The Route to Infinite Dilution

12 The Route to Infinite Dilution JAN J. SPITZER and H. P. BENNETTO

Downloaded by UNIV OF ARIZONA on March 11, 2017 | http://pubs.acs.org Publication Date: June 1, 1976 | doi: 10.1021/ba-1976-0155.ch012

Department of Chemistry, Queen Elizabeth College (University of London), Campden Hill, London, W.8

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hückel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hückel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log γvs. concentra tion curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed.

T

he theory of D e b y e a n d H u c k e l has survived m u c h c r i t i c i s m since the appearance of their celebrated paper (I). This is no doubt because of the

s i m p l i c i t y and essential correctness of the l i m i t i n g laws ( 2 , 3 , 4).

Nevertheless,

many modifications of their treatment have failed to provide a convincing picture of the interionic effects a n d " s t r u c t u r e " i n the concentration range of practical importance (5, 6).

T h e work presented here was stimulated b y the difficulties

of extrapolation encountered i n a mixed-solvent e m f study (7), a n d contradicts current trends suggesting that the inadequacy of the D H theory for a l l but very dilute solutions springs solely f r o m the crudity of the original model. T h e authors propose a more realistic m o d e l that allows the ions to be polarizable a n d leads to m a r k e d l y different results. Modification

of the Debye-Huckel

Theory

M a n y authors attribute the limitations of the D H treatment to a failure of the electrostatic approach. W e take the view, however, that the n o n c o u l o m b i c 197

Furter; Thermodynamic Behavior of Electrolytes in Mixed Solvents Advances in Chemistry; American Chemical Society: Washington, DC, 1976.

198

THERMODYNAMIC

BEHAVIOR

OF

ELECTROLYTES

aspects have been over-emphasized, and regard w i t h particular caution the i n clusion of D H formalism into models that treat the short-range forces explicitly. Some insights have come f r o m various elegant theoretical approaches (8, 9,10), but these are r e v i e w e d elsewhere (2, 3,6,11) a n d w i l l not be dealt w i t h here. T h e Structured I o n i c C l o u d . W e base our treatment on the general solution of the linearized P o i s s o n - B o l t z m a n n equation ( L P B E , E q u a t i o n 1). V ^ = xty

(1)

2

w h i c h is consistent f r o m the electrostatic, t h e r m o d y n a m i c , a n d statistical-mechanical points of v i e w . Downloaded by UNIV OF ARIZONA on March 11, 2017 | http://pubs.acs.org Publication Date: June 1, 1976 | doi: 10.1021/ba-1976-0155.ch012

1

It is usually w r i t t e n i n the r a d i a l l y s y m m e t r i c f o r m

on the assumption that the ionic atmosphere is spherical (2).

As shown later, this

choice is indeed appropriate for the D H model, but not for ours.

The penetrating

analysis of F r a n k and Thompson (16) demonstrated the implausibility of the D H radial solution for an average separation of ions (in water) of about 100 A ; and, following their suggestion that a more satisfactory picture is to be found i n a lattice distribution of ions, we show h o w the concept of a structured ionic c l o u d can be a c c o m m o d a t e d w i t h i n the D H f r a m e w o r k . A w e l l - k n o w n result of the D H theory is that the charge dq i n the v o l u m e element Airr dr 2

(Figure l a ) .

has a m a x i m u m value at a distance K ~

1

f r o m the central ion

T h e ionic c l o u d can be reduced to a charge on an i n f i n i t e s i m a l l y

2

thin shell placed at a distance (a + K~ ) f r o m the center of the central ion such 1

that the potential caused b y the reduced ionic c l o u d is

D ( l + ica), and the total f i e l d at r = fl is

A structure m a y be imposed on the ionic c l o u d b y supposing that dq i n the volume element dv = r sin Sdddipdr has a finite number, n , of m a x i m a similarly situated at K ~ f r o m the surface of the central ion ( F i g u r e l b ) . B y analogy, this non-radial atmosphere is reducible to a corresponding array of point charges, and this device later enables us to formulate the necessary boundary conditions. 2

1

The nature of approximations involved in the derivation of the LPBE remains obscure (2), and after the analyses of Fowler (12), Onsager (13), and Kirkwood (14), it appears that no more can be learned about them from a statistical-mechanical argument. Following the early pronouncement of Guggenheim and Fowler (15), supported by other analysis (4), we consider the LPBE the only logical choice for a model in which the ions obey the laws of electrostatics. 1

All symbols have their usual meaning in the c.g.s. system of units, as given in Ref. 3. The common interpretation that the central ion 'sees" its ionic cloud at a distance k~ away is valid for the point-charge model only. For the D H second approximation the ionic cloud can be reduced to a charge located on a spherical surface at K SO as to maintain a constant potential at the surface of the central ion. Therefore, it cannot be replaced by a point charge. 2

1

- 1



12.

SPITZER AND BENNETTO

Figure

la.

Left:

Route to Infinite

199

Dilution

a segment of the spherical ionic atmosphere. reduced ionic cloud of Debye and Huckel.

Right:

the

Figure lb. Left: the non-spherical ionic atmosphere. The excess charge d q has an absolute maximum in one volume element. Right: the reduced ionic cloud; non-spherical model (n = 1).

T h u s the contribution of the structured ionic c l o u d to the total potential at the surface of the central i o n w i l l not be as it is i n the D H theory, a n d because the electrostatic m o d e l requires a n equipotential surface to be m a i n t a i n e d there, a new m o d e l is needed.

W e therefore approximate a n i o n to a dielectric sphere

of radius a, characterized b y the dielectric constant of the solvent D, and h a v i n g a charge Q, residing o n a n i n f i n i t e s i m a l l y t h i n c o n d u c t i n g surface.

T h i s type

of m o d e l has been exploited b y previous workers (17,18) a n d m a y be reconciled w i t h a q u a n t u m - m e c h a n i c a l description (18). T h e m u t u a l polarization of ions is equivalent to the redistribution of surface charge o n the central i o n i n response to the nonhomogeneous f i e l d of the ionic cloud.

W e need not speculate here on the physical nature of the equipotential

surface, except to emphasize that it refers to a solvated species, a n d one of our


200

THERMODYNAMIC

BEHAVIOR

OF ELECTROLYTES

central hypotheses is that the self energy of such species is inevitably raised through polarization. This raising of ionic energy is relative to a " g r o u n d state" at i n f i n i t e d i l u t i o n , a n d is thus expected to be more p r o n o u n c e d as the concentration increases. T h e overall effect might be interpreted as a gradual de-solvation of the ions.

Calculations General Solution of the L P B E .

F r o m the p r e c e d i n g discussion it is ap-

parent that a general solution of the L P B E i n spherical polar co-ordinates is


needed, subject to the usual b o u n d a r y conditions.

E q u a t i o n 1 c a n be easily

separated into a product solution (19) w h i c h m a y be w r i t t e n i n the f o r m lKr,M = '"

1

/

2

£

E Ki+mM

? i ( c o s 0) [ A m

m l

sin (m 0; a n d m > 0, / > 0. E a c h t i m e the m u l t i p l i c a t i o n is p e r f o r m e d we integrate over the surface of a unit sphere, a n d on consideration of the orthogonal properties of spherical harmonics w e f i n d that C o n d i t i o n 5 determines a l l the constants C i , D i (except D o J a n d C o n d i t i o n 6 determines m

m

the r e m a i n i n g constants i n terms of k n o w n C i a n d D i . m

m

T h e f i n a l results are


given i n T a b l e I.

T a b l e I.

Constants of Integration forthe General Solution o f the Linearized Poisson—Boltzmann Equation

A i = —-——7-7(7— m

nDa Ei^ (Ka)\l. 2

B

+ KG)

/2

=— —

m l

)

—-[

[_

B

° ° ~ \ J

Bel -

m

( Pi)

m

m(

J.Tj

1

v

i ;

cos (mcpi) * v

l

l + KG

6

/ KG , , ( 7 — ) nDa E^y (Ka) \1 + Ka) 2 Q a » ( / - m ) ! / Ka

(2/+1)

n

2

^-(7^)!(r7^)

Doo = 0 ^ Qa / Ka

"

l

°oi =

(l + m)\

s

Ji= 1 \2LPi (X[)

|_

2

-Q

2

Cml=

l+

(I + m)\

m

j

nDa E y (Ka)\} + Ka/ /2/cV/*Qexp(Kg) 2

£ P/ (*i)

771—^

— (——)

SWi)-n(n^)

.

z

n D \1 + Ka)

A 2>i(*i) i= 1

w

t= 1

The function E i/ (/ca) is defined b y E^ (Ka) = — [r- *K (Kr)] 1+

2

v

/2

l+V2

r=a

T h e constant DQO vanishes as expected because the potential V^(r,6,(p) was d e f i n e d as due to o n l y the polarization of the central i o n ; the potential arising f r o m the charge Q itself was taken out as Q/Dr.

Thus the " L a p l a c e p o t e n t i a l , "

V , results f r o m a l l the multipoles i n d u c e d i n the surface of the central i o n b y L

the structured i o n i c atmosphere, a n d vanishes at i n f i n i t e d i l u t i o n as r e q u i r e d .

The Electrostatic Free Energy T h e C h a r g i n g Process.

T h e calculation of the non-ideal electrostatic energy

has sometimes led to difficulties (2), but the use of the L P B E here guarantees that


204

THERMODYNAMIC BEHAVIOR

OF

ELECTROLYTES

any charging process w i l l give consistent results. A modification of Guntelberg's method is developed below w h i c h takes account of the a d d i t i o n a l effects of the polarization of the ions and their non-spherical distribution, and it is a useful first exercise to consider briefly the D H m o d e l i n a rigorous manner w h i c h recognizes the interactive nature of the central ion a n d its ionic atmosphere. Thus w h e n the central ion a n d its ionic atmosphere are charged simultaneously, the total electrostatic energy, W , is given b y D


W

°

H

= \?r + I 2 Da 2

H

+ 2W

[P (r)tia (r)dv DH

DH

(14)

int

Jv

T h e first and second terms on the right are self-energies of the central ion a n d the ionic atmosphere, and the third contains the interaction energies of the central ion with its ionic cloud and vice versa. According to Green's reciprocal theorem, these energies are equal a n d are given b y W

< n t

= ^O^ (a) = i

{

a

(15)

{r)^a(r)dv

m

P

Since the ideal solution is d e f i n e d b y the absence of interactions between ions, the total energy W r e q u i r e d to charge the central i o n a n d its ionic c l o u d i n a n ideal solution is obtained f r o m E q u a t i o n 14 b y setting W = 0: I S

int

w

is

1 2 ! + I Cp (r)ypia {r)dv 2 Da 2 Jv m

=

(16)

m

This expression can be formally identified w i t h the chemical potential Ui of the central ion i n the ideal solution, where C{ is the ionic concentration on the mole fraction scale: ls

.is =

a

W

i s = .o + M

k

T

i

n c

(17)

.

A similar identification cannot be m a d e for E q u a t i o n 14 because it contains the interaction energy twice, but on rearranging w e derive M

. = (

W

D H _

W

)

=

1 2 !

2 Da

+

I

C

2

p

DH

{ r ) ] P i

DH

{ r ) d v

+

W

.

n

t

(1 ) 8

Jv

W h e n E q u a t i o n 16 or E q u a t i o n 17 is subtracted f r o m E q u a t i o n 18, w e obtain E q u a t i o n 15, the non-ideal part of the free energy. Thus the self-energies of the central ion a n d the ionic atmosphere cancel out i n the D H m o d e l . I n the nonspherical m o d e l , however, w e must expect three contributions to the non-ideal part of the free energy i n respect of the D H model: (a) the energy of interaction between the central ion a n d the ionic c l o u d w i l l be greater than the D H energy because of the polarization; (b) the internal energy of the central ion w i l l increase because of its polarization; and (c) the self-energy of the ionic cloud w i l l increase because of its structure, w h i c h prevents the charge f r o m smoothing out into a spherically s y m m e t r i c f o r m .


12.

SPITZER AND BENNETTO

205

Route to Infinite Dilution

F o r charging u p a central polarizable i o n a n d its ionic c l o u d we thus derive

a

. = (

- W

W

) = i f

i n t

The Route to Infinite Dilution

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