1038
Communicationsto the Editor
factor of 2s compared to those reported previously for low dose experiments. I t is believed that some of the observed differences may be, a t least in part, due to different methods of analyses. The percentages for the distribution of phenols in airand N2O-saturated systems are also reported in Table 11. Because it is not readily possible to estimate the degree of cross disproportionation relative to disproportionation between the same radicals, the ratios of the observed distribution of nitrophenols in N20-saturated systems without added oxidants cannot be related to the initial yields of corresponding radicals. I t is clear from our results that extreme caution must be exercised to interpret observed product distribution in terms of the yield of radical intermediates.
Acknowledgment. We thank our colleagues a t the Radiation Research Laboratories for their suggestions during the preparation Of this manuscript* Discussions with R*H* Schuler were particularly helpful.
References and Notes Supported In part by the U.S. Atomic Energy Commission. K. Bhatia, Anal. Chem., 45, 1344 (1973). K. Bhatia and R. H. Schuler, J. Phys. Chem., 77, 1356 (1973). K. Bhatia and R. H. Schuler, J. Phys. Chem., 77, 1888 (1973). M. A. Schuler, K. Bhatia, and R. H. Schuler, J. Phys. Chem., 78, 1063 (1974). (6) K. Bhatia, Radiet. Res., 59, 537 (1974). (7) K. Bhatia and R. H. Schuler, J. Phys. Chem., 78, 2335 (1974). ( 8 ) For a summary of previous work on this subject see J. H. Fendler and G. L. Gasowski, J. Org. Chem., 33, 1865 (1966). (9) M. K. Eberhardt and M. Yoshida, J. Phys. Chem., 77,589 (1973). (10) R. W. Matthews and D. F. Sangster, J. fhys. Chem., 71,4056 (1967). (11) M. K. Eberhardt, J. Phys. Chem., 78, 1795 (1974). (12) K.-D. Asmus, B. Cercek, M. Ebert, A. Henglein, and A. Wigger, Trans. Faraday Soc.,83, 2435 (1967). (13) M. Anbar, “Fundamental Processes In Radiation Chemistry”, P. Ausloos, Ed., Interscience, New York, N.Y., 1968. p 651. (14) M. Anbar and P. Neta, ht. J. Appl. Radiet. Isotopes, 18, 493 (1967). (15) T. I. Balkas, J. H. Fendler, and R. H. Schuler, J. Phys. Chem., 74, 4497 11970). (16) K. Eiben and R. W. Fessenden, J. Phys. Chem., 75, 1186 (1971). (17) K . 4 . Asmus, A. Wigger, and A. Henglein, Eer. Bunsenges. Phys. Chem., 70, 862 (1966). (18) J. H. Boyer, “The Chemistry of the Nitro and Nihoso Groups”, H. F e w , Ed., Interscience, New York, N.Y.. 1969, p 264. (19) A. Mantaka, D. G. Marketos. and G. Steln, J. Phys. Chem., 75, 3885 (1971). (20) L. M. Dorfman, I. A. Taub. and R . E. Buhler, J. Chem. Phys., 38, 3051 (1962). (1) (2) (3) (4) (5)
COMMUNICATIONS TO THE EDITOR
Comment on “The Debye-Bjerrum Treatment of Dilute Ionic Solutions”, by J. C. Justice’
Since f l I = fl,, this is derived from
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which is the mathematical statement of the physical fact that the radial component o f RELATIVE velocity (RCRV) vanishes when the center-to-center distance rll between two ions equals a. If a in (2) is replaced by another distance R, for example, R = q, then (2) clearly requires that the RCRV vanish at rll = q, because a component of velocity vanishes only at a rigid barrier normal to the direction of that component. However if the meaning of r is changed in the equation, a concommitant change must be made in the model; therefore Justice’s replacement of a in (2) by q requires a model in which charges are centered in spheres whose contact distance is q. In short, the Justice equation does not derive from the primitive model for which the contact distance is a, but rather for one with q as the contact distance. Justice repudiates this ridiculous model and asserts that the primitive model satisfies a boundary condition which he writes
Sir: In 1957, Fuoss and Onsager presented a conductance equation2 which was derived using as their model rigid charged spheres of diameter a in a continuum to represent the solution (the primitive model). Later: the integrations were repeated in order to obtain terms of order c3I2which had been neglected in the earlier derivation; the FuossHsia equation3 h = A(c; ho,KA, a ) was the result. In 1968, Justice4 replaced a by the Bjerrum distance q = e2/2DkT and used the equation A = A(c; Ao, K A ,q) in the analysis of conductance data.5 Fuoss6 protested that the Justice equation could be derived only if the original Fuoss-Onsager model were replaced by a model in which the ions were surrounded by rigid spheres of radius ql2. busticel then attempted to show that his equation was consistent with the primitive model. His arguments stem from his failure to understand the physical significance of one of the Fuoss-Onsager boundary conditions; he confused relative and absolute velocities. The boundary condition’ is
[(fllvl, - f , l v , l ) 4 r ==a0 The Journal of Physical Chemistry, Vol. 79, No. 10, 1975
(1)
Vll(vll- ~ ~ ~ ) =- 0r l ~ = ~
[(g,v,,
- g,lV,l)’rlrPR
=0
(2)
(3)
thereby seemingly justifying his “modification” of the Fuoss-Hsia equation. Since f,& = gll, (3) looks deceptively like (21, with a replaced by R. However the symbols for vec-
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
