to take into account. ihr act,ivity coefficient effect. (roulombic plus hard sphere parts) and the electrophoretic corrections. The HNC approximation and MSA treatment. give similar results at concentrations lower than 0.5M. In t.he ca.se of more concentrated solutions HNC theory is in better agreement with experimental data than MSA calculations. The approximation used in eq.(26) for the the total distribution functions hcSA (r) is t.oo poor in this range of concentration. LB acknowledges

support

from NSF grant CHE-9513588

and OSR 9452893.

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