J. Phys. Chem. 1994,98, 8713-8720
8713
Weak Collision Effects on Broadening Factors for Unimolecular Reactions N. Snider Department of Chemistry, Queen's University, Kingston, Ontario, K7L 3N6 Canada Received: March 25, 1994; In Final Form: June 21, 1994'
Broadening factors were obtained analytically and computationally for three models of thermal unimolecular reactions. All three models incorporate weak collision effects. In two of the models the exponential formula was assumed for the energy transfer. The Lindemann formula was assumed for the microcanonical rate constants in one of the models, the extended Kassel formula in the other. In the third model the extended Kassel formula was assumed for the microcanonical rate constants, and the stepladder formula was assumed for the energy transfer. The broadening factor F was expressed as a function of u, a bounded variable obtained from an algebraic transformation of the collision frequency. As a function of u, F has a single minimum. Weak collisions were found to shift the location of this minimum, call it Umy in the negative direction, but the shifts are small for the extended Kassel models. Weak collisions have more of an effect on F(u,), causing it to decrease. In no case, however, was F(um)found to decrease by more than 0.25. The decrease in F(um)as a function of average energy transferred per collision was found to be approximately the same for all three models.
I. Introduction
S K ( E )- K, Z=
It is standard practice in unimolecular kinetics to plot rate coefficient vs total pressure.' Typically, the resulting curve is linear at low pressure, and it levels off at high pressure. The intermediate pressure region, the so-called falloff region, is not well characterized in such a plot. Characterization of the falloff region is improved by a transformation from rate coefficient K to broadening factor F. The broadening factor is the ratio of the given K to the strict Lindemann rate coefficient KL which has the same high- and low-pressure asymptotic behavior.2 The broadening factor as a function of pressure, or equivalently, of collision frequency, has a minimum. The location of this minimum and the minimum value of the broadening factor are quantities which characterize the falloff behavior of a given reaction at a given temperature. If the reactant is dispersed in a gas of weakcolliders, the location of the minimum and the minimum value of the broadening factor also depend on the nature of the collider. It has been shown for strong collisions that transformation to an independent variable u results in a broadening factor F(u) which must, under fairly general conditions, be concave upward.3 If c is total concentration and if koc and K, are the asymptotic low- and high-pressure expressions for K , then u is given by u=-
k,c - K ,
koc + K ,
One sees that the physically relevant range of u is from -1 to 1 and that u = -1 corresponds to the low-pressure limit, u = 1 to the high-pressure limit. An F(u)not everywhereconcaveupward between-1 and 1 must pertain to a reaction which is in some way anomalous. It has also been shown for strong collisions3that F(u) can be expressed as an average over a distribution function pa,
In the strong collision limit z is given in terms of the microcanonical rate constant K ( E ) , ~~
0
Abstract published in Aduance ACS Abstracts, August 1, 1994.
JK(E)
+
K,
(3)
where
E is the internal energy, Et is the threshold energy for the reaction, and p,(E) is Boltzmann's distribution function. The second line of eq 3 defines 9. The distribution function pa(z) has the special property that its first moment is zero. In the strong collision limit the behavior of pa(z) can be related to the behavior of p,(E) and K ( E ) .Thus ~ pa is more closely related to molecular properties than is F. This article has two objectives. The first of these is the rederivation of eq 2 in a way which shows that it is valid not only for strong collisions but also for weak collisions as well. The derivation presented below is based on a formula for K due to Singh and Pritchard4which is valid under fairly general conditions. The second objective of this article is an elucidation of the effects of weak collisions on F(u). To this end, analytic and numerical investigations of selected models were carried out. In two of these models the exponential formula for energy transfer was assumed. In the third the stepladder model formula was assumed. These two formulas represent quite different energy transfer characteristics, and agreement between them is a reasonable test of the model independence of the quantities in question. 11. Basic Equations
A. Generalization to Weak Collisions. Let the reactant's internal states be partitioned according to energy. Let 6 be the width of any given energy interval, and let E , be an energy within the nth interval. Let cn(t) be the concentration of molecules with energies in the nth interval at time t. The c i s are, under fairly general gas phase conditions, solutions of an equation similar to the master equation of stochastic theory,
where the components of the vector IC) are the c i s and the elements of the matrix A are expressible in terms of rate constants k, for collisional transitions of molecules to interval n from interval m,
0022-3654 /94/2098-87 13%04.50/0 Q 1994 American Chemical Societv
8714 The Journal of Physical Chemistry, Vol. 98, No. 35,1994
Anm=
-I; k,,,
Snider
F(u) = - = 2(f1/'1[Cr + 1 - (C, - l)u]-')fl/')
for n = m
K
KL
=k,
formZn
(10)
where C, is defined by
The concentration CY is that of a bath gas which is assumed to be present in large excess. The matrix K is the diagonal matrix of the micrccanonical rate constants,
c, = (f'/'IC-'lf'/')C
(11)
Clearly, the inverse of C,satisfies the relation
Knm = K J n m where K,, denotes K(E.),6, is the Kronecker 6 and it is implicit that K,, is zero if Enis less than the reaction threshold energy Et. Let nnebe the fraction of molecules which would be contained in interval n if the distribution over energies were a Boltzmann distribution. For vanishingly small interval widths one has
nnc = q-'N(E,) exp( -
$)6
where q is a partition function, N(En)is the density of states, and kTis Boltzmann's constant times temperature. It is a well-known consequence of the principle of detailed balancing that the matrix A(#) with elements
A;:
= n;izA
where
K,(fl/'l(CM1
+ C)-'(f'/')c,
(5)
z = (C, - 1)(C, + 1)-1
(13)
Consulting eqs 6 and 7 and recalling that all of the X i s for I greater than 1 are negative, one concludes that C, and therefore C,, are positive definite matrices. It follows that the matrix 1 uZ is positive definite for all u between -1 and 1 and that its inverse exists. Thus the vector la'/') given by
la'/') = (1
+ Z)-'/2(f1/Z)
(14)
(a112la 1/2 ) = $f'/'l(l 1
+ C;')If'/')
=1
(15)
=0
(16)
and (a'/'(Z(a'/') = $f'/'((l 1 -q')lf'/')
From eqs 10 and 13-16 one finds that F(u) is given in terms of and Z by
F(u) = (a'/'l(l - Z')'/'(l - uZ)-'(l - Z')'/'la'/') (6) (7)
and 1 is the identity matrix. One sees that the vector If1/') has unit norm since the high-pressure limit of K is given by K,
(12)
is well-defined. From eqs 11-14 one readily derives
and C are defined by
If'/') = K2/'K'/Zn'/' l e )
=1
Equation 12 and the unit norm property of lfl/') may be used to verify that F(u) approaches 1 as u approaches f l . Let the matrix Z be defined by
nl/'
mn ne
is symmetric. Let Xland 141)be the eigenvaluesand the normalized eigenvectors of A(*). If the X i s are numbered in descendingorder, then XI is zero and 141) is Jnf/'), the vector with nth component equal to n:'. It was shown in ref 4 that, provided IX.lcMis very large compared to K for all non-zero A,, a good approximation to the rate coefficient which follows from eq 4 is given by K
(f'/'"'lf"')
Equation 17 is eq 2 recast in matrix form. Let ZIand 16) be the eigenvalues and the normalized eigenvectors of Z. From eqs 1 1-1 3 one sees that the 101)'s are the eigenvectors of C and C, as well. Let N be the rank of Z. One recovers eq 2 from eq 17 by spectrally decomposing Z and passing to the continuum limit. The distribution function pa is given by l(a1/'P,)l2
= (nf/'1Klnf/')
p,(z)
= lim N+m
The bimolecular rate constant which applies in the low-pressure limit is found from eq 5 by letting CY in the inner product go to zero,
k, = K,(f'/'I~'(f'/') Setting c equal to cMin eq 1 and using the two foregoing equations, one obtains for u,
Z/+l- Z /
= n!(a'/21(1- ZZ)'/Z[(I - ~z)-'zI" x (1 - uZ)-'(l -Z')'/'~a'/')
The strict Lindemann rate coefficient which yields the same ko and K., is given by (9)
Combining eqs 5,8, and 9, one obtains for the broadening factor as a function of u,
(18)
with z set equal to ZI. As is shown in Appendix A, the pa of eq 18 reduces in the strong collision limit to the pa derived in ref 3. B. Derivative Inequalities. It has already been pointed out3 that eq 2 implies that the second derivative of F(u)is everywhere positive for u between -1 and 1. From eq 17 one finds for the nth derivative of F(u)
dun
1 KL=-(~+I()K, 2
(17)
(19)
One sees by inspection of eq 19 that all of the even derivatives of F are positive for u between -1 and 1. Setting u and n equal to 1 in eq 19 and using eqs 15 and 16, one derives 1 -F'(I) = (al/'l(l - Z)-'(a'/') - 1 2
(20)
where the prime denotes first derivative. Equation 20 is of use
Weak Collision Effects on Broadening Factors
The Journal of Physical Chemistry, Vol. 98, No. 35, 1994 8715
in the interpretation of the numerical results which are presented below. Setting u equal to zero in eq 19 and using eq 16, one obtains dnF du
= -(a'/21~31a1/2) for n = 1 = n!(a1/21(Zn - Z"+')la'/')
In the continuum limit eq 25 becomes
It is a well-known finding' that BEl is appreciably smaller than for n > 1
(21)
Use of Schwarz's inequality along with eqs 15-17 and 21 yields the following inequalities for the first and second derivatives of F a t u = 0,
j3d. This finding has been cause for comment in a number of
articles.5-7 In terms of eq 27 this effect is seen to arise from a relatively heavier weighting of z-values near 1 by the weak collision Pa.
111. Models
-F(O) [ 1 - F(0)] IF'(0) I F(0)[ 1 - F(O)] h2 ( O ) IF(0)[ 1 - F(O)]
(22)
where the double prime denotes second derivative. The foregoing results show that eq 17 implies considerable restriction on the functional form of F(u). C. Broadening Factor and Collision Efficiency. A number of collision efficiency factors have been used to characterize weak collision effects on the kinetics of unimolecular reactions.5 The one which is adopted here, and which is henceforth denoted 3/, is defined as follows: K(W)
= Ks(Pc(J)
(23)
where w is collision frequency and subscript s refers to strong collisions. Since the weakcollision ratecoefficient always reaches a given value a t a higher w, 0, is always less than 1. Assuming that the same proportionality constant relates w and CM both for strong colliders and for weak colliders, one can show that eq 5 yields6
where @&and 0,' are the collision efficiencies in the limits of zero and infinite collision frequency. As seen from eq 23, the weak collision ko is smaller than the strong collision ko by the factor &. As seen from eq 1, u is therefore redefined in the weak collision case. This redefinition absorbs the factor B& so that differences between the strong collision F(u) and the weak collision F(u) reflect only deviations of /3, from its low-pressure limiting value. This property of F(u) should not be perceived as a drawback since the broadening factor is meant to characterize the falloff behavior of the rate coefficient, not its asymptotic behavior at low pressure. Equations 11 and 24 give for the ratio of pCl to 3&!,
Use of eqs 13-1 6 gives
A. Lindemann-Exponential. The exponential formula for the kn,'s has been much used in theoretical and computational studies5-9 of weak collision effects on unimolecular rate coefficients. It was recently showng that an analytic formula for K is forthcoming if one uses for K(E)a formula which gives strict Lindemann behavior in the strong collision limit,"J
where O(x) is the unit step function. For this model the strong collision broadening factor is, by definition, everywhere equal to 1. Thus, the rate equation derived in ref 9 offers a unique opportunity for the investigation of weak collision effects on the broadening factor. If E , and E, are sufficiently large, the k,,'s for the exponential model are given by
The parameter k* is a scale factor which plays no essential role in the theory of the broadening factor. Equation 29 in conjunction with the principle of detailed balancing implies that pc is given by
An equation of this form for pc implies the assumption that the density of states is an exponentially increasing function of E in the vicinity of some reference energy E,,
The average energy transferred per collision (A,??)is related to the parameters a and y by
Equation 3 1 also serves to define the parameter f . The limit of infinite f is the strong collision limit. The rate coefficient for the Lindemann-exponential model is given by eq 31 of ref 9. An alternate derivation is presented here in order to establish a connection with the material in section 11. The elements of the matrix C for the exponential model are known to be given by7
-Pc1 Pco
or, by using eq 20,
where the matrix Y has elements
and
q5
is given by
Snider
8716 The Journal of Physical Chemistry, Vol. 98, No. 35, 1994
$ =22k TL The matrices C and Y are N X N matrices. In the continuum limit N approaches infinity and 4 approaches zero in such a way that N4 is always equal to a large but finite constant; call it 0. For the Lindemann-exponential model the eigenvalues p/ of k*C are related to the eigenvalues yl of Y by
Equation 41 is equivalent to eq 31 of ref 9. In the low-pressure region G is equal to [-I, and eq 41 gives
since all of the K,,'s for this model are equal to KO. Clearly, the eigenvectors 10,) of C are the same as the eigenvectors of Y.Thus one may write eq 5 for this model as
= 8,w
(42)
where the second line of eq 42 is seen by noting that the coefficient of w on the right hand side of the first line approaches 1 as 5-1 goes to zero. The relationship between & and [ in eq 42 is equivalent to the well-known Troe formulas which relates the exponential model's /3& to C. Let the parameter Wbe defined by where the collision frequency w is equal to k * c ~ Passing . to the continuum limit, one obtains
w =8,F'
(43)
From eq 42 one finds
(35) where G and P&)
are given by
8, = (1 - w2
(44)
The limit W + 0 is the strong collision limit, and the limit W - . 1 is the limit of very weak collisions. The parameter W has no particular physical significance, but it does enable one to write the equations which follow in a more compact form. Equations 37, 41, 43, and 44 give for the ratio of K to K",
and
4[1- W ' ( 1 - w 2 G ]
-=
[m+ 112
K,
(45)
Equations 1, 37, and 41-44 give G in terms of u and W, with y understood to be equal toy,. Comparison of eqs 18 and 37 suggests a close connection between pa(z) and Pbb). From eqs 6,28, and 30 along with the definitions of 4 and 0, one obtains for the components of the vector
G=
1
W(1 - u ) u (1 - W2(1 - u )
+ +
From eqs 9,45, and 46 one obtains for the broadening factor as a function of u,
F(u) = Some analytic results pertaining to the eigenvalues of Y are known.11 These results in conjunction with results of numerical diagonalizations of Y's with large 0 and even larger N yield, for the factors which comprise Pb,
8
[dl+ u + (1 - W2(1 - u) +
d1 + u + (1 + W2(1 -u)]2 (47)
Equation 47 indicates considerable asymmetry of F(u) around u = 0. Differentiating eq 47 and setting the derivative equal to zero, one finds for um,the value of u at which F has its minimum
and
Equation 48 confirms the asymmetry of F(u). As the collisions become weaker, the minimum shifts to more negative values of u, approaching -1/3 in the limit of very weakcollisions. Equations 47 and 48 give for the minimum value of F,
Equations 37-39 give
F(um)= 1 - W2 4 = 0 otherwise
(40)
Substituting eq 40 in eq 35 and letting y be equal to 4 cos2(x/2), one obtains
It is noteworthy that F(um) is never the less than 3/4 for this model. From eqs 26 and 41 one derives for the ratio of the limiting collision efficiencies,
Weak Collision Effects on Broadening Factors
The Journal of Physical Chemistry, Vol. 98, No. 35, 1994 8717 TABLE 1: Broadening Factor Minima and Their Locations, Ratios of Limiting Collision Efficiencies, and Moment Ratios for the Extended Kassel-Exponential Model
&1 1 -=fld 1 + wz
In agreement with previous finding~,5-~ this ratio is less than 1, but it never falls below l / 2 . With the K.(S given by eq 28, C, is given by
c, = (1 - @)1+ WY
(49)
where eqs 24, 43, and 44 have been used to cast C, in terms of W. From eqs 13 and 49 one obtains Z in terms of Y,
z = [WY + ((1 - Hy- 1111 [WY + ((1 - w 2+ 1)1]-1 Thus the eigenvectors of Z are also the 18,)’s, and the eigenvalues of Z are given by z, =
[WY, + (1 m y , + (1 -
wz- 11
(50)
wz+ 11
t m
3.0 1.o 0.3 0.1 0.03 0.01 co
3.0 1.0 0.3 0.1 0.03 0.01
3.0 1.o 0.3 0.03 0.01
pa(z) =
d[ 1 + z - ( 1 - W2(1 - z ) ][( 1 + wy(1 - z ) - 1 + z ) ( l - z)3 (1 - wy-1 for < Z
