J. Phys. Chem. B 2004, 108, 5763-5770
5763
Ionic Solutions in the Binding Mean Spherical Approximation. Thermodynamics of Associating Electrolytes up to Very High Concentrations Teresa Vilarin˜ o,† Olivier Bernard,‡ and Jean-Pierre Simonin*,‡ Departamento de Quimica Fisica e Enxen˜ eria Quimica I, Facultade de Ciencias, UniVersidade da Corun˜ a, 15071 A Corun˜ a, Spain, and Laboratoire LI2C (UMR 7612), UniVersite´ P.M. Curie, Boıˆte no. 51, 4 Place Jussieu, 75252 Paris Cedex 05, France ReceiVed: December 23, 2003; In Final Form: March 4, 2004
A previous model, developed to describe the thermodynamic properties of associating electrolytes within the binding mean spherical approximation (BIMSA), is modified on the basis of a better representation of the pair. The ions are represented as charged hard spheres interacting through a sticky point potential, leading to the creation of pairs. In this version of the model, the species constituting the pair are allowed to have charges and sizes different from those of the free ions. New expressions for the osmotic and activity coefficients are proposed that account for these features. The equations were applied to fit the parameters in the model to osmotic data for a variety of aqueous solutions of inorganic acids, and for a base, NaOH, at 25 °C up to the highest concentrations for which data are available. In all cases, accurate representations could be obtained up to very high concentrations, such as 28 mol kg-1 for nitric acid, 45 mol kg-1 for hydrochloric acid, and 29 mol kg-1 for NaOH. The case of sulfuric acid could be treated consistently in the concentration range 0-27.7 mol kg-1 by considering the equilibrium between sulfate and bisulfate ions. In all cases, the new model represents an improvement over the previous BIMSA model.
Introduction (MSA)1,2
The mean spherical approximation is a theory for ionic solutions in which the solvent is regarded as a continuum (primitive model), as well as for models with molecular solvent.3 The main feature of the MSA is that, when the ions are modeled as charged hard spheres interacting through Coulomb forces, it gives explicit analytic relations for the thermodynamic properties such as the osmotic and activity coefficients. In contrast with many of the available analytic models for solutions, this model provides a “molecular” representation for the solute, with parameters having some direct physical meaning. Descriptions of the thermodynamic properties of real aqueous electrolytes using the MSA have been proposed for strong electrolytes.4-9 The case of associating electrolytes has also been considered within the MSA framework with the addition of a mass action law10-13 and within the binding mean spherical approximation (BIMSA)14 which combines15 the features of the MSA with the Wertheim model for association.16 The BIMSA provides an attractive framework for a statistical mechanical description of association, including ion pairing and chemical association. Besides ion-ion interactions, this model accounts for ion-pair and pair-pair electrostatic and excluded volume forces. However, it must be noticed that in our previous studies14 within the BIMSA a pair was depicted as two contacting spheres of sizes and charges identical to those of the free ions. Although this assumption might be acceptable in the case of electrostatic ion pairing, it is expected to be inadequate in the case of chemical association, e.g., for partially dissociated acids in water. * To whom correspondence should be addressed. E-mail: sim@ccr. jussieu.fr. † Universidade da Corun ˜ a. ‡ Universite ´ P. M. Curie.
This was noticed in ref 14a when attempting to consistently represent data for solutions of sulfuric acid over a wide concentration range. In a continued effort to obtain more realistic descriptions of ionic solutions at the McMillan-Mayer (MM) level,17a we give here a first approach to this issue by allowing the sizes and the charges of the species in the pair to be different (generally smaller) than those for the free (solvated) ions. This change is aimed at a better representation of the pair in which a covalent bond created between the reacting ions modifies (certainly lowers) the hydration of the two species, especially the cation, and their respective charges. New expressions accounting for these assumptions were derived by bringing ad hoc modifications to the original equations. For this purpose, a simple electrolyte solution is viewed as a three-species (3S) system composed of the two free ions and the pair, the latter being a separate species possessing its own chemical potential. This viewpoint is equivalent to the chemical model in which the pair is regarded as a separate entity and the amount of pairs is evaluated by using a law of mass action. Thus, the 3S picture is somewhat in contrast to that of the 2S original Wertheim framework in which the pair need not be considered as an independent species because it is merely defined as two ions being in contact. In the first section of this paper, we give a short summary of the original BIMSA theory and derive analytical equations for the new model. The latter are particularized to the case of a two-component electrolyte in which the cation and the anion can bind to form one single pair. The second section is dedicated to a discussion of experimental results. The new equations are applied to fit data for a variety of aqueous solutions of acids at 25 °C.
10.1021/jp037993i CCC: $27.50 © 2004 American Chemical Society Published on Web 04/09/2004
5764 J. Phys. Chem. B, Vol. 108, No. 18, 2004
Vilarin˜o et al. For a given K*, the pair density can be calculated using eqs 1 and 3 to give
FP )
2K*FCFA 1 + K*Ft + [1 + 2K*Ft + K*2(FC - FA)2]1/2
(5)
Figure 1. Sticky pointassociation model. In the present model, the properties of the species in the pair are different from those of the free ions. Property Pi is either the electric charge or the size of species i.
with
Theory
Relations for thermodynamic properties such as the osmotic and activity coefficients may be derived from an expression of the Helmholtz energy, F. Generally, the latter is computed by integrating the thermodynamic relation2
We consider an ionic solution containing one type of cation and one type of anion, and we denote by C and A, respectively, the atoms or chemical groups (such as H, Na, NO3, or ClO4) corresponding to these ions. The cation and the anion are supposed to form one single pair denoted by P. This association is assumed to result from a sticky point interaction15 (see Figure 1), which may be treated by using the Wertheim formalism.16 For convenience, the free ions will be denoted by 1 for the cation and 2 for the anion, whereas the same species in the pair will be denoted by 3 and 4, respectively. Each species i is modeled as a charged hard sphere of hard core diameter σi and number density Fi (number of species i per volume unit). The electric charge borne by species i is ezi, e being the elementary charge. Thus, z3 and z4 are supposed to be the charges on C and A in the pair. The model is developed at the primitive level of solutions where the solvent is regarded as a continuum. The pure solvent has a relative permittivity w; the solution is assumed to have a relative permittivity, , that may be different from that of pure solvent. Let us underline that is not assigned the experimental value of the relative permittivity of solution, sol, because the latter is a macroscopic quantity whereas the former is used to describe electrostatic interactions for microscopic ion-ion separations.9 Basic physical considerations point to a value for that is greater than that of sol, and this is exactly what was found in ref 9. The temperature of the system is T, Boltzmann constant is denoted by kB and we set β ) 1/kBT. Basic BIMSA Relations in the 2S Case. In the BIMSA (and in Wertheim theory16), the density of species i may be split into bonded and nonbonded parts. In the present case of a single ion pair formation, one has
F C ) F1 + FP
(1a)
FA ) F2 + FP
(1b)
expressing the conservation of C and A and
The BIMSA model includes a mass action law (MAL) introducing a thermodynamic association constant K as
∂β∆F ) ∆U ∂β in which ∆U is the excess internal energy, which may be calculated from the MSA pair correlation function. An expression found in earlier work for the Helmholtz energy14a,15 is HS + ∆FMSA F ) F id 2S + ∆F
0 0 βF id 2S ) FC(βµC + ln FC - 1) + FA(βµA + ln FA - 1) (8)
with µ0i being the standard chemical potential of species i. The expression of β∆FMSA in the BIMSA framework has the form14a
β∆F MSA ) β∆Uel +
Γ3 λ + β∆FMAL Y Y F (9) 3π σCA C A P
in which ∆Uel is the excess electrostatic internal energy (per volume unit), Γ is the classic MSA screening parameter (the analogue of κ appearing in the Debye-Hu¨ckel theory)
λ≡
r≡
yP y1y2
βe2 4π0
σCA ) (σC + σA)/2
(10)
(3)
(4)
yi denoting the activity coefficient of species i (at the MM level).
(11)
and Yi is defined by
(2)
with
FP F1F2
(7)
in which the terms on the right-hand side are the ideal, hard sphere (HS), and MSA contributions, respectively. Their expressions being rather large, they will not be detailed here; the complete expressions may be found elsewhere.8,9,14 Instead, we give below some main results that are next used for introducing the 3S representation. The first term in eq 7, F id 2S, representing the 2S ideal contribution, may be written as
Yi ) K* ≡
(6)
is twice the Bjerrum distance for a solution of relative permittivity (0 being the permittivity of a vacuum)
F3 ) F4 )FP
K ) K*r
Ft ) FC + FA
zi - ησi2 1 + Γσi
(12)
Moreover, the contribution originating from the mass action law (MAL) is15
β∆FMAL ) FC ln RC + FA ln RA + FP
(13)
in the dimerizing case, RX being the proportion of free ions X (unbound ion fraction).
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J. Phys. Chem. B, Vol. 108, No. 18, 2004 5765
In the BIMSA, the ratio of activity coefficients, eq 4, is expressed as14a
r2S ) g∞CA/gCA
F1 ≡ FCRC
(15a)
F2 ≡ FARA
(15b)
+ β∆µi + ln Fi
σCσA σ0 ) 2 σC + σA Xn )
(25)
π
Fkσnk ∑ 6 k ) C,A
(26)
and gelCA being the electrostatic contribution to the RDF, given in the MSA by2b
gelCA ) -
λ Y Y σCA C A
(27)
Therefore, since the limiting value of gHS CA is 1 at infinite dilution, eqs 14, 22, 23, and 27 lead to
β∆Fassoc ) β∆Fassoc0 + FP
(16)
[
λ∞ λ YCYA z z σCA σCA,∞ C A
]
(28)
with
where
β∆µi ) ln yi
(24)
with
and the classic definition for the chemical potential of species i is
βµi )
3X2 X22 1 + σ + σ2 1 - X3 2(1 - X )2 0 2(1 - X )3 0 3 3
(14)
with gCA being the cation-anion contact radial distribution function (RDF), the superscript ∞ denoting infinite dilution of solute. Notice that eq 14 may be obtained by combining the Bro¨nsted relation for chemical kinetics (with P the activated state) with a more modern representation in terms of RDF’s.19 Passage to the 3S Representation in the BIMSA Model. We now wish to introduce the pair P explicitly as a separate species. Passage to the 3S description may be done by first MAL, each term being given rewriting the quantity βF id 2S + β∆F by eqs 8 and 13, as follows. The proportion of free 1 and 2 ions is given by
βµ0i
gHS CA )
β∆Fassoc0 ) - FP ln gHS CA
(17)
(29)
the contribution for uncharged species and Moreover, the condition of chemical equilibrium between the free ions and the pair may be written as
µ1 + µ2 ) µP
(18)
with the thermodynamic equilibrium constant K satisfying the relation
ln K ) β(µ01 + µ02 - µ0P)
(19)
in which µ01 ) µ0C and µ02 ) µ0A. Then, using eq 1 and these relations, one obtains after rearrangement of terms MAL assoc ) βFid βFid 2S + β∆F 3S + β∆F
λ∞ ≡
βe2 4π0W
(30)
the value of λ for pure water (of dielectric constant W) or infinite dilution, and σCA,∞ the value of σCA at infinite dilution. So, by combining eqs 7, 9, 20, and 28, we obtain the result HS + β∆Fassoc0 + β∆F el βF ) βF id 3S + β∆F
λ∞ σCA,∞
zCzAFP (31)
with
(20)
β∆Fel ≡ β∆Uel +
Γ3 3π
(32)
where 0 β∆Fid 3S ) F1(βµ1° + lnF1) + F2(βµ2° + lnF2) + FP(βµP + lnFP) - (F1 + F2 + FP) (21)
is the ideal part of the Helmholtz energy for the 3S system (1 + 2 + P) and
β∆Fassoc ) FP ln r
(22)
is the contribution from association, with r being given by eqs 4 and 14. In eq 14, the exponential MSA approximation20 may be used to give an expression of the contact RDF, which yields el gCA ) gHS CA exp(gCA)
gHS CA
(23)
being the cation-anion contact HS-RDF, which with may be expressed using the Boublik-Mansoori-CarnahanStarling (BMCSL) relation21
the electrostatic contribution, the second term representing the entropic term in the MSA. Modification of Parameters for the Pair. We now wish to include the possibility that species 3 and 4 (the cation and the anion in the pair) have specific charges and sizes, different from those of the free ions. To this end, we propose here to modify the above expression of F, eq 31, as follows. The equation for β∆F HS 3S is expressed using the BMCSL relation22 for hard spheres
( )
3X1X2 X32 X32 π β∆FHS X ln(1 X ) + + 3S ) 0 3 6 1 - X3 X (1 - X )2 X23 3 3 (33) in which
Xn )
π
4
∑Fkσnk 6 k)1
(34)
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Vilarin˜o et al.
Besides, we replace subscripts C and A by 3 and 4, respectively, in the terms in which FP is factorized. This leads to the following expressions. For the HS association contribution, we get
β∆Fassoc0 ) - FP ln gHS 3S 34 with
gHS 34
known by using eq 5. However, K* can no longer be obtained using eq 14 in eq 2 in the 3S case. The new relation may be derived from the equilibrium condition between 1 + 2 and the pair as expressed by eq 18. This is achieved by first noting that, for each species i ) 1, 2, P, one has
(35) µi )
given by eq 24 in which now
σ3σ4 σ0 ) 2 σ3 + σ4
(36)
The electrostatic internal energy is
β∆Uel3S ) - λ
{
Γzk + ησk
4
∑Fkzk 1 + Γσ k)1
-
k
(
FP
z3Y4
+
2σ34 1 + Γσ3 z4Y3
)}
(37)
with differentiation at constant Γ because of the general relation20,24
σ34 ) (σ3 + σ4)/2
η)
π 2δ
{
FP
4
FkYk2 + ∑ σ k)1
{∑
with
δ ) 1 - X3 +
1 + Γσ3
zkσk
4
k)1
(
34
σ3
FP
+
σ4
) }
1 + Γσ4
Y3Y4
z4σ32 + z3σ42
Fk + 1 + Γσk 2σ34 (1 + Γσ3)(1 + Γσ4)
}
{∑ 4
(40)
}
(42)
λ∞ zzF σ34,∞ 3 4 P
(43)
with eqs 21 and 42 expressing the first and the third contributions, σ34,∞ being the value of σ34 at infinite dilution (in the case that σ3 and/or σ4 are allowed to vary with concentration), and with HS assoc0 β∆FEV 3S ≡ β∆F3S + β∆F3S
(47)
β∆µP,0 ) β∆µ3,0 + β∆µ4,0 - ln gHS 34 +
and we get the final equation EV el βF3S ) βFid 3S + β∆F3S + β∆F3S -
∂ ln gHS 34 - FP ∂Fi
(39)
We note that these formulas coincide with those for ionic mixtures.14b,15 We therefore have
Γ3 3π
+
β∆µeli
for i ) 1, 2 with β∆µHS and β∆µeli being given in ref 8 (the i term Mi of eq 30 and the first term of eq 19, respectively). For the pair, we find after some algebra
(41)
β∆Fel3S ≡ β∆Uel3S +
β∆µHS i
(38)
σk3 σ32σ42 FP Fk + 2 k)1 1 + Γσk σ34 (1 + Γσ3)(1 + Γσ4)
π
(46)
meaning that the effective value of the MSA parameter Γ corresponds to optimum screening for a given Helmholtz energy F. This relation applied to F3S (eq 43) yields eq 39. The differentiation of the excess terms in eq 31 leads to the expression of ∆µi,0, for constant parameters σi and
β∆µi,0 )
with
Γ2 ) πλ
(45) Γ
∂F )0 ∂Γ
1 + Γσ4
and
|
∂F3S ∂Fi
(44)
being the excluded volume (EV) contribution for the mixture of free ions and ion pairs, which agrees with the general result obtained using the thermodynamic perturbation theory.23 Chemical Equilibrium. The quantity Fp may be calculated for a given electrolyte concentration once the value of K* is
λY3Y4 λ∞z3z4 (48) σ34 σ34,∞
in which the first two terms, ∆µ3,0 and ∆µ4,0, are given by eq 47. In this equation, the term containing the product Y3Y4 comes from the derivative of ∆U el3S with respect to FP and the last term is simply the derivative of the last term of eq 43. Then, by combining eqs 2-4, 16-19, 47, and 48, one readily gets the new expression for K*
g34 K* ) K ∞ exp(β∆µ1,0 + β∆µ2,0 - β∆µ3,0 - β∆µ4,0) (49) g34 with g34 the 3-4 RDF given by the EXP-approximation, eqs 23, 24, 27, and 36. This equation replaces eq 2 (with r defined by eq 14) in the 3S case. It satisfies the relation K* f K as Ft f 0. We note that eqs 47-49 are not only valid for constant ion sizes and solution permittivity, but also in the case that these parameters are a function of the sole total density, Ft, or equivalently the salt concentration as assumed in previous work.8,9,14 Indeed, for any such parameter x, one has the following relation:
∂x ∂x ∂x ∂x + )0 ∂F1 ∂F2 ∂F3 ∂F4
(50)
from which eq 49 may be deduced by taking x ) σi or . Finally, eqs 5, 39, and 40 constitute three coupled equations in the unknowns Γ, η, and Fp. These three equations can be easily solved by numerical iterations with the starting values Γ ) κ/2 (κ being the Debye screening parameter) and Fp)0. Calculation of Excess Thermodynamic Properties. A quantity of interest is the MM osmotic coefficient defined as
φ)
βPosm Ft
(51)
in which Posm is the osmotic pressure, which may be calculated
Thermodynamics of Associating Electrolytes
J. Phys. Chem. B, Vol. 108, No. 18, 2004 5767
using the relation
Posm ) F1µ1 + F2µ2 + FPµP - F3S
(52)
In the present framework, the osmotic coefficient is obtained at the MM level of solutions17a at which the solvent chemical potential is constant, equal to its value for pure solvent. This condition is fulfilled by applying a suitable excess pressure on solution, the osmotic pressure, Posm, appearing in eq 51. Deviation of the MM osmotic coefficient (eq 51) from unity is caused by the departures from ideality for the gas of interacting solute particles, a notion originally introduced by van’t Hoff in his seminal paper.17b To account for hydration effects, the size of the free cation and the permittivity were allowed to vary with concentration as in previous studies.8,9,14 Because anions are generally weakly hydrated, their diameter may be kept constant, equal to their crystallographic value for simple anions or adjusted so as to yield an optimum fit in the case of complex ions. The diameter of the free cation and the inverse of the solution relative permittivity were taken as linear functions of the salt concentration (measured macroscopic solution permittivities, sol, are well described by this representation) (1) σ1 ) σ(0) 1 + σS CS
(53)
-1 ) -1 w (1 + RCS)
(54)
where CS is the molar concentration of the electrolyte and σ (0) 1 , (0) σ(1) , and R are model parameters. The first one, σ , is the S 1 diameter of 1 at infinite dilution. Contrary to the latter, the other two parameters reflect the influence of both ions and are characteristic of the nature of the salt S. Then, by virtue of eqs 43, 46, 51, and 52 and after some calculations, we get the osmotic coefficient in the form
φ ) φ0 + ∆φσ + ∆φ
∆φσ )
(56)
with
1
∑i
|
∂βF3S
Ft
∂σi
D(σi)
(61)
D(-1)
(62)
Γ
for which we define
∆φ )
|
1 ∂βF3S Ft ∂-1
Γ
∂ ∂ D ≡ Ft ) CS ∂Ft ∂CS Using eq 28 and after some cumbersome algebra, it turns out finally that, as in previous work8,14a
|
∂βF3S ∂-1
) β∆Uel3S
∂β∆FHS 3S ) Fi(F1 + 2σiF2 + 3σi2F3) ∂σi
FP Ft
(57)
the semi-ideal25 contribution to φ0 for an effective total number density F1 + F2 + FP ) Ft - FP HS assoc0 ∆φEV 0 ) ∆φ0 + ∆φ0
in which the expression of
∆φHS 0
)∆φassoc0 0
FP
was given in an earlier 4
∑ Fk
Ft k)1
(58)
|
∂β∆Uel3S ∂σ1
|
∂β∆Uel3S ∂σ3
{
Γ
) λFP Y3 Γ
Γ2(z1 + ησ12) - 2η 1 + Γσ1
) λF1Y1
∂ ln gHS 34 (59) ∂Fk
2λ η2 Γ3 π Ft 3πFt
(65)
Γ2(z3 + ησ32) - 2η 1 + Γσ3 Y4 Γ(z3 + ησ32) + 2ησ3 Y3Y4 σ34 (1 + Γσ )2 2σ 2 34
}
(66)
Taking σ3 ) σ1 and σ4 ) σ2 in these last two expressions and summing them, one recovers the relation obtained in previous work.14a The mean activity coefficient of electrolyte, ln y(, may be written in a way similar to eq 55 as
ln y( ) (ln y()0 + (∆ ln y()σ + (∆ ln y()
(67)
paper8
and8
∆φel0 ) -
(64)
with the expressions of F1, F2, and F3 given in eq 30 of this reference. For the electrostatic term, we find after some lengthy algebra
3
φid 0 )1-
(63)
Γ
which may be inserted into eq 62. The derivative of βF3s in eq 61 involves three contributions, coming from the HS, association and electrostatic interactions. The HS term was given in earlier work8 as
(55)
with φ0 the result for concentration-independent parameters, and ∆φ and ∆φσ the additional terms corresponding to the concentration dependence of ion size and relative permittivity, respectively. One has EV el φ0 ) φid 0 + ∆φ0 + ∆φ0
Moreover, as shown previously,8 one has the contributions originating from the variation of the parameters with concentration
in which (ln y()0 may be computed from the activity coefficients of the free ions 1 and 2 (eq 47) by using the classic relation between the latter and the mean salt activity coefficient,26 which yields
1 (ln y()0 ) [νC(β∆µ1,0 + ln R1) + νA(β∆µ2,0 + ln R2)] (68) ν (60) where νi denotes the total number of particles of type i released
5768 J. Phys. Chem. B, Vol. 108, No. 18, 2004
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TABLE 1: Results for Parameter Valuesa electrolyte
max. mb
HClO4
16
HNO3
28
HCl
45
H2SO4
27.7
NaOH
29
z3
σ3c
σ4c
d 102 σ(1) S
102 Re
Ke
Klite
AARD (%)f
0.427 1 0.418 1 0.258 1 0.388 1 0.758 1
3.870 5.040 3.723 5.040 3.726 5.040 2.781 5.040 3.797 3.870
4.20 4.53 3.10 3.40 3.20 3.62 4.20 4.40 3.40 3.55
-7.758 -9.550 -7.802 -7.259 -7.859 -7.759 -13.53 -12.39 -5.033 -5.591
14.11 0.04334 10.77 10.63 6.823 5.245 6.532 27.05 2.019 -3.838
0.00450 0.5458 0.02408 0.07455 0.002052 0.05829 86.19 0.1309 0.2805 1.145
0.026 0.026 0.036-0.1 0.036-0.1 6 × 10-7-0.089 6 × 10-7-0.089 79-98 79-98 0.17-0.29 0.17-0.29
0.26 0.86 0.38 1.1 0.73 1.3 0.81 13 0.58 3.5
The fitted parameters were σ3, σ(1) S , R, and K. For each electrolyte, the first line gives the results obtained from fit using the present model; the second line is the result of fit using the previous association model (in which z3 ) z1 and σ3 ) σ1). b In units of mol kg-1. c In Å. d In Å mol-1 L. e In mol-1 L. f Average absolute relative deviation. a
by one molecule of salt and ν ) νC + νA. The other two terms in eq 67 may be obtained using the relation8
(∆ ln y()X ) ∆φX
(69)
for X ) σ and . Results and Discussion First, it was checked numerically that the osmotic and activity coefficients given by eqs 55 and 67 accurately satisfy the Gibbs-Duhem (GD) relationship at the MM level for arbitrary values of the parameters. Next, for comparison with experiment, the computed osmotic coefficient values were converted from the McMillan-Mayer to the Lewis-Randall (experimental) reference system by using density data for the solutions.27 The densities were taken from a compilation;28 in the case of HCl at very high concentration (above 16 mol kg-1), some data were taken from ref 29. This conversion keeps the GD relation satisfied.27 Thermodynamic data for concentrations above 0.1 mol kg-1 have been considered for the following aqueous electrolyte solutions: Common inorganic acids HClO4, HNO3, HCl, and H2SO4 and one base, NaOH. Most of the experimental osmotic coefficient data at 25 °C were taken from the NIST databank,30 except for HCl between 19 and 45 mol kg-1.31 In the databank, the data for the 1:1 electrolytes come from the compilation by Hamer and Wu32 and those for sulfuric acid are from the work of Rard et al.33 The diameters of the free anions, σ2 (kept constant versus concentration), and the diameter of H+ ion at infinite dilution were taken from previous work:9,14a 3.62 Å for Cl-, 3.40 Å for NO3-, 4.53 Å for ClO4-, 4.40 Å for SO42-, 3.55 Å for OH-, and 5.04 Å for σH(0)+. The charges of the species in the pair, z3 and z4 (with z4 ) z1 + z2 - z3) , were taken from a database, developed on the basis of DFT molecular structure calculations.34 The size of the cation in a pair, σ3, was taken independent of salt concentration in order to reduce the number of adjustable parameters and because a concentration-dependent σ3 did not improve significantly the accuracy of fits. The sizes of the anions in a pair, σ4, were assigned reasonable values, smaller than those for the free ions. “Experimental” values for the association constant, Klit, were found in the compilation of Sillen.35 In the case of HNO3, the value36 of 0.045 L mol-1 was recalculated using more recent activity coefficients for this electrolyte,30,32 yielding a value of ca. 0.1 L mol-1. An alternative value for HCl was found in the literature.37 The values for the set of parameters are given in Table 1. Then, for a given electrolyte, the parameters that yield the osmotic coefficient (eqs 55-65) were adjusted to the experi-
Figure 2. Plot of osmotic coefficient vs molality for HClO4. Circles: experimental result; solid line: result of fit using this model; dashed line: result of fit using the previous BIMSA model. Inset: result at low concentration, between 0.1 and 1 mol kg-1.
Figure 3. Case of HNO3. Same legend as for Figure 2.
mental data by a least-squares procedure using a classic Marquardt algorithm. In this procedure, four parameters were adjusted, namely σ3, σ(1) S , R, and K. The results for the fits of the osmotic coefficient are shown in Figures 2-6, together with the result of fits using the previous BIMSA model. The results for the parameter values are collected in Table 1. It is seen that the osmotic coefficient values are very well represented to 16 mol kg-1 for HClO4 (vs 12 mol kg-1 in ref 14a), to 28 mol
Thermodynamics of Associating Electrolytes
Figure 4. Case of HCl. Same legend as for Figure 2.
Figure 5. Case of H2SO4. Same legend as for Figure 2. The (poor) result using the previous BIMSA model is shown with a dotted line.
Figure 6. Case of NaOH. Same legend as for Figure 2.
kg-1 for HNO3 (vs 10 mol kg-1), to 45 mol kg-1 for HCl (vs 16 mol kg-1 without association in ref 9), to 29 mol kg-1 for NaOH (vs 10 mol kg-1 in ref 14a) and consistently on the molality range 0-27.7 mol kg-1 for H2SO4. It is noticed that the fits in the case of HClO4 and HCl are also excellent at low concentration, between 0.1 and 1 mol kg-1. The discrepancy is
J. Phys. Chem. B, Vol. 108, No. 18, 2004 5769
Figure 7. Degree of association of HNO3 in solution. Symbols: experimental data, (b): ref 36; the dashed line is a fit of these data; (2): ref 38. Solid line: result from fit.
Figure 8. Degree of association of HClO4 in solution. Symbols (b): experimental data from ref 36. Solid line: result from fit.
smaller than 1% for HNO3 and NaOH in this range, and it is 1.4% on an average for H2SO4. For the acids HClO4, HNO3, and HCl, the adjusted values for σ3 and σ(1) S are of the same order of magnitude. The common value for σ3 is 3.8 ( 0.1 Å, which stands for the size of the proton in the pair. It may be interpreted as a size including a contribution from hydration because of the nonzero local charge on the H species. For sulfuric acid, the optimum value for σ3 is found to be significantly smaller, on the order of 2.8 Å. The calculated curves were found not to depend dramatically on the value of the charge z3. For example, changing its value by (20% resulted in an increase of the average deviation of fit (AARD) of ca. 0.5% for HClO4 and HNO3 solutions and 0.9% for H2SO4 solution. The values obtained for the association constant K are consistent with the “experimental” ones in the case of H2SO4 and NaOH and slightly lower than the minimum value observed in the case of HNO3 and HCl. It is much lower for HClO4. The degree of association (proportion of pairs), RP, found from the fits was compared with experimental determinations using NMR for HNO3 and HClO4 solutions.36,38 The result is shown in Figures 7 and 8. The calculated curve is bell-shaped in the case of HNO3 and lies below the experimental data. It is appreciably above the experimental data at high concentration in the case
5770 J. Phys. Chem. B, Vol. 108, No. 18, 2004 of HClO4. Therefore, the calculated RP shows opposite behaviors as compared to experiment in these two cases for which the adjusted association constant K is lower than the literature value. Further calculations showed that imposing a higher association constant and fitting the three remaining parameters did not result necessarily in an increase of the amount of pairs, the latter being a complicated function of all of the parameters involved in the model. Besides, it must be emphasized that “experimental” values of K are subject to large uncertainties. So, the experimental values for HClO4 and HNO3 were derived by extrapolation of plots for the activity coefficient of the pair between 0 and 4 mol L-1.36 The difficulty in the case of sulfuric acid has been stated by Pitzer et al.39 Moreover, except for H2SO4 and HClO4 solutions,36 the nature of the associated species is not quite clear, as shown by the following examples. Recent ab initio numerical simulations have suggested the existence of Cl-H‚‚‚Cl- species in concentrated HCl solutions.40 As far as we are aware, association in NaOH solutions has not been studied in detail. Therefore, the picture of two ions associating to form one single species might sometimes be a convenient simplification and such systems should be described by suitable models once the nature of the associated species is known. In subsequent work, it will be attempted to introduce the effect of hydration in the pair formation process, which certainly is the main missing ingredient in the present model at the MM level. Acknowledgment. One of the authors (T.V.) gratefully acknowledges financial support through a fellowship from the Spanish Ministerio de Educacio´n, Cultura y Deporte. References and Notes (1) Percus, J. K.; Yevick, G. Phys. ReV. 1964, 136, 290. Lebowitz, J. L.; Percus, J. K. Phys. ReV. 1966, 144, 251. Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1970, 52, 4307. (2) (a) Blum, L. Mol. Phys. 1975, 30, 1529. (b) Blum, L.; Høye, J. S. J. Phys. Chem. 1977, 81, 1311. (3) Blum, L. J. Stat. Phys. 1978, 18, 451. Blum, L.; Vericat, F.; Fawcett, W. R. J. Chem. Phys. 1992, 96, 3039. Blum, L. J. Chem. Phys. 2002, 117, 756. (4) Triolo, R.; Grigera, J. R.; Blum, L. J. Phys. Chem. 1976, 80, 1858. Triolo, R.; Blum, L.; Floriano, M. A. J. Chem. Phys. 1978, 82, 1368. Triolo, R.; Blum, L.; Floriano, M. A. J. Phys. Chem. 1977, 67, 5956. (5) Watanasiri, S.; Brule´, M. R.; Lee, L. L. J. Phys. Chem. 1982, 86, 292. Sun, T.; Le´nard, J. L.; Teja, A. S. J. Phys. Chem. 1994, 98, 6870. (6) Fawcett, W. R.; Tikanen, A. C. J. Phys. Chem. 1996, 100, 4251. (7) Molero, M.; Gonzalez-Arjona, D.; Calvente, J. J.; Lopez-Perez, G. J. Electroanal. Chem. 1999, 460, 100. Lopez-Perez, G.; Gonzalez-Arjona, D.; Molero, M. J. Electroanal. Chem. 2000, 480, 9. (8) Simonin, J. P., Blum, L.; Turq, P. J. Phys. Chem. 1996, 100, 7704.
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