Journal o f Solution Chemistry, VoL 16, No. 2, 1987

Simple Statistical Mechanics of Electrolytes with a Concentration Dependent Dielectric Constant. Part 1. The Pressure Equation

Jose L. Martin, ~ Juan Luis G6mez-Est6vez 1'2 and Manel Canales ~

Received April 30, 1986; In Final Form November 4, 1986

We have used an approximation of the Adelman's theory of solutions to take into account the non-pairwise additive effects on the interaction potentials for a real electrolyte solution. As a result, a solute concentration dependent dielectric con- stant ~p appears. The comparison with experimental data is done by means o f the pressure equation. The influence o f % on solution properties is analyzed in two molality ranges using as a reference simplified models but with the dielectric constant o f the pure solvent E instead of e p .

KEY WORDS: Electrolytes;- Adelman's theory of solutions; solute concenlra- lion dependent dielectric constant; pressure equation; osmotic coefficients.

1. INTRODUCTION

In 1976, Adelman (1~ proposed a new theory of solutions based on the Ornstein-Zernicke equation. (2~ This theory takes into account the non-additive effects (NAE) in the McMillan-Mayer (MM) potentials by means of an effective pair potential. When applied to electrolyte solu- tions, the non-pairwise additivity of the potential implies the existence of a solute concentration dependent dielectric constant % or, as referred to by Friedman, the thermodynamic equilibrium dielectric constant.~ The appearance of a solute concentration dielectric constant is not new in the classical electrolyte literature (46~ but, until recently, (3"26~ its introduction in several models for electrolyte solutions

1Departamento de Termologia, Facultad de Fisica, Urtiversidad de Barcelona, Diagonal, 645; 08028 Barcelona, Spain.

2"I"o whom correspondence should be addressed.

87 0095-9782/87/0200-0087505.00/0 �9 1987 Plenum Publishing Corporation

88 Martin, G6mez-Est~vez, and Canales

was without a firm theoretical basis. As Friedman has remarked, ~ it would be interesting to inves-

tigate the influence of this kind of effect on ionic solution properties and a comparison with the earlier approaches at the (MM) level.(:'8)

The objective of this work is to study how the introduction of % can affect the thermodynamic properties of the electrolyte solution. In this paper, we will consider the virial route to the thermodynamic properties by using the pressure equation. (9,~~ The compressibility equation (~,2,8~~ constitutes other approach that will be considered in a forthcoming work. All of the computations will be made within the framework of the MM theory. (2'8) As a consequence, several ther- modynamical corrections must be made (1~ in order to compare with experimental data.

In the next section, we briefly review Adelman's theory on how to construct approximate pair correlation functions including %. The pressure equation is obtained, indicating the approximations which are necessary in the derivation and how the results of the Adelman's theory can be incorporated. The osmotic coefficient data of the aqueous solutions of alkali halides at 25~ are analyzed using these equations. The experimental dielectric constant data employed in this paper are presented, and we discuss the results obtained in terms of % by means of the distance of closest approach a.

2. APPROXIMATE PAIR CORRELATION FUNCTIONS

Within the context of the MM theory (2,7,8) is necessary to give the potential of mean force among N solute molecules at infinite dilution Wff. This potential is usually assumed to be pairwise additive

= x (1) i<j

where W~ is the pair potential of mean force at infinite dilution be- tween ion i and j. The simplest model for an electrolyte solution is the restricted primitive model (RPM) defined by giving an expression for

Wi~' (r) = I'V:~ (r) + ZiZje2/er (2)

W~* = oo r < a

= 0 r > a (3 )

Concentration Dependent Dielectric Constant 89

where the ions are considered as point charges (z~e) within hard spheres of equal radii (a/2). The solvent only appears in the expres- sion of Wi~ by means of the dielectric constant ~. There are several approximation methods ~7'a~ for obtaining the ion-ion pair distribution functions gij for the RPM. One of them is the exponential approxima- tion (EXP) defined by (7)

gi~(r) ~ gi~Sexp [-/3 Wij(r)] (4)

where g~S is the pair distribution function for hard spheres,/3 = (kT) -~ and W~j(r) is the pair potential of mean force at finite concentration. Eq. (4) has been applied with Wij(r) obtained by means of the Poisson- Boltzmann (PB) equation ~ and the mean spherical approximation (MSA). ~13)

The approximate form of Eq. (3) written as

gij(r) ~ g~ij s [1 -/3 Wi~(r) + 1/~Wi~(r)] (5)

with g~ijS(r) = 0 (r- a) (6)

where 0 (r - a) is the step function and with W~j(r) given by

Wij(r) - ZiZJe2 (exp(Ka) -)exp(-Kr) (7) ~r 1 + ra

constitutes the basis of the Pitzer's semiempirical treatment.~4) The in- verse Debye screening length r is defined as

K 2 = (4rre2/EkT)Z CiZi2 (8)

where C~ = v iC = v ~r is the bulk number concentration (particles per unit volume) of species i. Equations (5), (6), and (7) can be com- bined giving results for the activity and osmotic coefficient data of real electrolyte solutions which are comparable with those obtained using more accurate approximations/11) Taking g~jS(r) = 0 (r - a) constitutes the zeroth approximation ~u) in the hard-sphere distribution function. The first approximation, corresponding to the second virial coefficient is(9A1,15)

0

g~ijS(r) = 1 + (4rra3/3)p [1 - ( 3 r / 4 a ) + ( l / 1 6 ) ( r / a ) 3]

1

r<~a

a < r<~2a

r > 2a (9)

90 Martin, Gomez-Estevez, and Canales

where p is the total bulk number concentration of hard spheres with diameter a (,o = C+ + C_ = v C in our case). The use of Eq. (9) in- stead of 0 (r - a) would improve the results. We will call the ap- proximation given by Eqs. (5)-(7), the Debye-Htickel plus hard sphere in the zero order approximation (DHHS0). It differs from that employed by Pailthorpe et aL (tx) because there is one more term than in their DHHS, i.e., the/3 2I, V~ ( r ) /2 term in the pair distribution function. The extra term makes no contribution to the internal energy (for sym- metrical electrolytes), but includes some nonelectrostatic effects and contributes to osmotic coefficients and free energy. The next ap- proximation given by Eqs. (5), (7), and (9) will be called the Debye-Ht~ckel plus hard sphere in the first approximation (DHHS1).

The DHHS0 and DHHS1 are approximations to the RPM which is pairwise additive by definition. When a comparison is made with real electrolyte solutions the non-pairwise additivity effects (2,s) are included in the fitted values of a. In order to elucidate their effect, we will use in this paper the DHHS0 and DHHS1 approximations as reference, pairwise additive, pair distribution functions before the NAE are incor- porated into the pair distribution functions.

Aldeman's theory, (13) takes into account the solvent effects in a rigorous way by means of an effective pair potential Wi~ ff that is de- pendent on the state of the system to which gij pertains (Le., on the temperature, pressure and solute concentration). The starting point is the exact Ornstein-Zernicke (OZ) equation for the solution at the Born-Oppenheimer (BO) leveL (s) By means of exact mathematical transformations, an effective OZ equation is obtained (~) involving only the solute components

hij(rl2) = Ci jeff(rl2 ) --t-~pkf hik(r13)C~eff(r32)dr3 (10)

where err �9 C~j is the effective direct correlation function, hij(r) ---- gij(r) - 1 and O k = Nk/V. When r ---, ao and for arbitrary solute concentration, Adelman finds (~3)

ZiZje 2 limitC~rr = -/3 ~ (11) r-.=o E or

where % is a solute concentration dependent dielectric constant men- tioned in the introduction to this paper. Using Eq. (11) and with the OZ equation, the behavior of hij for r---,oO, in the limit of dilute solu- tions as well as the non-pairwise additive effects, can be found (n)

Concentration Dependent Dielectric Constant 91

Zi Zj e 2 hij = - / 3 exp (-Kor) (12)

EpF

where K o is related to Debye's kappa by

K~ = K2(e/ep) (13)

An approximate for the dependence of h~j(r) for all r can be found by solving a modified PB equation. Taking Eq. (11) for Cff r and following the procedure given by Henderson and Blum, 06,17) it is possible to show that when a solute concentration dependent constant ep is considered, the PB equation has the same form as the usual PB equation but with % instead of e. This fact allows us to write an approximate expression for Wff r incorporating the diameter of the ions.

Wi~ fr= Zi_je2[exp(Kpa)Z l exp( 'K0r) (14) %r 1 + Kpa

From Eq. (14), we can construct an effective pair correlation function which takes into account, at least in part, the NAE as

gi~r ~ g~S (r) [1-/3 Wffr(r) +1/~2 (Wffr(r)) 2] (15)

The approximation given by Eqs. (6) and (15) will be called the non- additive Debye-Htlckel plus hard sphere in the zero order approxima- tion (NADHHS0). The combined use of Eqs. (9) and (15) gives the NADHHS1. The comparison of these two approximations with the ad- ditive approximations will give us some information about the impor- tance of the NAE. This task will be accomplished by means of the virial route to the osmotic coefficient adapted to this particular case.

3. THERMODYNAMICS: THE PRESSURE EQUATION

The solute-solute pair distribution functions gij(r) obtained by using the effective pair potential will be the same as gij(r) obtained directly at the BO level as long as the state parameters also match. (2) For this reason, the compressibility equation for the osmotic pressure derived by Kirkwood and Buff 0'2'1~ will be the same. These results will be presented in a forthcoming paper (Part 2 of this series). However, the compressibility equation gives the derivatives of the activity coef- ficient or the osmotic coefficient 4~ with respect to the concentration of the solution. If we are interested in the absolute values of ~b, we must

92 Martin, Gomez-Este'vez, and Canales

use the virial route (1~ to the thermodynamic properties. The problem which emerges now is that we do not have a pressure equation adapted for working in the context of Adelman's theory. (2) The usual pressure equation at the BO level cannot be used because of our lack of knowledge of solvent-solvent and solvent-solute pair correlation func- tions. Moreover the usual pressure equation depends on the real inter- molecular potential, not the effective potential.

The MM theory (2,s) can be used in an approximate way for this purpose. For this reason, we present here a brief derivation of the pressure equation using the MM theory. The MM theory shows that the grand partition function at the MM level (~: MM) for a system with component o- as a solute in the solvent w is (2's'~8)

~ ~M~l ---- E f exp[-/3 W~o"ld(No") (16)

No"~0 N~!

where Zo- is the activity of the solute, y2 the activity coefficient of the solute at infinite dilution (referred to an ideal gas scale), and W~o- is the potential of mean force among N solute molecules at infinite dilu- tion (Zo- ---, 0). (2'18) It must be stressed that W~o" depends on {No"}, T and Zw, the activity of the solvent. The activity coefficient of the solute at infinite dilution y 2 depends only on T and Zw.

The osmotic pressure zr is related with ~: MM by means of the relationship (19)

w = k T ( D lns MM/3 V)T4Lw4zo- (17)

where/z, , , /.t~ are the chemical potential for the solvent and solute, respectively. Using de Green's method, (~2,ls) we can write

with

zr = (1/~:MM) Z (Zo-/~2) No- ( ~ZN~ ~ 0 No-! - - ~ ) s~,,, (18)

( 3ZNO- )T.w

1 1 = No" VN~ "'" f0 e x p [-~ WN~ -] dx{ ... dZ/~ o-

1 1 ~ W~ ~176 (VNO-/kT)Y.... f. exp weo-I - v v ( D V ~ ) r'~wdXl"" dZN~

No- t t

(19) and where the solute coordinates q~ are related to the qi' by q~ -- V1/3q" . Until now, Eqs. (18) and (19) are exact. In order to evaluate the partial derivative of W~, we use the pair-wise additive assumption given in Eq. (1), but only in the partial derivative. Following the usual

Concentration Dependent Dielectric Constant

procedure (12'18) we have

93

=

- ( 1 / k T ) Z Nr 6 V(N~-2) !

�9 f exp [/3 W~ldr3 ... drN~]

(N~ / V) f exp [-/3 W~:,,] d{ N~}

d W~ (r12) �9 f f r l : drldr2

dr12 (20)

Using the exact relation, i.e., including the non-pairwise additive ef- fects, for the solute-solute pair correlation function g~(2) in terms of

MM (18)

P~go-,,2 (rl,r2,Zw,Z~)=(1/~MM) N~0Z (N~-2)! fexp['fl W~]dr3"'" drN~

we have for 7r (21) |

eft 2 (~r/p~kT) = 1 - (p~/6kT)s r( dr )T,~wg~(r,Zw, Z~)4~rr dr (22)

where we write ~rf g ~ (r) instead of g ~ (r) to stress that this pair correla- tion function is exact and depends on the thermodynamic state of the solution. In other words is the Adelman's solute-solute pair correlation function.

Usually, Eq.(22) has been employed after introducing a pair cor- relation function computed also under the pairwise additivity assump-

g;-o-(r) or simply g ~ (r). In this sense, the following equation for tion, add 7r

k | d W ~ )X~,wg~(r,Zw,Zc,)4'rrddr 1r~dd/p~kT= 1 - (p~./6 T)for( dr (23)

contains one more approximation than Eq. (22) because the NAE have not been considered in g ~ (r). A comparison between Eqs. (22) and (23) would reflect in part the NAE which, in our case, are introduced by means of %.

Using the definition of the osmotic coefficient (12,15) in the general multicomponent case,

(~ = 71"/qr ideal = T / " / Z C i k T (24)

we have for (b, after substitution of W~ [see Eqs. (2) and (3)]

94 Martin, Gdmez-Est~vez, and Canales

2~ Ce 2 | 2"n" Ca 3 4~ = 1 + 3 e k T ZiZJvivJ (r) tadr+ ~ ]~vivjg~ff(a +) (25) ', 3v i,i

where g~fr(a+) are contact values of the exact radial distribution func- tions. A similar equation can be written for the additive case with the adequate gij(r).

Introducing the DHHS0 and DHHS1 pair distribution functions in Eq. (25), we obtain for ~b in the 1-1 electrolyte case (u = 2)

K 3 K4a ~b DnHs~ 1- + C[(4/3)~ra3"F ] (26)

487r C(1 +K a) 96rr C 2 (1 +K a) 2

~b onnSl = ~b DartS0 + Iv / (3) (96) ]/~ (K a) exp (-K a) - h (K a) l + K a

K 4a +C[(5 /12)1rvCa3][(4 /3)~ 'a3"~ ] (27)

96rr C2(1 +Ka) 2

where the functions/x (x) and h (x) are defined by

/z (x) = 12x + 6 (28)

h (x) = 5x 3- 9x z + 6x + 6 (29)

Equation (26) is not new. It was obtained by Pitzer (t41 and constitutes the basis for his semiempirical approach to the thermodynamics of electrolyte solutions.

For the NADHHS0 and NADHHS1 approximations, the expres- sions for 4~ have the same form as Eqs. (26) and (27), respectively, after the substitution of K p for K.

4. ANALYSIS OF THE OSMOTIC COEFFICIENTS OF AQUEOUS ALKALI HALIDES AT 25~

In order to study the NAE, the experimental osmotic coefficients are necessary. In this work, we have considered the aqueous solutions of alkali halides at 25 ~ with the exception of LiF. The compilation of Hamer and Wu (z~ has been employed. They fitted the available ex- perimental data by means of an equation of the form.

Concentration Dependent Dielectric Constant 95

q5 = 1 -2.30259{ ~Z+Z_TA [(l+B,mV2). 1 (B* ) 3m 1 + B * m ~

- 4 .60521og( l+B*m' /2 ) ] - ( t 3 - / 2 ) m - ( 2 / 3 ) C m 2

- ( 3 / 4 ) D m 3 - ( 4 / 5 ) E m 4 - ( 5 / 6 ) F m 5 - ( 6 / 7 ) G m 6} (30)

where A is the DH limiting slope, B*, /3 , C, D, E, F, and G are ad- justable parameters which are dependent on solute, temperature and pressure, and m is the molality of the solution.

The experimental osmotic coefficients given by Eq. (30) must be converted to the theoretical coefficients at the MM level 6 MM by means of( H )

~exp = (1 - vCV2)4~ r~M (31)

before comparison with the theoretical predictions. In Eq. (31), C is the molarity and V 2 is the partial molar volume of the salt in the solu- tion. In this paper, we have used the expressions compiled by Millero (21) for the apparent molar volume V~

V~ = Vg + SvC" + b,,C (32)

where Sv is the limiting DH slope for V~ and V~, b v are known for each electrolyte. The partial molal volumes are calculated using the relation

V 2 = ( D m V e J D r n ) r , p (33)

The conversion from molarity to molality has been done by using the empirical relationship proposed by Harned and Owen (~2)

C = p o m + B ' m s + C ' m 3 (34)

where p o is the density of pure solvent and B ' , C' are adjustable parameters which depend on solute and temperature. The values of B ' , C' has been obtained using known density data (23'24) following a known procedure. (2s)

Finally, it is necessary to know the concentration dependence of %. Due to the complexity of the systems considered in this work, we have used the experimental dielectric constant e sot defined by (2s)

ffsol = limit(~ (r - 4arcr/~ ) (35)

where o- is the static conductivity and co = 27rf where f is the fre- quency. However, �9 SOL is not a true equilibrium property and really we have(3, 26)

96 Martin, Go'mez-Est~vez, and Canales

Table I. Values of a Which Best Fit $ Showing the Influence of

Short Range and Non-additivity Effects a

b Salt r a(DHHS0) a(NADHHS0) a(DHHS1) a(NADHHS1)

NaF 2.31 2.80 2.54 2.81 2.54 KF 2.69 3.35 3.28 3.32 3.27 RbF 2.84 3.75 3.69 3.72 3.66 CsF 3.05 3.95 3.91 3.91 3.87 LiCI 2.41 4.16 4.07 4.10 4.01 NaCI 2.76 3.54 3.45 3.52 3.43 KC1 3.14 3.37 3.29 3.36 3.27 RbCI 3.29 3.31 3.24 3.31 3.22 C sCI 3.50 3.15 3.09 3.15 3.09 LiBr 2.55 4.33 4.24 4.26 4.18 NaBr 2.90 3.81 3.72 3.78 3.68 KBr 3.28 3.56 3.47 3.54 3.46 RbBr 3.43 3.43 3.30 3.37 3.29 CsBr 3.64 3.21 3.14 3.19 3.14 LiI 2.76 4.83 4.75 4.78 4.64 NaI 3.11 4.20 4.12 4.15 4.06 KI 3.49 3.90 3.83 3.87 3.83 RbI 3.64 3.54 3.46 3.52 3.44 CsI 3.85 3.36 3.29 3.35 3.28

aThe molality range is 0.1 - 1 mol-kg -t. Units of a, angstroms, bSum of the bare ion radii (in A) calculated from Pauling's radii, Ref. 34.

Eso,~ -- eo + esoD (36)

where E ~DD is a dynamical contribution which persist at zero frequency. However, it seems that up to moderately concentrated aqueous solu- tions e~oo is only a small fraction of the quantity eo - ~ .(3,27) Neverthe- less, the question remains open as discussed recently by Winsor and Cole. (28)

In this paper, we assume that eSOL~eo. Only the experimental data (2932) up to 2 tool-din -3 were considered. The ESOL data for each electrolyte were fitted to

ep = e - 8 C + r 2 (37)

where ~ and tk are adjustable parameters. When only data up to 1 mol-dm -3 are available, the concentration behavior is practically linear, Le., y "- O.

Concentration Dependent Dielectric Constant

Table II. Values of a which Best Fit ~b in an Extended

Molality Range ~

97

Salt a (DHHS0) a (NADHHS0) a (DHHS 1) a (NADHHS 1)

KF 3.36 3.32 3.35 3.31 RbF 3.68 3.62 CsF 3.98 3.89 LiCI 4.20 4.08 4.07 3,96 NaCI 3.52 3.30 3.48 3.26 KC1 3.36 3.19 3.33 3.15 RbC1 3.36 3.18 3.33 3.16 CsC1 3.27 3.11 3.25 3.09 LiBr 4.44 4.34 4.28 4.20 NaBr 3.83 3.69 3.76 3.61 KBr 3.57 3.48 3.52 3.44 RbBr 3.43 3,40 CsBr 3.33 3.31 LiI 4.86 4.63 NaI 4.27 4.16 4.15 4.04 KI 3.94 3.88 3.86 3,80 RbI 3.65 3.60 CsI 3.48 3.45

aMolality range, 0.1 - 2 mol-kg -1. Units: A.

We note that the computation of the pair correlation functions using experimental dielectric constants is not new. A theoretical cal- culation, based on the MSA with a primitive model in which the dielectric constant of the solution is set at 25, has been carried out for a 14.9 molal LiC1 aqueous solution. ~33) The experimental value C29) of e SOL for LiC1 is nearly the same.

After the conversion of theoretical values of r to the experimen- tal ones [see Eq. (31)], the NAE are measured by considering a, in effect the cation-anion distance of closest approach, a parameter to be fitted to the data. We have adopted the criterion that the best a value is that which gives a min imum standard deviation s of the fit.

5. RESULTS AND DISCUSSION

In our study, we have considered two different molality ranges: 0.1 - 1 and 0.1 - 2 mol-kg -1 with the exception of NaF which saturates at 0.983 mol-kg -1. Usually MM models with solvent-averaged poten- tials which are pairwise additive are limited to sufficiently dilute solu-

98 Martin, Gomez-Estevez, and Cana|es

1 . 1 6 0 z "~ t '

~ 1 . 0 3 5

0 . 9 1 0 L , , t , , ~ , ,

0 0.4 0.8 1.2 1.6 2.0

m/mol kg -1 Fig. 1. Comparison among osmotic coefficient data and the DHHS0 and DHHS1 ap- proximations for LiCI: (__.__._), experimental data; ( . . . ) , DHHS0 with a = 4.20 A; (- - -), DHHS1 with a = 4.07 A.

1 . 1 6 0

a.o35 .~...~"

0 . 9 1 0 , ~ ~ ~ t ~ ~ , ~ - -

0 0 . 4 0 . 8 1.2 1.6 2 . 0

m / t o o l k g -1

Fig, 2, Comparison among osmotic coefficient data and the NADHHS0 and NADHHS1 approximations for LiCI: ( ), experimental data; ( . . . . _ ) , NADHHS0 with a = 4.08 A; ( - . - .D , NADHHS1 with a = 3.96 A.

tions so that the effects due to three or more ions are negligible. However, since 'dilute' is not well defined considering two different molality ranges, some information about this point can be extracted.

After fitting the osmotic coefficient data a for each alkali halide can be obtained jointly with s for the four approximations (DHHS0, NADHHS0, DHHS1, and NADHHS1). In Tables I and II the results

Concentration Dependent Dielectric Constant 99

1200 f

1.055

0.91( J , , ~ , ,. , ,. , 0 0.4 08 1 2 16 2.0

m/tool kg -~

Fig. 3. DHHS0, DHHSI , and NADHHS1 predictions of~b for NaI. ( ), experimental data; ( . . . ) DHHS0 with a = 4.27 A; ( _ _ -_) , DHHS1 with a = 4.15 A; ( . . . . _), NADHHS1 with a = 4.04 A.

'~ f

0.92!

0.851

\

I I I I I I I I I

0 0.2 0.4 0.6 0.8 1.0 m / t o o l kg -1

Fig. 4. DHHS1 and NADHHS I predictions of~b for NaF. ( ) experimental data; (- - -) DHHS1 with a = 2.81 A; ( . . . . _), NADHHS1 with a = 2.54 A.

are summarized, and in Figs. 1 - 6 several examples are shown. The s values are all about 10 -3 in the two molality ranges, which agrees with the usual experimental uncertainties. As can be seen in the figures, a comparable fit can be obtained with the four approximations, but the a values differ among themselves. This fact agrees with the well known success of the MM models in fitting the experimental data. (3) However, the physical meaning of the adjusted parameters is more difficult to

100 Martin, G6mez-Fst~vez, and Canales

1.0 ,,,$7 / , , ' /?" /

,//Y/ 0.95 , , / ~ . / ~ 4 ~ " "

0.90 , ~ . ~ ~ L.__._t..~_ L . , , 0 0.4 0.8 1.2 1.6 2.0

ml mot kg -1

Fig. 5. DHHS1 and NADHHS1 predictions ofq~for KF. (_-,_._) experimental data; (- - -) DHHS] with a = 3.35 A; ( . . . . _) NADHHS1 with a = 3.31 A.

1.00

0.92 ~%~~.~,,~,~..~,

%. ~ ~ . . ~ - . ~ . ~ . . . .

0.84 , , ~ ,____~. ,. , . ~

0 0.4 0.8 1.2 1.6 2.0

r n l m o I kg "1 Fig. 6. DHHS0, DHHS1, and NADHHS I predictions of& for CsCI ( . ) , experimental

data; ( . . . ) , DHI--IS0 with a = 3.27 A; (- - -), DHHS1 with a ~ 3.25 A; (_._._), NADHHS1 with a --- 3.09 A,

assess because the additive effects are incorporated in them. From Table I (the shorter molality range), we can see that the

NAE introduced by means of % are more important that the short range effects, which at these molalities seems reasonable. Always we obtain a (additive) greater than a (non-additive). The NAE contribu- tion to a is important for NaF (--10%) whereas, for the rest of alkali halides, it is between 1 - 4%. When the molality is increased (see

Concentration Dependent Dielectric Constant

NaF

KF

RbF

Cs F

Li CI

NsCI

KCI

Rb C!

CsC[

Li Br

N~Br K ~

Rb l l r

C s l l r

Li |

N i l

K I

Cs I

101

Fig. 7, Influence on a values due to the hard sphere part of the pair distribution func- tions: ( . . . ) DHHS0; (- - -), DHHS1; ( ) sum of the bare-ion radii. The covered molality range is 0.1 - 2 mol-kg -1 .

. . . . . ~ . . . . 7 . . . .

NaF

KF

~b F

CsF

L~ C1

NIC{

CI)~

RbCI

CsCI

LJ Br

NIBr

KBr

RbBr

CsBr Lf !

KI Rbl

Cs l

r a / A

Fig. 8. Change of a values after the inclusion of the non-additivity effects by means of~_: �9 P

( . . . ) , DHHS1; ( . . . . ._) NADHHS1; ( ) sum of the bare 1on radii. The covered molality range is 0.1 - 2 mol-kg -1 .

Table II), the effects o f the hard sphere part o f the g(r) are s tronger but always less than the % effects . Again , a (addit ive) is greater than a (addit ive) . T h e change o f a (non-addi t ive ) wi th concentra t ion is evi-

102 Martin, G6mez-Estevez, and Canales

Table III. Correlation Among the d• Parameters and the Differences a ( D H H S 0 ) - r a n d a ( N A D H H S 0 ) - r

DHHS0 NADHHS0 a a

Salt a - r a - r d • b d • b d • b

NaF 0.49 0.23 KF 0.66 0.59 0.2153 RbF 0.91 0.85 0.2039 CsF 0.90 0.86 0.2131 LiCI 1.75 1.66 0.90 0.62 1.092 NaCI 0.78 0.69 0.25 0,40 0.1748 KCI 0.23 0.]5 0.18 -0.0975 RbC1 0.02 -0.05 0.01 -0.1610 CsCI -0.35 -0,06 -0.20 -0.14 -0.2584 LiBr 1.78 1,69 0.90 0.71 1.060 NaBr 0.91 0.82 0.30 0.36 0.1924 KBr 0.28 0.19 0.14 -0.1099 RbBr 0 -0.13 -0.01 -0.2124 CsBr -0.43 -0.50 -0.25 -0.18 -0.2924 LiI 2.07 1.99 1.30 1.159 NaI 1.09 1.01 0.35 0.1845 KI 0.41 0.34 0.32 -0.1199 RbI -0.10' -0.18 -0.04 -0.2524 CsI -0.49 -0.56 -0.30 -0.19 -0.3356

a Values of r, a (DHHS0) and a(NADHHS0) in A taken from Table I. ~ d • values taken from references 35, 36, and 37, respectively.

dent if we compare the data of Tables I and II. The differences a (non- additive, 2m) - a (non-additive, 1 m) can be as high as 4.5% for NaCI but only about 0.3% for KCI and KBr.

In Fig. 7 are represented a for the DHHS0 and DHHS1 ap- proximations jointly with the sum of the bare-ion radii r for the dif- ferent alkali halides. With the exception of fluorides, the trend of a vs.

alkali halides is opposite to the sum of bare ion radii giving values of a lesser than r for cesium and rubidium halides. Whereas Fig. 7 shows the influence of the hard sphere part of the radial distribution functions on the adjusted a, in Fig. 8 the influence of the NAE is shown. The pattern is similar to that observed in the previous figure, but the in- fluences of the NAE are bigger. Again, a values lower than the cor- responding sum of the bare ion-radii are obtained for cesium and rubidium halides. Even though a better form for the hard sphere ef- fects and an approximate evaluation of the NAE have been included in gij(r), some aspects of the interactions in solution have been missed.

Concentration Dependent Dielectric Constant 103

A better model should include other short range effects (m apart of the hard sphere part in I,V~ [see Eqs. (2) and (3)]. A first step could involve using the d+_ parameter, the depth of Rasaiah extra- square well interaction. (3s) In Table III, we show the differences a(DHHS0) - r and a (NADHHS0) - r jointly with the d+_ parameters obtained from three different sources. (35-37) As can be seen, a good cor- relation exists among them, showing that non-hard sphere effects have been absorbed in the fitted a values. The potassium halides are the border among a - r > 0 and a - r < 0 values, in the same form that for d+_/kT parameters. Using the notion of equivalent hamiltonian models proposed by Justice and Justice,(36) a set of d+_ could be com- puted from our a values. We will present them in a future work but, as a first remark, we think that the d+_ computed by using non-additive a should be more realistic than those obtained by means of a (additive).

As a conclusion, we can affirm that the introduction of the non- additivity effects by means of % is relevant, giving a values closer to the sum of bare ion-radii. However, apart from the approximate nature of the radial distribution functions, there exist difficulties related to the approximate character of Eq. (22). In a forthcoming paper, we will try to obviate this problem by using the compressibility equation (~,2,8-1~ which is exact. As a consequence, a better analysis of the NAE effects could be accomplished.

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