Calculation of the Average Energy of Recoil Hot Reactions

Hot Hydrogen Replacement Reactions in Alkane Systems Moderated with Noble Gases

BY LEONARD D. SPICER"

Department of Chemistry, University of Utah, Salt Lake City, Utah 84112, U.S.A.

Received 6th July, 1977

The average energy of reaction for recoil hot species in a well scavenged system is defined explicitly and calculated for recoil tritium for hydrogen replacement reactions with cyclohexane and n-butane as a function of noble gas moderation. In general this average reaction energy is relatively indepen- dent of the composition of the system. However, in cases of high reactivity or low threshold energies for hot reaction where the primary reactive process competes significantly with thermal scavenging, more significant composition dependencies may be manifest. The generality of the method outlined is specifically dependent on the availability of information on the energy dependence of the cross sections for the reactions of interest.

The energetics of recoil hot reactions are of considerable interest since recoil processes can be used to explore the entire energy range over which reaction excitation functions normally extend.l. The energy dependence of the reaction process is manifest within the integral yield, most often illustrated using the Miller-Dodson equation 3*

5 = 11; JPi(E)n(E) dE

where5 is the collision fraction for recoil species with reactant i, Pi(E) is the probability of reaction on collision with i and n(E) is the collisional distribution function for recoil species.

Conventional kinetic treatments are in general inappropriate for recoil-stimulated reactions. Formalisms based either on the Miller-Dodson approach or statistical, non-Boltzmann theory have been used previously;4-10 however, in both cases a limited number of' experimental parameters can be adjusted to explore the kinetic behaviour of these reactions. One primary variable is system composition, which has the direct effect of diluting the potentially reactive collisions and a secondary effect of changing the recoil collision density. Other variables such as initial recoil energy, reduced mass effects and pressure or phase changes tend either to have a minimal influence or to increase the complexity of the overall reaction by opening new reaction channels. Obtaining quantitative information other than integral yields for hot reaction processes in general is difficult. We have recently outlined a method for determining the average energy for recoil hot reactions in the gas phase as a function of system composition.ll Here, that technique is amplified and applied to the hot tritium for hydrogen replacement reaction in cyclohexane and n-butane as a function of noble gas moderation.

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528 E N E R G Y OF RECOIL H O T R E A C T I O N S

RESULTS AND DISCUSSION

The rationale for the approach used has been addressed earlier.ll Basically, the trajectory of the recoil species is followed on a collision-by-collision basis, and the distribution of reactive collisions is established using the reaction excitation function, PR(E) and the collisional distribution function n(E). In the absence of reaction, n(E) is most frequently approximated by an inverse energy dependen~e.~ Thus

where the constant a can be defined exactly for elastic, hard spheres l2 as the average logarithmic energy loss per collision,

n(E)dE = dE/aE (2)

a = (In Ei/Ef), (3) Ei is the initial energy before collision and Ef is the energy after collision. In practice the hard sphere approximation overestimates the value of a and a soft sphere model appears to be more accurate.13

Consistent with this approach a constant fractional energy is lost on collision and thus

where F = e4. After n non-reactive collisions, a recoil species with initial energy Eo will possess an energy E," = EoFn. For a multicomponent system, a = c f i a i whereh

is the collision fraction for component i and cti is the corresponding average logarithmic energy loss on collision with i. In this case Fi = e-ai and the average fraction of energy retained, F, for a system containing two components i and j will be

Ef = EiF (4)

i

After n collisions En = EoFyfiFyf.'.

In a two component system consisting of a reactant R and a moderator M, the probability of reaction on collision with R as a function of energy can be represented as

where aR(E) is the reactive cross section and a,@) is the total collision cross section most frequently approximated by an energy-independent average cross section. For this recoil system,

N1 = l-fKPR(EO) (8) represents the probability that reaction does not occur on the first collision at an initial energy, Eo. For a single recoil species, the probability that reaction will occur on the second collision at energy El = EoFkFk is then

Likewise for the third collision at E2 the probability that reaction will occur is given by

For most reactions the function P,(E) < 1 at all energies; a typical value in all but the most reactive systems might be 10-1 near the maximum in the cross section

p2 = - fRPR(EO)lfRPR(E1). (9)

(10) p 3 = [l - fRPR(EO)] [l - fRPR(El)]fRPR(E2).

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L . D . SPICER 529

function, meaning that one collision in ten is effective in producing reaction at a specified energy. Therefore, eqn (10) can be approximated by

P 3 * -fRPR(EO) -fRPR(E1)]fRPR(EZ) (11)

where the terms of higher than squared order in the probability of reaction are considered to be negligible. This approximation is valid as long as the total number of collisions experienced by the hot atom in the reactive energy range is not large. The applications presented later confirm this expectation for realistic systems. Similar reasoning can be applied for each successive collision giving the probability of reaction at each energy simply by

r i t - 1 1

The average energy at which reaction occurs can then be defined in terms of the distribution of reactive collisions as

Since En = EoFiJRFhJM, eqn (13) can be written as r n - I 1

For an ensemble of hot atoms the average in eqn (14) must be summed over all the hot atoms and all initial energies, Eo values, if the nascent recoil species is born with a distribution of energies.

Eqn (14) gives the average energy at which hot reaction occurs in terms of the composition of the system and the reactive excitation function P R ( E ) . The result in eqn (14) reveals the basic factors which can be expected in general to affect the average energy of reaction, and the extent to which this average energy can be perturbed. The composition dependence for the average energy of reaction derives both from the dilution effect of the added moderator on potentially reactive collisions and from the fractional energy losses experienced by the recoil species in non- reactive encounters. The latter source gives an exponential functionality in the fractional energy retention values FR and FM. The combined effect is small, however, in systems where P ( E ) is < 1 for all energies since the increased collision density available for reaction on successive collisions due to the presence of moderator is offset by the decreased overall probability of reaction which is proportional to the collision fraction of reactant at any one energy.

The expression in eqn (14) is generally applicable to the average energy for all reactive processes, however, average energies for individual reaction channels j characterized by partial excitation functions Pi,(,!?) can be estimated directly provided the integral value of the reaction probability at high energies for all reaction paths not considered in a particular instance is small. Otherwise the reactive collisional distribution function will be significantly perturbed from that represented by the

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530 ENERGY OF RECOIL HOT REACTIONS

series of equations like (12) for an individual reaction path Pi,@) due to the influence of concurrent reaction processes.14 This, however, does not often provide a serious restriction since the yields of individual products are frequently only a few percent, and for cases where they are significantly higher such as hydrogen abstraction reactions by recoil hydrogen atoms, the major part of the excitation function is manifest at low relative collision energies. Nevertheless, care must be taken in applying these equations to partial excitation functions.

Estimation of the fractional energy losses on collision from average logarithmic energy loss parameters, a values, can be made based on some fraction of the hard sphere value. More realistic soft sphere values derived from intermolecular potentials for high energy scattering processes l3 frequently are found to be about one half to one tenth of the hard sphere values.

In practice, eqn (14) can be applied to a system at fixed moderation if the reactive cross section and the fractional energy losses can be estimated. Unfortunately precise relationships have been developed for the total reactive excitation function in very few systems. Estimates of the reactive cross section as a function of energy must, therefore, frequently be made based on analogy to those r e ~ 0 r t e d . l ~ While even a rough estimate will suffice, this does at present provide the most serious restriction to the general applicability of the approach outlined.

In applying eqn (14) to real systems the following considerations are important. If the probability of reaction on collision over any available energy range exceeds a value of 0.1, the neglect of higher order cross terms in the series of equations like (12) may introduce significant error and eqn (13) and (14) will be more complex than indicated. This is particularly true at low moderation wheref, approaches one. For most recoil systems this source of error is negligible if the total number of collisions is not large, if the hot reaction paths are distinct from thermal reactions and/or scavenger reactions and if the total yield of hot products is less than -30 %.

Since changes in the distribution of hot collisions over the reactive energy range as a function of moderation must be accounted for in applying eqn (13) and (14), the non-reactive scattering characteristics of the hot atom collisions with reactant and moderator are important. Finally, when the scavenging reaction used in a particular system is either competitively inefficient or merely represents an extension of a hot reaction process to thermal energies, the formalism of eqn (14) breaks down. This may occur, of course, only when the excitation function for hot reaction has a threshold energy in the thermal region. In such a case the collision density for hot atoms is perturbed by the thermal bath gas over the low energy region of the reaction cross section, and recoil atoms which reach the thermal energy range are not removed by scavenger, resulting in an unusual weighting of the low energy reaction probability. This tends to emphasize a marked decrease in the average energy of reaction with increased moderation, particularly in systems with relatively large hot reaction cross sections.

A P P L I C A T I O N S

The above formalism can be applied to hydrogen replacement reaction on cyclo- hexane l6 and n-butane I6-l * where the cross section for reaction has been determined in tritium ion beam experiments. The cross sections used are parameterized in the general form a(E) = C(E-EJU e-m(E-Et) suggested by LeRoy.15 Based on the additional experimental results for cyclopentane, n-hexane, and 1-chlorobutane,l it is thought that the model systems used here are representative of the general features of hydrogen replacement reactions in alkanes and cycloalkanes. The parameters

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L . D . SPICER 53 1

used in these calculations are listed in table 1, and the calculated results for the average energy of reaction at 0 and 99 mol % He, Ne, and Ar moderator are listed in table 2.

TABLE 1 .-MOLECULAR PARAMETERS FOR TRJTIUM FOR HYDROGEN REPLACEMENT REACTION WITH CYCLOHEXANE AND n-BUTANE

T+ c-C a 1 2 T+n-C4Hlo

CIAz eV-n 1 . 1 4 1 . 3 4 ~ 10-3 a 0.92 2.08 rn1eV-l 0 . 0 8 5 0 . 1 4 9 EtIeV 1.88 6.0

TABLE 2.-AVERAGE ENERGIES CALCULATED FOR RECOIL TEUTIUM FOR HYDROGEN REPLACEMENT REACTIONS IN CYCLOHEXANE AND n-BUTANE AS A FUNCTION OF NOBLE GAS MODERATION

average energy for reaction mole fraction reactant

reaction system 1 .oo 0.01

T+C-CgHiZ + C-CeH1 IT+ H He bath a Ne bath a

Ar bath a

Ne bath Eo = 500 eV Ne bath Et = 0 . 1 eV Ne bath a~ = 2.45

Ne bath

Ne bath

MM = 0.272 SR = 30 A"

T+n-C4Hlo --+ n-C4H9T+H

1 5 . 7 8 eV 1 5 . 7 8 1 5 . 7 8 1 5 . 7 7 1 2 . 2 1 2 1 . 5 7

1 6 . 5 0

22.20

1 5 . 2 8 eV 1 5 . 3 2 1 5 . 3 5 1 5 . 3 2 1 1 . 4 8 1 5 . 2 4

1 5 . 3 3

2 2 . 1 4

a Eo = 250eV, Et = 1.88 eV, S;'* 2 2 = 58.1 A2, S i t 9 2 2 = 20.3 A2, Sit* 22 = 21.9A2, Si,'* 2 2 = b Same as a,

CEO = 500 eV, Et = 6.0 eV, S;'. 22 = 44.2A2, S i i * 2 2 = 21.9 A2, aR = 27.5 A2, a~ = 0.665," aHe = 0.115,13, 16 aNe = 0.075,13, 16 a& = 0.045.13, 16

0.624," = 0.075,13* l6

except as indicated.

The results in table 2 quite clearly show the general insensitivity of the average reaction energy on moderation, on initial recoil energy as long as it is well above the maximum in the reaction cross section and on the non-reactive scattering cross section taken as an energy-independent constant. There is, however, a relatively marked dependence of the average energy of reaction on both the collisional energy loss characteristics (a values) for tritium scattering and the threshold energy for reaction as it approaches the average thermal energy limit.

The increased a values used to test the energy loss properties are the hard sphere value for Ne and the experimental ratio a(c-C,H,,)/a(Ne) determined by extrapola- tion. The dependence illustrated, while representative, does not accurately evaluate the average energy for reaction in pure reactant because the value calculated is based on only 2 collisions and thus depends significantly on the initial energy of the trajectory. In any case, where the energy loss of the hot atom on non-reactive collisions with either reactant or moderator is large, the collisional sampling of the excitation function for any one trajectory will be minimal. Under such circumstances, statistical data about the reaction dynamics can only be derived from an experiment or calculation in which the initial energy of the recoil species is broadly distributed. For the reaction in table 2 with aR = 2.45 and aM = 0.272 the average reaction energy

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532 E N E R G Y O F R E C O I L HOT R E A C T I O N S

calculated in pure cyclohexane for trajectories having initial energies ranging from 200 to 400 eV in 2 eV increments is calculated to be ( E ) = 21.9 eV.

The dependence on threshold energy is calculated in the absence of scavenger. In a real system an efficient scavenger may well provide apparent threshold behaviour which is independent of the excitation function threshold by successfully competing for low energy species. When this competition is significant, the observed reaction yields represent a composite of hot, moderator dependent kinetics and epithermal reaction processes competitive with scavenging reactions.

The fact that the average energy of reaction is essentially independent of the choice of noble gas moderator is not surprising in view of the small variation calculated between pure reactant and 99 % moderated systems. This conclusion is consistent with the experimental results reported earlier that the average vibrational excitation of t-cyclobutane formed in the recoil hot tritium reaction with cyclobutane is independent of moderation for several noble gas and polyatomic moderators up to 80 % dilution.20

The implication of the results in table 2 is clearly that dilution up to 99 % with non-reactive moderators does not appreciably lower the average energy of hot reaction in well scavenged systems exhibiting moderate reactivity at high energies. While dilution clearly has a direct effect on the absolute yields of hot reaction products, the increased density of potentially reactive low energy collisions is almost quanfita- tively counteracted by the decreased collision fraction for reactant.

This work was supported by the U.S. Energy Research and Development Adminis- tration. The author is grateful to one of the referees for helpful comments.

D. S . Urch, Radiochemistry, M.P.T. International Review of Sci., 1975, 8, 49 ; 1972, 8, 149. L. D. Spicer and B. S. Rabinovitch, Ann. Reu. Phys. Chem., 1970, 21, 349. J. M. Miller, J. W. Gryder and R. W. Dodson, J. Chem. Phys., 1950, 18,579. R. Wolfgang, J. Chem. Phys., 1963, 39,2983 ; P. J. Estrup and R. Wolfgang, J. Amer. Chem. SOC., 1960, 82, 2665. M. D. Kostin, J. Appl. Phys., 1965,36, 850; R. M. Felder and M. D. Kostin, J. Chem. Phys., 1965, 43, 3082. P. V. Thomsen, J. Chem. Phys., 1968, 49, 756.

J. T. Adams and R. N. Porter, J. Chem. Phys., 1973, 59, 4105 ; R. N. Porter and S . Kunt, J. Chem. Phys., 1970,52,3240; R. N. Porter, J. Chem. Phys., 1966,45,2284. C. Rebick and J. Dubrin, J. Chem. Phys., 1970,53,2079.

D. J. Stevens and L. D. Spicer, J. Chem. Phys., 1977, 66, 5253.

p. 147.

' M. Baer, J. Chem. Phys., 1969, 50, 3116.

lo J. Keizer, J. Chem. Phys., 1973, 58, 4524 ; J. Keizer, J. Chem. Phys., 1972, 56, 5958.

l2 S. Glasstone, Principles of Nuclear Reactor Engineering (Van Nostrand, New York, 1955),

l3 P. Estrup, J. Chem. Phys., 1964, 41, 164. l4 C . J. Malerich and L. D. Spicer, J. Chem. Phys., 1973,59, 1577.

l 6 M. Menzinger and R. Wolfgang, J. Chem. Phys., 1969, 50, 2991. R. L. LeRoy, J. Phys. Chem., 1969, 73,4338.

J. W. Beatty and S . Wexler, J. Phys. Chem., 1971, 75, 2417; S. Wexler, J. Phys. Chem., 1972, 76, 2939.

l8 M. Menzinger, R. L. LeRoy and A. J. Yencha, J. Phys. Chern., 1972,76,2937, l9 R. L. LeRoy, A. J. Yencha, M. Menzinger and R. Wolfgang, J. Chem. Phys., 1973, 58, 1741. 2o N. S. Nogar and L. D. Spicer, J. Chem. Phys., 1977, 66, 3624. 21 A. Dalgarno and F. S . Smith, Proc. Roy. SOC. A , 1962, 267,417. 22 J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, MoZecular Themy of Gases and Liquids (John

Wiley, N.Y., 1964).

(PAPER 7/1190)

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