Energy Transfer in Thermal Unimolecular Reactions
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 93
Intermolecular Energy Transfer Effects in Thermal Unimolecular Reactions D. C. Tardy* and R. J. Malins Department of Chemistry, University of Iowa, Iowa City, Iowa 52242 (Received June 28, 1978)
Traditional unimolecular reaction rate theories have incorporated the strong collision assumption primarily to obtain a simple expression for huni. However, it is known that many experimental studies do not provide the environment for strong collisions. Hence, weak collider corrections to the expression for huni have been empirically incorporated. We have derived a new self-contained expression for the many-level system which accounts for weak collision phenomena from first principles; the requirement for strong collision behavior is quantitatively described by this formulation. In this formalism the calculated rate constants are directly related to the average transition probabilities for energy transfer. Using this result it can be shown that p’ and not 0 is the correct collisional energy transfer efficiency to relate experimental data to model calculations. The prediction of energy transfer effects in model steady state thermal systems using temperature, falloff, dilution, transition probabilities, and reactant complexity as variable parameters is discussed. Methods (information theory and model calculations) for obtaining energy transfer information from experimental data are presented. Non-steady-state solutions for weak colliders are evaluated for the cyclopropane system.
I. Introduction Until 20 years ago the experimental and calculational study of collision effects in thermal and nonthermal gas-phase unimolecular reactions had proceeded along a somewhat disorderly path. A comprehensive review by Tardy and Iiabinovitchl on intermolecular vibrational energy transfer in unimolecular reaction should be consulted for historical and technical details not enumerated in this paper. All rationales for understanding and interpreting the observed phenomena rely on the Lindemann scheme,’ which states that the unimolecular reaction A products (1) proceeds via the mechanism -+
AE, + M AE,
AE,
k,
+M
kZJ
‘11
products
AEJ + M
(2)
AE, + M
(3)
( E ; 2 E,)
(4)
and the observed unimolecular rate constant by
for SC (p(E,)is the density of internal eigenstates at energy E,). Deviations from the behavior predicted by eq 5b and 6b are termed weak collider ( W C ) effects, and as shown by the equationtr are directly related to the steady state populations which in turn depend on collision model, temperature, collisional frequency (w), and ratio of substrate to collision partner (dilution). Experimentally, weak collider effects have been characterized by a relative collisional efficiency factor, /3.1,7,8 Many definitions of p have been used and as a result ambiguities have propagated. Since is an inherent collisional property a knowledge of w is necessarily involved. The experimentally determined pressure is related to w by the collision cross section (a) and the reduced mass of the colliding pair; thus errors (unknown or otherwise) in a will appear in w . ~ (3 is best defined from a particular experiment; the usual reference (assumled to be a strong collider) is the substrate. For the neat substrate a “traditional” falloff curve for hmi results; such plots are found in Figure 1 for the cyclopropane and methyl isocyanide isomerizations performed a t various temperatures. The addition of a diluent (M), which can be either a weak or strong collider, to a given pressure of substrate (equivalent w) causes a concomitant increase in hml. The magnitude of the increment depends on the nature of M and A, temperature, and dilution ( D
Here AE, is a reactant molecule with internal energy E,, M is any collision partner, h,, is the bimolecular rate constant for energy transfer, and h, is the microscopic rate constant for the unimolecular process. Typically the RRKM formulation is used for h,.3d The reference model for such a system is generally called the “strong collider” (SC) model, which is obtained as a solution to eq 2-4 by assuming that all reactant energy states are still “receiving” molecules at the same rate as they would if the system were a t Boltzmann equilibrium with no reaction o c ~ u r r i n g . ~ - l ~ = w,(w*). For this particular solution, the fractional populations (R,) As in thermodynamics integral and differential quanare defined by tities have been defined. Additionally can be defined for a given increment in W M or hml. Thus, four p quantities are defined.1,7,8 i. Differential quantities appropriate for small additions of M Aw(SC) = -, Ah(SC) = Ah(WC) (7a) for SC where Aw(WC)
P,(a
@,’(D) =
Ah(WC) Ah(SC)
--
0022-3654/79/2083-0093$01.00/00 1979 American Chemical Society
Aw(SC) = A w ( W C )
(8a)
94
D. C.Tardy and R. J, Malins
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979
ii. Integral quantities appropriate for high dilutions
For the second-order region (low pressure, Le., w that GJ = 2 )
-
0 so
Po(0) = PO'(0)
The connection between the experimental quantities and the microscopic inputs is made by solving the first-order master equation d[AEtl/dt = -a Cpji[AE,] -t. J
L1C11,j[AE,1 J
-
h,[AEtl
(9)
where pLJis the probability per collision of transfer from
El to E,. Since the pLJ'sare not known a priori the general method is to solve eq 9 for a particular set of transition probabilities. 'I'o reduce the number of parameters, models for p U ( i I j ) are described by their shape and average energy removed, ( AE).
Figure 1. Calculated plots of k l k , vs. log w (arbitrary units) for methyl isocyanide at (a) 353 K, and (b) 546 K for (1) strong collision and for a 360-cm-' SL model at dilutions of ( 2 ) 1, (3) 10, and (4) a;(c) similar plots for cyclopropane at 728 K for (1) strong collider and for a 352-cm-' SL model at dilutions of (2) 0.5, (3) 1, (4) 5,(5)10, (6) 50, and (7) m. Dashed lines represent experimental dilution paths.
into matrix notation so that the solution is obtained by an inverse or an iteration t e c h n i q ~ e , or ~ , ~a many shot expansion;12these techniques are equiva1ent.l Analytical approaches, mainly by Troe,13 Forst,14 and Pritchard,15 have also been used. A new method,9J6which is computationally efficient and is useful for any arbitrarily defined py's, involves the definition of an input parameter pi.
p, = C P , ~ R , ~ ~ (AE)=
Cpj:
J5i
Two examples of models which bracket extreme behavior are (i) step ladder Pi] =
*(I2 )
J
A brief description of this technique shows how model calculations are directly related to the p' (not p) quantities. The input parameter for the ith level (also equivalent to the average transition probability to that level under steady state conditions) and eq 11 define RLSs.
(AE) = E, - EJ+k
Ch,j+k
(ii) exponential p = ce-l$ 11
-
The meaning of the p , term can be elucidated by examining the total rate of molecules arriving at energy E,,
q/(W
t'iTC
Other models, Gaussian and Poission, have also been u ~ e d . The ~ , ~ensemble of p,'s for all i and j are found by applying the "detailed balance and completeness" constraints7 to the chosen model. Equation 9 can be solved either by assuming a steady state as is discussed in section I1 or for time-dependent systems the exact solution can be obtained as presented in section 111. A further dichotomy of section I1 is made by presenting two subsections on the prediction of thermal rate constants and the interpretation of experimental data. 11. Steady S t a t e Systems. Conventional T h e r m a l Reactions A. Prediction of Thermal Rate Constants. Calculation Technique. Thermal systems are, of course, steady state systems and are rigorously modeled by the steady state equation
(10)
By dropping terms second order in the R; for which Ei I E,, one can approximate the rigorous steady state equation by the quasi-steady-state equation
This approximation has been showngto yield values numerically very similar to the exact steady state expression. At least six methods of solving the coupled equations given in eq 11have been used. The equations can be recast
tiWC=
(14)
wp,[At,tI
This can be compared to the total rate under a reference condition; specifically the rate for the same set of molecules under Boltzmann equilibrium tlSC(which is an upper limit for tLWC) tiSC = WfB(E:)[Atotl
(15)
where f,(E,) is the normalized Boltzmann distribution function a t the temperature in question. Therefore, the relative input into the ith level, T,, is
The p,'s can also be related to the exact solution of the master equation under rigorous steady state c o n d i t i ~ n s . ~ J ~ More importantly it can be shown that the measured efficiency, /3', is an appropriate average of Ti hifi
h,,,,wC
Ch,R,"(WC)
ET,---
w f
-w
+ k,
Thus this equation relates the microscopic to t,he macroscopic world for energy transfer in unimolecular reactions and operationally defines a strong collider when p , = fs(E,). Equations 12 and 13 exhibit a cyclic relationship between R,""and the pi;the R,"" define the p , and vice versa. This cycle forms a natural iteration scheme for the cal-
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 95
Energy Transfer in Thermal Unlmolecular Reactions
1.0
0.1
0-01 10‘1
I
10
(AE)/(E’) Flgure 2. Plots of relative collisional efficiency vs. the reduced parameter €’for five systems at p =. 0 for the step ladder, Poisson, and exponential C3H6(V);C5H,0 (7) The models: N20 (0);NO,C:I ( 0 ) ;CH3NC (0); abscissa refers to the SL curve; Pand €have been arbitrarily displaced to the right by log 1.5 and log 2.25, respectively, to prevent crowding
culation of both the RLs*and p,. This scheme, recast in matrix format, is represented by = p (n) (jf @-lK)-l.p ( n ) = &jss(n+l) pgsdn)
Flgure 3. Quasi-universal summary plots of &’(m) vs. €’on a SL model for all molecules at reaction orders of (1) 1.10, (2) 1.25, (3) 1.50, (4) methyl isocyanide, 353 1.75, and (5) 2.00; cyclopropane, 728 K (V); K (0);and at 546 i< (O), I.o
(174
1 (1%~) Here the superscript n denotes the nth iterate. This method and those matrix techniques described earlier7~8J2 can be shown to be a modification of the Gauss-Jacobi17 iterative solution to the quasi-steady state master equation of eq n9 Results from Model Calculations. Extensive calculations for k,,, by Tardy and Rabinovitch7 have shown a universal relationship between p’, molecular complexity of reactant, reaction order, dilution, and temperature. For infinite dilui ion in the second-order region p’ (for a given model shape) is a universal function of E’ (E’ is defined where the latter quantity is the Boltzmann as (AE)/(E’) average energy of reacting molecules in the second-order region). The universal curves for various collision models are depicted in Figure 2. From these plots it can be seen that weak collider corrections are small when E’ 1 5 for a step ladder model or for E’ 1 10 with the exponential model. This definition is operationally related to the Lindemann--Hinshelwood criteria that a strong collider produces collision deactivation of a critically energized molecule in a single collision. Their analysis was extended to the falloff region;8 Figure 3 displays the weak collider behavior for various reaction orders. As expected the p’ quantity increases with a decrease in ch and as E ’ is increased. The temperature dependence of p’ is related to the temperature dependence of E’. The dependence of ( AE) on temperature is not known a priori and present day experiments have not provided such information; thus no dependence will be assumed. The dependence of ( E + )on temperature is directly calculated from the vibrational frequencies of the substrate and varies as T” ( x II). In the second-order region (ch = 2) and for low values of E’, it is found that p’ depends on P ( y 2 2). However, as either E’ increases or decreases y will monotonically decrease to zero. The dependence of p’ on temperature also affects the Arrhenius activation energy. In fact, it has been shown”J8 that the activation energy for a weak collider can be -2RT less than that observed for a strong collider. As the number of atoms in the reactant increases,
-
10
0
v
w
z
2”
05
U
Figure 4. Summary plots of population depletion function -yN(D)vs. dilution for (a) a step ladder model and (b) an exponential model, p = 0, as calculated for the N02CI, CH3NC,and C3H, systems at 476, 546, and 728 K, for the various values of €’shown on right-hand side of the graph. The individual computational points which these quasiuniversal curves suinmarize are not shown. The limiting values for D = also include computational results for the NO , and dimethylcyclopropane systems.
(E’) will increase causing a decrease in p’, thus increase in the molecular complexity of the reactant produces a change in p’ analogous to that by changing the temperature. The effect of dilution on p’ (a new set of quasi-universal curves result) can be quantitatively interpreted by factoring p’ into two parts: yN’(D) (related to the steady state populations) and yk’ (related to the transition probabilis determined so that ities). For a given weak collider y N ( D )produces the dilution effect. Figure 4 exhibits the universal behavior of yN’(D) as a function of D and E in the second-order region. It can be seen for low dilution that a weak collider will behave more like a strong collider than it does when at high dilution. Analogous curves result for the falloff region; however, the difference between the high and low diilution points decreases as ch decreases. R. Interpretaition of Expermental Data. For any given experimental thermal system, ideally we want to know the distribution of molecules and obtain a reasonable set of
96
The Journal of Physical Chemistry, Vol. 83,
0.0
-
-1.0
.
-1.0
-c . X
3.0
-
D. C. Tardy and R. J. Malins
No. 1, 1979
\
-
I-
k
W
.
b
I I
b
-4.0
I
43
B 0.2
0I
0.1
0.6
1.0
R‘
Figure 5. Plots of total surprisal vs 8’for the (i) exponential model (A), (ii) Gaussian model (0), (iii) step ladder model (O), (iv) least biased model (0).
transition probabilities. However, too few parameters are observed for this; commonly, only the temperature, pressure, and the rate constant are measure. Hence, one would like the values for the distribution and probability which are consistent with this information. Two methods are available. First, various model calculations can be performed and directly compared to the experimental observations or secondly the experimental data can be directly inverted by surprisal techniques.lg The former technique has normally been used; specifically the quasi-universal curves discussed in the previous section are used. The latter method is new and a brief description is in order. The optimum distribution is obtained by taking the total surprisal, It It = --A S = -R 5s 1n (RF/R,O (18)
R
and maximizing it under the constraints of (i) the normalization of the R,, (ii) the fixed given temperature (which is the same as fixed average energy, ( E ) , because the distribution is constant), and (iii) the observed rate constant. The resulting equation is
It = - XRlssIn ( R F / R :
ss) -
Xo(l -
X R t s )L
1
X1(CE1R,“‘- E ) - X,(p’Xk,R,O L
1
ss
-
Ck1RLS5) (19) 1
Here p’ is as in eq 7 and the R,O ss are chosen to be the SC values of eq 5b. This results in the following equation for the RF:
Rlss= R;
ss
exp(-1 - Xo - hlE, - A&,)
(20)
Three equations with three unknowns result when R,””as given in eq 20 is substituted into the three constraining equations. These nonlinear equations for Xo, XI, and hz are obtained by iterative methods. This “least biased” distribution can be compared to that obtained from the collision model dependent calculations of the previous section by evaluating the total surprisal, eq 18, for the respective distributions. This comparison is shown in Figure 5 for the somewhat arbitrary example of cyclopropane isomerization at 750 K and w of lo4 5-l and
~l
04
oa
LO
8/ Figure 6. Plots of surprisal for all levels above E o vs. 8’for the (I) exponential model (A), (ii) Gaussian model (0), (iii) step ladder model (D), (IV)least biased model (0).
where 8’ has been varied from 0.1 to 1.0. Clearly the two distributions are not equivalent. In Figure 6 the ratio of the surprisal above Eo (eq 18 where the sum runs over i such that E, 2 Eo)to the total surprisal is plotted. This figure shows that the model dependent calculations specify more than the minimum necessary information about the energy levels below Eo. Thus for thermal systems, no information about the energy states below Eo is obtained from the least biased method; this was to he expected from the p’ constraining equation. It is important to note that, because of the “quasi-universal” nature of the curve formed by all the iterative models examined, our comments about these models would appear to be valid for all models and therefore be a result of merely specifying a model (any model) prior to evaluating the distribution. Substituting eq 17a into 17b
(1f
w-lK)-1.p.fi1t39
= R1;s
(21)
produces an equation suitable for finding an optimized set of p,’s as in hp((AE))=
l(In
-
d K ) - l . P ( ( AE)).Rlt””- R S S l ( 2 2 )
For these calculations the P matrix is defined as a function of ( AE). This method has the advantage of easy calculation and, more importantly, it relates all ( A E ) ’ sfor all models to the same least biased distribution. When information concerning model type is irrelevant then the average energy transferred per collision ( ( AE)T) as used by Troe13 provides an adequate representation to present the results. The least biased method can be applied directly to the dilution data of I i n and RabinovitchZofor the isomerization of CH3NC. The computational results are displayed in Figure 7. In this figure, ( LLE),~ is plotted as a function of dilution. Clearly, the least biased distribution distinguishes between model type, that is to say the “dilution ( A E ) path” is not the same for the Gaussian and exponential models. TroeI3 has shown when considering only a single dilution experiment that when ( h E ) T is used no distinction between models can be made; in fact all models go to the diffusion limit. However in the methyl isocyanide example, sufficient experimental information is supplied so that the number of degrees of freedom in fitting the data is reduced.
The Journal of Physical Chemlstry, Vola83, No. 1, 1979 97
Energy Transfer In Thermal Unlmolecular Reactlons
TABLE I: Representative Collisional Efficiencies and (A E )" inert gas -NO,C1 He Ne Ar Kr Xe H, N, 0,
c1,
CO, N,O SiF,
SF,
CCl,F,
P(inert)/ P(parent)
Po W (i)/W
(P
Pobsd
NO,C1(476.5 0.61-2.8 0.37-1.9 0.20-1.5
0.11-1.0 0.026-0.26 1.0-5.0 0.23-2.0 0.071-1.8 0.25-0.67 0.29-1.9 0.39-2.0 0.25-2.0 0.20-1.7 0.37-1.6
CH ,NC 54-215 He 60-240 35-130 Ne 60-220 39-150 Ar 60-230 Kr 30-85 50-140 26-92 40-140 Xe 65-162 40-100 H, 21-71 30-100 N, 11-50 16-65 CO, 15-43 15-45 CH, 8-38 10-45 CD jF 8-28 10-35 CHF, 8-34 10-45 CF, 6-20 6-20 HCN 16-27 15-25 C,H,CN 8-34 7-30 n-C,H,CN 8-26 9-30 CFFN 19-52 20-55 CH,CCH 21-47 20-45 C ,H ,CCH a Taken in part from ref 7 and 8.
Po
SL E R,(E+)= 377 cm-')
1.00 0.3 7 -1.7 0.46-2.3 0.28-2.2 0.19-1.8 0.046-0.46 0.46-2.2 0.28-2.4 0.093-2.3 0.37-1.0 0.37-2.4 0.46-2.3 0.28-2.2 0.28-2.3 0.46-2.3
(-)
0.15 0.22 0.30 0.36 0.46 0.15 0.34 0.34 0.50 0.49 0.48 0.51 0.49 0.71
1-00 0.10 0.15 0.21 0.26 0.31 0.11 0.27 0.26 0.40 0.42 0.41 0.46 0.42 0.69
1.00 0.11 0.18 0.25 0.33 0.41 0.12 0.32 0.31
CII,NC ( 5 4 6 K, (E+)= 463 crn-') 1.00 1.00 1.00 0.14 0.16 0.17 0.18 0.15 0,15 0.24 0.60 0.35 0.46 0.52 0.38 0.53 0.66 0.84 0.51 0.51 0.66
It can be shown that the slopes from ( U ) T vs, dilution plots can be used to calculate (Ai?) for both weak (helium) and strong (CH,NC) colliders. Such an analysis for helium gives ( A E ) of 450 :h 30 cm-l and 620 f 50 cm-l with the Gaussian and exponential models, respectively. However, using the quasi-universal curve technique' ( AE)for helium was found to be -350 cm-l for the above collision models. This difference is attributed to the populations below Eo; the latter method allows for depletion while the former analysis maintains a quasi-equilibrium distribution below Eo.The least biased distribution also provides a lower limit of ( AE) 2000 cm-' for CH,NC; this value is in agreement with other systems.l In general, information (model type) pertinent to energy transfer probabilities are known from other experiments' which may include chemical or photochemical activation systems so that a further constraint on the least biased distribution can be invoked. This generally will involve the solution of a large set of simultaneous equations (nonlinear) so that a more practical solution is to apply the quasi-universal curves discussed earlier. A brief summary of the experimental observations follows. The isomerization of methyl isocyanide has provided the bulk of detailed information on energy transfer on unimolecular systems.l Table I shows some representative deactivators along with the experimental efficiency and the 'calculated values for ( A E ) ;( A E ) has a lower limit of -200 cm-l. Similarly, Figure 8 relates the measured efficiency with boiling point or molecular complexity of the deactivator in the methyl isocyanide system. As the molecular complexity of the deactivator
0.37
0
-
(0)
(AE), cm-'
SL
E
1.00 0.25 0.31 0.39 0.44 0.49 0.27 0.44 0.43 0.57 0.58 0.57 0.62 0.58 0.78
1.00 0.20 0.26 0.34 0.40 0.47 0.21 0.38 0.36
1.00 0.32 0.34 0.35 0.37 0.33 0.33 0.43 0.70 0.52 0.60 0.64 0.53 0.65 0.75 0.90 0.64 0.64 0.75
1.00 0.23 0.26 0.26 0.27 0.25 0.25 0.32 0.63 0.41 0.51 0.55 0,43 0.57 0.70 0.84 0.55 0.55 0.70
SL
E
190 240 310 360 410 2 10 360 350 500 520 510 560 520 820
200 300 420 530 680 220 520 500
295 320 330 345 310 310 420 840 540 650 730 560 740 930 1350 720 720 930
275 305 320 335 290 290 450 1850 700 1200 1400 800 1500 2300 3000 1400 1400 2300
L
-800
E
u A+
Lu
I/
-160C
I
I
I
I
I
1
5
10
15
20
25
xc
Flgure 7. Plots of average change in energy for ali collisions ((A€),) vs. collision dilution, xc (= w(He)/w(CH,NC)), for exponential (V)and Gaussian (A) models. The negative sign indicates energy lost by the reactant.
increases (boiling point), the value of p' increases toward its upper limit of unity. This correlation has been rationalized by saiying that as the collision partner becomes increasingly coimplex, a greater energy sink exists during the collision. Eventually, the energy sink becomes so large that on every collision of a reactive molecule (AE,with Ei > E,) with thie buffer gas, the reactant molecule is deactivated (has final energy after collision of E, < E,,). Once this situation is reached, further increase in the
98
D. C. Tardy and R. J. Malins
The Journal of Physical Chemistty, Vol. 83,No. 1, 1979
I
1
I
1
I
100
ux)
300
400
h l L l H Q POlPT
(%)
Figure 8. A plot of & vs. boiling point for CH3NCat 554 K: monatomic species (0);diatomic and small linear molecules (A);other molecules (e). Taken from ref 1.
energy sink has no effect. This line of reasoning has yielded the “intuitive” definitions that a strong collider is a model for which the average energy removed per collision, ( A E ) , is large (-2000 cm-l) whereas a weak collider is one for which ( AE)is small. These definitions, though lacking the rigor of the previously given definition of a strong collider, are operationally quite efficient. 111. Nonsteady-State Systems In some instances in which it is not certain that the system is in a steady state, a full scale simulation of the system and solution of the exact master equation (eq 23) is necessary. Specifically, a system in which the induction time (time to reach steady state) is comparable to the total reaction time qualifies for such an analysis. When eq 9 is recast into matrix notation the following equation results:
d--A = -[@(I- P)+ &]A = f,A dt This equation may be integrated numerically, or exactly as in (24) A(t) = exp(Et)-li(t = 0) For the case that L is time independent, this solution is quite easily and conveniently (and far cheaper than numerical integration) evaluated as in
A ( t ) = Fe+lI2Cexp(k)eTp;1/2A(t = 0)
(25)
where is the matrix used in symmetrizing while is the eigenvector matrix and h, is the eigenvalue matrix of the symmetrized form of L. We have found this technique to have many advantages over the traditional numerical i n t e g r a t i ~ n . ~ A specific example for which this technique may be applicable is the curvature of the Arrhenius plots for cyclopropane as obtained from single pulse shock tube experiments. Although much of this curvature is undoubtedly a result of boundary layer effects,’l it is still a relevant question to examine the collision effects which could produce such curvature. We have shown that this curvature cannot be accounted for by simple falloff calculations2z by calculating the Arrhenius curves as a function of w for step sizes of 400, 300, and 200 cm-’ (roughly the limit set on the value of the step size by conventional experiments conducted at temperatures a t or below 700 K). In all cases the experimental data crossed through the Arrhenius curves for w’s differing by several orders of magnitude. Alternatively, o-ne can fix the pressure at the experimental estimate (corresponding to an w i= 108-109) and extrapolate along the steps size to yield the result that at 1600 K, the step size must be 60 cm-’
1 2
I
I
I
I
4
6
8
10
tog w Figure 9. Plot of induction time vs. w for 400-cm-’ step ladder (SL) and variable step Gaussian set for 400 cm-’ at Eo (VG): (i) VG at 2000 K, (ii) VG at 1000 K, (iii) SL at 2000 K, (iv) SL at 1000 K.
in order to obtain the experimental results. While this result does not disprove the possibility that the system is in a steady state it does require a stronger temperature dependence for (AL3) than what has been experimentally observed. We have examined non-steady-state effects with two sets of calculations using the technique of eq 25. In the first, the master equation was integrated for a step ladder collision model with a step size of 400 cm-l. The data were analyzed by noting the time at which the rate constant reached 95% of its steady-state value. This time is referred to as the induction time and the values are plotted as a function of w in Figure 9. The general significance of the plot in Figure 9 is that a time point on the ordinate (the time duration of the experiment) and an w on the abscissa (the w of the experiment) can be used to determine if the system is in a steady state. If this designated point falls above the plotted line, then the system has sufficient time to reach steady state; similarly, if the point falls below the line, then the system does not reach a steady state. For the cyclopropane single pulse shock tube experiment (the dwell time is about 1ms), the boundary between transient and steady state regimes is w lo5. Thus for this model the system is in a steady state since w A further study of the transient phenomena was prompted by the conclusion made in the previous section that conventional thermal experiments (steady state) which yield values do not tell what is happening below Eo. For the shock tube experiments, the population distribution below Eo changes dramatically as the temperature rises from ambient to the final shock temperature. Thus it appears that the total P matrix is important; for low excitation levels Landau-Tellerz3 calculations predict that ( AE) should be about 20 cm-l. This fact coupled with conventional ( a E ) values a t Eo led us to perform the non-steady-state calculations for a collision model in which ( A E ) decreases below Eo. The energy transfer model is described by p v = N exp(-((IE, - Ell - &(E))/Q(E))’) (26)
(a)
where N is the normalization constant fixed by detailed balance and completeness and &(E)is the most probable energy step. Q(E)was varied linearly from 100 cm-’ at the zero point to 400 cm-l at Eo and was constant at 400 cm-’ for all energies above Eo. (It is noted when &(E)is held constant that this calculation generates the same steady-state rate constants as those produced by eq 16a
Energy Transfer in Thermal Unimolecular Reactions
and 17.) The corresponding induction time graph for this model is also given in Figure 9. The important point is that the boundary line between transient and nontransient behavior has been moved significantly upward. (It should be noted that this figure employs the instantaneous rate constant. The corresponding curve for the "observed' rate constant obtained from the initial and final total concentrations would be log (5) units above the appropriate line in Figure 9.) AJthough the boundary between transient and steady-state behavior is still below the experimental estimate, the curve can be shifted upward even farther by lowering the zero point energy value of &(E)to nearer the Landau--Teller limit. Any further changes in the cyclopropane calculations would be futile until the boundary layer question is settled. We have demonstrated that collisional effects can cause Arrhenius curvature and have calculated the induction times for various models of ( A E ) below Eo. For models in which (AE) a t low energy approaches the LandauTeller limit, induction times are significant and must be taken into account.
-
IV. Summary In this article we have attempted to review prior work and conclusions dealing with energy transfer in thermal unimolecular reactions and to show some of the state of the art methods for dealing with collisional phenomena in gas phase unimolecular reactions by master equation modeling. The three topics discussed (prediction, interpretation, and simulation) form a self-consistent foundation for understanding collisional effects in unimolecular systems. System analysis can be obtained by using quasi-universal curves or by direct modeling on a highspeed digital computer. The latter is a viable alternative due to the widespread availability of computers and the associated software. Acknowledgment. D.C.T. expresses his gratitude to Professor B. S. Rabinovitch, who introduced him to the area of energy transfer in unimolecular reactions, for the many stimulating and pleasurable discussions which formed the foundation for this and earlier (joint) publications. R.J.M. appreciated a 3M Graduate Fellowship during his last year of study.
References and Notes (1) D. C. Tardy and E. S. Rabinovitch, Chem. Rev., 77, 369 (1977). (2) F. A. Lindemann, Trans. faraday Soc., 17, 598 (1922). (3) (a)R. A. Marcus and 0. K. Rice, J. Phys. Colloid. Chem., 55, 894 (1951); (b) R. Marcus, J . Chem. fhys., 20, 359 (1952). (4) W. Forst, "Theory of Unimolecular Reactions", Academic Press, New York, N.Y., 1973. (5) P. J. Robinson and K. A. Holbrook, "Unimolecular Reactions", Wiley-Interscience, New York, N.Y., 1972. (6) N. B. Slater, "Theories of Unimolecular Reactions",Cornell University Press, Ithaca, N.Y., 1959. (7) D. C. Tardy and E. S. Rabinovitch, J. Chem. fhys., 45, 3720 (1966). (8) D. C. Tardy and B. S. Rabinovitch, J. Chem. fhys., 48, 1282 (1968). (9) R. J. Malins, Ph.D,, Thesis, University of Iowa, 1978. (10) R. J. Malins and D. C. Tardy, fhys. Chem., submitted for publication. (11) D. C. Tardy, Chem. fhys. Lett., 17, 431 (1972). (12) R. V. Serauskas and E. W. Schlag, J. Chem. fhys., 42, 3009 (1965).
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 99
(a)J. Troe, Ber. Bunsenges. Phys. Chem., 77, 665 (1973); 78, 478 (1974); (b) J Chem. Phys., 66, 4745, 4758 (1977). A. P. Penner and W. Forst, Chem. Phys., 11, 243 (1975). H 0. Pritchard, Acc. Chem. Res., 9, 99 (1976). R. J. Malins and D. C. Tardy, Chem. fhys. Leff., 57, 289 (1978). G. Dahlquist and A. Bjork, "Numerical Methods", Prentice Hall, Englewood Cliffs, N.J., 1974. R. C. Bhattacharjeeand W. Forst, Chem. Phys. Lett., 26,395 (1974). R. D. Levine and R. B. Bernstein, Acc. Chem. Res., 7, 393 (1974). Y. N. Lin and E. S. Rabinovitch, J. Phys. Chem., 72, 126 (1968). G. E. Skinner Int. J . Chem. Kinet., 9, 863 (1977). R. J. Malins and D. C. Tardy, Int. J . Chem. Kinet., submitted for
publication. K. F. Herzfeki and T. A. Litoitz, "Absorptionand Dispersion of Ultrasonic Waves", Academic Press, New York, N.Y., 1959.
Discussion J. TROE(Institut fur Physikalische Chemie der Universitat Gottingen). I would like to reiterate (see ref 1) that your representation of & against ( AE)d,,,/E+ is inconvenient since it suggests that in the diffusion limit (p,
