EXCITATION FUNCTIONS FROM
THE
PHOTOCHEMICAL RECOIL METHOD
An Average Value of the Cross Section for H
+ HBr
1321 -t
H,
+ Br
over the 0.35-1.?-eV Collision Energy Range*
by R. G. Gann and J. Dubrin* Department of Chemistry, Massachusetts Institute of Technology, Clcmbridge, Massachusetts 08139 (Received November 16, 1971) Publication costs assisted by the U.8.Atomic Energy Commission
An average cross section for the title reaction has been computed from certain previously reported hot atom and it is defined as XE(E)EB~ = (photochemical-recoil) data. The cross section is equal to 1.6 i 0.3 i2, h’(E)SR(E)dE,where &(E) is the corresponding microscopic quantity and h’(E) is a known normalized distribution function of energies.
JA:& Introduction
We have recently employed a photolytic bulb method to determine the excitation function or translational energy dependence of the cross section for reaction 1 from threshold (-0.35 eV) to -2 eV.2 Hydrogen H”
+ n-CrDlo
----f
HD
+ C4De
(1)
atoms of different well known initial kinetic energies between 0.3 and 2.1 eV were generated for reaction by the selective photodissociation of either HI or HBr in the presence of the reactant.a Those hydrogen atoms having a kinetic energy greater than the threshold energy for (1) are termed “hot” atoms and are designated H*. I n addition to reaction 1, the hot hydrogens also react with the HBr(H1)
H*
+ HBr
--t
HP
+ Br
(2)
Unreacted hydrogen atoms that have been translationally relaxed or moderated below the threshold energy for abstraction from butane
+ n-C4Dlo H + C4Ds H* + H B r + H + HBr
H”
---t
(3)
(4)
are preferentially removed or scavenged by reaction with the HBr
H $- HBr -+ Hz
+ Br
(5)
The activation energy for reaction 2 is -0.1 eV.4 In the limit of zero HBr, the steady-state HZ/HD product ratio is a measure of the competition between moderating (3) and reactive (1) encounters in pure butane. A knowledge of the nonreactive differential scattering cross sections for (3), together with the H2/HD product ratio as a function of initial H atom energy, allows one to calculate the excitation function for (1) over the energy range sampled experimentally. However, since very little is known about the magnitude and energy dependence of the nonreactive inelastic cross sections
for (3), yields were also measured in systems highly diluted with a rare gas, xenon. Then by suitable extrapolation of the yield data, the competition between translational energy degradation and chemical reaction could be referenced effectively to an “clastic scattering” medium.286 Thus the computation of the excitation function only requires a knowledge of the H-Xe potential. Here, we calculat,e from some of the moderator data an average or phenomenological cross section for reaction 2. The yield data2 from the reaction of H atoms of 1.7 eV initial energy are employed, and thus this cross section represents an average of the microscopic quantity over the 0.35-1.7-eV collision energy range.
Method and Calculations2~~~6 A quantity of special interest in a three-component HBr-C4Dl0-Xe system is the limiting integral reaction probability. It is experimentally defined at a particular (’) source kinetic energy as lRPo’(H*-C4Dl0)
lim
yo’(xc4D10)
XC&Dlo+O
XC4Dlo
(6,
YO’(Xc~,,)is the H D yield as a function of butane mole fraction, X C ~ D Since ~ ~ . the X H B ~ / X Cis~ norD~~ ~ X C ~tend D~~ mally maintained constant, both X H Band to zero with increasing Xe content. Expression 7 relates this experimental quantity to the excitation ~ D(1). ~~ function S R ( E ) Cfor (1) This work was supported in part by the U. S. Atomic Energy Commission through funds provided under Contract AT(30-1)-905. (2) R. G. Gann, W. M. Ollison, and J. Dubrin, J . Chem. Phys., 54, 2304 (1971). References to other photochemical-recoil studies are contained in this reference. (3) A. Kuppermann, J. Stevenson, and P. O’Keefe, Discuss. Faraday Soc., 44, 46 (1967); R. N. Porter, J . Chem. Phys., 45, 2284 (1966). (4) I. Amdur and G. G . Hammes, “Chemical Kinetics: Principles and Selected Topics,” McGraw-Hill, New York, N . Y., 1966. (6) R. G . Gann, W. M. Ollison, and J. Dubrin, J . Amer. Chem. Soc., 92, 450 (1970). (6) C. Rebick and J. Dubrin, J . Chem. Phys., 5 3 , 2079 (1970).
The Journal of Physical Chemistry, Vol. 76, N o . 9 , 1978
R. G. GANNAND J. DUBRIN
1322
4.J
and thus expression 11 is given by
Eo is the threshold energy, 0.35 eV, for (1). The quantity TLO’(EL)X~ is the H-Xe collision density function evaluated in the limit of zero butane, and EL) is the H-Xe total scattering cross section. G(EL,E) is a known, normalized distribution function of relative kinetic energies E between an atom of laboratory energy E L and a target gas having a Maxwellian velocity distribution. Sincc the source energy used here is substantially greater than Eo and since the masses of C4D10 and Xe are much greater than that of H, we can simplify (7) with negligible error6
IRPo’(H*-HBr)
=
The mass M is the sum of the H and Xe masses; p is the reduced mass of this pair; Q(E)xe is the H-Xe momentum transport cross section and is calculated from the H-Xe repulsive potential. The average cross section for reaction 2 then follows from (14) SR(E)HBr
L:::
SR(E)HBr [Q(E)xeE - */~kTQ(E)xel-’dE
I n these moderator studies, the Hz/HD product ratio was also determined as a function of the [HBr]/ [C4DIO]composition ratio at fixed moderator content,2 Le., IXe]/( [C4D10] [HBr]) = constant. The measurements were conducted at Xe mole fractions of 0, 0.94, 0.96, 0.98, and 0.985. The Hz/HD ratio was linearly dependent on the [HBr]/ [RD] ratio and within experimental error the slopes m’ of such plots were equal to one another. The finding that m’ is independent of Xe content suggests the energy dependence of the collision density is about the same for the moderated and unmoderated media. In any event, it is very reasonable to identify thc slope measured at very high moderation with that in the limit of pure rare gas.
+
L::X[Q (g)S,:
(154
(E)xeE - 3 / ~ k T (QE )xe]-’dE
-
IRPo’(H *-HBr)
(15b)
[Q(E)xeE - 3/2kTQ(E)~e]-’dE
Q(E)xe is computed from the potential measured by Fink and given in ref 2. Q(E)xe = 13.4E-0.37AZ Using (16) along with (12), we find from (15b)
-
SR(E)HBr = 1.6
f
0.3 A’
lyi:
SR(E)HBrh’(E)dE
The limiting slope mot is equal to the ratio of the limiting integral reaction probabilities for reactions 1 and 2 IRPo’(H*-HBr) mol = (10) IRPo’(H *-C4DIo)
The distribution function is shown in Figure 1.
and by analogy to (8)) we may write
Discussion
IRPo’(H*-HBr)
=
11.7
f
2.3
(12)
As discussed at length e l ~ e w h e r e , little ~ ’ ~ error is involved in representing no’(E)xein (11) by the asymptotic form given by The Journal of Physical Chemistry, Vol. 76, N o . 9,1978
(17)
(The bath temperature, T , is 296°K.2) For convenience, expression 15 is written in terms of a normalized distribution function of energies, h’(E) SR(E)HBr =
where SR(E)HB~ is the excitation function for (2). At the initial energy 1.7 eV, the slope m‘ (E mol) is 3.4 f 0.6 and the corresponding IRPO’(H*-C~DIO)is 3.45 f 0.10.2 Using (lo), we then find
(16)
(18)
- 0.041-’
= 0.65E0.37[E
Martin and Willard (MW)* have measured the relative amounts of H2and H D from reaction of 2.9-eV H atoms with HBr-DZ mixtures (H Dz --t H D D). From an estimate of the average number of moderating collisions a hot hydrogen undergoes with DZin the 0.42.9 eV interval, they determined an average reaction
+
+
(7) The rapid approach to asymptotic behavior is a result of the very small fractional energy loss in an H-Xe collision. A more exact form which gives the functional dependence of the “high energy tail” (no‘(E)xeabove 1.7 eV) is found in the thesis of R. Gann (also see ref 6). (8) R. M. Martin and J. E. Willard, J . Chem. Phys., 40, 3007 (1964).
EXCITATION FUNCTIONS FROM
1.50
THE
PHOTOCHEMICAL RECOIL METHOD
I
1323
Since the number of moderating collisions is inversely dependent on zi7 it can be reasoned that the MW calculation overestimates p(E) and thus &(E). Moreover, as calculated in their manner, these values are not that meaningful-the actual distribution of collision energies in the H-D2 system is essentially unspecified. lo The calculation of the steady-state collision density requires a prior knowledge of the excitation function for H D2 HD D. On the other hand, the average or phenomenological cross section derived here (0.35-1.7 eV), is defined in terms of a known distribution function h’(E). It is of interest to note that if an ;of 0.07 for the H-D2 system is used instead of the hard-sphere value, the MW treatment results in an average cross section of 1.5 A2 for (2). This is essentially equal to the value found here for 1.7-eV H atoms. This close agreement suggests that the cross section for ( 2 ) (on the average) does not rise in passing from 1.7 to 2.9 eV. Obviously this and any further interpretations require assumptions about the relative steadystate collision densities and excitation functions (see previous paragraph). It certainly would be desirable to determine SR(E)HB~, using the photochemical recoil method. For this purpose, one needs to measure the limiting slope, defined by (9), at various initial energies. The subsequent analysis is quite straightforward. Although we have reported measurements for 1.2 and 2.1 eV initial energy,2 the experimental errors are such that little can be reached conclusively about the energy dependence of ( 2 ) in this interval. More accurate measurements are planned. ~
+
0
0.6
1.2
1.8
E (eV)
Figure 1. T h e distribution function h ’ ( e ) given b y expression 19 in the text.
__
probability, p(E), for ( 2 ) over the same energy interval. When p ( E ) is multiplied by the H-HBr collision cross section employed by the authors in their calculation, ?ne obtains an average reaction cross section of 7.5 A2. A direct comparison of our cross section with MW’s is not possible since the initial energies are substantially different. However, me strongly believe that their value is too large. In the RIW analysis, a hard-sphere (elastic) potential was used to compute the average fractional energy loss that a hot hydrogen atom suffers in a moderating collision with D2. In actual fact, the H-D2 repulsive potentialg is much “softer,” and in the potential energy range of interest it can be represented adequately by an inverse power form
V(r)H--Dz =
C -*
r4’
C = 1.7 eV A4
(20)
With the above potential, we find the average fractional energy loss ;to be substantially less than predicted by a hard-sphere potential, e . g . , at E = 1.5 eV, zi -0.07 as opposed to the hard-sphere value of 0.32.
--f
+
(9) K. T . Tang and M. Karplus, J. Chem. Phys., 49, 1676 (1968); R. Nelson, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1968. (10) Given the average fractional energy loss, one may easily calculate the average number of moderating collisions a hot atom makes over a specified energy interval. However, the collision density is desired, if one wishes to compute the average cross section from a prior knowledge of S R ( E ) H B ~i.e., ; no meaningful comparison of experiment with theory is possible without this information.
The Journal of Physical Chemistry, “01. 76, N o . 9, 197B