Communications to the Editor
1039
tor velocities in (3) have a different meaning from those in (2)! Justice defines V i j as the mean velocity vector of a j ion, located with respect to an i ion by the vector rij, so eq 3 states that the difference of the scalar products of the mean velocities vij and vj; with r vanish at r = R. In the Fuoss-Onsager boundary condition (2), (vij - vji) is the relative velocity of ions i and j , and (vij - vji).r is therefore the radial component of their relative velocity; note that (vij - vi;) is a vector difference. Equation 3 therefore states that the difference of averages dotted into r vanishes at r = R, while ( 2 ) states that the difference dotted into r vanishes a t a specified value of r. Obviously (2) and (3) are ’ completely different statements regarding the behavior of the model. Furthermore, we shall show that (3), although mathematically correct, is a trivial statement, and worse than useless as a boundary condition. First, regarding triviality: consider the scalar product (V-rij) of any given vector with rjj (V-rij) = Iqlrijl
COS cp
(4)
where q is the angle between V and rij. Recall that rij is the vector which locates the j ion with respect to the i ion; since there is no directional correlation between the position of the two ions, the average of cos (o is zero and therefore (V-rjj) = 0
(5)
for all values of lrijl 1 a. If we set V = vij and lrijl = R, we have Justice’s equation [~ij(rij).rij]~O
(64
[vji(rji).rji]~= O
(6b)
and congruently
Justice’s eq 3 is obtained by adding (6b) to (6a), after multiplication by g , = g,, and using the fact that r,j = -r,i. Clearly, (3) says nothing about relative velocities, with which the Fuoss-Onsager boundary condition is concerned; it merely states that zero minus zero is zero. Second, regarding the possible role of (3) as a boundary condition: construct from (6) the linear combination gil(~Lprij)r+ Agji(Vji.rji)+ = 0
(7)
where A is an arbitrary constant and r and r! are any values of lrijl = lrjil. Equation 3 follows from (7) by setting A = -1 and r = r’ = R; there is, however, no physical justification for this choice. It is merely an efficient (albeit illegal) means of obtaining an equation which superficially resembles ( 2 ) . Or, looking at the situation from the mathematical point of view, (6a) and (6b), which are independent equations by Kohlrausch’s law of independent mobilities, together with the three electrostatic boundary conditions,s give five equations to determine the four constants of integration which appear in the Fuoss-Hsia development, an awkward dilemma indeed. Consequently Justice’s equations cannot rigorously lead to eq 3.5 of ref 2 with a replaced by g , and lacking this equation, the equation in which Justice would replace a by R = q cannot be derived. It has already been shown that condition ( 2 ) with a replaced by q demands a model in which ions are surrounded by rigid spheres of radius 912. In this sequence of comments on conductance equations, emphasis has been on mathematical detail, perhaps to such an extent that one fundamental feature has been obscured:
namely, that theory describes the behavior of a model and not that of a real physical system. The concept of relative velocity has, of course, no meaning for a pair of real ions rattling around in a swarm of moving solvent molecules. For two oppositely charged spheres moving in a continuum, however, velocities can be precisely described by the classical equations of motion (in which Brownian motion is replaced by the virtual force of the Einstein kT grad c term). When a model anion and cation drift together, they may stick for a while due to electrostatic attraction; during the dwell time of the pair, their relative radial motion must be zero, while the motion of the center of gravity of the pair is determined by the laws of conservation of energy and momentum. Formation of a pair of real ions is the consequence of the diffusion of the two ions into adjacent sites in the particulate solvent; details of this process are irrelevant for the calculation of the long-range relaxation and electrophoretic e f f e c t ~ . ~ J ~ We therefore iterate previous conclusions.6 The Justice equation is not derivable from the primitive model, and the model for which the equation can be derived using ( 2 ) with a replaced by q is physically unrealistic. References and Notes (1) J. C. Justice, J. Phys. Chem., 79, 454 (1975). (2) R. M. Fuoss and L. Onsager, J. Phys. Chem., 61,668 (1957). (3) R. M. Fuoss and K. L. Hsia, Proc. Natl. Acad. Sci. U.S., 57, 1550; 58,1818 (1967). (4) J. C. Justice, J. Chim. Phys., 65, 353 (1968). (5) J. C. Justice, Nectrochim. Acta, 16, 701 (1971). (6) R. M. Fuoss, J. Phys. Chem., 78, 1383 (1974). (7) Reference 2, eq 3.4; ref 6, eq 7. (8) Reference 2, eq 3.1, 3.2, 3.3; ref 6, eq 4-6. (9) R. M. Fuoss, Proc. Nati. Acad. Sci. U.S., 71, 4491 (1974). (IO) R. M. Fuoss. J. Phys. Chem., 79,525 (1975).
Sterling Chemistry Laboratory Yale University New Haven, Connecticut 06520
Raymond M. Fuoss
Received September 7, 1974
Reply to the Comment by Raymond M. Fuoss on “The Debye-Bjerrum Treatment of Dilute Ionic Solutions” Sir: The correct interpretation of the physical meaning of the velocity vectors vidQ and v ~ Qused ~ ~ by , various authors2-6 in the theory of electrolytes c o n d ~ c t a n c ecan , ~ be achieved only by coming back to the definition of these quantities which is given by
v~$Q=
+ wj(ejX - ejVg$ipQ -
U~PQ
ejVQ$jQ’Q- kTVg In fi$Q)
(1)
This obviously is not the actual velocity vector of an ion j at Q but a time-average8 velocity vector of ions of type j at Q when an ion of type i is present at P. This fact is plainly recognized by O n ~ a g e r and , ~ Falkenhagen.5 Consequently the boundary condition used by Fuoss and Onsager
- fjQiP.vjQiP).r= 0 at r = a
(fipjQ.vi$’Q
(2)
which is strictly equivalent to uipjQ
= vjg,iP at r = a
(3)
The Jownal of Physical Chemistry, Vol. 79,No. IO, 1975
Communications to the Editor
1040
where the subscript r means we are dealing only with the radial components of the vectors, cannot have the meaning claimed by Fuoss. I t cannot be applied to the actuallo collision of two specified ions given the time-average nature of the quantity involved. I t is easy to showll from eq 1that (4)
(13) Fuoss admits in his comment that if the involved velocities were timeaverage quantities, then (5)or (5’)would be true.
Laboratoire d’Electrochimie Eit. F University of Paris VI 75230 Paris Cedex 05, France
Jean-Claude Justice
Received December 27, 1974
for any value of r, so that the only possibility for (3) is now u&Pr . jQ = 0
1
or
(5)
atr=a
Thus, all our conclusions, formerly published,12 are exact and the present controversy s01ved.l~Moreover, condition (2), when applied t o definition (l),can be replaced by condition ( 5 ) or (5’). Since condition (5’) is derivable alone from the combination of condition (5) with eq 1, one of the two last conditions is sufficient. This gives the solution of the “awkward dilemna” raised by Fuoss. In conclusion, one must keep in mind that the vij vectors are not only average velocity vectors but also perturbation terms. As average velocity vectors their radial component may vanish when the particles encounter an infinite repulsion potential surface. As perturbation terms they can become negligible when the particles enter an area where predominates an attraction energy strong compared with the external perturbation cause. The initial condition proposed by Fuoss belongs to the first category. When used to fit into the Bjerrum model it belongs to the second category.
References and Notes (1) R. M. Fuoss, J. Phys. Chem., preceding paper in this issue. (2) E. Pitts, Proc. R. Soc., 217, 43 (1953). (3) R . M. Fuoss and L. Onsager, J. Phys. Chem., 81,668 (1957). (4) L. Onsager and S. K. Kim, J. Phys. Chem., 61,215 (1957). (5)H. Falkenhagen, W. Ebeling, and W. D. Kraeft, “Mass Transport Properties of Ionized Dilute Solutions”, in “Ionic interactions”, Vol. I, s. Petrucci, Ed., Academic Press, New York, N.Y. 16) T. J. MurDhv and E. G. D. Cohen. J. Chem. Phvs.. 53. 2173 (1970). These veciors are of paramount importance since, together with the distribution functions fiAo = f/o‘p. they define the continuity equation which, is at the heart of the conductance theory.
-a f@/a
t = V ofi$ovj,dO
+V
p$oip~iQip=
-a $oip/a t
In eq 1, all terms are already time average quantities: for #!Po, I),%~, and fi$O. (cf. an excellent demonstration of this in Fuoss monography ): uipQwhich is the hydrodamic velocity vector of the medium at 0 is also an average since it is derived later by use of the Navier-Stokes equation. As for the term containing the external field strength X it may be considered as constant during the time T of averaging. (If the field is constant then there is no limit to T , if not T must be small compared to the period of the alternative field but great compared to collision time in the solution for the continuity eauation of conductance to hold.) (9) R. M. Fuoss and F. Accasdina, “Electrolytic Conductance”, Interscience, New York, N.Y., pp 117 and 118. (10) It must be noted that if the actual collision of two ions in a fluid had to be described, not only the boundary condition of Fuoss should be used with another definition of the velocity vectors and with an other continuity equation but, more important, this boundary condition would be totally irrealistic for hard-spheres collisions because in that case the radial relative velocities never vanish. They undergo a discontinuity and the nature of the collision should be specified differently. As it Is formulated, it could only be applied to nonbouncing collisions (as the smooth landing of a plane, for instance (everlasting contact)). (11) This can be done in two ways. (1) The time-average velocity vectors vanish at equilibrium (X = 0) as is recognized by Fuoss3 so that they represent a perturbation due to the external field. These perturbations on ions of opposite charges ( i # 1) must be opposite In directlon. This was the line of demonstration adopted in the paper12 commented on by Fuoss.’ (2) The same conclusion can of course be reached by discussing the consequences, on each term of eq 1, of inverting the elements of each couple (i, 1) and (P, Q). All the necessary material will be found in ref 3 or 9. (12) J. C. Justice, J. Phys. Chem., 79, 454 (1975). The Journal of Physical Chemistry, Vol. 79, No. 10, 1975
Carbanion Solvation in n-Electron Donor, Aprotic, Low Dielectric Constant Solvents Publication costs assisted by the Universidadde Bilbao
Sir: Aprotic, n-electron donor solvents (n-DS) display excellent cation solvating properties, an effect obviously due to the lone electron pairs of their oxygen or nitrogen at0ms.l On the other hand, any anion-n-DS molecule interaction seems on the same grounds highly improbable. Many experimental results give support to this assumption. For example, recent studies of the conductivity of inorganic salts in hexamethylphosphoramide show that this strong n-donor solvent is unable to solvate the weakly basic anions c104-, N&-, Br-, and pycrate-.2 Experiments by Slates and Szwarc3 also point in this direction. These researchers found that the diffusion coefficients of the radical anions of aromatic hydrocarbons in tetrahydrofuran (THF) as derived from ionic mobilities are very similar to those of the parent hydrocarbons. The conclusions reached from the cited studies are in accordance with kinetic results on the anionic polymerization of styrene in n-electron donor, aprotic, low dielectric constant solvents. For example, Parry et al. found that k(-), the propagation rate constant of the “free” polystyryl anion in tetrahydropyran (THP) (i.e., the rate constant of the elemental reaction: “free” polystyryl anion + styrene monomer in THP) was, within experimental error, independent of the counterion? Table I collects these results together with data from Hirohara et al.5 and from Bohm and Schulz6 On the other hand, kinetic experiments on sodium polystyryl in a series of ethereal solvents (i.e., same counterion, different medium) have shown that the “free” carbanion propagation rate constant is solvent independent (see Figure 1).All these results are from the same laboratory (Institut fur physikalische Chemie, Universitat Mainz, W. Germany) in order to be sure of the self-consistency of the data. The reported conductometric and kinetic results seem to confirm the a priori assumption that no detectable carbanion-n-DS molecule interaction exists. However, recent experiments by Ise et al. on the electric field effect upon the anionic polymerization of styrene in THP5 and 2-methyltetrahydrofuran (2-MTHF)l’ have raised the question of the carbanion solvation by n-electron donor solvents, particularly ethereal solvents. These researchers found an acceleration of the propagation reaction, i.e., an increase in k,, the overall propagation rate constant, on increasing the electric field strength up to
