J . Phys. Chem. 1988, 92, 4080-4084
4080
Rate Parameters for the Reactions of C,H3 and C4H, with H, and C2H, Maja A. Weissman* McDonnell Douglas Research Laboratories, St. Louis, Missouri 63166
and Sidney W. Benson Loker Hydrocarbon Research Institute, University of Southern California, Los Angeles, California 90089- 1661 (Received: March 24, 1987)
The temperature-dependent rate parameters for the reactions of C2H3and C4HSwith H2 and C2H2were calculated. Transition-state theory estimation methods developed by Benson were used to estimate frequency factors at 300 K, A(300), and heat capacitiesof activation Ad;, over the temperature range of 300-1 500 K. Recently published experimental measurements of the rate constants at 300 K, k(300), were used together with the estimated A(300) to derive the activation energies at 300 K, E(300). The Ab:s were used next to compute E(T), A ( T ) , and k ( T ) . The k(T) values were well reproduced by modified Arrhenius expressions, log k‘= log A ‘ + n log T - E’/2.3RT, with n = (AC?,);$/R. The reactions with H2 have small Arrhenius plot curvatures. The expression derived for a reverse reaction, H + C2H4 H2+ C2H3, fits the high-temperature data published previously. The reactions of addition to C2H2exhibit significant Arrhenius plot curvatures as expected for these three-electron, three-center transition states. The activation energies are lower than those of similar reactions of alkyl radicals, in accord with the lower ionization potentials of vinylic radicals. The preferred rate parameters for the modified Arrhenius expressions are given.
-
Introduction The reactions of addition of vinyl and butadienyl radicals to acetylene are thought to play a crucial role in the formation of the first aromatic ring in the soot initiation A quantitative demonstration of their role could not be made because the appropriate rate parameters had not been determined. Experimental measurements were unavailable and estimations of rate parameters for this type of reaction cannot be made with sufficient accuracy. Preexponential factors and heat capacities of activation can be estimated with sufficient confidence3 but the estimation methods for the activation energies are not sufficiently a c c ~ r a t e . ~ We can only expect the activation energy to be lower than that for alkyl radicals because, in the current representation of addition reaction^,^ a correlation exists between the ionization potential (IP) of radicals and the activation energy, and the I P of vinylic radicals is lower than that of alkyl radicals. In the literature concerning the role of these reactions in soot i n i t i a t i ~ nvarious ,~~ rate constant estimation methods were used. Recommended A factors vary between los2 and 1010.2L/(mol-s) while the activation energies vary between 0 and 62.8 kJ/mol. Methodology In a recent publication, Callearlo reported the first experimentally measured rate constants for the addition reactions of vinyl and 1-butadienyl radicals to acetylene. The rate constants were measured at low temperatures (300-500 K ) relative to the reactions of these radicals with H2.
H2
+ C2H3
+
H
+ C2H4
(1)
(1) Cole, J. A.; Bittner, J. D.; Longwell, J. P.; Howard, J. P. The Chemistry of Combustion Processes: ACS Symposium Series 249; American Chemical Society: Washington, DC, 1984; p 3. (2) Toqan, M. Ph.D. Dissertation, Massachusetts Institute of Technology, 1984. (3) Benson, S. W. Thermochemical Kinetics; Wiley: New York, 1976. (4) Benson, S. W.; Haugen, G. R. J. Phys. Chem. 1967, 71, 1735. (5) Tanzawa, T.; Gardiner, W. C., Jr. J. Phys. Chem. 1980, 84, 236. (6) Cole, J. A. M.A. Thesis, Massachusetts Institute of Technology, 1982. (7) Frenklach, M.; Taki, S.; Durgaprasad, M. B.: Matula, R. A. Combust. - . Flame 1983, 54, 8 1, (8) Weissman, M. A.; Benson, S. W. Int. J. Chem. Kinet. 1984, 16, 307. (9) Colket, M. B., 111. Progress in the Physics and Chemistry of Combustion; Proceedings, 1985 Technical Meeting, Eastern Section of the Combustion Institute: Philadelphia, Nov. 1985. (10) Callear, B.; Smith, G. B. J . Phys. Chem. 1986, 90, 3229
0022-3654/88/2092-4080$01.50/0
H2
+ C4H5
-+
C2H3 + C2H2 C4H5 k1(300K ) =
C2H2
+ C4H6
(2)
C4H5
(3)
C6H7
(4)
-
H
---+
L/(mol.s),
E = 23.0 kJ/mol
k2/k1 = 6, k3/k1 = 30, k4/k2 = 11.6
(at 300 K)
In the present work, previous transition-state theory (TST) estimations of preexponential factors]’ were used together with the k( 300) values measured by Callear to derive the E(300) values. (Since we are not aware of any evidence regarding the nature of the transition states of vinyl radicals’ reactions, we carried out the present calculations in the assumption that they are identical with the transition states developed for alkyl radical^.^) Next, the estimated heat capacities of activation were used to derive A( T), E( T ) , and k( T ) in the 300-1500 K temperature range to evaluate Arrhenius plot curvatures. =
Stc,T,
+
(T2/T1)
AT = (e2kT/h)estc.TlR
= ET,+ (AC3,)?(T2 - Ti) log k( TI) = log A( Ti) - E( Ti)/RT] log k(T2) = log A(T2) - E(T2)/RT2
= log A ( T , ) - E(T1)/RT2 + ( A b , ) ?
7-2
T2 -
Tl
(Ad,includes
the heat capacity of the reaction coordinate). Rate constants were also described with modified, three-parameter Arrhenius relationships, log k’ = log A’ + n log T E’/2.3RT. According to the definitions given in ref 3 and taking 800 K as an average temperature log A’ = log A(800) - n log (e.800)
E’ = E(800) - nR.800
n = (AC+,):#~/R (11) Weissman, M. A.; Benson, S. W. Symp. ( I n t . ) Combust., [Proc.], 20th 1983, 58.
0 1988 American Chemical Society
Reactions of C2H3and CIHS with H2 and C2H2
The Journal ofphysical Chemistry, Voi. 92, No. 14, 1988 4081
TABLE I: Transition-State Theory Estimations of Rate Parameters for the Reaction'
degree of freedom model reaction (TS = C,H4) symmetry spin translation external rotation (C-H)3,oc reactn coordn (H-C-H) ,450 (C-C=C) 700 (H*H),,, 2( H*H.C) (linearlooo) Z(H.H.C) (bent 135') (H*C-H),O% (H*C=C) gw total linearlooo bent 135'
-
A (300), L/ (moles)
ASot3w
300
500
ACO+",T 800 1000
1500
-35.8
-4.9
-3.9
-2.2
1.8
-1.1
2.8 1.4 0.11 1.39 0
0 0 0 0 1.98
0 0 0 0 1.98
0 0 0 0 1.76
0 0 0 0 1.52
0 0 0 0 1.05
0 -0.3 0 0.2
-0.10 -0.95 0 0.76
-0.56 -1.43 0.05 2.08
-1.16 -1.74 0.34 3.06
-1.40 -1.83 0.56 3.36
-1.70 -1.91 1.08 3.68
1
0
4.0 0.1 0.2
0.76 0.38 0.66
2.08 1.04 1.30
3.06 1.53 1.68
3.36 1.68 1.78
3.68 1.84 1.89
-29.9 -25.3
-2.17 -1.55
-0.14 -0.18
3.27 2.74
3.87 3.19
4.83 3.99
(linear TS) 109,6(bent 135' TS)
Log k (I/mol.s)
COP.T
radical C2H3 l-C,H, I-CGH,
AH~0300 66.6b 80.2 93.7
300 57.0 69.8 52.0
10.8 19.7 28.6
500 13.9 27.4 40.9
800 17.2 33.9 50.6
1000
1500
19.0 37.0 55.0
21.9 41.9 61.8
'Values derived b,y group additivity in kcal/mol and cal/(moLK) (ref 3). The group [Cd-H] was derived from the above values of C2H3. Reference 19.
I
I
I
I
I
1.0
1.5
2.0
2.5
3.0
3.5
1ooofr(1/K)
il 5.0
Entropies and heat capacities corresponding to vibrational frequencies as listed in cal/(mol.K) in ref 3.
TABLE 11: Thermochemistry of C2H3, l-C4H, and 1-C,H7 Used in Estimations'
I 0.5
4.0
3.0
2'ol 1 .o
I
I
I
I
I..,
I
The same type of estimation seemed appropriate for the reactions with H2, (1) and (2), because rate expressions reported earlier for reaction 1 l 2 9 l 7 based on high-temperature measurements are scattered" and some are in serious disagreement with Callear's low-temperature data.]" We are not aware of any other rate measurements for reaction 2.
-
Results H2 C2H3 H C2H4. Until the recent rate constant measurements at low temperatures (300-500 K) made by Callear,'" only rate data at high temperatures for the reverse reaction were available.'2-16 We have plotted rate constants for the forward reaction obtained from these high-temperature data for the reverse reaction and the equilibrium constant corresponding to the thermochemistry listed in Table I1 in Figure l a . The heat of formation and the entropy of the vinyl were recently19 finally
+
+
(12) Peeters, J. Symp. (Int.) Combust., [Proc.],16th 1977, 969. (13) Just, A. Th.; Roth, P.; Damm, R. Symp. (Int.) Combust., [Proc.], 16th 1977, 961. (14) Baldwin, R. R.; Simmons, R. F.; Walker, R. W. Trans. Faraday SOC. 1966, 45, 806. (15) Skinner, G. B.; Sweet, R. C.; Davis, S. K. J . Phys. Chem. 1971, 7 5 , 1. (16) Peeters, J.; Mahnen, M. Combust. Inst. Eur. Symp. 1973, 5 3 . (17) Warnatz, J. In Combustion Chemistry: Gardiner, W. C . , Jr., Ed.; Springer: New York, 1984; p 197. (18) Tsang, W.; Hampson, R. F. J . Phys. Chem. Ref Data 1986.15, 1087.
(19) Parmar, S. S.; Benson, S. W., submitted for publication in Int. J . Chem. Kinet. (20) Sharma, R. B.; Semo, N. M.; Koski, W. S. Int. J . Chem. Kinet. 1985, 17, 831.
The Journal of Physical Chemistry, Vol. 92, No. 14, 19E18
4082
review by Warnatz" and considered seriously scattered. Warnatz considered that more experimental data are needed for this important reaction. The expression he recommends, k = lo1' ' - l o 2 / ' L/(mol.s) has average values and slope. An earlier expression estimated by Benson and Haugen4 by analogy of the reverse reaction with other abstraction reactions, k-l = 10108-'0'1' L/ (mobs), favors the higher values with smaller slopes.16 In the recent NBS review" a rate expression, based on the high-temperature data,', was derived for the temperature range of 300-2500 K. This expression, however, predicts rate constants at 300 K lower by 2 orders of magnitude than Callear's recent low-temperature measurements.1° This discrepancy is not unexpected since an error of a factor of 2 in the measurement of the rate constant at 1700-2000 K, attributed to the activation energy results in an error in the activation energy of 12.6 kJ/mol. Such an error in the activation energy may result in an error of -2 orders of magnitude in the rate constant at 300 K. Extrapolations to low temperatures of all high-temperature expressions deviate seriously from the actual measurements made in ref 10. We have used transition-state theory estimation methods developed by Benson to estimate A-factor values at 300 K for a linear configuration of the C.H.H half-bonds and also for a configuration bent by 135' (Table I ) . These A(300) values together with the rate constants at 300 K measured by Callearlo yielded the activation energy at 300 K, E(300). Next, we have used the heat capacities of activation corresponding to the same TS models to obtain A( 7')and E(7') in the temperature range of 300-1500 K (Table I). The rate constants k( T), computed with the relationship k( T ) = A( T)e-€(nIRT,are plotted on Figure 1. It is apparent that the linear transition-state model comes closer to reproduce an average value of the high-temperature measurements. The bent transition state predicts higher rate constants like those reported in ref 16 or higher values for the equilibrium constant, K,, Le., a higher value for the heat of formation of C2H3 or a lower value for the entropy of formation of C2H3. The Arrhenius plot curvature is quite small and does not justify the much larger slopes of some high-temperature data12*13~15 versus low-temperature data.I0 The rate constant k(T) (in L/(mol.s)) can be cast in a modified, three-parameter Arrhenius relationship ( E is in kJ/mol) linear TS:
bent (135')
+ 0.7 log T - 21.4/8 log k-l = 8.5 + 0.7 log T - 33.5/0 TS: log k+, = 7.3 + 0.7 log T - 27.2/8 log k-1 = 9.5 + 0.6 log T - 59.0/0
TABLE 111: Transition-State Theory Estimations of Rate Parameters for the Reaction'
HZ
+
C4H5
- k>=c(H.] 'H
degree
of freedom
ASotjm
model reaction -34.5 (TS = CdH,) symmetry 1 . spin 1 . 0.03 translation external rotation 0.42 0 (C-H)31OO reactn coord (H-C-H) 1450 0 (H-C=Choo -0.3 (H*H)2800 0 2(H.H.C) lineariOOOo 0.2 2(H.H-C) bent 135 3.4 (H-C-H) loo0 0.1 (H-C=C) 0.2 total linear -31.1 bent -27.9
-
log A(3L'O)
500 -4.1
300 -5.3 4 0 4 0 0 0 1.98
0 0 0 0 1.98
ACO+", 7800 1000 1500 -2.9
-2.2
0 0 0 0 1.76
-1.3
0 0 0 0 1.52
0
0 0 0 1.05
-0.10 -0.56 -1.16 -1.40 -1.70 -0.95 -1.43 -1.74 -1.83 -1.91 0 0.05 0.34 0.56 1.08 0.76 2.08 3.06 3.36 3.68 1.38 2.04 2.53 2.68 2.84 0.38 1.04 1.53 1.68 1.84 0.66 1.30 1.68 1.78 1.89 -2.55 -1.93
0.34 0.38
2.58 2.05
log k(300)
3.47 2.79
4.63 3.79
E(300)
linear
bent 135'
linear
bent 135"
linear
bent 135"
8.3
9.0
5.0b
5.0b
18.8
23.0
Entropies and heat capacities corresponding to vibrational frequencies as listed in cal/(mol-K) in ref 3, E in kJ/mol, A and k in L/(mol.s). bExperimental measurements (ref 10).
TABLE I V Transition-State Theory Estimations of Rate Parameters for the Reaction"
-
-+
log k+l = 6.5
where 8 = 2.3RT and R is the gas constant. These expressions constitute a considerable improvement over those available previously. H2 C4Hs H + C4H6. The TST estimation for the above reaction is given in Table 111. The curvature in the Arrhenius plot is very small. The bent TS yielded rate parameters that are the closest to the parameters estimated by Benson and Haugen4 in 1967 by analogy with similar abstraction reactions (log kz (L/(mol.s)) = 9.4 - 5 . 9 / 0 , log k2 (L/(mol-s)) (300) = 5.1; this is in amazing accord with the recent measurement of Callearlo (1986)). Our preferred expression for a linear TS ( E in kJ/mol) is
+
Weissman and Benson
-
log k2 (L/(mol.s)) = 6.6
+ 0.5 log T - 15.5/8
log k-z (L/(mol.s)) = 7.8
+ 0.7 log T - 25.1 / e
-
C2H3 + C2H2 C4Hs. Benson and Haugen4 have estimated by analogy with similar abstraction reactions that log A , = 8.3 L/(mol.s). Cole6has derived k3 using AH,,,,,,, and a linear plot containing data of all known alkyl radical of log k3 vs AHreaCtlOn addition reactions to acetylene. Then, assuming A , = l o a 2L/ (moles), an average value for all alkyl radicals, Cole has derived
L
degree of freedom model reaction (TS = I-C,H,) symmetry spin external rotation (C-C)1mo reactn coord (C=C),,,, (C = c,,,, 2(H-C-C),,
