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Hydrogen-deuterium isotope effect in the λ-initiated chain reaction between carbon tetrachloride and cyclohexane in the liquid phase
Radiat. Phys. Chem. Vol. 15, pp. 223-230 Pergamon Press Ltd., 1980. Printed in Great Britain
HYDROGEN-DEUTERIUM ISOTOPE EFFECT IN THE y-INITIATED CHAIN REACTION BETWEEN CARBON TETRACHLORIDE AND CYCLOHEXANE IN THE LIQUID PHASE T. Q. NGUYEN, T. M. DANG and T. GAUMANN Physical Chemistry Department, Federal School of Technology, CH-1015 Lausanne, Switzerland
(Received 5 February 1979) Abstract--The hydrogen-deuterium kinetic isotope effect for the chain reaction between carbontetrachloride and cyclohexane was measured in the liquid phase, in the temperature range from 302 to 413 K. The rate expression for the two following reactions was obtained by competitive experiments in ternary solutions of CC14 in c-C6Ht2/c-C6Dt2 mixtures (2)
• CC13+ c-C6H12
~'CHCI3 + c - C 6 n l i.
(2')
• CCI3+ c-C6DI2
:~CDCI3+ c-C6DI I-
log~0 (k:/k2,)=-0.38-+0.04-(10.1-+0.3) kJ. tool-J/2.303 RT. The high value for the difference in activation energies between (2) and (2') suggests that tunneling is important in the proton abstraction reaction by "CCI3.
1. I N T R O D U C T I O N A CONSIDERABLE amount of experimental data on the transposition kinetic isotope effect exists now in the literature for metathetical reactions both in the gas phase "-3) and in the liquid phase. <4'5> Despite the simple nature of these reactions, there have been few studies on hydrogen atom abstraction from cycloalkanes, t2'6'7~ particularly in the condensed phasefl ~ Moreover, similar systems were studied over a limited temperature range and were not always free of side-reactions/9~ We chose to study carbon-tetrachloride and cyclohexane as the first of a series of experiments on isotopic effects in CCL/cycloalkane mixtures. Our choice was dictated by the two following considerations: (1) The fully deuteriated c - C 6 D 1 2 is readily available in our laboratory with high isotopic and chemical purity (10). (2) The mechanism and kinetics of the 3'radiolysis of carbon-tetrachloride and hydrocarbon mixtures are well understood, and it was shown ~'~'~2) that the formation of chloroform proceeds mainly via a free radical chain reaction at a high enough temperature (1)
c-C6Hll • + CCL-'-> c-C6H.C1 + • CC13
(2)
"CCI3 + c-C6HI2--->CHCI3 + c - C 6 H , .
Any parasitic reactions will be negligible compared to the main chain sequence. 2. E X P E R I M E N T A L
2.1 Materials The chemical purity of deuteriated cyclohexane is 99.9% with an isotopic purity of 99.6% as determined by mass spectrometry. Protiated cyclohexane was from Phillips 'Research grade' with 99.95% purity. Both were dried over molecular sieves and used without further purification. Commercial grade CHCI3, CCI4 and c-C6H, were separated from added stabilizer and impurities by preparative gas chromatography. CDCI3 (CIBA) has a degree of deuteriation of 99.8% and was used as received. All the samples were prepared from a stock solution containing 8433~,mol/g of c-C6DI2, l1981zmol/g of cC6H~2 and 570/tmol/g of CC14. Aliquots of 0.75 ml in Pyrex ampoules of 1.5 ml were degassed by the usual freeze-pump-and-thaw technique. The irradiations were done in a Gammacell 220 at a dose-rate of 0.15 Mrad/h. Irradiation temperatures were held constant to within-+ 0.5° in an ethylene glycol bath. 2.2 Analysis All the irradiation products were determined by quantitative gas chromatography, c-C6HIjCI/cC6D.CI were separated on a 6 m DC-710 packed column at 50°C and CHCI3/CDCI3 on a 100m capillary column of S q u a l a n e at 8°C, In accord with the results of Cartoni, <13> we found that the best separation of CHCI3 and CDC13 was obtained
223
224
T. Q. NGUYEN et al.
on a non-polar stationary phase. However the separation was not complete. To insure good. accuracy, a two parameter calibration curve depicted in Fig. 1, was used. The total amount of CHCI3 and CDC13 w a s obtained by surface integration and the ratio CHCi3/CDC13, from the ratio of their heights multiplied with the correction factor f. By chosing the calibration curve as close as possible to the experimental results, it was possible to keep the precision for CHCI3/CDCI3 within 1%. The non-chain products hexachloroethane, bicyclohexyl and 1,1,1-tris-chloro-methyl-cyclohexane, were separated on a 4 m Apiezion L column programmed from 80 to 150°C at a rate of l°/mn. 3. R E S U L T S As shown previously for the system CC14/nhexane (11) the stationary concentration of "CC13 under irradiation increases steadily with the mole fraction of CCIo and will eventually outpass the concentration of all the radicals; a plateau is reached at above 300/xmol CCL/g. In this range, the concentration dependence (and indirectly, dose dependence) is minimized and the reaction sequence greatly simplified c-C6Hm2-----','*c-C6HII" + . . .
Initiation c-C6Di2----~-*c-C6Dll. + . . .
g
(1)
Ic-C6HI 1"-~ CCI 4
~" c-C6HI ,CI + "CCI3
(1') Ic-C6DI l" + CC1 Propagation
>c-C6Dt ICI + "CC13
(2)
I'CCI3+ c-C6HI2------~ c-C6HI1.
(2')
[ ' C C i 3 + c-C6DI2-------~ CDCI2 + c-C6D1,.
(3)
Termination
{ 2.CC13
>C2C16.
According to this scheme, the primary kinetic isotope effect is given by
d[CHCl~]/dt × [c-C6D,2] k2/kz- ~
(I)
[Go(CHCI3)]. [c-C6D12]o - ~ x [c-C6H,2]o where Go is the radiolytic yield extrapolated to zero-dose and [c-C6D~2]o,, [c-C6H12]o the initial concentrations of these compounds. Four to five samples were irradiated at different doses for each temperature. At most 25% of the initial amount of CCL was consumed at the highest dose. The relation between reagent and product isotopic compositions, extent of reaction and the isotope effect is described by Melander. {14~ Under our experimental conditions, the radiolytic yields did not
!
I
f
CHCI 3 DCI 3
1.1
60..
s,o.,,~
1.0
0
I
I
I
I
2
4
6
8
pmol ( C H C I 3 + C D C I
10
3)
FIG. 1. Calibration curve for the ratio [CHCI3]/[CDCI3] as a function of the absolute amount of
chloroform injected on the column. Insert shows the gas chromatogram of CHCI3 and CDCL3. O, [CHCI3]/[CDCI3] = 8.7; O, [CHCI3]/[CDCI3] = 3.5; l , 1.1.
[CHCI3]/[CDCI3] = 2.0; [], [CHCI3]/[CDCI3] =
The y-initiated chain reaction between
carbon tetrachloride
TABLE
T/K
Dose- lO-tg/ eV.cm_ 3
G(C6H.C1) + G(C6DIICI)
413.2
0.17
835
0.33
850
0.50
826
0.66
823
0.83
811
0.38
312
0.77
308
1.15
307
1.53
301
1.92
297
0.99
I 18
384.2
357.2
336.2
317.7
301.7
1.98
118
2.98
106
3.97
111
4.96
105
3.02
48.7
5.66
45.8
9.02
50.4
12.0
47.4
15.1
47.9
6.66
23.7
13.3
23.0
20.0
23. I
26.7
22.7
12.2
12.4
24.5
I 1.9
36.7
12.1
49.0
I 1.7
*s = standard deviation.
R P C Vol. 15 No. 2-3---H
G(C6H~2CI) G(C6DI~CI)
786
1.325 1.327 1.329 1.329 1.323 1.313 1.3 lO 1.311 1.296 1.302
302
G(CHCI3) G(CDCI~)
[C6D,2]o - I[C6D~,CI] [C6HI2]0- ~[C6HIICI]
I.I 12 1.105 1.098 1.096 1.107 I. 110 I. 103 1.109 I. 120 I. 108
7.081
1.391 1.388 1.377 1.392 1.413 1.386 1.387 1.384 1.400 1.401
7.077
7.077
132
1.834 1.83O 1.842 1.853 1.823 1.826 1.804 1.841 1.798
1.692 1.704 1.610 1.594 1.489 1.496 1.418 1.407 1.368 1.370
46. I
2.268
7.089
44.8
2.245 2.294 2.305 2.337 2.253 2.226 2.260 2.285
7.125
1.663 1.670 1.426 1.439 1.323 1.282 1.170 1.184
21. I
2.838 2.748 2.856 2.824 2.777 2.751 2.751 2.765
7.901
-1.237 1.040 1.045 0.871 0.876 0.851 0.840
9,2
3.245
7.080
8.7
3.092 3.094 3.015 3.029 2.905 2.905
850
852 860 857
300 306 305 305
114 1.603 1.578 1.567 1.552 1.537 1.548 1.533 1.501
133 131 141
49.9 47.9 49.6
21.7 21.1 21.9
9.5 9.3
225
I.
G(CHCI3) + G(CDC13)
1.087 1.086 1.085 1.078 1.073 1.078 1.064 1.073 1.055 1.056
and cyclohexane
kJkr ± s* 7.9 ± 0.1
7.124
7.163 7.204 7.242 I0.0 ± 0. I
7.114 7.150 7.185 7.219
13.1 ±0.I
7.115 7.144 7.186 7.212 16.4 ± 0.3
7.184 7.218 7.266 19.9 ± 0.3
7.132 7.171 7.171
7.112 7.139 7.164
23.5 ± 0.2
226
T. Q. NGUYEN et al.
depend on dose and the ratio k2/kr, can be approximated by
k2/k2, = G(CHCi3) × [c-C6D,2]o- ~[c-C6D,,CI]
(II)
G(CDCI3)
[c-C6H,2]0-~[c-C6H,,CI]"
The above expression was found to be dose-independent (Table 1) and extrapolation to zero-dose is trival except for the lowest irradiation temperature (28.5°). The data for six irradiation temperatures are shown in Table 1. The values for k2/kz in Table 1 are averaged over the dose, except at 28.5 ° where extrapolation to zero-dose by linear regression is given. The Arrhenius plot of kJk2, is shown on Fig. 2. Exponential least-square regression of the experimental data, used with the Yntema's approximation "~ to account for the change of the variable k to In k, gives the following values for the isotope effect
log,o(k2/k2,)
=
0.38 -+ 0.04 - (10.1 -+0.3) kJ.mol-~/2.303 RT.
I
(5)
c-C6Ht, .(or c-C6D1,.) + CHCi3 --) c-C6H,2(or c-C6DI,H) + "CCI3 c-C6Hij.(or c-C6D, i.) + CDCI3 ~ c-C6H, ,D(or c-C6D,2) + 'CCI3
(b) CHCI3 and CDCI3 can be formed by dismutation between "CCI3 and c-C6H,l.(or c-C6D,,.)
(6c)
/~
c-C6H,ICC13
•CCI3 + c-C6HII.--~x (6d)
\ - - ~ c-C6H w + CHCI3
(6C')
/---> c-C6Dt ICC13 "CCI3 + c-C6D, I . - -
(6d')
~ - - - ~ c C6Dw + CDCI3
-
Its quantitative interpretation needs some knowledge of the sources of error to be expected, such as (a) The products relevant to the isotope effect can undergo back reactions "6>
25
(4)
I
I
2O
15 ¢4
Rate constants for reaction (l) is over 3 orders of magnitude higher than rate constant for reaction (2) in the range of temperatures investigated. "2''7) Back reactions (4) and (5) are endothermic by 7-11 kJ-moi -~ (Table 4) and therefore, their competition with reaction (l) can be disregarded. That (4) and (5) are indeed negligible was shown by the absence of a dose dependence for expression (II) at all temperatures except the lowest one. Care was taken to avoid complications by reaction (6d) by maximizing the chain length and minimizing the concentration of cyciohexyl radicals. From known rate constants for reactions (1) and (2), O2J7) extra yields of chloroform by reaction (6d) can be estimated to amount to less than 0.1% of the yield of CHC13 and CDCI3. Isotopic scrambling between the starting products, catalysed by the hydrogen chloride formed under irradiation may be another source of complication. "8~ The overall reaction can be formulated as following (7)
lO
HCI or D C I c - C 6 H I 2 + c-C6D12
)
c-C6H,~D + c-C6D,,H.
I 2.4
I 2.8
I 3.2
103/T
FIG. 2. The Arrhenius plot of kflk2. The vertical bars denote the standard deviation.
Within the concentration limits of hydrochloric acid produced by radiolysis (G(HCl) ca. (5)) competition of (7) with the chain sequence (1) and (2) is inefficient and can be neglected according to the results of Fletcher and Freeman. "8) Even if this were not true, the replacement of c-C6HI2 and c-C6D12 by c-C6Hf~ and c-C6DnH would be unlikely to drastically change the experimental ratio
The y-initiated chain reaction between carbon tetrachloride and cyclohexane
k2/k2, given by expression (II) since only secondary kinetic isotope effect are involved"9~
i
I
I
I
I
I
I
227 I
I
I
G ,300
(7)
\c/H+\c/D /
\H
(k2) In
/
\D
(k21k2,)
contrast
to
,2\C/H
the
/
.240
\D 42(3
(knlkD intra. = k21kr). dose-independent
~
ratio
(III)
k , / k , , - Go(c-C6H,~CI) x [c-C6D,,.] Go(c-C6D,,CI)
[c-C6H,,']"
The ratio [c-C6H~.]/[c-C6D,:] can be computed from the distribution of the recombination products, either from bicyclohexyl or from trichIoromethylcyciohexane products. The yields of the former contain a large contribution from reactions in the spur ~2°' and thus, are unsuitable for our purpose. The trichloromethyl-cyclohexane products result from the cross-recombination between .CC13 and cyclohexyl radical (reactions 6d) and (6d')). TABLE
CC14 tz mol/g
%c-C6Di2
Dose.10-19 eV.cm_3
626 100 314 600 680 780 590 670 600
0 0 52.77 60.29 70.27 81.51 87.56 94.89 100
0.37-1.12 0.42-1.25 1.1 I-3.33 1.37--4.11 1.63-4.88 2.68--8.03 1.77-5.31 3.17-9.50 3.4%10.5
G(C6H.C1) a + G(C6DnCI) 550 550 370 314 221 152 114 82 55
180
x~.\
30C
G(CHC13)/G(CDC13), G(c-C6H,~CI)/G(c-C6D,~CI) decreases with dose and this effect is particularly obvious for irradiations at low temperature and at low concentrations of CC14 (Table 3). However, the effect is minimal under our condition and consequently does not affect the chloroform yields. Thus, it is unlikely to spoil the precision of our results. The dependence of k2/kz., on the mole fraction of the deuteriated compound was investigated. The data presented in Table 2 and Fig. 3 demonstrate the validity of our assumptions. Experiments were carried out to measure the secondary kinetic isotope effect at the first step of the chain reaction given by kdkv. Under homogeneous kinetic assumptions, an expression analogous to (II) can be derived
CHCI3
•.
.12o
18C
COCl 3
~
Ok, ~" 1 ; 4 60
50
100% C6H12
100%
C6D12
FIG. 3. Dependence of the yields of chain radiolytic products on the mole fraction of c-C6D~2. Right-hand scale refers to the G values of protiated compounds, left-hand scale to the G values of deuteriated compounds. Since .CC13 are formed at random by secondary reactions outside the spur, especially at low concentrations in CC14,"~) trichloromethyl-cyclohexane are more representative of the bulk distribution of radicals. After substitution, the following expression was derived
kdkv = Go(c-C6H,CI) × Go(c-C6DIICCI3) Go(c-C6D,CI)
Go(c-C6H,CCI3)
1 + k6a,/k6~, x l + k6a/k6~"
(IV)
For this series of measurements, it is desirable to have a high yield of trichloromethyl-cyclohexane 2.
G(C6H.CI)a G(C6DnCI) --10.1 7.58 5.05 2.64 1.60 0.63 --
"The results given are the average value over three different doses.
G(CHCI3)a + G(CDCI3)
G(CHCI3)" G(CDCI3)
k2/k2'a
600
--
--
550 350 278 218 165
-12.3 8.98 6.03 3.08
-13.8 13.6 14.2 13.6
81
0.75
14.0
45
--
122
1.90
13.4 --
T. Q. NGUYEN et al.
228
TABLE 3.
T/K
Dose. 10-19/ eV'cm -~
G(C6HHCI) + G(C6DllCI)
G(C6HNCI) G(C6DNCI)
G(CrHIICCI3)* G(C6HHCC13) + G(C6DNCCI3) G(CrDNCCI3)
kl/kv
357.9
0.45 0.91 1.36 1.81
66.2 57.2 47.0 37.3
3.98 2.96 2.54 2.33
1.09 1.28 1.30 1.22
3.50 2.59 2.18 1.76
0.93 0.94 0.96 1.09
335.7
0.68 1.36 2.04 2.72
34.1 3 !.8 30.1 25.6
3.33 2.67 2.02 1.82
0.66 0.81 1.00 1.01
2.83 2.67 1.89 1.55
0.97 0.82 0.88 0.97
317.7
0.91 1.81 2.72 3.63
19.2 19.0 18.5 17.0
2.42 2.23 1.54
0.64 0.73 0.75 0.97
1.95 2.10 1.97 1.34
-0.95 0.93 0.95
--
*Trichloromethyl-benzene was used as standard for calibration. a n d t h e e x p e r i m e n t a l c o n d i t i o n s w e r e modified appropriately: a low c o n c e n t r a t i o n in C C L ( 1 1 . 0 / z m o l / g o f C C L in 7 5 5 4 / z m o l / g c-C6D12 a n d 3 2 2 8 / z m o l / g c-C~Hz2) w a s u s e d c o n c u r r e n t l y w i t h a high d o s e - r a t e (0.67 M r a d / h ) . R e s u l t s at t h r e e t e m p e r a t u r e s a r e g i v e n in T a b l e 3. T h e m o l e c u l a r formation of chloro-cyclohexane precludes the use o f d a t a o b t a i n e d at i r r a d i a t i o n t e m p e r a t u r e b e l o w 40 °. T o c o m p u t e kJkv, w e u s e d f o r t h e ratio k6a/k6~ t h e s a m e v a l u e o f 2.4 o b t a i n e d f r o m t h e r e c o m -
b i n a t i o n r e a c t i o n b e t w e e n .CC13 a n d n-CrHz3, in t h e liquid p h a s e , °~) k6a,/kr¢, is u n k n o w n b u t m o s t likely a b o u t 30--40% smaller t h a n ksd/k6c (2"21) a n d w e c h o s e a v a l u e of 1.8. T h e a v e r a g e v a l u e o f kdkv is 0.95 __+0.06. T h u s , n o s e c o n d a r y i s o t o p e effect c a n b e d e r i v e d f r o m o u r data. 4. D I S C U S S I O N T h e d i f f e r e n c e in t h e a c t i v a t i o n e n e r g y o f r e a c t i o n s (2) a n d (2') a t t r i b u t e d to a n i s o t o p e effect is
TABLE c-CrHH-H a DoN (kJ.mo1-1) ~ (cm -1) De (kJ'mol -~) R (/~) F (mdyn-/~ -)) P=9 /3 (A-~) h
393.0 2933 410.5 1.12 4.68 1.05~ 1.85
4,
CCI3-H 399.6 3019~ 417.6 1.07e 4.96 1.05g 1.89
c-C6HH-CCI3 326.3 ° 920a 331.8 1.54e 2.99 -1.65
"Reference (6) and citations therein. bs. W. Benson Thermochemical Kinetics. Wiley, New York, 1968. cPs for CDCI3 = 2256cm -~. L. D. Spicer and C. D. PouRer. In Physical Chemistry, an Advanced Treatise, (Edited by H. Eyring), Vol. VII. Academic Press, New York, 1975. dG. Herzberg. Infrared and Raman Spectra of Polyatomic Molecules. I lth Edn. Van Nostrand, Amsterdam, 1964.
eTables of lnteratomic Distances and Configuration in Molecules and Ions. Special publication No. 18. The Chemical Society, London, 1965. fThe force constant F is estimated from the relation F = (2/zv)2. ~ where /z is the reduced mass calculated from the masses of end atoms. ~Following Johnston (Ref. [25], p. 346), the bond energy index p is set equal to q and used as an adjustable parameter to fit V:~ (Table 6) to the observed activation energy of 46.4 kJ.moi -~ for k2 (12). h/3 is the Morse parameter calculated from the relation /3 = rr~ (2~/D~) I/2.
The y-initiated chain reaction between carbon tetrachloride and cyclohexane
229
TABLE 5. (c-C6HII) . . .
n' m* V*(kJ-mol 1) R),B(fk) R~c(,~.) F),a(mdyn-,~-t) F~c(mdyn.~ ~) F~c(mdyn.A ~) F,(10"erg-rad -2) ~(cm i)~ ~(cm-I) ~ ~(cm-~) ~
H ... (CC13)
(c-C6DII) . ..
0.530 0.470 46.4 1.29 1.27 1.25 0.86 1.63 0.055 622 491 1425i
D . . . (CC13)
0.530 0.470 46.4 1.29 1.27 1.25 0.86 1.63 0.055 586 354 I092i
n t, m*: Bond orders of the A-B and B-C bond, respectively; A refers to the group and C, to (CC13). R*: Bond length obtained from the Pauling's bond-order-bond-length relationship R ~ = R -0.26 In (bond order). "The frequencies are calculated by using the linear triatomic model C . . . H (or D ) . . . C.
( c - C 6 H l l ) o r (c-C6DII), B to (H) o r (D)
T A B L E 6.
°C
(k~/kz) exptl,
(k2/kr) ~ calc.
1]iH/1/ie
(F~t/FD)*
(Fm/Fso)*
(Fb:q/Fbo) 2 2 :~
(F~o/rm)"
140.0 111.0 84.0 63.0 44.5 28.5
7.93 10.0 13.1 16.4 19.9 23.5
5.00 5.61 6.58 7.61 9.21 11.1
1.31 1.31 1.31 1.31 1.31 1.31
1.60 1.70 1.83 1.99 2.25 2.53
0.980 0.977 0.973 0.970 0.%7 0.964
0.893 0.878 0.861 0.845 0.829 0.814
2.67 2.94 3.26 3.57 3.91 4.26
~kJk~_, = V-~D'\F--D/'\F~o F-~oJ" \r~-u/' F = (½u)/sinh(~u) with u = hv/kT. F* represents the tunneling correction factor; F ~ and Fr are quantum correction factors for the real vibrations of the activated complex and reactant respectively; subscripts s and b stand for stretching and bending modes. (see Ref. [23] for more details).
10.1_0.3 kJ-mol 1. T o a first approximation this c o r r e s p o n d s to the zero-point energy difference of the reactants. If we a s s u m e that a transition state involves a C - H stretching f r e q u e n c y (Table 4), we should o b s e r v e an activation energy difference of 4 . 3 k J . m o l -I. T h e possibility to include one or several bending m o d e s in the transition state has b e e n discussed by Bell. (22) H e estimates that such a contribution can a m o u n t at most 1.1 kJ-mo1-1 for C-H bonds. A quantum mechanical correction for tunneling tends to increase the apparent difference in activation energy. T h e results calculated for a linear three-atomic activated c o m p l e x model, based on the figures given in Table 4, are p r e s e n t e d in Tables 5 and 6, using the B E B O m e t h o d for calculations. (23~ Although the results with such a simple model c o r r e s p o n d only to a crude approximation, they do give an estimation of
the magnitude of the tunnel effect. Thus, we estimate with 4.3 kJ.mo1-1 for the contribution of the stretching mode, l . l k J . m o i -1 for tunneling, a value of 9 . 4 k J . m o l -I for the difference in the activation energies. The a g r e e m e n t with the e x p e r i m e n t data is perhaps fortuitous, but it shows that tunneling may account for an important contribution in the activation energy. T h e ratio of the corresponding Arrhenius factors is 0.42-+ 0.04. E x t e n d e d calculations by S c h n e i d e r and Stern (24~ for isotope effect without tunneling limited the range of this ratio b e t w e e n 0.71 and 1.67. T h e low experimental value is again an indication for a contribution by a tunnel effect to the abstraction reaction (2). H o w e v e r it should be kept in mind that an underestimation of this ratio could be c o m p e n s a t e d for by an o v e r e s t i m a t i o n in the difference of the activation energies.
230
T. Q. NGUYEN et al.
Acknowledgements--We thank the Swiss National Science Foundation for a grant and Prof. L. A. Rajbenbach for helpful discussions. REFERENCES 1. A. F. TROTMAN-DICKENSON and G. S. MILNE. Tables o f Bimolecular Gas Reactions. National Bureau of Standards, U.S. Government Printing Office, Washington, D.C., No. NSRDS-NBS 9, 1967. 2. P. GRAY, A. A. HEROD and A. JONES, Chem. Rev. 1971, 71,247. 3. F. S. KLEIN, Ann. Rev. Phys. Chem. 1975, 26, 191. 4. S. E. SCHEPPELE, Chem. Rev. 1972, 72, 511. 5. E. BUNCEL and C. C. LEE (Editor), Isotopes in Organic Chemistry Series, Vol. 2. Elsevier, Amsterdam, 1976. 6. N. FUJISAKI and T. G,~UMANN, Ber. Bunsenges. Physik. Chem. 1977, 81,544. 7. (a) E. G. SPITTLER, S. J. P. JORDAN, M. DORFMAN and M. C. SAUER, JR., J. phys. Chem. 1963, 67, 2235. (b) R. R. HENTZ, J. Y. CHANG and M. BURTON, J. phys. Chem 1965, 69, 2027. 8. J. R. McNESBY, A. S. GORDON and S. R. SMITH, J. Am. Chem. Soc. 1956, 78, 1287. 9. S. HAUTECLOQUE, C. R. Acad. Sci. Paris (C), 1971, 272, 2094.
10. T. GAUMANN, H. Oz and O. PIRINGER, Heir. Chim. Acta 1978, 61, 258. 11. T. Q. NGUYEN and T. G~UMANN. Radiat. Phys. Chem. 1978, 11, 183. 12. M.G. KATZ, A. HOROWITZand L. A. RAJBENBACH,Int. J. Chem. Kinet. 1975, 7, 183. 13. G. P. CARTONI,Anal. Chem. 1967, 39, 1618. 14. L. MELANDER, Isotope Effects on Reaction Rates, p. 48ff. Ronald Press, New York, 1960. 15. R. J. CVETANOVI~ and D. L. SINGLETON,Int. J. Chem. Kinet. 1977, 9, 481. 16. T. Q. NGUYEN and T. GA,UMANN, Radiat. Phys. Chem. 1977, 10, 263. 17. M. G. KATZ, G. BARUCH and L. A. RAJBENBACH, Int. J. Chem. Kinet. 1976, 8, 131. 18. J. W. FLETCHER and G. R. FREEMAN, Can. J. Chem. 1966, 44, 2645. 19. M. WOLFSBERG, Ann. Rev. Phys. Chem. 1969, 20, 449. 20. H. WIDMER and T. GAUMANN, Heir. Chim. Acta 1963, 46, 2766. 21. N. FUJISAKI and T. GA.UMANN, to be published. 22. R. P. BELL, Chem. Soc. Rev. 1974, 3, 513. 23. H. S. JOHNSTON, Gas Phase Reaction Rate Theory. Ronald Press, New York, 1966. 24. M. E. SCHNEIDER and M. J. STERN, J. Am. Chem. Soc. 1972, 94, 1517.
HYDROGEN-DEUTERIUM ISOTOPE EFFECT IN THE y-INITIATED CHAIN REACTION BETWEEN CARBON TETRACHLORIDE AND CYCLOHEXANE IN THE LIQUID PHASE T. Q. NGUYEN, T. M. DANG and T. GAUMANN Physical Chemistry Department, Federal School of Technology, CH-1015 Lausanne, Switzerland
(Received 5 February 1979) Abstract--The hydrogen-deuterium kinetic isotope effect for the chain reaction between carbontetrachloride and cyclohexane was measured in the liquid phase, in the temperature range from 302 to 413 K. The rate expression for the two following reactions was obtained by competitive experiments in ternary solutions of CC14 in c-C6Ht2/c-C6Dt2 mixtures (2)
• CC13+ c-C6H12
~'CHCI3 + c - C 6 n l i.
(2')
• CCI3+ c-C6DI2
:~CDCI3+ c-C6DI I-
log~0 (k:/k2,)=-0.38-+0.04-(10.1-+0.3) kJ. tool-J/2.303 RT. The high value for the difference in activation energies between (2) and (2') suggests that tunneling is important in the proton abstraction reaction by "CCI3.
1. I N T R O D U C T I O N A CONSIDERABLE amount of experimental data on the transposition kinetic isotope effect exists now in the literature for metathetical reactions both in the gas phase "-3) and in the liquid phase. <4'5> Despite the simple nature of these reactions, there have been few studies on hydrogen atom abstraction from cycloalkanes, t2'6'7~ particularly in the condensed phasefl ~ Moreover, similar systems were studied over a limited temperature range and were not always free of side-reactions/9~ We chose to study carbon-tetrachloride and cyclohexane as the first of a series of experiments on isotopic effects in CCL/cycloalkane mixtures. Our choice was dictated by the two following considerations: (1) The fully deuteriated c - C 6 D 1 2 is readily available in our laboratory with high isotopic and chemical purity (10). (2) The mechanism and kinetics of the 3'radiolysis of carbon-tetrachloride and hydrocarbon mixtures are well understood, and it was shown ~'~'~2) that the formation of chloroform proceeds mainly via a free radical chain reaction at a high enough temperature (1)
c-C6Hll • + CCL-'-> c-C6H.C1 + • CC13
(2)
"CCI3 + c-C6HI2--->CHCI3 + c - C 6 H , .
Any parasitic reactions will be negligible compared to the main chain sequence. 2. E X P E R I M E N T A L
2.1 Materials The chemical purity of deuteriated cyclohexane is 99.9% with an isotopic purity of 99.6% as determined by mass spectrometry. Protiated cyclohexane was from Phillips 'Research grade' with 99.95% purity. Both were dried over molecular sieves and used without further purification. Commercial grade CHCI3, CCI4 and c-C6H, were separated from added stabilizer and impurities by preparative gas chromatography. CDCI3 (CIBA) has a degree of deuteriation of 99.8% and was used as received. All the samples were prepared from a stock solution containing 8433~,mol/g of c-C6DI2, l1981zmol/g of cC6H~2 and 570/tmol/g of CC14. Aliquots of 0.75 ml in Pyrex ampoules of 1.5 ml were degassed by the usual freeze-pump-and-thaw technique. The irradiations were done in a Gammacell 220 at a dose-rate of 0.15 Mrad/h. Irradiation temperatures were held constant to within-+ 0.5° in an ethylene glycol bath. 2.2 Analysis All the irradiation products were determined by quantitative gas chromatography, c-C6HIjCI/cC6D.CI were separated on a 6 m DC-710 packed column at 50°C and CHCI3/CDCI3 on a 100m capillary column of S q u a l a n e at 8°C, In accord with the results of Cartoni, <13> we found that the best separation of CHCI3 and CDC13 was obtained
223
224
T. Q. NGUYEN et al.
on a non-polar stationary phase. However the separation was not complete. To insure good. accuracy, a two parameter calibration curve depicted in Fig. 1, was used. The total amount of CHCI3 and CDC13 w a s obtained by surface integration and the ratio CHCi3/CDC13, from the ratio of their heights multiplied with the correction factor f. By chosing the calibration curve as close as possible to the experimental results, it was possible to keep the precision for CHCI3/CDCI3 within 1%. The non-chain products hexachloroethane, bicyclohexyl and 1,1,1-tris-chloro-methyl-cyclohexane, were separated on a 4 m Apiezion L column programmed from 80 to 150°C at a rate of l°/mn. 3. R E S U L T S As shown previously for the system CC14/nhexane (11) the stationary concentration of "CC13 under irradiation increases steadily with the mole fraction of CCIo and will eventually outpass the concentration of all the radicals; a plateau is reached at above 300/xmol CCL/g. In this range, the concentration dependence (and indirectly, dose dependence) is minimized and the reaction sequence greatly simplified c-C6Hm2-----','*c-C6HII" + . . .
Initiation c-C6Di2----~-*c-C6Dll. + . . .
g
(1)
Ic-C6HI 1"-~ CCI 4
~" c-C6HI ,CI + "CCI3
(1') Ic-C6DI l" + CC1 Propagation
>c-C6Dt ICI + "CC13
(2)
I'CCI3+ c-C6HI2------~ c-C6HI1.
(2')
[ ' C C i 3 + c-C6DI2-------~ CDCI2 + c-C6D1,.
(3)
Termination
{ 2.CC13
>C2C16.
According to this scheme, the primary kinetic isotope effect is given by
d[CHCl~]/dt × [c-C6D,2] k2/kz- ~
(I)
[Go(CHCI3)]. [c-C6D12]o - ~ x [c-C6H,2]o where Go is the radiolytic yield extrapolated to zero-dose and [c-C6D~2]o,, [c-C6H12]o the initial concentrations of these compounds. Four to five samples were irradiated at different doses for each temperature. At most 25% of the initial amount of CCL was consumed at the highest dose. The relation between reagent and product isotopic compositions, extent of reaction and the isotope effect is described by Melander. {14~ Under our experimental conditions, the radiolytic yields did not
!
I
f
CHCI 3 DCI 3
1.1
60..
s,o.,,~
1.0
0
I
I
I
I
2
4
6
8
pmol ( C H C I 3 + C D C I
10
3)
FIG. 1. Calibration curve for the ratio [CHCI3]/[CDCI3] as a function of the absolute amount of
chloroform injected on the column. Insert shows the gas chromatogram of CHCI3 and CDCL3. O, [CHCI3]/[CDCI3] = 8.7; O, [CHCI3]/[CDCI3] = 3.5; l , 1.1.
[CHCI3]/[CDCI3] = 2.0; [], [CHCI3]/[CDCI3] =
The y-initiated chain reaction between
carbon tetrachloride
TABLE
T/K
Dose- lO-tg/ eV.cm_ 3
G(C6H.C1) + G(C6DIICI)
413.2
0.17
835
0.33
850
0.50
826
0.66
823
0.83
811
0.38
312
0.77
308
1.15
307
1.53
301
1.92
297
0.99
I 18
384.2
357.2
336.2
317.7
301.7
1.98
118
2.98
106
3.97
111
4.96
105
3.02
48.7
5.66
45.8
9.02
50.4
12.0
47.4
15.1
47.9
6.66
23.7
13.3
23.0
20.0
23. I
26.7
22.7
12.2
12.4
24.5
I 1.9
36.7
12.1
49.0
I 1.7
*s = standard deviation.
R P C Vol. 15 No. 2-3---H
G(C6H~2CI) G(C6DI~CI)
786
1.325 1.327 1.329 1.329 1.323 1.313 1.3 lO 1.311 1.296 1.302
302
G(CHCI3) G(CDCI~)
[C6D,2]o - I[C6D~,CI] [C6HI2]0- ~[C6HIICI]
I.I 12 1.105 1.098 1.096 1.107 I. 110 I. 103 1.109 I. 120 I. 108
7.081
1.391 1.388 1.377 1.392 1.413 1.386 1.387 1.384 1.400 1.401
7.077
7.077
132
1.834 1.83O 1.842 1.853 1.823 1.826 1.804 1.841 1.798
1.692 1.704 1.610 1.594 1.489 1.496 1.418 1.407 1.368 1.370
46. I
2.268
7.089
44.8
2.245 2.294 2.305 2.337 2.253 2.226 2.260 2.285
7.125
1.663 1.670 1.426 1.439 1.323 1.282 1.170 1.184
21. I
2.838 2.748 2.856 2.824 2.777 2.751 2.751 2.765
7.901
-1.237 1.040 1.045 0.871 0.876 0.851 0.840
9,2
3.245
7.080
8.7
3.092 3.094 3.015 3.029 2.905 2.905
850
852 860 857
300 306 305 305
114 1.603 1.578 1.567 1.552 1.537 1.548 1.533 1.501
133 131 141
49.9 47.9 49.6
21.7 21.1 21.9
9.5 9.3
225
I.
G(CHCI3) + G(CDC13)
1.087 1.086 1.085 1.078 1.073 1.078 1.064 1.073 1.055 1.056
and cyclohexane
kJkr ± s* 7.9 ± 0.1
7.124
7.163 7.204 7.242 I0.0 ± 0. I
7.114 7.150 7.185 7.219
13.1 ±0.I
7.115 7.144 7.186 7.212 16.4 ± 0.3
7.184 7.218 7.266 19.9 ± 0.3
7.132 7.171 7.171
7.112 7.139 7.164
23.5 ± 0.2
226
T. Q. NGUYEN et al.
depend on dose and the ratio k2/kr, can be approximated by
k2/k2, = G(CHCi3) × [c-C6D,2]o- ~[c-C6D,,CI]
(II)
G(CDCI3)
[c-C6H,2]0-~[c-C6H,,CI]"
The above expression was found to be dose-independent (Table 1) and extrapolation to zero-dose is trival except for the lowest irradiation temperature (28.5°). The data for six irradiation temperatures are shown in Table 1. The values for k2/kz in Table 1 are averaged over the dose, except at 28.5 ° where extrapolation to zero-dose by linear regression is given. The Arrhenius plot of kJk2, is shown on Fig. 2. Exponential least-square regression of the experimental data, used with the Yntema's approximation "~ to account for the change of the variable k to In k, gives the following values for the isotope effect
log,o(k2/k2,)
=
0.38 -+ 0.04 - (10.1 -+0.3) kJ.mol-~/2.303 RT.
I
(5)
c-C6Ht, .(or c-C6D1,.) + CHCi3 --) c-C6H,2(or c-C6DI,H) + "CCI3 c-C6Hij.(or c-C6D, i.) + CDCI3 ~ c-C6H, ,D(or c-C6D,2) + 'CCI3
(b) CHCI3 and CDCI3 can be formed by dismutation between "CCI3 and c-C6H,l.(or c-C6D,,.)
(6c)
/~
c-C6H,ICC13
•CCI3 + c-C6HII.--~x (6d)
\ - - ~ c-C6H w + CHCI3
(6C')
/---> c-C6Dt ICC13 "CCI3 + c-C6D, I . - -
(6d')
~ - - - ~ c C6Dw + CDCI3
-
Its quantitative interpretation needs some knowledge of the sources of error to be expected, such as (a) The products relevant to the isotope effect can undergo back reactions "6>
25
(4)
I
I
2O
15 ¢4
Rate constants for reaction (l) is over 3 orders of magnitude higher than rate constant for reaction (2) in the range of temperatures investigated. "2''7) Back reactions (4) and (5) are endothermic by 7-11 kJ-moi -~ (Table 4) and therefore, their competition with reaction (l) can be disregarded. That (4) and (5) are indeed negligible was shown by the absence of a dose dependence for expression (II) at all temperatures except the lowest one. Care was taken to avoid complications by reaction (6d) by maximizing the chain length and minimizing the concentration of cyciohexyl radicals. From known rate constants for reactions (1) and (2), O2J7) extra yields of chloroform by reaction (6d) can be estimated to amount to less than 0.1% of the yield of CHC13 and CDCI3. Isotopic scrambling between the starting products, catalysed by the hydrogen chloride formed under irradiation may be another source of complication. "8~ The overall reaction can be formulated as following (7)
lO
HCI or D C I c - C 6 H I 2 + c-C6D12
)
c-C6H,~D + c-C6D,,H.
I 2.4
I 2.8
I 3.2
103/T
FIG. 2. The Arrhenius plot of kflk2. The vertical bars denote the standard deviation.
Within the concentration limits of hydrochloric acid produced by radiolysis (G(HCl) ca. (5)) competition of (7) with the chain sequence (1) and (2) is inefficient and can be neglected according to the results of Fletcher and Freeman. "8) Even if this were not true, the replacement of c-C6HI2 and c-C6D12 by c-C6Hf~ and c-C6DnH would be unlikely to drastically change the experimental ratio
The y-initiated chain reaction between carbon tetrachloride and cyclohexane
k2/k2, given by expression (II) since only secondary kinetic isotope effect are involved"9~
i
I
I
I
I
I
I
227 I
I
I
G ,300
(7)
\c/H+\c/D /
\H
(k2) In
/
\D
(k21k2,)
contrast
to
,2\C/H
the
/
.240
\D 42(3
(knlkD intra. = k21kr). dose-independent
~
ratio
(III)
k , / k , , - Go(c-C6H,~CI) x [c-C6D,,.] Go(c-C6D,,CI)
[c-C6H,,']"
The ratio [c-C6H~.]/[c-C6D,:] can be computed from the distribution of the recombination products, either from bicyclohexyl or from trichIoromethylcyciohexane products. The yields of the former contain a large contribution from reactions in the spur ~2°' and thus, are unsuitable for our purpose. The trichloromethyl-cyclohexane products result from the cross-recombination between .CC13 and cyclohexyl radical (reactions 6d) and (6d')). TABLE
CC14 tz mol/g
%c-C6Di2
Dose.10-19 eV.cm_3
626 100 314 600 680 780 590 670 600
0 0 52.77 60.29 70.27 81.51 87.56 94.89 100
0.37-1.12 0.42-1.25 1.1 I-3.33 1.37--4.11 1.63-4.88 2.68--8.03 1.77-5.31 3.17-9.50 3.4%10.5
G(C6H.C1) a + G(C6DnCI) 550 550 370 314 221 152 114 82 55
180
x~.\
30C
G(CHC13)/G(CDC13), G(c-C6H,~CI)/G(c-C6D,~CI) decreases with dose and this effect is particularly obvious for irradiations at low temperature and at low concentrations of CC14 (Table 3). However, the effect is minimal under our condition and consequently does not affect the chloroform yields. Thus, it is unlikely to spoil the precision of our results. The dependence of k2/kz., on the mole fraction of the deuteriated compound was investigated. The data presented in Table 2 and Fig. 3 demonstrate the validity of our assumptions. Experiments were carried out to measure the secondary kinetic isotope effect at the first step of the chain reaction given by kdkv. Under homogeneous kinetic assumptions, an expression analogous to (II) can be derived
CHCI3
•.
.12o
18C
COCl 3
~
Ok, ~" 1 ; 4 60
50
100% C6H12
100%
C6D12
FIG. 3. Dependence of the yields of chain radiolytic products on the mole fraction of c-C6D~2. Right-hand scale refers to the G values of protiated compounds, left-hand scale to the G values of deuteriated compounds. Since .CC13 are formed at random by secondary reactions outside the spur, especially at low concentrations in CC14,"~) trichloromethyl-cyclohexane are more representative of the bulk distribution of radicals. After substitution, the following expression was derived
kdkv = Go(c-C6H,CI) × Go(c-C6DIICCI3) Go(c-C6D,CI)
Go(c-C6H,CCI3)
1 + k6a,/k6~, x l + k6a/k6~"
(IV)
For this series of measurements, it is desirable to have a high yield of trichloromethyl-cyclohexane 2.
G(C6H.CI)a G(C6DnCI) --10.1 7.58 5.05 2.64 1.60 0.63 --
"The results given are the average value over three different doses.
G(CHCI3)a + G(CDCI3)
G(CHCI3)" G(CDCI3)
k2/k2'a
600
--
--
550 350 278 218 165
-12.3 8.98 6.03 3.08
-13.8 13.6 14.2 13.6
81
0.75
14.0
45
--
122
1.90
13.4 --
T. Q. NGUYEN et al.
228
TABLE 3.
T/K
Dose. 10-19/ eV'cm -~
G(C6HHCI) + G(C6DllCI)
G(C6HNCI) G(C6DNCI)
G(CrHIICCI3)* G(C6HHCC13) + G(C6DNCCI3) G(CrDNCCI3)
kl/kv
357.9
0.45 0.91 1.36 1.81
66.2 57.2 47.0 37.3
3.98 2.96 2.54 2.33
1.09 1.28 1.30 1.22
3.50 2.59 2.18 1.76
0.93 0.94 0.96 1.09
335.7
0.68 1.36 2.04 2.72
34.1 3 !.8 30.1 25.6
3.33 2.67 2.02 1.82
0.66 0.81 1.00 1.01
2.83 2.67 1.89 1.55
0.97 0.82 0.88 0.97
317.7
0.91 1.81 2.72 3.63
19.2 19.0 18.5 17.0
2.42 2.23 1.54
0.64 0.73 0.75 0.97
1.95 2.10 1.97 1.34
-0.95 0.93 0.95
--
*Trichloromethyl-benzene was used as standard for calibration. a n d t h e e x p e r i m e n t a l c o n d i t i o n s w e r e modified appropriately: a low c o n c e n t r a t i o n in C C L ( 1 1 . 0 / z m o l / g o f C C L in 7 5 5 4 / z m o l / g c-C6D12 a n d 3 2 2 8 / z m o l / g c-C~Hz2) w a s u s e d c o n c u r r e n t l y w i t h a high d o s e - r a t e (0.67 M r a d / h ) . R e s u l t s at t h r e e t e m p e r a t u r e s a r e g i v e n in T a b l e 3. T h e m o l e c u l a r formation of chloro-cyclohexane precludes the use o f d a t a o b t a i n e d at i r r a d i a t i o n t e m p e r a t u r e b e l o w 40 °. T o c o m p u t e kJkv, w e u s e d f o r t h e ratio k6a/k6~ t h e s a m e v a l u e o f 2.4 o b t a i n e d f r o m t h e r e c o m -
b i n a t i o n r e a c t i o n b e t w e e n .CC13 a n d n-CrHz3, in t h e liquid p h a s e , °~) k6a,/kr¢, is u n k n o w n b u t m o s t likely a b o u t 30--40% smaller t h a n ksd/k6c (2"21) a n d w e c h o s e a v a l u e of 1.8. T h e a v e r a g e v a l u e o f kdkv is 0.95 __+0.06. T h u s , n o s e c o n d a r y i s o t o p e effect c a n b e d e r i v e d f r o m o u r data. 4. D I S C U S S I O N T h e d i f f e r e n c e in t h e a c t i v a t i o n e n e r g y o f r e a c t i o n s (2) a n d (2') a t t r i b u t e d to a n i s o t o p e effect is
TABLE c-CrHH-H a DoN (kJ.mo1-1) ~ (cm -1) De (kJ'mol -~) R (/~) F (mdyn-/~ -)) P=9 /3 (A-~) h
393.0 2933 410.5 1.12 4.68 1.05~ 1.85
4,
CCI3-H 399.6 3019~ 417.6 1.07e 4.96 1.05g 1.89
c-C6HH-CCI3 326.3 ° 920a 331.8 1.54e 2.99 -1.65
"Reference (6) and citations therein. bs. W. Benson Thermochemical Kinetics. Wiley, New York, 1968. cPs for CDCI3 = 2256cm -~. L. D. Spicer and C. D. PouRer. In Physical Chemistry, an Advanced Treatise, (Edited by H. Eyring), Vol. VII. Academic Press, New York, 1975. dG. Herzberg. Infrared and Raman Spectra of Polyatomic Molecules. I lth Edn. Van Nostrand, Amsterdam, 1964.
eTables of lnteratomic Distances and Configuration in Molecules and Ions. Special publication No. 18. The Chemical Society, London, 1965. fThe force constant F is estimated from the relation F = (2/zv)2. ~ where /z is the reduced mass calculated from the masses of end atoms. ~Following Johnston (Ref. [25], p. 346), the bond energy index p is set equal to q and used as an adjustable parameter to fit V:~ (Table 6) to the observed activation energy of 46.4 kJ.moi -~ for k2 (12). h/3 is the Morse parameter calculated from the relation /3 = rr~ (2~/D~) I/2.
The y-initiated chain reaction between carbon tetrachloride and cyclohexane
229
TABLE 5. (c-C6HII) . . .
n' m* V*(kJ-mol 1) R),B(fk) R~c(,~.) F),a(mdyn-,~-t) F~c(mdyn.~ ~) F~c(mdyn.A ~) F,(10"erg-rad -2) ~(cm i)~ ~(cm-I) ~ ~(cm-~) ~
H ... (CC13)
(c-C6DII) . ..
0.530 0.470 46.4 1.29 1.27 1.25 0.86 1.63 0.055 622 491 1425i
D . . . (CC13)
0.530 0.470 46.4 1.29 1.27 1.25 0.86 1.63 0.055 586 354 I092i
n t, m*: Bond orders of the A-B and B-C bond, respectively; A refers to the group and C, to (CC13). R*: Bond length obtained from the Pauling's bond-order-bond-length relationship R ~ = R -0.26 In (bond order). "The frequencies are calculated by using the linear triatomic model C . . . H (or D ) . . . C.
( c - C 6 H l l ) o r (c-C6DII), B to (H) o r (D)
T A B L E 6.
°C
(k~/kz) exptl,
(k2/kr) ~ calc.
1]iH/1/ie
(F~t/FD)*
(Fm/Fso)*
(Fb:q/Fbo) 2 2 :~
(F~o/rm)"
140.0 111.0 84.0 63.0 44.5 28.5
7.93 10.0 13.1 16.4 19.9 23.5
5.00 5.61 6.58 7.61 9.21 11.1
1.31 1.31 1.31 1.31 1.31 1.31
1.60 1.70 1.83 1.99 2.25 2.53
0.980 0.977 0.973 0.970 0.%7 0.964
0.893 0.878 0.861 0.845 0.829 0.814
2.67 2.94 3.26 3.57 3.91 4.26
~kJk~_, = V-~D'\F--D/'\F~o F-~oJ" \r~-u/' F = (½u)/sinh(~u) with u = hv/kT. F* represents the tunneling correction factor; F ~ and Fr are quantum correction factors for the real vibrations of the activated complex and reactant respectively; subscripts s and b stand for stretching and bending modes. (see Ref. [23] for more details).
10.1_0.3 kJ-mol 1. T o a first approximation this c o r r e s p o n d s to the zero-point energy difference of the reactants. If we a s s u m e that a transition state involves a C - H stretching f r e q u e n c y (Table 4), we should o b s e r v e an activation energy difference of 4 . 3 k J . m o l -I. T h e possibility to include one or several bending m o d e s in the transition state has b e e n discussed by Bell. (22) H e estimates that such a contribution can a m o u n t at most 1.1 kJ-mo1-1 for C-H bonds. A quantum mechanical correction for tunneling tends to increase the apparent difference in activation energy. T h e results calculated for a linear three-atomic activated c o m p l e x model, based on the figures given in Table 4, are p r e s e n t e d in Tables 5 and 6, using the B E B O m e t h o d for calculations. (23~ Although the results with such a simple model c o r r e s p o n d only to a crude approximation, they do give an estimation of
the magnitude of the tunnel effect. Thus, we estimate with 4.3 kJ.mo1-1 for the contribution of the stretching mode, l . l k J . m o i -1 for tunneling, a value of 9 . 4 k J . m o l -I for the difference in the activation energies. The a g r e e m e n t with the e x p e r i m e n t data is perhaps fortuitous, but it shows that tunneling may account for an important contribution in the activation energy. T h e ratio of the corresponding Arrhenius factors is 0.42-+ 0.04. E x t e n d e d calculations by S c h n e i d e r and Stern (24~ for isotope effect without tunneling limited the range of this ratio b e t w e e n 0.71 and 1.67. T h e low experimental value is again an indication for a contribution by a tunnel effect to the abstraction reaction (2). H o w e v e r it should be kept in mind that an underestimation of this ratio could be c o m p e n s a t e d for by an o v e r e s t i m a t i o n in the difference of the activation energies.
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