The linear extension of the Debye-Hückel theory of electrolyte solutions

Volume 5. number 2

THE

CHEMICAL PHYSICS LETTERS

LINEAR

EXTENSION

OF

THE

1 March 1970

DEBYE-HUCKEL

OF ELECTROLYTE

THEORY

SOLUTIONS

C. W. OUTHWAJTE Department

of Applied

Mathematics

and Computing Science,

The University,

Sheffieid

SIO 2TN. UK

Received 8 December 1969

The recent linear extension of the Debye- HUckeltheory predicts the breakdownof the Debye-Hitickel theory at Ka= 1.2412.

1. INTRODUCTION Recently a linear extension of the Debye-Hifckel theory of electrolyte solutions [l] has been given [2] which takes into account the fluctuation potential. An approximate differential equation for the mean potential P(1;2) at the field point r2, for an ion fixed at rl, was found for ~12(= ]rl - r2l) 3 2a, where a is the radius of the exclusion volume for ions of equal size. This analysis predicted that for symmetrical electrolytes the Debye-Htickel theory broke down at approximately ua = 1.495. In this Ietter we derive the exact equation satisfied by @(1;2) for 7-12 a 2a for this linear extension of the Debye-HUckel

theory. The derived equation predicts that the Debye-HUckel theory breaks also briefly indicate the relation to Kirkwood and Poirier’s work [3].

down at tia = 1.2412. We

2. THEORY

We consider only symmetrical electrolytes, each ion type having ionic diameter a. Using the notation of ref. [2] we have from eqs. (38) and (42) of ref. [Z] that the modified Poisson-Eoltzmann equation for the mean potential @(1;2) is

fCI

1+-T

%1,,=,] ar2

sin02d021

‘12 > 2a ,

(1)

where 7 = f(c1. Linearising and writing the integral as a surface integral over S23 gives v2%(1;2) = K2

haa2

Transforming

J[ S23

*(1;3) + i$

a!Jji3)]dS23

.

(2)

the surface integral to an integral ever the volume of ~23 eq. (2) becomes

For the first term this transformation is effected by using Green’s reciprocal relation on the functions *(1;3) and l/r2 and for the second term by using the divergence theorem. Kirkwood and Pokier’s theory arises on neglect of the second integral in (3). To simplify eq. (3) we substitute 4(1;2) =u(~)/Y where ‘=~12 and express the integral over ~23 in terms of the dipolar coordinates ?-l,~ centered at ‘1, ‘2. Eq. (3) then becomes

77

Volume 6. number 2

CHEMICAL PHYSICSLETTERS

1 March 1970

(4)

where the limits of integration-are (i)

7-Yl<

72


Y-cI
;

So integrating first w.ith respect to 3’2 then ential equation d2,(

~~ 2(1+7)

d7_2 =

(ii) 9-l - r‘:r2
,

7<71<7+a.

and simplifying

71

yields the integral difference-differ-

1

r+a I((?‘+

6’) + tl(Y

- a) + KS

u(R)d.i?

7

2

(5)

20.

7-a

The correct

physical

solution satisfying

this equation is 50 f~(r) = C A, exp(-.z,z;a)

having positive

real parts,

zcoshz+rsinhz

Preliminary investigation being given approximately 3

,

(6)

72=1

. where 2, are the roots,

of the transcendental = z3(1 tr) ‘TV . Ka

of eq. (7) indicated that for small by

27 l’%

N ln(2 ,‘r2)

The root 21 is the one instrumental

equation

where 7: =

pfa2

=

(7)

there are two real positive

roots,

these

2 3 6~~(1+ 7)/(6+ 67 - 37 -7 ).

in obtaining the Debye-Htickel

result.

As

Ka

increases

(8)

21 and 22

gradually merge and coalesce at KU = 1.2412. For KC-I >1.2412 the roots of eq. (7) are complex conjugates so that eq. (5) predicts a damped oscillatory soluticn, however the validity of eq. (5) is then suspect because of the linearisation of eq. (1) and the linearisation involved in the derivation of the fluctuation potential. As yet we cannot derive the constants A,, of eq. (6). This is because we do not yet know the equation satisfied by U(Y)(i.e. rk(1;2)) in the region (5 s r s 20 so that we do not have a complete potential problem. This equation for fl(?‘) in the region a s 3’ c 2a awaits the solution of the intersecting sphere problem for the fluctuation potential problem [2]. Kirkwood and Poirier obtained their integral equation by using an expansion in powers of the charging parameter &’and retaining all terms at least of equation form of their integral equation for Y 2 2a can be simply order E2. The difference-differential obtained from eq. (5) by expanding in powers of K and retaining terms to order K2. Eq. (5) thus represents a generalisation of Kirkwood and Poirier’s results and we would expect activity coefficients derived using eq. (5) to display similar qualitative trends to those calculated from Kirkwood and Poirier’s theory. The relation of eq. (5) to the classical Debye-Hifckel theory can be seen by expanding each term on the right-hand side of eq. (5) about 7. This gives

2 d2”=K2fC+K 1 dv2

722a.

A simple extension of the Debye-Hiickel lecting the derivatives

(3

1

theory containin a natural breakdown is now obtained by negof higher order than d%/dr2 or d5U/dr4 [2].

78 ‘.

Volume 5, number 2

CHEMICAL

PHYSICS

LETTERS

1 March 1970

REFERENCES [1] P.Debje and E.Hiickel, Z.Physik 24 (1923) 185. [2] C.W. Outhwaite, J.Chem.Phys. 50 (1969) 2277. [3] J.G.Kirkwood

and J.C.Poirier.

J.Phys.Chem.

58 (1954)

591.

79