7556
J . Phys. Chem. 1993,97,7556-7563
Strong Collision Broadening Factors from Theories of Unimolecular Rate Coefficients S.H. Robertson, N. Snider,' and D. M. Wardlawt Department of Chemistry, Queen's University, Kingston, Ontario, Canada K7L 3N6 Received: October 13, 1992; In Final Form: April 15, 1993
The broadening factor representation of falloff behavior of unimolecular rate coefficients as a function of pressure is essentially an expression of deviations from a referential strict Lindemann falloff curve. There are advantages in recasting the broadening factor as a function of a new variable u, a bounded function of pressure. It is shown analytically that, as a function of u, F has a relative minimum and is always concave upward. The important features of F(u) are therefore the location and the value of its minimum. Broadening factors for a wide range of temperatures were calculated, in the strong collision limit, from standard RRKM theory for aziridine inversion and from a variational version of RRKM theory for methane dissociation. An extension of the Kassel model yields analytic results for F which are compared to the RRKM results for F and to the standard Kassel model results for F. The extended Kassel model is found to be better than its standard version in describing the temperature dependence of the RRKM-based broadening factors but cannot fully reproduce the diverse behavior which the latter display. These differences are interpretable in terms of energy dependence of transition-state sums of states and reactant molecule densities of states given by the theories.
I. Introduction The pressure dependence of chemical reaction rates remains one of the central problems of gas-phase kinetics. Techniques of increasing sophistication are being applied to the problem. It may happen that technicaldetails tend toobscure the relationships between the observed rates and the underlying molecular dynamics. In other words, computational considerations often take precedence over the matter of theoretical interpretation. It has become widespread practice to express falloff behavior of the unimolecular rate coefficient k in terms of a function F, known as the broadening factor.' This factor expressesdeviations of the k of interest from a reference k which is calculated from Lindemann-Hinshelwood theory. The advantageof this recasting of the pressure dependence of k is that it provides a more descriptive representation of this dependence than do plots of k versus pressure. Such a recasting does not provide a replacement for, or simplification of, the calculation of rate constants. It is instead a means of providing a more informative representation of their pressure dependence. In ref 1, strong collision falloff curves for 20 unimolecular reactions were calculated using specific rate constants k ( E ) from RRKM theory in conjunction with the standard strong collision formula for k. Broadening factors were extracted from the calculated falloff curves, and these were approximated by fitting to an empirical function characterized by numerous parameters. As yet, no theoretical interpretation of the broadening factors which emerge from the various RRKM calculations has been forthcoming. One of the objectives of this article is to provide some such interpretation. In section 11, the broadening factor is examined in some detail. It is argued that the broadening factor is best expressed as a function of a variable u instead of the reduced collision frequency as in previous work. The variable u has the advantage of being a bounded function of the collision frequency, the latter quantity being unbounded. It is shown that insight into the interpretation of F is provided by expressing F as an average over a distribution. In the strong collision limit, this distribution is over the specific rate constants k ( E ) of the system (E is the total energy). When F(u) is expressed in this way, it is readily seen that this function
f
Author to whom correspondence should be addressed. NSERC of Canada University Research Fellow.
is between zero and one, and that its second derivative is everywhere positive. It follows that F(u) has one and only one minimum. In section 111,a detailed examinationof a simplemodel, chosen because it provides analytic results for F, is given. This model is a straightforward extension of the Kassel model. It has been shown2 that the standard Kassel model, even with optimal adjustment of its parameters, does not give good fits to strong collision falloff curves from standard RRKM theory. Nevertheless, extensive calculations of broadening factors for the standard Kassel model have been reported.' The material in section I11 extends and illuminates these calculations. The extended Kassel model does not admit the full complexity of Fs which arise from RRKM theory. This point is made in sections IV and V, where broadening factors obtained from standard RRKM theory and from variational RRKM theory are considered. The aziridine inversion system was chosen since, at first glance, it appears to lend itself to a standard RRKM treatment: there is a tight transition state at a fixed location which can be reasonably represented as a collection of independent harmonic oscillators. Methanedissociationwas chosen because it definitely does not lenditself toa standard RRKM treatment: the transitionstate location is not fixed but must be determined variationally since there is no potential energy barrier (relative to separated products) on the minimum energy path connecting reactants and products. Furthermore, some degrees of freedom undergo large amplitude, anharmonic, and coupled motion at the transition state, rendering a harmonic oscillator description inappropriate. Results for F for both systems show some deviation from the general pattern of the Fs for the extended Kassel model. This deviation is discussed in section VI. Concluding remarks are collected in section VII.
II. Broadening Factor A. Reformulation. The rate coefficient k for a unimolecular reaction as a function of collision frequency has a limiting form given by the Lindemann theory. Defining kg (c = bath gas concentration) and k, as the asymptotic low- and high-pressure expressions for k,one may write the strict Lindemann formula
0022-365419312097-7556$04.00/0 0 1993 American Chemical Society
Broadening Factors in Unimolecular Rate Theory
The Journal of Physical Chemistry, Vol. 97, No. 29, I993 I551
as
is given by5
-k=
kocIkm 1 koc/k, Since k~ is directly proportional to the collision frequency w and since k, is constant, one is free to define a reduced collision frequency w, by k,
+
w, = koc/ k ,
(2) It is found both experimentally and theoretically that unimolecular rate coefficients as a function of collision frequency deviate from eq 1 at intermediate collision frequencies. Deviations from strict Lindemann behavior are expressed by the broadening factor F, which is given by
(3) where w, is determined via eq 2 from k~ and k , for the reaction of interest. It has been shown that, subject to conditions which almost certainly apply to unimolecular reactions at low temperatures, F is always less than 0ne.3 In other words, the falloff region under these conditions is broader than the strict Lindemann falloff region with the same kg and k,. This is the original sense in which F became known as the broadening factor.' Another sense in which F may be called the broadening factor emerges in the context of its expression as an average over a distribution; this expression is derived below. Only the strong collision case is considered in what follows. The strong collision integral formula for k may be found in any one of severalstandard textsa4 It is expressed in terms of collision frequency w, the unimolecular rate constant k(E) for reaction at energy E, and the Boltzmann distribution function p,(E), (4)
where E, is the threshold energy for the reaction. The Boltzmann distribution function is given by P,@) = q-'NE) e x ~ ( - P ) (5) where q is a partition function, N(E) is the density of states of thereactant moleculein energyspace,andBis (kg7')-l. Inspection of the asymptotic behavior of the right side of eq 4 shows that it gives for k g and k , the following formulas:
In the integrand on the right side of eq 4, k(E) plays two distinct roles. In thenumerator, k(E)expresses the reactivity of molecules at energy E, in the denominator, k(E) is an essential term in h(E), the depletion factor
h(E) =
w
w
+ k(E)
(7)
where EAis the solution of the equation
k(EA)=
(9) Note that eqs 8 and 9 imply a change in the mathematical role of k from a variable dependent on E to an independent variable X of which the reactivity distribution PR is a function. One can interpret p ~ ( hdh ) as the fractional contribution to k, from k(E) between X and X + dX, i.e., p~ is the integrand of eq 6b expressed as function of X = k(&) and divided by k,. From eqs 4,7, and 8, one sees that k l k , may be written as an average over the reactivity distribution
= (h)R (10) where the second line of eq 10 follows from eqs 7 and 9 with the understanding that ( ) R denotes an average over the reactivity distribution. From eqs 2, 6, and 7, one sees that w, is given by = w( (11) In strict Lindemann theory, p~ is the 6 function ~(X-XL), and ( X - l ) R is then XL-1. Substitution of eq 11 into eq 10 and use of eq 3 give for the strong collision broadening factor 0,
= (h)R/hL (12) where hLis the strict Lindemann depletion factor obtained when XL-~ is taken to equal the (X-')R for the reaction of interest. A better characterization of F is obtained if new variables u and z are introduced
Note that u and z are simply different ways of expressing the collision frequency and the reactivity. The strict Lindemann depletion factor is given by
where eq 13a has been used to eliminate w, in favor of u. Where there is a range of reactivities, there is a dependenceof the depletion factor on z
The chemical reaction depresses the populations of molecules in reactivestatesrelative to their Boltzmann populations. Thestrong collision formula for the extent of this depression is given by eq 7.
The difference between Lindemann theory and general strong collision theory is that in Lindemann theory there is only one @ E ) , whereas eq 4 makes allowance for a range of k(E)'s, i.e., for a range of reactivities. It is useful to introduce a distribution function ~ R ( X )of reactivities A, which in strong collision theory
where eqs 7,9, and 13 have been used to obtain eq 14. For u and z near -1, h can deviate enormously from hL. On the other hand, as z approaches -1, the reactivity and the reactivity distribution function both approach zero. Hence, p~ does not reflect the possible importance of the contributions to ( h ) from ~ z values close to -1. To remedy this deficiency, a distribution function
Robertson et al.
7558 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
quantity 1 - F(0) is greatest when pa is broadest, but a skewed pa can only be moderately broad because the first moment of pa
pa is introduced
is always zero.
B. Reactivity Distribution in Strong Collision RRKM Theory. In RRKM theory, k(E) is found to be related to G(E), the sum of states for the transition state, and to N ( E ) by the formula4
where E, is defined by )R k(Ez) -
(
)R k(E,)
+
=z
From eq 13b, one sees that, in terms of A, pa and p~ are related by
The second term on the right of eq 15b remains finite as X goes tozero. Hence,p,doesgivenonzeroweight tovery low reactivities. From eq 15b, one sees it appropriate to name pa the augmented reactivity distribution function. Letting ( )a denote an average over pa, one sees from eqs 13b and 15 that the first moment of pa(z) is zero
where h is Planck's constant. Let the average (X),of a function of E be defined by
Straightforward integration by parts shows that (G)o is the partition function for the transition state.4 One also sees that ( N ) , is equal to FEN(&), where FE is a quantity of interest in the theory of unimolecular reactions in the low-pressure limit.6 From eqs 2, 5 , 6, 11, and 21, one finds that k, and (h-1)~in RRKM theory are related to ( G ) o and ( N ) o
( z ) ,= 0
(16) From eqs 12, 14, and 15a, one sees that the broadening factor, expressed as an average over pa, is given by
Equation 17 is the central result of this article. From eq 13b, one sees that pa is the 6 function 6(z) if the molecule has only one reactivity. From eq 17, one sees that pa equal to 6(z) implies F equal to one for all values of u between -1 and + l . The extent to which pa weights values of z different from zero determines the extent to which F deviates from one. The other sense in which F is a broadening factor is now clear: it reflects the broadening of the distribution function pa. Twice differentiating both sides of eq 17 with respect to u gives
In this notation, ( X-')R is given by
From eqs 15a and 21-23, one obtains the strong collision RRKM formula for the augmented reactivity distribution function
du2 Since u and z always lie between -1 and +1, the right side of eq 18 is everywhere positive in the interval (-1,l). Thus F(u) is concave upward everywhere in this interval. Making use of eq 16, one verifies that F is equal to 1 at the end points of this interval, as it should be. A continuous function F which has the same value Fe at both end points of an interval and a positive second derivative everywhere within that interval is less than F, within the interval, and it has a single minimum there. Thus, F is seen to be less than one for finite values of the collision frequency. Furthermore, F has one and only one minimum at some finite value of collision frequency. By differentiating both sides of eq 17 once with respect to u and setting u equal to zero, one sees that F and its first derivative at u = 0 are simply related to the second and third moments of
In the following section, eq 24 will be employed within the context of a simple model.
III. Extended Kassel Model In this section, explicit formulas for G and N will be adopted with a view to making more concrete the ideas of the previous section. These formulas pertain to a model for which the name extended Kassel model is an appropriate one. For this model, N and G are given by
N ( E ) = cN(EN- Et
Pa (z2), =
1 -F(O)
( z ' ) , = -F'(O)
+ E)'-';
cG(EG-
(19)
(20)
Equation 19 confirms the foregoing assertion that the extent to which F deviates from one is an indication of the extent to which Pa is broadened. Equation 20, in conjunction with the fact that F(u) is everywhere concave upward, enables one to conclude that the minimum in F occurs at positive u if pa is skewed toward positive z and at negative u if pa is skewed toward negative z.The
G(E) =
Et + E)'-' (26)
The parameters CN and CG are scale factors which are of no consequence in what follows. The physical interpretation of the parameters has been discussed extensively in the l i t e r a t ~ r eThe .~ energy parameters EN- Et and EGare zero in the standard Kassel model and positive in the extended Kassel model. The physical interpretation of these energies will be addressed in section VI. It is convenient to introduce a new variable x given by X
= @(E- Et)
(27)
Broadening Factors in Unimolecular Rate Theory
The Journal of Physical Chemistry, Vol. 97, No. 29, I993 7559 1 .oo
In terms of x, N , and G, are given by
0.96
where Ps is a polynomial of degree s - 1 if s is an integer s-1
B"
P,(B) = (S - l ) ! x "-0
0.92
n
3
n!
W
lJ-
The parameters BN and BG are given by
BN = @ E N ; BG = B E G (30) If s is equal to one or if BGis equal to BN,then the strict Lindemann formula holds, which means that F is everywhere equal to one. The more likely situation is for s to be greater than one and BG to be much less than BN. It then follows that x is a monotonically increasing functionof z. Substituting eqs 28 into eq 25 and solving for x, one obtains
0.88
1
Od4
0.80
/
-1.0
I
I
-0.5
0.0
I
0.5
1
U Figure 1. Extended Kassel model (EG = 0.005E~, s = 2) broadening factor curves for different values of kBT/EN: 0.0005 (A),0.005 (W), 0.05
(A),0.5 ( O ) , 5.0 (0).Symbols are plotted at every fifth data point; solid lines result from linearly connecting adjacent data points.
where the y t are given by
and z+ is given by
One sees from eq 3 1 that xis only positive and finite for z between z- and z+ where I+ is given by eq 32 and z- is the value of z for which BNJJ-is equal to Bey+
From eqs 24, 28, and 31, one obtains the strong collision pa for the extended Kassel model
pa = 0; otherwise
(34)
As T approaches zero, the B's increase without bound. Thus, zapproaches 0 as T approaches 0, unless BGis identically zero, in which case z- approaches -1. This result indicates that the lowtemperature behavior of the extended Kassel model is qualitatively different from that of the standard Kassel model (vide infra). As T increases without bound, the B's go to zero, which implies that z+ goes to zero. Since (z), is zero, the approach of either z- or z+ to zero implies the approach of pa to 6(z). Moreover, the limiting behavior of z- and z+ implies that pa is skewed toward positive z at low T and toward negative z at high T . Thus, it is characteristic of the extended Kassel model that F approaches one for all u in both the high and low Tlimits. Furthermore, the minimum in F(u)occurs at positive u when Tis low, and it shifts to negative u as T increases. These trends have also been found for Fs determined from standard RRKM calculations' for a variety of systems. For s equal to 2, one obtains relatively simple expressions for pa and for F. These are derived in Appendix A. Graphs of F(u) based on eq A5 are shown in Figure 1. Parameter values used to generate the curves in this figure are given in the caption. The
curves illustrate the above-mentioned assertions that F goes to one at both limits of temperature for the extended Kassel model and that the minimum in F shifts from positive u to negative u as T increases. Inspection of eq 34 and numerical calculations show that pa is bimodal at intermediate temperatures if s is greater than two. The consequences for F(u)are that it deviates more from one and thatitsminimumfallsat avalueofuclosetozero. At theextremes of temperature, one of the peaks in pa subsides. The remaining peak grows and shifts toward z = 0. Such behavior implies approach of F to one. It has been shown numerically' that for the standard Kassel model with fixed B and increasing s F eventuallyapproaches one. The implication is that pa narrows under these conditions. It is of some interest to investigate this narrowing. Such an investigation was carried out analytically and is outlined in Appendix B. Because the first moment of pais zero, this distribution function must become centered at z = 0 as it becomes progressively narrower. Let Eo be the value of E corresponding to z = 0. The analysis presented in Appendix B shows that EOincreases with increasing s. This increase implies a decrease in the derivative of In k at E equal to EO. As seen from eq 15a, the approach of this derivative to zero implies approach of pa to a 6 function at z = 0.
IV. Aziridme Inversion Aziridine, C ~ H S Nundergoes , inversion at the N atom. This process has been studied by Borchardt and Bauer' and by Carter et a1.8 The transition state for the reaction is well defined, it being the configuration at which the two C atoms, the N atom, and the H atom bonded to the N atom are coplanar. Harmonic frequencies of the molecular vibrations for both the equilibrium and the transition-state configurations were obtained from ab initio calculations.9 These frequencies where used to determine the sum of states, G ( E ) , at the transition state and the density of states, N ( E ) , for the equilibrium configuration using direct count.10 Values of k(E) were obtained using eq 21. Rotation was not included in the analysis. The change in moments of inertia is comparativelysmall. Hence, couplingbetween rotation and vibration was assumed to be negligible. The high-pressure limit rate coefficient, k,( T), and the lowpressure limit second-order rate coefficient, ko, were determined using standard formulas.' The Lindemann-Hinshelwd factor FHLwas then computed from eq 1. Using eqs 4 and 5 , values of
Robertson et al.
7560 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 1 .o
1 .o
I
0.8
0.9
-
n
n
0.6
I0.8
W
LL
LL
0.4
0.7
0.6
-
0
I
-0.5
I
I
0.5
0.0
n? ! -1’ 0 Y
1
L
U
lines result from linearly connecting adjacent data points.
k were obtained from calculated k(E)’s by straightforward numerical integration. From calculated values of k, ko, and k,, or, F, and u were calculated using eqs 2,3, and 13b, respectively. A plot of F against u is shown in Figure 2. One sees that the behavior of F is somewhat, but not entirely, similar to that shown in Figure 1. It can be seen that the minimum in the F curve decreases as temperature increases until about 900 K, at which point it begins to increase. As the temperature increases, there is a drift of the location of the minimum in F, first in the positive direction and then in the negative direction. This behavior contrasts with that seen in Figure 1, where the drift is always in the negative direction. V. Methane Dissociation
CH,
-.CH,+
00
05
10
U
Figure 2. Aziridine inversion broadening factor curves for different temperatures: 300 K (0),600 K (n),900 K (A), 1200 K (B), 1500 K (A),1800 K (0). Symbols are plotted at every seventh data point; solid
The reaction
-0 5
H
(35)
is important because of its role in combustion. It has been widely studied.11.12 From a theoretical standpoint, reaction 35 exhibits a number of interesting features: it has no well-defined transition state and so cannot be treated by conventional RRKM theory. A related problem is that certain vibrational degrees of freedom of methane go over to free rotors on dissociation. Hence, the transition state is expected to be poorly described as a collection of independent harmonic oscillators. For this reason, a number of sophisticated theoretical treatments have been developed to investigate this and similar reactions. For the present calculations, the variational version of RRKM known as flexible transitionstate theory (FTST) developed by Wardlaw and MarcusI3 was employed. Aubanel and Wardlaw11 applied FTST to reaction 35 and determined the sum of states for a limited set of total energies, E, and angular momenta, J , from an analytical potential energy surface obtained by fitting ab-initio points calculated by H i r ~ t . 1The ~ location of variational transition states for reaction 35 on the HirstI4 potential energy surface is, to a good approximation, independent of J , so G ( E ) does, in this case, correspond to the sum of states for a single, variationally determined transition state a t each energy. The density of states was obtained by the standard direct count techniques10 assuming the equilibrium methane molecule can be described in rigidrotor-harmonic-oscillator terms. The remainder of the calculation followed that for the aziridine inversion, and the results are shown in Figure 3. It can be seen from Figure 3 that this reaction follows a pattern somewhat different from that exhibited by aziridine. The temperature dependence of the value of the
Figure 3. Same as Figure 2 except for methane dissociation.
minimum in F is roughly the same as in Figures 1 and 2, but said value has the additional feature of being nearly constant in the temperature range 900-1200 K. The location of the minimum as a function of T behaves roughly the same in both Figures 2 and 3, but again in Figure 3 this quantity is also very nearly constant in the temperature range 900-1200 K. Moreover, the minimum appears to shift slightly in the positive direction in the temperature range 1200-1800 K.
VI. Discussion In the intermediate pressure (falloff) range, plots of unimolecular rate coefficients vs pressure are not informative. Herein lies the motivation for establishing a standard of comparison. The standard which has been widely adopted is the strict Lindemann falloff curve with the same asymptotes. The broadening factor Fexpresses deviations from this standard. Distinctive features of F are exhibited when it is expressed as a function of the variable u defined in eq 13a. Under not very restrictive conditions, F(u) lies everywhere between 0 and 1 and is concave upward. If the minimum value of F is close to zero or to one, then this minimum must fall a t a value of u very near to zero. As seen from Figures 1-3, different falloff curves are distinguished by the position of the minimum in F and by the value of F a t said minimum. It is important to note that reformulating the falloff behavior of the rate coefficient in terms of F and u ultimately neither simplifies nor complicates the theoretical treatment of the rate coefficient. What it does do is provide quantities on which a theoretical analysis may better focus. Equations 19 and 20 show that F decreases as the distribution function pa(z) broadens and that the position of the minimum in F is related to the skewing of pa. From eq 15a, one sees that the strong collision pa depends on the thermal distribution function pc and on the derivative of k(E). Within the context of RRKM theory, the magnitude of dk/dE is related to the difference in the derivatives of In G and of In N , as shown in eq 24. Theoretical models often predict that the derivative of k is small a t E just above the threshold and again a t large E. (Near the threshold, the derivative of In k may be large even though the derivative of k is very small since, near the threshold, k itself may be even smaller. This point should be borne in mind when perusing Figures 4 and 5.) At sufficiently high T,E, for positive z is in a range where dk/dE is small, but pa eventually falls sharply because of the Boltzmann factor in pc. Thus at high T,pa has a peak a t positivez. This peakgrows and shifts toward z = 0 as Tincreases. Because of the sharp fall in pa a t sufficiently high z, there is a net skewing toward negative z . The consequences for F(u) are shown by the high-Tcurves in Figures 1-3: F(u) approaches one as T increases, and its minimum is located at negative u.
Broadening Factors in Unimolecular Rate Theory
-I
The Journal of Physical Chemistry, Vol. 97,No. 29, 1993 7561
I
_-
n 20.0
40'0
I
30.0
r'\ 7
I
-
y -20.0
m
\
20.0
n
W
j::
W
1
W
I -60.0
- i n-.-n I 0.0 ,
~
0.0
0.2
0.4
0.6
0.'1
0.2
0.3
0.4
0.5
Figure 4. Plot of In &(E) vs ln(E/Et) for aziridine inversion: RRKM theory (-), standard Kassel model with s = 18 (- -), extended Kassel model with s = 18 and EG = 0.2E1(- -).
-
At intermediate T, values of E, around z = 0 are those for which dk/dE is large. Here pa may be bimodal, and deviations of F from one are largest. At low T, the form of pa in most cases is one in which there is a sharp peak at negativez. As Tdecreases, this peak grows and shifts toward z = 0 because only E, values close to Et are significantly weighted by pc and because k(Et) is nonzero. The extended Kassel model is a fairly realistic model which illustrates the just-mentioned behavior of pa(z) and the related behavior of F(u). The extended Kassel model has two parameters over and above the s parameter of the standard Kassel model. The physical interpretation of the s parameter has been much d i s c ~ s s e d . *It~ is ~ understood to be equal, or somehow related, to the number of active modes of the reactant molecule. A physically reasonable value of ENis Et. Broadening factors are virtually insensitive to variations in EN as long as this parameter is much larger than EG. Broadening factors are much more sensitive to changes in EGif EGis relatively small. It has been argued that EGaccounts for quantization and that it is equal to the zero-point energy of the transition state.15 Such a view has been modified by subsequent work16 wherein EGis replaced by a function of E which approaches the zero-point energy of the transition state in the limit of infinite E. One might treat EGas an adjustable parameter if a semiquantitativefit to k(E) is desired. Setting ENequal to Et in eqs 26 and combiningthese equations with eq 21, one obtains
Figure 4 shows log-log plots of k(E) for aziridine inversion from standard RRKM theory compared with curves obtained from eq 36 with EG equal to zero (standard Kassel model) and with EG set equal to some small but finite value. Figure 5 shows the curve of k(E) for methane dissociationfrom FTST compared with curves obtained from eq 36. The finite EGcurves are in closer agreement with the RRKM curves, as would be expected since there is an extra adjustable parameter. The agreement in Figures 4 and 5 is not entirely satisfactory, especially just above threshold, where k(E) curves from RRKM theory increase less rapidly than those for the standard Kassel model but more rapidly than those for the extended Kassel model. No attempt was made to optimize the parameters of the extended Kassel model to obtain best fits to the standard RRKM (aziridine) and FTST (methane) rate coefficient curves in Figures 4 and 5. Doing so would not remove
thequalitative differencebetween the Tdependenceof thelocation of the minimum in F(u)for the extended Kassel model and that for the RRKM calculations. One sees from Figures 4 and 5 that the k(E) curve from FTST has considerably more structure at ln(E/Et) < 0.2 (Le., E < 1.2Et) than the one from standard RRKM theory, which is a smooth function of energy. We attribute these fluctuations to the sparser density of states for methane compared to aziridine. The actual scale of these fluctuations, which is expected to be smaller than the scale of fluctuations in Figure 5, is undetermined because the FTST data are only available for a limited set of energies whose (nonuniform) spacings exceed the mean spacing between methane energy levels in the energy range of interest. In any case, the observed differences in the temperature dependence of F(u)for methane versus aziridine are most likely due to differences in the energy dependence of the respective k(E)'s on a wider energy scale than the fluctuations in Figure 5. From inspection of the FTST curve in Figure 5 , one may conclude that smoothing the fluctuations will almost certainly produce a curve with dk/dE as a function of E quite different from that of the standard RRKM curve in Figure 4. Apart from the halt in theF(u) curves for methane dissociation between 900 and 1200 K, these curves are similar to those for aziridine inversion. This similarity is striking in view of the differencein the k(E) curves for these reactions as seen in Figures 4 and 5. On the other hand, the difference in the k(E) curves is not surprising in view of intrinsic differences in the nature of the respective transition states. Aziridineinversion has a potential energy barrier along the reaction coordinate, a situation which dictates the identification of a (tight) transition state atop this barrier and use of standard RRKM theory. Methane dissociation, on the other hand, has no known potential barrier along its reaction coordinate, dictating the use of a variational version of RRKM theory to locate the energy and angular momentum dependent transition states. The particular version used here, FTST, takes proper account of transitional modes as they evolve from vibrational degrees of freedom in reactants to free rotational degrees of freedom in the products. It has been foundll that the (canonical) transition state becomes increasingly tighter with increasing temperature. At low energies the transition state is essentially loose, so the transitional modes approach a twodimensionalfree rotor which would contribute a 1 to the exponent s in a Kassel-type model. At high energies, the transition state
Robertson et al.
1562 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
is tighter, and the transitional modes approach two (degenerate) vibrations which would contribute a 2 to the exponent s. It has, however, been established17that a harmonic oscillator treatment of the transitional modes for this system within the framework of variationaltransition state theory yields a rate coefficient whose magnitude and temperature dependence are in qualitative agreement with the FTST prediction. This is a surprising result whose origin is not understood and which indicates that methane dissociation may not be representative of the class of reactions having no potential energy barrier. A more precise interpretation of the differences in the falloff behavior predicted by standard RRKM theory and variational versions of RRKM theory, such as FTST, is clearly premature and awaits a systematic study of a variety of unimolecular dissociation reactions.
MI. Summary and Conclusions Falloff behavior of unimolecular rate coefficients is apparently best characterized by converting the data to broadening factors, F. A new expression for F (eq 17) is the central result of this paper. The derivation of this formula presented here applies only in the strongcollision limit, but it can be extended's to include weak collisions if in eq 10 one adopts a somewhat different interpretation of the variable of integration. l9 This reformulation facilitates a rigorous analytical treatment which reveals that, under quite general conditions, F(u) must conform to several restrictions. Data which yield an F(u)which does not meet these restrictions must pertain to a reaction which is in some sense exceptional. A major consequence of these restrictions is that F(u) has a single minimum and may therefore be characterized by the location and the magnitude of this minimum. The shape and breadth of the distribution function painfluence,via moments of z (eqs 19 and 20), the location and magnitudeof this minimum. Within the context of strong collision unimolecularrate theory, broadening factors have been obtained for the standard Kassel model, the extended Kassel model, a full standard RRKM treatment of aziridine inversion, and a full variational RRKM (FTST) treatment of C-H bond fission in methane. The temperature dependence of the location udaof the minimum of F(u) and the magnitude of this minimum Fdn = F(udn)have been analyzed and found to be different for each of the four cases considered. For the falloff curves generated from standard RRKM theory and from FTST, both u h and 1- Fd,, increase and then decrease with increasing T. The results presented here show that FTST gives a location and a value of the minimum of F which are nearly constant in an intermediate temperature range of several hundred Kelvin. Ascertaining whether or not this latter feature is a general one within the context of FTST (or other variational versions of RRKM theory) requires systematic evaluation of a set of dissociation reactions having no barrier to the reverse association process. The standard Kassel model completely fails to reproduce this behavior: u- shifts to more negative u, and 1 - Fdndecreases as T increases. The extended Kassel model does better but falls short of a complete description: 1 - Fminfirst increases and then decreases as T increases, as it does for aziridine and methane, but the Tdependence of I(is qualitatively the same as that for the standard Kassel model. In the strong collision limit, the temperature dependence of ni,# and that of 1 - Fd,, are related to the behavior of the density of states N ( E ) and of the detailed unimolecular rate constants k(E). This relationship has been illustrated analytically (section 111) in the context of a simple model which is an extension of the Kassel model. The extended Kassel model containstwoadditional parameters (eq 26). The extension was motivated by the fact that the standard Kassel model, even with optimal adjustment of its parameters, does not give good fits to strong collision falloff curves from RRKM theory. The additional parameters in the extendedversionofferthepossibilityof better fits, but thediversity of behavior of F(u)'s from RRKM theory is apparently too wide
to be captured by the simple analytic formulas for N(E) and k ( E ) provided by the extended Kassel model. The utility of analytic functions, which do admit the full range of behavior of F(u) and yet contain few parameters, is unquestionable. This utility would be further enhanced if the parameters in these functions could be related straightforwardly to molecular parameters. The search for such functions continues. Finally, weak collision effects on F(u) are almost certain to be considerable. To some extent, these are eliminated by the use of the reduced collision frequencydefined by eq 2. What remains are effects related to the interplay between k ( E ) , N(E), and the energy-transfer distributions at different E . These effects are currently under investigation.
Acknowledgment. This work was supportedby operatinggrants from NSERC Canada and by a grant from the Queen's University Advisory Research Committee. Appendix A Extended Kassel Model with s = 2 One sees from eq 29 that Pz is given by P2(B) = B
+1
(All Substitutings = 2 intoeqs 32 and 33 andusingeqA1, oneobtains for zk the equations
Substitution of eqs A2 into eq 31 gives for x the equation
Finally, substitution of s = 2 into eq 34 along with eq A1 for P2 and eqs A2 for gives for pa the formula Pa
=-
; for z - < z pa = 0;
otherwise
< z+ (A41
Equation A4 shows explicitly that pabecomes progressively more sharply peaked at z-as z- approaches 0 and progressively more sharply peaked at z+ as z+ approaches 0. From eqs 17 and A4, one obtains the following formula for F(u) for the extended Kassel model with s = 2
where w(a) and a are given by
and y(0,a) is the complement of the incomplete y function. The function w is monotonically increasing for positive a and has the following asymptotic forms: w = -a2(ln a
+ y + 1)
a
-
o
w=1-4/a a-m (A71 where y is Euler's constant. At low temperature, a becomes very large for the extended Kassel model sincez- approaches0. Hence, the low-T approximation to F is
[ ez+'(1 - z+u)]
F = 1 - z - ~ 1( - u2) 1 +
(A8)
At high temperature, a approaches 0 because z+ approaches 0.
Broadening Factors in Unimolecular Rate Theory
The Journal of Physical Chemistry, Vol. 97, NO.29, 1993 7563
Hence, the high-T approximation to F is
For s large compared to B, one may make the approximation
Equations A8 and A9 confirm the above-mentioned assertions that F goes to one at both limits of temperature for the extended Kassel model and that the minimum in F shifts from positive u to negative u as T increases. For the standard Kassel model, eq A5 simplifies to
Equation A10 shows that F at fixed u increases monotonically to one with increasing T for the standard Kassel model with s = 2. Further analysis shows that the minimum in F given by eq A10 comes at u = -0.16 in the limit of zero temperature (z+ 1).
Substitution of eq B7 in eq B4 followed by use of the resulting formulas for R,(BN)and R,(BG)in eq B3 gives
x0=s-l Substituting this result into eqs B5 and B6 and making use of the requirement that s - 1 be large compared to BN and BG, one obtains
-
Appendix B Extended Kassel Model at Large s Given the constraints on parit is reasonable to assume that the magnitude of p,(O) is closely correlated with the narrowness of pa. From eq 15a and the definition of k(E,), one obtains
Substitution of these results into eq B2 gives pa(())
= [2(S - l)/T] ' l 2 ( B-~BG)-'
Thus, the growth in pa(0) with increasing s is brought about by the decrease in the derivative of In k with respect to E at E equal to Eo. This decrease in turn is brought about by the increase in EOwith increasing s.
References and Notes
From eqs 27 and 3 1, one sees that for the extended Kassel model eq B1 becomes
BN - BG
'('0
- Et) = xo = R ~ ( B G ) R ,-( R,(B,) ~~)
-BG
(B3)
where R,(B) is defined by
R,(B) = [P,(B)]('-')-' From eqs 21, 22, 23, and 26, one obtains
(B4)
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