J . Phys. Chem. 1989, 93, 1681-1683
1681
Monoenergetic Unimolecular Rate Constants and Their Dependence on Pressure and Fluctuations in State-Specific Unimolecular Rate Constants Da-hong Lu and William L. Hase* Department of Chemistry, Wayne State University, Detroit, Michigan 48202 (Received: September 9, 1988; In Final Form: December 2, 1988)
The Porter-Thomas distribution function P ( k ) is used to represent fluctuations in state-specific unimolecular rate constants for states that lie within a narrow energy interval E E + dE. The sensitivity of concentration of reactant molecules versus time N ( r , E ) ,the time-dependent rate constant k(r,E),and the collision-averaged rate constant k(w,E) to the width of P ( k ) is considered. The collision-averaged rate constant k(w,E), found from an analysis of a Stern-Volmer plot, is found to be rather insensitive to P ( k ) . -+
A number of theoretical studies'" have indicated that molecules with covalent intramolecular potentials will exhibit significant state specificity in their unimolecular decomposition rates, when they are excited to individual quantum mechanical vibrational/rotational levels. As a result of recent experimental advances, it has become possible to observe this phen~menon.~-'~ Many molecules have a large number of states within an energy interval E E dE, and the above results indicate that states within this energy interval may exhibit a broad fluctuation in unimolecular rate constants ki. The topic of this Letter is to study how these fluctuations in ki,represented by the distribution P ( k ) , affect the time-dependent monoenergetic unimolecular rate constant k(7,E) and the phenomenological collision-averaged monoenergetic unimolecular rate constant k(w,E). The latter rate constant is measured from a Stern-Volmer analysis of chemical activation, radiationless transition, or overtone excitation experiments. The work presented here complements the recent study by Miller," where the effect of fluctuation in ki on the Lindemann-Hinshelwood pressure-dependent unimolecular rate constant kUi(w,E) was considered. The Porter-Thomas distributionI2 is used for P ( k ) ; i.e.
-
+
-
For v
> 2 there is a maximum in P(k) located at15 (3)
There is not a maximum for v equal to 1 and 2, and the largest value for P ( k ) is at k = 0. The population of reactant molecules within the energy interval E E d E is given by
-
+
N(T,E) = Noxmexp(-kT) P ( k ) d k
(4)
where exponential decay is assumed for each state and No is the total number of reactant molecules. For the Porter-Thomas P(k) distribution N(T,E)/No = (1 -k 2ET/V)-v/2 (5)
-
Miller" has shown that, in the limit u m, the right-hand side of eq 5 becomes the simple exponential exp(-ET), as predicted by RRKM theory.16 For v finite, the decay is nonexponential. exp(-&)] is This is illustrated in Figure 1, where N(7,E)/[No plotted versus ET. (A similar analysis has been given by Miller.") m limiting value of the term N(T,E)/[N~ exp(-&)] is The 7 infinity for finite v and unity for infinite v. The 7 = 0 limiting value of this term is one and independent of v. The unimolecular lifetime distribution P(T,E) is related to N ( T , E )vial7
-
where is the average state-specific unimolecular rate constant E dE within the energy interval E
+
which, for the Porter-Thomas distribution, is P(T,E) = &(I 2kT/V)-[(Y/2)+11 and v is the "effective number of decay channels". This distribution is borrowed from the field of nuclear physicsI2 and has been used in other studies of state specificity in unimolecular decomposi(1) One of the earliest theoretical analysis of quantum mechanical state specificity in unimolecular decomposition was by Mies and Kraus: Mies, F. H.; Kraus, M. J . Chem. Phys. 1966,45,4455. Mies, F. H. J. Chem. Phys. 1969, 51, 787, 798. (2) Waite. B. A.: Miller. W. H. J . Chem. Phvs. 1980. 73. 3713: 1981. 74. 391b.' Waite,'B. A.;'Gray, S. K.; Miller, W. H. j . Chem. Phys. 1983, 78, 259: (3) Miller, W. H. Chem. Reu. 1987, 87, 19. (4) Bai, Y. Y.; Hose, G.; McCurdy, C. W.; Taylor, H. S. Chem. Phys. Lett. 1983, 99, 342. ( 5 ) Hedges. Jr.. R. M.: Skodie. R. J.: Borondo.. F.:. Reinhardt. W. P. ACS Symp. Ser.-1984, No. 263, 323: (6) Swamy, K. N.; Hase, W. L.; Garrett, B. C.; McCurdy, C. W.; McNutt, J. F. J . Phys. Chem. 1986, 90, 3517. (7) Dai, H. L.; Field, R. W.; Kinsey, J. L. J. Chem. Phys. 1985,82, 1606. (8) Schubert, U.; Riedle, E.; Neusser, H. J.; Schlag, E. W. J. Chem. Phys. 1986, 84, 6182. (9) Guyer, D. R.; Polik, W. F.; Moore, C. B. J. Chem. Phys. 1986, 84, 6519. (10) Engel, Y. M.; Levine, R. D.; Thoman, Jr., J. W.; Steinfeld, J. I.; McKay, R. I. J . Chem. Phys. 1987, 86, 6561. Thoman, Jr., J. W.; Steinfeld, J. I.; McKay, R.;Knight, A. E. W. J . Chem. Phys. 1987, 86, 5909. (11) Miller, W. H. J . Phys. Chem. 1988, 92, 4261. (12) Porter, C. E.; Thomas, R. G. Phys. Reu. 1956, 104, 483.
0022-3654/89/2093-1681$01 S O / O
-
+
(7) For the v limit, P(7,E) becomes the RRKM exponential form16 P ( T , E ) = E exp(-&). The time-dependent unimolecular rate constant k(7,E) is related to N(7,E) by
For the Porter-Thomas P ( k ) distribution k(7,E) = E/(1 2kT/V)
+
(9) At the limit 7 = 0, k(7,E) equals E. For finite v, the 7 - 0 3 limiting value of k(7,E) is zero. If v is infinite, k(7,E) is independent of 7 and equals E. (13) Lu, D.-H.; Hase, W. L. J . Chem. Phys., accepted for publication. (14) For a review see: Levine, R. D. Ado. Chem. Phys. 1988, 70, 53. (1 5)-For the modified Porter-Thomas distribution given by kP(k), ,k equals k for all v. See ref 14. (16) (a) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley: New York, 1972. (b) Forst, W. Theory of Unimolecular Reactionr; Academic Press: New York, 1973. (c) Hase, W. L. In Dynamics of Molecular Collisions; Part B; Miller, W. H., Ed.; Plenum Press: New York, 1976; p 121. (d) Hase, W. L. Chem. Phys. Lett. 1985, 116, 312. (1 7) Bunker, D. L. Theory of Elementary Gas Reaction Rates; Pergamon: Oxford, 1966; p 48.
0 1989 American Chemical Society
1682 The Journal of Physical Chemistry, Vol. 93, No. 5, 1989
Letters
10.0 1 8.0 -
1.0
0.0
2.0
3.0
0.0
1 4.0 5.0
I
~
-2.0
-1.0
k-tau m
(solid curve).
Studies of unimolecular reactions involving chemical activation,I8 radiationless transition^,'^ and overtone excitationm are often interpreted by the mechanism18 A*
k(w,E)
-exp(-x)
where A* is monoenergetically excited. The phenomenological collision-averaged monoenergetic unimolecular rate constant k ( w , E ) is given by D k ( w , E ) = w(10) S
... +
[-i+l+(-;
(11)
where P(7) is the lifetime distribution, eq 6 , and W(7)= exp(-wr) is the probability the reactant avoids a collision for time 7 . 2 3 S is simply equal to 1 - D. Inserting these expressions for D and S into eq 10 yields (12)
limiting values of k(w,E) have The general w 0 and w been given p r e v i o ~ s l y . ~ ~ For the Porter-Thomas P ( k ) distribution the analytic form of k(w,E) depends on whether v is odd or even; i.e. 2R exp[-vw/(2R)] even v: k(w,E) = -w (13)
x(-u/2)+2
(18) Rabinovitch, B. S.;Setser, D. W. Adu. Photochem. 1964, 3, 1 . (19) Hippler, H.; Luther, K.; Troe, J.; Wendelken, H. J. J . Chem. Phys.
1983. 239. -.- -, .79. ., -- .
(20) Crim, F. F. Annu. Rev. Phys. Chem. 1984, 35, 657. (21) See ref 16b, p 228, and ref 16c, p 150. (22) Hase, W. L.; Duchovic, R. J.; Swamy, K. N . ; Wolf, R. J. J . Chem. Phys. 1984,80, 714. (23) Slater. N. B. Theory of Uimolecular Reactions; Cornell University Press: Ithaca, NY, 1959; p 1 9 . (24) Marcus, R. A,; Hase, W. L.; Swamy, K. N. J . Chem. Phys. 1984,88, 6717. (25) Handbook of Mathematical Functions; Abramowitz, M., Stegunm, I . A., Eds.; Dover Publications: New York, 1965; p 228.
(- ;+
1 ) ...(-
I+
-
+
l m e - p p 1 / 2dp
;) (-;
(15)
+ 1)...(-
L m e - @ p - 1dp / 2 = x1l2[1 - erf (x112)]
-
;)
(16)
--
Equations 13-16 result in rather simple expressions for k(w,E) in the w m and w 0 limits: For the w limit, k(w,E) is independent of v and equals k . At the w 0 limit, k ( o , E ) depends upon the value of v and the following expressions are obtained: v = 1 and 2: v
k(0,E) = 0
> 2 and finite:
k(0,E) = (
v-2 ~
(17) -
)
k
(18)
-
It should be noted that k(0,E) in eq 18 is the same as k,,, in eq 3. That is, for v > 2 and finite, the value of k(w,E) in the w 0 limit is the same as the value of k where the P ( k ) distribution has its maximum. For v = 1 and 2, where there is not a maximum in P ( k ) , k(0,E) equals the value of k where P ( k ) has its largest value; i.e., k = 0. In recent experimental studies26 it has been found that state-specific rate constants for H2C0 H2 CO dissociation are well-described by the Porter-Thomas distribution with v = 4. For this case k(w,E) varies from L to 1 / 2 between and w 0 limits, respectively. This is not a substantial the w change in k ( w , E ) . To find k(w,E) in the v m limit, it is convenient to begin with eq 10 and 11 and insert the Porter-Thomas-N(~,E) for the v m limit. One then finds that D = k / ( w k ) and S = w / ( w + R ) , so that
-
-
In eq 13 EVl2[(v/2)(w/R)] is the exponential integral.25 The
x-W
where x = vw/(2k) and
and is often determined experimentally from a Stern-Volmer plot. If one makes the strong-collision assumption of unimolecular rate theory,I6 D is given by21*22
’
2.0
+ l)(-; + 2)
2 stabilized reactant A (S)
k ( w , E ) = N o / [L m N ( 7 , E )exp(-w~) d r ] - w
7
integral in the denominator of eq 14 is evaluated by using integration by parts to give x(-v/2)+1
D = xm 0 W ( 7 ) P(7) d r
,
Figure 2. Plot of k(w,E)/E versus natural logarithm of w / f . The curves are defined in Figure 1.
l m p - y / 2 e - pdp =
decomposition products (D)
- -
I
1.0
Ln ( w h
Figure 1. Plot of N(7,E)/[N0 e x p ( - k ) ] versus Lr for v equal to 1 (long-dashed curve), 2 (dotted curve), 3 (dashed curve), 10 (chain-dashed curve), and
I
0.0
-
-
+
v
-
m:
k(w,E) =
k
+
-
(19)
The sensitivity of k(w,E) to u, “the effective number of decay channels”, is shown in Figure 2. The remarkable aspect of these plots is the rather small deviations of k(w,E) from k . Only for w
