HETEROGENEOUS ACTIVATION IN THERMAL UNIMOLECULAR REACTION
Heterogeneous Activation in Thermal Unimolecular
4483 Reactionla
by Kenneth M. Maloneylb and B. S. Rabinovitch Department of Chemistry, University of Washington, Seattle, Washington 98106
(Received May $8, 1968)
The contribution of simple heterogeneity to the observed rate of reaction for thermal unimolecular systems has been examined. The equation of continuity was solved with appropriate boundary conditions. It is shown for spherical reactors that the magnitude of heterogeneous activation-deactivation effects is a function of the reactor size, the degree of falloff, temperature, and collision cross section. The variation of the concentration of activated substrate molecules as a function of the radial distance, r, of the pressure and collision diameter of the reactant and of the specific reaction rate of the reactant is illustrated, along with the behavior of a heterogeneity function H(E)' that ranges between the limits of zero (no heterogeneity) and unity (complete heterogeneity). Application is made to the ethyl isocyanide thermal isomerization.
Introduction Homogeneous thermal unimolecular gas-phase reactions are frequently studied in a spherical reactor as a function of pressure and temperature. It is desirable that the contribution of heterogeneity to the observed rate be quantitatively assessed. Hudson and Heicklenz (HH) have made an important contribution to the treatment of this problem. They examined the steady-state solution of the nonhomogeneous differential equation
DV2C(r)
+ R'(r) - k,C(r) = 0
(1)
subject to the boundary conditions
-D- dC (rd dr
=
IC,C(ro)
(r = ro)
and
(r
C(r) = finite
=
0)
(3)
where C(r) is the concentration of reacting molecules at radial distance r, D is the diffusion coefficient, R'(r) is the rate of homogeneous production or introduction of reactant molecules, k., is the first-order rate constant for homogeneous reaction, ro is the radius of the spherical vessel, and k , is the velocity constant for the heterogeneous wall reaction. Condition 2 takes into account reaction at the wall; condition 3 ensures a solution at r = 0 and is introduced because eq 1has a singularity. The sQlution of eq 1 for a sphere was given by HH, for R'(r) a constant, as
DC(r) - 1 R'ro2 K , ( K , / K , ) sinh { K v ' / Z r / r/(Kv'/'r/ro) ~) (4) [cosh { K.,'/') ( K , - 1) sinh { KV1/') /Kvl/']
+
where K.,
=
k,ro2/Dand K,
=
k,ro/D, and
where R , and R , were defined by H H to be the total homogeneous and heterogeneous rates, respectively. Equation 1 may be made very general, but some applications to a range of energy states (see below) will necessitate its restatement in the form of a set of equations that span the range of energy states.
Equation of Continuity for Activation-Deactivation H H considered as an illustration the simple but important example of activation and deactivation of excited molecules. In this case, C(r) becomes CE*(r), the concentration of activated molecules in unit range of energy at E; R' becomes RE', the rate of homogeneous activation of molecules of reactant CAto states at E , and may be taken as independent of r; k, is the specific rate constant for collisional deact> w(10 mm), then the falloff behavior occurs at such high pressures that the wall effects characteristic of lower pressure studies do not enter; if k E < lo-^ mm), then a significant degree of falloff is not observable and such a system only can be examined close to the “high-pressure” region. An important aspect of the behavior of C,*(r)’ is not only the magnitude of the limiting value CE*(ro)’, which increases with decreasing pressure and with increasing value of k,, but also the distance rh at which
Figure 4. Concentration profiles of CE~*(T)‘ us. r for ~ C E corresponding to 10 mm with u = 4.0 A @ E = 5.83 X 107 sec-1). For all pressures above 10-3 mm the proximity of C E * ( ~ to ) ’ unity everywhere, except at r = TO, is such that graphical representation of the area under the curves cannot be shown; the heads of the dashed arrows, however, indicate the limiting concentration CE*(TO)’ in each instance. Above 10 mm CE*(T)’becomes unity. The system is in the second-order region a t all pressures.
the quantity CE*(r) - CE*(ss) first becomes greater than zero. As the value of k, increases, for a given pressure and reactor size, rh approaches closer to r0 and the change in CE*(r0)’ becomes very large. This behavior is for the reason that the rate of wall activation is independent of k E , while CE*(ss) varies inversely with k,; also, as k, increases, the activated molecules react before they can diffuse any appreciable distance from wall, and the concentration perturbation at the wall is not propagated inward. Figure 5 illustrates the variation of H(E)’ as a function of pressure in 1-, 12-, and 200-1. reactors, respectively, for the several k E and collision diameters. Volume 72,Number 13 December 1968
KENNETH M. MALONEY AND B. S. RABINOVITCH
4486 When C E * ( T ) becomes equal to C,*(SS) for all r , H(E)‘ = 0. The arrow markers designate the respective threshold pressures below which each reaction (with characteristic rate constant k E ) is in its own second-order region; the arbitrary but practical criterion was adopted that k E > 100 ( ~ , C A 3k,/r0) for secondorder behavior. For each reaction, the magnitude of H(E)’ at values of the pressure below the markers depends simply on the relative rates of wall and homogeneous collision (activation) of A. At pressures above threshold, collisional deactivation as well as activation of AE* can occur at the wall so that the net rate oE wall activation is reduced. It should be understood from the nature of Lindemann falloff that no effect on the
+
1.0
,
IO-^
10-2
where RT(E)/R,(E) is the ratio of total rate in an energy range at E (due to the homogeneous plus heterogeneous contributions) to the rate if only the first contribution were to enter. Then
10-1
I
k,.
lOmm
I
I
10-2
16’
P (mm)
Figure 5. Summary plots of H ( E ) ’ us. pressure for u = 4.0 and 7.0 A. These plots illustrate the dependence of heterogeneity on pressure, reactor volume, k ~ and , collision cross section. The top scale is for the 4.0-A case and the bottom scale is for the 7.0-A case. That portion of each respective curve to the left of the arrows (positioned a t the top of the figure for the t.0-A case and a t the bottom of the figure for the 7.0-A case) corresponds approximately to the second-order region for the values of k~ in question. The pressures a t which the curves for the lowest ks (corresponding mm. to 10-3 mm) enter the second-order region are below
observed rate constant due to “heterogeneity” may raise it above IC, for the reaction in its high-pressure equilibrium region-even though all activation were to take place at the wall. By contrast, the heterogeneity effect is a maximum for a given reaction when it is in the second order. The H(E)’ curves given in Figure 5 were c$culated for the assumed collision diameters of 4 and 7 A. For the former case the homogeneous collision rate is reduced by one-third, all curves are moved to higher (by a factor of 3) pressures, and wall effects are enhanced. The correct form (but not quantitative magnitude) of the pressure dependence of wall effects may be readily seen from the formulation of a simplified ratio of rates in terms of Lindemann-type expressions The J O U T T U of~Physical Chemistry
The first factor in the ratio of eq 12 describes the relative importance of wall and homogeneous collisions; the second factor relates to the region of falloff in question, in the form of the relative importance of total and homogeneous rates of removal of activated molecules. For k, > ( k v C ~3k,/ro), , the ratio has the full value of the first term, corresponding to maximum wall effect in the second-order region; for k , > k , >> ~ , C A as , the pressure of the system decreases the second factor decreases more slowly than the first factor increases and H’ goes to unity-also, the observed first-order rate constant is in this circumstance equal to k,. The simplification that entered the formulation of eq 12 was the (implicit) premature spatial averaging of CE*(r) and its replacement by an (unspecified) average. Figures 1-5 provide a sufficient illustration of the dependence of H(E)’ on various parameters SO that semiquantitative estimates of H(E)’ may be readily made for other systems. The variation of H’ with temperature enters through the dependence of D and of the range of k E on T , as well as of CA if pressure is kept constant. I n the second-order region, H(E)’ does not vary with T if CA is kept constant. I n the falloff region, increase of T leads to increase of H(E)’. The magnitudes of these effects are illustrated in Figure 6 for the case of a 12-1. vessel, with ICE = lo-’ mm and u = 4 8. An actual system in which the average effective value of kEl and the region of falloff, changes with temperature is more complex, but the expected variation of H(E)‘ over a considerable range of temperature is not large.
HETEROGENEOUS ACTIVATION IN THERMAL UNIMOLECULAR REACTION
4487
____cL
200
400
300
SO0
600
T ('C)
10-4
10-3
10-2
p(mm1
Figure 6. Illustration of the variation of H(E)' with temperature in a 12-1. vessel at several constant pressures for k~ = 0.1 mm and u = 4 b. The rate of change of H ( E ) ' with temperature passes through a maximum with decreasing pressure. The variation of H(E)' at lo-* mm, which is in the second-order region, is due to the variation of C A with temperature at constant pressure.
Figure 7. Variation of the average heterogeneity function, (H(E)'),with pressure for the ethyl isocyanide system a t 231 O in three reactors. The arrow shows the pressure below which the reaction would be effectively in the spond-order region in the largest vessel for the case u = 7 A ; if u were less, ( H ( E ) ' )would be increased.
The Cyclopropane Reaction
geneity functions N ( E )' were calculated and averaged over the distribution function for the reacting molecules
As a practical example, H H considered the cyclopropane isomerization data which had been obtained by one of US;^ they used the limiting second-order formulation for a pressure of 1.53 X loe3 mm in a 12-1. reactor at 500". The reduced rate, k/k,,4 under these conditions is 6 X lo+, which corresponds to a reaction order of6 1.75. One may estimate5 that a representative value of k, for this system is 5 X 105 sec-I at the degree of falloff in question. The limiting calculation given by H H overestimated the heterogeneous contribution by only -15%.
Illustration : Ethyl Isocyanide Isomerization Owing to the increased molecular complexity of ethyl isocyanide as compared with methyl isocyanide,6 the accessibility of the second-order thermal region decreases significantly and makes rate measurements in the pressure regions where heterogeneity becomes important unavoidable. The effect of heterogeneity was calculated for this system for the several reactor volumes (0.5-200 l.), temperatures (190, 231, and 260°), and pressures used. The Marcus-Rice7 formulation was used to generate kE at 50-cm-1 intervals.8 The corresponding hetero-
Summary plots of ( H ( E ) ' ) vs. pressure at the central temperature of 230.9' for the 0.5-, 12-, and 200-1. reactors with QA = 7 A are given in Figure 7. Simple heterogeneity is estimated to become important below mm in the 12-1. reactor and below 1.5 X 5X in the 200-1. reactor. The experimental results are in reasonable agreement (3) E. S, Rabinovitch, P. W. Gilderson, and A. T.Blades, J . Amer. Chem. SOC.,86,2994 (1964). (4) E. W. Schlag and E. S. Rabinovitch, ibid., 82, 5996 (1960). (5) D. C. Tardy and B. S. Rabinovitch, J . Chem. Phys., 48, 1282 (1968). (6) F. W. Schneider and B. S. Rabinovitch, J . Amer. Chem. Soc., 84, 4215 (1962). (7) R. A. Marcus and 0. K. Rice, J. Phys. Colloid Chem., 5 5 , 894 (1951). (8) K. M. Maloney and E. 8. Rabinovitoh, submitted for publication.
Volume 72. Number 18 December 1968
