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Thermal unimolecular reactions in the fall-off regime:A master equation analysis
Volume 76, number
I
CHEMICAL
THERMAi.a UNIMOLECULAR
PHYSICS
15 November
LEll-ERS
REACTIONS IN THE FALLOFF
1980
REGIME:
A MASTER EQUATION ANALYSIS Wnmen
D. LA\\ RANCE,
41a1-1E.W.
KFuiGHT
School of Science. GnfJirfl Cln~~ersx_~,Kaarhan.Qucerrslarrd4/i I. tlusrralra
Robert
C. GILBERT
Departmenr of i’leorerrcal Chetnrsrry, Unn erslr_vof S)dncy
S~dtw_~ IXSIV2006
A usrrakn
and Keith D. KING of Clrcmrcal Engmcernzg, Lhrerslry of -4delarde Adelaide S-l 5001. -lustralro
Deparrnlent
Reccrvcd 27 July 1980
A trchnlque
molecular
for combmmg
RREU
data m the fall-ofi rcyme
fhcor) \\Ith n master cqunrmn approach Xl~croscop~c rates for SF6 decomposltIon
1. Introduction
To date most analyses of thermal ununolecular reactlon rate data have used either h&-pressure hmtt data or, alternatrvely, fall-off pressure data with strong colhslons assumed. An mcreasing number of reactrons are beutg studled m which only fall-off data are available, the hrgh-pressure Imnt bemg expmunentally inaccesnble. It IS thus desirable to have a means of anfysmg fall-off data rapIdly and accurately. We here demonstrate the use of RRKM theory combined with an energy-gramed master equation (ECME) approach to analyse efficiently and prectsely thermal data m the fall-off regune. From such an analysis one can obtain the cntical energy (E,,), the h&-pressure Arrhemus parameters, microscopic reactron rates [k(E)] , and the average downward
energy transferred
m col-
hslons Our method IS particularly sulted to data significantly far from the high-pressure hmit, it is complementary to the iterative method of Astholz et al [I], wtuch is best suited for data close to the hghpressure limit. As an ~Uustrative example,
the analysis
of the sulphur
we present
hexafluortde
results
for
shock-tube
IS opphed
IO the anal) us oi therm4
are obramcd
uni-
m an lllustrstlve calculation.
data of Bott and Jacobs [21. Our rationale for choosing these data in particular IS that a detailed description of the thermal behavtour of SF6 plays a central role m the theoretical modelhng of infrared multiphoton unrmolecular decomposrtion. A previous analysls of these data has been carned out by Lyman [3] who also used an RRKM master equation method. Recent advances in the solution of the EChlE [4] have paved the way for a number of improvements to thus earher work. These improvements are discussed as they appear throu& the text.
2. Theory The form of the EChlE whrch we use has been derived previously
Rg(E)=w
[4] and IS as follows:
s NE’, E)zdE) -
PW. E’ME’)I
dE’
E’+& + wY&wh where
g(E)
(1) is the total
population
energy E. R is the macroscopic
of the states at
rate coefficient
for 113
Volume 76. number 1
CHLhlICAL
the reaction, w is a reference colhston frequency (usually hard sphere). and P(E, E’) IS the probabrltty function
descnbl’ng
collrslonal
transfer
X
[JE
[J
E
-1 1 1
(E’ - E) P(E, E’) dE P(E, E’) dE
RRKhl
theory
IS used as a convenient
the only measure of P required to determme R. Here we have chos2n an evponenttal form for P when E IS less than E’, P(E, E’) values for E greater than E’ bemg obtamed from the mlcroscoplc reverslbdtty relation. (The us2 of thus form for P(E, E’) removes a restrtctton of Lyman’s work m whtch IS essentrally
a functional form was denved spectfically for the molecule under study Furthermore. Lyman then parametenzed this functton usmg rela\atton data obtamed at temperaturrs sevrral hundreds of degrees below the temperatures reached in the thermal decomposrtton.) For reasons explnmed previously [4]. tn the RRKhl and EGME calculatrons all evtcrnal and internal moments of Inertid of reactant and compku are asstgned the sdme values. This eff2ctively mcorporates all angular momentum effects in an average way into the angular-momentum-Independent RRKRl parameters and k(E) curv2s. Our method for the numertcal solution of the EGhlE offers several advantages comparzd with varrous alternahve procedures [ 1,3]. We use a raped monotontcally converging etgenvalue-etgenvector method. the details ofwhrch have been described prevtously [5] TIus method has the dual advantage of requtring considerably less computer tune (allowing a wade range of parameters to be tested) and of readily factitating the check for convergence which is necessary stnce the EGhlE IS computed as a frrutedtfference approximatton to the Integral of eq. (1) [4] Precrse defimtions of X-(E) and of the densrty of states functron p(E). whrch tnclude angular momentum effects, are given in ref. [4].
calculated for use m the solution of the master equahon With grven k(E) values the EGME IS used to calculate rate coefficients for any temperature and pressure. Pressure effects are mcluded through the o term [eq. (1)] whtle temperature effects are incorporated both m w and rn UE), the average energy transferred per colhston (A&) IS treated as a varrable parameter. Prevtous workers have found that theu results could be interpreted by a temperature dependence for
suits
calculatton and (4E) varied rn the EGhlE cakulatlon, the whole process berg repeated untrl reasonable agreement with the experrmental results IS obtamed.
4. l?esults and discussion The results of our calculattons are compared with the expenmental data of Bott and Jacobs In fig. 1 and the associated RRKM parameters listed in table 1. Ftg. S_shows the k(E) versus E curve obtamed The hrgh pressure parameters at 1000 K are log Am = la.65 5
= 88.59
(A,
n-t s-l),
kcal mol-r .
The average downward SF6 collisions was found for both T Ii2 and T-l/2
114
means by
which the microscopic rate coefficients k(E) may be
from previous analyses of thermal unimol2cular reactrons that,the precrse functronal form for P(E, E’) is not mlportant, the average downward energy transf2rred p2r colllston =
3. Methodology
of energy
from E’ to E. It IS well known
UE)
15 November 1980
PH\I SICS LETTERS
energv transferr2d in Arto be 435 cm-l at 1750 K, forms of(u).
. \
26-30alm
.38-45
\
\ \ % ‘\ \ \ \\ . 0 ,‘,
.14-17 oos-0s q
0 13-O 16
2
rig
1
Plot
0
I
21 48
50
oi
52
mxroscoptc
54
56
104/T
(K-I)
rat2
56
60
wrsus
mrersr
the RREM
Table 1 parameters used to caiculatc the b(E) values for the of fig. 2 Both molecule and complex are assrgned the same moment of mertla Ar/STs colbsron drameter taken as 4 5 A The R(L) values obtamed from these RRKM paramet2rs are used m the ECME for both temperature varrat:ons of (AD RRIA
tune
Complex
frequent>
degeneracy
frequency
dezcncracy
917 770 640 615 512 345
3 1 2 3 3 3
901 735 711 610 584 497 487 392 345 330
7 1 1 1 2 1 1 2 2 1
symmetry
number.
10
external
symmetry
number:
0.167
p.rramctcrs
The experunental
all but
external
12
08 energy/c&x
16
Z”
lo4
FIN 7 Plot oi XT(E) versus cncry? (as E - &o) obt.und
coettlclent
temperature The pomts correspond to the data of ref [ 1 ] Our tit to thus dsta IS shoan for Tt’* (sohd curve) dnd T-l’* (dashed curve) varr3trons of (~5)
Molecule
04
62
the
tiom
hstcd m table 1
results are well reproduced for
i-ughhest pressure.
where N.Z predict
a faster
rate at high temperatures than that observed. The expenmental method of momtormg the reaction was changed for thus pressure and consequently we feel that thus discrepancy between the experimental and calculated rates is possibly due to a systematrc experlmental error. Comparison of our calculations with rhose of Lyman (fig. 3) shows our curves to consistently have a smaller slope, which results in a better fit to the data. The limited temperature range and the spread of the expenmental results makes it inappropriate to comment on the temperature dependence of the average energy transferred per colhaon. The T-O 5 dependence fits the lower pressure data better, whereas the To 5 dependence provides a shghtly better fit at the higher pressures. The cntlcal energy requrred to fit the data is 87.0 kcal mol-1, a value some 4 3 kcal mol- 1 less than Lyman’s His figure 18 closer to the 93 f 3 kcal mol-t estimated by Benson usmg thermodynamic considerations [7]. However more recent experimental data [8] shows our result to be quite reasonable: these measurements yielded lower bounds to the F5S-F
Volume
76, number
CHEMICAL
1
-
26-30alm
.
38-45
PHYSICS
LE’ITERS
15 November
1980
ferred per colhslon relative to the value of 176 cm-l for the average energy transferred per collision obtamed by Lyman
* 14-17 005-06 0 13-O
16
5. Conclusion The use of RRKhl rheory wth current EGME techniques enables thermal ummolecular reaction rate data III the fall-off regune to be analysed easi!y to obtam the microscopic rate coefficients wthout resort to the strong colliston assumption
Acknowledgement Fundmg by the Australian Research mlttee IS gratefully acknowledged.
Grants
Com-
References I
50
TIE. 3. Comparison tittmg usmg a Tt’* L,
man’s
tit
52
of
54
56
104/T
(K-I)
cxperuncnt~
(AE) dcpcndcncc
58
60
[ 11 D C. Astholz, (1979)
data
(pomts) wltb our (sohd curve) and wth
cunc)
[?I
i8\,“ott
[3j
J L.
i-l]
bond energy of 89.6 2 2 and 85 2 2 kcal mol- L with an overall estimate of 89 9 i- 3.4 kcal mol-t . The lack of energy transfer data for comparable systems m this temperature regune prevents us from making any comment as to the merits of our value of 435 cm-t for the average downward energy trans-
116
J Troe
and W 14w.ters.
J Chem
Phys
70
5107.
L>maa.
and
T.A
Jacobs.
J. Chem
Phls
J. Chcm
67
(1977)
Phls.
50
(1969)
1868.
R G. Gdbert and K D. Kmg, Chem Phys 49 (1980) 367 [5 ] B J Caynor, R G. Gllbrrt and E D Kmg, Chem Phys Letters 55 (1978) 40 [6] H Endo. K;. Glinzer and J. Tree. J. Phys. Chem 83 (1979) 2083 [7j S W Benson, Chem Rev 78 (1978) 23 [S] T Kunp. R-C Estlrr and R N. Zare, 1. Chcm Ph> s 70 (1979) 5925
I
CHEMICAL
THERMAi.a UNIMOLECULAR
PHYSICS
15 November
LEll-ERS
REACTIONS IN THE FALLOFF
1980
REGIME:
A MASTER EQUATION ANALYSIS Wnmen
D. LA\\ RANCE,
41a1-1E.W.
KFuiGHT
School of Science. GnfJirfl Cln~~ersx_~,Kaarhan.Qucerrslarrd4/i I. tlusrralra
Robert
C. GILBERT
Departmenr of i’leorerrcal Chetnrsrry, Unn erslr_vof S)dncy
S~dtw_~ IXSIV2006
A usrrakn
and Keith D. KING of Clrcmrcal Engmcernzg, Lhrerslry of -4delarde Adelaide S-l 5001. -lustralro
Deparrnlent
Reccrvcd 27 July 1980
A trchnlque
molecular
for combmmg
RREU
data m the fall-ofi rcyme
fhcor) \\Ith n master cqunrmn approach Xl~croscop~c rates for SF6 decomposltIon
1. Introduction
To date most analyses of thermal ununolecular reactlon rate data have used either h&-pressure hmtt data or, alternatrvely, fall-off pressure data with strong colhslons assumed. An mcreasing number of reactrons are beutg studled m which only fall-off data are available, the hrgh-pressure Imnt bemg expmunentally inaccesnble. It IS thus desirable to have a means of anfysmg fall-off data rapIdly and accurately. We here demonstrate the use of RRKM theory combined with an energy-gramed master equation (ECME) approach to analyse efficiently and prectsely thermal data m the fall-off regune. From such an analysis one can obtain the cntical energy (E,,), the h&-pressure Arrhemus parameters, microscopic reactron rates [k(E)] , and the average downward
energy transferred
m col-
hslons Our method IS particularly sulted to data significantly far from the high-pressure hmit, it is complementary to the iterative method of Astholz et al [I], wtuch is best suited for data close to the hghpressure limit. As an ~Uustrative example,
the analysis
of the sulphur
we present
hexafluortde
results
for
shock-tube
IS opphed
IO the anal) us oi therm4
are obramcd
uni-
m an lllustrstlve calculation.
data of Bott and Jacobs [21. Our rationale for choosing these data in particular IS that a detailed description of the thermal behavtour of SF6 plays a central role m the theoretical modelhng of infrared multiphoton unrmolecular decomposrtion. A previous analysls of these data has been carned out by Lyman [3] who also used an RRKM master equation method. Recent advances in the solution of the EChlE [4] have paved the way for a number of improvements to thus earher work. These improvements are discussed as they appear throu& the text.
2. Theory The form of the EChlE whrch we use has been derived previously
Rg(E)=w
[4] and IS as follows:
s NE’, E)zdE) -
PW. E’ME’)I
dE’
E’+& + wY&wh where
g(E)
(1) is the total
population
energy E. R is the macroscopic
of the states at
rate coefficient
for 113
Volume 76. number 1
CHLhlICAL
the reaction, w is a reference colhston frequency (usually hard sphere). and P(E, E’) IS the probabrltty function
descnbl’ng
collrslonal
transfer
X
[JE
[J
E
-1 1 1
(E’ - E) P(E, E’) dE P(E, E’) dE
RRKhl
theory
IS used as a convenient
the only measure of P required to determme R. Here we have chos2n an evponenttal form for P when E IS less than E’, P(E, E’) values for E greater than E’ bemg obtamed from the mlcroscoplc reverslbdtty relation. (The us2 of thus form for P(E, E’) removes a restrtctton of Lyman’s work m whtch IS essentrally
a functional form was denved spectfically for the molecule under study Furthermore. Lyman then parametenzed this functton usmg rela\atton data obtamed at temperaturrs sevrral hundreds of degrees below the temperatures reached in the thermal decomposrtton.) For reasons explnmed previously [4]. tn the RRKhl and EGME calculatrons all evtcrnal and internal moments of Inertid of reactant and compku are asstgned the sdme values. This eff2ctively mcorporates all angular momentum effects in an average way into the angular-momentum-Independent RRKRl parameters and k(E) curv2s. Our method for the numertcal solution of the EGhlE offers several advantages comparzd with varrous alternahve procedures [ 1,3]. We use a raped monotontcally converging etgenvalue-etgenvector method. the details ofwhrch have been described prevtously [5] TIus method has the dual advantage of requtring considerably less computer tune (allowing a wade range of parameters to be tested) and of readily factitating the check for convergence which is necessary stnce the EGhlE IS computed as a frrutedtfference approximatton to the Integral of eq. (1) [4] Precrse defimtions of X-(E) and of the densrty of states functron p(E). whrch tnclude angular momentum effects, are given in ref. [4].
calculated for use m the solution of the master equahon With grven k(E) values the EGME IS used to calculate rate coefficients for any temperature and pressure. Pressure effects are mcluded through the o term [eq. (1)] whtle temperature effects are incorporated both m w and rn UE), the average energy transferred per colhston (A&) IS treated as a varrable parameter. Prevtous workers have found that theu results could be interpreted by a temperature dependence for
suits
calculatton and (4E) varied rn the EGhlE cakulatlon, the whole process berg repeated untrl reasonable agreement with the experrmental results IS obtamed.
4. l?esults and discussion The results of our calculattons are compared with the expenmental data of Bott and Jacobs In fig. 1 and the associated RRKM parameters listed in table 1. Ftg. S_shows the k(E) versus E curve obtamed The hrgh pressure parameters at 1000 K are log Am = la.65 5
= 88.59
(A,
n-t s-l),
kcal mol-r .
The average downward SF6 collisions was found for both T Ii2 and T-l/2
114
means by
which the microscopic rate coefficients k(E) may be
from previous analyses of thermal unimol2cular reactrons that,the precrse functronal form for P(E, E’) is not mlportant, the average downward energy transf2rred p2r colllston =
3. Methodology
of energy
from E’ to E. It IS well known
UE)
15 November 1980
PH\I SICS LETTERS
energv transferr2d in Arto be 435 cm-l at 1750 K, forms of(u).
. \
26-30alm
.38-45
\
\ \ % ‘\ \ \ \\ . 0 ,‘,
.14-17 oos-0s q
0 13-O 16
2
rig
1
Plot
0
I
21 48
50
oi
52
mxroscoptc
54
56
104/T
(K-I)
rat2
56
60
wrsus
mrersr
the RREM
Table 1 parameters used to caiculatc the b(E) values for the of fig. 2 Both molecule and complex are assrgned the same moment of mertla Ar/STs colbsron drameter taken as 4 5 A The R(L) values obtamed from these RRKM paramet2rs are used m the ECME for both temperature varrat:ons of (AD RRIA
tune
Complex
frequent>
degeneracy
frequency
dezcncracy
917 770 640 615 512 345
3 1 2 3 3 3
901 735 711 610 584 497 487 392 345 330
7 1 1 1 2 1 1 2 2 1
symmetry
number.
10
external
symmetry
number:
0.167
p.rramctcrs
The experunental
all but
external
12
08 energy/c&x
16
Z”
lo4
FIN 7 Plot oi XT(E) versus cncry? (as E - &o) obt.und
coettlclent
temperature The pomts correspond to the data of ref [ 1 ] Our tit to thus dsta IS shoan for Tt’* (sohd curve) dnd T-l’* (dashed curve) varr3trons of (~5)
Molecule
04
62
the
tiom
hstcd m table 1
results are well reproduced for
i-ughhest pressure.
where N.Z predict
a faster
rate at high temperatures than that observed. The expenmental method of momtormg the reaction was changed for thus pressure and consequently we feel that thus discrepancy between the experimental and calculated rates is possibly due to a systematrc experlmental error. Comparison of our calculations with rhose of Lyman (fig. 3) shows our curves to consistently have a smaller slope, which results in a better fit to the data. The limited temperature range and the spread of the expenmental results makes it inappropriate to comment on the temperature dependence of the average energy transferred per colhaon. The T-O 5 dependence fits the lower pressure data better, whereas the To 5 dependence provides a shghtly better fit at the higher pressures. The cntlcal energy requrred to fit the data is 87.0 kcal mol-1, a value some 4 3 kcal mol- 1 less than Lyman’s His figure 18 closer to the 93 f 3 kcal mol-t estimated by Benson usmg thermodynamic considerations [7]. However more recent experimental data [8] shows our result to be quite reasonable: these measurements yielded lower bounds to the F5S-F
Volume
76, number
CHEMICAL
1
-
26-30alm
.
38-45
PHYSICS
LE’ITERS
15 November
1980
ferred per colhslon relative to the value of 176 cm-l for the average energy transferred per collision obtamed by Lyman
* 14-17 005-06 0 13-O
16
5. Conclusion The use of RRKhl rheory wth current EGME techniques enables thermal ummolecular reaction rate data III the fall-off regune to be analysed easi!y to obtam the microscopic rate coefficients wthout resort to the strong colliston assumption
Acknowledgement Fundmg by the Australian Research mlttee IS gratefully acknowledged.
Grants
Com-
References I
50
TIE. 3. Comparison tittmg usmg a Tt’* L,
man’s
tit
52
of
54
56
104/T
(K-I)
cxperuncnt~
(AE) dcpcndcncc
58
60
[ 11 D C. Astholz, (1979)
data
(pomts) wltb our (sohd curve) and wth
cunc)
[?I
i8\,“ott
[3j
J L.
i-l]
bond energy of 89.6 2 2 and 85 2 2 kcal mol- L with an overall estimate of 89 9 i- 3.4 kcal mol-t . The lack of energy transfer data for comparable systems m this temperature regune prevents us from making any comment as to the merits of our value of 435 cm-t for the average downward energy trans-
116
J Troe
and W 14w.ters.
J Chem
Phys
70
5107.
L>maa.
and
T.A
Jacobs.
J. Chem
Phls
J. Chcm
67
(1977)
Phls.
50
(1969)
1868.
R G. Gdbert and K D. Kmg, Chem Phys 49 (1980) 367 [5 ] B J Caynor, R G. Gllbrrt and E D Kmg, Chem Phys Letters 55 (1978) 40 [6] H Endo. K;. Glinzer and J. Tree. J. Phys. Chem 83 (1979) 2083 [7j S W Benson, Chem Rev 78 (1978) 23 [S] T Kunp. R-C Estlrr and R N. Zare, 1. Chcm Ph> s 70 (1979) 5925