Osmotic and activity coefficients of strongly associated electrolytes over large concentration ranges from chemical model calculations

journalof NIOLECULAR

LIQUIDS

ELSEVIER

Journal of Molecular Liquids 87 (2000) 191-216

www.elsevier.nl/locate/molliq

O s m o t i c and activity coefficients of strongly associated electrolytes over large c o n c e n t r a t i o n ranges f r o m c h e m i c a l m o d e l calculations H. Krienke and J. Barthel

Institut ffir Physikaiische und Theoretische Chemie der Universit~t Regensburg, D-93040 Regensburg, Germany

M. Holovko, I. Protsykevich, and Yu. Kalyushnyi

Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii St.,UA-290011, Lviv-11, Ukraine

Abstract

Association in electrolyte solutions is investigated in the framework of chemical models. Several ways for the calculation of the association equilibria are proposed to explain the measured thermodynamic excess functions. The MSA-MAL (mean spherical approximation - mass action law),an extended Bjerrum model, yields a reliable description of the measurements for moderate association mainly of electrostatic origin. In connection with expressions for the MSA activity coefficients and with the use of the Ebeling association constant a reliable explanation is possible with a single parameter which is the minimum contact distance R of the free ions. For stronger association (low degrees of dissociation a) the solution of the associative mean spherical approximation (AMSA) reproduces satisfactorily the measured data up to high concentrations. The results of these two approaches are compared to experimental data and discussed with regard to their interconnections and limitations. Comparison is also made with other approaches of the literature. © 2000 Elsevier Science B.V.All rights reserved.

1

Introduction

Physical models of electrolyte solutions assume complete dissociation [37, 38] In a chemical model of associated electrolytes in solution the formation of electrically neutral ion pairs [C~+A~-] ° determines the concentrations of the free ions, c+ = c_ = ac, a is the degree of dissociation and c [tool~din 3] is the analytical electrolyte concentration, c = p / 2 N A where t

t

0167-7322/00/$ - see front matter © 2000 Elsevier Science B.V. All fights reserved. PII S0167-7322(00) 00121-5

192 p = p+ + p_ is the total number density and NA is the Avogadro constant. The equilibrium constant KA of ion-pair formation is given by the expression I- c~ _ KA (y'~)2v c°~2 Y0

(I)

where y', is the mean activity coefficient of the free ions in solution, and Y'0 is that of the ion pairs. In statistical theories the defining concept for KA starts from pair correlation functions g+_ (r) only depending on the interactions between oppositely charged ions at infinite dilution, W~_(r). This concept is used in the low concentration chemical model approach [1,2], or in the MSA-MAL concept, (see e.g. [3] and references there), both starting from Bjerrum's theory [9] . Another route is the associative mean spherical approximation (AMSA), [29], based on the Wertheim theory of associating fluids [53]. In the present paper we show the interconnections of these two approaches and use them to explain the concentration dependence of the vapor pressures of nonaqueous solutions of associating electrolytes. The low concentration chemical model and data analysis by the use of Pitzer's equations serve for further comparison.

2

Experimental data

Osmotic coefficients for the comparison of the theoretical and experimental data are available from measurements of our laboratory with equipment known to yield precise absolute vapor pressures [2]. Experimental osmotic coefficients were chosen from this data collection for ethanol and 2-propanol solutions of LiCl04 [4], and Pr4NBr [6], ethanol solutions of Bu4NBr and BispipBr [7], and acetone and acetonitrile solutions of LiCl04 [5] and Bu4NBr [7]. The selected electrolyte solutions with solvents of relative permittivities in the range 20 < e < 36 cover the range 20 < KA/[rnol-ldm a] < 2000 of association constants. Osmotic coefficients ~LR w e r e obtained by measurements of vapor pressures p at molalities m and temperature T from vapor pressure lowering Ap = p* -- p, with p* as the vapor pressure of the pure solvent, using the relationship

oLn_

2raM8 1 [ln(~)Ap(B

~V**(0) ]

(2)

In eq. (2) Ms is the molar mass of the solvent, B is the second virial coefficient of the gas phase solvent, and V,*(l) is the molar volume of the liquid solvent. R = kNA is the gas constant and k is the Boltzmann constant. For the experimental details see ref. [2]. The experimental osmotic coefficients are values at temperature T in the Lewis - Randall (LR) system. For the comparison with theoretical data they must be converted to the

193 Table 1 Solvent p r o p e r t i e s solvent

I acetone

acetonitrile

ethanol

2-propanol

2VI~[g/mol]

58.080

41.053

46.069

60.069

B [cm3/moI]

-2132

-6190

-2981

-3424

d* [g/cm 3]

0.78421

0.77675

V*(0 [cm3/mol] 70.995 p* [Pa]

30803

0.785016 0.780716

52.852

58.685

76.971

11745

7870

5777

Table 2 Molar

mass

ME o f

the salts

Et4NBr [ Pr4NBr

salt

ME [g/mol]

210.16

266.27

Bu4NBr

BispipBr

LiCl04

322.38

234.19

105.39

McMillan - Mayer (MM) system and molarity c c~ : ff~LR(1 -~ IO-3mME)~

c:

md (1 + lO-3mME)

(3)

(4)

In eqs. (3) to (5) d and d* are the densities of solution and solvent, respectively, and lYls is the molar mass of the electrolyte. The densities are available in form of polynomials 3

d = d* + ~ A~m i

(5)

i=1

The data used for the calculations of osmotic and activity coefficients for the solutions studied in this paper are taken from refs. [4-7] and are given in Tables 1. to 3..

The experimental A p data underlying the calculation of the osmotic coefficients in Figs. 1 to 5 are represented in the quoted literature by polynomials of the type 5

Ap = ~ Bim i-1 i=1

The coefficients of eq. (6) are summarized in Tab. 4.

(6)

194 Table 3 Coefficients of the density polynoms, eq. (4) system

A1 *

10 2

A2 * 102 I A3 * 103 t

acetone

Bu4NBr

8.4597

-4.3788

30.122

LiCl04

8.2131

-1.5952

10.844

acetonitrile

LiCl04

7.1513

-0.64268 0.8780

ethanol

Et4NBr

6.82404 -1.05912 1.16521

Pr4NBr

6.74128

Bu4NBr

7.03586 -1.58245 1.94468

BispipBr

8.5183

-1.4846

0

LiCl04

7.2479

-0.9627

2.3760

-1.13343 1.00378

2-propanol

Pr4NBr

7.0297

-1.7230

3.6417

LiClO,t

7.0936

-0.8510

1.972

T h e m a s s a c t i o n law ( M A L )

The association constant KA, eq. (1), is based on the pair correlation function g+_(r) of two ions forming the ion pair under consideration [35, 37, 38]

' P+P-

--

(1 L_a) cc~2

f°°

(y:)2

- 4~rNA J g+_(r)w+_(r)r2dr = KA ~ o Yo

(7)

In eq. (7) a weight function w+_(r) is introduced which serves to define that part ofg+_(r) which contributes to the ion-pair formation. For the calculation of KA it is useful to divide the potential of mean force W+_(r) = -kTln(g+_ (r)) into its concentration-independent part at infinite dilution W~_(r), and the concentration dependent part T+_(C~C,r)

g+_(r) ----exp[-•W+_(r)] = e x p [ - Z W ? _ ( r ) - ~T+_(c~c,r)]

;

1 Z = k--T

(8)

W~_ (r) contains the short range repulsive terms and the attractive dispersion interactions as well as the solvation contributions due to the averaging process over solvent molecules. A simple description of the ionic interactions in the chemical model treats the ions as charged particles in a continuum of relative permittivity e and uses a maximum distance

195 Table 4 Vapor pressure lowering Ap = p* -- p of various electrolyte solutions system

conc.range

B1

B2

B3

B4

B5

acetone, e=20.70 LiCl04

0.02-0.58

0.030530

17.7091 -12.3816

22.8147

-14.5306

Bu4NBr

0.01-0.83

0.067388

15.2414 -17.2042

22.4815

-10.9669

1.36384

-0.433865

acetordtrile, e = 35.95 LiCl04

0.06-1.15

0.027440

5.67914

-1.43816

ethanol, e=24.36 LiCl04

0.03-1.45

0.020398

3.90564

-0.038299

1.14958 -0.280582

Et4NBr

0.04-2.30

0.038743 3.009947

-0.416629

0.407903 -0.081779

Pr4NBr

0.04-2.72

0.045417

2.94803

-0.088710

0.286696

-0.055415

Bu4NBr

0.05-2.5 0.039623 3.09796

-0.291257

0.468935

-0.090505

BispipBr

0.05-0.68 0.016029

3.28833

-2.16043

2.02113

-0.458073

2-propanol, e = 19.39 Pr4NBr

0.03-1.25 0.018778 2.44496 -1.59381

1.83340 -0.598477

LiCl04

0.05-1.48 0.015177 2.74737

1.24942 -0.232199

-1.01672

R up to which two ions are counted as an ion pair. In turn, this distance R is the minimum distance up to which two free ions can approach each other. This idea leads to the representation of g+-(r) in eq. (7) in terms of interactions of charged hard spheres when choosing

wiT(~) = w"s(~) +

w,~(~)

(9)

with the hard sphere potential w H S ( r ) = oo

;

r < R

and

wHS(T) = 0

;

r > R

(10)

and the Coulomb potential between the charges zi and zj WiC(r) = z~zJe2 47r~0er'

(11)

A natural choice of the weight function w+_(r) in eq. (7) for the construction of an ion pair with a contact distance R then should be (Bi strength parameter, 5(x): Dirac's delta function)

~+_(~)

=

Ba(~ - R)

(12)

196 This leads to the MAL in the form (1 - c~) _ 4~rNAB exp[-flW~_ (R) - flr+_(ac, R)]R 2 = KA (y')____2z

ca2

y,o

-

(13)

The exponential of "r+_ (ac, R) then is identified as

exp[-flr+_(ac, R)] - (y')2 _ (yHS)2 (y~')2 y'o yyS ,,ev

(14)

~o

with hard sphere and electrostatic contributions of the activity coefficients corresponing to the division of the potential in eq. (9). From eq. (13) follows the association constant I(A for charged hard spheres

(15)

KA = 4rrNa B R 2 exp [-flW~_ (R)] = 4~rNABR 2 exp(b)

where b is the Bjerrum parameter characterising the Coulomb interactions between ions at contact distance R Z2C2

b - 4rreoekTR

(16)

The strength B of forming an ion pair must be estimated from reasonable assumptions. For B = R/3 the association constant of Fuoss [26] is derived. In general B should also depend oi1 the electrostatic interactions, e.g. on the Bjerrum parameter b. In section 5 Ebcling's way [20, 23] is discussed to estimate the association constant based only on electrostatic interactions. On the other hand, in chemical models the activity coefficients are found from an expression of r+_(~c, r), which is based on the generalized virial expansions [3, 27, 36]

-flr+_(r) = ln[yHS(r)] + G+_(r) + flwC_(r) - flAr+_(r)

for

r _> R

(17)

yHS(r) is the cavity correlation function of the hard sphere reference system for the ions, defined by

(18)

~'~(~) = exp[-#W'~(r)]V"~(~)

flat+_ (r) includes three- and more-particle correlations which may be neglected in a first approximation. G+_ (r) is a screened Coulomb potential under the influence of short range interactions the so-called chain sum. For charged hard spheres of diameter R the contact value yHS(r = R) in the MSA equals the Percus-Yevick value yHS'PY(R) [3, 48, 52]

yHS,PV(R) _= 1 + 0.5r/ -(1-~7) ---7

rr

;

~-pR3

rl=-6 (p++p-)Ra= 6

(19)

197 and for the screened potential at r -- R the MSA-solution

G +_ MsA ( R ) [11, 28] may be used

1

GMS_A(R) = b(1 + pR)2

(2O)

to yield the relationship

--~7+M_SA(R)=- ln[yHS'PY(R)] + G +_MSA(R)+ zwc_(R) =

(21)

[1+0.57] [(1+1rR)2 1]

In [(1 _,q)2j + b

where F is the screening parameter of the MSA, introduced by Blum [10] FR = 1[@ + 2~'R- 1]

(22)

In eq. (22) ~' is the Debye - Hfickel screening parameter for the free ions in solution

(n')2=as2=a\

( e2 Zl PlZ~~ = c~(SrbRNAc) eoekT /

(23)

From eq. (21) follows the relationship exp[--3T•U_SA(F,_ R)] =

yHS'PY(R)(y~')2vev--

(24)

~o

FR b (FR)2 ] 1(1+ - 0.5~ ~)2 exp _ 2 b ~ + F R ) 4 - (1 + FR) ~] leading to the well known MSA expression for the electrostatic contribution of the activity coefficients of the free ions [10, 50, 51]

FR

ln(y~') = - b (FR~------~ 1 +

_

b(l+~'R-x/l+2~'R)

~'R

(25)

and to an expression for the electrostatic contribution of the activity coefficient of ion pairs

ln(y• t')

(rR)2 -b(1- + rR) 2

(26)

The new expression derived for y~' permits the consequent treatment in the ionic picture ans differs in this point from the expression previously used by us [37] , which was obtained on the basis of a model of interactions of dipolar hard spheres and ions [12, 56]. The comparison of eq. (24) and eq. (14) shows that ygS(R) corresponds to the nonelecHS~2~yHS trostatic part of the ratio of activity coefficients kYa) / o •

198

With these expressions the MAL reads

(1 - a) _

( el"~2

KAyHS(R ) ~

[-2b

= 47rNABexp(b)R 21 + 0.57] exp [

FR ] (1 + FR)J

(27)

L- ~ - ~ J with the association constant KA given by eq. (15). Another possibility to express the ratio of activity coefficients Carnahan - Starling formula for yHS(R) [15]

ygs(R) = yHS,CS(R) _ 1 -- 0.5r]

fyHS~2/yHS is the use of the k -I- ) / (28)

(1 - ~)3 For dilute solutions

4

yHS(R) = 1 is the usual approximation.

T h e r m o d y n a m i c s using the M A L

The mass action law, eq. (27), is the basis for the calculation of the degree of dissociation o when the association constant KA is known and information is available on the contact distance R of ion-pair formation. In the chemical picture the total mean ionic activity coefficient Yl combines the contributions from the mass action law, yMAL = a, repulsive core interactions of all ions, the mean activity coefficient for the hard sphere interactions, yHS,( which must be not confused with the contact value yHS(R) of the cavity pair correlation function, eq. (19)), and the electrostatic interactions of the free ions, ye~' ±

(29)

H S y± a' y+ = c~y±

Eq. (29) with yHS = 1 is the mean electrolyte activity coefficient of the low concentration chemical model [1, 3] where

in(y::')--

b

~'R

2 (1 + ~'R)

(30)

and ln(y: l') =

Boc~m"

(31)

If the repulsive core interactions between ions are represented by hard spheres of diameter R, the Percus-Yevick theory provides the following expression for yHS,

yHS,PY _ (1 + 27?)2 (1 -- ?7)4

(32)

199

One could also use the Carnahan - Starling expression for In (yHS :i: ) [15]

tn(yHS,CS) =

(87 -- 972 + 373) (1 -- 7) 3

(33)

It is assumed that yHS ± is not affected by the formation of ion pairs. Using eqs. (25) and (32) from eq. (29) follows

[(1+27) 2]

[(I+~'R-,~)]

(34)

or the alternative expression when eqs. (25) and (33) are used Y+ = a e x p [ (87 -(1-972 ----,~ + 373) -

b(l

+ ~'R - / , R ~).]~

J

(35)

The osmotic coefficient (I) = po~m/pkT contains the contributions from the mass action law, repulsive forces between the ions pHS/pkT, and from the electrostatic interactions between the free ions, pel'/pkT

pMAL/pkT, from the

pMAL pHS ,I~

-

pkT + ~

pet'

(36)

+ pk----T

The MAL contribution from free ions and ion pairs is given by

pMAL pkT-

1

a)

(37)

5(1-

The hard

sphere excess contribution may be calculated by the Carnahan - Starling expression for hard spheres at contact distance R [15]. Again it is assumed that the ion-pair formation has no influence on this contribution

pHS,CS __ pkT

1+7+72-73

(38)

(1 - 7) 3

In this theory the electrostaticexcess contributiondue to the free ions follows from the MSA of charged hard spheres of equal diameter R [10, 50, 51] as pC,' _

(FR)~ __

pkT

3~rpR3

(X/1 + 2 ~ ' R - 1) 3 24~rpR 3

(39)

Then the osmotic coefficient reads ¢:

1+7+72"73_ (1 - 7) 3

(x/i-+2~'R-1) 3 241rpR 3

1-a 2

(40)

Ebeling and Grigo [21] were the first who used the MSA expressions eqs. (25) and (39) for the free ions in the chemical picture.

200 5

The association constant

The pressure equation in the chemical picture, eq. (40), yields in dilute solutions the density expansion P .... - ~'~P'a + ~'~p'~p~bab + O[(p'~)~] + kT "'" a ab

(41)

where the sum is taken over free anions, cations, and ionic pairs ( a -- +, - , 0 ). From the MAL follows the connection with the association constant and the concentrations of the ions and the ion pairs as p2 P0' = ( 1 - - a ) ~ P = ~ K A + O ( P ~ )

~ ;

p+=ap+--

p 2

p2 -~KA+O(p~)=p'_

(42)

At the limit of vanishing ion concentration the degree of dissociation tends towards unity, -+ 1, and the chemical picture becomes equal to the physical picture at complete dissociation. Then the cluster expansion for a system of interacting charged hard spheres with contact distances R is given in terms of the analytical density p and the Bjerrum parameter b as [23, 34] -- P°~"~ -- 1 p- -k T

bxR

x~2R2

6

2b

ko(b)

(43)

with ko(b)

1

b2

3

2

b2m

.,~ (2.~)!(2m- 3)

(44)

Thermodynamic functions must be independent of how the system is presented with the help of models. Therefore the pressures of the charged hard sphere systems must be equal in the chemical and in the physical pictures Posm(P'~, K A) = Pos,~(P, 13W~+, ~3W_~_,j3W~_)

(45)

Equating the density expansions of the physical and chemical pictures, eq. (43) and eq. (40), one obtains Ebelings expression for KA [20] b 2m

I4~ = 87vNAR 3 ~

(2m)!(2m - 3)

(46)

r~>2

Together with eq. (15) this result allows to identify the constant B of the weight function eq. (12) in the case of pure electrostatic ion association as b2ra

B = 2 exp(-b)R ~ m_>2

(2rn)!(2m - 3)

(47)

201 However, in real systems the mean ionic force potential in infinite dilution Wi~(r ) contains the solvent averaged solvation interactions W~°l~(r), which can be taken into account by an expansion of eq. (9) .

.

.

.

.

.

.

,o,,

(48)

A simple model assumes constant potential W~°*'(r) = W~*~for R < r < R', where R' > R is now the distance up to which the two ions are counted as an ion pair. This model yields the association constant [39] (49) In eq. (49) xi means the ratio xi = Pi/p, tij and lij are defined as . tgj = exp[-j3W~j] - 1

;

z~zjd lij = 47reoekT

(50)

and B ( x ) is the Kirkwood function xm--3

B(x) = ~

rn> 4

( m ) ! ( m - 3)

(51)

Eq. (49) can be written as KA = K E + A K A with

For the simplification of the description we will use the sticky limit in which R' --+ R and tij --+ cc in such a way that we fix A / ( A as A K A = 87rNAR 3 exp(b)K~ °tÈ

(53)

This leads to b2m ] tea -= 87rNAR 3 m~>2(2m)!(2m - 3) + exp(b)K~A°'"

(54)

and suggests the following expression for the constant B

B

=

2R exp(-b) E

b2m ] (2m)!(2m - 3) + K~°'~

(55)

rn>2

where the contribution K~ °~ is specified e.g. by additional short range interactions given in eq. (52).

202 6

O n e p a r a m e t e r fits i n the B j e r r u m - NISA - M A L t h e o r y

A theory which uses the Ebeling association constant K E, eq. (46), eq. (27), and the expressions for activity and osmotic coefficients, eqs. (34) and (37), is a simple description of thermodynamic excess properties of electrolytes, which demands only one parameter, namely the contact distance R of the ions. This theory has its origin in the Bjerrum concept of electrostatic ion association, and it changes over for a -+ 1 to the MSA description of nonassociating electrolytes [37]. Therefore this type of theory will be called Bjerrum MSA - MAL theory. Fig. 1 shows, that it is able to describe osmotic coefficients for L i C l 0 4 in acetonitrile at 25°C with a single parameter of R = 0.42 n m (broken lines). The association constant is calculated from eq. (46) for the system discussed in Fig. 1 as KA~ = 10.45 dmSmo1-1.

1.0

0.8 (t), F R 0.6

0.4

0.2

0.0

0.0

0.2

0.4 0.6 c (mol/I)

0.8

1.0

F i g u r e 1. Osmotic coefficients • for L i C l 0 4 in acetonitrile at 25°C. The experimental - values (o) are converted from the LR to the MM system. Theoretical ~ - and FR - values according to eq. (22)from the Bjerrum - MSA - MAL theory (broken lines). Theoretical • - and F B R - values according to eq. (97) from the AMSA theory (solid lines) In both theories R = 0.42 n m .

Due to its construction the Bjerrum - MSA - MAL theory is especially useful for the case that a deviates not too strong from unity. Contributions due to the interactions of the pairs with each other and with free ions are treated in this theory only in a minor way, e.g. in eq. (26). Especially the case a --+ 0 cannot be treated in the developed scheme, since in that case also F R --+ 0 (see eqs. (22) and (23)), and we have no electrostatic contributions, e.g. for ~, eq. (40). Therefore, a generalized version of the theory is now discussed, the

203 associative mean spherical approximation ( A M S A ) [8, 13, 29, 31, 33], which is based on Wertheims treatment of associating systems [53, 54], , and it is shown, how the Bjerrum MSA - MAL theory is embedded in this AMSA variant.

7

The AMSA-

theory

The AMSA theory represents the two-density version of the traditional MSA for an ionic fluid of associative particles [29, 31, 53]. Recently, analytical expressions for the thermodynamic functions of the multicomponent mixture of charged hard spheres with sticky-point associative interaction were derived by Bernard and Blum [8]. Their derivation is based on the direct integration of the internal energy, which was possible due to the availability of the analytic solution of the AMSA [13, 29, 31] and the use of the exponential approximation in the MAL [8]. We present here the way of solution only for the simplest case of the restricted primitive model (RPM) given by interactions according to eqs. (9) to (11) and z+ = - z _ . It is supposed that the cations and anions form ion pairs and the density of the ionic species i splits into two parts, namely the density of free ions api, and the density of ion pairs (1 - a)pi. They are connected by the MAL, eq. (1). The total pair correlation function hij(r) between two ions i and j, can be represented as a sum of four terms: -

h,j(,-)

=

hi°°(,-) +

+

i4°(r) +

(56)

where the upper index 0 announces that the corresponding ion is free , and the index 1 indicates, when it is bounded in an ion pair. In order to treat corrcctly the limit a --+ 0 it is advantageous to represent h~j (r) in the following form [18]

h j(r) = %°°(r) + h°)(r) +

+

(57)

where

h°)(,r) = alz°~(r)

; hij20(r)= alz~°(r) ; h~)(r) = a2h~)(r)

(58)

In the case of orientationally averaged association interactions the MAL may be stated as [291 oo

1Z-a -- 47rNA f f,~(r)g°+°. (r)r2dr CO~2

(59)

0

At the sticky limit the weight function f,,~(r), which characterizes the association, may be given by a delta function BS(r - R). Then

1

-

c~

C~2

_

47rNnBR2gO+O. (R)

(60)

204 where gOO(R) is the contact value at r = R of the pair distribution function of the unbound oppositely charged ions. In order to account correctly for the saturation effects due to ionic association in the ionic pair correlation function we use instead of the usual Ornstein-Zernike (OZ) equation Wertheim's modification of this equation, the WOZ equation, [53, 54] for the calculation of hij (r) h=c+c.p.h

(61)

The star denotes a convolution, the matrices h and c have the elements h~#(r) and c~#(r) , and p is the matrix of number densites. Pi p+

p=

P+ 0 0 0

00 p_ p_

00 p_ 0

(62)

Because of the symmetry of the RPM we define sum and difference functions

•

=

"

2

'

- h+_)

2

(63)

eald corresponding expressions for the c - functions. As a result the WOZ, eq. (56), decouples into a set of two matrix equations h. = c~ + pcs * X * h~

(64)

and hd = Cd + flCd * X * hd

(65)

where p = p+ + p_ = 2p+ . The corresponding matrices are ,o~,dzh°l\

/cOO c ° I \

hs,d -= ( hs,d

Cs,d

(~

~)

s,d

The AMSA closures for the electroneutral sum problem (index s) are the same as for the associative hard sphere PY approximation [54] h°°(r) = - 1 ,

h°l(r) = h~° = h]l(r) = 0 =

;

r < R

= ~ cs (~) = T f o . ( ~ ) g ° ° ( ~ )

;

~ > n

(67)

The last closure in the sticky limit, f~s(r) = B6(r - R), according to eq. (60) can be written in the form cy(r)

-

(1 - ~ ) ~ ( ~ _ n )

47rp

;

~ > R

(68)

205 or in an equivalent manner h °° = - 1 ,

oo

Cs

~

o 1 _- -

Cs

1 = h: ° = o,

lO

Cs

-~

11

Cs

0

~

hy =

•

;

<_ R

r>R

(69)

In an analogous way we have for the difference case (index d)

cOO __ ---,bR

c °1 = c 1° = cy = o

The function g ° ° ( R )

,.

r > R

(70)

, introduced in the MAL, eq. (59), is defined by

g°°_ (n) = 1 + h°°_ (R) = 1 + h°~°(n) - h°°(R)

(71)

In an EXP - approximation one also uses [8] g°+°(R) ---- [1 + h~°(r)] exp[-h°°(r)]

(72)

This leads to a MAL of the form 1 - oz _ 4 g N A B R 2 [ 1 + hOO(R)] exp[_hoo(n)]

(73)

C(R 2

which must be compared, e.g., with the corresponding equations of Section 3 after establishment of explicite expressions for the h functions. The solution for h~ formally coincides with Wertheim's solution, [54] for dimerizing hard spheres with (~ calculated from eq. (60). The solution for hd was obtained by the use of the Wertheim - Baxter factorization technique [29]. This solution leads to a system of five algebraic equations for five unknown constants J0, J1, a0, al, and a. The general solutions of the AMSA for ions with different charges and radii ( also for multiple binding) were developed by Blum and coworkers [8, 13, 14] and in the case of binary association applied to the description of thermodynamic excess functions of aqueous electrolytes [46]. Here we iocus on the general idea of the solution for binary association. We will show, that the system of algebraic equations can be reduced in the RPM to a system of only two equations for J0 and a, and that J0 is connected with Blum's screening parameter F8 [8]. As usual, we introduce the functions o~

J2~ = 2~p / sh~ ' ( s ) d s

(74)

2O6 and equivalent relations for the direct correlation functions oo

S~~ = 2~rp.f sc~(s)ds

(75)

r

In terms of these functions the eq. (65) is transformed into two equations

- / dtQ(r + t)XQT(t)

Sd(r) = Q(r)

(76)

0

and

+ f dtJd(Ir - tl)XQT(t)

Jd(r) = Q(r)

(77)

0

which after differentiation read 27rprca(r) = - Q ' ( r )

+ f dtQ'(r + t)XQT(t)

(78)

0

and

2np'rhd(r) =

-Q'(r) +

2~pfdt(r - t)ha(lr - t])XQT(t)

(79)

0

From the AMSA closures we have the form of Q(r) Q(r) = q(r) -

lim A e x p ( - # r )

(80)

/*--+0÷

Here q(r) is a linear matrix function of r q(r)=qo+ql(r-R)

;

r_
and

q(r)=0

;

r>R

(81)

The matrix A has nonzero values only in the first row

=( ooo

o a, 0)

(82)

Taking into account the structure of h~(r) for r _< R according to eq. (70), we have from eq. (76) for the matrix elements of ql

'±7tdth~X~5.4~

q~ = 2~rp~

~,=0 5 = 0 0

(83)

207 Due to the structure of the matrices A and X this leads to a form

q~# = J~a#

;

~,13=0,1

(84)

where oo

•] ~ - - 2 ~ - p f tdt[h~ ° ( t ) + h ~ l(t)]

;

~=0,1

(85)

0

From the boundary conditions we have

qOO = qOZ = q]O = o

1-o~

" q~Z _ '

(86)

2

From eq. (78) in the asymptotic case r --+ oc two equations are obtained (~R) 2 = a 2 + 2aoal

(87)

and

al = ao(K 1° + K 11) + a l K 1°

(88)

The functions K ~# are defined as R

(89)

I£ ~ = f q~# (t)dt o

and given by h.i0 =

1

- 2 JlaO

;

KU

=-2

1j a 1 i 1- 5(1-ol)

(90)

Therefore eq. (88) transforms to

(91)

2al + (aR)2j1R + a0(1 - a) = 0 In the asymptotic limit r --+ 0 we find from eq. (78)

q~# = ~ ~ f dtQ'~(t)X~Q#a(t) = 7=0 ~=0 b

-

- qVx~(~,~

_ q0~ ) + q~X~A ~ /

(92)

~:0 5 : 0

leading to two other independent equations: (~R)2(1 + J0R) 2 + 2aoJoR : O

(93)

208 and 1

aoJ1R + alJoR + (1 + JoR)(t~2R2J1R --{-~ao(1 - c~)) = 0

(94)

The eqs. (87), (91), (93), and (94) make up a system of four equations for the four parameters Jo, J1, ao and al. After some algebra this system can be presented as only one nonlinear equation for the parameter Jo (nR)2(1 + J0/l~)4[Ol -- J0n(1 - a)] = 4(JoR) 2

(95)

The eq. (95) can be written in another form, when we introduce Blum's screening parameter FB instead of J0 FBR

J0/~ -

1 + r.n

(96)

Then a simple nonlinear equation for this scaling parameter FB is derived which was first given by Bernard and Blum [8]:

4(rBR)2(1 + FBR)2 = (~R)2 (~ + rBR) (1 + rBR)

(97)

Equation (97) contains the degree of dissociation a and should be considered together with the MAL eqs. (60) or (73). From eq. (79) we have for the contact value h°e°(R) = g°O(R) (see eq. (75) h°°(n) _

qoo

2~pR -

Joao

2~pR

_

b(1 + JoR)" -

b

(1 + r~R)2

(9s)

As a result we have for the contact value the same form as for the usual (nonassociating) MSA - ha(R) -value [11, 28], but with FB instead of the usual F. Using eq. (98) in connection with eq. (73) one finds for the MAL: 1 - a = 4~rNABR2[1 b ca2 + h°°(R)] exp[(1 + FBR)2]

(99)

1 + h°°(R) is given by ygS(R). The approximation h°°(R) = 0 is identical with the assumption of yHS(R) = 1, and eq. (99) has the same form as eq. (15) in connection with eq. (27) with the only exception that the new scaling parameter FB is used instead of the F, defined by eq. (22). Bernard and Blum derived for the electrostatic parts of the osmotic and activity coefficients of the free ions expressions like eqs. (25) and (39) [8]

pe~'

(FBR)3

pkT --

3~rpR 3

(100)

209 In' ~l', FBR LY± ) = - b ( 1 + FBR)

(101)

Taking these results into account and comparing eq. (99) with eq. (27) we see that the electrostatic part of the activity coefficient of the ion pairs, y~V, is given by

(1-'BR) 2 ln0/;") = - b ( 1 + r s R ) 2

(102)

The c~ - dependence should be discussed in two different regimes, namely, in the weak ( -+ 1 ) and in the strong ( a ~ 0 ) association regimes. In the regime of weak association it follows from eq. (95) that n2~ --

4J3

(103)

(1 + JoR) 4

For very small ion concentration, J0 ~ 0, and J0 =

-~x/r~

(104)

In the more general case this regime realizes if

FBR 1 > a > > (1 + 2FBR)

(105)

This is the case of electrostatic ion association described by the Bjerrnm - MSA - MAL theory as can be shown using the relation (96) in eq. (103) leading to

4 ( r . n ) 2 ( 1 + r ~ n ) 2 = (~n)2~

(106)

The calculation of FB according to eq. (106) and the MAL, eq. (99), corresponds to the calculation of F according to eqs. (22) and (23). The effect of changing the calculation of FBR according to eq. (106) instead of eq. (97) is shown in Fig. 1 for LiC104 in acetonitrile at 25°C using R = 0.42 nm. In the strong association regime follows from eq. (95)

4JoR --(t~R)2(1 -- a)] -- (1 + JoR)4

(107)

For very small ion concentrations follows another expression for J0

J0R = - 4 ( ~ R ) ~ ( 1 - 5)

(108)

The strong association regime is realized if O
-

FBR (1 + 2r'sR)

(lO9)

210 Then another connection between FB and a is obtained from eqs. (107) and (96)

(110)

4F~R(1 + r . R ) 3 = (~n)e(1 - a)

The limit of complete association ( a = 0 ) can also be treated [18, 32, 33], leading to the MSA- Chandler-Silbey-Ladanyi (CSL) equations [16, 40, 45]. In the regime of weak association ( c~ --> 1 ) according to eqs. (102) and (106) it is found for small ionic concentrations that

In(yf) =

-b(~R)2~ 1 (~R)2a 4 (I + FBR)4 ~ -b 4

(iii)

Equation (111) is the theoretical confirmation of the empirical form of the 1on-pair activity coefficient, eq. (31), used in the chemical CM2 model for the interpretation of experimental data in [4-7]. In the regime of strong association ( a --+ 0) it follows according to eq. (110) that

In(y°~")----b(gR)416 (I(I+-FBR) sa) 2 ~ _b~6)4 (i _ ~)2

(112)

Thus at increasing association the concentration dependence of l n ( y f ) changes from linear to quadratic. The expressions for the activity and osmotic coefficients again are given by eqs. (29) and (36), but now with eqs. (100) and (101) instead of eqs. (25) and (39). The influence of the formation of ion pairs on the hard sphere contributions can be calculated with the help of the thermodynamic perturbation theory [54] and is included into the MAL - terms ln(y Hs)

ln(yMAL) = ln(a) _ I_~(i - a)p 0 - -Op pMAL _ pkT

8

(57 - 2~ 2) - ln(~) - ~(1 - ~)(1 - 7)(1 - 0.57)

1 (1 - oz) 1 + p ~aln(ygs)] --~---p J = -1(12

~ (1+~-

0.572)

(113)

(114)

'(1 - r/)(1 - 0.5,?)

O n e p a r a m e t e r fits w i t h the A M S A - t h e o r y

In the case of stronger association a one - parameter fit in the framework of the AMSA is possible, when the Ebeling association constant KAE is connected with the association parameter B of the AMSA two density theory according to eq. (47). r'B is calculated from eq. (97) and from the MAL, eq. (99). Osmotic and activity coefficients are calculated with the help of eqs. (100), (101), and of eqs. (113), (114) respectively. As a first example we compare the thermodynamic excess functions of LiCl04 in different solvents, derived from vapor pressure measurements, with the predictions of the AMSA theory. The only adjusted parameter - the contact distance R - is in an acceptable region between 0.42 nm and 0.48 nm.

211 In Fig.3 osmotic coefficients and mean ionic activity coefficients for tetrabutylammonium bromide in acetone and in ethanol are shown. The only adjusted parameter, the ion-pair contact distance R, is different in both solvents ( R = 0.34 n m in acetone, and R = 0.49 nm in ethanol). In Fig. 4 the osmotic coefficients and mean ionic activity coefficients y± of tetraethylammonium bromide E t 4 N B r and bispiperidinium bromide, ( B i s p i p B r ) in ethanol are shown. W i t h the adjustment of only the contact distance R ( R = 0.39nm for E t 4 N B r , and R = 0.34'n.m for BispipBr) the experimental values are described up to a concentration of c = 0.6 tool~din 3.

1.0

0.8

~

~

-

~

5

~~ ~o

~ (2)

dp , Y

0.6

~:~3~.q3.~.~--- (I)

~ - (3)

0.4

0.2

0.0 0.0

0.2

0.4 c

0.6

0.8

1.0

(mol/I)

F i g u r e 2. Experimental and calculated osmotic coefficients • and mean ionic activity coefficients y . of LiCl04 at 25 ° in different solvents at 25°: The experimental • - values (corrected to MM level) are given by symbols: (circles: acetone; squares: acetonitrile; diamonds: 2-propanot). Calculations according to the AMSA theory are represented by solid lines (1): acetone, R = 0.48nm; (2): acetonitrile, R = 0.42 nm; (3): 2-propanol, R -- 0.43 nm. The broken lines in the lower part of the figure show the corresponding mean ionic activity coefficients.

212

1.0

0.8

¢,y

o ~

(2)

0.6

o2111:

--

0.4

........... "" ......................

0.0 0.00

0.20

(1)

(2) (l)

0.40 c (rnol/])

0.60

0.80

F i g u r e 3. Osmotic coefficients • (full lines) and mean ionic activity coefficients y+ (broken lines) of tetrabutylammonium bromide Bu4NBr in different solvents at 25°: The presentation is as in Fig. 2.: (circles: acetone, squares: ethanol). Calculations (AMSA): (1): acetone, R = 0.34 nm, (2): ethanol, R = 0.49 rim.

1,0

0.8

¢,y 0.6

~ ~ o u ~-~ (1)

(2)

0.4

0.2 (2) 0.0

0.00

0.20

0.40

0.60

0.80

C (rnol/I)

F i g u r e 4. Osmotic coefficients • (full lines) and mean ionic activity coefficients y:L (broken lines) of tetraethylammonium bromide Et4NBr and bispiperidinium (azoniaspiro[5,5]undecan) bromide, ( B i s p i p B r ) in ethanol at 25 °. The presentation is as in Fig. 2.: (circles: Et4NBr; squares: BispipBr).

213 Calculations (AMSA): (1): E Q N B r , R = 0.39nm; (2): BispipBr, R = 0.34nm.

In Fig. 5 osmotic coefficients and mean ionic activity coefficients of tetrapropylammonium bromide P r 4 N B r solutions in ethanol and in 2-propanol are shown. The adjusted contact distance R of the ion pair is R -- 0.45 n m in the ethanol solution, and R -- 0.30 n m in the 2-propanol solution.

1.0

0.8

¢,y 0.6

~

0.4

"

~

~

(2)

\\ 0.2

'"- Y

o.o

0.00

..................

........ 0.20

(1)

(2)

0.40

0,60

0.80

C(tool/I) F i g u r e 5. Osmotic coefficients • (full lines) and mean ionic activity coefficients y+ (broken lines) of tetrapropylammonium bromide P r ~ N B r in different solvents at 25 °. The presentation is as in Fig. 2.: (circles: ethanol; squares: 2-propanol). Calculations (AMSA): (1): ethanol, R = 0.45 nm; (2): 2-propanol, R - - 0 . 3 0 nm.

Fig. 2 ( LiCl04 in acetone, acetonitrile and 2-propanol), Fig. 3 ( B u 4 N B r in acetone and ethanol), Fig. 4 ( E Q N B r and BispipBr in ethanol) and Fig. 5 ( P r 4 N B r in ethanol and 2-propanol) show the satisfactory agreement of one-parametric AMSA calculations with experimental results. Distance parameters around 0.45 n m for LiCl04 in protic solvents are reasonable, they are larger than the sum of the ionic radii, indicating contributions of contact and solvent shared ion pairs. The distance parameters for tetraalkylammonium halides must be chosen smaller than the sum of ionic radii. This is a well known feature, because spectroscopic and neutron diffraction measurements indicate partial impenetration of halide ions into the space occupied by tetrapropyl- and tetrabutylammonium ions [3]. The osmotic and activity coefficients of the low concentration chemical model (lcCM) calculations which yield association constants (CM1) or association constants and Bo coefficients (CM2) [4-7] are equal to those of this paper. The basic concept of the lcCM approach is given in [1, 3], in ref. [3] it is embedded into the framework of statistical

214 thermodynamics. One parametric CM1 calculations are limited to concentrations below 0.15 tool kg -1, CM2 calculations are possible up to 0.6 molal solutions. Pitzer equations [42, 43] yield correct osmotic and activity coefficients of all investigated solutions for the up to saturation [4-7], however up to seven parameters are needed to represent the excess functions, and physical information is not available from these plots which have their importance in the field of engineering data.

9

Conclusions

A unified approach to the description of the excess functions of nonassociating and associating electrolytes is proposed which leads to numerically simple final formulae, able to represent experimental data with a minimum of adjustable parameters. These parameters show realistic values in their definition range. Other properties such as conductivity or diffllsion coefficients can also be treated in the framework of this theory and will be the subject of subsequent articles.

10

Acknowledgment

This work was completed in the framework of a cooperation between our institutes at Regensburg and at Lviv, registered under UKR-028-96 at the 'Internationales Btiro des Bundcsministeriums ffir Bildung, Wissenschaft, Forschung und Technologie (BMBF) bei der Deutschen Luft- und Raumfahrtgesellschaft (DLR)', Germany, and the Ministery on Science and Technology of Ukraine. The financial support of this contract is greatfully acknowledged.

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