938
NOTES
should also be reflected in the electrostatic behavior of ions. Undoubtedly, the dielectric constant and the viscosity of a solvent in the vicinity of the ion are considerably different from the macroscopic values, and thus the ion does not move, or form ion pairs, in a medium of such properties as described by the bulk parameters; what is puzzling to us is the fact that the experimental data on conductance can still be described satisfactorily well by the equations, obtained from the most rigorous theoretical treatment on electrolytic conductance, using the macroscopic values for the solvent properties.
We conclude from this study that LaFe(CN)6.4H20 is moderately associated in formamide. I n acetoneformamide mixtures the ion association is very well described by the Fuoss-Bjerrum equation. I n dioxane-formamide mixtures the dependence of ion association is strongly affected by the ion-solvent and possibly solvent-solvent interactions.
Acknowledgment. I wish t80thank the Department of Engineering, Princeton University, Princeton, N. J., for providing me the computer time for these calculations.
NOTES
Alkyl Radical Disproportionation by R. L. Thommarson McDonncll Douglas Astronautics Company- Western Division Santa Monica, California 90406 (Received March 3.5, 1969)
Alkyl radical disproportionation reactions
R.
+ R’.
4RH
+ R’(-H)
(1)
are observed when an alkyl radical larger than methyl is present in the reaction system. They are extremely efficient, proceeding with rate constants comparable to those for alkyl radical recombination. They have no apparent activation energy in contrast to the 8 f 3 kcal/mol expected of the hydrogen abstraction reactions of alkyl radicals. Moreover, the A factors are about a factor of 10 higher than the “typical” A factors for alkyl radical hydrogen abstraction reactions. The negligible activation energy js rationalized by the high exothermicity of disproportionation and the generally observed trend of decreasing activation energy with increasing reaction exothermicity. However, the high A factors have led to some controversy. I t has been argued that the high A factors and small temperature dependence of disproportionation rates suggest that the process is not closely related to ordinary radical Habstraction reactions. Radical recombination and disproportionation were envisaged as proceeding through a common transition state involving initial formation of the excited dimer RR’ which could decompose unimolecularly to give alkane and a1kene.l Benson2has pointed out that such a process is incompatible with experimental results and presented arguments T h e Journal of Physical Chemistiy
favoring the “head-to-tail” model, which is similar, though necessarily looser, than the transition state assumed for normal H-abstraction reactions. Recently, Johnston3 has offered the empirical bond energy-bond order method as a tool for calculating the activation energies and rate constants of elementary bimolecular H-transfer reactions without using adjustable parameters from kinetic data. This method has been highly successful when applied to a variety of H-transfer reaction^.^ The purpose of this paper is to examine alkyl radical disproportionation using a rather direct modification of the bond energy-bond order method.
Computational Procedure The generalized hydrogen abstraction reaction can be written
A-H
+ B = A...H*..B= A + €€-B nl
n2
(2)
where nl and n2 are the bond orders in the transition state of the breaking bond A . - H and the forming bond H . .E, respectively. The total bond order is assumed to be conserved throughout the reaction so that nl nz = 1. The potential energy for forming the transition state is postulated on a trial basis3 t o be equal to
-
+
(1) J. A. Kerr and A. F. Trotman-Dickenson, Progr. Reaction Kinefics, 1, 113 (1961); P. 5. Dixon, A. P. Sefani, and M. Sewarc, J. Amer. Chem. Soc., 85, 2851 (1963). (2) S.W. Benson, Advan. Photochem., 2, 1 (1964). (3) H. S. Johnston, “Gas Phase Reaction Rate Theory,” Ronald Press Company, New York, N. Y . ,1966. (4) (a) H. S. Johnston, Adzan. Cliem. Phys., 3, 131 (1961); (b) H. S. Johnston and C. Parr, J . Amer. Chem. Soc., 8 5 , 2544 (1963); ( c ) S. W. Mxyer, L. Schieler, and H. S. Johnston, J . Chem. Phys., 45, 385 (1966); (d) S. W. Mayer and L. Schieler, J . P h y s . Chem., 72,2628 (1968).
NOTES
939
the energy required to break the bond A-H to A. * * H of bond order nl, less the energy supplied by forming He * . B to order n2, plus a repulsion energy V , arising from the parallel spins on A and B. Considering alkyl radical disproportionation specifically, the bond order increases by one in the overall reaction owing to a-bond formatibn in the product molecule A, and it is this factor that renders disproportionation reactions highly exothermic. This can be accounted for by assuming that when bond A-H is broken to A. * .H of bond order nl incipient n-bond formation of order nz occim. Then the potential energy of formation of the transition state can be expressed as
V
=
constants of A and B in the transition state is smalL4b This should be particularly true of disproportionation reactions since the steric factors are ca. 0.1. The equations for the force constants of the triatomic complex have been given by Johnston (ref 3, pp 339344). The only change required by the modifications introduced here is in the force constant along the reaction path, F,, which is the second derivative evaluated at the point of maximum V*, where dV/dnl is zero
DA-H - DA-HnlP - D B - - H ~ ~ ’D,nZQ4- Vr (3)
where D is the bond dissociation energy, and p and y are the slopes of the log (dissociation energy) vs. log (bond order) lines for C-H and C-C bonds, respectively. The bond indices have the values p = 1.082 and q = 1.191. Since incipient n-bond formation is being explicitly accounted for, DA-H is taken to be the same as the corresponding C-H bond dissociation energy in the AH, molecule. The triplet repulsion is assumed to be given by an anti-Morse function,4b and is expressed in terms of bond orders as
Vr = D A - - F I [ ( ’-PAR /~)~
where y = 0.26 p and B = 1/2 exp(-QAR). F , is in dyn/cm and the dissociation energies are in cal/mol. I n terms of local bond properties, the rate constant expression takes the form (ref 3, p 224)
k
=
BeB,B,
(-”-)‘I2
[(%)2 A-H
F+
A-H
]
X
X
[I
+ (~/2)e-PAR(nln2)0~26p] (4)
where p is the A-B bond Morse constant and AR(& is the sum of the A-H and B-H bond equilibrium internuclear distances less the A-B bond equilibrium internuclear distance. To a first approximation the effect of incipient n-bond formation on the triplet repulsion energy manifests itself as DA-B = DA-BHnlq. This correction is negligible for the bond orders of the transition state, vide infra, and the expediency of neglecting it was adopted for the kinetic calculations. The calculated rate constants refer to the linear triatomic model. Some justification for using this model comes from (1) the observations2 that for most atom-molecule reactions the A factors fall in the range * 0*5 l./mol sec; ( 2 ) for radical-molecule reactions 1010.5 I./mol the A factors may be represented by 108-5*0.5 sec; and (3) for radical-radical disproportionation the rate constants (presumably equal to the A factors) are clustered about 109Jl./mol sec. This relative constancy of the preexponential factor for a given class of reactions has been attributed to the probability that most of the force constants for polyatomic A and B are not extensively altered in the transition state.3 Also, the estimation of bending force constants in the transition state is not dependable when A and B are not monatomic. Although this modification cannot be completely correct, it is believed that in many reactions the net correction arising from changes in bending force
10’0 exp( - V*/RT) l./mol sec
(6)
where R represents internuclear distances in A, F represents the stretching force constants in dyn/cm, Zi represents classical vibrational amplitudes (It = ( ~ T ~ T / F , ) in ” ~units ) of cm for the B-H stretch and radians for the A-H-B bend (denoted by &); a* is the imaginary frequency (cm-l) associated with the reacB , is tion coordinate, and W = (1 - F12/F~~F22)1’2. the ratio of quantum corrections ((1/2u)/sinh ( u / 2 ) ; u = hv/lcT) for the real vibrations in the transition state to those for the reactants, the tunnelling correction being omitted. Be represents the ratio of the electronic partition function of the transition state to the product of the electronic partition functions of the reactants. The value of Be is taken to be 1/4; the Table I: Bond Propertiesa De,
Bond
CHa-H Primary (C-CH) Secondary (C-H) Tertiary (C-H)
c-c
koal/mol
106.0 102.0 99.0 95.0 83.0
Vibrational wave number, om-’
Bond length,
3100
1.09 1.09 1.09 1.09 1 .
