5308
J. Phys. Chem. B 2007, 111, 5308-5315
Relaxation Effect on the Onsager Coefficients of Mixed Strong Electrolytes in the Mean Spherical Approximation Steven Van Damme* and Johan Deconinck Computational Electrochemistry Group (CEG), Department of Electrotechnics and Power Electronics (ETEC), Vrije UniVersiteit Brussel (VUB), Pleinlaan 2, 1050 Brussel, Belgium ReceiVed: February 28, 2007; In Final Form: March 9, 2007
The Fuoss-Onsager continuity equations are solved by using the equilibrium pair distribution functions of the mean spherical approximation in the case of equal diameters. An analytical expression is obtained for the relaxation effect on the Onsager coefficients of mixed strong electrolytes. This work also extends the existing expressions for the conductivity of binary and ternary electrolytes to any number of ions.
1. Introduction Irreversible thermodynamics postulates a linear coupling between fluxes and driving forces. For ionic transport in electrolyte solutions these linear laws are known to hold up to steep gradients of concentration and electric potential. Unfortunately little is known about the values and the concentration dependence of the proportionality factors lij, called Onsager coefficients. For some common binary electrolytes the necessary experimental data to extract the Onsager coefficients are available, but for mixed electrolytes the data are mostly incomplete. Semiempirical mixing rules can be used in this case to estimate the Onsager coefficients, but still only if the Onsager coefficients in all the constituting binary electrolytes are known. The easiest approach, followed by commercial electrochemical simulation programs, is to take the infinite dilution limit, i.e., lij ) 0 for the cross coefficients and lii ) ωini for the main coefficients. To compensate for the errors at finite concentrations, the mobilities ωi are adjusted for each electrolyte composition. The validity of this approach is questionable since the cross coefficients rise rapidly with concentration. For 1-1 electrolytes they often reach 10% of a main coefficient at 0.1 N and 20% at 1 N, and for 2-1 electrolytes this is respectively 30% and 60%.1 Clearly there is a need for theoretical expressions of the Onsager coefficients. Already in 1957 Onsager et al.2 provided a mathematical description based on the Debye-Hu¨ckel equilibrium distribution functions. Because the Debye-Hu¨ckel theory only accounts for the long-range electrostatic interaction, the expressions are only valid at low concentrations. The shortrange interaction becomes increasingly important at higher concentrations and tends to moderate or even reverse the effect of the electrostatic interaction.3 In the mean spherical approximation (MSA) the short-range interaction is modeled by a hardsphere repulsion, assigning an effective diameter to each ion. The electrophoretic effect on the Onsager coefficients has recently been expressed in the mean spherical approximation by Dufreˆche et al.4 So far the relaxation effect has only been expressed for the conductance of binary5 and ternary electrolytes.6,7 The aim of this paper is to derive an expression for the relaxation effect on the Onsager coefficients in mixed electro* Address correspondence to this author. E-mail: vub.ac.be.
stvdamme@
lytes in the mean spherical approximation. The procedure is analogous to that of Onsager et al.2 Compared to the procedure of Durand-Vidal et al.,6 no matrix inversion is required, because the differential equations are first decoupled. The advantage of describing the relaxation effect on the Onsager coefficients and not just for one particular property like the conductivity is that the Onsager coefficients are more general. Any transport property can be written in terms of the Onsager coefficients, so the relaxation effect on it follows immediately from the expressions given in this paper. 2. General Theory A mixed electrolyte solution is considered, containing n1, n2, ..., ns ions (per m3) of species 1, 2, ..., s with charges e1, e2, ..., es, diameters σ1, σ2, ..., σs, and mobilities ω1, ω2, ..., ωs. At equilibrium every ion is surrounded by an ionic atmosphere of opposite charge. When an external force is applied, the atmosphere is perturbed, giving rise to an additional counteracting internal force. This is the relaxation effect. The basic equation for the relaxation effect is the stationary hydrodynamic continuity equation
∇ B ‚(fijb V ij) - ∇ B ‚(fjib V ji) ) 0
(1)
where fji is the two-particle density, which satisfies the symmetry relation
r ) fij(-b) r fji(b)
(2)
and is related to the pair distribution function gji by
r ) njnigji(b) r fji(b)
(3)
b Vji is the velocity of a an i-ion located at b r in the atmosphere of a j-ion, and is defined by
r ) ωi(K Bji(b) r - kBT∇ B ln fji(b)) r b V ji(b)
(4)
In this formula the hydrodynamic motion of the fluid surrounding ion i caused by the presence of the moving ion j has been neglected. In doing so the relaxation effect is decoupled from the electrophoretic effect. The force K Bji acting on an i-ion in the atmosphere of a j-ion is
r )B k i + δk Bi - ei∇ B ψj (b) r K Bji(b)
10.1021/jp071651l CCC: $37.00 © 2007 American Chemical Society Published on Web 04/24/2007
(5)
Relaxation Effect on Onsager Coefficients
J. Phys. Chem. B, Vol. 111, No. 19, 2007 5309
where B ki is the applied external force, δk Bi is the force due to the i-ion’s own asymmetric atmosphere caused by the external force, and -ei∇ψ B j is the force exerted by the j-ion and its atmosphere. The electric potential ψj around ion j is related to the pair distribution functions by Poisson’s equation
∇ B ψj(b) r )2
4π
this can be cast into a more convenient form
∇ B∇ B ψ′j 2
(6)
Substituting eqs 4 and 5 into the continuity eq 1 leads to a nonlinear differential equation. In a first approximation only the linear terms are retained
B 2gji(b) r + eiωi∇ B 2ψj(b) r + ejωj∇ B 2ψi(-b) r ) (ωj + ωi)kBT∇ k i - ωjB k j)‚∇ B gji(b) r (7) (ωiB
niei2ωi
s
∑
∑
s
nieigji(b) r ∑ i)1
4π
∇ B 2ψ′j kBT i)1 ωj + ωi ωiB k i - ωjB kj 4π s 4π s nieiej ωi 2 ∇ B ψ′i ) niei ‚∇ B goji kBT i)1 ωj + ωi kBT i)1 ω j + ωi (15) 2
∑
which is the Fuoss-Onsager continuity equation. With the definitions of the ionic strength
I≡
∑ niei2 2 i)1
(16)
the relative ionic strength
2.1. First-Order Perturbation of the Ionic Atmospheres. If the applied forces B kj are small in the sense
|k Bj| , κkBT
s
1
njej2
µj ≡
(17)
s
∑ i)1
(8)
niei2
where κ is the Debye screening parameter
κ≡
x
a mean mobility
4π
s
niei2 ∑ k T i)1
s
(9)
B
then the problem can be treated by the first-order perturbation method
nj ej2ωj
tj ≡
(10)
Bk)R(r) (1 Br‚1
(11)
where the B 1 stands for a unit vector, and satisfy the symmetry conditions
r ) -ψ′j(b) r ψ′j(-b) g′ji(-b) r ) -g′ji(b) r
(12)
Substitution of eq 10 into the linearized continuity equation yields
B g′ji(b) r + eiωi∇ B ψ′j(b) r - ejωj∇ B ψ′i(b) r ) (ωj + ωi)kBT∇ 2
k i - ωjB k j)‚∇ B goji(b) r (13) (ωiB
and a matrix A with elements
s
kBT
niei∇ B 2ψ′i ) 0 ∑ i)1
ω j tk
s
aji ≡ δji
∑ω +ω
k)1
j
+ k
ω j ti ωj + ωi
(20)
where δij is the Kronecker symbol, the Fuoss-Onsager continuity equation can be written in more compact form s
(∇ B 2δji - κ2aji)∇ B2 ∑ i)1
() ψ′i
)-
ei
κ2 ej
µi ωiB k i - ωjB kj ‚∇ B goji ω + ω i j i (21)
s
∑ i)1 e
2.2. Decoupling of the Set of Differential Equations. With the proper linear combinations the set of differential eqs 21 can be decoupled. The set of differential equations is then replaced by an equivalent set2 s
()
s
tj χjp ∑ (∇ B 2δji - κ2aji)∇ B2 ∑ j)1 i)1 s
-κ where the equilibrium terms on the left have canceled out and the perturbed term on the right has been neglected. By means of Poisson’s equation and the identity
(19)
ω j
niei ωi ∑ i)1
where ψoj (r) and goji(r) are the potential and pair distribution function at equilibrium, i.e., if B kj ) B 0. The perturbed potential ψ′j(r b) and pair distribution function g′ji(r b) are linearly related to the applied forces, so they are of the type
ej
µjωj
2
r ) goji(r) + g′ji(b) r gji(b)
4π
)
s
ψi(-b) r ) ψoi (r) + ψ′i(-b) r
2
(18)
the limiting transference number
ψj(b) r ) ψoj (r) + ψ′j(b) r
2
µiωi ∑ i)1
ω j≡
2
∑ j)1
ψ′i
)
ei
tj χjp ej
s
µi ωiB k i - ωjB kj ‚∇ B goji (22) ω + ω i j i
∑ i)1 e
with the coefficients χjp satisfying the set of equations s
(14)
aji(tj χjp) ) qp(ti χip) ∑ j)1
(23)
5310 J. Phys. Chem. B, Vol. 111, No. 19, 2007
Van Damme and Deconinck
or alternatively, since ajitj ) aijti, s
aji χip ) qpχjp ∑ i)1
(24)
In other words, the χip are the components of the eigenvector of the matrix A corresponding to the eigenvalue qp. The eigenvalues and the eigenvectors are given by
∑ i)1 ω + R i
χjp )
ejei r
VjiC )
(34)
and the hard sphere potential
ω j ti
s
qp )
2.3. The Mean Spherical Approximation. The equilibrium pair distribution functions goji of the MSA8 are used in the right side of eq 30. In the MSA the effective pair potential Vji is the superposition of the Coulomb potential
VjiHS(r>σji) ) 0
(25) p
VjiHS(r
