J . Phys. Chem. 1993, 97, 6830-6834
6830
Non-Steady-State Dynamics in High-Temperature Systems V. Bernshtein and I. Oref Department of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel Received: January 26, 1993; In Final Form: April 6, I993
Values of rate coefficients of uni- and bimolecular reactions at high temperatures are important for understanding basic processes in shock tubes and combustion systems. The usual procedure for deciphering a kinetic scheme of a high-temperature process is to solve a matrix of coupled differential equations representing the rate equations with rate coefficients which are functions of temperature alone. Time-independent rate coefficients for unimolecular (or bimolecular) reactions imply a steady-state distribution of reactants during the reaction. This is certainly true at low temperatures but, as the present work shows, fails badly a t high temperatures. The consequence of this failure is decay curves which deviate significantly from the ones obtained from constant rate coefficients. We show samples of such behavior for cyclobutene isomerization and cyclobutane fission.
Introduction
Theory
Since 1922, when Lindemann proposed a mechanism for unimolecular reactions, significant changes have been made in our basic understanding of the kinetics of uni- and bimolecular reactions. Rice and Ramsperger and later Kassel have developed a statistical theory (RRK) for calculation of individual energydependent ratecoefficients,k ( E ) ,which was recast intoa quantum statistical form by Marcus to give the now well-known RRKM theory. It soon became clearZthat the dynamics of a unimolecular (or bimolecular) reaction depends not only on the energy and angular momentum dependent specific rate coefficients, k(E,J),but also on the population distribution of the reactant molecules at a given internal energy state and on the rate of collisional pumping into and out of the energy state. When the population distribution is in a steady state, the unimolecular reaction is characterized by a single, time-independent rate coefficient. This overall rate coefficient, kuniris identical to the individual rate coefficients which control the population decay from each of the energy states of the moleculeo2The rate of pumping in and out of a given state depends on the energy-transfer transition probability, a function of the internal energy of the excited m ~ l e c u l e . ~That . ~ is to say, the average energy transferred per collision, ( A E ) d , depends on the internal energy. Experimental results indicate that ( u ) d can be expressed by the e x p r e ~ s i o n ~ , ~
The processes taking place during a chemical reaction can be described by a master equation which for a substrate diluted in a bath gas at temperature Tis
( AE)d= a
+ b(E)"
(1) At low levels of excitation n can be larger than unity, at intermediate levels of excitation n = 1, and at high levels of excitation n = 0. This, of course, affects the dynamics of the system. Evaluation of the overall rate coefficient, kuni,therefore requires the solution of a master equation (ME) which includes the population distribution, energy-transfer probabilities, and k(E,J).IJThe techniques for evaluating kuniat low temperatures are now well established,and, problems and uncertaintiesin energy transfer notwithstanding,there are production programs available which give reasonable fits to experimental data.5 To facilitate the solution of the ME, steady states are assumed in n ~ m e r i c a l ' Jand ~ ~analyticallOJlcalculations. This assumption is no doubt valid at low temperatures. However, at high temperatures and low pressures there is good reason to believe that things are not that simple and that kuniis time-dependent, which is tantamount to saying that the system does not attain a steady state. The purpose of this work is to probe the hightemperature region and to evaluate kunifor two sample molecules under a variety of experimental conditions.
dNi/dt = w6Ex(PjiNj - P i p i ) - kiNi j
where Ni is the time-dependent population of state i, w is the collision frequency,6E is theenergy interval over whichsummation is done, P is the probability of transferring a given amount of energy from state i to statej (ij)or the reverse (ii),and ki is the energy and angular momentum dependent (RRKM) rate coefficient. The first term indicates pumping into state i from all other statesj, the second indicates depletion by collisions of state i to all other states j , and the third is depletion of state i by reaction. The collisional energy transfer transition function Pij is given, for strong collisions, by the Boltzmann expression
where p(E) is the density of vibrational-rotational states, Q is a normalization factor, and E stands for energy level i. For weak colliders, the exponential transition probability
pii = c';
exp(-hE/a)
(4) is used where ci is the energy-dependent normalization factor. AE is the amount of energy transferred from state i to statej and a is the average energy transferred in a down collision. ( M ) d and its energy dependenceare given by eq 1. All values of P obey conservation of probability, 6E&mPij = 1, and detailed balance. The two contraints are obeyed when ci is calculated by a recursion equation12
"
where n is the upper bound for the states of the substrate above which no up collision occurs,IZ B is the Boltzmann population distribution, and r is the unnormalized exponential transition probability Pi, = rij/Ci. The value of the highest term, c,,, is calculated first, then cWl,and so on down the energy states ladder. The master equation, eq 2, can be recast in a matrix form6 dN/dt =H*N (6) where N represents the population vector and H is the n X n
0022-3654/93/2091-6830%04.00/0 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 26,1993 6831
Non-Steady-State Dynamics
TABLE I: Values of the Parameters Used in RRKM and Master Equation Calculations of Cyclobutane Fission and Cyclobutene Isomerization
a o
Cyclobutane Fission EO= 62.5 kcal/moP log A = 15.6" u = 4.75 A (in Ar)b and 5.54 A (in SFs)b normal modes and degeneracies of molecule 29802, 2950 2, 28904, 1445 4, 12604, 12204, 10002, 926 1, 9002, 7502, 741 1, 627 1, 197 1 normal modes and degeneracies of complexC 29808, 14454, 12602, 12502, 11002, 1000 1, 898 1, 850 1, 7402, 6002, 380 1, 3502, 150 1 Cyclobutene Isomerization EO= 32.7kcallmold log A = 13.3d u = 4.70 A (in Ar)b and 5.49 A (in SF6)b normal modes and degeneracies of moleculec 3101 6, 1651 1, 1451 3, 1151 3, 1001,6, 801 2, 701 3 normal modes and degeneracies of complexc 31006, 1650 1, 14503, 11503, 10005, 8002, 7003 Constants for Eq 1 b u = 55 (in Ar) and 190 (in SFs) b = 0.013 (in Ar) and 0.015 (in sF6)
0.024
COLLISIONS
c too
I,
d 4600 e 8800 f 13150 q woo
!$
O.OI8 0.012
11 18
II I1
0.006
0.000 0
20
40
€0
100
80
120
ENERGY (kcal/md)
Figure 1. Population distribution of cyclobutane in Ar at 1500 K and 1 Torr as a function of the number of collisions.
7T-l 0.6
From ref 13. From ref 14. From ref 15. From ref 16.
matrix of transition probabilities. The time-dependent solution of eq 6 is
(7) where ai, is the ij eigenvector component and the j3, values are constants which are determined from boundary conditions. The set of (negative of the) eigenvalues A i s are real numbers which determine the values of the unimolecular rate coefficients, as discussed in eqs 9 and 10. We have calculated the values of A, and used them in the calculations reported in the next section.
I .Ol
0.6 O
I[\?[:
Results and Discussions The purpose of this work is to explore the dynamicsof a reactive system: the dependence of the population distribution f ( E ) and the average energy transferred per collision ( AE)on temperature, activation energy,molecular size, and type of bath gas. As typical examples, the dynamics of cyclobutane fission and cyclobutene isomerizationwas studied as a function of temperature, pressure, and type of bath gas collider.
-
C4H6
CH2=CH-CH=CH2
(R2)
The advantage of using these molecules is that they have similar cross ' sections and numbers of0modes but a large difference . in their activation energies. Thus, the effects of activation energy and number of modes on the value of the rate coefficient can be separated from each other. Table I lists the molecular parameters used in the calculations. For a strong collider, the population maintains its Boltzmann distribution independent of time or degree of reaction. This is, actually, one definition of a strong collider. For a weak collider, however, the high-energy end of the population distribution is depleted faster than the low-energy end. This is well-known, and it is a manifestation of the weakness of the collision, Le., the inefficiency of the bath to pump the upper levels around Eo fast enough. That is to say, the molecules cross the dividing surface for decomposition faster than the rate of replenishment of the levels near that surface. To demonstrate this point, here and in the rest of the calculations, the master equation was solved and the eigenvalues and eigenvectors found by solving the matrix eq 6. The energy increment was taken to be 30 cm-*,and the upper
S
8
h
\
I
0.0
T- l oI .
I+:
0.2
0
0
20 40 60 80 100 ENERGY (kcal/moll
Figure 2. Ratio of the population distribution of cyclobutane in Ar at 1500 K to that of the initial Boltzmann distribution as a function of energy for various numbers of collisions. Part a: 1 Torr,number of collisions (a) 5,(b) 700,(c) 4600,(d) 8800,(e) 13 150. Part b: lo3Torr, number of collisions (a) 4900,(b) 37 200,(c) 72 500, (d) 1 1 1 OOO. Part c: lo7 Torr (upper line) 5 X 108 collisions, (centerline) 109 collisions, (bottom line) 2 x 109 collisions.
value of the energy was chosen by taking the ratio of the Boltzmann population distribution at its maximum to that at the upper energy value to be >lo'. This ratio was found by calculations to ensure that all the conservation relations hold. In all the calculations, the initial ( t = 0) population distribution was Boltzmann, which implies a stepwise heating. This is shown for cyclobutane in Ar (at 1500 K and 1 Torr) in Figure 1, where the population profile
Bernshtein and Oref
6832 The Journal of Physical Chemistry, Vol. 97, No. 26, 1993
- 80
1 40
I\
-0
$00 0.61
: :1
0.41
{
SF'
80
dkk -
?m 0
c
NUMBER OF COLLISIONS Figure 3. Left coordinate, the ratio of the unimolecular rate coefficient to that at steady state vs the number of collisions. Right coordinate, the total population as a function of the number of collisions and the steadystate (S.S.) total population as a function of the number of collisionsfor cyclobutane in Ar at 1500 K at (a) 10-4 Torr, (b) 1 Torr, and (c) 103
Torr.
is given as a function of energy and number of collisions (i.e,, time). A better picture is obtained when the ratio of the actual population distribution,f(E), to that of the Boltzmann distribution at time 0 is plotted versus energy. Figure 2 shows the population ratio for cyclobutane in Ar at 1500 K and at 1, IO3,and lO7Torr as a function of energy and the number of collisions. The depletion of the high levels at low pressures and the complexity of the depletion process are clearly seen. If a pseudosteady state is assumed,*
then the rate coefficients kLnifor depletion from individual levels are identical and equal to the overall rate coefficient kzd. NT is the total number of reactant molecules, NT = clN1.
k'uni. = k z = ~ XjZl
(9)
That is to say, there is one (absolute value of) eigenvalue, hill, which has a value much smaller than that of all other eigenvalues. We denote this eigenvalue k z i to indicate its steady-state value.
NUMBER OF COLLISIONS Figure 4. Left coordinate, the ratio of the unimolecular rate coefficient to that at steady state vs the number of collisions. Right coordinate, the total population as a function of the number of collisionsand the steadystate (S.S)total population as a function of the number of collisions for cyclobutane at 2000 K (a) Ar, 1Torr; (b) Ar, IO3Torr; (c) sF6,1 Torr. In the non-steady-statecase, more than one eigenvaluecontribubs to the population depletion, and Ni has a multiexponential dependence on time given by eq 7. We exhibit the non-steady-state behavior by plotting kui/ vs the number of collisions, which is equivalent to elapsed time. kuniis defined by the equation
e
Figure 3 shows the dependence of l o g ( k ~ / e on ) the number of collisions for cyclobutane fission at 1500 K at three pressures. In addition, the population depletion as a function of the number of collisions is given for the actual case and for the one where steady state is assumed. At low pressure, the value of k h at the beginning of the reaction is 5 times its steady-state value. It obtains its steady-state value after 40% of the reaction is over. At higher pressures, the discrepancy is smaller, and it obtains its steady-state value very fast at the high-pressure limit. The situation is much more dramatic at 2000 K,Figure 4,where, at 1 Torr, the reaction is almost over before the system approaches a steady state. Theeffectofthestrongercollider,SF6,ascompared to that of the weak collider, Ar, can also be seen in Figure 4. As expected, the system obtains a steady state much faster, but so
The Journal of Physical Chemistry, Vol. 97, No. 26, 1993 6833
Non-Steady-State Dynamics
E
,
0.4ba AFTER 20 COLL.
0.2
IO0
-0.0
0 0
200
400
0.o
600
- 4 - 2
NUMBER OF COLLISIONS
Figure 5. Left coordinate, the ratio of the unimolecular rate coefficient to that at steady state vs the number of collisions. Right coordinate, the total population as a function of the number of collisions and the steadystate (SS)total population as a function of the number of collisions for cyclobutane in Ar at 1500 K at (a) 1 Torr and (b) lo3 Torr. 0.8
a
100
- 60
Es
0
200
400
600
NUMBER OF COLLISIONS
Figure 6. Left coordinate, the ratio of the unimolecular rate coefficient to that at steady state vs the number of collisions. Right coordinate, the total population as a function of the number of collisions and the steadystate (SS)total population as a function of the number of collisions for cyclobutane in SF6 at 1500 K at (a) 1 Torr and (b) lo3 Torr.
is the rate of decomposition; so again most of the reaction is over before a steady state is obtained. Cyclobutene isomerization, Figure 5,shows the situation much more vividly. A steady state is obtained only after the reaction is almost over. The effect of the strong collider (Figure 6 ) SF6, ascompared with thatoftheweakone,Ar (Figure5),istoenhance
0 2 4 6 Log (Pressure) (Torr)
8
Figure 7. Ratio of the unimolecular rate coefficient for cyclobutene isomerization to that at steady state vs pressure for various number of collisions.
both the rate of reaction and the aDDroach to steady state. The .. net effect is that the reaction in the non-steady-state regime is the predominant one. The effects of pressure and collider are summarized in Figure 7 for cyclobutene isomerization at 1500 K for Ar and SF6 as bath gases. The deviation from steady state is larger for Ar (weak collider) than for sF6 (strong collider). It also shows that the maximum deviation occurs at 1000 Torr. The shape of the curve is probably a manifestation of the fact that the maximum deviation occurs at 3/2 order in the falloff curve. We have also used a reverse exponential model for transition probability distribution f ~ n c t i o n ,but ~ the model failed at high temperatures. The dynamics obtained had nonphysical features, thus excluding this particular functional form from possible energy-transfer models at high temperatures. To enhance the generality of our conclusions, we have chosen to present the results as a function of the number of collisions instead of a function of time. This way, the results are presented on a per collision basis, and differences in rate coefficients and populations due to differences in collisions cross sections are eliminated. To convert to time units, one needs to divide the number of collisions n by the collisional frequency w,which, at lSOOKand1 Torr,is7.42X 106ss-l(Ar)and7.74X106s-*(SF6) for cyclobutene and 7.54 X lo6 s-I (Ar) and 7.78 X 106 s-1 (SFs) for cyclobutane. The population evolution in the steady state is given by N = N“, exp( - k z i ( n / w ) ) . Simple Lindemann-type considerations fir show right away that the higher the pressure the lower the degree of reaction for a given n. This can clearly be seen in Figures 3-6. In conclusion, at high temperatures, steady state obtains only at high pressures. The terms “low” and “high” are a function of the activation energy: the higher the activation energy, the higher the temperature range in which, pragmatically, non-steady-state effects obtain. For cyclobutane, this range will be 1500-2000 K and for cyclobutene 1200-1500 K. The weaker the collider, the more noticeable are the non-steady-state effects. Maximum deviations occur at the intermediate pressure of 1000 Torr. The
ei
6834 The Journal of Physical Chemistry, Vol. 97, No. 26, 1993 deviations can be as high as a factor of 2 in the level of reaction and will diminish at low temperatures or high pressures.
Acknowledgment. This work is supported by the P. and E. Nathan Research Fund and the Fund for the Promotion of Research at the Technion (to LO.), by the Center for Absorption in the Science Ministry,of Immigrant Absorption, State of Israel, and by the Wolfson Family Charitable Trust Program for the Absorption of Immigrant Scientists (to V.B.). References and Notes (1) (a) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley-Interscience: New York, 1972. (b) Forst, W. UnimolecularReacrionss: Academic Press: New York, 1973. (2) Oref, I.; Tardy, W. C. Chem. Rev. 1990,90, 1407.
Bemshtein and Oref (3) Shi, J.; Barker, J. R. J. Chem. Phys. 1988,88,6219. H.;Troe, J. In Bimolecular Reactions; Baggott, J. E., Ashfold, M. N., Eds.; The Royal Society of Chemistry: London, 1989. (5) Gilbert, R. G.; Smith, S.C. Theory of Unimolecular and Recombination Reactions; Blackwell Scientific Publications: Oxford, 1990. (6) Tzidoni, E.; Oref, I. Chem. Phys. 1984, 84,403. (7) Gaynor, B. C.; Gilbert, R. G.; King, K. D. Chem. Phys. kft.1978, (4) The subject is reviewed by: Hippler,
55,40. Dove, J. E.; Troe, J. Chem. Phys. 1978,35, 1. Penner, A. P.; Forst, W. J. Chem. Phys. 1980, 72, 1435. Schranz, H.W.; Nordholm, S.Chem. Phys. 1983, 74, 365. Gilbert, R. G.; Luther, K.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983.87, 1983. (12) Gilbert, R. G.; King, K. D. Chem. Phys. 1980,49, 367. (13) Carr, R. W.; Walters, W . D. J. Phys. Chem. 1963,67, 1370. (14) Rossi, M.J.; Pladziewicz, J. R.; Barker, J. R. J. Chem. Phys. 1983, 78, 6695. (15) Pawlowska, Z.; Oref, I. J. Phys. Chem. 1990, 94, 567. (16) Hauser, W. P.; Walters, W. D. J. Phys. Chem. 1963,67, 1328.
(8) (9) (10) (11)
