J . Phys. Chem. 1992, 96, 8084-8089
8084
(22) C h e n t , E.; Sander, L.; Kopelman, R. Phys. Rev. A 1989.39.6455. (23) Lindenberg, K.; West, B.; Kopelman, R. In Noise and Chuos in Nonlinear DyMmical System; Moss,F., Lugiato, L., Schleich, W., Eds.; Cambridge University Press: Cambridge, 1990; p 142. (24) Li,'L.; Kopelman, R. J. Phys. Chem., submitted for publication. (25) Hoeben, J.; Kopclman, R. J. Chem. Phys. 1976,65,2817. Kopelman, R.; Argyrakis, P. 1.Chem. Phys. 1980, 72, 3053.
(26) Havlin, S.;Ben-Avraham, D. Adv. Phys. 1987, 36, 695. (27) Argyrakis, P.; Kopelman, R. Phys. Rev. A, in press. (28) Hertz, P. Murh. Ann. 1909, 6, 387. (29) LI. L.; Clh", E.; Argyrakis, P.; Harmon, L. A.; Parus, S. J.; Kopelman, R. In Fractal Aspects ofMureriuls: Disorderedsysrems; Weitz, D. A.; Sander, L. M., Mandelbrot, B. B., Eds.; Material Research Society: Pittsburgh, 1988; p 21 1 .
Rate Equations for the Photochemical Isomerization of Bicyciobutane Gary D. Bent Physics Department, University of Connecticut, Stows, Connecticut 06269 (Received: May 8, I992; In Final Form: June 22, 1992)
This paper derives and solves the rate equations for several excited states forming several different products. These equations are then applied to bicyclobutane in which the 3P Rydberg states of bicyclobutane were determined in a previous paper to be the progenitors for the photochemicalformation of butadiene and cyclobutene. The llBzstate is deduced to form &butadiene and cyclobutene, while the 3'Al and l'Bl states form fruwbutadiene. The rate equations are used as a theoretical framework in which to compare computed values with experimental data. Under a set of assumptions, the reaction rates for the formation of these isomers are either calculated or bounded.
lntrodpetioa Bicyclobutane has been an intriguing molecule to study for years because it is the simplest of a series of highly strained molecules and because it forms the isomers cyclobutene and butadiene. The thermal isomerization of bicyclobutane has been extensively investigated. In recent years, the photochemical isomerization has also attracted attention. Becknell et al.' irradiated bicyclobutane at 185 nm and observed the production of butadiene and cyclobutene in the ratio 10:1. Becknell, in his dissertation,* revised this ratio to 13:l. Adam et al.? using the same solvent and wavelength, produced butadiene and cyclobutene as primary products but essentially in the ratio 1:l. Becknel12 suggested that these workers may have been working with a bicyclobutane sample contaminated with cyclobutene. Both sets of investigators postulated the sequence of reactions in Figure l a as providing the pathway by which butadiene was formed while the sequence for the pathway of cyclobutene postulated by Adam et al.' is shown in Figure lb. In addition, Becknell et al.' postulated the pathway shown in Figure IC as an alternative possibility for the formation of butadiene. By labeling positions in bicyclobutane with deuterium or carbon 13, they were able to show that both reactions a and c of Figure 1 provide pathways for forming butadiene in the ratio 2:l. The possibility for more than one excited state participating in thae reactions is clearly present and was mentioned by Becknell et al.' Recently, Walters et a1.4 and Bent and Rossi5published ab initio calculations of the ground state and low-lying excited states of bicyclobutane. There is substantial agreement between the two sets of calculations. This paper follows up important conclusions of the Bent and Rossi paper. In the next section, the calculations of Bent and Rossi on the low-lying excited states are summarized. In the following sections, the rate equations for the photolysis of bicyclobutane are derived and solved. The experimental data from Beclmel12and the &tor strengths calculated by Bent and R w i 5 are used in the rate equations to predict the reaction rates for a particular excited state to form a particular isomer.
experiments of Becknell et al.,' Becknell,2and Adam et al.3 It is not surprising that these states are clustered close together since the 2lA1state is essentially a Rydberg state of S symmetry derived from excitation to a 3s orbital while the l1B2,3'AI, and l1B1states all have symmetry of P Rydberg states involving excitation to 3p, 3p, and 3px orbitals, respectively. In an atom, all these states would bc degenerate, and, in bicyclobutane, the states of P symmetry are nearly degenerate. The 2lA1state was eliminated from consideration as one of the states leading to the products in Figure 1 because its oscillator strength (ft(r) in Table I) is too low, and it is farther off-resonance than the other states. The results of Bent and Rossi indicate that the 1lBZ,3.1Al, and l'Bl all could be involved in the formation of butadiene and cyclobutene. They used the orbital plots of the highest occupied molecular orbitals and the Mulliken population analyses to predict that the l'Bz state would form butadiene via the pathway of Figure l a and cyclobutene via the Figure l b pathway and that the 3IAI and l1B1states would form butadiene via the pathway of Figure 1c. Figure 2 shows the scheme predicted by Bent and R w i for the photochemical isomerization of bicyclobutene to butadiene and cyclobutene. Figure 2 includes the symbols that will be used in the photochemical rate equations.
The Low-LyingExcited States of Bicyclobutane Table I shows the vertical excitation energies and the oscillator strengths of the highest occupied molecular orbitalsfor the ground state and the four lowest singlet excited states as calculated by Bent and Rossi.5 As can be seen in Table I, all four excited states are accessible from irradiation at 185 nm (6.7 eV) used in the
Rate Equtiom The rate equations will be derived for the experimental conditions used by Becknell.2 However, those conditions should be typical for most photochemical experiments done in solution. The bicyclobutane sample is an optically thick target. Then, the intensity of the 185-nm line, I(x,t), that passes though such a
0022-3654 f 92f 2096-8084503.00 f 0
TABLE I: Vertical Exdbtioa EJIW&.S (in eV) rad Oscillator St", 6(r), for tbe Ground rad Low-LyhgRydberg States of B i W C W
state designation
state name
g
1'Al 2'A1(38) 1'B2(3py) 3'A1(3pz) 1'Bl(3px)
1 2 3
mi
A(r)
0.0 6.42
6.89 6.91 7.02
3.9 X 3.1 X 1.7 X 3.5 X
10-4 lo-' (4.5 X lo-') lo-' (2.4 X lo-') lo-' (5.0 x
a Values are from ref 5. The values in parentheses are the adjusted oscillator strengths so that Cifi = 7.4 X lo-' which is the value measured by Wiberg et al. (ref IO).
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 8085
Photochemical Isomerization of Bicyclobutane
ki, the reaction rate for forming butadiene according to Figure IC, k,,the reaction rate for forming cyclobutene, and kix is the rate of formation of any other product which depopulates the ith state. In the above equations, it is assumed that each excited state can contribute to all photochemical products. In eqs 1-3, Z ( t ) is the average light intensity over the path length, L,at time t given by
= (L)-l
Z(t)
Figure 1. Possible reaction pathways for the photochemical transformation of bicyclobutane to (a) cis-butadiene, (b) cyclobutene, and (c) tram-butadiene.
J-'Z(X,t) 0
dx
(4)
The 11B2,3IAI, and llB1states have excitation energies within about 0.1 eV of each other. Hence, the intensity Z(t) is taken to be the same for each state. The Au appears in eqs 1 and 2 h u e I ( t ) is the intensity of the complete 185-nm line coming from the mercury lamp used in the experimcnh2 What is needed in the above equations is the intensity at 185 nm per unit frequency interval, I(v*,t). However, Z(t)
=
~ v z Z ( u , tdv )
= Z(v*,t)Au
(5)
VI
Hence I(v*,t) = Z(t)/Au. These equations can be solved by using the steady-state assumption' and setting dn,/dt = 0 which yields ng(0 = Ng exp[-G(t)l
A Figure 2. Energy level diagram for the photochemical isomerization of bicydobutane according to the model given in the text. The B;s are the
Einstein absorption coefficients; the A;s are Einstein spontaneous emission coefficients; klb is the reaction rate for state 1 (1'B2) to form cis-butadiene according to the pathway of Figure la; kl, is the reaction rate for state 1 to form cyclobutene according to the pathway of Figure lb; kh is the reaction rate for state 2 (3'AI) to form tram-butadiene according to the pathway of Figure IC; and k3,is the reaction rate for state 3 (1 IBI) to form tram-butadieneaccording to the pathway of Figure 1c. sample is a function of time, t, and of the position along the path length in the sample, x. The number of bicyclobutane molecules in the ground state in a unit volume centered on x at a time t is n,(x,r). However, the sample solution is magnetically stirred2 so that, on the order of seconds, the distribution of ground-state reactant molecules, excited-state reactant molecules, and product molecules becomes homogeneous. The time steps of interest are on the order of minutes and hours. Thus n,(xl,t) = ng(xz,t)for any xIand x2 and n,(x,t) = n,(t), a function independent of x. Thus, the rate equations for the photochemical reactions in bicyclobutane can be written as dng/dt = -(CB/CAV)Z(~) n g ( t ) C f i + C ( A i + kig)ni(t) (1) i
dni/dt = (C~/cAv)fiZ(t) n,(t)
i
where Ngis the number of bicyclobutane molecules per unit volume at t = 0 and C(t) = ( C B / ~ A v ) F0S ' Z ( tdt' I)
(7)
where F = CfiKi/(Ai i
+ kig + Ki)
(8)
The other solutions are ni(t)
= (CB/cb)NgZ(t)W/(Ai+ kig + Kill exp[-G(t)l
(9)
for i = 1, 2, and 3 and nq(t) = (Fq/F)Ng[l - ex~(-G(t))l, Q
b, si c
(10)
where Fq = W;'kiq/(Ai + kig + KO i
(1 1)
The exponential function, G(t), is discussed in the Appendix. The general formula for quantum efficiency for each state is given by8
+ kig + Ki)
kiq/(Ai
4iq
(12)
where q = b and s represent values for forming butadiene by the two different pathways and q = c represents cyclobutene formation. Equation 11 can be recast as
- (Ai + kig + Ki)ni(r) (2)
for i = 1, 2, and 3, and
(6)
Fq
(13)
Z M i q i
The quantum yield for each product is
dnq/dt = Ckiqni(t), q = b, s, c i
(3)
In eqs 1-3, ni is the number of molecules per unit volume in the ith excited state of bicyclobutane, nb the number per unit volume that have formed the isomer butadiene in accordance with Figure la, n, the number per unit volume which have formed butadiene in accordance with Figure IC, and n, the number per unit volume which have formed cyclobutene. The parameterf, represents the oscillator strength for the ith state, CB is a constant that converts oscillator strength to the absorption coefficient, Au is the frequency interval at half-maximum for the 185-nm line, c is the velocity of light, Ai is the coefficient for spontaneous emission for the ith state, and kig is any other path, such as internal conversion or intersystem crossing, that returns the ith state to the ground state. The quantity Ki is given by Ki = kfb ki, + kic + kix,where kh is the reaction rate for forming butadiene according to Figure la,
+
@q
= Udiq/Xh 9 = b, i
i
8.C
(14)
sincefilzfr is the fraction of absorbed photons in the ith state. The oscillator strengths, fi, should include the Franck-Condon factors for the vibrational overlaps in the transitions from the ground to excited states. Bent and Ross? showed that the Stokes shifts are approximately the same for the 2lAl, l'Bz, and llB1 states of bicyclobutane which have similar geometrical structum. Thus, the Franckxondon factors should be nearly the same for all the Rydberg 3P states. The same Franck-condon factor would appear in the numerator and denominator of eq 14 and, therefore, cancel. The ratio of oscillator strengths can be used as the ratio calculated from electronic wave functions independent of Franck-Condon factors. From eq 10, the ratio of the number of butadiene molecules formed from Figure la, h(t),and Figure IC, n&), to the number
8W6 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
of cyclobutene molecules, n,(t), is [nb(t) + %(t)l/nc(t) = (Fb + Fs)/Fc
(15)
and the product ratio for the two possible pathways for forming butadiene is nb(t)/%(t) = F b / F s
(16)
However, from eq 14, the quantum yields can be written as @q
= Fq/n,,
= b, S, c
(17)
I
Thus (ab + *s)/*c
= (Fb + Fs)/Fc
(18)
= Fb/Fs
(19)
and *b/*s
Becknel12measured the quantum yields to be @b = 0.33, CP, = 0.17, and CP, = 0.035. From eqs 18 and 19, (Fb
+ F,)/F,
14.3
(20)
= 1.9 (21) From eqs 15 and 16 and the measurements of Becknel12 (22) (Fb + F,)/F, = 13 Fb/F,
Fb/F,
=2
(23)
Hence the data and the equations are not entirely consistent, but the differences are within the cited experimental error2of &lo%. In the following, the ratio (Fb + F,)/F, will be taken as 13.6 and the ratio Fb/F,as 2.
Analysis of the Rate Equations As an extension of his bicyclobutane work, Becknel12measured the products from the photolysis of exo,exo-2,4-dimethylbicyclobutane (i.e., bicyclobutane with a methyl group substituted for a hydrogen on each of the nonbridging carbons). He found that the quantum yield for the analog of the cyclobutene product increased and that the quantum yield for the analog of the cisbutadiene product of Figure l a decreased, while the sum of the two quantum yields was approximately the same as when bicyclobutane was photolyzed. The quantum yield for the analog of the trans-butadiene of Figure IC was not affected. Methyl substitution obviously perturbs the excited states so that production of cyclobutene is more likely and the production of cis-butadiene less likely. However, methyl substitution does not noticeably perturb the excited states for the production of trans-butadiene. These observations point to different excited states being involved in the production of trans-butadiene than in the production of cis-butadiene and cyclobutene. For the moment, let us label as A and B the excited states that form cis-butadieneand cyclobutene and as C and D the excited states that form trans-butadiene. Since there are only three excited states, one of these states does not exist or is the same as one of the others. It certainly is possible that two excited states, C and D, could form trans-butadiene (Figure IC) and not be noticeably perturbed by methyl substitution. It is easy to believe that one excited state (A or B) could be perturbed by methyl substitution so that the production of cyclobutene is enhanced with a corresponding decrease in the production of cis-butadiene (Figure la). It is less likely that two excited states are perturbed so that these conditions are met. Hence, the methyl substitution suggests that only one excited state (call it A) forms the isomers cyclobutene and cisbutadiene. State B then does not exist. These conclusions are very similar to the ones of Bent and Rapsis who reasoned from orbital plots and population analyses. From the work of Bent and Rossi, state A was identified as the 1 'Bz state and states C and D as the 3'A1 and I'BI states. Becknel12 measured the gas-phase molar absorptivity, c, of bicyclobutane at 185 nm to be 4816 L mol-' cm-I. The oscillator strength, J of the transitions at 185 nm are related to t by9
Bent f = 4.32 x 1 0 - 9 € ~ ~ (24) where Aii is the full width of the wave number interval at half of c. From the bicyclobutane absorption spectrum of Wiberg et a1.,I0 Aii E 6000 cm-'.Thus, eq 24 yields f = 12.5 X 1W2. This should be compared to the C& from Table I; C& = 5.2 X and the measured value of Wiberg et al.,1° f = 7.4 X lo-*. The value of Wiberg et al. is probably the most accurate since it was obtained by integrating over the bandwidth. All the photochemical work has been done in solution. Backnel12 measured the molar absorptivity at 185 nm of bicyclobutane in solution to be c = 3840 L mol-' cm-'. From the absorption spectrum of Becknell? Aii E 7000 cm-' for bicyclobutane in solution. Equation 24 gives an oscillator strength in solution of f, = 11.6 X Thus the oscillator strength calculated from the molar absorptivity should be taken a s f = 12 X 10-2 whether in the gas phase or solution. The oscillator strengths,&, which will be used in the equations below are the ones given in parentheses in Table I. These have been adjusted so that CJ; agrees with the value measured by Wiberg et a1.I0 From eq 13 and the above assignment of states, Fb F, =f,dlC;and F, =fi& The ratios (Fb + F,)/F, and Fb/Fs can be expanded as f l 4 l b +f242s +f343s
= 13.6fdlc
(25)
h4lb = 2(f242s +f343s)
(26)
From eq 14 and the measurements of Becknell?
=
[email protected] 9
= [fl(4lb + 4lc) + f 2 6 2 s +h43sl/u, i
(27)
Using the oscillator strengths given in parentheses in Table I, eqs 25-27 yield 4 l b = 0.547, &,= 0.060, and 2442,
+ 543, = 12.3
(28)
From eq 12, 4 l b and 41care given by
+ klg + klb + klc + klx) = 0.547 4lc = kl,/(Al + klg + klb + klc + k~x)= 0.060
4lb
= klb/(Al
(29) (30)
Equations 29 and 30 can be solved for klb and k l , if k l ,
