NOTES
proportionation of ethyl radicals gave small values” ( k ~ / k=~ 1.7, 2.3). Also, the k d / k , ratio was not greatly affected by changing from two tritium-labeled isopropyl radicals to mixed radicals where only one contained tritium. Activation Energy. Three different bath temperature were used to measure any difference in the activation energies of disproportionation and combination. The product distribution was practically the same a t all three temperatures. The propene (m.p. 88°K.) was a liquid a t 90°K. but was solidified a t the other two temperatures. On the basis of our experiments, an activation energy difference as large as 100 cal./mole is unlikely. This is in contrast to a value of 260 cal./ mole reported by Klein, Scheer, and Kelley.’ Both the m a l l primary isotope effect in disproportionation and negligibly small temperature effect on IC&, support the proposition that the transition states for disproportionation and combination are loose and similar. However, the k d / k , ratio is different for different phases (gas and solid), and a distinct difference between the two transition states is therefore indicated. (6) K.W. Watkins and H. C. Moser, J.Phys. Chem.,69,1040(1965). (7) R. Klein, M. D. Scheer, and R. Kelley, ibid., 68, 598 (1964).
1061
It then follows, as is explicitly stated by Forster, that an individual donor molecule decays exponentially, Le. -
P ( t ) = poe
of Excitation Energy‘
by Mira Leibowitz Department of Physics, The Hebrew University, Jerusalem, Israel (Received October 8 , 1964)
In a well-known paper by Forster12a law is derived for the dependence of the fluorescence decay of donor molecules on the concentration of acceptor molecules in solution. A dipole-dipole interaction between donor and acceptor is assumed. The probability p(t) for an individual donor molecule to be in an excited state a t time t after excitation is subject to the equation
where T O is the lifetime of the excited donor molecule in the absence of acceptor molecules, Rk is the distance of the donor molecule from the kth molecule of the acceptor, Ro is a constant depending on the nature of the n’olecule’ is the number Of acceptor nlolecules around the donor molecule.
1
+ k -zN1
( R o / R k ) a j( t / r o )
(2)
It is however, not the decay of a single molecule which is observed, but that of the whole ensemble of molecules with varying distributions of distances R,. Accordingly, Forster averages over all possible values of Rk and obtains
where w(R,)dR, is the probability of finding an acceptor molecule a t a distance between R, and R, dR, from the excited donor molecule. From this expression, the final approximated result is derived. Sveshnikov3 rightly criticizes this approach. He remarks that the above averaging procedure does not account for the different distributions of acceptor molecules around different donor molecules. The proper procedure, according to Sveshnikov, is described by the relation
+
,,(t) = 1 -
no Remarks on Forster’s Theory of Transfer
{
no
N
i-1
k-1
e-(Ro/Rds(t/ro)
e--l/~~
(4)
where nois the number of excited donors and R,, is the distance between the kth acceptor and the ith donor. Sveshnikov treated the problem in a subsequent article4 and arrived by an approximated method a t results similar to those of Forster. I n reply to Sveshnikov’s criticism, Galanins remarks that Forster’s method is justified by the following argument: since a great number of molecules is assumed, one is allowed to convert the sum in eq. 4 into an integral by weighing the integrand by w(Rk)dRk (the distribution function of R,). Then, by changing the order of multiplication and integration, eq. 3 is obtained. Galanin is right in that one may convert the sum in eq. 4 into an integral, but then one must apply the distribution function of the product in (4) (which may be rather complicated) and not that of the individual Rk’s. If one were to follow Galanin’s suggestion, one would average over the donor molecules with the dis(1) Supported by the u. s. Atomic Energy ~ ~ ~Division ~ of i and Medicine. ( 2 ) T.FGrster, 2.Naturfor8ch., 4a3 321 (1949). (3) B. Ia. Sveshnikov, Dokl. Akad. Nauk S S S R , 111, 78 (1956). (4) B. Ia. Sveshnikov. ibid., 115, 274 (1957). (5) V. V. Antonov-Romanovsky and M ,D. Galanin, O p t . i Spektroakopiya, 3 , 389 (1957).
Volume 69, Number 3 March 1966
~
~
1062
SOTES
tribution function of the acceptor molecules, which is not a plausible procedure. Roznian6 indeed applies the proper method, which involves some extended calculations. Remarkably enough, he arrives a t Forster’s final results. Galanin hiniself5J treats the problem by a different approach, averaging already in the differential equation (1). The resulting formula is identical with that of Forster. The question now arises: how is it that Forster’s method, which is open to the above criticism, still yields the correct result? It is the purpose of this communication to clarify this question. Following Galanin’s a p p r o a ~ hthe , ~ decay with time of the number of excited donor molecules n(t) is given by
(say, by some additional mechanism) the integration of the corresponding new equation ( 7 ) would have yielded an integral, not necessarily equal to that obtained by Forster’s approach. I t thus seems, that the proper result obtained by Forster nevertheless does not justify his statistical reasoning. Acknowledgment. The author wishes to thank Dr.
A. Weinreb for helpful discussions. (6) I. M. Rozrnan, Opt. i Spektroakopiya, 4, 536 (1958). (7) M. D. Galanin, Zh. Ekaperim. i Teor. Fiz., 28, 485 (1955).
The Temperature Dependence of the cis-trans Photoisomerization of Azo Compounds : Theoretical Considerations
where f(R) = 1/To(Ro/R)6and N(R,t) is the number of acceptor niolecules whose distance from an excited donor molecule at time t is R. It is assumed that an acceptor molecule, once excited, can exert no further quenching eftict on a donor molecule. Therefore, N(R,t) decays exponentially with time with a rate determined by f(R).
Substituting (6) in ( 5 ) he obtains =
--{:
+ No
lm
e-“”‘f(R)w(R)dR)
n(t) (7)
In integrating this equation, one may change the order of integration with respect to the variables R and t. The result is
This relation developed by Galanin, however, contains an integral which is identical with that of Flirster (eq. 3), and which, after proper approximations, leads to an identical result. The fact that Galanin’s correct approach coincides niatheniatically with Forster’s incorrect approach is a “coincidence,” due to the assumption that the acceptor molecules decay exponentially with the coefficient f(R). This causes f(R) to appear in eq. 7 both as a fartor and as an exponent, and this again causes the result upon integration to be equal to that of Forster. Had we assunled N(R,t) to decrease a t a differerit rate, The J o l ~ f ’ 7 of d Physacul Chrmaatry
Department of Chemistry, University of California, Riverside, California (Received January 18. 1964)
Recent of the temperature and wave length sensitivity of the cis e trans photoisomerization of aromatic azo compounds have revealed that (i) trans, and, &, for the quantum yields, &, for the cis the trans cis photoisomerizations decrease with decreasing temperature, indicating that an activation energy is required for photoisomerization; (ii) the activation energy for the cis + trans transformation is smaller than for the trans + cis transformation; is less than unity a t all tem(iii) the sum of 4c and peratures studied; and (iv) excitation of an azo molecule to its lowest (n-r*) singlet state results in a greater photoisomerization quantum yield than does excitation to the higher lying (r-r*) state. I n an attempt to understand some of these observations we have applied molecular orbital theory (with overlap included) to a calculation of the dependence of the ground state and of the excited (n-r*) and (r-r*) state energies of simple aliphatic azo compounds on the angle of rotation about the N-N bond axis. I n this note we present the results of these calculations and show how they offer a possible theoretical explana-
-
N ( R , ~= ) Noe-f(R)l
k?d!dt
by David R. Kearns
-
(1) P. B. Birnbaurn and D. W. G. Syles, Trans. Faraday ~ o c . 50, , 1192 (1954). (2) G . Zirnmerman, L. Chow, and E. Paik, J . A m . Chem. SOC.,8 0 , 3525 (1955).
(3) E. Fischer, ibid., 82, 3249 (1960). (4) J. Malkin and E. Fischer. J . Phya. Chem., 66, 2482 (1962).
