J. Phys. Chem. lQ83, 87,2683-2699
2683
Nonequilibrium Effects in Chemical Kinetlcs. Straight-Line Paths for Homonuclear Diatomic Dissociation-Recom bination Process Carmay Llm and Donald G. truhlar’ Department of Chemistry and Chemical Physlcs Program, University of Minnesota, Minneapolis, Minnesota 55455 (Received: December 8, 1982; In Final Form: February 7, 1983)
An iterative numerical method is used to simulate the nonequilibrium distributionsfor nonlinear kinetic systems.
This method is much simpler and more stable than the full time-dependent solution of the master equation, yet it yields both the phenomenological rate constants and the nonequilibrium internal-state distributions. The method should be general but it is illustrated here by applying it to the dissociation-recombination problem of a diatomic molecule in excess inert gas. In particular, we studied the dissociation of O2 and the recombination of 0 atoms in excess Ar at 293, 600, 3000, 4000, and 18000 K. We find that, after an induction period, a quasisteady state is reached for which the phenomenological rate law holds with temperature-dependent but concentration-independent rate coefficients over the entire temperature range from 293 to 18000 K. The phenomenological rate law is valid even when there is appreciable back-reaction, and the phenomenological rate constant can be predicted by an eigenvalue analysis even where nonlinear processes play a significant role over the whole steady kinetics regime. The upper and lower limits of the ratio of forward to reverse phenomenological rates for which the rate coefficients are constant are obtained numerically at each temperature. We also calculate the nonequilibrium internal-state distributions at high temperature, and we find that they can be characterized by an invariant vector in concentration space over the entire range for which the phenomenological rate law applies. This is also shown analyticallyby applying singular-perturbation techniques. The numerical results also confirm the essential correctness of the predictions of the singular-perturbation method of Brau, Hogarth, and McElwain, even at the very high temperature of 18000 K, where the perturbation parameter is not very small.
I. Introduction A fundamental problem in chemical kinetics concerns the conditions under which a phenomenological rate law is obeyed. For instance, consider the elementary reaction 1
O 2 + M- T O + O + M
(1.1)
where M is the third body, for example, 0 or O2 or Ar. After an initial transient period we expect to observe the rate law -(d[O,l/dt) = k1[02l[M] - k-1[012[Ml
(1.2)
where kl and klare the forward and reverse phenomenological rate constants. The conditions of concentration and pressure for which eq 1.2 applies will be called the regime of steady kinetics or the quasisteady state (it is sometimes called the steady state by other investigators, but “steady state” has more than one meaning and this sometimes leads to confusion). The phenomenological rate constants are also called the steady rate constants. The forward equilibrium rate constant is given by lz1e
= CZ,ekM”,c
(1.3)
U
where Z; is the thermal probability of state u of O2 being populated at temperature T and kMu,cis the state-specific rate constant for O Z ( v )+ M 0 + 0 + M (1.4)
-
at translational temperature T. Strictly speaking kIe
should be called the forward local-equilibrium rate constant since equilibrium rate constants refer to situations in which the states of the reactants are at equilibrium with each other and the states of the products are at equilibrium with each other, but chemical (global) equilibrium of reactants with products is not achieved.’ Equation 1.3 (1) R. K. Boyd, Chem. Rev., 77, 93 (1977).
illustrates that equilibrium rate constants can generally be computed from the energy levels and the rate constants for state-specific reactive processes. If local equilibrium is maintained in both reactants and products, the phenomenological rate constants are simply equal to the localequilibrium rate constants. In general, however, the equilibrium distributions of internal and translational energy are not completely maintained in both reactants and products. In this case the phenomenological rate constants differ from the local-equilibrium rate constants since the former depend on the transition rates for all possible state-to-state processes, nonreactive as well as reactive. The systematics of the nonequilibrium internal energy distributions and phenomenological rate constants are not well understood, except for the case of first-order systems. For a first-order system (linear kinetics) the master equation can be solved analytically in terms of the eigenvalues and eigenvectors of the transition m a t r i ~ . l - ~ This brings out many interesting features of nonequilibrium kinetics. For nonlinear systems a generally applicable method is to solve the master equation numerically,”’ but this leads to results that are hard to generalize. In some cases important progress has been made by using singular-perturbation In the present article we use an iterative numerical method, similar to that employed in various contexts by previous workers,12-15to simulate (2) J. Wei and C. D. Prater, Adv. Catal., 13, 203 (1962). (3) N. S. Snider, J . Chem. Phys., 42, 548 (1965). (4) A. W. Yau and H. 0. Pritchard, J . Phys. Chem., 83, 134 (1979). (5) D. L. S. McElwain and H. 0. Pritchard, J . Am. Chem. SOC.,91, 7693 (1969). (6) J. E. Dove and D. G. Jones, J . Chem. Phys., 55, 1531 (1971). (7) H. 0. Pritchard, SrJec. Period. Rep.: React. Kinet., 1, 243 (1975). (8) C. A. Brau, J . Chem. Phys., 47, i153 (1967). (9) C. A. Brau, J . Chem. Phys., 47, 3076 (1967). (10) W. L. Hogarth and D. L. S. McElwain, J . Chem. Phys., 63,2502 (1975). (11)W. L. Hogarth and D. L. S. McElwain, Chem. Phys., 19, 429 (1977). (12) T. A. Bak and P. G. Ssrensen, Adu. Chem. Phys., 15, 219 (1969).
0022-365418312087-2683~~ 1.5QJQ 0 1983 American Chemical Society
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The Journal of Physical Chemistty, Vol. 87, No. 15, 1983
the nonequilibrium distributions for non-first-order systems. This method is much simpler and more stable than the full time-dependent solution of the master equation, yet it yields both phenomenological rate constants and nonequilibrium distributions. The method should be general, but it is illustrated here by applying it to the dissociation-recombination problem of a diatomic molecule in excess inert gas where analytic solutions obtained by singular-perturbation theory are available for comparison with our numerical results. The actual case that we consider is a multi-vibrational-state model of reaction 1.1at two high temperatures, 3000 and 4000 K, and two low temperatures, 600 and 293 K. We also present some results for the very high temperature of 18000 K. We assume that the reaction occurs in a large excess of Ar and that the translational degrees of freedom are completely equilibrated. We neglect vibration-vibration energy transfer, and rotational degrees of freedom are not included explicitly. For these reasons, especially the last, the model is not a quantitatively realistic model of oxygen dissociation, although Kiefer and Hajduk16and Ramakrishna and Babul7-lghave concluded that it is qualitatively realistic. In any event, our main interest is in establishing techniques and theoretical constructs for studying the systematics of steady rate constants and internal energy distributions associated with the regime of steady kinetics for nonlinear systems, and the actual results predicted by the model for reaction 1.1 are only of incidental interest in the present study. When Ar is present in large excess, the forward reaction 1.1 is pseudo first order. Thus, if recombination is negligible, the master equation is linear (first order) and the eigenvalue solution may be used. The main focus of the present paper is on concentration regimes where recombination is not negligible and where techniques applicable to general nonlinear reactions are required. We do, however, compute the eigenvalue solution for the linear-dissociation case to provide further understanding of the results that we obtain in the nonlinear regime. Section I1 gives the total pressures and the state-selected rate constants used as input in this work. Section I11 defines the notation used for reaction rates and rate coefficients and the quantities used to characterize the extent of nonequilibrium. Section IV presents the method used to numerically calculate steady rate constants and the concentration profiles in the steady-kinetics regime. Section V presents the results of the numerical calculations. Section VI gives a brief review of the analytic solution for linear dissociation kinetics and the eigenvalue solutions a t 3000,4000, and 18000 K. Section VI1 gives singular-perturbation-theory results for the present nonlinear example. Section VI11 compares the analytic results to the numerical ones, section IX discusses the very high temperature case of 18000 K, and section X is a summary of the answers to various questions that we have addressed for reaction 1.1. 11. System The parameters of the model system are the ones used by Kiefer and Hajduk.16 Ar is present in excess and only O2 + Ar and 0 + 0 + Ar collisions are considered. O2 is modeled as a truncated harmonic oscillator with 27 vi(13) H. S. Johnston and J. Birks, Acc. Chem. Res., 5, 327 (1972). (14) H. D. Kutz and G. Burns, J . Chem. Phys., 72, 3652 (1980). (15) H. D. Kutz and G. Burns, J . Chem. Phys., 74, 3947 (1981). (16) J. H. Kiefer and J.-C. Hajduk, Chem. Phys., 38, 329 (1979). (17) M. Ramakrishna and S. V. Babu, Chem. Phys. Lett., 57, 557 (1978). (18) M. Ramakrishna and S. V. Babu, Chem. Phys., 36, 259 (1979). (19) M. Ramakrishna and S. V. Babu, Chem. Phys., 42, 325 (1979).
Lim and Truhlar
brational levels spaced by w = 2255.7 K. (We find it convenient to use temperature units for all energetic quantities.) The dissociation energy of level u is called D,. The gap D26 between the highest level and the continuum is taken as 272.0 K so that the ground-state dissociation energy, Do, is 58 920.2 K. Dissociation rate constants are given by
kMu,c= ZMexp[-(X/D,
+ 1/T)D,]
(2.1)
where the collision rate constant is ZAr = (7.7088 X 1012)F/2cm3mol-l s-l, and the vibrational bias parameter is X = 3.5. Recombination rate constants kMc,L, are calculated by detailed balance
kMC,"= kM,,c/K,
(2.2)
where K, is the equilibrium constant for the state-specific dissociation process 1.4. The equilibrium constant for reaction 1.1 is
Kle= CK,
(2.3)
U
Energy-transfer rate constants are given by the LandauTeller model: kMu,,-l = UkM1,o
kMU-l,U= kMu,,-l exp(-o/n
(2.4)
(2.5)
where the initial state is the first subscript and the final state is the second. Thus, all energy-transfer rates are determined by kA'l,O
= ZArpAr1,o
(2.6)
with pP"l,otaken from the experimental study of Camace20 For 3000 and 4000 K we use his tabulated values PArl,o(4000 K) = 6.3 X lo4 and pP",,,(3000 K) = 1.6 X For 293 and 600 K we made Landau-Teller extrapolations of his other tabulated values to obtain PArl,0(600K) = 1.2 X and PArl,0(293K) = 1.8 X The reactive rate constants for the very high temperature study at 18000 K were given by eq 2.1 and 2.2, and the energy-transfer rate constants were given by the Landau-Teller model, eq 2.4 and 2.5. However, khl,o was not determined by eq 2.6 since P1,&8OOO K) was obtained by Landau extrapolations of the tabulated values of Camacm resulted in some energy-transfer rate constants that were greater than the collision rate constant, ZAr(18000 K). Thus, we determine kArl,Ofrom
kArl,O= ( ~ , ) - l ( l- e-"/T)/[Ar]
(2.7)
where the vibrational relaxation time T, of O2by Ar is given by CamacZ0 (?,)-I = (1.2 x 10-7)T1/6(1- e-2228/T)e-(1.04x10-'/~1'3[Ar] (2.8) These rate constants serve as our standard ones at 18000 K. We then made artificial variations, as discussed in section IX, to study their effect on the quantities characterizing the very high temperature quasisteady state. The two conserved concentration quantities, [Ar] and C, where the latter is defined by c = 2[02] + [O] (2.9) were set equal to typical experimental values. For the high-temperature runs at 18000,4000, and 3000 K, we set [Ar] = 2.125 X mol cm-3 to agree with the experimental conditions for the shock-tube study of Wray,21and (20) M. Camac, J. Chem. Phys., 34, 448 (1961). (21) K. L. Wray, J . Chem. Phys., 37, 1254 (1962).
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
Nonequilibrium Effects in Chemical Kinetics
TABLE I : Equilibrium Constants and Forward and Reverse Local-Equilibrium Rate Constants'
Kle, mol
T,K
cin-
18000
6.56'
4000 3000 600 293
k : , cm' mol-' s-'
k.:, cm6 mol-' s - '
3.65" 6.43' 5.036 3.17." 4.67.''
5.56" 2.50' 3.34" 1.67" 3.4212
1.51.' 1.90-40 1.37-85
&,e
=
CK",cX"e
2885
(3.8)
U
(3.9) U
a In this and the following tables a superscript denotes a power of ten by which the entry is multiplied.
we set C = 0.02[Ar], the same dilution factor as studied by Kiefer and Hajduk.16 For 600 and 293 K, we set [Ar] = 1.184 X lo-' mol cm-3 and C = O.O126[Ar]. These mimic the conditions under which Morgan and SchifP2 studied the recombination reaction. Thus, we shall sometimes call the high temperatures the dissociation temperatures and the low temperatures the recombination temperatures. The equilibrium constants and local-equilibrium rate constants for the five temperatures studied are given in Table I. Since Ar is present in excess, the forward reaction is pseudo first order and the reverse reaction is pseudo second order. The pseudo rate coefficients are denoted k where & = kh[Ar] (2.10)
111. Rate Equations and Rate Coefficients We denote the concentration of O2 in level u by nu = [ 0 2 ( u ) ] .Then the master equation becomes the coupled set 26
dn,/dt = C (kU,,unul - ku,unu)+ Kc,u[0]2 - &,,,nu (3.1)
Since the products have only one state for the present reaction and since we assume throughout this paper that translational degjees of freedom are in thermal equilibrium, k-lf equals kleand hence k-Lf is independent of time and concentrations. In contrast klf depends strongly on these variables. Hence, although -(d[O,]/dt) = k l f [ 0 2 ] - K-1f[0]2
(3.10)
klf and klfare not in general the observable steady-rate constcnts. Unjer certain circumstances, however, kIf or both klf and klfmay become equal to the observable phenomenological rate constant; these special cases are discussed in section V. The phenomenological rate constants are defined here by requiring that they simultaneously satisfy 1.2 and the rate quotient law: kl/k-1
= KIe
(3.11)
Using eq 1.2,2.10, and 3.11, we can then compute kl from the nu and [O] at any time by the relation R1 = -(d[O2l/dt)/[O2lG (3.12) where G is the unitless reaction affinity defined by25
G = 1 - [O12/(Kie[O~l)
(3.13)
Having obtained kl we calculate k1from eq 2.10 and 3.11 and Kle. Another interpretation of the unitless reaction affinity is perceived by rewriting it as
G = 1 - R-I/RI
u '=O
(3.14)
U'ZU
'/,(d[O]/dt) = C(kut,,nUt - Rc,u~[o12)(3.2) U'
where t is time and the time dependence of n,(t) and [O](t) is not shown explicitly, and where any sum without explicit limits runs from 0 to 26. Three sets of reaction rate coefficients are of interest: the one-way flux coefficients k,f and k-?, the equilibrium rate constants k? and kle, and the phenomenological rate constants Itl and k-l.23p24 Again we deal with pseudo rate coefficients defined by eq 2.10. The one-way flux coefficients are defined at any time by K,f = CR,&" (3.3)
where we define R1 and R-l by analogy to eq 3.6 and 3.7 as Rl = m 2 1 (3.15)
R-l = k-1[032
Thus, R1 and Rdl are phenomenological measures of the forward and reverse rates. According to eq 3.13 G is a thermodynamic measure of deviation from e q ~ i l i b r i u m , ~ ~ but according to eq 3.14 it is a measure in terms of the phenomenological kinetics. The deviations of the one-way flux coefficients from the local-equilibrium ones are represented as24
U
K-,f
=
Ck,,+
klf = K1"l + ql) k1f= K-,e(l + q-1)
(3.4)
U
where x u is the time-dependent fraction of O2 molecules in level u: (3.5) X " = n,/(Cn,) = nu/[021 U
The flux coefficients are defined such that the one-way reactive fluxes at any time are given by Rlf = Klf[O2] (3.6) R-lf = k-lf[O]z (3.7) in the forward and reverse directions, respectively. The equilibrium rate constants are given by eq 1.3, which yields (22) J. E. Morgan and H. I. Schiff, J. Chem. Phys., 38, 1495 (1963). (23) B. Widom, Science, 148, 1555 (1965). (24) J. Ross, J. C. Light, and K. E. Shuler in "Kinetic Processes in Gases and Plasmas", A. R. Hochstim, Ed., Academic Press, New York, 1969, p 281.
(3.16)
(3.17) (3.18)
Here q-l = 0 as explained above. Another useful quantity is the deviation from unity of the ratio of actual net rate to what it would be for the same values o f ~ [ 0 2and ] [O] if local equilibrium were maintained, i.e. (3.19) where
Rle = Kle[02]
(3.20)
R-le = kle[0I2
(3.21)
(25) The usual definition of the reaction affinity is -RT In G. See K. Denbigh; "The Principles of Chemical Equilibrium", 3rd ed., Cambridge University Press, London, 1971, p 140.
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The Journal of Physical Chemisiv, Vol. 87, No. 15, 1983
In the present case, R-le equals R-lf since kleequals k-lf. RINturns out to be particularly useful for the discussion that follows so we note here some purely algebraic consequences of the definitions presented in this section, namely R1- R-1 RIN = -1 (3.22) Rle - R-le
=
=
(k1 - k , e ) / k l e
(3.23)
la1
- 77-1(1 - G)I/G
(3.24)
Another important interpretation of RIN,according to eq 3.23, is that RINis a measure of the relative deviation of the phenomenological rate constant kl from its local equilibrium limit kle. For the case where q-l = 0 we obtain RIN = Vl/G (3.25) If we suppose that the steady-kinetics regime is defined by the constancy of kl which in turn implies the constancy of RIN through eq 3.23, then ql is propor_tional to G. Consequently, eq 3.25 and 3.17 imply that klf is a linear function of G. One simple feature of the first-order system is the straight-line reaction paths given by233326 n(t) = ne + a ( t ) A (3.26) where neis the equilibrium vector, A is a time-independent vector, and a ( t ) is a scalar coefficient which decreases to zero as time approaches infinity. In order to find such straight-line reaction paths in our nonlinear system, we make use of eq 1.3, 2.10, 3.3, 3.17, and 3.25 to derive klf - k l e R,N = ___ kleG
Ck,&,
- x,,~)
(3.27) From eq 3.27, we see that RIN will be constant if (x, -
x:)/G is a constant for all O2states, u = 0, ..., 26. Hence,
if there is no nonequilibrium effect in the back-reaction = 0), the nonequilibrium distributions in the quasisteady regime may be characterized by an invariant vector with components d, = G-'(x, - x:) (3.28) Thus, possible straight-line reaction paths of the form of eq 3.26 for diatomic dissociation-recombination processes are given by x(t) = xe + G(t)d (3.29) where d is a time-independent (and hence concentrationindependent) vector. If d is constant, then so is the unit vector d defined by d = d/(d( (3.30) The unit vector d is sometimes more convenient than d because it involves only x and xe: ci, = ( x u - X U " / [ C ( X " - X,e)2]1'2 (3.31) 0
Another useful quantity for discussing the numerical results presented below is +L,= d u / x u e G-'[(x,/x,") - 13 (3.32) ( 2 6 ) J. Wei, Ind. Eng. Chem. Fundam., 4, 161 (1965).
For the present example, if we neglect the reverse reaction, eq 3.1 and 3.2 become first order. For any firstorder system the quasisteady state is defined by the constancy of the ratio of the deviations from equilibrium for the concentrations of any two states3 Thus, for this case the steady-kinetics regime is characterized by the constancy of the unit vector A with components
A, = (n, - n,e)/[C(n, , - n,e)2]1/2
(3.33)
where n: = [ 0 2 ( u ) ]at equilibrium. When the back-reaction is neglected, not only does the system become pseudo-first order p u t also all the n; become zero. Then the constancy of A implies the constancy of x, and the quasisteady state can be characterized by the constancy of x , or of the relative fractional level populations
f, = X,/X,e
(3.34)
We shall test the constancy of the various internal-statedistribution vectors given by eq 3.28, and 3.30-3.34 numerically for the quasisteady state obtained for the present nonlinear, reversible example.
IV. Numerical Determination of the Phenomenological Rate Constants Our procedure consists of starting with an arbitrary set and then iteratively adof concentrations (n, and [O]), justing the concentrations acsording to the master equation (as discussed below) until 12, computed from the instantaneous concentrations becomes suitably constant. Since we are only interested in characterizing the regime of steady kinetics, we can use a large time step in altering the concentrations; this ia computational advantage over actually solving the master equation. With a suitable but large time step the profile of concentrations as a function of iteration will not necessarily mimic the profile of concentrations as a function of time that would be yielded by a solution of the master equation; however, if the procedure does converge to the regime of steady kinetics, the concentrations that characterize this quasisteady state will be the same as if this state was obtained by solving the master equation with a small time step. The essence of the method then is actually converging to a quasisteady state, and the algorithm for achieving this is given by the following two equations:
,'#V
(4.1)
where the superscript (;I denotes the ith interation and At is the time step. First we assume a set of nJoland [O](Oi, corresponding to a given initial value (Rl/R-JIo' of the phenomenological rates defined by eq 3.15 and 3.16. The initial values n,loj and [O]'ol,along with the state-specific rate constants and an appropriate time step, are then substituted into the right-hand side of eq 4.1 and 4.2 to find the second approximation to the concentrations; and this proctss is repeated to any desired degree of convergence of 12, computed from eq 3.12 and 3.13. The final value of R1/R-l of the ratio of phenomenological rates is sometimes, but not always (see below), close to (RI/R-Jlo1. The above algorithm is then repeated for a wide range of (Rl/R-l)lo! to map out the entire steady-kinetics regime. To illustrate the procedures used to obtain a set of nulol and [O]'OI corresponding to a given (Rl/R-l)lol,we consider
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983 2687
Nonequlllbrium Effects in Chemical Kinetics
the calculations carried out at 4000 K. We examined whether the quasisteady state that is reached for some final value of R1/R-' is independent of the starting distribution by using two different initial distributions x,(OI. First, we set x,(OI to x / at 4000 K; second, we used a Boltzmann distribution corresponding to 300 K. However, for (Rl/ R-,)lo1 values of 1.001 and 0.999, the second starting distribution was the Boltzmann distribution at 4100 K or 3900 K, respectively. (When we tried using initial Boltzmann distributions at 300 K this close to chemical equilibrium, the calculations did not converge.) We then determined [O2]1Oland [O]lolby calculating a, the degree of dissociation of O2molecules, for a given (RJR-J'OI. Defining a constant 0 by p = Kle[2C(Rl/R-l)~o~]-' (4.3) a was determined by solving a quadratic equation of the form
+ pa - p = 0
(4.4)
[OI'Ol = aC
(4.5)
[O,]'O' = (1 - a)C/2
(4.6)
a2
We then calculated
The value of [O2]f01 and the values of x,(OI then yield n,(OI. The choice of time step is arbitrary, and for the present study we find $hat a reasonable time step is given by (k,,)-' where k, is the largest pseudo-first-order rate constant for the depopulation rate of nu and is given by (4.7) U'ZU
(k-)-' is evaluated to be 1.0 X lo-* s at 4000 K. However, when (Rl/R-JIol1 1.0 X lo4,we find that the time step can be increased by a fator of 10 and we can still converge to a quasisteady state. For some iterations this larger time step results in negative populations of some higher energy u states of O2 in which case their populations are set to zero, and the entire population distribution is renormalized. Eventually the algorithm converges to a set of positive populations. For the studies at 3000 and 18000 K choices of niol, [0]@1, and the time step are made by the same procedures as used at 4000 K. We started the iterations for these temperatures with Boltzpann distributions at 3000 and was evaluated to be 1.2 X 18000 K, respectively. lo* s at 3000 K and 4.6 X lo4 s at 18000 K. The optimum time steps used to converge to a quasisteady state at 3000 K were 1.0 X lo4 s when (Rl/R-l)~ol 1 1.0 X lo4 and 1.0 X s when (Rl/R-l)~ol < 1.0 X lo4, and at 18000 K, At = 5.0 x 10-9 8. For the studies at 600 and 293 K, the iterations were begun with Boltzmann distributions at 600 and 293 K, respectively. To determine [02]101and [0]1°1 we first calculated the degree of recombination a' defined by [OI'OI = (1 - a')C
(4.8)
[O,]'O'= a'C/2
(4.9)
This was obtained by solving a' = 1 - P'jZ + (P/2) - ( P 3 I 2 / 8 )
(4.10)
The optimum time steps used to converge to a quasisteady state at 600 and 293 K are 1.0 X lo4 s when (Rl/ R-l)Iol 1 1.0 X lo4,but, when (Rl/R-l)lol< 1.0 X lo4,a time step of 5.0 X lo4 s was used at 600 K, whereas a larger time step of 3.0 X s could be used at 293 K.
The numerical calculations were performed for a wide range of R1/R_,,which is used to parametricize the extent of nonequilibrium during the quasisteady state. For 4000 K, starting with a Boltzmann distribution at 4000 K results in a quasisteady state after about 1300 iterations for a time step of 1.0 X s and after about 13 000 iterations for the smaller time step of 1.0 X s; the phenomenological rate constant obtained is 1.470 X lo8 cm3 mol-' s-'. For the 300 K starting Boltzmann distribution, the quasisteady state is reached within approximately 3000 iterations for a time step of 1.0 X s and about 33000 iterations for a time step of 1.0 X s; the phenomenological rate constant obtained is 1.469 X lo8 cm3 mol-' s-'. Thus, for we find that the quasisteady state a given initial (R1/R-l)iol, is independent of the starting distribution to at least three significant figures. Hence, although two starting distributions were used, the values recorded in Tables 111, VII, and VI11 are for the initial Boltzmann distribution at 4000 K. Our value of kl at 4000 K is also in good agreement with kl obtained by Kiefer and HajdukI6 for the case of linear dissociation. Based on the success of the method for 4000 K, independence to starting distributions of the steady rate constants at other temperatures was verified for only one R1/R-' at each of the other temperatures. We chose the fmt starting distributions as local-equilibrium distributions at 18000, 3000,600,and 293 K for the studies of these four temperatures. For the second initial distributions at 18000, 3000, and 293 K we used local-equilibrium distributions at 17000, 3100, and 300 K, respectively. However, the second starting distribution at 600 K differed from the first only in that the values of [Ar] and [O] + 2[02] were set equal to their values in the high-temperature runs. In all tests we found that the quasisteady state for a given R1/R-, is independent of the starting distribution. The results for the first starting distributions are recorded in Tables 11, IV-VI, VIII, and IX, and the approximate number of iterations needed to converge to a quasisteady state for the first starting distribution are as follows: At 18000 K, the quasisteady state is reached after approximately 4250 iterations for a time step of 5.0 X s; the value of the phenomenological rate constant is 1.621 X 10l2 cm3mol-' s-'. At 3000 K, the quasisteady state is reached after about 500 iterations for a time step of 1.0 X lo4 s and after about 28 000 iterations for a time step of 1.0 X lo* s; the phenomenological rate constant obtained is 1.372 X lo6 cm3 mol-' s-', At 600 K the number of iterations cm3 mol-l s-l is needed for kl to first reach 2.043 X roughly 1600 with a time step of 1.0 X lo4 s and about 35000 with a time step of 1.0 X s. At 293 K the quasisteady state is reached after about 4000 iterations with a time step of 1.0 X lo* s and after approximately 12000 iterations with a time step of 3.0 X s; k l obcm3 mol-' s-l. tained is 3.542 X
V. Discussion of Numerical Results We shall first discuss the results tabulated in Tables 11-VI. The values for the tables were recorded when k l first became steady to four significant figures. Total concentrations of O2 and 0, the forward phenomenological rate constant kl,the forward one-way flux coefficient kif, and their deviations RIN and ql, respectively, from the local-equilibrium estimate are tabulated for each ratio R,/R-' of the forward to reverse phenomenological rates over the entire steady-kinetics regime. The results show that, after an initial transient period, the system does obey the phenomenological rate law (eq 1.2) since k, remains constant all the way up to chemical equilibrium (R1/R-' = 1) as it is approached from either direction. The first
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The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
Lim and Truhlar
If,
TABLE 11: Forward Phenomenological Rate Constant k Forward One-way Flux Coefficient k and Measures R I N and , o f t h e Deviations from Local Equilibrium as Functions of t h e Ratio R I/R.' of t h e Forward to Reverse Phenomenological Rates and Total Concentrations of 0, and 0 at 18 000 K
q
[O,l, R,/R., 1.7' 1.000 9.999.'
mol cm-3 4.5-13 2.8-'@ 2.8-,'
[OI, mol cm-3 4.3-9 4.3-9 4.3-9
k , , cm3 mol-' s - ' 1.6212 1.62" 1.6212
kif, cm' mol-' s-' 1.6212 3.65" 3.6513
R 'N
77'
-9.56-' -9.56.' -9.56.'
-9.56.' 2.99- ' 1.4-4
TABLE 111: Forward Phenomenological Rate Constant k , , Forward One-way Flux Coefficient k , : and Measures R," and 71 of t h e Deviations from Local Equilibrium as Functions of t h e Ratio R JR-' of t h e Forward t o Reverse Phenomenological
Rates and Total Concentrations of 0,and 0 a t 4000 K
R ,/R-' 1.8'@ 8.4' 1.06 1.04 1.0' 2.0 1.001 0.999 0.5 0.1 5.2-3 1.0-3 a
[O,l,
[OI,
mol c m - 3 2.1-9 2.0.~ 1.2-9 6.5." 7.0-'3 1.4-14
mol cm-3 1.7-I' 2.5.''
7.0-15 7.0-15 3.5-15 7.3 l 6 7.3-'' 2.8.''
1.8-9 4.1-9 4.3-9 4.3-9 4.3-9 4.3-9 4.3-9
k , , cm3 mol-' s - ' 1.47' 1.47' 1.47' 1.47' 1.47' 1.47' 1.47' 1.47' 1.47' 1.47' 1.47' 1.47'
k If, cm3 mol-' s - ' 1.47' 1.47' 1.47' 1.47' 1.52' 3.958 6.42' 6.43' 1.14' 4 . 9 3 9" 4.76" a 1.19'' a
R'" -7.71.' -7.71.' -7.71.' -7.71 ' -7.71-' - 7.7 1.' -7.71-' -7.71.' - 7.71.' -7.71.' -7.71.' -7.71.'
r)'
-7.71.' -7.71.' -7.71.' -7.71-' -7.64-' -3.85.' - 8.82." 6. 54-4 7.65.' 6.66' " 7.31' " 1.84' "
These values are not converged.
TABLE IV: Forward Phenomenological Rate Constant k,, Forward One-way Flux Coefficient kif, and Measures R," and 71, of the Deviations from Local Equilibrium as Functions of t h e Ratio R,IR., of t h e Forward t o Reverse Phenomenological Rates and Total Concentrations of 0, and 0 a t 3000 K [ 0 2 L
R,/R., 2.4'' 1.06
2.0 1.0002 9.993.' 0.5 5.9-2 1.1-3
8.5 a
mol c m - 3 2.1 2.0.~ 2.4-12 1.2-', 1.2-12 6.0-j3 7.0-14 1.3-15
101, mol c m - 3
k,,c m 3 mol- ' s- '
kif, em3 mol- ' s- '
1.0-16
RIN -7.27.' -7.27.' -7.27.' -7.27.' -7.27.' - 7.27.' -7.27.' - 7.27.' -7.27.'
77'
-7.27.' - 7.27.' -3.64.' - 1.32-4 6.6T4 7.27.' 1.17'
These values are not converged.
TABLE V : Forward Phenomenological Rate Constant k , , Forward One-way Flux Coefficient k I f , and Measures R," and 71 I of t h e Deviations from Local Equilibrium as Functions of t h e Ratio R '/R., of t h e Forward t o Reverse Phenomenological Rates and Total Concentrations of 0, and 0 a t 600 K
R,/R., 4.043 1.0~~
2.0 1.001 0.999 0.5 1.0.~' 4.0-3s a
[O'l, mol 7.5.'' 7.5-1° 7.5-1° 7.5-'O 7.5- I o 7.5-1° 1.2-13
4.7-16
[OI, mol cm-3 6.0-47 3.8-45 2.7-" 3.8'25 3.8-25 5.3-25 1.5-9
1.F9
k , , cm3 mo1-Is-l 2.04-2' 2.04-28 2.04-" 2.042.04-28 2.04.'' 2.04.'' 2.04-"
kif, c m 3 mol-' s - ' 2.04-28 2.04-2' 2.61-28 3.17-28 3.1 7- 2 ' 4.29-28 1.13'" 2.819"
R'N - 3.56-
111
'
-3.56.' -3.56.' -. 3.5 6- ' - 3.56.' -3.56.' - 3.55.' -3.55.'
-3.56.' -3.56-'
-1.78.' -3.68-4 3.46-p 3.56 3 . 5 5 3 4"
8.8836 "
These values are not converged.
and last entries of R1/R..l represent the upper and lower limits of the steady-kinetics regime. These values were verified in each case by using at least three different values of (Rl/R-l)lolto converge to a quasisteady state. For example, at 4000 K initial conditions where (Rl/R-l)'ol= 1.0 X 10l8, 1.0 X 10l6,and 1.0 X 1014 all resulted in a quasisteady state for which R1/R-l = 1.8 X 10'O whereas initial conditions with (Rl/R-l)lol= 1.0 X 10-lo, 1.0 X and 1.0 X lov5all resulted in a quasisteady state with R1/R-l = 4.0 X In general, when (Rl/R-l)lolexceeds the top equals value of R1/R-, in a given table, the final R1/RW1
the top tabulated value, and, when (R1/R-,){OIis less than the lowest value of Rl/R-l in a given table, the final R1/R-l equals the lowest tabulated value. In other cases the final R1/R-lis usually close to (Rl/R..l)~o~. The regime of steady kinetics or the quasisteady state is found to hold for values of Rl/R-l ranging f r o m 1.7 X lo7 to about 1.000 at 18000 K, from 1.8 X 1O'O to 4.0 X at 4000 K, from 2.4 X lo1' to 8.5 X at 3000 K, from 4.0 X to 4.0 X at 600 K, and from 1.9 X los7to 1.5 X at 293 K. These values can in turn be related to the total concentrations of O2and 0 from the second and third columns of Tables
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983 2889
Nonequilibrium Effects in Chemical Kinetics
TABLE VI: Forward Phenomenological Rate Constant k , , Forward One-way Flux Coefficient k l f , and Measures R I N and 71 of the Deviations from Local Equilibrium as Functions of the Ratio R JR., of the Forward to Reverse Phenomenological Rates and Total Concentrationsof 0, and 0 at 293 K [O*I, 101, k , , cm3 k l f , cm3 R JR-, mol cm-3 mol cm-3 mol-' s-l mol-' s-' Rl '71 2.4-91
1.gS7 1.040
1.0-67
1.0-57 1.0-52 7.1-48 1.0-47
1.OZ0 1.0'O
2.0 1.001 0.9998 0.5
a
1.0-47
1.4-47
l.O-1°
1.0-42
1.0-20 1.0-40 1.5-82
1.0-37 1.0-27 1.5-9
-2.42.' -2.42.' -2.42.' - 2.42.' -2.42.' -2.42.' -2.42.' -2.42.' -2.42-' -2.42.' -2.42.' - 2.42.'
3.54- 7 3 3. 54-73 3.54-73 3.54-73 3.54- 7 3 3.54-73 3.54-73 3.54-73 3.54-73 3.54-73 3.54-73 3.54-73
-2.42-' - 2.42.' -2.42.' -2.42-' -1.21.' - 1.57-4 5.88-5
2.42.' 2.429 2.4219 2.4239 1.57"
These values are not converged.
11-VI where we note that 2[0J + [ O ] is a constant. It is clear that nonequilibrium effects are observed since kl differs from kle at each temperature. The ratio kl/kle is 0.04,0.23, 0.27,0.64, and 0.76, or, in terms of RINdefined by eq 3.22-3.24, the magnitude of the nonequilibrium effect for vibrational disequilibrium is -0.96, -0.77, -0.73, -0.36, and -0.24 at 18000, 4000, 3000, 600, and 293 K, respectively. We observe that, as the temperature decreases, the magnitude of these nonequilibrium effects diminishes and concomitantly the extent of the steadykinetics regime broadens. This is consistent with the fact that the steady-kinetics regime would extend from --03 to + m if there were no nonequilibrium effects. The fifth and last columns of Tables 11-VI show that k,f equals k , and q1 equals R I Nwhen R1 >> R-,. However, when the back-reaction is no longer negligible, klf and q1 increase as R1/R-Idecreases. The fact that k , remains constant under these conditions demonstrates that iterating to a quasisteady state does provide a practical way to calculate nonequilibrium steady rate constants for nonlinear systems, which is one of the main results of this study. At equilibrium, klf reaches its equilibrium value kle, and q' passes through zero. In the region where recombination dominates, k,f and q1 increase rapidly and it takes many more iterations before k,f and q1 tend to steady values; for this reason, as indicated in the tables, some of the klf and q , values are not yet converged when k1 has become steady to four significant figures. The behavior of klf and thus q1 will be rationalized below when we look at the fractional level population in steady dissociation. It should be clear from the previous paragraph that there are two sets of conditions under which klf becomes an observable rate constant. In circumstances where there are no nonequilibrium effects klf = kle = kl and klf becomes an observable. Alternatively, if a quasisteady state exists when the back-flux rate is negligible, then klf equals kl as shown numerically in Tables 11-VI. The equality of k,f and kl when R1lR-l >> 1 can be proved as follows. From the definition of kl in eq 3.15, we can write kl as El
= R,/[021
(5.1)
Equations 1.2, 2.10, 3.6, 3.7, 3.10, 3.15, and 3.16 yield the equality
R1 - R-1 = Rlf - R-lf
(5.2)
which may be rearranged to give
Rlf = R1 - R-1
+ R-If
(5.3)
Since k-, is usually of the same order of magnitude as k-lf or k-le, eq 3.7 and 3.16 imply that R-lf and R-, are also of
the same order of magnitude. Hence, if R-l is negligible compared to R,, then so is R-lf and eq 5.3 reduces to
Rlf = R1 Substituting eq 5.4 into eq 5.1, we obtain
(5.4)
= Rlf/[02l (5.5) Equation 5.5 is just the definition ofJlf given in eq 3.6 and we have thus shown the equality of kl and klf when Rl/R-I >> 1. The conditions under which k-,f becomes observable are more restrictive. If nonequilibrium effects are negligible, then k-lf = k-le = k-, and both kIf and kFlf become observable. However, even if a quasisteady state exists whtre the forward reaction is negligible, k..,f does not equal k1 and is thus not me_asurablefor the following reaspns. From the definition of k-l in eq 3.16, we can write k-, as Rl
L-l = R - 1 / [ 0 ] 2 When R1 > 1, the solution is
[o2I(t)= [021eb$oe-’o’
(6.14)
i.e. k , = Xo/[ArI
VII. Analytic Solutions to Nonlinear Dissociation-Recombination Kinetics In this section we shall summarize the stochastic singular-perturbation theory of dissociation and recombination of diatomic molecules in inert diluents as first developed by Braus and extended by Hogarth and McElwain.1° This will allow for comparison to our numerical results in section VIII, and in section VIIIE we will extend the theory to explain the straight-line paths that we found in the iterative calculations. To make the development as clear and systematic as possible we present the theory in terms of the dimensionless quantities I W ( t ) )and V ( t ) defined in section VI, and we will introduce two different scalings of the time variable. Since the lowest eigenvalue ho is usually very much smaller than all the others (see ref 27 and section VI), we introduce a dimensionless parameter
(6.16)
Eigenvalues X, and eigenfunctions Ix,) which satisfy eq 6.8 and 6.9 were found by using the subroutine RSG (a subprogram of the EISPACK system) for 18OOO, 4000, and 3000 K. A similar eigenvalue analysis a t 600 and 293 K failed because the EISPACK routines are numerically inaccurate for eigenvalues as small as kle (see Table I) at the recombination temperatures. The eigenvalues, Xo, X1, X2, and as well as No calculated from eq 6.13 are given in Table X. The quantity c2 in the table will be used in section VI1 and discussed in section VIII. The last column of Table X gives values of NO3Oo defined by NO3O0= (11O3Oo1x0) (6.17) and Woois a diagonal matrix whose elements are the normalized Boltzmann distribution over the internal states of O2 molecules at 300 K, and (xo)is determined not at 300 K but at the temperature of interest. These values will be used to calculate the incubation time for the forward reaction in section VIII. Dividing b(l8000 K), X0(4000K), and X0(3000 K) from the Table X by the value of [Ar] at high temperature gives k1(18000 K) = 1.621 X 10l2cm3 mol-l s-l, k1(4000 K) = 1.470 X lo8 cm3 mol-’ s-l, and k1(3000 K) = 1.372 X lo6 cm3 mol-l s-l, which are in excellent agreement with the numerical values of k, listed in Tables 11-IV. We note that X,’s, p > 0, span over 3 orders of magnitude for all three dissociation temperatures. The dimensionless parameter t is always less than unity, although its magnitude is temperature dependent, and it approaches unity at 18000 K where Xo and X1 are the same order of magnitude.
t
(7.1)
= (Xo/X1)’/2
Hogarth and McElwainlO used the boundary-layer technique and showed that the quasisteady state may be characterized by two time scales instead of the infinite number introduced by Brau. Physically this means that the internal degrees of freedom relax nonreactively on a single time scale of order and dissociation and recombination occur on a time scale of order X0-’. To incorporate these time scales mathematically it is convenient to define the ratio A, =
>0
X,/X1
(7.2)
/ . l
Our working assumption that there are only two time scales means that A,, is treated as zeroth order in t . We will now derive some useful relationships that follow from this assumption. The equilibrium rate constant expressed in matrix form is (7.3) By expanding the vector 11) in eigenvectors of W’L &le
= -(llLll)
11) =
CN,lx,)
(7.4)
P
and using eq 6.9 we establish the relation (l(\kll)=
CxUe=
= il
1
(7.5)
L’
Substituting 7.4 into eq 7.3 and using eq 6.8 and 6.9 we obtain &e
= -CCN,N,(x,lLlx,) e u
=
CX,N,Z
(7.6)
u
Then eq 2.10, 6.16, 7.1, and 7 . 2 yield kle/kl = No2+ E-’ C AgNP2
(7.7)
P>O
( 2 7 ) J. T.Bartis and
B. Widom, J . C h e n . Phys., 60, 3474
(1974).
Nonequiiibrium Effects in Chemical Kinetics
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
The two-time-scale assumption that A,, is zeroth order in c and the further assumption that the observed dissociation rate constant is of the same order of magnitude as the equilibrium dissociation rate constant then imply
No = 1 + t2NO* N , = EN,,*
p>
(7.8)
*(d/dt)lW(t)) = L[IW(t)) - VYt)ll)l
(7.10)
where symbols are defined in eq 6.3-6.6. To solve the nonlinear problem, the molecular distribution function IW(t)) is written as a sum of two terms
IW))= V ( t ) l l ) + IW))
(7.11)
where IU(t)) is zero at equilibrium. IU(t)) is then expanded in terms of the eigenvectors of eq 6.8 as IUW) = Cc,(t)Ix,) (7.12) P
where p
= 0, ..., 26
(7.13)
We now defiie a dimensionless time variable 8 such that O = t/f (7.14) where f will be specified below. If we substitute eq 7.11 and 7.12 into eq 7.10 and 6.2 and make use of the relations 6.8,6.9, 6.13, 7.8, 7.9, and 7.14, the coefficients a,(O) must satisfy the master equations (d/d8)[co(8) + Vz(O)] = -Xof~o(O) - ~~No*[dVZ(d)/dOl (7.15) [dc,(O)/d8]
+ X,fc,(B)
= -tN,*[dVZ(O)/dO]
(7.16)
P>O
and the conservation law E-lV(t9) 2VZ(S) 2[1 t2No*]co(0) 2t N,*c,(O) = D (7.17)
+
+
+
+ tc,(l)(~)+ 0 ( € 2 ) = 0, ...,26 v(e) = Vo)(e)+ Ev(l)(~) + o(2)
+
P>O
where we have introduced two unitless constants D = C/[02le = E-'V(O) + 2V(O) 2[1 + e2No*]co(0) 2t N,*c,(O)
+
(7.23) (7.24)
The initial conditions are expressed as c,(')(O) = ~ , ( 0 ) 6 , ~
(7.9)
0
where No* and N,* are zeroth order in t. We will use these relations in the singular-perturbation-theory solution of the master equation. When both dissociation and recombination are considered, the nonlinear master equation is given by
c,(t) = (x,l*lU(t))
c,(e) = C,TO)
2693
= 0,
p
..., 26
V q o ) = V(0)6,,
(7.25) (7.26)
Substituting expansions 7.23 and 7.24 into eq 7.21, 7.22, and 7.17, we first obtain the equations to zeroth order in t, and applying the initial conditions 7.25 and 7.26, we get the zeroth-order solutions which can be used to express the first-order equations. The latter can in turn be solved for the first-order solution. The solution to first order is then given by co(0) = co(0)- 4tV(O)E
N,*c,(O)(l - e-*,')
(7.27)
>0
(7.28)
>,O
c,(8) = c,(0)e-*P8
V(O) = V(0)
p
+ 2tE
N,*c,(O)(l - e-"@) (7.29) P>O
Thus, for times of order of A = 1 + ( ~ 0 ,- 1)G(t)
(8.34)
Comparison to eq 8.29 gives an analytic result for the components d, of the invariant vector, viz. d, = x U e ( ~ o -U 1)
(8.35)
Equation 8.35 gives d to first order in t. We can also show that the quasisteady state is characterized by an invariant vector even to second order in t. Proceeding as above, eq 8.28-8.31 remain valid as a result of eq 7.47. However, to second order in t, co(t) is obtained by rearranging eq 7.49 and substituting with eq 7.8 and 7.50 to give
From eq 8.36 we see that co to second order in t is a factor of N0-l times its value to first order; eq 8.34 is perturbed to (8.37) x , ( t ) / x > = 1 + (x0JVo-l - l)G(t)
so that the second-order result for the invariant vector is d, =
x,~(xOJVO-~
- 1)
(8.38)
From the above results we see that the invariant vector originally obtained in the iterative calculations can also be calculated analytically from two quantities, xovand No, obtained in the eigenvalue analysis. The values of xouand No at 3000,4000, and 18000 K were obtained as discussed in section VI and were used to calculate d from eq 8.35 and 8.38. We find that d obtained analytically to second order in t agrees to at least two signifcant figures with d obtained numerically. At 3000 and 4000 K, the first-order expression (eq 8.35) for d differs from the numerical d for some lower u states, but it is in agreement to two significant figures for the rest of the states. However, eq 8.35 fails to predict the numerical dU)sat 18000 K. We were not able to calculate d analytically at 600 and 300 K since the
eigenvalue programs failed to give the correct Ixo)at these low temperatures.
IX. Calculations at 18000 K In the previous sections we found that, after an induction period, a quasisteady state is reached for which the phenomenological rate law is valid even when there is appreciable back-reaction. If the induction time for dissociation is so short that negligible recombination has occurred, then kl = klf = A,,/[Ar]and one can predict the rate constant from an eigenvalue analysis alone, to lowest order in t. If, however, a quasisteady state is first reached only when the back-reaction is significant, then the phenomenological rate constant kl differs from the forward flux coefficients klf, and kl is not necessarily equal to the smallest eigenvalue, hoe To illustrate this, we attempted to find a quasisteady state that exists only when the back-reaction is significant. Since we observed that the steady-kinetics regime becomes narrower as the temperature is increased, we started our calculations at 18000 K, the highest temperature at which 02-Ar rates of dissociation and vibration relaxation have been studied experimentally.,l In order to ensure that we obtain a quasisteady state that is first reached when the back-reaction is no longer negligible, we lowered the internal relaxation rate constants from their standard values given in section I1 by factors of 10, lo2, lo3, and lo5, respectively, without altering the reactive rate constants. The calculations were then carried out as described in section IV and an eigenvalue analysis was made for each case. The results of the calculations at 18000 K are tabulated in Table XI, which also summarizes some of the results at 3000 and 4000 K. The upper limit of R1/R-l and the corresponding [O,], kl, klf, Xo[Ar],the concentration-dependent term of eq 7.53,8(No - 1)(Kle))-'[OI,71TD,7TD= 4.6(X1 - A,,)-', and T~~~ are tabulated for each temperature at 3000,4000, and 18000 K. The second column in Table XI, k,u,/kC,ul,gives the ratio of the actual set of energytransfer rate constants used in the numerical calculations to the set of energy-transfer rate constants defined by eq 2.4-2.6 at 3000 and 4000 K or eq 2.7 and 2.8 at 18 000 K. R1/R-l, [O,], and klf were recorded when k , first became steady to four significant figures. The upper limit of R1/R-l was converged using three different values of (R1/R-l){oifor all the case studies at 18000 K. Thus, initial conditions where (Rl/R-l)'oi = 1.0 X 1.0 X 1020,and 1.0 X 1015 all resulted in a quasisteady state with approximately equal upper limits of R1/R-l. klf did not become steady but increases to its equilibrium value as R1/R-l decreases to unity. The studies at 18000 K show that, as the internal relaxation rates decrease, so do the upper limit of R1/R-l and the phenomenologicalrate constants. In particular, when k,,~/kCud= the upper limit of R1/R-l = 2.7 where the back-reaction is clearly not negligible. As expected under these conditions, klf differs appreciably from kl; however, we see that A,,/ [Ar] gives kl to four significant figures even when the quasisteady state is first reached where there is significant recombination occurring. Since the "perturbation parameter" t (given in Table X) is no longer much less than unity, as required in the singular-perturbation theory of Brau, the first-order-in-t relation between k1 and Xo in eq 7.45 would not be expected to be valid. Nevertheless, to second order in t, eq 7.53 would also predict kl to be equal to Xo since the concentration-dependent term 8(No - 1)(Kle)-'[O] in eq 7.53 was estimated to be 0(1O-l1) for all cases at 18000 K. The validity of the second-order expression is consistent with the fact that
Lim and Truhlar
The Journal of Physical Chemistty, Vol. 87,No. 15, 1983
2698
TABLE XI: Upper Limit of the Forward to Reverse Phenomenological Rates R , / R _ ,and the Corresponding [O,], Forward Phenomenological Rate Constant h ,, Forward One-way Flux Coefficient kif, the Smallest Nonzero Eigenvalue A , of the Transition Matrix .I.-'L, the Concentration-Dependent Term 8(N, - 1 ) ( K I e ) - ' [ 0 ] in Eq 7.53, the Transient Decay Times at r I T D Calculated Numerically and r T D Determined Analytically from E q 8.26, and the Dissociation Incubation time 3000, 4000, and 18 000 K'
mol cm-3
mol-' s - '
k,', c m 3 mol-'s-'
2.12481-9 2.116-9 4.5." 3.6-16 2.9." 7.49.9-20
1.3726 1.4706 1.621" 1.268" 1.194" 1.184" 1.183'*
1.3726 1.4706 1.621" 1.27112 1.537'' 1.45113 1.45013
kuu'/
T, K 3000 4000
18000 18000
lzcuu, R,/R., 1 1 1
lo..'
18000 18000 18000 a
LO219
lo-'
2.4" 1.8"' 1.61.34 1.02 2.7 3.6
k , , cm3
,AAr 1 , cm3 mol-'
8(N, - 1)
s-'
(K'')-'[o]*
1.3726 1.4706 1.621'' 1.268'' 1.194'' 1.184" 1.183"
[0J over the steady-kinetics regime for 3000, 4000, and 18 000 K is G4.25
d,( 18000 K) obtained numerically can also be predicted analytically from a second-order expression, although not from a first-order one. The major assumptions of the singular-perturbationtheory treatment are not satisfied a t 18000 K: the parameter t is not very much less than l , as required for the validity of low-order solutions, and Xo is not of the order of kle, as_required for eq 7.8 and 7.9. In fact, t = 0.7-0.9, and Xo/kle ranges from 0.04 to 0.03. Hence, the fact that the second-order expressions for the rate coefficients and the d vector do agree with the numerical results a t 18000 K implies that the results have a greater range of validity than the available derivations would suggest. Next, we shall compare the values of the transient decay times calculated analytically from eq 8.26 to those obtained in the iterative solutions. To obtain the latter we solve the rate equation
(9.1) assuming that R1 and klare constants a t all times; for t =T ~ and~ this ~ yields ,
