On the excess thermodynamic properties of aqueous electrolyte solutions—I

t inorg, nucL Chem. Vol. 43, pp, 1005-1010, 1981 Printed in Great Britain

0022-190218110510054)6502.CB]0 Pergamon Press Ltd

ON THE EXCESS THERMODYNAMIC PROPERTIES OF AQUEOUS ELECTROLYTE SOLUTIONSmI FREE ENERGIES INCLUDING EXCESS SOLVENT ISOTOPE EFFECTS GERALD DESSAUGES and W. ALEXANDER VAN HOOK* Chemistry Department, University of Tennessee, Knoxville, TN 37916, U.S.A. (Reveived 6 February 1980; received[or publication 13 July 1980)

Abstraet--A new approximate folmulation describing the electrostatic contribution to the excess free energy of electrolyte solutions is presented. The final equation contains only one parameter and reduces to the Debye-Huckel limiting law at low enough concentrations. It represents the experimental data nicely for the series of alkali metal halides out to quite high concentration, and clarifies certain difficulties in interpretation of solvent isotope effect data with extended theories now in widespread use. INTRODUCTION Consider as an example the concentration dependence of the logarithmic activity coefficients of the alkali metal chlorides. An initial rapid decrease at low concentrations is followed by a more or less pronounced minimum after which In y crosses to the ++ quadrant (at least for the lower cations), and may increase to very large positive values unless the solubility limit is first exceeded. The sharpness of the minimum increases in the order Cs > Rb>K>Na>Li. The initial slope of the In y vs m curve was explained by Debye and Huckel[1] who introduced the concept of the ionic atmosphere in order to arrive at a solution of the Possion-Boltzman equation. Their solution has been extended to finite concentrations by numerous authors who have made empirical or semiempirical extensions to be theory[2] proposing equations which are typically of the form: Iny=-Sfl'/Z/(l+Al'/2)+Bl+Cl2+ ....

(I)

In this equation S-, is the theoretical limiting slope, A is related to an empirical size (excluded volume) parameter, I is the ionic strength, and B,C, etc are empirical constants to be evaluated from fits to the data. It is important to recognize that any model which retains the DH expression derived from the single ion atmosphere model, then corrects it with some kind of a power series, must be regarded as empirical at higher concentrations where the linearized Poisson-Boltzman equations is no longer valid. Although expressions of the type of eqn (1) constitute an eminently practical approach for data complation and interpolation[3], they suffer from the limitation that the physical phenomena(on) responsible for the marked minimum may be lost in the tangle of correlation errors which necessarily exist between the parameters of fit, A, B, C, etc. Another important reason for dissatisfaction with eqn (1) stems from attempts to apply it to the rationalization of solvent isotope effects[4--6] all of which show phenomenologically well behaved Ir(r ~- 1/2) dependencies out to relatively high concentrations with slopes that are much larger than those calculated from the known isotope effect on S-, or *To whom correspondence should be addressed.

its derivatives~ If the observed isotope effects ~,re developed as linear combinations of AB=Bh-Ba, &C, etc., they are not well behaved in the context of normal expansion coefficients (AB> :z A C > ~AD, etc.) and this is puzzling. Certainly the correct extended theory should account for the properties of D20 solutions in the same framework used to describe H20 solutions. Excess solvent isotope effects are small and the expectation that they should be understandable in terms of first order corrections to the theory appears reasonable. For all of these reasons we have concluded that examination of new approaches to the propblem is called for. Thus Lietzke, Stoughton and Fuoss[7] have successfully parametrized the problem around a two state model. In the present contribution we attempt to clarify the situation by making appropriate changes in the electrostaticpart of the model thereby arriving at simple empirical or semi-empirical expressions containing fewer parameters than eqn (1). We suggest approximate solutions to ILhe Poisson-Boltzman differential equation based on boundary conditions assumed to reflect the electrostatic contribution to the nature of solutions at moderate concentrations. Bennetto and Spitzer[8] have also considered electrostatics of moderately concentrated solutions but from a different point of view. In their approach 'the assumption of a spherically symmetric electrostatic potential was abondoned, one result being a considerable increase in mathematical complexity. A MODIFIED DEBYE MODEL

In the Debye theory of electrolyte solutions it is assured that the ionic atmosphere about a central ion displays spherical symmetry so that solution to the linearized Posson-Boltzman equation is readily given Ae --,

B e -~

~ ( r ) = - - ~ - r + Dr"

(2)

In eqn (2) tb(r) is the total electrical potential and Z = 41re2 S' n,z~ Dk T xT'

(3)

with E being the charge of the electron, D the dielectric constant, z the ionic charge and n~ the number density of 1005

GERALD DESSAUGESand W. ALEXANDERVAN HOOK

1006

ions. For physical reasons it is necessary that the potential O(r) remain bounded even at large values of r and/or r. Previous authors/2/have assured this by inserting boundary conditions appropriate to very dilute solutions, concluding since lim (O(r)) = 0, that B = 0. That this is not the r~A general case has been pointed out by Moelwyn-Hughes (9); "Particular solutions are selected by imposing appropriate boundary conditions. Thus for example, to make ~ zero when r is infinite, we may assume that B is zero, t h r o u g h o t h e r m e a n s o f i m p o s i n g the s a m e limit exist." (Emphasis added). We have proceeded by applying the Huckel zero field condition to ions of finite size directly to eqn (2), as outlined by Harned and Owen[10]. In that case the field attributed to the ionic atmosphere (at r = b) is zero and the total field equals the field of the ion alone. ( 6 ' is the potential of the ionic atmosphere, ~k the total potential, 6"= $* + el(self), and b is a parameter defining the radius of the excluded volume.)

dr/~b

dr

~=b = -- D b 2"

Using eqn (4) and solving 2 for

B,

(4)

ion the potential is given approximately by eqn (2), where the use of the term B# 0 acts to approximate the complicated nature of the superposition in the region of intermediate r. The situation arising from the simple model described at the end of the last paragraph is illustrated in the following few lines. For the two ion, two atmosphere, system the approximate superposed potential, ~b'(app), is given $,.

,

Ale -~

tapp.I = T

. B~e ~ _ A 2 e - " ' , + B2e "~2

+ ~

-,,~1 m2e--(,J+a~> ~¢rl n2e~(rl +at) ~_Bte _~ ~b'(app) = Atelr "~ rl + Ar r~ rl + Ar (9) The second term on the right, that containing A2 is negligible at large enough Ar, and the equation can be rewritten in a form not too different from eqn (2). ~'(app) = A l e - " " + B~e "'~ 1 -~ ( a d a , ) e ~ , ]

one obtains

l)e -"b + B ( r b Db 2

-

I)C b

(5)

or B =

- ze + A ( K b + 1)e-'b ( x b - l)e "b

(6)

We now consider the behavior of B (eqn 6) as a function of K keeping in mind that the total potential must remain bounded. The condition of Debye may be applied at the lower limit, ~ = 0; ZE

6,,=o = ~

(8)

At the boundary r~ = r2 = ~ spherical symmetry is restored and therefore the condition bounding the potential is Bl + B2 = 0. In the vicinity of either one or the other of the ions, however, (say Ion 1), A r = ( r ~ - r 2 ) > r ~ SO

rl ze - A(Kb + Db 2 =

~-~ - - ~ r 2 '

(7)

and it follows that A,=o = zE. Substitution in eqn (6) shows B,=o = 0. At the other limit, (super) high concentration, Kb ~ , the exponential in the denominator of eqn (8) predominates, and B,== = 0. Now consider intermediate values of K. If attention is focused on a single ion in solution surrounded by an ionic atmosphere which we specify by a continuous charge density (as did Debye), eqn(2) clearly demonstrates that B = 0 for all values of K. Otherwise the potential diverges at large r. If, on the other hand, we refine the model by recognizing that the ionic atmosphere in the immediate region of the central ion consists of discrete ions the simple sphericalized Debye approach fails. Even so it is clear that at very large r the potential remains bounded as evidenced by the physical condition of electroneutrality over any macroscopic region in the solution (X vi$i,=®= 0). To qualitatively explore this, consider a model for the solution which is one step more complicated than the Debye model. This consists of an ion and its far-enough removed counterion, each with an ionic atmosphere. At any point in the system the net potential, ~,', may be considered as a superposition of the potentials due to the individual ions. Under the boundary condition r~ ~ ~, $' is strictly zero. Even so in the vicinity of each individual

rl

1 + Ar/rl

(10) J"

A proper development would involve the summation over a whole set of countrions, each with its own ionic atmosphere and migh well result in the cancellation or partial cancellation of the bracketed term/Ill. In any case we have chosen to approximate the complicated angular dependent superposed potential with eqn (2). The demonstration presented here has certainly not been rigorous, but it has outlined a physical argument which supports the adoption of eqn (2) as a suitable, albeit empirical, working equation. Equation 2 (employing B as defined in eqn(6)) is obviously the better closer one is to the parent ion. Fortunately it is in just that region that the principle contibution to the excess free energy arises. By employing the approximate relation 2 in the G~intelberg charging process/12/ thereby defining #% interesting questions concerning the precise form of the potential at intermediate values of r are ignored (but perhaps partially accounted for later by inserting an empirical concentration dependence for the parameter b in the equations for/ze~). A# "x=

fO~e

¢*de

ze~,* -(ze)2K [ 1 - 2 e -"b] = 2 = 2D t K-~-ZT-.IJ (11)

6 ' is the potential of the ionic atmosphere; from eqn (2) and (7), ~b* = - ze

ze . . . .

---D-r-+ --D~- +

1)C b e "" (Kb- 1)e"b Dr

ze + z e ( x b +

(12)

and at the lower bound of the integration r = b. The procedure exactly parallels that given by Harned and Owen[10] and yields In y +

= - SvI 1/21 - 2e -~b

rb-

1 "

(13)

On the excess thermodynamic properties of aqueous electrolyte solutions--I In eqn (13) the concentration of ions per unit volume and the ionic strength have been assumed to be realated through the solvent density, an approximation shared by most other authors[2]. Equation (13) is in an interesting form. It reduces to the limiting law (In ,/± = - S~I ';2) at very low concentrations, has a natural zero at Kb = In2 and diverges toward infinity as Kb approaches 1. It is therefore of a form to qualitatively and perhaps quantitatively explain the observed concentration dependence of the activity coefficients. Some further remarks on eqn (13) are in order. Examination of eqn (4) shows the left hand side is concentration dependent because ~* is a function of K. The contribution of the ionic atmosphere is measured in terms of the Debye radius, l/K, which is converging to zero at high concentrations although geometric packing factors preclude it from ever attaining that value. It is lherefore reasonable to expect "b" to vary from a value characteristic of the dilute system, bo, to one characterizing limiting high concentrations, b~. We have examined several alternate routes of introducing the concentration dependence of "b". The most rational prodedes as follows. In this method we count the ,otal number of positive, z+n+, and negative, z n_, charges that are contained in the sphere of radius "b" while admitting that at high enough concentrations there is a geometrically limited upper value, say M 3, because of the intrinsic size of the ions. The number of positive ions is In? is the bulk cation concentration). n. = f b n ~e ~'*/DKT47rr2dr. )

(14)

A similar equation can be written for the anions. After expansion and integration we have through first order (z+ and z are both positive) 4~b3[ :.n+ = :+n ° - - ; -

3E2z+2]. 1 2DbKT]' 3~-2Z+z ]

2 n =z-n°~

-

1 + 2D--Dyg~j.

05)

The development follows Moelwyn-Hughes (9). The condition of electro-neutrality, z + z ° - z n_° =0, holds over the bulk solution but over the range of integration, 0 to "b" z+n+ is not necessarily equal to z_n_ except in the limit of limiting low concentration. At the high concentration limit, m*, z+n, ,- z.n_ = M 3 and lira b = b, : M(3/f41r~n, (max)zi)) ~/3 i

= M(3(55.508M~)/(47rdom*~uizD) I/3

(16)

i

where m* is the maximum aquamolity and M~ is the molecular weight of the solvent. According to this scheme, b = M 3(55"508M01/3

or b = b~

(17)

4zrdom ~ ulz,

whichever is larger. In the present calculations m* is never approached and it is therefore unnecessary to numerically define b,,.

1007

COMPARISONS WITH EXPERIMENT

The ideas set forth above are tested in a series ,of calculations reported in Table 1 where parameters of fit for the alkali metal and hydrogen halides (298.15k, O to 2 m) together with variances (o-2) are found. Plots for three representative compounds (LiC1, NaCI and CsC]) are shown in Fig. 1. The present comparisons have been made against the Hamer-Wu (3a) correlating functions based on eqn (I) and containing enought parameters to place the smoothed line comfortably within the experimental dispersion. Our examination of the original experimental data shows scatter generally between ().~{)3 and 0.006 unit (but sometimes as big as 0.01 or larger) for these different salts (see, for example, columns R-Z. Table 1, Ref.[3a]). The present standard deviations with respect to Ref. [3a] are mostly less than 0.015 mail and we conclude that the present fits deviate no more than 1.5 to 3 or 4 times the scatter in the experimental data. Considering the range of tha data and the fact that our equation contains only one adjustable parameter the fits are quite satisfactory. In Table 2 fits of 298.15K data (NaCI, 15CI) extending all the way to six molal are presented. At 6 molal the LiCI activity coefficient is already near 3 as compared with its minimum of about 0.75 near 0.4m The change for NaC1 is not as marked. Values of m parameters for these fits are compared with the 0--2 m fits in Table 2. The dependence of M on the selected concentration range is modest but the variances of fit (Table 2) increase markedly as ths range is extended. This is only to be expected; any model which includes nothing but an approximation to electrostatic effects should begin to fail somewhere well before 6m. Even so the qualitative agreement between calculation and experiment at high concentration is gratifying. The effect of temperature on the parameter M is explored in the first part of Table 2. M parameters for the 0--2 m NaC1 data at 0(4a), 25 (3) and 100°C(4b)are reported. There is very little change in M with temperature. The values at 0 and 100° are a little 'smaller ~:han at 25°C. Figure 2 and the last part of Table 2 compare experimental and calculated isotope effects on osmotic coefficients at 3.82 and 100°C. The data are from (!raft and Van Hook (4a) and Dessauges, Miljevic and 'can Hook (4b) as deduced from freezing and boiling point measurements respectively. The freezing point data show significantly less scatter for experimental reasons. The boiling point elevation experiments were difficult to execute. The fits to the isotope effects are most usefully presented in terms of the calculated effects on the M parameter, A M [ M = ( M H - M D ) / M H . These are small, amounting to only 0,017 _+0.002 (3.82°C) and 0.005 + 0.003 (100°C) respectively. The magnitude of these effects is reasonable especially when compared to the alarmingly large and compensating isotope effects which must be assigned to AB/B and AC[C in the extended formalism presented in eqn (1).

DISCUSSION

An approximate extension to the Debye-Huckel theory of the electrostatic contribution to the excess free energy of solutions of electrolytes has been presented. It is qualitatively and quantitatively more successful than earlier one parameter extensions because it does not

1008

GERALD DESSAUGESand W. ALEXANDER VAN HOOK Table 1. Fits of eqns (15) and (21) to activity coefficientsof alkali metal halides, 0-2 m, 298.15k[I, 21 Salt

Csl

CsBr

CsCI

M

0.222

0.226

0.230

i04c 2

2.6

2.7

3.5

Salt

Rbl

RbBr

RbCI

M

0.244

0.246

0.248

i04c 2

2.2

2.0

2.1

Salt

KI

KBr

KCI

M

0.275

0.264

0.258

i04~ 2

1.6

1.4

1.4

Salt

Nal

NaBr

NaCI

M

0.306

0.291

0.280

i04c 2

2.0

1.6

1.4

Salt

Lil

LiBr

LiCI

M

0.341

0.325

0.317

ii.

2.1

2.8

Salt

El

HBr

HCI

M

0.350

0.337

0.325

3.3

2.6

i04~ 2

i04c 2

13.

i.

Experimental data from reference

2.

Error in M is +0.001

attempt to force a limiting solution by applying the low concentration boundary condition at intermediate concentrations. The present approach is formulated in terms of a concentration dependent distance parameter, b, and that concentration dependence is deduced by an approximate model calculation. The resulting fits nicely explain the qualitative features of the free energy-concentration diagrams but fail to represent the experimental data to within experimental precision. (Variances of fit are 3 to 5 times observed experimental scatter). The cause for this failure might well lie in the approximate bounding of the potential, in an improper description of the concentration dependence of b, in errors introduced in the approximate Giintelberg charging process, or in a combination of these reasons. We feel that the drawbacks are compensated in major part by the simplicity of the approach. An examination of the M parameters (Table I) shows definite systematics with ion size, the value of M increasing markedly as the cation size falls along the series Cs > Rb > K > Na > Li > H. The observation is in qualitative agreement with the model which interprets M 3 as the maximum number of ions which can be contained inside the sphere of radius b~. The observation that M is

3.

not very sensitive to temperature or concenti'ation range (Table 2) also supports a relatively simple geometric interpretation. We regard the improved interpretation of the isotope effect data as important. The earlier observations of Van Hook et all4--6] that observed concentration dependences of isotope effects on the thermodynamic properties of electrolytes could not be rationalized by eqn (1) or similar equations without invoking large and compensating isotope effects in the higher order virial terms led to the development of the procedure reported in this paper. Here rationalization of the observed effects in terms of an isotope effect of reasonable size on the one parameter of the theory affords good agreement between calculation and experiment. A detailed interpretation of the physical origin of this effect awaits further analysis.

Acknowledgements--This research was supportedby the National Science Foundation,ChemicalThermodynamicsProgram,and by the National Institutes of Health, Institute of General Medical Sciences. G. D. extends thanks to Stiftungfiir Stipendienauf dem Geboete der Chemie, Basel for partoal support. The comments of one referee have markedlyimprovedthe presentation.

On the excess thermodynamic properties of aqueous electrolyte solutions--I

.2

.6

1.0

1009

1.4

•

LiCI

-.1

-,,2 i

•

o

~

-.4

-.5

0

NaCI

/

I!

,

-.6

DH

~•

CsCI

Fig. 1. Comparison of experimental [3] and calculated activity coetficients (eqns 15, 21) for three salts in H20,298.15 k, 0-2 m. Upper Curve, LiCI: Hammer and Wu (3a), O; Calculated M = 0.317, ×. Middle Curve, NaCh Hammer and Wu(3a), O; Calculated M = 0.280, +. Lower Curve, CsCI: Hammer and Wu (3a), A; Calculated M = 0.230, A. Debye-Huckel Limiting Law = DH. Table 2. Effect of concentration, temperature and isotope effect on the parameter M, eqns (15) and (21)[I] Activity Coefficient Salt

NaCI

NaCI

NaCI

TemFerature , k

273.15

298.15

373.15

Concentration ranBe of fit, m

0--2

0--2

0--2

M

0.272

0.280

0.269

104o 2

0.8

1.4

2.7

Reference (Exp. data)

2b,4a

3

2b,4b

Activity Coefficient Salt

NaCI

NaCI

LiCI

LiCI

Temperature, k

298.15

298.15

298.15

298.15

Concentration Range of fit, m

0--2

0--6

0--2

0--6

M

0.280

0.271

0.317

0.302

104o 2

1.4

16

2.8

81

Reference (Exp. data)

3

3

3

3

Isotope Effect on Osmotic Coefficient Salt NaCI

NaCI

Temperature, k

276.97

373.15

Concentration Range, m

0--2

0--2

(MH-MD) /~

0.017__+O.O02

Reference (Exp. data)

4a

i.

Error in M is +0.001

0. OO5_+O. 003 4b

1010

GERALD DESSAUGES and W. ALEXANDER VAN HOOK .5

~.o ~

REFERENCES

1,,9

0 '10

i

.<3 eq

o

v.-

2.5

.5

1.0

1,5

Fig. 2. Isotope effects on osmotic coet~cients of NaCI solutions at two temperatures. Upper curve 3.820, Experimental: from Freezing point depression (4a); Calculated, AMIM = 0.017_+0.002. Lower Curve 100°, Experimental: from boiling point elevation (4b) []; Calculated, AM/M = 0.005 _+0.003.

I. P. Debye and E. Huckel, Physik. Z. 24, 185 (1923). 2. See for example, (a) H. S. Harned and B. B. Owen, Physical Chemistry of Electrolytic Solutions, 3rd Ed. Rheinhold, New York (1958); (b) R. A. Robinson and R. H. Stokes, Electrolyte Solutions. Academic Press, New York 1955; (c) K. S. Pitzer et at. Accounts o[ Chem. ges. 10, 317 (1977) and references cited therin. 3. See for example, W. J. Hamer and Y. C. Wu, ./. Phys. Chem. Re[. Data 1, 1047 (1972); B. R. Staples and R. L. Nuttall, NBS Technical Note 928, U. S. Dept. Commerce 1976; R. N. Goldberg, B. R. Staples, R. L. Nuttall and R. Arbuckle, NBS Special Publication 485, U.S. Dept. Commerce 1977. 4. (a) Q. C. Craft and W. A. Van Hook, J. Solution Chem. 4, 923 (1975); (b) G. Dessauges, N. Miljevic and W. A. Van Hook, Ibid, submitted; (c) O. D. Bonner, J. Chem. Thermodynamics 3, 837 (1971). 5. Q. C. Craft and W. A. Van Hook, Z Solution Chem. 4, 901 (1975). 6. G. Dessauges, N. Miljevic and W. A. Van Hook, J. Phys. Chem. In press O. D. Bonner and Y. S. Choi, J. Phys. Chem. 78, 1723 (1974). 7. M. H. Lietzke, R. W. Stoughton, P. R. M. Fuoss, Proc. Natl. Acad. Sci., 58, 39 (1968). 8. H. P. Bennetto and J. J. Spitzer, J. Chem. Soc., Far. Trans. I 72, 2108 (1976); ibid 73, 1066 (1977); J. J. Spitzer, J. Solution Chem. 7, 669 (1978). 9. E. A. Moelwyn-Hughes, Physical Chemistry Chap. 18, p. 866. Pergamon Press, London (1957). 10. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, p. 65. Academic Press, New York (1955). 11. Thus for rl>>Ar and r A r ~ 1, l +[(B2/BOe~ar/(1 + Ar/rO] approaches 0 since B2 ~ - BI, Ar/q ~ 0 and e Ka, ~ 1.