Solution of Onsager's equation in closed form

Physica A 152 (1988) 459-468 North-Holland, Amsterdam

SOLUTION

OF ONSAGER’S

EQUATION

Douglas 6 Brandy

Court,

IN CLOSED

FORM

K. McILROY Calamvale

Received

4116, Australia

12 May 1988

The problem of the Wien dissociation of a weak electrolyte to which a uniform electric field X is applied is analysed by direct solution of the partial differential equation governing the distribution of pairs of associated ions in the electrolyte (Onsager’s equation). The method of solution is based on the correct choice of the characteristic numbers of the weak electrolyte which leads to a transformation of Onsager’s equation into two parts both of which are identically zero in this problem. The solution reproduces Onsager’s result (see L. Onsager, J. Chem. Phys. 2 (1934) 599) in the form of a definite integral for the distribution function describing associated ion pairs. The mathematical details of Onsager’s solution have apparently never been published previously for this result which leads directly to his well known equation for the relative increase, K(X)IK(O), in the dissociation constant of the weak electrolyte.

1. Introduction In 1934 Onsager’) published his equation for the relative dissociation constant of a weak electrolyte to which a uniform intensity X is applied in the form

K(X) = 4Pww”*1 2(-pqy2 K(O)

increase in the electric field of

(l-1)

.

J, is the ordinary Bessel function of order one, q = -eiej12DkT > 0 is the Bjerrum association distance and p = IX(e,wi - ejoj)/2kT(wi + wj) where ei, ej are the charges of the ions, oi, wj are their mobility coefficients, D is the dielectric constant of the medium, T is absolute temperature and k is Boltzmann’s constant. Eq. (1.1) is a simple deduction from Onsager’s solution’) for the distribution function f(~-, 0) describing associated ion pairs of the weak electrolyte, 29

f(r, 0) =

9e

pr(cos

e-l)+zq/r

I [

J, (-S@)“’

cos i] ems”’ ds

0

0378-4371/88/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

,

(1.2)

D.K. McILROY

460

where J, denotes the ordinary Bessel function of order zero, r is the distance of separation of an i, j ion pair, 0 is the angle between r and X and K is a constant equal to v,K(O) where vij is the number of “bound” ions. The derivation of eq. (1.2) “involves elaborate analysis” ‘). Unfortunately Onsager apparently never published the details of his analysis which has led to considerable uncertainty amongst

researchers

in this field as to the generality

and applicability

of eqs.

(1.1) and (1.2). In this article we give an independent analysis of the partial differential equation governing f(r, 0) (Onsager’s equation) arriving at eq. (1.2) as our solution. Our analysis involves a transformation of Onsager’s equation using the appropriate forms of the characteristic numbers of the weak electrolyte and we show that the transformed equation may be divided into two parts both of which are identically zero in this problem.

2. Mathematical 2.1.

formulation

The governing

Onsager’s is’)

r2

differential equation

equation,

the equation

governing

df

d2f

the distribution

function

d’f

f(r, O),

df

~+(1+2r-2Erlcos8)~+~+(cot~+2ersin~)~=O,

(2.1) where E = 2pq. In eqs. (1.2) and (2.1) r and 0 are spherical polar coordinates, there being no dependence on the third coordinate, 4, due to symmetry about the applied electric field vector X. In eq. (2.1) the variable r has been made dimensionless on division by 2q which will serve temporarily as the only characteristic number for the system and we have renormalized f by dividing by v,K(O). lii

The distribution eC”y(r,

0) = 1

function

,

f(r,

f3) must

!im_ f(r, 0) = 0,

satisfy the boundary

conditions

(2.2)

the first being a manifestation of the Maxwell-Boltzmann (equilibrium) distribution at the distance of closest approach of the i, j ions and the second the requirement of complete dissociation of ions pairs at infinite separation of the ions. The relative increase in the dissociation constant of the weak electrolyte is obtained through the flux integral of associated ion pairs across a closed surface surrounding the origin as

SOLUTION

OF ONSAGER’S

EQUATION

461

$$$=-~j($+(+2ecosB)~)r2sinBdB. 0 2.2.

(2.3)

The Jield-free solution

In zero applied electric field a true equilibrium applies everywhere in r, 8 space. The solution for the total distribution function, f,,,, is therefore the Maxwell-Boltzmann distribution &(r,

0) = el”

(2.4)

The total distribution function is the sum of f(r, 0) and the distribution function describing dissociated ions and since the latter is unity (r+(O) in non-normalized units) we have f(r, f3) = el” - 1

(25)

which of course satisfies eqs. (2.1) and (2.2) when E = 0. 2.3. Asymptotic behavior of f(r, 0) Firstly we observe that by defining h(r, 0) as h(r, 13) = f(r, 0) e-” coss

eq. (2.1) is rendered

(2.6)

in separable form: 2

a2h r2 _ + (1 + 2r) $ 61r2

- e2r2h = -

$

+ cot 8 2

+ E cos 8h

,

(2.7)

and if we seek a solution of this equation in the form h(r, 0) = R(r)@(e), where R is a function of r only and 0 is a function of 8 only, we obtain for the R equation 2 d2R

r -g

+ (1 + 2r) z

where (Yis the separation r

2 d2R + 2r g dr2

-

- (e2r2 + cx)R = 0,

constant.

l2r2R =

0

(2.8)

For large r eq. (2.8) reduces to (2.9)

and if, in the standard way, we set R = S(r) exp(- i J (2/r) dr) = S(r)lr we find that

462

D.K. McILROY

d2S

l2s=o,

dr2-

(2.10)

which has the solution

S(r) = A eer + B emEr ,

(2.11)

where A and B are constants. Since the second boundary condition (2.2) requires that A = 0, we have from eqs. (2.6) and (2.11) f(r,

0)

_

;

e4cosH-‘)

as r-00.

of eqs.

(2.12)

Our method of solution will be to transform eq. (2.1) via transformations of the independent variables to a form in which the new dependent variable is not explicitly dependent on E. Thus we remove the e-dependent factor given in eq. (2.12) from f(r, 0) and define the function g(r, x) such that g(r, x) = f(r, f3) epE’+‘) where

x A cos 0. From

r’ 9

lii 2.4.

eqs. (2.1)

+ (1 + 2r - 2er2)

and the corresponding e-“‘g(r,

,

$

boundary

x) = 1 ,

!iir

(2.13) and (2.13)

+ (1 - x2) 2

conditions

we then

- 2x 2

have

+ E(X -

1-

2r)g = () , (2.14)

are from eqs. (2.2)

g(r, x) = 0.

(2.15)

Characteristic numbers of a weak electrolyte

We have previously described 2q as the characteristic length of the weak electrolyte. This length adequately describes the situation in the case of zero applied electric field because in this case the electrolyte is isotropic, the mean ionic trajectories of dissociation being directed radially from the origin. However, application of a uniform electric field bends these trajectories towards the direction of the applied field’) and an isotropic characteristic length such as 2q may be inadequate in describing this distortion. To investigate the structure of a weak electrolyte in the presence of a uniform applied electric field consider the total electrical potential @(r, x) of an i,j ion pair. This is the sum of the Coulomb potential and the potential due to the applied field:

SOLUTION

OF ONSAGER’S

(2.16)

- 2EYX )

@(r, x) = -;

which has been non-dimensionalized reference direction $, for potential @(Y, X) will

463

EQUATION

attain

a maximum

by division by kT and which has as the 19,= 7r/2. For given values of E and X, at r = rmax, say,

value

which

is given

by

Ymax = (26X)-l’*. This is finite for 0 c 8 < ~r/2 but is complex for rr/2 < 0 c n and so would give rise to an unsatisfactory characteristic length, 2qrmax, for the electrolyte. This difficulty may be overcome by introducing an “effective potential” Ge defined as follows. The mean relative velocity, U, of the oppositely charged ions of a pair is given by’) u = -grad

@ - grad(log

(2.17)

f)

for both processes of ionic association and dissociation. function g(r, X) defined by eq. (2.13), eq. (2.17) is u = -grad(@

+

lr(x

- 1)) - grad(log

Since @(r, x) is the potential eqs. (2.17) and (2.18) that

associated di, defined

Qe = @ + er(x - 1) = -5

-

lr(x

g) .

with the function by + 1)

In

terms

of the

(2.18) f(r, 0) it follows from

(2.19)

is the “potential” which is associated with the function g(r, x). Qe(r, x) has the same value as @(r, x) at 8 = 0 and has a maximum value for given values of E and x at r = rmax where r max= (E(l + x))-1’2 .

(2.20)

The reference direction for @Jr, x) is $ = n whilst rmax is finite for 0 < 8 < n, has the minimum value (2~))“~ at 8 = 0 and increases monotonically as 8 increases from zero. This definition of rmax may therefore lead to a suitable radial characteristic length 2qr,,_ possessing properties as prescribed above in the first paragraph of this section. Accordingly, we next transform the r coordinate by division by rmax and define

(2.21)

464

D.K. McILROY

where y L 1 + X. Eq. (2.21) y coordinate

suggests

that the appropriate

transformation

of the

is

y = (Eyy which implies

(2.22)

, that the characteristic

number

for the y coordinate

is ych where

(2.23) at 8 = 0 and ych decreases monotonically The maximum value of yC, is (2/e)“’ as 8 increases from zero and therefore is anisotropic as required. Eqs. (2.21) and (2.23) lead to the following simple relations: rly = r , Setting

ry = et-y.

(2.24)

g(r, x) 2 G(F, y) eq. (2.14)

becomes

(2.25) with the boundary

conditions

(2.15)

becoming (2.26)

3. Mathematical

solution

If we next assume that G(F, y) is not an explicit have from eq. (2.25)

and

function

of E then we must

SOLUTION OF ONSAGER’S

EQUATION

465

We first consider the solution of eq. (3.1) which is a parabolic equation whose characteristic directions are defined by (3.3) This has the solution ~7= CF where C is a constant, and we therefore canonical coordinates tJ(l, 9) =

c r

)

77=

77(r,Y>

define the

(3.4)

and H(t, 7) 2 G(F, 7) whence eq. (3.1) becomes +

57

2

a77_ +

ar ay

72

- 4(@ + y2)H = 0.

(3.5)

- However for a parabolic equation the choice of n(r, y) is arbitrary. n = Y eq. (3.5) reduces to

If we take

(3.6) and on setting H( 5, y) = Z( 5) Y(y)

in this equation we have

(3.7) and 4dz+ d5

(

y-;

>

z=o

where y is the separation constant. Eq. (3.7) is the modified Bessel equation

(3.8)

of order zero with the solution

466

D.K. McILROY

Y(Y) = EZJ(4

+ y)“2y]

+ FK,[(4

+ #“Y]

,

(3.9)

where E and F are constants and Z,, K,, are respectively the modified Bessel function of the first kind and modified Bessel function of the second kind both of order

zero.

Eq.

(3.8)

is easily

shown

to have the solution

Z( 6) = L( emy5’4 ,

(3.10)

where L is constant. Since H( 5, ~7) must be finite at y = 0 (i.e. at 8 = n) and the function K,[(4 + Y)~‘*?] contains the factor log[(4 + r)“‘Y/2]Z0[(4 + Y)“~Y] we must set F = 0. Thus the solution to eq. (3.6) becomes

H(t,Y)=Mte

7t’4z,[(4

M is constant.

where

G(f

>7)

=

M

z

The solution

e-Y94r

Z”](4 +

r

or, in terms

+ y)“*y]

of the original

(3.11)

)

of eq. (3.1)

is then

(3.12)

YY2Yl7

variables,

-y/4r g(r,

x)

=

M !?.--r

ZlJ{E(4 + Y)(l +

If we next set y = 4(s - 1) eq. (3.13) g(r, x; s) = M $

Z,[{4~s(l

(3.13)

W’l

may be rewritten

+ x)}“‘]

e-“”

.

as (3.14)

Clearly eq. (3.14) satisfies the second boundary condition of eq. (2.15) provided only that s is not large and negative, but it does not satisfy the first boundary condition. However, consider b

b

I

a

l/r

g(r, x; s) ds = M >

I a

Z,(2s”*j)

e-“” ds ,

(3.15)

which is also of course a solution of eq. (3.1) for all values of the constants a and b. On substitution of this function expressed in terms of F and 7 we find that the left-hand side of eq. (3.2) equals

467

SOLUTION OF ONSAGER’S EQUATION

*

r

ej(l_b)‘i{ b”2( 1 - b)z;(2b”2~)

+ b( 1 - b)z,(2b”*y)

/r>

2kly2 - eY(1-a)‘r{a1’2(1 - a)Z~(2a1’*~) + a(1 - a)Z0(2a1’*~)lY} , ?

(3.16)

(where the ’ denotes differentiation with respect to the argument) which is identically zero if and only if a = 0, 1 or ~0and b = 0, 1 or ~0.Thus eq. (3.15) is a solution of eq. (3.2), and eq. (2.25), only when a = 0, b = 1 or a = 0, b = 00 or a = 1, b = 00. However on integrating by parts we have from eq. (3.15) b

M ell’ r

I

Z0(2s1’2y) eeslr ds = M{e(1-a)‘rZo(2a”2~)

+ O(r e(l-o)‘r)}

a

(3.17)

as r+O, so that eq. (3.15) satisfies the required boundary condition in this limit if and only if a = 0, M = 1. Furthermore considering the case a = 0, b = 00we have using the identity3) cc

I

x.ZO(ax)e -P2x2,-Jx

=

21

e

-l?i4p=

(3.18)

2P

0

that

_.

Zo(2s112j9 e-Y””

&

=

__ __

eYlr+rY

(3.19)

0

which does not satisfy the second boundary condition of eq. (2.26). (This solution could have been obtained more directly by choosing the arbitrary function ~(7, j) = ‘y in the analysis of eq. (3.1).) Clearly the remaining - solution, i.e. the case a = 0, b = 1, satisfies the boundary condition as r/y + 00. Thus from eqs. (2.13) and (3.15) the solution of Onsager’s equation which satisfies the boundary conditions of eq. (2.2) is 1

rr(cos0-1)+1/r

f(r, 0) = e

r

1 Zo[(R~~)1’2 cos i]

emsir ds ,

(3.20)

0

which agrees with Onsager’s solution, eq. (1.2), since Z,(x) = J,(k). Since 5 = y/r = 1 /r, the transformation r to r turns out to be unnecessary in the foregoing analysis; all that is needed is the transformation y to 7. This

D.K.

468

implies length

that a weak electrolyte

2q and the number

McILROY

may be completely

characterized

by the radial

yC, = ( yl~)"~.

References 1) L. Onsager, J. Chem. Phys. 2 (1934) 599. 2) D.P. Mason, D.K. McIlroy and C.J. Wright, Proc. Roy. Sot. Lond. A 371 (1980) 413. 3) I.N. Sneddon, Special Functions of Mathematical Physics and Chemistry, 3rd ed. (Longman, London, 1980), chap. 4.