2362
J . Phys. Chem. 1990, 94, 2362-2367
Moderation and Absorption Effects on Hot Replacement Reactions of 38CiAtoms in Mixtures of o-Dichlorobenzene and Hexafluorobenzene K. Berei, J. Gadti, A. Kereszturi, Z. Szatmiiry, and Sz. Vass* Central Research Institute f o r Physics of the Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary (Receioed: April 5, 1989; In Final Form: September 18, 1989)
Conditions are given for the equivalence of the Estrup-Wolfgang description of the hot atom reaction kinetics with the first-order Hurwitz approximation in the neutron slowing down theory. Conclusions are drawn for the applicability of this approach for describing hot atom replacement processes in reactive mixtures. Analytical and numerical calculations were carried out to explain an unusual concentration dependence of 3sC1-for-CIsubstitution, found experimentally in liquid binary mixtures of o-dichlorobenzene and hexafluorobenzene.
1. Introduction In an earlier experimental study’ of hot 38Cl atoms reacting in the binary mixtures of o-dichlorobenzene (o-C6H4CI2)and hexafluorobenzene (C6F6), it was found that the yields from 38CI-for-CI replacement corrected for dilution varied with the component concentrations of the mixture-growing to relatively high values with increasing C,F6 content-whereas the yields from 3sC1-for-F replacement corrected for dilution were found to be constant over the whole dilution range. This finding was interpreted in terms of liquid cage reactions of the thermal 38C1atoms masking the results of the true hot However, it cannot be excluded that the different behaviors are caused by the different moderating and absorption properties of the two components which may result in changing the energy spectra of the hot 38Clatoms along with the varying concentration of the components, thus leading to different probabilities of the replacement p r o c e ~ s e s . ~ , ~ The aim of the present paper is to elucidate this problem in neutron slowing down calculations carried out in hypothetical model systems chosen such that their parameters correspond to o-C6H4C12 and C6F6 mixtures of different concentrations.
2. Formulas Used 2.1. Classical Estrup-Wolfgang Theory of Hot Atom Reaction Kinetics. Studies on the chemistry of energetic atoms produced by nuclear reactions date back to 40 years ago, when the first papers of Libby5 and of Miller et aL6s7were published. The final formulation of the kinetic theory of hot atom reactions was done 10 years later by Estrup and W ~ l f g a n g . ~The , ~ basic concept in the kinetic theory is the “total” probability PI that a hot atom of initial energy Eo reacts with component j of a mixture in the (nonthermal) energy range of interest E , , E2 (Eth 1 in a range much wider than the mean energy loss per collision (cf. the shape of the triangular excitation functions in ref 15 and 16), the low-reactivity condition does not hold, z f g i ( E ) 1. but (3 remains constant near to zero and thus the Hunvitz approximation can be applied. Probably this is the reason why the Estrup-Wolfgang theory was found to be valid up to higher reactivities than e x p e ~ t e d . ' ~ , ' ~ Results of a demonstrational calculation of reaction probabilities by utilizing the Hurwitz approximation to the collision density function in a two-component mixture may thus be assumed qualitatively characteristic for the highly reactive experimental systems in ref 1 and 2. In order to avoid complicated formulas, cross sections of the most simple curvature were chosen. Let aej # 0 be independent of the hot atom energy and let aaj(E)-and thus T,(E)-also be described by constant functions differing from zero in disjunct kinetic energy intervals (as plotted in Figure 1 ) ; thus -+
- fi -
-
=
aa,(E)/aeJ
=
I
= const, if E,, 5 E 5 EJ2,j = 1, 2 (17) 0, otherwise
Z%,(E)[ t x : S t k ( a l - '
I
=
+ F,rJo)]-l= 4, if EllS E 5
0. otherwise
-;
Figure 1. Schematic representation of T ~ ( E=) uaj(E)/uej,the absorption cross section function normalized to the corresponding elastic scattering cross section, used in approximate analytical calculations of radiochemical yields; see eq 17 and the text.
This property permits all integrals to easily be evaluated, and after elementary steps, the collision density function can be expressed as n ( E ) = ( t W [ m a x (E,E21)/E221@2 x
1
[max (E,EII)/EI2]@l,if E 5 .El2
I , if E 1 2< E I Ezz 0, if Ez2 I E
= 1, 2
(18)
(19)
By substituting now n(E) from eq 19 into eq 10, the probabilities R1 and R2 of hot reaction with components 1 and 2, respectively, are given as Rl = (1 - 62)4'2[1 - (1 - 6,)4l] (20.1) and R, = 1 - (1 - 62)4'2 (20.2) where 6, = (Ej2- Ej!)/Ej2is the relative width of function T ~ ( E ) , j = 1, 2; see also Figure 1. Some experimental results can be qualitatively explained by analyzing R, in terms of 4j and For example, in several systems Rj was found to be-within the experimental uncertainties-a linear function of X,,at least in a limited range of the component mole fractions. In order to explain this behavior, let us assume that the absorption peaks are narrow enough to approximate (1 - 6,)+~ by 1 - C$j6j in eq 20.1 and 20.2; in this case the reaction probability with component j can be written in the simple form R, C$,Jp This approximation does not require extremely sharp (resonance-like) peaks. If C$J remains below 0.2, the error is less than 20% in the calculated values of reaction probability up to 6j 0.4. In the low-reactivity limit the conditions for this approximation are even better. The effect of component concentration on reaction probability is determined by 4j and ,$,and it is obtained from eq 13 and 18 as the analytical expression Rj F , T " ~ ~ ~ [4-[ (Fjsoj)]-] I
-
-
-
= A'jOejToj6j[ 1
+ Xja,is"j/CXkaek]-'[CA'kaekEk]-'= X j q k
7,"
It can be shown that this simplification does not affect the validity of the conclusions. It follows from this choice of cross sections and energy intervals that in the exponent of the collision density function we have the form
F,T,"[[(]
I----=
(16)
k
(21) Equation 21 defines q,the radiochemical yield corrected for dilution (as called in experimental studies). If the mole fraction A,' 0, U, is constant in all systems where the previously mentioned conditions hold for the widths of the absorption peaks. In these systems Rj is a linear function of the corresponding mole fraction. Kontis and Urch developed a linear empirical model18 for describing hot yields of 38CIreactions from different liquid systems,18-*' in terms of yields from the pure components. The
-
(18) Kontis, S. S.; Urch, D. S. Radiochim. Acta 1973,20, 39. (19) Kontis, S. S.; Malcolme-Lawes, D. J.; Urch, D. S. Radiochim. Acra 1977, 24, 87.
The Journal of Physical Chemistry, Vol. 94, No. 6, 1990 2365
Hot Replacement Reactions of 38CI Atoms
I
A
01 1
1
2
3
I
4
5
7
50t
*
1
6
A
Figure 2. Logarithmic energy loss per elastic hard-sphere collision vs the mass ratio of the colliding partners. The arrows mark the points which correspond to A = 186.1/38 = 4.90, the C6F6/'*C1 mass ratio, and A = 147/38 = 3.87, the C6H4C12/'*CImass ratio.
coefficients of yields were expressed by rational expressions of reaction cross sections of all components present,I8J9 thus resembling the present treatment. In low-reactivity systems the concentration dependence of yi is determined by the factor (xkXkcrekEk)-l.For demonstration, one of the parameters in the sum, the mean logarithmic energy loss per (elastic hard-sphere) collision with a particular component, is plotted in Figure 2 vs the reactant molecule to hot atom mass ratio A. The other parameter, the elastic scattering cross section cr,, is assumed to increase with molecular size. If the values of molar mass and of molecular size of the components do not change rapidly, &Xku&k is expected to be a slowly varying function of the mixture composition. For example, assuming that the encounters of hot 38Cl atoms with C6F6 and o-C6H4C12molecules are elastic hard-sphere collisions, we find that t(C6F6) 0.36 and t(C6H4Cl2) 0.44. If Ue(C6F6) cre(C6H4C12),then (CkXkuek(k)-'varies with the mole fraction in their binary mixtures by less than 22%; cf. Figure 2. Considering now the problem of hot 38C1-for-Fand hot 38CIfor-CI substitution in binary mixtures of C6F6 and o-C6H4C1,, Y(C6F6) was found to be constant (or, at least, a slowly varying function) within the experimental error over a wide (0 5 X(C6F6) I0.9) range of C6F6 mole fractions X(C6F6), while Y(o-C6H4CI2) increased with increasing values of X(C6F6); see Figure 3. This significant difference between the corrected radiochemical yields of the two replacement processes enables us to assign these processes to the appropriate ones among eq 20.1 and 20.2. The constancy of Y(C6F6) vs X(C6F6) indicates that the hot 38C1-for-F substitution can be described in terms of the model discussed above, Le., by any of eq 20.1 and 20.2. The strong nonlinearity of Y(C6H4C12)suggests that the hot 38C1-for-C1substitution can only be related to the equation of the more complex structure, Le., to eq 20.1 R(O-C&4C12) [ 1 - X(C6Fs)Y(C,jF6)] [ 1 - (1 - SI)"] (22.1) and R(C6F6) X(C6F6) Y(C6F6) (22.2) In eq 22.1 and 22.2 the mixture composition is characterized by the mole fraction of C6F6. If the hot 38Cl-for-C1substitution in o-C6H4C12is a highly reactive process, then 701 >> 1, apart from small o-C6H4CI2(Le., high C6F6) mole fractions; 4l and thus 1 - (1 - 6,)@1 = y are constant or slowly varying functions of the mixture composition in a wide range of mole fractions. From eq 22.1 the corresponding radiochemical yield corrected for dilution is obtained by dividing both sides by X(O-C~H.,C~~) = 1 - X(C6F6): Y(O-C6H4C12) = y [ 1 - X(C6F6)Y(C6F6)][1 - X(C6F6)I-I (23) In the given range of component mole fractions, the qualitative agreement of eq 23 with the experimental results is satisfactory.
-
-
/
-
-
-
(20) Chandrashekar, N.; Bhave, R. N.; Rao, B. S. M. J . Radioanal. Nucl. Chem., Lett. 1985, 93. 73. (21) Kontis, S . S.;Gasparakis, E. A.; Petrou, J. K. J . Radioanal. Nucl. Chem. 1986, 102, 321.
/
/
01
03
05
A
09
07
x'csFE)
Figure 3. Experimental radiochemical yields' (corrected for dilution) from hot '*CI-for-Cl (A) and 38C1-for-F(0)replacement in binary mixtures of o-C6H4CI, and C,F6, vs the mole fractions of C6F6.
-
For X(C6F6) 0, Y(O-C6H4C12) 7, < 1 ; Since Y(C6F6) 0.15 > Eb, the duration of the encounter becomes too short for any interaction: in both cases T ( E ) 0. Between the two extreme energies, T ( E )is assumed to have a smooth curvature with a single maximum at a certain E,; cf. ref 15 and 16. By comparing two absorption peaks, from the relation E b l < Eb2,one may assume that E,, < Em2. The binding energy of CI in oC6H4C1, is Eb(C1) 3.7 eV and that of F in C6F6 is E,(F) 5.0 eV.22 The relation Eb(C1) < Eb(F) is an independent physical argument which supports the idea that the absorption peak of hot 38C1-for-C1substitution in o-C6H4C12is located at lower energies than that of hot 38C1-for-Fsubstitution in C6F6, in agreement with conclusions drawn on the assumption derived purely from the concentration dependence of radiochemical yields. 3.2. Numerical Analysis of Radiochemical Yields in Highly Reactive Binary Mixtures. Results obtained for properties of radiochemical yields in highly reactive binary mixtures can be confirmed by numerical calculations. In such calculations eq 8 is solved for the collision density function by utilizing a computer code.23 Partly because of the mathematical formalism used in the computer code and partly because of our insufficient knowledge of microscopic processes, the computations are made under the following conditions: 1. The energy loss of hot atoms is due to their elastic hardsphere collisions with the molecules, thus purely determined by the corresponding mass ratios. 2. The microscopic absorption cross section curves vs hot atom kinetic energy are described by the Breit-Wigner formula -+
-
-
uJE) = u O , ( l
+ 4 [ ( E - E o , ) / r 0 , ] 2 ) - 1for j
-
= 1, 2
(24)
(22) Vedeneyev, V. I.; Gurvich, L. V.; Kondratyev, V. N.; Medvedev, V. A.; Frankevich, Y. L. Bond Energies, Ionization Potentials and Electron Affinities; Arnold: London, 1966. (23) Ishiguro, Y.; Takano, M. Nippon Genshiryoku Kenkyusho, [Rep.] 1971, JAERI-1219.
Berei et al.
2366 The Journal of Physical Chemistry, Vol. 94, No. 6,1990 TABLE I: Interdependence of Resonance Maxima 7,' and of Half-Widths r,O of Breit-Wigner Type Absorption Cross Section Functions Assumed To Describe Hot "CI-for-CI and "CI-for-F Reohcement in Neat o-CrHACI, and CAF,. Respectively
r, 2 ° ,
eV
72'
0. I 0.5
2.1 0.26
5.5 0.52
1.0
0.1 18
0.299
ri.20, eV
7,'
2.0
0.055
5.0
0.027
'2O
0.104 0.049
with maximum location E o j , full width Pi, and amplitude a',.. The microscopic parameters are chosen such that the system resembles binary mixtures of o-dichlorobenzene and hexafluorobenzene. The parameters are denoted respectively by subscripts 1 and 2: A , = 147/38 = 3.87, a, = 0.347, and El = 0.437 as well as A, = 186.1/38 = 4.90, a, = 0.437, and E2 = 0.358. It has to be pointed out that realistic encounters of hot 38Clatoms with o-C6H,C12and C6F6 molecules are probably not hard-sphere collisions. In a correct description the intermolecular potentials, the distribution of the collision energy among the molecular segments, etc., should be taken into account. Due to our insufficient knowledge of the realistic hot processes taking place in the condensed phase, we cannot propose any other simple approximation that would more reliably reflect the nature of the energy loss. Considering a 38C1recoil energy spectrum constructed from measured data on the 37Cl(n,r)38C1nuclear the average kinetic energy of recoil 38CIatoms is 294 eV and the recoil energy distribution shows that less than 0.5% of recoil 38Clatoms have initial kinetic energy below 20 eV. Thus, d ( E ) in eq 8 can be replaced by a discrete line resulting in a l / E hot atom kinetic energy spectrum in the (hot) absorption region. Since published data on reaction cross sections are not available (at least to our best knowledge), we assumed that the maximum locations of the absorption cross section functions are at the corresponding binding energies of chlorine (EO, 3.7 eV) and of fluorine ( E a 2 5.0 eV) in aromatic hydrocarbons.22 The present choice of the E o j values means that the absorption cross section curves are shifted along the kinetic energy axis toward the thermal region, causing some distortion in the ratio of the radiochemical yields arising from the two compounds. The remaining parameters are the amplitudes and widths of the absorption cross section functions, as well as the elastic scattering cross sections. The values of the parameter combination 7 O j = aoaj/uejwere estimated from hot 38C1-for-C1and hot 38C1-for-F replacement yields obtained experimentally in single-reactant system^.',^^-^^ In this case, the amplitude Pican numerically be determined as a parametric function of the width r, for any experimental radiochemical yields ( R , 12% in o-C6H4C12for the hot 38C1-for-Cireplacement, and R, 18% in C6F6 for the hot 38CI-for-Freplacement; cf. ref 26). In the neat systems the same set of values rj= 0.1, 0.5, 1.0, 2.0, and 4.0 eV has been chosen for the widths of the two absorption cross section functions j = 1 , 2; the results are collected in Table I. The calculation of hot atom replacement yields (corrected for dilution) in the mixture of the two components has been carried out by using the 5 X 5 possible pairs from Table I at 0.3, 0.5, and 0.9 mole fractions of C6F6. The results of these calculations are summarized in T a b l e 11. The group of most interesting data in this table belongs to rl = 0.1 eV; these results are plotted in Figure 4. I n this case, the radiochemical yield from hot 3sC1-for-C1 replacement in o-C6H4CI2(corrected for dilution) increases with the mole fraction of C6F6 for all values of r2.However, by varying r,, some values can be found where the radiochemical yields from hot 38CI-for-Freplacement in C6F6 do not change significantly and thus are comparable with the experimental results in ref 1
TABLE 11: Calculated Radiochemical Yields (Corrected for Dilution) YI = Y(o-C&CI2) from Hot %X-for-CI Replacement and Y 2= Y(C6F6) from Hot 38C1-for-FReplacement in Different Binary Mixtures of o-CnHdC1,and C,Fn"
X, = X(C6F6) r,o
0.1
0.5
I .o
2.0
4.0
0.3
0.7
0.9
0.3
0.7
0.9
0.1 0.5 1 .o 2.0 4.0 0. I 0.5 1 .o 2.0 4.0 0. I 0.5 1.0 2.0 4.0 0. I 0.5 1 .O 2.0 4.0 0.1 0.5 1 .0 2.0 4.0
12.71 12.86 13.00 13.00 13.14 11.71 11.86 12.00 12.00 12.14 11.43 11.71 11.71 11.86 11.86 11.29 11.57 11.71 11.71 11.86 11.29 11.43 11.57 11.57 11.71
16.00 16.33 16.33 16.67 17.00 11.67 13.00 13.00 13.00 13.33 12.00 12.33 12.33 12.67 12.67 12.00 12.00 12.00 12.33 12.33 11.67 12.00 12.00 12.00 12.33
19.00 19.00 19.00 20.00 20.00 13.00 13.00 13.00 14.00 14.00 13.00 13.00 13.00 13.00 13.00 12.00 12.00 12.00 13.00 13.00 12.00 12.00 12.00 12.00 13.00
25.00 18.33 17.33 16.33 16.00 25.00 18.33 17.33 16.33 16.00 24.67 18.33 17.33 16.67 16.33 24.67 18.33 17.33 16.67 16.33 24.67 18.33 17.33 16.67 16.33
19.71 18.00 17.57 17.14 17.00 19.57 18.00 17.57 17.29 17.14 19.71 18.00 17.57 17.29 17.29 19.71 18.00 17.57 17.43 17.29 19.57 18.00 17.57 17.43 17.29
18.45 17.88 17.78 17.67 17.67 18.45 17.89 17.89 17.78 17.78 18.45 17.89 17.89 17.78 17.78 18.45 17.89 17.89 17.78 17.78 18.45 17.89 17.89 17.78 17.78
'Variable parameters are X, = X(C6F6), the C6F6mole fraction, as well as rIoand rl0,half-widths corresponding to Breit-Wigner type absorption cross sections assumed to describe the hot replacement process.
-
-
1;
/
Ferro, L. J.; Spicer, L. D. J . Chem. Phys. 1978, 69, 1320. Berei, K.; S t k k l i n , G. Radiochim. Acta 1971, 25, 39. Berei, K.; VasBros, L. Radiochim. Acta 1974, 21, 75. Brinkman. G. A,: Kaspersen, F. M.; Visser, J. Radiochim. Acta 1980,
Brinkman, G. A.; Veenber, J. Th.: Visser, J.; Kaspersen, F. M.: Lindner, L. Radiochim. Acta 1979, 26, 85.
01
05
\
to
---
20--40
\ \ \
20
\
1)
--
(24) (25) (26) (27) 27, 7. (28)
y2 = Y(C6F6)
Y , = Y(o-C,H,CIJ
r20
Y(0- C&
10'
'
0.1
I
0.3
0.5
0.7
0.9
,qc8F6i
Figure 4. Calculated radiochemical yields (corrected for dilution) from hot 38CI-for-C1 and 38CI-for-F replacement in binary mixtures of oC6H4CI2and C6F6,vs the mole fractions of C6F,. The calculations were made by using a numerical neutron slowing down algorithm.23 Hot replacement processes are described by Breit-Wigner type absorption cross section functions, and their half-widths are the parameters of the presented curves. For the choice of the other parameters see the text.
(see Figure 3) and with the results of the approximate analytical calculations discussed in section 3.1. Trends found in experimental results' could be reproduced by the present numerical calculations. Two problems, however, need further discussion. First, cross section functions, as compared with expectations, are too narrow, and second, the slope and values of calculated hot 38CI-for-CI replacement yields turned out to be essentially less than those obtained from experiments (cf. Figures 3 and 4). Theoretical cross section functions are available from (semiclassical) trajectory calculations, made by Karplus et al.29 in
The Journal of Physical Chemistry, Vol. 94, No. 6,1990 2367
Hot Replacement Reactions of 38Cl Atoms
+
systems 'H jHkH and by Kuntz et in systems 3H + 'HR (mainly for R = CH,), assuming hydrogen isotopes i , j , k = 1, 2, 3. All calculations resulted in broad cross section functions for the 'H-forJH r e p l a ~ e m e n t . ~ ~Similar , ~ ' results were obtained by making use of a simple, diatomic hard-sphere model elaboratored by S ~ p l i n s k a and s ~ ~by M a l c ~ l m e - L a w e s . ~This ~ , ~ model ~ was later developed by Malcolme-Lawes for describing hot reactions of 3H and I8F with five-atom molecules as CH4, CH2F2, and CF4.35 Model calculations in these systems resulted again in broad cross section functions of hot I8F-for-F replacement. The mechanism of hot I8F gas-phase reactions with CH3CF3 was investigated in detail by Krohn et al.36337 Their experimental findings were analyzed in terms of thermochemical energetic^.^^ According to this analysis, the excited (CH3CF2I8F)*molecule, the primary product of the hot I8F-for-F replacement in CH3CF3, can lose its energy in three channels. The deexcitation without chemical changes proceeds in a relatively narrow, 1 eV wide channel; in the other two, -7 eV broad channels CH2CF2+ Hi8F and CH2CFI8F+ H F form. Gas- and liquid-phase hot reactions of 38Clwith C6H5Fwere studied by Coenen et Their conclusion is that hot 38C1-for-F replacement in C6H5F proceeds exclusively through a one-step process. Deexcitation of (C6HS38Cl)* and (c6F538c1)*molecules through carbon-bond scission or formation of C6H4 + H38C1 and CsF4 F38C1 is expected to be less probable. Thus, based on the results of and considerations made by Krohn et al.,37a relatively narrow r2 1.0-4.0-eV excitation function of hot 38C1-for-Fsubstitution in C6F6 is not physically unrealistic. In agreement with the experimental findings of Berei and Ache1 and with the present numerical results, hot 38C1-for-Freplacement yields from binary mixtures of C6H5Fand n-C5H12,corrected for dilution, were found to be independent of mixture composition.38 Very limited information is available on fine details in the mechanism of hot 38C1reactions with chlorinated benzenes. Due to the lack of gas-phase experiments, no conclusion can be drawn on the deexcitation energetics of (c6H538cI)*and (C6H4C138C1)* molecules, the primary products of hot 38C1-for-C1replacement in mono- and dichlorobenzene. Thus, the parameter value rl 0.1 eV, obtained from numerical calculations for the hot 38CIfor-cl replacement in o-C6H4C12,cannot be compared with experimental data. A conclusion drawn from liquid-phase experi m e n t ~ ' . ~ unequivocally * ~ ~ - ~ ~ , ~suggests ~ that hot 38C1-for-C1replacement in chlorinated benzenes is generally a complicated multistep process. The (probably) most recent study on the however, claims that hot 38C1-for-Clreplacement in certain C6H5C1 mixtures may proceed through a single-step reaction. Provided that physical evidence for a narrow excitation function is found, the essential difference between experimental and calculated yields still remains and suggests that a process different from direct hot replacement should also be taken into consideration when interpreting hot 38CI-for-C1replacement yields from mixtures O f O-C6H4C12 and C6F6.
-
+
-
-
4. Summary and Conclusions The interpretation of experimental hot replacement yields in mixtures, in terms of neutron slowing down theory, requires that (29) Karplus, M.; Porter, R. N.; Sharma, R. D. J . Chem. Phys. 1965,43, 3259. (30) Kuntz, P. J.; N h e t h , E. M.; Pollnyi, J. C.; Wong, W. H. J . Chem. Phys. 1970, 52, 4654. (31) Karplus, M.; Porter, R. N.; Sharma, R. D. J . Chem. Phys. 1966,45, 3871. (32) Suplinskas, R. J . J . Chem. Phys. 1968, 49, 5046. (33) Malcolme-Lawes, D. J. J . Chem. Soc., Faraday Trans. 2 1972, 68, 1613. (34) Malcolme-Lawes, D. J . J . Chem. Phys. 1972, 57, 5522. (35) Malcolme-Lawes, D. J. J . Chem. SOC.,Faraday Trans. 2 1974, 70, 1942. (36) Krohn, K. A.; Parks, N. J.; Root, J. W. J . Chem. Phys. 1971, 55, 5771. (37) Krohn, K. A.; Parks, N . J.; Root, J. W. J . Chem. Phys. 1971, 55,
the concepts used in the hot atom chemistry are translated into the language of the neutron slowing down problem. This translation is given in eq 15 and 16 of section 2.3, by utilizing elastic scattering and absorption (reactive collision) cross sections of the hot atom with the components present. Under the conditions set by eq 15 and 16, the Estrup-Wolfgang theory is equivalent with the first-order Hurwitz approximation. This equivalence means that the low reactivity is not a necessary condition for the validity of the Estrup-Wolfgang formalism. All analytical approximations, leading to the collision density function used by Estrup and Wolfgang, require the ratio of the elastic scattering to total (elastic scattering + absorption) cross sections to be a slowly varying function of hot atom kinetic energy. As discussed by D r e ~ n e r , lthe ~ * present ~~ EstrupWolfgang (Weinberg-Wigner-Corngold and first-order Hurwitz) approximations are essentially good up to total reaction probability R 0.4-0.5; the validity of the second-order Hurwitz approximation is extended up to R 0.849. Thus, the Hurwitz conditions provide a wider applicability to the same formalism, as indicated by the results obtained from numerical ~nodeling.'~-'~ Hot replacement probabilities were calculated versus the composition of a binary mixture by utilizing specially chosen absorption (= replacement) cross sections in the approximate collision density function. Absorption cross sections were assumed to be constant functions in nonoverlapping intervals of the hot atom kinetic energy. On the condition that component 1 has a high absorption peak at lower energies and component 2 a narrow one at higher energies, the hot replacement yield from component 1, corrected for dilution, may drastically increase with the mole fraction of component 2, while the replacement yield from component 2, corrected for dilution, remains a slowly varying function of the same variable. Let component 1 be associated with o-C6H4C12 and component 2 with C6F6. Provided that the conditions set for their absorption peaks hold, this analytical approach properly reflects the concentration dependence of experimental yields from hot 38C1-for-C1and hot 38C1-for-Freplacement. Experimental absorption cross section data for these hot replacement processes, which could justify the above picture, are not known to us. For binding energies of c1 in C6H4C12and of F in c6F6 the relationship Eb(C1) < Eb(F) holds, and this relationship suggests that the absorption peaks are positioned as follows from the concentration dependence of the yields. Numerical calculations with parameters resembling binary mixtures of o-C6F4CI2and C6F6 have been carried out as well. In these calculations the collision density function was exactly determined by assuming Breit-Wigner type absorption cross section functions with maximum locations at the corresponding binding energies. The amplitude and width of the Breit-Wigner function are interdependent. The widths in the two absorption peaks were freely varied; corresponding amplitudes were determined from adjusting theoretical reaction probabilities to experimental yields arising from neat o-C6H4CI2and C6F6. Values of logarithmic energy loss per collision were calculated from mass ratios of colliding partners. All properties of experimental hot 38C1-for-Freplacement data can be reproduced by numerical slowing down/absorption calculations. Apart from the narrow cross section function of 38C1-for-C1replacement, numerical yields remain systematically and essentially below experimental results. In accordance with liquid-phase experiments, this difference may be ascribed to process(es) different from direct hot replacement.
-
-
Acknowledgment. This work has been supported by the AKA and OTKA foundations of the Hungarian Academy of Sciences. Registry No. 38Cl,14158-34-0; o-C,H,CI,, 95-50-1; C6F,, 392-56-3; neutron, 12586-31-1.
5785. . ..
(38) Coenen, H. H.; Machulla, H.-J.; Stiicklin, G. J . Am. Chem. SOC. 1977, 99, 2892.
(39) Dresner, L. Oak Ridge Natl. Lab., [Rep.]1958, ORNL-2594. Bibliographical data taken from ref 13.
