Nonequilibrium Time-Dependent Unimolecular Reactions
The Journal of Physical Chemistry, Vol. 83, No. 8, 7979
1017
Model for Nonequilibrium Time-Dependent Unimolecular Reactions. Single Collision Approximation R. J. Mallnst and D. C. Tardy” Department of Chemistfy, UniversiYy of Iowa, Iowa City, Iowa 52242 (Received May 22, 1978)
Mathematical equations which represent nonthermal unimolecular reactions (e.g., those initiated producing highly vibrationally excited reactants in a short time) are derived and analyzed. Solutions based on master equation integrations and a “single collision approximation” (SCA) are compared for reactive and nonreactive systems with a range of collisional energy transfer models. The SCA solution is qualitatively correct and provides a benchmark for the behavior of nonequilibrium systems; for weak colliders “up” transitions must be included. Four strong collider models were tested and if the criteria of equilibrium below Eo is required then the probability of goiing to the jth level must necessarily be given by the Boltzmann factor for the jth level. This requirement does not involve the artificial separation of reacting and nonreacting levels that is often used.
components: the known thermal populations and the Introduction nonthermal populations. In eq 2, Rirepresents the steady The stochastic treatment of gas phase thermal unimolecular reactions is well e~tablished;l-~ techniques and Ni(t) = N,t(t)Rj + sj(t) Ntot = CNj(t) (2) J formulae by which rate constants may be calculated for any model of collisional behavior are well d o c u m e n t e d l ~ ~ ~ state fractional population of molecules a t Ei (Ri= Ni/ and have been successfully used. Nonequilibrium effects CjNj= Ni/Ntotat steady state) and si(t) is the nonsteady originating from non-RRKM behavior have recently been state component of the population. di~cussed.~ However, the situation for nonthermal systems By substituting eq 2 into both sides of eq 1, eq 3a and with “weak” colliders has not been thoroughly developed. Except for the treatment of chemical activation systems? few equations exist to predict the reaction rate under nonequilibriurn condlitions. In particular, the rate for a d reaction with both a thermal and a nonthermal component --NL(t) = Ntot(t)[-oCpj,R, + aCP$j - k,R,1 + dt J I has not been sufficiently analyzed. Until recently this type [-wCp,,&(t) + wCp,,J,(t) - k,s,(t)l (3b) of calculation would only have academic merit. However, 1 J with the advent of high power pulsed lasers the experiments are now feasible. b result. Because of the relationship of the R, to the steady In this paper we present a “single collision state master equation (i.e., eq 1 set equal to zero) the first approximation” (SCA) treatment for this general type of term of eq 3a and the first term in brackets are equal to unimolecular reaction. The derived equations for pre~ e r o . From ~ , ~ the nonzero terms in eq 3a and b, a pseudicting the rate will be directly compared to the results domaster equation for the nonthermal components, J,(t), obtained from integration of the master equation. The is obtained: physical meaning of our SCA will be described and comments will be made on how the calculated results relate to the conventional treatment of the general concepts of weak and strong collider. Thus, the solution of eq 4 provides the key in determining the time dependence of the populations for these systems. I. Equations for the SCA Solution of this equation can be independently developed The traditional treatment of gas phase unimolecular for two collisional models: (i) the strong collider and (ii) reactions begins by assuming a Lindemann mechanism8 the weak collider. and can be represented by a set of coupled first-order Derived Equations for Model Systems. “Strong” master equations similar to Collider Equations. Traditionally a strong collider is one which produces a Boltzmann equilibrium distribution of (d/dt)N,(t) = -oCp,,N,(t) + wCpLjNj(t)- k,N,(t) (1) J J populations below E,. To achieve this an artifical dichotomization is made into reactive and nonreactive states. where N,(t) is the population of molecules of energy E, at When molecules in the reactive category undergo collisions time t, pLJis the probability per collision of transition from they are “shuffled” to a nonreactive level; while only a Ej to E,, o is the collision frequency, and h, is the microrate small fraction (Boltzmann factor) of nonreactive molecules constant for product formation at E,. (Throughout this which undergo collision produce reactive molecules. Aldiscussion we shall use the convention that k, is defined ternatively it is often assumed that a strong collider model for all E, with k, 0 for E, < Eo,Eo the critical energy for transfers large amounts of energy on collision with the reaction.) Previous s t u d i e s l ~ have ~ - ~ produced formulae chaperone. for calculating the populations of reactants under the However, a strong collider can also be defined6>‘jas one condition of a thermal steady state of each level. Hence in which the average probability of transition to energy we will set the total time dependent populations into two E, at steady state is the Boltzmann factor, fB(Ei): Department of Chemistry, Kansas State University, Manhattan, Kan. 66506. 0022-3654/79/2083-1017$01.00/0
0 1979 American Chemical Society
1018
R. J. Mallns and D. C.Tardy
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
when p(Ei)is the density of eigenstates at E+ The simplest model which directly obtains this result is the detailed strong collider (DSC) mentioned by N ~ r d h o l m ;he ~ demonstrated that this model reproduces the collisional effects discussed by Slateralo In the DSC model, the probability term pij is set equal to f B ( E i for ) all j . (It can be shown6 from master equation integration that the DSC model reproduces the strong collider limiting behavior.) Due to mathematical simplicity we will assume that the DSC model represents strong colliders. Substitution of the DSC formula for the pi;s in eq 4 produces eq 5a and 5b for the 6,(t). In obtaining eq 5a we (d/dt)di(t) = -(w
+ ki)6i(t)
(5b)
have made use of the fact that the sum of the R;s is unity, so that the sum of the &(t)’smust be zero in order for the sum of the Ni(t) to equal Ntot(t).The time dependence of N&) is found by solving eq 6 and substituting for 6,(t) (d/dt)Nbt(t) = C k i N i ( t ) = CkiNtot(t)Ri + Ch$i(t) (6) i
i
resulting in eq 7. In this equation kunithrmis the thermal
unimolecular rate constant for the temperature (used to define the Ri)and pressure of the experiment. If the nonthermal aspect of the experiment consists of the “insertion” of molecules at a single energy, e.g., El, to form a monoenergetic perturbation to the steady state, then only one 6i will be significantly nonzero and eq 7 simplifies to form eq 8. Note that in eq 8, the second term is zero for
< Eo (because hl = 0 for El < E,). This equation predicts that for strong colliders, perturbations to the levels below Eo should have no effect on the overall rate other than that produced by a minor temperature modification which would directly effect kunithrm. “Weak” Collider Equations. General colliders will also be treated by assuming a monoenergetic input (perturbation) to the steady state. Carringtonll has shown that the evolution of a general distribution can be represented by the sum of the evolution vectors, one for each energy level, of initially monoenergetic distributions and that the coefficients weighting these vectors in the sum are time independent. We shall assume that this reasoning also holds for eq 4 so that the generalized version of the problem may be considered as a sum over the appropriate monoenergetic solutions. We formalize our monoenergetic distribution with the equation p2,6,(t=0) = 0 for all j # 1, where El is the energy of the perturbation. At this point we introduce our SCA (eq 9) to simplify eq 4. This approximation has been used p,6,(t>O) z O only if i = j or i # j = 1 (9) El
by other authors12and further justification is unwarranted a t this time. Insertion of this approximation into eq 4 yields eq 10a and 10b for the 6,(t). These equations may 6 d t ) = -(41 - Pn) + k,)6,(t) (104 6 , # l ( t ) = ~ ( 4-1P A
+ h,)6,(t)+ p d d t )
(lob)
be readily solved as follows:
61(t)= Cle-(w(l-Pd+kdt
The total concentration as a function of time can be determined from N~~~ = Noe-kunitht
(W
+
c
Cihi
i w ( l - pii)
+ ki - hunithrm
e-(w(l-PtJ+ki)t+
Qbt
w ( 1 - pa) + hl
(54
ai(t) = Cie-(w+kJt
i
(1lb)
e-(w(”Pll)+kl)t -
(12)
hunpm
where
From the equation it can be seen that if the spike (perturbation) is placed below E,, both the summation term and Qbt term remain. Of these terms, the Qbt term is the largest because it contains the factor Cl which dominates the set of C,. For this case, the magnitude of the effect will depend directly on the magnitude of the up collision transition probabilities contained in Qbb These terms will be negligible for strong colliders but significant for weak colliders. 11. Application to a Model System
The utility of the equations developed in the preceding section has been examined by application to a model molecular system. A model system was chosen over a real system in the hope that the properties of the equations could be more easily examined if the effect due to molecular complexity was minimized. The model was based on a hypothetical molecule with four vibrational modes chosen to be degenerate at 500 cm-l. The transition state was chosen to have three degenerate vibrational modes of 450 cm-l. Microrate constants were calculated using the RRKM expression;2 overall rotations were considered adiabatic. The energy grid (spacing between the quasi-quantized levels) was 1000 cm-l and the energy range was from 0 (the zero point energy) to 40 000 cm-l; the critical energy was 16 000 cm-’. Seven collision models were studied. The weak ~ o l l i d e r s were l ~ ~ of the exponential form (EWC(1) and EWC(2) with average steps of one and two grid units, respectively: p I , e-(lEJLEgl)/(m) for (E, < E,)) and the stepladder form (SLM(1) with a average step of one grid 6,+(hE), ). Four strong colliders were investiunit: p , gated: (1) the DSd model previously mentioned, (2) the statistical model (SM) of Serauskas and Schlag,13(3) the traditional strong collider-constant (TSC(C)) defined in eq 13a and 13b6
-
-
p,, = 0 for E,, E, 2 Eo TSC(C)
(134
p,, = constant otherwise
U3b)
and (4) the traditional strong collider-Boltzmann (TSC(B)) defined in eq 14a and 14b6 p,, = 0 for E,, E, I Eo (14a) p,, = fB(E,) otherwise
(14b)
All matrices were forced to meet the requirements of detailed balance and completene~s.~ The computations were performed by treating the master equation as an initial value problem of a set of
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
Nonequilibrium lime-Dependent Unimolecular Reactions
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1019
-
-10-
-12
t
I ’ 10 20 30
40
Figure 1. Plot of level deviations after 0.333 X 1O-’s for nonreactive model system with excitation at E,= 16000 cm-‘, a = 103/kT= 0.9, and w = 1.59 >< 10’. Collision models, as described in text, are represented as iollows: (0)DSC, TSC(B); (A) TSC(C); (V)SM; (0) EWC(2); (0) EWC(1); (0) SLM(1).
coupled first-order differential equations with constant coefficients. (The assumption of constant coefficients is equivalent to assuming a large excess of inert buffer gas acting as a heat sink.) For such a system of equations, the exact solution may be written explicitly and evaluated at any set of time p ~ i n t , s . ~ J ~The J ~ time J ~ zero distributions were Boltzmann distributions to which a “spike” of 1% of the total number of molecules was added. All distributions were normalized to the same total number of molecules. The average energy was then calculated and used to define a Boltzmann temperature which was used to calculate the pY’s, For the computation of eq 12a, the method of ref 5 and 6 was used to calculate the R,’s. Nonreactive Calculations. In order to properly assess the nonthermal calculation results, a series of integrations were performed for the relaxation of the spike distribution in the absence of reattion. The results were analyzed by examining the level deviation, LD(i,t,to),as a function of time:
LD(i,t,LO=m):= (N,(t=a)- N,(t))/N,(t=m) (15) As expected tlhe calculations confirmed that (1)the weak collider models “spilled” the spike population into adjacent levels, both higher (and lower in energy, as the system returned to Boltzmann equilibrium, and (2) the strong colliders spread the spike uniformly among the levels and no level other than the “spiked” level was ever seriously perturbed from its equilibrium value. A sample of the plots of LD(i,t,tO=m) is given in Figure 1. Nonthermai Reaction Calculations. For the nonthermal reaction simulations, the energy of the “spiked” level was varied sequentially from two grid points below Eo to four grid points above Eo. The results were analyzed by examining the enhanced rate factor Mi) as a function of “spike” level: X(i) = [(ln N,,,,(t=O)- In N , , ( t ) ) / t ] / h ( S S E ) = h(obsd)/h(SSE) (16)
(k(SSE)is the rate constant obtained using the steady state eigenvector of the master equation as a distribution vector and calculating the expection value over the microrate
In ( p
16 15’ Figure 2. Plot of XM(15)/XDSC(15) in reactive model system vs. Pje,15 for a = 1.4, w = 3.18 X lo7, and collision models as represented In Figure 1. INTEGRATION 1
1
EQUATION
12a
w ”
0
V
0
rr
.“
0
::
1 -
8
4
A 2 -
i ” V
SPIKE
LEVEL
Figure 3. Plots of A(/? as a function of spike excitation level i ; left-hand plot is from full master equation integration and right-hand plot is from eq 12, except for DSC and TSC(B) which are from eq 7. Symbols are as defined for Figure 1.
constants16 and h(obsd) is the observed macroscopic rate constant). Equation 12 indicates that X(i)should be dependent on C,p,i, j > Eo,thus the ratio X(i)/hDSC(i)for i = 15 (the last nonreacting level) was plotted as a function of the transition element p16,15(the leading term of the Qtot sum in eq 12). This plot is shown as Figure 2. It is apparent from this figure that the rate increase is directly linked to the up transition probability. The full set of h(i)’s is shown in Figure 3. Similarly, Figure 4 shows these results normalized to the DSC model. Since the DSC is the “assumed” strong collider, normalization to it reflects the degree of “weak collider behavior”. Weak collider effects are manifested only for “spike” levels within a few grid units of Eo. A comparison of the results of eq 12 with that of an exact master equation integration is poor in the region of interest, i.e., for spikes just above and below Eo. However,
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The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
R. J. Maiins and D. C. Tardy
-32 I
In SPIKE
LEVEL
Figure 4. Plots of XM(i)/XDSC(i) vs. spike excitation level i ; see Figure 1 for symbol identification.
qualitatively the “single collision approximation” used to derive eq 12 predicts the correct increase in rate and the correct dependence on the up transition probability for spikes just below E,,. Also an interpretation of the SCA can be made. By zeroing all terms in the master equation except those which connect with the spike level, we have essentially limited the lifetime of molecules which leave the spike to a single collision. After a single collision, the molecule is either in a reacting level and becomes product or is in a nonreacting level and is thermalized; it cannot make a second collisional jump because the connecting terms, pL.8jJ*E, have been removed. This means that the error in the predicted rate obtained via eq 1 2 (referenced to the exact integration) is a measure of “second collision” effects. In fact, the effect of second collisions for a spike just below Eo can be related to (AE)up,the average change in energy on up transitions. If the ( AE)upis large, then on the first collision, a large fraction of the spike will be transferred above Eo. Equation 12 forces these molecules to remain above Eo until they react whereas, in reality, many are deactivated on the “second collision”. The result is an overestimate of the rate. Similary, for small the second collision brings many molecules above Eo which eq 1 2 forces to remain below Eo after the first collision; the result is an underestimate of the rate. Figure 5 plots the deviation between eq 1 2 and the integrated master equation as a function of (AE),, for a spike placed in the last nonreacting level. Previous workers12have rationalized their experimental results using essentially the same philosophy as our SCA. As seen from the above results the approximation works well only for spikes well above Eo where the collision process is a minor perturbation to the reaction process. Thus when analyzing systems with monoenergetic inputs of molecules the SCA should only be made when the input level is sufficiently far away from Eo.
(a),,,
111. Conclusion In this article we have derived a set of equations for analyzing nonthermal reaction rates. We have labeled these as the “single collision approximation” because of the simplifications of the master equation required to obtain them. When compared to the fully integrated master equation results, these equations are qualitatively correct in that they have the proper dependence on the elements of interest (the up transition elements of the P
Figure 5. Deviation of eq 12 (Xntqr“(i) - Xw ’*(i)/A“wn(i)) vs. average energy transferred “up” ( ( A € )up); see Figure 1 caption for symbol identification. DSC and TSC(B) models are not included since they did not involve eq 12.
matrix) and, as a “first approximation”, should be useful in providing a starting point for further experimental and theoretical work. Hence the derived equations can be used in analyzing nonthermal unimolecular reactions. In addition, we have provided some interesting data on the distinction of weak and strong colliders. The data show that for weak colliders, even at high internal energies, up transitions are still important. That is to say, that one cannot treat these systems as a series of small energy jumps which only decrease the internal energy. The up transition elements must be included for an adequate representation, particularly under nonthermal conditions. For strong colliders we have examined three qualities assumed by various authors to denote strong collider behavior: (1) large decrease in energy on an “average” collision, (2) deactivation of reacting molecules on every collision, and (3) that the average transition probability at steady state to any level be given by the Boltzmann factor for that l e ~ e l . All ~ , ~of the strong collider models used in section I1 match requirement 1. The SM matches only 1 and none of the others. The TSC(B) and TSC(C) also match requirement 2; in addition TSC(C) matches 3. The TSC(B) and DSC match 3 and DSC does not match 2 . (TSC(B) actually matches requirement 3, only approximately, but the approximation is to four significant figures.) Under thermal conditions, all of these models will yield p”s, the ratio of observed rate constant to the strong collider upper bound,’ which are unity to within experimental error. Hence, one cannot distinguish the models under thermal conditions. However, if one wishes to adhere to the concept17 that in strong collider systems, what happens below Eo does not affect what happens above Eo (that is to say, that a spike below Eo should not significantly affect the rate), then one finds that only the models matching requirement 3 are acceptable. The fact that meeting requirement 3 is unaffected by decoupling of reacting levels (requirement 2 ) shows that this requirement, (3),is sufficient. Acknowledgment. R.J.M. thanks the 3M Company for a Graduate Fellowship and the Los Alamos Scientific Laboratories for a summer fellowship at Los Alamos. Consultations with J. L. Lyman while R.J.M. was at LASL were stimulating and helpful in the preparation of this work. Support for computer calculations from the University of Iowa Graduate College was greatly appreciated.
Collisional Efficiencies in External Activation Systems
References and Notes (1) D. C. Tardy and B. S. Rabinovitch, Chem. Rev., 77, 369 (1977). (2) (a) W. Forst, “Theory of Unimolecular Reactions”, Academic Press, New Yolk, 1973; (b) P. J. Robinson and R. A. Holbrook, “Unimolecubr Reactions”, Wiley-Interscience, New York, 1972. (3) I. Procaccia, S. Mukamel, and J. Ross, J . Chem. Phys., 88, 3244 (1978). (4) D.C. Tardy and B. S. Rabinovitch, J. Chem. Phys., 45, 3720 (1966). (5) R. J. Malins and D. C. Tardy, Chem. Phys. Lett., 57, 289 (1978). (6) R. J. Malins, Ph.D. Thesis, University of Iowa, 1978. (7) (a) B. S. Rabinovitch,and D. W. Setser, Adv. photochem., 3, 1 (1964); (b) B. S. Rabinovitch and M. C. Flowers, (3. Rev. Chem. Soc., 18, 122 (1964). (8) F. A. Lindemann, Trans. Faraday Sac., 17, 598 (1922).
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(9) S. Nordholm, Chem. Phys., 10, 403 (1975). (10) N. B. Slater, “Theories of Unimolecular Reactions”, Cornell University Press, Ithaca, N.Y., 1959. (11) T. Carrington, J . Chem. Phys., 35, 807 (1961). (12) N. C. Lang, J. C. Polanyl, and J. Wanner, Chem. Phys ., 24,2 19 (1977). (13) R. V. Serauskas and E. W. Schlag, J. Chem. Phys., 42,3009 (1965). (14) I. Oppenheim, K. L. Shuler, and G. H. Weiss, Adv. Mol. Relaxation Processes, 1, 13 (1967). (15) R. G. Gilbert and I. G. Ross, Aust. J . Chem., 24, 1541 (1971). (16) B. Widom, Adv. Chem. Phys., 5, 353 (1963). (17) This idea is inherent in almost all strong collider derivations of a unimolecular rate constant, particularly those derivations which include the idea of “pseudoequilibrium” between reacting and nonreacting levels; see ref 2 for an example of such reasoning.
Behavior of Collisional Efficiencies in External Activation Systems. 2. Competitive Decomposition and Competitive Stabilization D. C. Tardy Depafitment of Chemistry, University of Iowa, Iowa City, Iowa 52242 (Received August 9, 1978)
Model calculations show that reversible isomerization systems (competitive stabilization: S1/S2) can provide useful and unique information on energy transfer probabilities. The S1/S2 technique is most sensitive in the low pressures region when the critical energy for isomerization is low and the equilibrium constant is large. The low pressure results are independent of excitation level so that transition state properties other than the critical energy are not required to deconvolute the experimental data. In terms of energy transfer properties the low pressure results for low critical energy systems are dependent on average step size (( AE))and weakly dependent on the “shape” of the energy transfer probability distribution function. Model calculations were also performed for multichannel decomposition systems (competitive decomposition: Dl/D2); these systems are most sensitive to small excess energies in the high pressure region. The Sl/S2 and D1/D2 techniques are compared to the conventional S/D competition.
Introduction Energy transfer from highly vibrationally excited species (molecules, radicals, or ions) to heat bath molecules via bimolecular collisions is a process that occurs in many complex reactions, e.g.
Hence an understanding of these energy transfer processes and a delineation of the fundamental parameters governing energy transfer probabilities is necessary to predict and deconvolute complex chemical systems. The purpose of this paper is to introduce and evaluate a new method for obtaining suclh information. For polyatomic systems of moderate complexity, external activations (chemical, photochemical, etc.) as compared to internal activation (thermal) techniques have provided the largest amount of direct data on collisional energy transfer probabilities.’ This in part is due to the large amount of averaging which occurs in thermal systems where numerous internal energy states contribute to the experimentally observed quantities. For an external activation system a distribution of internal energy levels, f,, is produced by the reaction:
f,
RE) If the internal energy of R is in excess of a critical energy for reaction, Eo, them: (i) products (D) can result from unimolecular decomposition or isomerization, or (ii) collisional processes can transport the species to another internal energy state. Eventually the species either react 0022-3654/79/2083-1021$01 .OO/O
(D) or are collisionally stabilized (S) to levels below Eo; by conservation of the normalized reactant: S + D = 1. This deactivation cascade is illustrated in Figure 1. The first use of the external activation technique in obtaining transition probabilities was published in 1960 by Rabinovitch and c o - ~ o r k e r s . ~By - ~measuring S and D as a function of pressure (collision number w ) they were able to extract information on the shape of the transition probability distribution function (model type) and the average energy removed per collision Henceforth this technique will be labeled S/D. The low pressure region (D >> S)gives rise to “turnup” for weak colliders (colliders in which more than one collision is necessary for stabilization). When the observed rate constant (defined as ha = w ( D / S ) )or S/D is plotted vs. S/D or w; the curve exhibits either “turnup” or “turndown” a t low values of S/D. The latter is exhibited in Figure 2. This “turnup” or “turndown” is due to an enhanced probability of producing D which in turn is due to the stepwise cascading nature of a weak collider in the stabilization process. The increased probability for weak colliders increase as ( AE) decreases. Thus a strong collider (traditionally one that transfers or removes large amounts of energy per collision does not exhibit “turnup” while as the collider becomes weaker (smaller ( AE))“turnup” is enhanced. From the shape and location of the “turnup”, ( A E ) and model type can be deduced independent of the collision cross section of reactant and heat bath molecules. The latter parameter should be taken as a variable since it is not known a priori. In the high pressure region (D
