Chemical Physics Letters 376 (2003) 381–388

www.elsevier.com/locate/cplett

Rate theory of methyl recombination at the lowtemperatures and pressures of planetary atmospheres

Gregory P. Smith *

Molecular Physics Laboratory, SRI International PS067, 333 Ravenswood Avenue, Menlo Park, CA 94025, USA

Received 10 April 2003; in final form 3 June 2003

Published online: 3 July 2003

Abstract

The recombination of methyl radicals in low pressures of hydrogen, helium, and nitrogen to form ethane controls

the concentrations observed for methyl and the photochemical synthesis of higher hydrocarbons above the Jovian

planets. Few measured or theoretical rate constants are available to provide reliable model predictions. RRKM and

master equation calculations are reported here, using three levels of transition state detail, to describe existing data and

provide consistent and reliable expressions for this rate constant at 65–300 K and any pressure. This gives k1 ¼3:59� 10�10 T�:262 e�37=T cm3/molec/s and k0ðH2Þ ¼ 3:32� 10�15 T�4:28 e�131=T cm6/molec2/s.

� 2003 Elsevier B.V. All rights reserved.

1. Introduction

Methyl radical concentration profiles in the at-mospheres of Saturn and Neptune have been de-

termined recently from emission measurements by

the infrared space observatory (ISO) [1,2]. Methyl

is produced from solar VUV photolysis of meth-

ane, and mainly lost by recombination reactions

with itself or hydrogen atoms. The reaction

forming ethane begins the kinetic sequences that

synthesize the larger hydrocarbons [3], many ofwhose vertical profiles have been determined for

the four giant planets and Titan by ISO observa-

tions [4] and Voyager spacecraft UV solar occul-

* Fax: +1-650-859-6196.

E-mail address: smith@mplvax.sri.com (G.P. Smith).

0009-2614/03/$ - see front matter � 2003 Elsevier B.V. All rights res

doi:10.1016/S0009-2614(03)00991-6

tation measurements [5]. In addition, transport

parameters for these planetary atmospheres in the

form of eddy diffusion coefficients have been de-rived from methyl and hydrocarbon profiles using

photochemical models [5]. It is clear from this

discussion and modeling studies [3,4,6] that accu-

rate rate coefficients for methyl recombination are

required.

Methyl emissions from Saturn were found to be

about 10 times weaker than model predictions,

leading to proposals to reduce the eddy diffusioncoefficient (thereby accessing faster low-altitude

chemistry) or to increase the recombination rate

for its removal [1,2]. The apparent flexibility to

increase this rate constant results from a lack of

recommended expressions applicable to the low

pressure and temperature conditions involved;

extrapolating two expressions [7–9] designed for

erved.

382 G.P. Smith / Chemical Physics Letters 376 (2003) 381–388

use above 300 K to 140 K presents a range of 300

for this rate.

The reaction is pressure dependent, with the

limits at high and low pressures determined by the

respective steps below

CH3 þ CH3 $ C2H6 k1; k�1

C2H6 þM ! C2H6 þM k2

where C2H6* is an activated ethane molecule

above the dissociation energy, and M is the plan-

etary bath gas, H2, He, or N2. Most theoretical

work has focussed on the high pressure limit,whereas in the low-pressure, high-altitude plane-

tary regions where photolysis occurs this reaction

is in the so-called pressure falloff regime where k�1

and k2 compete. Measurements also become very

difficult for these pressures. The forward methyl

recombination and reverse ethane decomposition

are related by the equilibrium constant.

The theoretical formulation of RRKM theoryfor describing, codifying, and predicting such rate

processes consists of a transition state model for

k�1 as a function of temperature or energy, and a

representation for the stabilization step (e.g., how

much energy is removed from an activated mole-

cule per collision). While much work has been done

on this system due to its importance in combustion,

lower temperatures were not examined. The goal ofthis work is to examine several theoretical models

for methyl recombination and its pressure depen-

dence, to obtain reasonable fits of existing data,

and to provide reasonable extrapolated expressions

for this rate constant under planetary conditions

for the appropriate bath gases M. This should

improve and constrain the models used to interpret

current and future observations.Extensive experimental data on the pressure

dependence in argon and the high pressure limit

ðk1Þ at 300–906 K are available in Walter et al.

[10], which includes the results from Slagle et al.

[9]. At lower temperatures, two studies provide k1values at 200 K [10,11]. Cody et al. [11,12] also

observed some falloff at lower pressures (in 0.6

Torr helium), and similar k1 values at 155 and 202K. Shock tube studies of the recombination at high

pressures [13] with some extrapolation [14] provide

a basis for extending the transition state theories to

higher temperatures. Several groups [15–17] report

pressure dependent rates in the 1200–1700 K

range. These form the most recent and reliable of

many studies of this reaction, on which to base the

theoretical models. Unfortunately, work in he-

lium, nitrogen, and hydrogen bath gases is rare,and the reaction is at the high pressure limit for

most room temperature experimental conditions.

A similar long history of theoretical investiga-

tions is available. The recent ab initio potential

surface/variational transition state/master equa-

tion study of Klippenstein and Harding [14] cites

about 20 of these studies. A fairly complex surface

description and transition state search is possible,given the large number of degrees of freedom. The

studies vary considerably in complexity and ease-

of-parameterization. One very simple approach is

our version [18] of the hindered Gorin model,

which assigns a temperature-dependent hindrance

parameter to restrict the rotation of the methyl

radicals in the transition state. A series of papers

by Wardlaw and collaborators [19–21] traces thedevelopment of a flexible/variable transition state

theory, an approach featuring several interpretable

parameters and convenient, successful use. The

Variflex code [22] contains one recent version

(1997) of this transition state formulation, in the

example provided for the CH3 +CH3 reaction.

2. Rate theory methods

The goal is to employ RRKM and related rate

theories to fit measured high pressure methyl re-

combination rate constants accurately, to fit the

argon pressure dependent data consistently, and

thereby to calculate sound extrapolated values at

low temperatures and hydrogen or helium pres-sures. The approaches are semi-empirical in that

parameters will be adjusted within reasonable

values to best match the data, and are chosen to be

transparent. Several methods are applied to both

the temperature dependence of k1 and the pres-

sure dependence, to investigate what degree of

sophistication is required and to evaluate the likely

reliability of the extrapolation.Several particular difficulties apply to RRKM

calculations for radical recombination reactions.

G.P. Smith / Chemical Physics Letters 376 (2003) 381–388 383

First, there is no intrinsic energy barrier to the

attractive potential between the fragments, and so

the energy maximum and position of the transition

state depends on adding the rotational energy to

produce an effective potential. This location is

distant, depends on an accurate long range po-tential, and differs greatly with energy and angular

momentum. A serious corollary problem is a lack

of knowledge of the potential, frequencies, and

internal rotations of many of the looser modes at

such distant fragment separations, but these fac-

tors and their variations play a critical role in de-

termining state densities near the transition state,

and in determining its location. They are alsoharder to compute ab initio.

The low pressure limit rate constant depends on

how efficiently the excess energy above the disso-

ciation energy is collisionally removed from the

activated ethane. The efficiency factor b or the

amount of energy removed per collision DE will

vary with bath gas M, and will need to be esti-

mated for He, H2, and N2 since the measurementsare mostly in Ar. The temperature variation is

determined from the pressure dependences of the

data at and above 300 K. Finally, the effects of

rotation also complicate the low pressure regime

[23]. Consider ethane as a separating pseudo-dia-

tomic with average rotational energy kT. Upon

approaching the large transition state distance, the

J value is conserved, but since the moment of in-ertia has increased and rotational B value de-

creased, the rotational energy must decrease. The

extra energy becomes available for dissociation;

considering the reaction in the decomposition di-

rection, less energy needs to be provided colli-

sionally. The approximate correction factors

employed become very large at low temperatures

and large separations, with resulting concerns re-garding their accuracy.

Three levels of transition state theory (TST)

were applied to predict k1ðT Þ and provide

the k�1ðEÞ values for the pressure dependent cal-

culations. The simplest is our restricted Gorin

parameterization [18,24], which locates a temper-

ature dependent (canonical) transition state at

the maximum of an effective potential composedof a Lennard-Jones attraction plus the average

thermal rotation ðErot ¼ kT Þ. This gives rþ=rCC ¼

ð6D0=RT Þ1=6. The Gorin model, the phase space

limit for this rate constant, uses free methyl radical

vibrations and rotations for the transition state

modes, while our restricted version and code ap-

plies an empirical hindrance parameter g to the

two two-dimensional methyl rotors to account forbumping into each other.

Canonical TST can be applied to more sophis-

ticated pictures of the CH3 +CH3 potential sur-

face. Consider defining a Morse potential for the

approaching CH3 fragments, developing an inter-

polation formula for how the CH3 frequencies

vary, and also compute how the phase space of the

rotating CH3 fragments is restricted as a functionof separation r. At each of several values of r, onecan compute a state density for the potential

transition state structure (ethane minus the C–C

stretch) averaged over thermal energy (E) and

angular momentum (J) distributions. The mini-

mum value defines the transition state (bottle-

neck), according to the minimum density of states

criterion. Variflex (http://chemistry.anl.gov/chem-dyn/VariFlex/) [22] and the Diau–Smith–Gilbert

[25] codes (with their defined rotor restrictions)

were used for these CVTST calculations. The

Variflex formulation uses the parameter �a� to in-

terpolate vibrational frequencies between the CH3

and C2H6 values via the formula vðrÞ ¼ vðCH3ÞþðvðC2H6Þ � vðCH3ÞÞe�aðr�reÞ.

Microcanonical variational transition statetheory (MCVTST) takes this approach one step

further by defining a transition state for each

(E; J ). The same potential may be used, and r is

varied until the minimum in state density from the

other modes is found. Then for k1 at any tem-

perature one only need perform the appropriate Eand J averaging. Variflex [22] was used for these

calculations.Three methods were used in computing low

pressure limit and falloff rate constants. The mas-

ter equation formulation is the most sophisticated,

as it takes into account explicitly the competing

decomposition and specific energy transfer steps

between different energy levels of activated ethane

molecules. Calculations used the Multiwell code of

Barker [26] with an exponential energy transferprobability model. The collision frequency is

Lennard-Jones ðkLJÞ between C2H6 and M, the

Fig. 1. High pressure limit rate constants for methyl recombi-

nation versus temperature. Experimental values (diamonds)

from Cody et al. [11,12] and Walter et al. [10] at low temper-

ature, Slagle et al. [9] near 800 K, and Hwang et al. [13] at high

temperatures. Five theory lines are shown. The Lennard-Jones

potential phase space limit and the restricted Gorin transition

state (1) are simple temperature dependent models. The two

canonical variational transition state calculations (2) use dif-

ferent frequency interpolations (�a�) versus fragment separation

distance. The microcanonical variational theory (3) evaluates

transition states for each energy and J value.

384 G.P. Smith / Chemical Physics Letters 376 (2003) 381–388

variable parameter is the energy transferred per

collision DEðT Þ, and any J dependence (MCVTST)

must be thermally averaged before proceeding. To

get k0 from this code, which outputs k=k1, one

must run at very low pressure. A rotational cor-

rection term is added to the energy.RRKM theory [23], our second method, com-

putes the low pressure limit by

k0 ¼ kLJbFWR

ZNðEÞe�E=RTdE=Q2; ð1Þ

where NðEÞ is the ethane density of states above

the dissociation limit, Q is the methyl partition

function, FWR is the Waage–Rabinowitch rota-tional correction factor, and the collisional effi-

ciency factor b can be calculated by b=ð1� b1=2Þ ¼DE=FERT (FE is a small term describing the energy

dependence of the state density). For intermediate

pressures, the falloff rate constant is given by the

Boltzmann weighted integral over energy of

k1ðEÞ=f1þ k1ðEÞ=k2½M g.Finally, the method of Troe [27] computes k0

and the falloff using a series of semi-empirical

factors from molecular parameters. Explicit kðEÞvalues are not required, only k1. This approach

also provides a convenient format to express the

falloff behavior for later use in rate constant

computation without repeating the more detailed

rate calculations. One computes a broadening

factor F to the Lindemann falloff behavior suchthat k=k1 ¼ F =ð1þ k1=k0½M Þ, where log F ¼logFc=½1þðlogPrþCÞ2=ðN � 0:14�ðlogPrþCÞÞ2 ,N ¼ 0:75� 1:27 log Fc, C ¼ �0:4� 0:67 log Fc, andPr ¼ k0½M =k1.

3. Results

Fig. 1 shows the results for k1, with compari-

sons to experimental values. The simple restricted

Gorin model provides a good fit, with our usual

simple functional form for the temperature de-pendence of the hindrance parameter giving g ¼246� 526 T�1=6. Note that at 100 K this model

reaches the unhindered Gorin phase-space limit,

and the rate constant will decrease at lower tem-

peratures. Using more sophisticated canonical

variational TST approaches produces similar

decreases at low temperature, reversing the usual

trend of a negative temperature dependence. The

data show that k1 stops increasing only below 200K, so the CVTST rate constant decrease below 300

K fails to provide the proper temperature depen-

dence for extrapolation. The CVTST results for

a ¼ 1:0 �AA�1 consistently overpredict the rate con-

stants, and a more rapid tightening of the transi-

tion state frequencies ða ¼ 0:7Þ is required for the

best fit. This behavior is consistent with the results

of Pesa et al. [21] and predecessor studies [20], allfor 300 K and above. Ab initio calculations [28]

support values of a ¼ 1:2–1.8, larger than the

standard 1.0, further detracting from this choice.

Another CVTST calculation was performed using

a potential supplied in the Diau–Smith code [25].

The methyl interactions are described by non-

bonded Lennard-Jones forces, and the coupled

cosine-squared model of Wardlaw and Marcus[19] is used. A good fit is obtained at 300 K and

above, slightly lower than the a ¼ 0:7 Variflex re-

sult shown in Fig. 1. However, the same unsup-

ported decrease at lower temperatures – though

less rapid – occurs.

Fig. 2. Selected fits of the pressure dependent results from

Walter et al. [10] (300 K) and Slagle et al. [9] (474, 700, and 300

K points above 2 Torr) for methyl recombination in argon,

using the J-averaged microcanonical rates in a master equation

calculation with DE ¼ 200� ðT=300Þ cal/mol. The 300 K the-

ory value is corrected for a 16% overprediction of k1.

G.P. Smith / Chemical Physics Letters 376 (2003) 381–388 385

The MCVTST calculation with standard a ¼1:0 provides excellent prediction of the k1 data

for all temperatures. The low temperature de-

crease only occurs below 140 K. As expected, the

microcanonical result is lower than the canonical

calculation for the same potential, until the lowtemperature decline sets in at what may be the

collision rate limit. Clearly this level of theory

should be used for predicting the low-tempera-

ture, pressure-dependent, master equation rate

constants.

The pressure dependent rate data in argon from

300 to 900 K was well matched by master equation

calculations, using kðEÞ values from the MCVTSTmodel and averaging over the thermal molecular J

distributions. Results at three temperatures are

shown in Fig. 2. The average energy transferred

per collision DE, the only adjustable parameter, is

200(T/300) cal/mol. This value at room tempera-

ture is about twice that observed for deactivation

of azulene or toluene [29,30], and the temperature

dependence is stronger than might be expected[31]. Lower pressure measurements at lower tem-

peratures are needed to confirm the implications of

this latter point; if energy transfer does not decline

as much, the extrapolated rate constants should be

faster. Substituting the CVTST a ¼ 0:7 or Gorin

kðEÞ values, or employing a rotationally hot J

average of the MCVTST kðEÞs, did not apprecia-

bly alter the falloff behavior.Low pressure limit rate constants from the Troe

and RRKM calculations, using the same DE, arenearly identical. Computed values of b � 0:2 are

nearly temperature independent. The predicted k0values are similar to the above master equation

results at 300–500 K, but twice as large at 800 K

and half the size at 100–200 K. Thus the falloff fits

at high temperature are poorer, and the experi-mental results would suggest a more typical decline

in b with increased temperature (less variation in

DE) for the RRKM calculations. In general, the

RRKM/Troe falloff curves show somewhat more

arc than the master equation results – Fc is higherand k0 is lower. Differences of +30% can occur in

the lower pressure range.

To derive recommended rate constant resultsfor lower temperatures and pressures from the

MCVTST/master equation results in argon, rela-

tive k0 values are needed for He, H2, and N2. These

depend on known differences in reduced collision

mass and Lennard-Jones cross section, and in

386 G.P. Smith / Chemical Physics Letters 376 (2003) 381–388

estimated values for DE. Averaging azulene and

toluene energy transfer results [29,30] for He, H2,

and N2 relative to Ar (0.43, 0.7, and 1.07) gives

k0ðHeÞ=k0ðArÞ ¼ 0:7, k0ðH2Þ=k0ðArÞ ¼ 1:7, and

k0ðN2Þ=k0ðArÞ ¼ 1:2. Limited falloff data in helium

at 577 and 810 K [9] suggest similar rates to argon,but recombination rates at 298 K in 1 Torr He [11]

are slower than in Ar [10]. By repeating our master

equation calculations for H2 instead of Ar, we

determined our final k0ðH2Þ=k0ðArÞ ¼ 1:4. Likelyuncertainties in relative values are 25–35%.

A plot of the computed low temperature pres-

sure dependent results in hydrogen is shown in

Fig. 3. For argon, multiply the X-axis values (di-vide k0) by 1.4, for helium by 2, for nitrogen by 1.7.

Pressure-axis-adjusted results from Cody et al. [11]

in helium are also displayed, and appear better

predicted at 200 than 300 K. The resulting rec-

ommended rate constant expressions for methyl

recombination in hydrogen are:

k1 ¼ 3:59� 10�10T�:262e�37=T cm3=molec=s; and

k0ðH2Þ ¼ 3:32� 10�15T�4:28e�131=T cm6=molec2=s

ð65–300 KÞ;

with a Troe falloff formula parameter Fc � 0:3from fitting the master equation results at thepressure where k0½H2 ¼ k1. This low pressure

Fig. 3. Recommended bimolecular methyl recombination rate

constants versus hydrogen pressure at various temperatures,

from the MCVTST/master equation calculations. Argon pres-

sures for equivalent rate constants are estimated to be 1.4 times

larger, and helium pressures twice those shown. Experimental

values are from Cody et al. [11] in helium at 298 K (circles) and

202 K (squares).

limit rate constant at 200 K is about four times the

value used in some recent models [6], which was

based on numerical extrapolation of experimental

and theoretical results in helium [9]. The falloff

appears lower and flatter than that predicted byRRKM theory or Troe calculations, for which

Fc � 0:56. There can also be 10% error or more

incurred in using the fit formulas to approximate

the master equation results shown in Fig. 3, with

the most significant difficulties (30%) occurring for

k=k1 ¼ 0:005–0.05.Given the similar importance for planetary at-

mospheres and lack of low temperature results forthe H+CH3 recombination that regenerates CH4,

and the reasonable success of the restricted Gorin

RRKM model for fitting and extrapolating the

ethane results, similar calculations (resembling our

previous work [24]) were performed for H+CH3 +

He. Our choice of a constant hindrance of 85%

produces a small positive temperature dependence,

to accommodate the 300 K k1 measurement ofSeakins et al. [32] and the recent Su and Michaels

[33] value at 1500 K; but like other theories [34],

this overpredicts the 200 K data point. (The

present choice produces the largest positive tem-

perature dependence a physically plausible hin-

dered Gorin model can have.) Falloff calculations

at 300 and 500 K using DE ¼ 125ðT=300Þ cal/mol

ðb ¼ 0:13Þ fit the data of Brouard et al. [35] ade-quately. Pressure dependent rate constant recom-

Fig. 4. Restricted Gorin model RRKM theory predictions for

H+CH3 recombination in helium, for a constant 85% hin-

drance and DE ¼ 125� ðT=300Þ cal/mol. Hydrogen pressures

giving similar rates are half as large. Symbols are for visual

guidance and do not represent data.

G.P. Smith / Chemical Physics Letters 376 (2003) 381–388 387

mendations are plotted in Fig. 4. Note the more

rapid falloff and smaller temperature dependence

of k0 for this smaller molecule compared to the

ethane system. The fit expressions for 100–500 K

in helium are:

k1 ¼ 5:91� 10�10T�:09e12=T cm3=molec=s;

k0ðHeÞ ¼ 2:14� 10�26T�1:06e17=T cm6=molec2=s

ð65–300 KÞ; and Fc ¼ 0:56:

4. Conclusions

Available rate theory approaches with straight-

forward potential surface parameters have been

applied to derive low-temperature, low-pressure

rate constant expressions for methyl and H atom

recombination with methyl radicals, for a rangeneglected by previous calculations but critical for

modeling giant planet atmospheric photochemis-

try. By fitting existing data, the method can provide

reliable extrapolations. Nonetheless, additional

measurements at low pressure and temperature are

needed to reduce uncertainty. Relative values for

H2, He, and N2 bath gases versus Ar are also

required. Although reasonable consistency wasfound among the various theoretical approaches,

the large rotational corrections applied at low

temperature merit further investigation – perhaps

using a two-dimensional (E and J) master equation

approach. Theory for k1 in the CH3 +CH3 case

appears to approach the phase space limit and

begin to decrease at low temperature. The canoni-

cal variational method performs the poorest ofthe three approaches tried in modeling the data.

Acknowledgements

This research was supported by the NSF Plan-

etary Astronomy Program, Grant AST-0074140,

and the NASA Planetary Atmospheres Program,Grant NAG5-9908. The author thanks Drs. David

Golden and David Huestis for useful discussions,

and Drs. John Barker, Eric Diau, Sean Smith,

Stephen Klippenstein, and Al Wagner for provid-

ing their codes.

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