J. Phys. Chem. 1983, 87, 4008-4012
4008
Normal-Mode Analysis of Diffusion in Multicomponent Electrolyte Solutions. 2. Associating Electrolytes Pierre Turq, + Loulslane Orcll, Jean Chevalet, Marlus Chemla, Laboratoire d'Electrochimie. Universiti Pierre et Marie Curie, ERA 3 10, 75005 Paris, France
and Reginald Mills Diffusion Research Unit, Research School of Physical Sciences, Australian National University, Canberra. Australia (Received: October 26, 1982; In final Form: May 13, 1983)
The diffusional and kinetic modes for a self-associatingelectrolyte are calculated, taking into account successive desolvation processes for the associated entities (pairs). The first result of this study concerns the coupling between the kinetic and electrostatic relaxation modes which it is found cannot be separated for unsymmetrical electrolytes. The second result is related to the fact that the diffusion coefficient of an associating electrolyte cannot be exmessed in terms of the diffusion coefficients of the different entitites (free ions and pairs) according only to their relative equilibrium proportions The local equilibrium condition will include the kinetic flow coming from the gradients of concentration.
Introduction The determination of transport properties in unsymmetrical electrolyte solutions has become of considerable interest in recent years because of the neutron experiments of Enderby' on NiC12. These experiments have shown the effect of the fundamental character of solution desolvation processes, especially in concentrated solutions, on the observable or macroscopic properties of a system. In another treatment,2the determination of pair formation and disassociation kinetics in electrolytes has been found to be closely correlated to the kinetics of desolvation of these entities. It has been well proven3 that, in most cases, the succession of kinetic steps is as follows: (a) formation of pairs from solvated unassociated ions, (b) desolvation (in one or more steps) of the pairs. The present work is devoted to the study of the influence of the desolvation kinetics of the associated entities on the diffusion coefficient of the bulk electrolyte. It follows a previous study4 on the influence of pair formation upon the diffusion coefficient of a symmetrical electrolyte. The present contribution considers unsymmetrical electrolytes such as NiC1, and polyelectrolytes. The pair formation and desolvation process can be described by the following successive steps where the k, are the rate constants for the transformation of a particle in the state i to the state j 4 2
,423
k2,
k32
Mm++ Ln- z=t Mm+(S2)Ln?Mm+(S)Ln-z=z k34 k4,
ML(m-n)+(1) (which is known as the Eigen mechanism3) and where Mm+ (species 5 concentration C,) is the metallic cation, Ln(species 6 concentration C,) is the ligand anion, Mm+(Sz)Ln(species 2 concentration C,) is a bisolvated complex pair, (1) (a) A. K. Soper, G. W. Neilson, J. E. Enderby, and R. A. Howe, J . Phys. C, 10, 1973 (1975); (b) R. A. Howe, W. S. Howells, and J. E. Enderby, ibid., 7, L l l l (1974). (2) M. Eigen and R. G. Wilkins, Adu. Chem. Ser., No.49, 55 (1965). (3) M. Eigen, Z. Phys. Chem. (Frankfurt am M a i n ) , 1, 176 (1954). (4)P. Turq, L. Orcil, M. Chemla, and J. Barthel, Ber. Runsenges. Phys. Chem., 85, 535 (1981).
Mm+(S)Ln-(species 3 concentration C,) is a monosolvated complex, and Mm+Ln-(species 4 concentration C,) is a desolvated complex ion. We assume that the uncharged solvent species is in large excess and that the concentration of any species can be separated into a (static) equilibrium value plus a heterogeneity which decays during the transport process: ci
=
c: + 6Ci
(2)
The present work will be devoted to the formulation and the solution of the equations for the decay of these heterogeneities. The next section will present the general equations and the methods of solution. The following one will concern the significance of the different modes, so obtained.
Kinetic and Transport Equations In the succession of reactions 1, the hydrodynamic continuity equation, including kinetic (or source) terms, is dCi/dt
+
=
ui
(i = 5, 6, 2, 3, 4)
(3)
where Ciis the concentratip of the i-th species, ciis the kinetic source term, and Ji is the flow expressed as a function of the diffusion coefficients Di, the charge Zi, and the bulk concentration C: of the i-th species. In general, eq 3 involves nonideality terms. These can be simply expressed, at least in terms of the limiting law."7 In the case of self-associating electrolytes, the most interesting phenomena occur from the association process and not from the long-range electrostatic interactions giving the ordinary nonideal contributions. We will therefore neglect here the effect of direct Coulomb interactions and consider only the association terms. It should be noticed that electrostatic association, such as the Bjerrum type, can be described as a particular case of the (5) R. H. Stokes, J. Chem. Soc., Faraday Trans. I , 73, 1140 (1977). (6) H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions", 3rd ed., Reinhold, New York, 1958. (7) H. Falkenhagen, 'Theorie der Elektrolyte", 5th ed., Hirzel, Leipzig, East Germany, 1971.
0022-3654/83/2087-4008$01.50/00 1983 American Chemical Society
The Journal of Physical Chemistry, Voi. 87, No. 20, 1983
Diffusion in Multicomponent Electrolyte Solutions
4009
unnecessarily complicate the already fairly complex equations. By use of the electroneutrality condition for the equilibrium concentrations
ccioz, =0 i
(6)
there remain only the fluctuating terms
9.2 = (4?re/t0)CZi6Ci i
X
Flgure 1. Diffusion coefficient D,, for a two-pair system. The upper plateau is D,, = 150 in arbitrary units. The intermediate vaiue is D = 100, and the lowest plain D, = 75. x corresponds to log K , ( C , C t ) varying from -3 to +4. y corresponds to log K , varying from -6 to 4-5.
a
+
(7)
and the set of kinetic equations given in eq 8. When one a c , / d t + a.32 = k&5C6 - k2lC2 - k23C2 + k&3
+ 9.3, = k 2 3 ~ 2- k& - k3,C3 + k,,C4 a C , / d t + 9.3, = k3,C3 - k4$* a c , / a t + a.3, = - k 1 2 c 5 c 6+ k Z 1 c 2 dC~/a+ t 9.~76= -k,&,C, + k2lCz
ac3/at
sets k52 = k12C2and k62 = k12C50,the system 8, for the first order in 6Ci, becomes a s c 2 / a t + 9.3, = k526C5 + k626C6 - k216C2 - k236C2 + k326C3
K3 Flgure 2. Dlffusion coefficient for a three-pair system (twosuccessive desolvations for the pairs). D, = 200 in arbitrary units, D, = 50, D, = 100, D, = 150, for the particular choice of K , = lo-', as a function of (a) K , C : , the constant of formation for the first-pair (completely solvated) multiplied by the concentrationof the free ions, and (b) K , , the constant of formation of the completely desolvated pair. Both axes K , C e Oand K , are on a logarithmic scale with log K , C : varying from -6 to +2 and log K , varying from -9 to +7.5.
+ a J 3 = k236C2 - k326C3 - k346C3 + k436C4 a 6 c 4 / a t + 9.3, = k3,6c3 - k 4 3 6 ~ 4 a s c , / a t + 9.3, = -k526c5 - k626C6 + kZ16c2 a 6 c 6 / a t + 9.36 = k526C5 - k626C6 + k216c2 (9)
a6C,/dt
general association process with suitable values of the askociation constants. Ji becomes = -Di9Ci z i e ( D i / k T ) z C i (4)
si
By taking the space coordinates Fourier transform
+
Ci(q) = 1 d 3 r e i T i ( r )
where the electrical field 2 can be expressed after Stephenavgby the Poisson equation
9.2 = ( 4 r e / c Oi ) E ~ i ~ i
(8)
(10)
and the time Laplace transform
(5)
Ci(q,s) = l m 0 d te-stCi(q,t)
In the derivation of eq 4 cross-term diffusion terms between the ions have been neglected. The activity coefficients of the ions have been assumed also to be independent of concentration. At a future time these effects could be included in the derivations but at this stage would
(11)
System 9 becomes (12)
{6Ci(q,s))(MJ = (Jci(qtt=o))
where (MI is a matrix given by (13) z 4
v -
2 2 2
z
Zl
v4>
v -
- k,,
s
+ 9'0, +
k,
z,
v -
,
z 4
z4
v -
z,
z
Z 6
+
z 4
4 2 4
v -L
v4
-
k,,
s
+ q2D,+
z v
hZ6
+
k,,
us
+
k,,
z v
s
izS
+
k,,
+ q2D,+
v6
+ k,,
4010
Turq et al.
The Journal of Physical Chemistry, Vol. 87,No. 20, 1983
b = bo + blq2 + b2q4
with = q,'D,
The normal modes are the roots in s of eq 15. det (MI = 0
c = c,q' (15)
This equation is generally of fifth order and, of the s corresponding roots, one can be described as the diffusional mode. It can be obtained simply by setting s = x q 2 and taking the limit of x for q 0. The four other modes are relaxation modes of the form s = so a($) with a finite nonzero limit for q 0. These four modes come from the three chemical reactions (kinetic modes) plus the electroneutrality condition (electrostatic or Debye mode). In fact, we shall see that for unsymmetrical electrolytes the processes of relaxation to chemical equilibrium and to electrical equilibrium are always coupled and cannot be simply described as kinetic or Debye modes. This is a consequence of the fact that the pairs are now charged and contribute to the charge balance of the solution. The main aim of this work is to obtain the diffusional modes. The general expression with two successive desolvation processes will be treated later. We will now introduce the two simpler cases, i.e., with either one or zero desolvation steps.
-
(20)
When only the first-order terms in q2 are kept, there remains only
+ c1q2 = 0
(21)
-Dap = x = -cl/bo
(22)
boxq'
+
-+
+ c2q4 + c3q6
where D,, is the diffusion coefficient of the ionic system. By introducing the transference number ti =
Ui/CVj
(23)
I
for each species, and the equilibrium constant
K , = kldk21
(24
for the formation of the pairs, we get the following ex pression for D,,:
Three-Component System (Desolvation) This case was treated for a symmetrical electrolyte in a previous papere4 The general equation for the corresponding normal modes is simply derived from (13) by taking the appropriate minor involving only species 2, 5 , and 6. The two relaxation modes are given by I
For 2' # 0 (unsymmetrical electrolytes) these two roots cannot be separated into an electrostatic relaxation frequency (involving only the uI's) and a kinetic relaxation frequency (involving only the k,'s). For a symmetrical electrolyte sR is simply the kinetic relaxation mode s R + = sk =
-(kzl
+ kj, + k62)
(17)
+ vg)
(18)
and sR-the Debye mode SR-
= SD =
The diffusional mode corresponds to a solution of the form s = xq'. The development of eq 15 leads in this case t o an equation in s3 s3 + as2 + bs + c = 0 (19)
This equation involves the equilibrium constant K , and not the separate values of the rate constants. It must be noticed however that the way in which this constant appears is not as normally expected from the law of mass action. The diffusion coefficient of the bulk electrolyte cannot be expressed in terms of the diffusion coefficients of the different entities (free ions and pairs) according only to their relative equilibrium proportions. The local equilibrium condition will include in any case the kinetic flow coming from the local gradients of c~ncentration.~ In the case of no association, eq 25 reduces to the well-known Nernst-Hartley result5 DNH
a =
a0
+ a,q'
(8) M. J. Stephen, J. Chem. Phys., 55, 3878 (1971). (9) B. J. Berne and R. Pecora, "Dynamic Light Scattering". Wiley, New York, 1976.
t5D6
(26)
+ t6D5
In the case of a symmetrical electrolyte eq 25 reduces to the previously obtained value: 1 Dap
with
=
= DNH 1
+0
+ K,(CSo + cfjO)
2
K,(CeO+
c,jo)
1 + K,(C50
+ CO, ) (27)
In this expression, clearly the apparent diffusion coefficient is a combination of the Nernst-Hartley diffusion coefficient and of the diffusion coefficient of the pairs, by means
The Journal of Physical Chemistry, Vol. 87, No. 20, 1983 4011
Diffusion in Multicomponent Electrolyte Solutions
of a function of the adimensional variable Kl(C,O + C,O) = 2KlCF,where CF is the concentration of the free ions.
with
K3 = k34/k43 Four-Component System (One Desolvation Process or Two Pairs) The three relaxation modes (kinetic and electrostatic) are coupled and can be obtained by the solution of a third-degree equation which is analytical according to Cardan's meth0d.l' The diffusional mode involves, as predicted, the diffusion coefficient of the desolvated pair D,, which can be assumed to be greater than D2. Here also, it is possible to express D,, in terms of the association constants of the pairs
K i = kiz/kzi
K2 = k23/k32
(28)
The complete expression for an unsymmetrical electrolyte is given in the Appendix. This expression is, as before, not reducible to a simple law of mass action analysis involving only the proportions of the different entities. Its asymptotic behavior leads to the expected values (NernstHartley). It must be noticed that the association proportions appear not only in terms of the association constants but also indirectly in the values of the transference numbers ti. This last behavior is typical of an unsymmetrical electrolyte and disappears for a symmetrical one where t z = t 3 = 0 and where t , and t 6 are independent of the concentration. This property occurs also for higher order neutral pairs in unsymmetrical electrolytes. Expression A1 takes a simpler form for symmetrical electrolytes: D,, = [D5t6 + D6t5 + DzKl(C5' + C6') + D,K,K2(C,' + C,O)l/[l + Ki(C5' + Cs')(l + Kz)1 (29) This expression is, as required, a combination of DNH, D,, and D3. The coefficients of this combination involve the two adimensional variables K1(C2 + c,') = 2 K l C ~(CF is the concentration of the free ions) and K2. In order to explore the variations of D,, with the association parameters, we have assumed arbitrary values of the diffusion coefficients D N H , D,, and D3 and studied the variation of D,, with K 2 and 2K1CFin a wide range for these parameters. The results are summarized in a three-dimensional diagram in Figure 1.
Five-Component System (Two Desolvation Processes or Three Pairs) The four relaxation modes of this system (kinetic and electrostatic coupled) can be evaluated as the roots of a fourth-degree equation taking the limit q2 0 in eq 15. (A fifth root is obviously in this case s = 0, corresponding to the diffusional mode which will be examined now). The general expression for D, of such an unsymmetrical electrolyte is very intricate and cannot be evaluated because it has too many parameters many of which are physically unavailable. It takes a much simpler form for a symmetrical electrolyte:
-
(10) I. N. Bronstein and K. A. Semendiaev, Eds., "Aide MBmoire de MathBmatiques", Eyrolles, Paris, 1976.
This expression is a combination of DNH,D2,D3,and D4 involving the adimensional variables Ki(C5' + C2), K,, and K3. It cannot be represented in a three-dimensional diagram, and for fixed values of Dm, D4,D3,and D, we have chosen to represent the variation of D,, as a function of K l ( C 2 + C,') and K 3 for a fixed value of K,. In this case D,, appears as a function of the form
which presents three plateau values (Figure 2 )
If the upper value and the intermediate value are "pure" one-species diffusion processes (respectively free ions and totally desolvated pairs), the lower plateau is a combination of the two-pair diffusion coefficients for the nondesolvated and monodesolvated pair complexes, as required by the limitation of a three-dimensional representation. In the limiting domains (plateaus) are independent of the kinetics, the transition between the different ranges occurs in a completely different manner as for thermodynamically governed systems. It has been shown4 that, in kinetically controlled systems, diffusion is faster than under equilibrium conditions.
Conclusion The present treatment allows the analysis of experimental diffusion measurements in self-associating systems with the assumption of ideality for the different components. This limitation is not too drastic in regard to the electrophoretic corrections (effects of hydrodynamic interactions) but clearly restricts the validity of the present treatment to systems where the excess properties coming from association processes are more important than the activity terms coming from long-range Coulomb interactions. It will not be too difficult to include these effects in a valid way for symmetrical electrolytes, and a t least in a semiempirical way for unsymmetrical ones. The present status of the work shows clearly the effect of the kinetic regime a t the transition between the different domains. Another interesting result is that, in unsymmetrical electrolytes, the kinetic relaxation times cannot be separated from the electrostatic relaxation process (Debye). In particular, for polyelectrolytes, any attempt to measure the Debye time will give also the kinetic parameters for the fluctuation of the charge on the polyion, coming from condensation-decondensation processes of the counterions. Light-scattering experiments (photon-beating techniques) could give information on this matter."
(11) H. Magdelenat, P. Turq, P. Tivant, M. Chemla, R. Menez, and M. Drifford, J. Chem. Educ., 55, 12 (1978).
4012
J. Phys.
Chem. 1983,87,4012-4014
Relative Contributions of Inner- and Outer-Shell Reorganization in Electron-Transfer Reactions in Solution Shahed U. M. Khan and John O'M. Bockrls" Department of Chemistry, Texas ABM University, College Station, Texas 77843 (Received: March 30, 1983)
Some recently published data on photoemission from solution has been used to compute the relative contributions of the inner to the outer solvation shell on the activation energy of electron-transfer reactions. One finds that, using the experimental values of force constants, the inner-shell contribution is about 3.2 times the outer-shell one, for the results of five redox systems.
Recently, the free energy of reorganization for the electron-transfer reaction in solution has been determined by using photoemission from solution experiments.' In this article we analyze the experimental results's2 of the free energy of reorganization with the object of determining the relative contribution or"the inner- to the outer-shell activation. From photoemission experiments' the reorganization energy, RPhoto(exptl)has been found by means of calculations via a Born-Haber cycle for the reactions of the type
If one accepts that this energy is a combination of one can obtain RE::: from the values of Rphoto(exptl)by substituting for R-,(calcd) from the theoretical expression of free energy of activation AG?&,,,(calcd) (for one reactant i0np-5 (1) P. Delahay, V. Burg, and A. Dziedzic, Chem. Phys. Lett., 79, 157 (1979). ( 2 ) P. Delahay, Chem. Phys. Lett., 87, 607 (1982).
= l/iinner(calcd)
(3)
where n is the number of water or ligand molecules oriented orthogonally to an ion in the first shell when the ion is in solution, f, and f, are respectively the force constants for the ion-solvent vibration in the oxidized and reduced ions, and Aq is the equilibrium ion-solvent bond length difference of the oxidized and reduced ions. (There is a degree of artificiality in the implied breakdown of the model into two sharply defined regions: an inner region which contains actively vibrating molecules and is treated molecularly and an outer one which is treated as a continuum. However, this is the formalism of the most used m~del.~) However, for the exchange reactions involving two reactant ions the corresponding inner-sphere reorganiza(3) P. George and J. S. Griffith in "The Enzymes",Val. 1,P. D. Boyer, H. Lardy, and K. Myrback, Eds., Academic Press, New York, 1959, Chapter-8, p 347. (4) R. A. Marcus, J. Phys. Chem., 67, 843 (1963). (5) R. A. Marcus, J. Chem. Phys., 43, 679 (1965).
0022-3654/83/2087-40 12$01.50/0 0 1983 American Chemical Society
