1246
J. Phys. Chem. 1996, 100, 1246-1255
Brownian Dynamics Derivation of the Association Formulation for the Conductance Equations Jean-Claude Justice Laboratoire d’Electrochimie, UniVersite´ Pierre et Marie Curie, 75230 Paris Cedex 05, France ReceiVed: May 9, 1995; In Final Form: July 10, 1995X
From the bases of statistical mechanics a conductance equation is obtained which includes the Bjerrum association concept without making use of any ad hoc chemical model. The Hamiltonian model used is general enough to permit the a posteriori introduction of more specific Gurney cosphere ionic interactions. The former analytical results of Fuoss-Onsager and Falkenhagen et al. become a limit case formulation of the restricted primitive model applied to ions of lower electronic charges in solvents in higher dielectric constants.
1. Introduction
KA )
Recently important contributions have been reported concerning the problem of transport properties of electrolyte solutions.1-5 Regardless of these modern developments in the underlined theories, it is of interest to treat the Onsager problem with increased resolution and accuracy. Up to the present all the analytical derivations of the variation with concentration of the molar conductance Λ of a binary symmetrical electrolyte in a solvent of dielectric constant D are based on the restricted primitive model (RPM), where all ions are hard spheres of diameter σ. The approximations introduced in these works with the integration of the continuity equation ultimately lead to the now widely used general formulation of Fuoss and Onsager6 (FO) after expansion in powers of the concentration c of the electrolyte
Λ ) Λ0 - Sc1/2 + E′c ln c + J(σ)c - J3/2(σ)c3/2
(1)
Differences in the results of the various authors are associated with the formal expressions for the coefficient E′ and the functional forms of J(σ) and J3/2(σ).7 Following a critical review8 the most reliable formulation available at present for these coefficients is given in the appendix. However the above result is often inadequate and it is now widely recognized that a much better formulation is obtained by grafting the Bjerrum concept of association on the above result. This is analogous to Bjerrum’s original introduction of the association in the formulation of the mean activity coefficient9 f(DH derived by Debye and Hu¨ckel.10 This leads to the following set of equations
Λ ) γ(Λ0 - Sc1/2γ1/2 + E′cγ ln cγ + J(R)cγ - J3/2(R)c3/2γ3/2) (2) 2 KA γ ) 1 - cf(
(
f( ) γ exp -
X
κqγ1/2 1 + κRγ1/2
(3)
)
(4)
Abstract published in AdVance ACS Abstracts, December 1, 1995.
0022-3654/96/20100-1246$12.00/0
4πNA R 2 ∫ r exp 2qr dr 1000 σ
( )
(5)
with
κ ) [(4π/DkBT)∑niei2]1/2
(6)
q ) z2e2/2DkBT
(7)
i
where ni and ei are the density and the charge of the ions of type i. In eq 2 γ is calculated by solving the mass action law eq 3, which makes use of Bjerrum’s original formulation of the activity coefficient eq 4. The association constant KA is to be identified with the short-range integral eq 5 of the Boltzmann function for the anion-cation direct potential energy
U+-(r) )
{
-2q/r if r > σ ∞ otherwise
(8)
in which the upper limit distance parameter R is usually taken equal to the Bjerrum length q. Of course, for the sake of selfconsistency, this value of R is also substituted to σ in the J and J3/2 functions of eq 6 in the same way as it was substituted by Bjerrum in the original Debye-Hu¨ckel formulation of the activity coefficient eq 9.
(
f(DH ) exp -
κq 1 + κσ
)
(9)
The progression from eq 1 to eqs 2-5 may lead to numerically very important changes, as shown in Figure 1. The numerical treatment of conductance data is carried out according to the following method. Initially the three parameters Λ0, KA, and J3/2 of eqs 2-4 are adjusted through a least-square procedure; then the value of the apparent distance of closest approach σ is evaluated by solving eq 5. The system of eqs 2-5 proves to be quite useful in the numerical treatment of conductance data for a range of electrolyte concentration such that
0 < κRγ1/2 < 1/2
(10)
Numerical tests show that, within the concentration range, the calculated values of Λ and of f( are insensitive (relative © 1996 American Chemical Society
Association Formulation for the Conductance Equations
J. Phys. Chem., Vol. 100, No. 4, 1996 1247
Figure 1. (long-dash-short-dash curve) limiting law equation; (longdash curve) eq 1; (solid curve) eqs 2-7; for a 1-1 electrolyte with σ ) 3.70 Å and Λ0 ) 59.62 in a solvent mixture H2O-tetrahydrofuran 91.80 mass % at 25.000 C (dielectric constant D ) 11.50 and viscosity coefficient η ) 0.6075 cP). The experimental points are those observed for CsCl in that mixture. Cf.: Renard, E.; Justice, J.-C. J. Solution Chem. 1974, 3, 633.
variation of a few hundredths of a percent) to changes in the value of R so long as
q/2 e R e 2q
(11)
This is evidence that the two different approaches used in the two regions defined by the distance R are complementary provided conditions 10 and 11 are satisfied. It seems that there must exist a general unified theory that incorporates the two separate cases. Let us underline the fact that if R is set equal to σ (which means that one forces the Debye-Hu¨ckel treatment to apply in the entire range 0 e r e ∞), we return to the original DebyeHu¨ckel formulation for f( and to the Fuoss-Onsager formulation for Λ. Indeed, KA is then equal to zero and γ ) 1. However, one observes that setting R ) σ leads to unrealistic numerical values for the calculated Λ, the more so when σ is small compared to q. Thus the statement that is often seen in the literature, that the Debye-Hu¨ckel limiting law is the limiting case as σ f 0, is totally irrelevant and misleading. For as σ tends to zero, KA tends to ∞ and thus γ tends to zero and so do f+- and Λ. This is consistent with the logical expectation that the solution must collapse in the limit of zero ion size. The Bjerrum approach allows a straightforward generalization11,12 of the Hamiltonian model of ionic interactions making use of more realistic short-range ion pair-solvent specific functions W*(r) in the KA integral
KA )
(
)
4πNA R 2 W*(r) 2q r exp dr + ∫ 0 1000 kBT r
(12)
The Rasaiah-Friedman13 square mound (or well) solvation cosphere overlap model with W*(r) given by
{
0 if r > σ + s (13) W*(r) ) d+- if σ < r < σ + s ∞ otherwise is an example of such a function where s is generally taken equal to one or two solvent molecule diameters. One must acknowledge that as attractive as the concept of ionic association is, grafting such a chemical model onto an equation resulting from the integration of the Onsager continuity equation remains, after all, purely empirical. The purpose of the present work is to show that the set of eqs 2-5 can be
derived from the basic equations which govern the dynamic behavior of electrolyte solutions. It was thought for a long time that generalizing the conductance equations, by taking the Onsager approach, was doomed to failure because of the inherent assumptions. This however does not seem to be the case, since the Onsager continuity equation has been shown14 to be directly linked to the fundamental equations of statistical dynamics, as will be reviewed in the next section. When Onsager15 derived the continuity equation in 1927, his purpose was to derive the conductance limiting law, a problem which had been erroneously treated by Debye in 1923. Later, an extension of that continuity equation was used by Fuoss and Onsager6 to extend the limiting law with higher terms in a series expansion in the electrolyte concentration. This led to the result summarized by eq 1, which was also obtained independently by a number of other authors. The extension of the limiting law required a formidable series of calculations that gave rise to additional terms that could no longer be neglected, as had been the case for the derivation of the limiting law. It then seemed that it was hardly possible to go beyond the FuossOnsager type extension of the limiting law. The difficulty was increased by some confusion concerning the chemical model of association. This led to polemical discussions which concluded finally with the recognition that the Bjerrum approach to the association, as summarized above, is the most efficient and the most self-consistent treatment. Interestingly enough, Fuoss, who objected most to the Bjerrum model, had contributed, in collaboration with Onsager and Skinner,16 to show that a more elaborate treatment, which basically made use of a nonlinearized Poisson-Boltzmann equation for electrolyte solutions at equilibrium, led to a result which favored the Bjerrum formulation. This important result, which unfortunately was overlooked at the time, was the starting point of the author’s conviction that a general, unified solution could be found. Such was the case indeed, but other difficulties had still to be overcome. Clearly, the use of the Debye and Hu¨ckel ion pair distribution functions was insufficient. Fuoss, Onsager, and Skinner had precisely shown that a better set of functions denoted as DHX (for the exponential extension of the original DH formulas) leads to a linear contribution versus concentration, which is in agreement with Bjerrum formulation. This was an important first result. Indeed the mass action law eq 3 can be expanded as
γ ) 1 - cKA + O(c3/2)
(14)
which is precisely the new term obtained by Fuoss, Onsager, and Skinner. A further advance, summarized in section 4, was achieved when a new correction, denoted as the echo effect, was proposed for the integration of the Onsager continuity equation.17 The result then read
Λ ) γ′(Λ0 - Sc1/2 + E′c ln c + J(R)c - J3/2(R)c3/2)
(15)
with
γ′ )
1 1 + KAc
(16)
Equation 16, an intermediate result between eq 14 and eq 3, is not yet the full mass action law formulation of Bjerrum but undoubtedly an encouraging result. Obviously the DHX distribution functions are still insufficient when used with the echo effect correction to reach the full set of eqs 2-5. A better formulation for the anion-cation equilibrium distribution function was the next step. It came from a
1248 J. Phys. Chem., Vol. 100, No. 4, 1996
Justice
reconsideration of the Mayer activity expansion equation.18 The set of eqs 4 and 5 was then proved to be a limiting case for the activity coefficient of binary symmetrical electrolytes. In other words the exact Bjerrum formulation was found for equilibrium properties. This will be reviewed in section 5. It also happens that in the mean time Friedman19 had derived the distribution functions which are consistent with the Bjerrum formulation. The question now obviously is: What is the result of introducing the Friedman distribution function equations both in the echo effect correction and in the Onsager continuity equation? As the work presented in this contribution will show, this leads to the formulation summarized in eqs 2-5. However the answer can be obtained only through a complete rederivation of the Onsager continuity equation, as will be done below.
1. Total Velocities
b V )b V eiP + ωi(eiB X+B FiP) (20) iP
iP iP iP b V jQ )b V ejQ + ωi(eiB X+B FjQ ) (21)
...etc.... 2. Internal Forces
B FiP ) -kBT∇P ln giP -
We shall in a first step adopt a system of notation due to Debye and Falkenhagen,21 which is of great help for understanding in a straightforward way how the various statistical quantities (such as distribution functions, mean interionic forces, mean velocities, ..., etc.) involved in describing the physical properties of electrolytic solutions are interconnected. This notation system follows the following rules: (i) A pair of lowercase and uppercase characters such as iP represents an ion of type i located at point P. (ii) Superscripts indicate the type(s) and the location(s) of the ions to which the time-averaged quantity belongs. (iii) Subscripts indicate any eventual set of restricting conditions, related to the presence of given types of ions at given locations, imposed during the time averaging. For iP is to be used as the aVerage interionic force instance B FjQ exerted on an ion of type i located at P when an ion of type j is present at Q. The meaning of the various symbols used is the following: kB is the Boltzmann constant; T is the temperature in Kelvin; dVP is the volume element around point P; ni is the density of ions of type i; ei is the electrical charge of ions of type i; ωi is the friction coefficient of ions of type i in the solution; B X is the applied external electric field; U(iP,jQ) is the potential energy of interaction between two ions at the indicated locations in the pure solVent taken as a dielectric and hydrodynamic continuum; b V are mean total velocities; b Ve are mean electrophoretic velocities; B F are mean interionic forces; g are distribution functions; ∇P is the gradient or divergence vectorial operator applied at location P; and the hydrodynamic tensor
χ(P,S;ωi,ωl) ) χL(P,S) + χS(P,S;ωi,ωl)
(17)
χL(P,S) ) (1/8πηr)(δ + rr/r2)
(18)
with
(19)
and r ) rS - rP and δ the unit tensor allows the evaluation of the electrophoretic velocity contribution transmitted through the solvent viscosity to an ion of type i and mobility ωi located at P by an ion of type l and mobility ωl located at S (cf. ref 22). Let us note that so far the short-range term χS(P,S;ωi,ωl) has always been neglected. First the set of the three hierarchies will be described. From them the continuity equation will be developed, and this will ultimately allow us to evaluate the changes g′ relative to the equilibrium values g° encountered by the ion pair distribution functions once a steady state is reached after an external electric field is applied in the electrolytic solution. These three hierarchies are related to the mean Velocities, interionic forces,
∑n ∫∇ U(iP,lS)g l
lS iP
P
dVS
(22)
l
iP iP B FjQ ) -kBT∇P ln gjQ - ∇PU(iP,jQ) -
∑n ∫∇ U(iP,lS)g l
lS iP,jQ
P
l
iP B FjQ,kR
2. Basic Set of Equations
χS(P,S;ωi,ωl) ) (1/8πηr3)(ωi-2 + ωl-2)(δ/3 - rr/r2)
TABLE 1
) -kBT∇P ln
iP gjQ,kR
dVS (23)
- ∇PU(iP,jQ) - ∇PU(iP,kR) -
∑n ∫∇ U(iP,lS)g l
lS iP,jQ,kR
P
dVS (24)
l
...etc.... 3. Electrophoretic Velocities
b V eiP )
∑n ∫χ(P,S;ω ,ω )(e BX + BF l
i
l
lS lS iP)giP
l
dVS
(25)
l
iP jQ ) χ(P,Q,ωi,ωj)(ejB X+B FiP )+ b V ejQ
∑n ∫χ(P,S;ω ,ω )(e BX + BF l
i
l
l
lS lS iP,jQ)giP,jQ
dVS (26)
l
...etc....
and electrophoretic Velocities of one type of ion at a specified location. One additional level in the hierarchy is reached when one more condition is imposed during the time averaging. This leads to a new pair of subscripts in the notation. It is obvious that the phenomenological aspect remains unchanged at each level of a given hierarchy so that only the the number of subscripts changes from one level to another (only the contributions specifically due to the direct interaction with the ions mentioned as subscripts must be added). In Table 1 the first equations of each of the three hierarchies which are needed to solve the conductance problem are given. This is also enough to make obvious the generalizations to any levels. let us briefly summarize the phenomenological content of each quantity. The total velocity b V (eqs 20 and 21) is the sum of three contributions: (1) the electrophoretic velocity b Ve, (2) the velocity imparted by the external field B X, and (3) the velocity imparted by the interionic force B F. The interionic force B F (eqs 22-24) is the sum of (1) the so-called diffusion force, that is the pseudoforce exerted by the gradient of concentration of the ion under consideration (This leads to the ∇P ln g term.), (2) the direct force exerted by the ions which are eventually present at fixed locations during the averaging and accounted for by subscripts (This leads to the ∇PU term(s).), and (3) the force exerted by all the other ions of the solutions, the action of which must be evaluated through the probability of their presence and summed up over all possible locations (This leads to the integral term.) The electrophoretic velocity b Ve (eqs 25 and 26) is also the sum of (1) the contribution of the ions noted in the subscripts and (2) that of all the other ions. Let us note that at equilibrium all these quantities vanish and the hierarchy of the internal forces B F becomes the well-known BornBogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy and that the second equation of the BBGKY hierarchy leads directly, after introduction of the Kirkwood approximations (described in section 5) for the three-ion distribution function, to the Poisson-Boltzmann equation, which is the basis of the famous Debye and Hu¨ckel treatment.
Association Formulation for the Conductance Equations
J. Phys. Chem., Vol. 100, No. 4, 1996 1249
3. Conductance Problem and Three Hierarchies The solution of the conductance problem rests on the evaluation of b ViP. This in turn requires the evaluation of both iP iP Ve . Rewriting eq 20 as B F and b
)
(
FiP VeiP b V ) ωieiB X 1+ + eiX ωieiX iP
(27)
differently expressed as
(
V iP b V iP ) b 0 1+
)
∆Xi VeiP + iP X V 0
(
)
∆Xi λei + i X λ 0
(29)
(30)
we obtain
(
Λ ) Λ0 1 +
∆X Λe + X Λ0
)
iP iP jQ jQ jQ ∇P‚gjQ (V biP + w bjQ ) + ∇Q‚giP (V b +w biP ))0
Restricting ourselves to the case of an homogeneous system with a linear response to the external perturbation (that is, eliminating terms made of cross products of perturbation quantities), one can extract the monitoring term of the continuity equation, which leads to
∂ jQ ∂ iP iP iP jQ jQ ) ∇P‚gjQ b V jQ + ∇Q cegiP b V iP ) - giP (32) - gjQ ∂t ∂t which expresses the conservation of the number of ions in the solution. 4. Echo Effect 4.1. Short Survey. Let us make use here of an obvious yet very important property of any conditional mean quantity and iP apply it in particular to b V jQ and b V jQ iP : If a conditional particle (designated by the subscript) is at a large distance from the particle (designated by the superscript), the mean quantity tends
(34)
which can be compared with the usual approximation iP iP iP jQ jQ V 0jQ)∇Pg°jQ ) -∇P‚gjQ w bjQ - ∇Q‚giP w biP (35) (V b0iP - b
VjQ is known, eq 35 must Because in eq 34 neither b ViP nor b V0jQ at zero approximate them by their limiting values b V0iP and b concentration. It had been believed that the only way to handle the problem was to carry out the integration of the continuity equation (eq 35) to obtain an initial set of values for b V(1)iP and ViP b V(1)jQ and then to integrate eq 34 using the initial values of b jQ and b V . There is however a simpler and much more efficient method for handling this otherwise infinite iteration problem in just one single step and also in an exact way. Let us notice first that in the case of a binary symmetrical electrolyte the continuity equation (eq 34) can be rewritten as
(
(31)
where Λe ) λie + λje. The term in parentheses is called the conductance coefficient denoted as fΛ. Its formal evaluation as a function of concentration and the characteristic parameters of the ionic interaction model used is the aim of the conductance theory. This coefficient also plays an important role in the derivation of the echo effect. The evaluation of both ∆X/X and Λe/Λ0 requires (through eqs 22 and 25, respectively) the prior evaluation of the perturbation g′ brought to the ion pair d.f. once a steady state is reached for the electrolyte solution after an external field has been applied. This evaluation proceeds through the integration of the Onsager15 continuity equation
(33)
where w bjQ iP is now a quantity which tends to zero as the distance between P and Q becomes large. Relations such as eq 33 relate quantities of consecutive levels in a hierarchy. After substitution in the continuity eq 32, one gets for the steady state
iP iP iP jQ jQ V jQ)∇Pg°jQ ) -∇P‚gjQ w bjQ - ∇Q‚giP w biP (V biP - b
where the two terms in parentheses are dimensionless. One obtains, from eq 29, the usual presentation of the ionic conductance coefficient where ∆Xi is the relaxation field acting on ions of type i and λei the electrophoretic ionic conductance imparted to that ion. Note that the location P of the ion can been dropped, since one-particle unconditional quantities do not depend on location in a uniform homogeneous system. Restricting ourselves to the case of a binary symmetrical electrolyte (1-1, 2-2, ..., etc.) for which
∆Xi ∆Xj ∆X ) ) X X X
jQ jQ b V iP )b V jQ + w biP
(28)
and converting velocities to conductance quantities leads to
λi ) λi0 1 +
to the Value of the unconditional mean quantity. We can thus express the velocity vectors as
V 0jQ) 1 + (V b0iP - b
)
∆X Λe ∇ g°iP ) + X Λ0 P jQ iP iP jQ jQ w bjQ - ∇Q‚giP w biP (36) -∇P‚gjQ
or
(
B 1+ (eiωi - ejωj)X
)
∆X Λe ∇ g°iP ) + X Λ0 P jQ iP iP jQ jQ w bjQ - ∇Q‚giP w biP (37) -∇P‚gjQ
One now understands why the term on the LHS of eq 37 can be called the monitoring term of the continuity equation, since it is the only term in which the external perturbation (the external field B X) explicitly appears. Should this term vanish, all perturbed quantities (ionic conductance among them) would vanish too. But one also readily observes that the real perturbation is not just the external field but rather the product B X(1 + ∆X/X + Λe/Λ0). One also observes that the LHS becomes zero if the friction coefficients ωi and ωj become infinite. This corresponds to the physical situation where the ions cannot move. Under this circumstance their motion could not be perturbed. Last but not least, for a pair of ions of the same type (eiωi ) ejωj) the pair distribution functions are unperturbed. The first step in the integration of the continuity equation will still consist of neglecting the conductance coefficient 1 + ∆X/X + Λe/Λ0 in eq 37, but the second step will use a much more efficient correction based on the following reasoning: Let us denote as
1250 J. Phys. Chem., Vol. 100, No. 4, 1996
Justice
g′(1) the first-order value of the perturbation of the pair d.f. The corrected value g′ is obtained by the simple formula
(
∆X Λe g′ ) g′(1) 1 + + X Λ0
γ′ )
)
but, since ∆X/X is also proportional to g′, the same correction holds for the relaxation term, so that
(
)
∆X ∆X ∆X Λe ) (1) 1 + + X X X Λ0
(38)
and, as a consequence, the correct evaluation of ∆X/X is obtained by solving eq 38 for ∆X/X.
∆X ) X
where
( )
Λe ∆X(1) 1 1+ Λ0 ∆X(1) X 1X
∆X(1)Sc R iP dr1 ) -ni∫σ g°jQ X
5. Conductance Continuity Equation We are now ready to build the conductance continuity equation. We will simplify, approximate, and reformulate the equations until a mathematical formulation is reached that can be integrated to obtain appropriate formal expressions for the pair distribution functions necessary to calculate ∆X and Λe. After the appropriate substitution of eqs 20-26 into eq 32, one obtains the following integro-differential equation
∂ iP iP iP ) ∇P{gjQ [ωi(eiB X - kBT∇P ln gjQ - gjQ ∂t
lS ∇PU(iP,jQ) - ∑nl∫∇PU(iP,lS)giP,jQ dVS) + l
The dominant term in conductance is ∆XSc /X, which can become numerically very large, since it is controlled by Coulombic forces between ions at very short distances. For this reason it is important to use an analytical expression that has a minimum number of approximations. This is the reason why the echo correction, described above, will be applied to this term while the first-order approximation is generally sufficient for all other contributions. This correction becomes
(
)
∆XSc ∆XLc ∆Xh Λe ∆XSc ∆XSc ) (1) 1 + + + + X X X X X Λ0
lS - ∇SU(lS,iP) ∑l nl∫χ(P,S;ωi,ωl)(elBX - kBT∇S ln giP,jQ kR lS ∇SU(lS,jQ) - ∑nk∫∇SU(lS,kR)giP,jQ,lS dVR)giP,jQ dVS + l
jQ X - kBT∇Q ln giP - ∇QU(jQ,iP) χ(P,Q,ωi,ωj)(ejB lS dVS)]} + ∇Q{ ) ∑l nl∫∇QU(jQ,lS)giP,jQ P h Q}
ihj
∂ jQ (45) - giP ∂t Let us express eq 45 in terms of the unconditional distribution functions using the following set of relations
from which one can extract the next order correction
)
(
)
∆XLc
∆Xh Λe 1 1+ + + S X X Λ0 ∆Xc (1) 1X
iP gjQ )
(40) lS ) giP,jQ
The conductance equation (eq 31) can be rewritten as
(
Λ ) Λ0 1 +
∆XSc ∆XLc ∆Xh Λe + + + X X X Λ0
)
kR giP,jQ,lS )
)
∆XLc ∆Xh Λe Λ ) γ′Λ0 1 + + + X X Λ0
giP,jQ gjQ
(46)
giP,jQ,lS giP,jQ
(47)
giP,jQ,lS,kR giP,jQ,lS
(48) (49)
(41) Let us note that
and substituting eq 40 for ∆XSc /X leads thus to
(
(44)
This will also prove to be very useful later.
S L ∆X ∆Xc ∆Xc ∆Xh ) + + X X X X
X
(43)
which will also prove to be very useful later since ultimately γ′ will identify with γ in the factor of eq 2 as defined by eq 3 (cf. eq 78). Let us note at this level that γ′ is a dimensionless parameter such that 0 < γ′ < 1, since ∆XSc (1)/X is negative. Moreover, it was shown formerly8 that for values of the cutoff distance R ≈ q the short-range Coulombic part of the relaxation contribution can be evaluated with a high degree of approximation by
(39)
More will be found concerning this topic in ref 17. 4.2. One Important Application. The relaxation field ∆X is made up of two contributions: a purely Coulombic contribution denoted ∆Xc and a hydrodynamic contribution denoted ∆Xh. Since the diffusion term in eq 22 vanishes in a uniform homogenuous system, one sees that any contribution to the relaxation field is the result of an integration over the whole volume occupied by the solution. One can consequently divide the integration volume of ∆Xc into two parts: a short-range part ∆XSc , integrating from r ) 0 to any value r ) R, and a long-range part ∆XLc integrating from r ) R to ∞. Then
∆XSc
1 ∆XSc (1) 1X
iP ) ∇P ln ∇P ln gjQ
(42)
giP,jQ ) ∇P ln giP,jQ jQ g
(50)
since gjQ is a constant in a uniform homogenuous system. It
Association Formulation for the Conductance Equations
(perturbed) excess correlation functions, which tend to zero as PQ f ∞.
follows that
-
1 ∂
iP,jQ
g
g ∂t jQ
) ∇P
{ [( giP,jQ jQ
ωi eiB X - kBT∇P ln g
iP,jQ
giP,jQ,lS
∇PU(iP,jQ) - ∑nl∫∇PU(iP,lS)
(
giP,jQ
l
giP,jQ ) giPgjQ(1 + h° iP,jQ + h′ iP,jQ)
-
)
g
dVS +
giP,jQ,lS,kR
∇SU(lS,jQ) - ∑nk∫∇SU(lS,kR)
iP,jQ
giP,jQ,lS
(
l
)
dVR ×
dVS + χ(P,Q,ωi,ωj) ejB X - kBT∇Q ln gjQ,iP -
g
iP,jQ,lS
g
∇QU(jQ,iP) - ∑nl∫∇QU(jQ,lS)
dVS
giP,jQ
{ }
l
)]}
+
1 ∂ jQ,iP iSj )(51) g PhQ giP ∂t
∇Q
This result is identical to the equation obtained by Ebeling et al.14 with giP ) giQ, as should be the case for symmetrical electrolytes. We retain however giP and giQ explicitly, since it will be important later to identify the type of ions to which each of these two quantities belong. Expanding eq 51 with respect to that
giP,jQ
(53)
giP,jQ,lS ) giPgjQglS(1 + h° iP,jQ,lS + h′ iP,jQ,lS)
∑l nl∫χ(P,S;ωi,ωl) elBX - kBT∇S ln giP,jQ,lS - ∇SU(lS,iP) giP,jQ,lS
J. Phys. Chem., Vol. 100, No. 4, 1996 1251
and taking into account
giP,jQ∇P ln giP,jQ ) ∇PgiP,jQ
giP,jQ,lS,kR ) giPgjQglSgkR(1 + h° iP,jQ,lS,kR + h′ iP,jQ,lS,kR) We shall now proceed to several simplifications: (1) Use of gi for giP for all the one-particle distribution functions. (2) Elimination of the equilibrium terms, that is the terms which remain in the equation when the external field is turned off. In that case the equation becomes
0 ) gi∇P{ωi(-kBT∇Ph° iP,jQ - h° iP,jQ∇PU(iP,jQ) -
∑l nlgl∫∇PU(iP,lS)h° iP,jQ,lS dVS) + ∑l nlgl∫χ(P,S;ωi,ωl) × (-kBT∇Sh° iP,jQ,lS - h° iP,jQ,lS∇SU(lS,iP) h° iP,jQ,lS∇SU(lS,jQ) - ∑nkgk∫∇SU(lS,kR)× l
h°
i,jQ,lS,kR
dVR) dVS + χ(P,Q,ωi,ωj)(-kBT∇Qh° jQ,iP -
h° iP,jQ∇QU(jQ,iP) - ∑nlgl∫∇QU(jQ,lS)h° iP,jQ,lS dVS)} + l
gj∇Q
{
}
ihj ) 0 (54) PhQ
which can thus be subtracted from eq 52. (3) Elimination of all cross-product terms of B X with h′ functions and of all crossproduct terms of h′ functions with one another, so that only the terms of first order in the external field B X (Ohm’s law) are kept in the differential equation. The result is
∂ X(1 + h° iP,jQ) - kBT∇Ph′ iP,jQ -gi h′ iP,jQ ) gi∇P{ωi(eiB ∂t
h′ iP,jQ∇PU(iP,jQ) - ∑nlgl∫∇PU(iP,lS)h′ iP,jQ,lS dVS) +
one now obtains
l
-
1 ∂
iP,jQ
g
gjQ ∂t
)
1
∇P{ωi(eiB Xg
iP,jQ
gjQ
- kBT∇Pg
iP,jQ
∑l nlgl∫χ(P,S;ωi,ωl)(elBX(1 + h° iP,jQ,lS) - kBT∇Sh′ iP,jQ,lS -
-
giP,jQ∇PU(iP,jQ) - ∑nl∫∇PU(iP,lS)giP,jQ,lS dVS) + l
∑l nl∫χ(P,S;ωi,ωl)(elBXgiP,jQ,lS - kBT∇SgiP,jQ,lS giP,jQ,lS∇SU(lS,iP) - giP,jQ,lS∇SU(lS,jQ) -
∑l nk∫∇SU(lS,kR)g
iP,jQ,lS,kR
h′ iP,jQ,lS∇SU(lS,iP) - h′ iP,jQ,lS∇SU(lS,jQ) -
∑l nkgk∫∇SU(lS,kR)h′ iP,jQ,lS,kR dVR) dVS + X(1 + h° iP,jQ) - kBT∇Qh′ jQ,iP χ(P,Q,ωi,ωj)(ejB h′ iP,jQ∇QU(jQ,iP) - ∑nlgl∫∇QU(jQ,lS)h′ iP,jQ,lS dVS)} + l
dVR) dVS + χ(P,Q,ωi,ωj) ×
{
gj∇Q
}
∂ ihj ) -gj h′ jQ,iP (55) PhQ ∂t
(ejB XgiP,jQ - kBT∇QgjQ,iP - giP,jQ∇QU(jQ,iP) -
∑l nl∫∇QU(jQ,lS)g
iP,jQ,lS
dVS)} +
1
{
∇Q giP
-
}
ihj ) PhQ 1 ∂
gjQ,iP (52) g ∂t iP
When the distance between the involved species is large, the multiparticle d.f. approaches the product of the single d.f.’s. For instance, giP,jQ f giPgjQ as PQ f ∞. We shall now express the g multiparticle d.f. in terms of h° (equilibrium) and h′
(4) Introduction of the classical Kirkwood approximations which consist first in expressing each N-particle d.f. when N > 2 in terms of a sum of N two-particle d.f.’s plus an excess term. For example
h′ iP,jQ,lS ≡ h′ iP,jQ + h′ iP,lS + h′ jQ,lS + ′ iP,jQ,lS h° iP,jQ,lS ≡ h° iP,jQ + h° iP,lS + h° jQ,lS + ° iP,jQ,lS Those terms that represent contributions of order greater than 2 in number of the particles are neglected.
1252 J. Phys. Chem., Vol. 100, No. 4, 1996
Justice
The continuity equation now reads
∂ -gi h′ iP,jQ ) gi∇P{ωi(eiB X(1 + h° iP,jQ) - kBT∇Ph′ iP,jQ ∂t
h′ iP,jQ∇PU(iP,jQ) - ∑nlgl∫∇PU(iP,lS)(h′ iP,jQ + h′ iP,lS +
that does not contribute either because of symmetry considerations in integral terms or invariance with respect to a derivative operator and (2) underlined all terms that contribute to the echo effect. This can be checked after substituting eq 22 for B FiP and eq 25 for b VeiP in the current set of approximations:
l
h′ jQ,lS) dVS) + ∑nlgl∫χ(P,S;ωi,ωl)(elB X(1 + h° iP,jQ +
B FiP ) -∑nlgl∫
l
l
h° iP,lS + h° jQ,lS) - kBT∇S(h′ iP,jQ + h′ iP,lS + h′ jQ,lS) (h′ iP,jQ + h′ iP,lS + h′ jQ,lS)∇SU(lS,iP) - (h′ iP,jQ + h′ iP,lS + h′ jQ,lS)∇SU(lS,jQ) - ∑nkgk∫∇SU(lS,kR)(h′ iP,jQ + h′
S + h′
iP,kR
+ h′
+ h′
jQ,lS
jQ,kR
+ h′
lS,kR
gkR,lS,iP
) dVR) dVS +
X(1 + h° iP,jQ) - kBT∇Qh′ jQ,iP χ(P,W,ωi,ωj)(ejB h′ iP,jQ∇QU(jQ,iP) - ∑nlgl∫∇QU(jQ,lS)(h′ iP,jQ + h′ iP,lS + l
h′ jQ,lS) dVS)} + gj∇Q
{
}
∂
ihj ) -gj h′ jQ,iP (56) PhQ ∂t
Let us make use now of a more conventional notation: Positions will be represented by vectors according to the following rules:
P f r1
Q f r2
S f r3
R f r4
Functions will be abbreviated in the following way
The tensor notation becomes
χ(P,S;ωi,ωl) f χil
The volume elements are written as
dVQ f dr2 ∂ ∇Q f ∂r2
∂ ∇S f ∂r3
{(
∂h′ij ∂Uij ∂ ωi eiB - h′ij X(1 + h°ij) - kBT ) gi ∂t ∂r1 ∂r1 ∂r1
∂h′ij
)
(
∂Uil (h′ij + h′il + h′jl) dr3 + ∑nlgl∫χil elB X(1 +
∑l nlgl∫ ∂r
1
l
∂ h°ij + h°il + h°jl) - kBT (h′ij + h′il + h′jl) - (h′ij + h′il + ∂r3 ∂Uli h′jl)
∂r3
∂Ulj
- (h′ij + h′il h′jl)
∂r3
- ∑nkgk∫
) ( l
∂Ulk (h′ij + h′il + ∂r3
∂Uji
∂Ujl - h′ij - ∑nlgl∫ (h′ij + h′il + h′jl) dr3 k BT ∂r2 ∂r2 ∂r2 l
{ }
)}
+
∂h′ji ∂ ihj ) -g gj (57) j ∂r r1 h r2 ∂t 2
giP
dVS (59)
After eliminating the terms contributing to the echo effect, that VeiP, and those giving no contribution, is those leading to B FiP and b one finally obtains
(
)
(
)}
It is not necessary to carry analytical developments any further. At this level, integration is a question of pure mathematical computing once the direct interaction functions Uij and the equilibrium distribution functions g°ij that match with the Uij functions chosen are given. We shall see below that this new equation is ready made to make use of new ion pair distribution functions which will prove quite useful. It will also allow a simple generalization of the analytical results of integration obtained by other workers and summarized in ref 7. 6. Pair Distribution Function at Equilibrium
X(1 + h°ij) h′ik + h′jl + h′jk + h′lk) dr4 dr3 + χij ejB ∂h′ji
)
glS,iP
)
{ }
The equation now reads
-gi
(
)
dVS f dr3 dVR f dr4
The ∇ operators become
∂ ∇P f ∂r1
glS,iP
dVR
∂Uli ∂ ) ∑nlgl∫χil elB Xh°il - kBT h′li - h′li ∂r3 ∂r3 l ∂Ulk n g ∫ ∑l k k ∂r (h′kl h′ki + h′li) dr4 dr3 (60) 3
{(
h′ jQ,iP f h′ji(r2,r1) f h′ji
dVP f dr1
∑l nk∫∇SU(lS,kR)
∂h′ij ∂h′ij ∂Uij ∂ ωi eiB Xh°ij - kBT - h′ij ) gi -gi ∂t ∂r1 ∂r1 ∂r1 ∂Uil ∂ n g ∑l l l∫ ∂r h′jl dr3 + ∑l nlgl∫χil elBXh°jl - kBT∂r h′jl 1 3 ∂Uli ∂Ulj (h′ij + h′jl) - (h′ij + h′il + h′jl) ∂r3 ∂r3 ∂h′ji ∂Ulk n g h′ dr dr + χ e B X h° k T ∫ ∑l k k ∂r jk 4 3 ij j ij B ∂r 3 2 ∂Uji ∂Ujl h′ij - ∑nlgl∫ (h′il + h′jl) dr3 + ∂r2 ∂r2 l ∂h′ji ∂ ihj gj ) -g (61) j ∂r2 r1 h r2 ∂t
U(iP,jQ) f Uij(r1,r2) f Uij
χ(P,Q;ωi,ωj) f χij
(
(58)
b V eiP ) ∑nl∫χil elB X - kBT∇S ln glS,iP - ∇SU(lS,iP) l
l
iP,l
∂Uil h′il dr3 ∂r1
in which we have (1) placed a double underline under each term
If the mean potential energy of ion pair interactions at equilibrium Ψ°ij is known, the corresponding ion pair distribution functions then read
( )
g°ij ) exp -
Ψ°ij kBT
(62)
This was the line followed by Debye and Hu¨ckel, whose result,
Association Formulation for the Conductance Equations
J. Phys. Chem., Vol. 100, No. 4, 1996 1253
after linearizing the Poisson-Boltzmann equation, is
Ψ°ij )
eiej exp(κ(σ - r)) Dr (1 + κσ)
(63)
One can now legitimately ask the following question: What are the equilibrium distribution functions that are consistent with the Bjerrum treatment? This problem has been solved exactly by Friedman,19 and the result is
{
if r < σ 1 + κ′(R r) 2q if σ < r < R γ2 exp g°+-(r) ) r 1 + κ′R 2q exp(κ′(R - r)) if r > R γ2 exp r 1 + κ′R (69)
The corresponding distribution functions are obtained by linearizing eq 62
eiej 2q exp(κ(σ - r)) g°ij ) 1 |eiej| r (1 + κσ)
(64)
which are commonly referred to as the DH distribution functions, whereas the nonlinearized versions
[
g°ij ) exp -
]
eiej 2q exp(κ(σ - r)) |eiej| r (1 + κσ)
ci ) ai + ∑ niKnan
(66)
n>1
which relates the activities ai of ionic species and their concentrations ci in a solution via the direct energy potential functions Uij imbedded in the Kn integrals, can be rewritten in the case of a solution of a binary symmetrical electrolyte without any approximation whatsoever as S L S × f( (R,cf( ) f( ) f( S 1 - f( S 2 L 2 (f( ) c(f( )
) kA + kS -
n 8π 3 (R - σ3) + ∑ KnS an-2 3 n>22
(67)
] ]
κ′ ) [(4π/DkbT)∑niγei2]1/2
(70)
i
The above distribution function is continuous in value and in derivative at the distance r ) R, but it is not normalized, since
g°+-(r) f γ2 as r f ∞
(71)
Grigo and Ebeling20 have discussed this drawback when comparisons with results of other theories for equilibrium are made thru numerical tests which require that normalization be achieved. Let us just note here that they have shown that once the normalization quantity 1 - γ2 is added, the above distribution functions lead to numerical evaluations of the running coordination numbers, which, in the case of the restricted primitive model, are as good as HNC, MSA, and Monte-Carlo evaluations in a wide range of concentration. However the present problem of conductance is not sensitive to this drawback, since it does not depend on any constant. Indeed any constant term would cancel out whether through spatial derivations or through spherically symmetrical vectorial integrations. As a consequence, it is allowed to use the above distribution functions as originally derived by Friedman without adding empirically any constant. In the general notation adopted in this work this now reads
g°iP,jQ f giPgjQ as PQ f ∞
(68)
The definitions of the undefined quantities and the derivation for the steps that give eqs 67 and 68 from eq 66 are found in ref 18. Though this new formulation is more complicated than the original version, it contains a great deal of new, useful information. It shows that the problem of deriving a theory for the mean activity coefficient can be separated into two distinct problems, a long-range problem, that of finding a good L in eq 67, and a short-range one, that of approximation for f( S in eq 68. In this respect it finding a good approximation for f( is interesting to observe that if one neglects all terms on the S reduces to the RHS of eq 68 except the first term, then f( Bjerrum fraction γ of free ions given by eq 3. We are free to L any choose for the derivation of the long-range quantity f( theory that fits over that range, and we shall use here the Debye-Hu¨ckel theory. The set of eqs 3-5 can be considered to be a basic formulation, independent of any ad-hoc chemical model assumption, since the formulation can be derived from fundamental statistical mechanics. We thus have a full justification of the Bjerrum treatment for activity coefficients at low concentrations but, nevertheless, over a concentration range much larger than that of the original Debye-Hu¨ckel result given by eq 9.
[ [
where
(65)
are called the DHX or sometimes Meeron d.f. functions. These sets of equations have often been used to solve the Onsager continuity equation. There is however another approach for deriving the distribution functions. It consists in first obtaining an expression for the evaluation of the mean ionic activity coefficients and then deriving the distribution functions which are self-consistent with this expression. This is the line we shall now summarized. We were recently able to show18 that the Mayer activity expansion equation
0
(72)
One observes that giP and gjQ can be identified with γ and that as a consequence eqs 69 can be rewritten as follows
[
]
2q 1 + κ′(R - r) r 1 + κ′R 2q κ′r 2 2 ) γ exp 1) f( exp(2q/r) r 1 + κ′R
S (r) ) g+g- exp g°+-
[ (
)]
(73)
and
[
g°ijL(r) ) gigj exp -
]
eiej 2q exp(κ′(R - r)) |eiej| r 1 + κ′R
(74)
where f( is given by eq 4, with the following new definition of the Debye reciprocal length κ′ -1
κ′ ) [(4π/DkbT)∑nigiei2]1/2
(75)
i
replacing the original definition given by eq 6 and setting all gi ) γ. This formal transformation makes the Friedman equilibrium distribution function suitable for introduction into eq 61. By setting gi ) γ ) 1 and R ) σ in eq 74, we recover the DHX eq 65 and hence the DH eq 64, which will be of great
1254 J. Phys. Chem., Vol. 100, No. 4, 1996
Justice
help in the derivation based on the new set of distribution functions in the new continuity equation.
So far we have reached the following formulation
(
Λ ≡ γΛ0 1 +
7. Evaluation of the Conductance Equation shown14
that, after making explicit the Ebeling et al. have hard core term in the direct interaction potential Uij and in the correlation functions hij following a suggestion due to Kremp,23 continuity equations such as eq 61 are easily transformed into a boundary-value problem. Likewise the formulation we have introduced in this work can also be converted to a boundaryvalue problem. For this reason we will not extend the analytical development further, since the methods used in earlier solutions of the Onsager equation are equally applicable to our new formulation. The solution of the boundary-value problem is carried out in several steps. For our present needs it will be necessary to describe only two of them. In a first step all the hydrodynamic terms (that is all those which depend explicitly of the hydrodynamic tensor χ) are neglected. This ultimately leads to an evaluation of the purely Coulombic part of the relaxation field term ∆Xc/X. In the second step of the integration the hydrodynamic terms are retained, but in those the initial solutions for the perturbations of the distribution functions obtained in the first step are introduced as known quantities. This ultimately leads to an evaluation of the hydrodynamic part of the relaxation field term ∆Xh/X. We are now ready to evaluate each of the four interionic terms of the conductance equation (eq 42)
Λ)
(
∆XLc ∆Xh Λe 1 Λ 1 + + + 0 X X Λ0 ∆XSc (1) 1X
)
R
(76)
where
(
)
(77)
Substituting this result in eq 43, which summarizes one possible application of the echo effect, leads to
γ′ )
1 1 ≡γ ) S 2 ∆Xc (1) 1 + cγf′( KA 1X
[
h°ijL ) exp
]
2q exp(-κ′(r - R)) -1 r 1 + κ′R
(79)
This can be verified by comparison with eq 53. Once this function is substituted into the continuity equation (eq 61), the latter becomes a differential equation in the unknown function h′+-(r), which can then be evaluated. We will not carry out the solution of this rather complex equation, since the solution has already been carried out by Ebeling et al. using
[2qr exp(-κr)] - 1
h°ijL(r) ) exp
(80)
which is an approximation of eq 79. Representing the Ebeling result for the Coulombic part of the relaxation coefficient as ∆Xc,E(σ,C)/X and generalizing it to our formulation gives
∆X(1)Lc ∆Xc,E(R,Cγ) ) X X
(81)
giglh′ lS,iP
2 ) -4πnigif′( ∫σ r2 exp(2q/r) dr
κqγ1/2 1 + κRγ1/2
2 with γ ) 1 - cf( KA. 7.2. Long-Range Part of the Coulombic Relaxation Contribution. The long-range part of the excess distribution function h°Lij, which corresponds to the d.f. of eq 74 is
B Xi ) -∑nl∫∇PU(iP,lS) B FiP ) ei∆
S ∆X(1)Sc Rg°ij(r) ) -ni∫σ jQ dr X g
f′( ) exp -
)
This can be derived by observing that the relaxation force given by eq 22 can now be rewritten as
which we have derived in section(4.2) after taking the echo effect into account. 7.1. Short-Range Part of the Coulombic Relaxation Contribution. The evaluation of the short-range part of the Coulombic contribution to the relaxation coefficient ∆XSc /X, which contributes to the γ′ factor, is now straightforward. Introducing eq 73 into eq 44 gives
2 ) -cγf′( KA
∆XLc ∆Xh Λe + + X X Λ0
(78)
The coefficient γ′, which was factorized in eq 42 through application of the echo effect, can now be identified exactly with the coefficient factor γ of eq 2.
l
gi
dVS (82)
and finally
B FiP ) ei∆ B Xi ) -∑nlgl∫∇PU(iP,lS)h′li dVS
(83)
l
which shows that Ebeling’s original result can be applied to the present problem if the following substitutions are carried out: (i) c f cγ (This is due to the presence of gl ) γ in the factor of the ion concentration nl in eq 83 and to that of κ′ in eq 79 instead of κ in eq 80.; (ii) σ f R (The validity of the substitution of σ by R requires the following comment. As long as σ is smaller than values of R comparable to the Bjerrum distance q, the relative radial velocity component of two ions of opposite charges remains negligible as long as the distance r between the two ions remains smaller than R. This property must in fact be considered as one of the fundamental reasons why the Bjerrum concept is valid in dilute solutions, that is, in solutions where the probability for a third ion to come simultaneously within the same distance range of each of the first two ions remains small enough. This is why it is customary to use the value R ) q in most numerical evaluations.) We have neglected the term exp(κ′R)/(1 + κ′R) in eq 79. It gives only a small contribution in (cγ)3/2 in the final conductance equation. The exact result is
∆X(1)Lc ∆Xc,E(R,Cγ) exp(κ′R) ) X X 1 + κ′R
(84)
7.3. Hydrodynamic Relaxation Contribution. The only treatment of the hydrodynamic part of the relaxation term ∆Xh/X is based on the Fuoss approximation, also followed by Quint
Association Formulation for the Conductance Equations
J. Phys. Chem., Vol. 100, No. 4, 1996 1255
and Viallard25 and Chen,26 where many terms among those which depend explicitly on the tensor χ in eq 61 were neglected (cf. ref 7, section 5.1.1 for a more detailed discussion). Nevertheless, a simple analysis of that contribution shows that, denoting the result of the Fuoss-Onsager treatment as ∆Xh,FO(σ,C)/X, the new result now reads
∆Xh ∆Xh,FO(σ,Cγ) ) X X
ln
( )] 2κq c1/2
)
∆Xc(R,Cγ) ∆Xh(σ,Cγ) Λe(σ,Cγ) + + X X Λ0
which can be applied to any previous evaluation of the various functions involved in parentheses. We have already discussed in the literature8 how the terms can be estimated using the best functions available. The fundamental result is the formal derivation of eqs 2-5. Let us note that the two functions ∆Xh(σ,Cγ)/X and Λe(σ,Cγ)/ Λ0 retain their dependence on σ and not on R as formerly proposed. It is a matter of experimental fact that these two functions do not vary numerically so long as σ is smaller than q (cf. respectively refs 8 and 24 for numerical studies concerning these quantities). This should be considered as a posteriori evidence that the strong coupling approximation introduced in ref 8 is valid. As a consequence, within the application range given above it is valid (and simpler) to use R as the distance parameter that controls both terms. Acknowledgment. The author wants to express his gratitude to Professors M. A. Coplan, W. Ebeling, and H. L. Friedman for their interest, their patience in our long discussions, and their determined help and continuous encouragement along the elaboration of this contribution. Appendix
S ) RΛ0 + β R)
(87)
0.8204 × 106 (T)3/2
(88)
82.501 η(T)1/2
(89)
β)
E′ ) E′1Λ0 - 2E′2
( )]
[
ξ)
2q R
1 + x + x2/2 2(1 + x + x2/2 - ex)
0.9571 1.1187 0.1523 + + x x3 x2 -2 7.0572 0.5738 + ∆2(x) ) 3 + - 0.6461 x 3x x2 ∆1(x) )
(86)
(90)
E′1 )
2.9422 × 1012 T
(91)
E′2 )
0.43329 × 108 η(T)2
(92)
( )]
κR 1 31 + - 1.179 - ln 1/2 2 4ξ ξ c 2 κR 2E′2 + 0.76685 + ln 1/2 (93) 3ξ c
with
f1(x) )
We have reached the following result
)
E′2β κq J3/2(R) ) -2 1/2[4E′1Λ0∆1(ξ) + 2E′2∆2(ξ)] + ∆ (ξ) Λ0 3 c (94)
8. Conclusion
(
[
(
eξ 1 2 1 + + 2 - 0.036 + f1(ξ) + ξ ξ ξ
+ 2E′2 -
(85)
7.4. Electrophoretic Contribution. The evaluation of the electrophoretic conductances λie and λje are based on eq 25, iP and B FjQ which in turn requires the evaluation of B FjQ iP and thus indirectly involves the echo effect. This problem has already been discussed in some detail in ref 17. Inspection of eqs 60 in this reference shows that the earlier evaluation of the electrophoretic term denoted as Λe(σ,C) becomes Λe(σ,Cγ).
Λ ) γΛ0 1 +
[
J(R) ) 2E′1Λ0 -Ei(ξ) +
∆3(x) )
4 - 2.2194 3x
t x e Ei(x) ) ∫-∞ dt t
(95) (96) (97) (98) (99) (100)
References and Notes (1) Altenberger, A. R.; Friedman, H. L. J. Chem. Phys. 1983, 78, 4162. (2) Friedman, H. L.; Zhong, E. C.; Raineri, F. O. Transport and Mixing Rules in Electrolyte Solutions, Proceedings of the Reuniones de Invernio, Statistical Physics, Oaxtepec, Mexico, January 1989. (3) Kremp, D.; Ebeling, W.; Krienke, H.; Sa¨ndig, R. J. Stat. Phys. 1983, 35, 1. (4) Ebeling, W.; Grigo, M. J. Solution Chem. 1982, 11, 3. (5) Ebeling, W.; Rose, J. J. Solution Chem. 1981, 10, 9. (6) Fuoss, R. M.; Accascina, F. Electrolytic Conductance; WileyInterscience: New York, 1959. (7) Justice, J.-C. Conductance of Electrolyte Solutions. In ComprehensiVe Treatise of Electrochemistry; Conway, B. E., Bockris, B. O’M., Yeager, E., Ed.; Plenum Press: New York, 1983; Vol. 5, pp 223-337. (8) Ebeling, W.; Justice, J.-C. J. Solution Chem. 1990, 19, 945. (9) Bjerrum, N. Mat.-Fys. Medd.sK. Dan. Vidensk. Selsk. 1926, 7, 7. Or in Selected Papers; Einar Munksgaard Publisher: Copenhagen, 1949. (10) Debye, P.; Hu¨ckel, E. Phys. Z. 1923, 24, 185. Or in Collected Papers of Peter J. W. Debye; Wiley-Interscience: New York, 1954. (11) Justice, M.-C.; Justice, J.-C. J. Solution Chem. 1976, 5, 543; 1977, 6, 819. (12) Justice, M.-C.; Justice, J.-C. Discuss. Faraday Soc. 1978, 64, 265. (13) Rasaiah, J. C.; Friedman, H. L. J. Chem. Phys. 1968, 48, 2742. (14) Falkenhagen, H.; Ebeling, W.; Kraeft, W. D. In Ionic Interaction; Petrucci, S., Ed.; Academic Press: New York, 1971. (15) Onsager, L. Phys. Z. 1926, 27, 388. Onsager, L. Phys. Z. 1927, 28, 277. (16) Fuoss, R. M.; Onsager, L.; Skinner, J. F. J. Phys. Chem. 1965, 69, 2581. (17) Justice, J.-C. J. Solution Chem. 1978, 7, 859. (18) Justice, J.-C. J. Solution Chem. 1991, 20, 1017. (19) Friedman, H. L. J. Solution Chem. 1980, 9, 371. (20) Grigo, M.; Ebeling, W. J. Solution Chem. 1984, 13, 321. (21) Debye, P.; Falkenhagen, H. Phys. Z. 1928, 29, 401. Or The Colected Papers of Peter J. W. Debye; Interscience Publications, Inc.: New York, 1954. (22) Yamakava, H. J. Chem. Phys. 1970, 53, 436. Felderhof, U. Physika 1977, 89A, 373. (23) Kremp, D. Ann. Phys. (Leipzig) 1966, 17, 278. (24) Justice, J.-C.; Pe´rie´, J.; Pe´rie´, M. J. Solution Chem. 1980, 9, 583. (25) Quint, J.; Viallard, A. J. Solution Chem. 1978, 7, 137. (26) Chen, M. S. J. Solution Chem. 1978, 7, 675.
JP951284Q
