Chemical Physics 53 (1980) 279-291 (0 North-Holland Publishing Company

A HIGH PRESSURE RATE CONSTANT FOR CHs + H AND AN ANALYSIS OF THE KINETfCS OF THE CHs + H -+ CH4 REACTION

Roger PATRICK, Michael J. PILLING and Graham J. ROGERS Phystaal chemistry Laboratory, South Parks Road, Oxford, OX1 3Q.Z UK

Received 21 December 1979; in final form 4 August 1980

The rate of reaction of CHs and H has been measured at 300 K over the pressure range SO-1000 Torr with Ar and SF6 as diluent gases. Flash photolysis of azomethane/ethylene mixtures was employed, coupled with gas chromatographic analy- sis and numerical simulation of the product yields. The rate data were compared with a RRKM analysis, using a minimum density of states criterion for locating the activated complex. The limiting high pressure rate constant so obtained is (1 .S * 0.7) X lo”0 cma molecule-ts-l. The RRKM model was found to give high temperature, high pressure rate constants for the dissociation of CH4 which are significantly higher than those found experimentally.

1. Introduction

There have been numerous studies of the rate of dis- sociation of methane [l-- 161 and the reverse associa- tion reaction [ 17-273 :

CH4; CHs+H. (0 m

The interest stems from the importance cf this reac- ticn in pyrolysis systems and its key position in the development of models for unimolecular reactions. However, in spite Jf the importance of this reaction, accurate rate data over a full temperature and pressure range are not available and little has been done to rationalize and collate that which is. The reverse reac- tion kiss been studied mainly at low pressures in flow systems, often in reactions initiated by H f &Ho. Only two investigations have been at sufficiently high pressures to permit extrapolation to the ihniting high pressure rate constant. Pilling and Robertson [25] deduced a value for kz,.. of 1 X !O-1Qcm3 molec-’ s-‘in a study of the flash photoiysis of ketene/hydro- gen mixtures. The reaction system was complex and the measurement of this rate constant only incidental to a study of methylene reactions. Cheng and Yeh [26,27] employed low intensity mercury photosensi- tized decomposition of ethane and found kl_- to be

2.49 X lO-‘O cm3 molec-‘s-l ; their system was also complex with methyl radicals reacting in four possible ways. This paper reports a further investigation which employs flash photolysis so that radical and atom con- centrations are high and reaction (II) thus optimized. Details of RRKM calculations based on the use of the criterion of minimum state density for locating the _ critical configuration are also presented.

2. Experimental

The er.perimental tech:Gques ar:d methods of numerical analysis have previously been described

[281.

3. Results

Azomethane and ethylene were employed as sour- ces of methyl radicals and hydrogen atoms respectively. Care was taken to ensure that the pressures of both photolytes were well within the range of low optical densities such that the light absorbed is proportional to the photolyte pressure. This ensured that any interference of the absorption by one photolyte due to increasing pressures of the other was well within experimental error, Ethylene is photolysed at the

280 R parrick et aL /High pressure rate constant for CH3 f H

experimental wavelengths to give acetylene

GH, k% +-Hz > OW

C,H, ": CaH, +2H. OIIb)

The relative yields of reactions (Ma) and (Mb) were determined by photolysing azomethane in the presence of variable amounts of ethylene. The acetylene yield was used to monitor the degree of ethylene photolysis and the methane yield, after correcting for the small

amount of CH3 photolysis [28], was used as a moni- tor for H atom production in reaction (IIIb). At low ethylene pressures, all the hydrogen atoms were scaven- ged by CHa ; the limiting low pressure methane yieId, after correction for CH3 photolysis, was divided by the acetylene yield and equated to 2@,,/(&, i- Qsb),

giving &J(& + &) = 05 + 0.05 for the experimental wavelength range. The photolysis of azomethane/ethyl-

ene mixtures was then studied over a range of condi- tions (100’mTorr azomethane, O-100 mTorr ethylene) in the presence of either Ar or SF6 at total pressures of

between 50 and 1CHlO Torr. Typical results are ihustra- ted in fig. 1_ The data were analyzed by comparison with numeriml simulations using the reaction scheme

presented in table 1. The total methyl radical yield in reaction (IV) of table 1, WJ, was evaluated by mass balance based on the full reaction scheme shown. WJ was equated to the yield of acetylene after cor- recting for the very small amount of acetylene pro-

14

\

12 - -1

0’

IO - E / 1

4 z

a-

t

/ 0

0 “0 - 6- /

0

0 20 40 60 I30 100

10%2H43/Torr

Fig. 1. Dependence of the &me and methane yields on the ethylene pressure in the flash photolysis of azomcthanelethyl- ene mixtures (single shot, argon pressure = 200 Tort, azome- thane pressure = 100 mTorr). l , CzH6; 0, CHq; - computer fits.

Tab!e 1 Reaction scheme for the analysis of the 5sh photolysis of azomethane/ethylene mixtures

Reaction k300 (cm3 moIec

--I *-I 1

hv +C2H4 -QH2 + 2H (-Ma) - hv + Czb +C2H2 + H+ (IIIb) - kv + CH3N2CH3 +2CH3 + N2 hv + CH3 +CH, +H _gy I

a3 f CH3 -+C2H6 (VU 4.2 x IO-” =) CH3 +H +CH+ w Ye text C2Hq +H +C2Hs (VII) k= 1.6 X IO-” [21] b,

(pressure dependent) CH2 + CH3 +&Ha f H (VW CH2 + CHz +C2H2 + HP 0

;:; ; ;$:: :!t#)

a) A discussion of the rate constants employed may be found in ref. [25] _ b, These reactions have a very small effect on the simulations and their omission changes the optimal values for reaction (11) by

6%. H + H + M + HZ + M has an even smaller effect.

281

a 12 16 20

IO3 TorrlPressure PiE. 2. Lindemann plot of I/& versus I/(argon pressure) for the association of Cl-is and I-S. 0, hl = A.r, X, hl = SF, (rescaled, see text), 0. data of Cheng and yeh L26.271. -, RRKM fits udng anharmonic model and n = 0.55; pc = (a) 0.006. (b) 0.035, (c) 0.070, (d) 0.105, (c) 0,140, (4 0.174. - - - -harmonic model for p, = 0.035.

duced by reaction (V) (table 1). IV,I was evaluated using the arguments and results discussed previously [28] _ AlI of he rate constants in the kinetic scheme are known with the exception of k2 which was treated as an adjustable parameter in the sirmdations. Typical iits are shown in fig. 1. A f&rdemann plot (l/k* versus l/[Ar]) is given in fig. 2. Data obtained using SF, as n coilisional stabiiser have also been inchrded by resca- ling the apparent pressure to give the best fit to the Ar data; this required that the ratio of the stabiirion efficiencies for Ar and SF6 be 1 : 2.84 which com- pares satisfactorily with a mean value of 1 : 2.56 which may be obtained by analysis of the data presented by Tardy and Rhbinovitch [29]-

It is not straightforward to m&e a systematic error analysis in an indirect measurement of a rate constant. ‘Ibe most sensitive experimentai observable, and the one which determined most directly ihe opti- mal fits, is the methane yield (fig. 1). Combining th@ scatter in methane, with that in acetylene (which is a measure of the photolytic hydrogen atom yield) and with the uncertainty in &&&, gives a total uncer- tainty of &-110/o. A variation ink, of &20% is required to match this magnitude of error at high pressures (~600 Torr Ar), but the CMa yield is more sensitive

to rhanges in k, at low pressures, and at 50 Tars of argon this uncertainty Falls to *lO%. The value of Xr, also depends on the vaiue taken for ke . Parkes and Quinn 1301 carefully discussed the errors in the latter, which arise not only from the random errors associated with a real time radical decay, or a fit to molecular modu$tion data, but also with the large uncertdmty in the methyl extinction coef5cient. They estimated the error inks to be +-23%. Calculations show that this is not reflected linearly in k, , but produces instead an uncertainty of about f14% at high pressures, rising to **20% at 50 Ton Ar. The other reactions in table 1 are of only minor importance in the fit and produce a combined uncertainty oi65%.

4. Discussion

In order to anaiyse the data fully and obtain a value for kz ,m, the high pressure limit of the association rate constant, it is necessary PO extrapolate the data using a model based, to a greater or lesser extent, on RRKM theory 13 I]. The main difficulty in applying such a model to a bond dissociation reaction lies in defining and characterising the transition state. Quack and

282 R Patrick et al. /High pressure me constmt for C/Is + H

Troe [32] developed and applied a model based on an adiabatic correlatiou of reactant states with the energy levels of the products. Although, in principle, this method requires few assumptions, in practice it is not easy to implement fully. Bunker and Pattengill 1331 sugg,ested an alternative statistical formulation in which the critical configuration iies at the position of minimum state density along the reaction coordinate. The rationale for this suggestion derives from the form of the potentisl culye over which the reaction occurs as illustra: 2d in fig. 3. Addition of zero-point energy to a basic Morse-type potential function results in a maximum occurring at a position which loosely desribes the critical configuration at zero excess energy. Since, at this position, the amount of the ini- tial en orgy, E,“, which is fixed is maximized, it follows that the number of states accessible to the system at the critical confIguration must be minimized. As E,’ inereascsp the separation, of some of the energy levels decreases as the bond is extended (vide infra) and this CZiUB8S tile POSitiOn of min_imunl state density to move ro @nailer vahres of 7, the reaction coordinate. In con- F~qWnce, the position of the activated complex is not invariant, but depends on El.

In order to implement the minhnutn state density model it is nccesjarJ to have information on:

(I) the variation of the vibrational frequencdes with r, the distauce along the reaction coordinate,

<ii) the variation of the principal moments of iner- tia of the molecule with r and,

(iii) the potential energy function ever which the reaction occurs.

The general technique for evaluating tile rate con- slant for an association reaction, is wah established [3i ] . The ovea!i meohanism is

where E * is the internal energy of CR; relative to the grouad state of CH, , Applying the principle of micro- s~opie reversibibty and using the RRKM expression for A&?‘) lead to the following expression for ks,.. :

where C$&I&) is the sum of the rotation-vibration quantum states in the activated complex, Q: and rZ, are the translational and rotational partition functions of the activated COmph, Q,-HJ and f& are the total partition functions of the reactants and symmetry num- bers have been included in th? rotational partition func- tions. tJJ$ is the enthalpy of the dissociation of CH4 atOK.

In order to defme the activated complex, the poten- tial surface over which the reaction occurs was assumed to be based on a Morse poteutial given by

V(r) = De { 1 - exp [ -p(r - r,)] p , (2)

where r is the distance along the reaction coordinate, r, is the equilibrium bond distance in CHe , De = AH,, f Ez(CH4) - Ea(CHs), (Ez refers to zeta point ener- gies), and p is a parameter. The Morse potential has been shown to model effectively the vibrational over- tonespectraofC~with~=186X10’0m-’ [34].

Herzberg [35,36] determined the rotational param- eters of CHs (table 2). If it is assumed that the CHs fragment has already achieved a planar methyl geome- try in the activated complex and that the reaction coor- dinate corresponds to extension of *he c’-H bond lying ducg the z-axis, then it may be shown that the variation of the moments of inertia of the complex with r, in the region of interest, reduces to

Is =~orJ3 * * *

Ix = ly = (2.92 X 10m4’ + 1 S7 X 10W4’ [r/d] *) kg m2 ,

(3)

The nine vibrational frequencies of methane 1361 and the six vibrational frequencies of CHs [35-371 are given in tabIe 2, which also illustrates the correla- tion between the various forms of motion in the com- plex and in the separated reactants. Although it is pos- sible to invoke a narmd coordinate analysis 1361 and Badgers rule [39] to model the dependence of the vibrational frequencies on r the following expressions are best regarded as empirical [32]

Y = ugH4 exp l--o@ - re)] ,

Y = uzH4 exp[--& - rJ

(4)

+ lJF”s {I - exp[-& - r,)] ] , (9 where Q is an adjustable parameter. The changes in

R. Patrick et al. /High pressure rate constant for CH, + H 283

‘fable 2 Molecular parameters employed in the RRKM model, showing correlations between CH4 and CH3 + 13 for exlension of the C-H, bond

---- specrroscopic conslanfs

CH4 rotation (x, v) r&-Ii) = 1.094 A L (H-C-H) = 109.5” fr = Jy = Iz = 5.34 X 1O-47 kg m* C-H stretch (~1 = 29169)(reaction coordinate) + rocking vibrations (2) (~2 = 1533.6) + rotation (2) +

deformation (1) (~4 = 1305.9) -+ remaining vtbrations in CH4 3 “3 = 3019.5 (3) +

“4 = 1306.2 (2) -f

Morse poiential &fg = 431.67 kJ mal-‘; D, = 468.86 kJ mol-’ 1381. p= 1.86h--’ [34]

relative translation (2) rotation Or, v) rotation (2) r&-H) = 1.08 A L <H-C-H) = 120° fz = 5.86 x 1fJ-47 kg mz Ix = I,, = 2.98 X 10da7 kg m2 bend (1) (vz = 580) remaining vibrations in CH3 Y, =3044(l) ug = 3162 (2) “4 = 1396 (2)

Lemwrd-Jones parameters ok”,, = 3.758 A, oAAr = 3.542 &CL, (CHh/k= 148.6 K.q, (Ar)/k = 93.3 K

CH34H relative translatioo Dr. y)

Fig. 3. Morse potential energy curves for CHJ-H,fl = 1.86 x 10” m-‘. - - - , vibmtionless curve; - zero point encrgy curve (see text).

284 R Pariick er al. /High presqre rate Constant for CH3 + H

TabIe 3

T$ dependence of the properties of the activated complex on E,_ T= 298 K, a = OS

E,’ (kJ rn01-~) f (_Q E,’ gCr rn01-~) k,(E:)(s-‘)

433.63 4.11 432.50 8.00 x lo* 454.59 3.73 430.03 4.89 x 109

.474.93 356 427.90 1.73 x 10’0 495.30 3-44 425.93 456 x 10’0 516.05 3.34 423.96 9.92 x IO’O 536.01 3.27 422.12 1.89 X IO”

the other vibrational frequencies are so smatl that they can be neglected.

Fig_ 3 shows the Morse potential curve for CHQ as a function of the CH3-H bond length and also illustra- tes the zero-point energy curve V,,(r) defmed by

V,,(r*) = V(r+) - E,(CHa , r = r,) f E;(CH, , r) , (6)

where E:(CH_,) is the zero-point energy for CHq less that of the reaction coordinate. It should be noted that V,(r), which refers to the total zero point energy

of CH4 as energy zero, shows a maximum, the magni- tude of which is determined by a, the parameter des- cribing the dependence of the vibrational frequen- cies on r. This maximum corresponds to the effective activation barrier to reaction at 0 IS, e.

As was discussed earlier, a further consequence of the variation of vibrational frequencies with r is that the position of the activated complex, as defmed by the minimum density of states, moves to smaller values of r as E* increases.(table 3). This implies that the effective activation energy, E; 9 is also energy dependent, as are the moments of inertia of the com- plex, and so averaging over adiabatic rotational states should be considered explicitly. This is best achieved by changing the energy variables (fig. 3). Following Marcus [3 I 1, when rotational energies are considered explicitly, with an adiabatic correlation; E’ is replaced by E,*, the energy in the active degrees of freedom:

E,C=E+ fE;+E;-.- EJ,

where E+ is the non-Tied energy in the active degrees of freedom of the activated complex, EJ is the rota- tional energy in CH4 and EJ’ that in the complex.

ThUS

X exp[-(El +Et + E,‘- Al@/kT] cl.5: , (7)

where &$ = EJ’ - EJ. The lower limit of the integra- tion arises because as EC-f 0, El -fEt. The total rate

constant is obtained by summtig the contributions frcm all the adiabatic rotational states:

kz,=- -1 Qt h Qcn3 QH 7 $+ WA)

E:+AEf Em=0

X x (U+ 1) exp(-E$kT) 1 X exp [-(Ez f Ef- Mo)/kT] aE,* , (8)

where the summation ZJ”=,,(U + 1) exp(-ET/W) is simply equal to Q:, the rotational partition function of the complex. The expression can be greatly simpli- fied if AE; is replaced by !AE> = (1 - I+/I)/kT

(where I* and I are moments of inertia for the complex and energized molecule respectively) which corresponds to the average value of AE; for all the rotational states of interest. (The justification for this step is that k2,_

is comparatively insensitive to fluctuations of Al?; about this mean value [31], although the approxima- tion is possibly more questionable in the present con- text than in its more common application with a pre- scribed configuration.) The integration variable then transforms as E+ f E,’ and the lower limit of the inte- gration reverts simply to Eg , thus

X exp [-(El + E* - L&)/kT] d(EC +E;) . (9)

In the pressure dependent regime

x exp[-(Ei t i? - AH~)/kT] d(Ef +Eo’)

1 + k(E,‘)/@&WI) 3 (10)

where k(E,f) is the microscopic rate constant given by $$ =* P(E~)/trN(E,‘).N(E:) is the density of active stat& at energy Ei, Z is the collision frequency and 0, is a collisional deactivation efficiency.

In the limit of Iow pressure,

X exp[-(E’ f .Ez - Mi)/kT] d(E* + E,’ ) _ (11)

Assuming that the Morse parameter, c, is fmed, eq. (9) contains one implicit adjustable parameter, (Y, which determines the dependence of the vibrational frequencies on r and hence the shape of the zero-point energy curve and the position of the minimum density of states, whilst eqs. (loj and (11) contain both 01 and PC, the collisional deactivation efficiency.

Following Troe [40] Z was equated to the Lennard- Jones collision frequency defmed by

&I =NA&-t&RT/w,-d 1~Z**2(A-M), (12)

where NIL is the Avogadro number, p*_hl: the reduced molar mass and oA_nr the Lennard-Jones collision diam- eter given by (oA + cr~)/2_ The reduced coliision inte-

gral CP,‘(A-M) may be expressed as an approximate

24-

20 -

16-

5 s 12- z -

8-

function of kT/eA_-M, where E*__M is the Lennard- Jones well depth defied by eA__M = (eue~~)~“:

s2***(A-M) = CO.697 f 0.5185 log(kT/cA_-M)] -’ _ (13)

The parameters required are given in table 2. PC may be taken as the stepwise deactivation parameter des- cribed by Troe [41] and expressed as

&/(l - ,B;“) = -_(AE)/FEkT, (14)

where (AE’) is the average energy removed per collision and FE is the energy dependence of the density of states, which may be approximated by

(19

where E, is the zero-point energy and a the iihitten- Rabinovitch factor, related to the frequency disper- sion parameter [31]. In order to implement eqs. :;$I 1) it is necessary to evaluate N(Ez) and

l ~s=o P(EL). Direct count methods, such as the extended Beyer--Swinehart algorithm [43] offer exact methods, but are very time consuming at high energies. A comparison of this method with the well known Whitten-Rabinovitch semiclassical technique

0 16000

Fig. 4. A comparison of densities of states.N(E), for CH4. using direct count 1401, -, and Whitten-Rabinovitch [30] - - -, methods.

[3 11 (fig. 4) reveals good agreement at high energies, but at lower rlalues the semiclassical approach overesti- mates IV@:). Since IV@:) Is only required for EJ > 300 kJ mol-r , the Whitten-Rabinovitch approxima. tion was employed for its evaluation, whilst a direct count technique was used for I$=_ P(EG).

There calculation are only stri%?~ valid for har- monic oscillstors and since, at the energies required for dissociation, the CHa potential function must be lrdgbly anharmonic, some corrections must be made. Haarhoff [43] developed suitable expression based on inverse Laphce transformation of partition functions for a system of Morse oscillators. The correction fac- tor, y, is given by:

7 = ((1 + 2/#~‘/WE’PJ

Xexp[-@’ - 1)/3(1 t E’)] )‘z’D

YeXp[(-,~(~)2 + (&$$)($j~] , (16)

where E’ is the reduced energy E,‘/E,, E, is the zero paint energy, LJ the average dissociation energy given by {n-‘CiDT’)-‘, n is the number of bonds, s the number of oscillators and 5’ is the Whitten-Rabino-

35-

30 -’

: ;* 25 -

y E” 20-

h 15-

IO -

t /

/+

8’

/ /

I& 1 I

20 40 60 80 100 120

IO” Torr/Pressure 1‘11. 5. ’ 4 t 01‘ Ilk vwws I/(ufgon pres~~ro) extended to LOW pressures: o. this work, M = AI; X, this work, M = .lFe (rc~uled;; +, :bL read ci iai. f 171. M = AI. RHKM fits, cf fig. 2 wiih p, = 0.035.

vitch frequency dispersion parameter. In the region of interest, E.’ = 400-600 kJ mol-‘, -y varies from = 1 to 3.

The microscopic rate constant, k(E,‘) and various parameters describing the activated complex were evaJ- uated as a function ofEi using the criterion of mini- mum state density. A modified form of the program of Bunker and Hase 1441 was employed and typical results are shown in table 3. The overall rate constant was obtained by numerical integration of eqs. (9)-/l 1) using third-order finite difference formulae [45] and fits obtained with the experimental data by adjusting (Y to match ks,, and (3, to reproduce the experimental fall-off. The optimal fit, shown in fig. 2, was obtained using the parameter set (Y = 0.55,4-l, 0, = 0.035,Z = 3.44 X JO-‘a cm3 molec-‘s-’ , which gives ka,_ = 1.54 X IO-‘lo cm3 molec- ‘s-r. When the pressure dependent errors in our estimates of ks are allowed for, the fitted parameters become ks (_ = 1 .S + 0.7 X lo-*’ cm3 molec-’ s-r and & = 0.035 f 0.015. This value for p, is unusally small, and the origin of this is not clear. However, the rate constant was studied to sufficiently high pressures for there to be no additional uncertainty in k, w arising from this abnormal & value. The dashed curve’illustrates the result of a calculation based on a harmonic approximation, using the optimal

/ -8’

R. P&rick rt al. /High pressure rate constant for CHs f H 2x7

parameters. Fig. 5 shows an extended extrapolation of the calculated fall-off curve and shows that the model is in good agreement with the flata of Halstead et al. [ 171. Their data may be compared directly with our measurements and model, since they used argon as a diluent. Comparison with the only other systematic high pressure study, by Cheng and Yeh [26,27], is more difficult, since they used ethane as a buffer gas. Their data have been included in fig. 2 after scaling their pressure by a factor of 1.94 to allow for the greater stabilization efficiency of CaHc [29]. Their rate con- stants are significantly higher than those reported here and show an abnormal pressure dependence.

It is of interest to compare the predictions of the present model with experimental rate constants obtained for the high pressure dissociation of methane. Chen et al. [ 121 studied the static pyrolysis of CH4 at =lOOO K whilst a number of studies have been made of both the high pressure [l-l l] and low pres- sure [ 13-161 dissociation reaction at -2000 K using shock tubes. The results are summarized in table 5.

The high pressure, intermediate and low pressure rate constants for the dissociation reaction are:

kr,m

Xexp[-(E+ + E,‘)/kT] d<E+ + Ei) , (17)

exp[-(Et + EG)/kT] d(Ec + Ei ) ’ ---i-?%?F&M] ’

X exp [ -(E+ + Ei )/kT] d(E + + E:) . (19)

The temperature dependence of & was based on eq. (14). Experiments indicate that the temperature depen- dence of UE) is weak and lies between Toa’ and T-O.’ [39] ; for the present purposes, it was assumed tempera- ture independent. Since at 298 K & = 0.035, FE = I Xl4 and (LIE) = -0.11 kJ mol-r , f3, becomes 0.013 and 0.005 at JO00 K and 2000 K, respectively. The tempera turc dependence of Z is adequately described by

by eq. (12), whilst additional temperature dependent terms arise from the partition functions and the depen- dence of the position of the activated complex. on i:;‘,

The apparent activation energy, A&, is given by

A&, = KT’ d In k/dT . (20) However, the resultant analytical expressions are com- plex and an alternative and simpler approach is to eval- uate the rate constant directly over a range of tem- peratures and to extract the apparent activation ener- gies by a least-squares Arrhenius analysis. These results are given in table 4, along with vahres of (E. ), the true, limiting high-pressure activation energy, assuming a temperature dependent n factor. For the dissociation reaction, (Et, 11 is given by

X exp[-(Z+ + Ez)/kr] d@‘* + S,‘) I

+

X [ Ef Q& WE&) In-

Xexp[-(Ef + E;)/kT] d(,!? + Eo) *I-’ . (‘I) J

(&,)r represents the apparent, energy-dependent, microscopic activation energy, I’:, averaged over the Boltzmann distribution of reacting molecules at a par- ticular temperature.

The temperature dependence of the high pressure association rate constant, k, _ is illustrated jn lig. 6. It is weak, but, surprisingly, is positive, i.e. the tempera- ture dependence reflects a positive apparent activation energy, which is abnormal for an association reaction. The temperature dependence is closely linked to LY and becomes less positive as LY jncreases. A possible source of error in the model is the fixed p value we have employed in the Morse potential. If the potentjal func- tion approaches the limiting dissociation energy more slowly than we have assumed, then a larger a value will be required to match k, ,_ and this will lead to a less positive temperature dependence. Increases in rate con- stants estimated using the midmum density of states criterion have been found before. It should be noted that in contrast the adiabatic charmel model gencrully produces s less steep and often negative tecnperaturc

i88

Table 4.

R. Patrick et al. /High presswe rate consrant for CH, + H

T(K) k’*% (cm molec -ls-1 k1.B (s-5 kz o

1 (&I” m&c-2 5-1) &0.)2 ~o,h (kJ mol-‘) (kJ mol-*)

I50 298 500 800

1000

1200

2000

2200

2400

_ - - 358 x lo**

1.84 X lo-So 5.49 X m-3 1.11 x 10-33 1.46 x lo-‘3

3.25 x IO-‘* 857 X10-a

1.20 x lo-= 6.45 x IO4

5.46 x IO--‘8 353 x lo4

352 X 10-l’ 3.97 x 10s

156 x lo-t6 3.01 x 106

2.61 x 10-29 6.17 x lo-ll- 1.70 433.38

1.19 x 1o-2g 154 x lo-‘O 150 433.17

896 X 1O-3o 2.67 X IO-” 1.29 432.96

321 x 10-30 4_23 x IO_‘0 0.92 43258

1.65 x 10-30 4.35 x lo-‘0 0.19 431.87

8.82 X IO-” 4.76 x lo-to -053 431.12

9.22 x ICI-32 5.96 x lo-to -4.79 426.89

552 x 10-32 6.23 X lo-” -5.96 425.68

3.41 x 10-3’ 659 X IO-” -7.07 424.59

150-500 - - -1.92 255

800-1000 415.34 443.04 -25.49 2.24

2000-2400 334.67 44352 -112.94 9.96

dependence [46]. Clearly precise measurements of pressure limit and compares the predictions with experi- kz,- are required over a range of temperatures to mental results. The RRKM values are consistently refme the moGe1. smaller than those obtained by experiment. This

Fig. 7 illustrates the temperature dependence pre- probably derives, at least partly, from the small value dieted for the decomposition of methane at the low of p, obtained by fitting the experimental association

Table 5

Experimental andtheoretical Arrhenius pznmeters for the dissociation of methane

kl,O Tmnge (K) k(cm’ molec-’ s-t) n) A-factor (cm3 rnoIe~-~s-~) &E(kJ mol-‘) Ref.

Hart& et al. 1850-2500

Heffmgon et al. 2000-2700

Roth and Just 1700-2300

Gardiner et al. 2000-2700

Tabayashi and Bauer 1950-2770

RRKM model 2000-2400

cben et al. 995-1103

RRIW model X00-1200

r range (K)

8.34 X IO-I6 3.32 X 1O-7

4.08 x lo-l6 3.65 X 10-T

4.31 x 1o-*6 7.85 X IO-’

1.48 x lO-‘6 3.82 x 10-t’

5.03 x lo-‘6 1.68 X IO+

352 X IO-” 3.11 x lo+

2.69 x lo-” 153 x 10-s

3.25 X lo-‘* 1.61 x lo+

Pressure (an-n) A-factor

362.2 [ill

377.0 1131

390.0 1141 270.0 ll5l 359.0 [I61 334.7 435.6 1121 415.3

aE(k3 mK1) Ref.

Palmer and Hirt 1 loo-14GO

Kondrat’ev 1590-1750

Skinner and Ruebrwein 1200-1800

Kevarkian et al. 1700-2000

Kozlav and Knorre 1850-2500

Hartig et al. 1850-2500

RRKM mod-1 2000-2400

then et al. 995-1103 RI+KM made1 800-1200

? =20 5 6.75 2-8 5

1

1.25 X IO” 422.6 [41 1.00 x 10’5 430.9 151 5.13 x lOI 4225 I71 1.32 X lOl4 389.1 181 450 x 10’3 380.7 191 1.26 X IO” 435.1 1.35 x lots

1111 4435

2.80 x lOI6 450.2 1ol6

[I21 1.19 x 443.0

a) Refers to 2200 K and IO00 K.

Fig. 6. Temperature dependence of the limiting high pres.Wre rate constant for the association reaction (k2,d. predicted by the RRKM model.

fall-off plot. Although anharmonicity was included in the calculation of N(E,‘), no correction was made when

evahrating Z$ =,P(E~), on the grounds that the excess energies are s&ll. However, since the non-dissociative

vibrations in the activated complex might be expected to be strongly coupled to the dissociative mode, a high degree of anharmonicity may result at even low values ofEt. Quite modest increases in C$,Op(E~) would have large effects on k2 and the valugof pc required

to tit the experimental falI off curve would be corre- spondingly increased_ Increasing E$_, &‘(_I?:) would also reduce the value of Q required to fit I&.

At the high pressure limit, the RRKM model pre- dicts rate constants which are consistently higher than those obtained experimentally, as is illustrated in fig. 8, by a comparison with the data of Hart& et al. [ 1 I]. Quack and Troe 1321 commented similarly on the dif- ficulty of modelling the high pressure, high tempera- ture dissociation data and were only able to do so with Q = 0.1 A-‘. Such a value Ieads to a large barrier to reaction because of the residual zero point energy maximum and this produces an improbably high appa- rent activation energy. It is possible, however, that the simple extrapolation technique used by Hart& et al. [I l] is inadequate and that none of the experh-nentai determinations at the higher temperature represent true estimates of k I,W. This interpretation is supported by calculations based on the present model (fig. 9) which show the calculated pressure dependence of the disso- ciation rate constant at 2200 K, eq. (18). At 1 atm, the rate constant is only 0.23% of its limiting high pressure value whilst at 5 atm, the pressure employed by Hart& et al. [I I], the rate constant is predicted to be 4.3 X

I I I

4.0 5-o 6.0

I04K/T

% -600 %# 0 %* -56-O

/ ’ i 2 ‘c

-52-O

- IO.0 12.0

Fig- 7. A comWn of the RRKM predictions of the limiting low pressure rate constnnt for CH4 dissociation with experimental results: 0 RRKM ulculations; - - Hartiget al. Ill];+++, Chen et 111. [12];-., Heffiigton et al. [13]:--- Roth and Just [14]: - .- - C&diner et al. [ 15 I; - - - Tabayashi and Bauer [ i6] _

290 R. Parrick er al. /High pressure rate constant for Ctfa + ff

.eo,o

,15- 0

10.0 \

\ \

\ 6.0

I I I I I 4.0 4.5 5.0 5.5 6.0

104K/T

Pia. 8. A comparison of the RRKPd predictions of the limiting high prcssuro rate conslant for CHQ dissociation with experimental results. 13, RRKM calculations; -- -, Hartiaetal.[ll];--- -Chenetai.[12].

.25,0

10s s-’ , compared with their value of 6.31 X JO4 s-l. Once again, an increase in & would improve the agree- ment, but the significant fact is that the theoretical value is now lower than the experimental one. This discrepancy could arise if additional reactions were involved in the decomposition of CHI, and reaction (!Y) &me would double the rate of removal of meth- me if no other H atom removal chLannels were opera- tive ,

H+CHs.+ Hz+CH3. (JW

I’&!. 9. Tin dcy.w,dcncc on pressure of the culculatcd retc con- ~1s~~ Cot CI34 dl,;socfntion,ut 2200 K.

Tabayashi and Bauer 1161 suggest that a reaction scheme involving at least 12 steps is necessary to repro- duce their experimental methane decomposition pro- files. The poor agreement between the different sets

of experimental data may also be seen to support the suggestion tit the high pressure limit has not been achieved. The only reported attempt to examine the pressure dependence of k, is that of Kozlov and Knorre [9] who observed no pressure dependence over the range 2-8 atm.

5. Conclusion

Experimental results on the association of CM3 f H are well described by an RRKM model based on the minimum density of states criterion, with two adjust- able parameters, a! and &. Further experimental work is planned to study the kinetics directly and to inves- tigate the temperature dependence of the rate con- stant.

The model, although not op! Lmised in view of the scatter in the experimental data, is only moderately successful in predicting the rate corlstants for CH4 decomposition. The lack of agreement argues forcibly for a more thorough study of the pressure dependence of the dissociation reaction.

Acknowledgement

We thank Cr. M. Quxck for helpful discussion and the Science Research Cmncil for research studentships to R.P. and G.J.R.

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