J. Phys. Chem. 1995, 99, 822-827
822
Conductance in Associated Electrolytes Using the Mean Spherical Approximation Pierre Turq,*,t Lesser Blum? Olivier Bernard: and Werner Kunzt Luboratoire d’electrochimie, Tour 74, Universitt Pierre & Marie Curie, URA 430, 8 Rue Cuvier, 75005 Paris, France, Department of Physics, University of Puerto Rico, Rio Piedras, Puerto Rico 00931 -3933, and Laboratoire de Physicochimie Thtorique, URA 503, Universitt Bordeaux, I, 351 Cours de la Libtration, 33405 Talence, Cedex, France Received: September 13, 1994@
We discuss the theory of electrolytic conductance in weak electrolytes using the concept of ionic association for nonideal electrolytes within the mean spherical approximation (MSA). The chemical model approach is tested for solutions of MgS04 in water and LiBr in acetonitrile.
1. Introduction The conductance of electrolyte solutions is one of the oldest subjects in physical chemistry, which was started by Kohlrausch,’ who introduced, at the same time as Arrhenius, the concept of incomplete electrolyte diss~ciation.~,~ Since Debye and Onsager, it is ~ e l l - k n o w n ~that - ~ limiting laws exist for the variation of conductance of completely dissociated electrolytes with concentration. These laws have been extended to higher concentrationsas a power series in Cln, plus nonanalytic terms in C log C.’s8 The classical theory was based on the Debye-Huckel equilibrium distribution functions. Recently we proposed an approach in which Onsager’s continuity equations were combined with MSA (mean spherical approximation) or HNC9Jo equilibrium correlation functions. This results in new extended laws, which will be published elsewhere, and which are valid over a larger range of concentrations.“ As mentioned in our previous papers, Ebeling and co-workers established a similar combination of the Fuoss-Onsager theory and MSA.12-15 Our approach essentially differs in two points: (1) As far as the contribution from the electrophoretic effect is concemed, our theory explicitly takes the different ion radii into account. Furthermore, we also include higher order terms from the Fuoss-Onsager theory. ( 2 ) The contribution of the relaxation effect is treated in a completely different way. We use a Green function approach which gives analytical expressions for the integral which defines the fiist-order relaxation effect from the MSA radial distribution functions. The next order is calculated using Debye-Hiickellike correlation functions as an approximation for the MSA correlation functions. In contrast, Ebeling and co-workers used this last approximation to calculate the relaxation effect even to the first order. In our previous work our approach was restricted to charged hard-sphere models of nonassociating ions. The contribution of the relaxation effect is obtained by a perturbation resolution of the continuity equation in the Fuoss-Onsager theory. When the HNC approximation is combined with transport theory, only the first terms of the solution of the continuity equation can be easily computed. Then the possible accuracy of the results for associating electrolytes in the HNC approximation is truncated by the perturbation resolution for transport theory. Other
approaches such as the echo or feedback effect7*16in transport theory are useful to obtain a Bjermm-like association term in the relaxation. These expressions are only valid at low concentration(up to 0.01 M) for purely electrostatic association. For higher concentrations, chemical models can be used to compute conductance and equilibrium properties conjointly. The purpose of the present paper is to show how chemical association can be taken into account in a simple way. Incomplete dissociation and nonideality effects can be included in our theory by using the so-called chemical models. The nonideality is introduced for incomplete dissociation both in equilibrium and nonequilibrium processes such as conductance. Consider the simple salt:
A-
+ B+ - A B
(1)
If @A, @E, and @AB are the concentrations of the “free” ions A and B, and of the undissociated salt AB, then the equilibrium activities aA, aB, and aAB are given by the law of mass action,
where K is the thermodynamic association constant of the electrolyte. From mass conservation we know that
eA= eE= C(l - a), Q A B = Ca
(3)
+
where C = @A @AB is the initial salt concentration and a is the dissociation fraction. The activities and concentrations are related by the activity coefficients defined by
=YpeAE =Y i
2 @AeB
(4)
Here y* is the mean activity coefficient of the electrolyte in the molar scale and y p is the mean activity coefficient of the pair. Putting it all together, we get
* To whom correspondence should be addressed. t Universiti. Pierre & Marie Curie.
* University of Puerto Rico. 5 @
Universitk Bordeaux I. Abstract published in Advance ACS Absrrucrs, December 1, 1994.
Only the “free ions” A- and B+ are conducting; the pairs do not participate in the electrical transport phenomena (at least for frequencies lower than the rotational frequencies of the
0022-365419512099-0822$09.00/0 0 1995 American Chemical Society
Conductance in Associated Electrolytes
J. Phys. Chem., Vol. 99, No. 2, 1995 823
solvent dipoles). The other nonideality effects are deduced from the values at infinite dilution by considering only the contribution dA' of the free ions:
A = A,(l - a) = (A,,
+ dA')(l - a)
(6)
From eq 6 the conductance of associated, nonideal electrolytes can be calculated, if one has a way of computing nonideality effects both in equilibrium and in transport processes and a, using the law of mass action, eq 5 , for example. This simple theory was pioneered and extensively used by Fuoss17 and later on by Ebeling and co-workers. From the theoretical point of view, the introduction of an association constant is a drawback. The value of the association constant will cover, to a certain extent, the approximations made both in the potential model and in the theory. However, from the practical point of view, it is of great interest to have simple but physically well-grounded equations which correctly describe both the thermodynamic and the transport data with a small number of adjustable parameters. In section 2 we summarize the basic results of our analytical theory forfree ions. In section 3 we evaluate the proportion of undissociated pairs using the traditional approach originally introduced by Bjer".'* In section 4 we compare theory and experiment for an aqueous and a nonaqueous electrolyte solution.
ei is the electric charge of the ith ion, E is the electric field, Tis the absolute temperature, k~ is the Boltmann constant, and D? the diffusion coefficient of the ith ion at infinite dilution, related to the equivalent conductance by the relation
where 9%the Faraday and R the gas constant. The final expressions for the quantities defined in eqs 10 and 11 arelo (A)
where
2. The MSA Approach for Electrolytic Conductance
In eq 6 the equivalent conductance of the free ions is written as and where dA' is computed for the actual value of the free ion concentration. The nonideality factors are separated into electrophoretic (hydrodynamic interactions) and relaxation effects. The total conductivity Ap of the free ions i is given by
In these expressions a; is the diameter of the ith ion, gi is the particle density (molecular concentration) of the ith ion, and 0, = (Ui Uj)/2.
+
E
= 432€0E,
(18)
where
n Fi -- n i O(
1+, :e)l[
y)
1+-
(9)
is the individual contribution of the ions of type i. In this expression Gviel/v? is the free ion electrophoretic correction and d X / X is the free ion relaxation effect. We have
where EO and E, are the permittivities of the vacuum and of the pure solvent, respectively. In the MSA the integrals can be performed analytic all^.^^-^* The electrostatic part can be extracted from the Laplace transformation of rh,,MSA(r)using the following approximations:
& r h j y S A ( r )exp(-Kqr) d r =
where 6Xlre1/Xis the first-order (major) contribution. 6XzE1/X and GXlhyd/Xare usually less important second-order corrections. The electrophoretic effect can also be divided into two contributions:
with
where
and
Turq et al.
824 J. Phys. Chem., Vol. 99, No. 2, 1995
4ne a2 --
2
(22)
ckBT
r is the screening parameter of the MSA.
(31)
(C) The hydrodynamic relaxation term of the conductivity is
-
In the limit C and ai 0, r approaches ~ / 2 .We recall that in the calculation of r and K~ we use the concentration of thefree unpaired ions. P, and A are expressions from MSA equilibrium theory.lg
Kq)U,) -
x exp(-lc,u,) X
+
Kq
I+
exp(-K
a,.)
x 2a,2 " (1
+ (b- Kq)U,) -
In this expression 17 is the viscosity of the solvent. (D) The first-order electrophoretic correction term is
x In our calculation of the relaxation part we neglect the P, terms, which are very small in most cases. (B) The MSA expressions for the second-order terms of the relaxation contribution are
(33)
(E) The second-order electrophoretic correction can be split into two terms:
dvi2e1= J
+I
(34)
The MSA expressions are
exp(-K u ) ij
4x2a,2
[1
exp(xu,)E, [(x
] + -[ KqC,
+ (h- Kq)Uij]
X
2A, X2
I=
- 2Kq2
+
- Kq 2Kq(X Kq)
X
+ r q ( i + ruj)
24nq(x2 - KU ):1(:
X
1
exp( -2Kqu,) + ~ K ~ ) U-, ]2Kq(X + Kq)U,2( 1 + xu,) + and
7exp((x - Kq)u,)El(xo,)( 1 + K X2
+
J=
~ u ~ )
12nq(i
4%
(
+ ru,)(i + ruj) Kq
K t
3
COSh(KqUij) -
sinh K ~ U , KqU,
exP(-Kqqj)
+ 2rKq + 2 r 2 - ( 2 r 2 / a 2 ) ~ @ l p :exp(-Kquk)
(36)
k
where
+
X
Cii = cosh K ~ U , - sinh K ~ U , Kq
3. Evaluation of the Proportion of Pairs in the Chemical Model (28)
3.1. Law of Mass Action. Equation 5 yields explicitly the value of the pair proportion a:
-exp( -Kqr) d r = J:
Knowing the initial values of the concentrations of the different
J. Phys. Chem., Vol. 99, No. 2, I995 825
Conductance in Associated Electrolytes species, we compute the activity coefficients yp of the pairs and that of the electrolyte y*. This is done using the free ions, from which the equilibrium concentrations can be found, and then a new value of the activity coefficient is obtained, etc. This generates then an iterative procedure allowing a rapid estimation of a. Without a particular model for the pairs, we assume that yp = 1. In our case we use the MSA (neglecting the small P, terms) to compute the activity coefficients of the free ions. We get In yi = In ye1
+ In y?
(38) (39)
and r is computed with the eq 23 for the actual value of the free ions’ concentration. The hard-sphere contribution is obtained from the PY theory for hard-sphere mixtures, using the approximation 22.
2. We can introduce short range effects by considering what happens in an interval u, 5 r 5 Ro where
+
R, = uv nu,
with n = 1, 2 the number of solvent molecules and us the diameter of the solvent. In this range, supplementary interactions occur, which corresponds to the solvation equilibrium of the ion-pair. Short range forces can be introduced into the theory by superimpositionof a step function to the electrostatic mean force potential.23
w,*
= const, ifu, Ir IR,
Wt=O,ifr?R, The association constant is then
r;-
K = 4nNAlO3S a+R+- r exp -- 2f)dr
with
3.2. Short Range Solvation Effects. The above treatment does not assume any a priori value for the association constant K . It can be very specific for the particular ion-pair under consideration (for example in the case of weak acids), or it can be evaluated by assuming a particular model. The most popular model for association constants is the Bjemm model, where’
K = 4nNA103JR r2exp(--) 29,- dr a+-
(42)
2
e 4+- - IZ+Z-12Ek,T
(43)
The cutoff distance R s is normally taken as the inflexion point of the coordination number. R
N+-(r) = Ju+-4nrzg+-(r) d r
(44)
In (44) g+-(r) is the asymptotic low-density approximation:
The corresponding inflexion point for N+-(R) is at R s = q+-. Expression 42 can be generalized in two ways: 1. We can take the general expression
K = 4 n N ~ l o ~ JU+-~ ” ‘ ? g + - ( rd) r
(46)
The problem in this case is that K depends on concentration and has only the simplest limit 42 for infinitely dilute solutions.
(47)
(48)
Notice that in (48) R+- is no longer the inflexion point of the coordination number as in Bjemm’s expression, but a shorter distance corresponding to the radius of the solvated ions. The corresponding association constant is then much smaller than Bjemm’s association constant. We remark also that R+can be taken as the average distance parameter, alternatively to a+- in the expression of the MSA part of the activity coefficients (electric part) and in all the equations for the electrolytic conductance. Then the MSA correlation functions used to calculate these quantities describe only the distribution of the free ions. Other estimates can be made for the association constant: for example, the contact association constant used by F U O S S ~ or the mass action constant proposed by Ebeling.14*15 The choice of a model of association constant depends on the experiment in order to optimize the fit for a particular situation.
4. Results and Discussion Since the aim of our approach is to give a simple but physical way of describing transport data, we compare our theory to experimental data. However, it is clear that only those systems that are in reasonable agreement with our basic assumptions can be chosen as test systems. (1) Our approach considers systems which deviate from ideality but which remain “ordinary” electrolyte solutions. For example, solutions of lithium salts in i - o ~ t a n o are i ~ ~extremely associated and are quite similar to solutions such as ntetrabutylammoniumbromide in CC4.25 In this kind of solution, the salt is nearly completely undissociated and the systems resemble a molten salt.26 Therefore, these “solutions” are not convenient candidates for testing our approach. Nevertheless, we applied our theory to LiCl in l-octanol in order to compare with Ebeling’s results. With the same input parameters we obtained an association constant of K = 62 OOO Umol which is similar to Ebeling’s result: K = 63 500 Umol. However, from what we discussed previously, it is clear that the physical meaning of these values is doubtful. All we can say is that the number of free ions is very small and hence the number of ion-ion dipoles is probably very high, a fact which is not taken into account in the two approaches discussed. (2) On the other hand, systems which are completely dissociated at moderate concentrations are not good candidates
826 J. Phys. Chem., Vol. 99,No. 2, 1995
Turq et al.
-dMIsOI
@.@
@ . @ @ . 2 a 4 @ 6 u u
(3
m e ) Figure 1. Equivalent conductance (in 9-L cm2 mol-[) of aqueous MgS04 solutions as a function of the square-root molarity at 25 "C: (W) experimental values taken from ref 27; (-) theoretical values predicted by the MSA.
either. The association constant will be very low and will be of significant importance only at very high concentrations, for which our theory is not adapted. Ebeling and co-workers13 described experimental data for these systems with the help of association constants. In our previous worklo restricted to nonassociating ions, we obtained a good description of conductance measurements up to 1 M for some of these systems without association constants. Magnesium sulfate in water and lithium bromide in acetonitrile are electrolyte solutions which are appropriate to our theory. We consider these two systems in detail. The association constants and the average distance parameter R+- which are used in our conductivity calculations are not the result of a pure theory, but they are inferred from a fitting process either of osmotic coefficients (for lithium bromide) over the same concentration range or directly of conductivity and then applied to activity coefficients which are less precise data. The iteration process for the calculation of the concentration dependence of the conductivity data is given in section 3.1. The input parameters are the diameters of the ions, ui, their diffusion coefficients at infinite dilution, D?, the relative dielectric constant cs, and the viscosity q of the pure solvent. All these parameters are taken from the literature. The Dp values of associating electrolytes are, in general, the result of an extrapolation of transport data and hence depend somewhat on the theory used to describe the data. However, this dependence is normally relatively small. Therefore, we used the literature values as input data in order to reduce the number of adjustable parameters. MgS04 in Water. Figure 1 shows the experimentally determined electrolytic conductancez7at 25 "C and its description with MSA and the chemical model concept. The fixed parameters are u+ = 1.3 A, u- = 5.2 A, D+O = 0.706 x cmz/s, D-o = 1.065 x cmz/s, cs = 78.3, and 17 = 0.89 cP.28 K = 170 Umol and uu = R = 7 8, in the MSA part of the activity coefficients and in the equations of the electrolytic conductance. Using the same parameters leads to a satisfactory description of the concentration dependence of mean activity coefficient up to 1 M (Figure 2). However, a description within the error bars of the experimental values (%0.01%) cannot be expected. The thermodynamic data are much less precise than the conductivity data. Furthermore, the considered concentration range is very wide. It should be noted that the theoretical description fails already at very small concentrations when ion association is neglected.
M m e )
@A
u
lb
Figure 2. Activity coefficient of aqueous MgS04 solutions as a
function of the square-root molarity: (W) experimental values taken from ref 27; (-) theoretical values predicted using the MSA expressions of this text. 200
cooductivlty
1
i
50
-
0 0
0.05
0.1
0.15
0.2
0.25
0.3
sqrt(c) Figure 3. Equivalent conductance (in 8 - I cm2mol-') of LiBr solutions in acetonitrile as a function of the square-root molarity at 25 "C: (0) experimental values taken from ref 29; (-) theoretical values predicted by the MSA.
LiBr in Acetonitrile. To further test our approach, we study a nonaqueous electrolyte solution: LiBr in acetonitrile is a good case because of its significant but not extremely strong associationz9~30(Figure 3). The thernodynamic data of this system were published elsewhere.30 In the present paper we use the same parameters as in ref 30: u+ = 1.56 A, u- = 3.92 A, D+O = 1.843 x cmz/s, D-O = 2.681 x cm2/s, cs = 35.95, 17 = 0.344 c P , K ~ ~= 183 L/mol, and R = 8.54 A. With these data, a satisfactory description of the electrolytic conductivity is obtained. Unfortunately the conductivity is only studied over a limited concentration range. However, the description of the experimental data is a good test since there is a strong concentration dependence of the electrolytic conductivity in the low-concentration range. The good agreement between experiment and theory in these two systems gives us confidence that our combined MSAchemical model approach takes into account the main effects which determine the variation of electrolytic conductance, even of moderately concentrated and associated electrolytes. Acknowledgment. L.B. acknowledges with gratitude the kind hospitality of the Universitk de Paris, and partial support through Grants NSF-CHE-89-01597 and EPSCOR RII-8610677. References and Notes (1) Kohlrausch, F. Wied. Ann. 1879, 6, 1; 1885, 26, 161. (2) Arrhenius, S. 2.Phys. Chem. (Leipzig) 1887, 1, 631.
Conductance in Associated Electrolytes (3) Kohlrausch, F. Ann. Phys. 1897, 62, 209. (4) Debye, P.; Hiickel, E. Phys. Z. 1924, 24, 49, 185, 305. (5) Onsager, L. Phys. Z. 1926, 27, 388; 1927, 28, 277. (6) Onsager, L.; Fuoss, R. M. J . Phys. Chem. 1932, 36, 2689. (7) Justice, J. C. In Comprehensive Treatise of Electrochemistry; Conway, B. E., Bockris, J. O’M, Yeager, E., EMS.; Plenum Press: London, 1983; Vol. 5, Chapter 3. (8) Fuoss, R. M.; Accascina, F. Electrolytic Conductance; Wiley Interscience: New York, 1959. (9) Bemard, 0.;Kunz, W.; Turq, P.; Blum, L. J . Phys. Chem. 1992, 96, 398. (10) Bemard, 0.;Kunz, W.; Turq, P.; Blum, L. J. Phys. Chem. 1992, 96, 3833. (1 1) Chhih, A.; Turq, P.; Bemard, 0.;Barthel, J.; Blum, L. Ber. BunsenGes. Phys. Chem., in press. (12) Ebeling, W.; Rose, J. J . Solution Chem. 1981, 10, 599. (13) Ebeling, W.; Grigo, M. J. Solution Chem. 1982, 11, 151. (14) Ebeling, W.; Grigo, M. Ann. Phys. (Leipzig) 1980, 37, 21. (15) Ebeling, W.; Grigo, M. Z. Phys. Chem. (Leipzig) 1984,265, 1072. (16) Ebeling, W.; Feistel, R.; Kelbg, G.; Sandig, R. J . Non-Equilib. Thermodyn. 1978, 3, 11.
J. Phys. Chem., Vol. 99, No. 2, 1995 827 Fuoss, R. M.; Kraus, C. A. J. Am. Chem. SOC.1957, 79, 3304. Bjemm, N. K.Dan. Vidensk, Felsk. Math. Fis. Medd. 1926, 7,9. Blum, L.; H@ye,J. S. J. Phys. Chem. 1977, 81, 1311. Lebowitz, J. L. Phys. Rev. 1964, 133, A895. Sanchez-Castro, C.; Blum, L. J . Phys. Chem. 1989, 93, 7478. Salacuse, J. J.; Stell, G. J. Chem. Phys. 1982, 77, 3714. Barthel, J. Ber. Bunsen-Ges. Phys. Chem. 1979, 83, 252. Fuoss, R. M. J . Am. Chem. SOC. 1958, 80,5059. Denning, K. F.; Plambeck, J. A. Can. J. Chem. 1972, 50, 1600. (26) Schreiber, D. R.; de Lima, M. C. P.; Piker, K. S.J. Phys. Chem. 1987, 91, 4087. (27) Lobo, V. M. M. Electrolyte Solutions, Data on Thermodynamic and Transport Properties; Coimbra Editora: Lisbon, Portugal, 1984; Vol. I, II. (28) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed. revised; Buttenvorth: London, 1959. (29) Moshtev, R. V.; Zlatilova, P. Electrochim. Acta 1982, 27, 1107. (30) Kunz, W.; Barthel, J.; Klein, L.; Cartailler, T.; Turq, P.; Reindl, B. J . Solution Chem. 1991, 20, 875. (31) Krumgalz, B. S.J . Chem. SOC.,Faraday Trans. 1 1983, 79, 571.
(17) (18) (19) (20) (21) (22) (23) (24) (25)
