1518
NOTES
found analytically. However, the kinetics would be the same if we used zwitterions in the mechanism. Paris2 presented a completely different mechanism which fit his chemiluminescent kinetics. His work on TMAE2+in strong alkali suggested carbene radicals as intermediates. We consider the carbene mechanism unlikely on the following grounds. First, Urry and Sheeto's results in alkali media indicate that carbenes are not involved. Second, Carpenter2s was unable to find unsymmetric tetraaminoethylenes when a mixture of two different tetraaminoethylene dications reacted with alkali. Finally, Paris has not shown the existence of any of his reaction intermediates and must
use a large number of mechanistic assumptions in order to achieve a kinetic relationship which does not fit the present experimental data.
Acknowledgments. Many discussions of this work with Professor W. H. Urry and Dr. R. H. Knipe are gratefully acknowledged. Mrs. Edith Kirk performed much of the experimental work. Dr. C. H. Shomate performed the calorimetric work, and E. M. Bens performed the gas chromatographic analyses. This research was supported by the Bureau of Naval Weapons and the Army Munitions Command. (26) W. R. Carpenter, unpublished data.
NOTES
The rate expressions are
The Relative Importance of Homogeneous
R,'
and Heterogeneous Reactions'
Aerospace Corporation, El Segundo, California (Received October 7, 1966)
I n many problems of gas-phase kinetics, both gasphase and wall reactions play a role. I n order to understand the detailed chemistry, it is necessary to know the relative importance of the homogeneous and heterogeneous reactions. If both reactions are first order in some species C, then a crude approximation for the ratio of the gas-phase reaction R , to the wall reaction R , is N
k,Ro2/4D (for an infinite cylinder) (la)
R,/R,
-
k,Ro2/9D (for a sphere)
(lb)
where k, is the first-order homogeneous rate constant, D is the diffusion coefficient, Ro is the radius of the vessel, and Ro/2 and Ro/3 are, respectively, the ratio of volumeto-surface area for infinite cylindrical and spherical reaction vessels. Although eq 1 is useful for crude approximations, it may be considerably in error for many problems. Therefore, for more accurate results, it is necessary to derive and tabulate the correct expressions. The Journal of Phyacal Chemistry
k,C(r}
(2d
R,' = k,C{Ro) (2b) where the primes on R,' and R,' indicate local reaction rates, C ( r ] is the concentration of C a t some radial distance r, and C( Ro ] is the concentration of C at the
by J. L. Hudson and Julian Heicklen
R,/R,
=
wall. The wall reaction rate constant k , has the dimensions of velocity. The applicable diffusion expression is
0 = DV2C
+ R' - k,C
(3)
and is subject to the boundary conditions
-D dC/dr C
=
k,C
= finite
(for r = Ro)
(for r
=
0)
where R' is the uniform homogeneous rate of production of C throughout the vessel. In order to solve the diffusion equation, it is convenient to work with dimensionless quantities. Therefore, we define
K,
k,Ro2/D
(4%)
K, kwRo/D (4b) Direct evaluation of eq 3 leads to expressions for C ( T ): for an infinite cylinder _ _ _ ~ ~
(1) This work was supported by the U. S. Air Force under Contract No. AF 04(695)-669. The authors wish to thank Mrs. V. M. Armstrong for her assistance with the manuscript.
NOTES
1519
Table I: Parameters for R,/R, K, fc
fa
0.01
0.04
0.09
0.25
1.0
4.0
9.0
0.00125 0.0007
O.OO50 0.00266
0.0113 0.00598
0.0309 0.0165
0.1201 0.0648
0.4331 0.2407
0.8519 0.4889
D c I T J-- - 1 R ' R ~ ~K ,
+
and for a sphere
R'Ro2
1
-
K, ( K , / K , ) sinh { K , ' / ' ~ / R O/ (}K v ' / * ~ l R ~ ) (5b) ( K , - 1) sinh { KV1/']/KV1'' cosh { Kvl/'}
+
where Io(XI and 11{X } are modified Bessel functions. The over-all reaction rates are found from the expressions: for an infinite cylinder
Ez,
=
R,
2 s L k v l R a CT{) rdr
=
2aLkwRoC{Ro}
(Ob)
where L is the length of the cylinder; for a sphere
R,
RQ = 4 r k , L C{r)r2dr
(6c)
R , = 4rRo2kwC{Ro} (64 Thus the ratios R , / R , become : for an infinite cylinder R,/R,
=
( K v / 2 K w ) fo
+
(74
=
(Kv/3Kw)
+ f.
(7b)
and for a sphere
Rv/Rw where
= [ K , ' ~ ' l o ( K v ' ~ ' } / 2 L ( K], '-~ ' 1} [ ( K v / 3 ) / ( K v ' /coth ' { KV1"]- 1) 1 - 1
fc
f.
E
k , = S/4
(K,,'K,)~o{K,'~~~/Ro} (54 K , ' ~ ~ I ~ { K , ' K~ ,~I}~ { K , ~ ~ ~ }
(8a) (8b)
The functions f o and f. depend on K , only, and they are tabulated in Table I for interesting values of K , . From the table, accurate values of R , / R , can be computed easily from eq 7a and 7b. One case of particular interest to gas kineticists is where C is an energetically excited species that can be deactivated on every collision, either in the gas phase or on the wall. For this case, k , becomes the collision frequency 2 and k , is comparable to one-fourth the mean velocity 5. In particular
IC,
=
Z = 4pu2(mT/m)"'
(94
D
=
(~T/2am)'/'
3(~T)'/'/8pu~(rm)'~'
=
25
1.798 1.083
(9b) (9C)Z
where p is the total particle density, u is the mean molecular diameter, K is the Boltzmann constant, T is the absolute temperature, and m is the molecular mass. Then
K, K,
kvRo2/D= 32rRo2p2u4/3 E
k,Ro/D
=
(104
4 ( 2 ) ' / ' R o p ~ ~ ~ / 3 (lob)
Thus for this case both K , and K , and consequently R,/R, are independent of T and m but increase with p and u. Furthermore, K , = 9.46KW2. The simple expressions (eq l a and lb) indicate that the ratio of homogeneous to heterogeneous reactions is unity when K , is 4 and 9, respectively, for an infinite cylinder and for a sphere. For these values of K , , the exact expressions give values of R,/R, of 3.51 and 3.57, respectively, for an infinite cylinder and for a sphere. Thus the simplified expressions overpredict the importance of the wall reaction. Finally, we apply the special case to an important example in the literature, Rabinovitch, Gilderson, and Blades4 found that the experimental hydrogen isotope effect in cyclopropane isomerization deviated torr at 511' from theory at pressures below 7 X in a 12-1. spherical vessel. A possible explanation was that wall deactivation was important at low pressure. From eq 7 and 10, the pressure can be calculated (at 511') when R , is comparable to R,, using RO = 14.2 cm and the reasonable value of 2.55 A for u. For K , = 1.0, R,/Rw is 1.09; and the pressure at 511' is 1.53 X torr. Thus at 1.53 X torr, the data should fit the theoretical value predicted for about 3 torr, in reasonable agreement with the findings of Rabinovitch, et aL4 ~~
(2) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids,'' John Wiley and Sons, Inc., New York, N. Y., 1954. (3) J. E. Mayer and M. G. Mayer, "Statistical Mechanics," John Wiley and Sons, Inc., New York, N. Y., 1950. (4) R. S. Rabinovitch, P. W. Gilderson, and A. T. Blades, J . Am. Chem. Soc., 86, 2994 (1964).
Volume 71, Number 6 April 1967
