COMMUNICATIONS TO THE EDITOR
3364 Experimental Method for Determining the Intersystem Crossing Rate Constant from Lowest Excited Singlet to Lowest Triplet State
Sir,: One of the most important, and experimentally elusive, parameters of intramolecular energy transfer is the rate constant for intersystem crossing from lowest excited singlet to lowest triplet state, k18c.l The usual experimentally determinable parameters such as the fluorescence quantum yield, $F, or experimental fluorescence lifetime, 7Fe, contain the rate constant for internal , the conversion from excited to ground singlet, k ~ c and natural rate constant for fluorescence, ~ F Oas well as ~ I S C . Therefore, a t best these experimental data can give only a maximum value of k m . This follows from
k ~ "= = l/(kFo
7Fe
(1)
$F/TF'
+ kISC +
(2)
kIC)
and ~ I S C= ~ / T F :
-
Then, since k10 limit on k18c
~
F
-~
O
I = C
> 0, setting < (1 -
k~sc
(1 k10
$F)/TF~
=
-~
I C (3)
0 gives the upper
d[Til/dt = I&'ISC?F~ - [Ti1/7pe - IT
[ T , ] / ~ F ~(6) '
where I,, is the rate of absorption into the So 4SI band, ITis the rate of absorption from TI --t T,, T, is some higher triplet state, T F ~ 'is the experimental fluorescence lifetime for T, --t TI emission, and rpe is the experimental phosphorescence lifetime. Both of the last two terms in eq 6 are negligible and/or cancel one another under the assumed conditions of low T, monitoring light. I n this case, after an excitation time, t >> 7 p e [Tllss = T P ~ T F ~ ~ ' I S C I ~
(4) Based upon quantum yield data, this approximation seems to be a good one for some aromatic hydrocarbons,2 but a more accurate and quantitative method of obtaining k ~ s cis desirable in order to study its dependence upon temperature, solvent media, and molecular properties. We wish to propose an experimental method of obtaining numerical values of k ~ s oinvolving optical density changes occurring during triplet-triplet absorption under steady-state conditions. Since many aromatic hydrocarbons and heterocyclic compounds exhibit triplet-triplet absorption, a this method should be applicable to a wide range of compounds. The equations derived here yield a rate constant ~ ' I S C which we will call the apparent k 1 8 ~constant. This apparent rate constant is equal to kIsc if only one triplet level lies below the lowest singlet. It is a sum of lowest singlet to triplet rate constants for other cases, which will be described in a later paragraph. The analysis given here is not directly applicable to heterocyclic compounds where n,n* states lie below the n , ~ *states. However, the approach given here may easily be extended to such cases. The optical density change, p(X), due to triplettriplet absorption from the lowest triplet state, TI to a higher triplet state, T,, at a wavelength X is +F)/TF~
P ( X ) = (T(X)I[TI]
(5)
where ET(X) is the triplet-triplet extinction coefficient a t wavelength A, I is the optical path length, and [TI] is the concentration in the lowest triplet state, TI. We The Journal of Physical Chemistry
shall assume4that steady-state conditions hold with respect to both SI and TI, (excitation time= 10 phosphorescence lifetimes), that no photochemical decomposition occurs, and that no bimolecular quenching occurs. We further assume that all of the exciting light (corresponding to ground singlet-lowest excited singlet absorption only) is absorbed by the sample, that direct population of T1by absorption from the ground state is negligible, and that the rate of absorption of the optical density measuring light, IT, i s very small compared to rate of absorption of exciting light, I,. The differential equation for the lowest triplet state population with respect to time is then
(7)
then P (A) 88 = ET( X)TPeTFek'ISCIal
(8)
and
7PeTFeIal (9) IC'ISC may be determined from eq 9 since all quantities on the right can be experimentally measured. The most difficult quantities to obtain are ET@) and T B ~ . However, recently, methods of obtaining T R ~ and ~ ~ E T ( X ) ~ have been improved. It should also be noted that ~ ' I S C can be calculated from measurements of ~(X)SS and ET(X) at one selected wavelength. Even if only a maximum value of ET is available, use of the eT(max) in eq 9 will yield a minimum value of ~ ' I S C . This minimum, together with the maximum from eq 4, will define the range of ~ ' I S C . k'IBC
=
P(X)SS/ET(X)
(1) See, for example, N. J. Turro, "Molecular Photochemistry," W. A. Benjamin, Inc., New Y o r k , N. Y., 1965, p 74, and S. K. Lower and .M.A. El-Sayed, Chem. Rev., 66, 199 (1966). Usually, estimates of krsc are made by assuming k ~ is c negligible or by comparing $F
+ $T (triplet state quantum yield) with unity.
(2) E. Lim and J. Laposa, J . Chem. Phys., 41, 3257 (1964). (3) (a) D. P. Craig and I. G. Ross, J. Chem. Soc., 1589 (b) B. R. Henry and M. Kasha, J . Chem. Phys., 47, 3319 (0) G. Porter and M. W. Windsor, PTOC. Roy. SOC.,A245, 238 (d) D. S. McClure, J . Chem. Phys., 19, 670 (1951). (4) All of the assumptions are controllable by experimental
(1954); (1967); (1958);
design, use of proper filters, light intensities, etc. ( 5 ) (a) See I. B. Berlman, "Handbook of Fluorescence Spectra of Aromatic Molecules," Academic Press, New Y o r k , N. Y . , 1965, p 231, for a bibliography of methods; (b) W. R. Ware and B. A. Baldwin, J . Chem. Phys., 43, 1194 (1965). (6) (a) R. A. Keller and S. G. Hadley, {bid., 42, 2382 (1965); (b) P. G. Bowers and G. Porter, Proc. Roy. Soc., A296, 348 (1967); (0) W. R. Dawson, J . Opt. SOC.Am., 58, 222 (1968).
e.g.,
,
~
COMMUNICATIONS TO THE EDITOR -
3355
~
~~
Table I: Values of Intersystem Crossing Rate Constants for Aromatic Hydrocarbons a t 77°K P (A)
einstein-1, sec cmz
x
Compound
Naphthalene Naphthalene-& Phenanthrene Triphenylene
10-
0.18
€T,a
mol-1 om-’1. x 10-4
1.3% 0.87 2.16 0.78
0.97 0.65 1.OS
me,d
k&IBC,@
TPe!e
nsec
sec
OF!
96‘ 96‘
2.25 18.3 3.68 14.8
0.39 0 . 40e 0.14 0.06
71’ 37’
aec-1
X 10-8
6 . 4 f2 . 0 6 . 3 i:2 . 0 12 2 ~ 3 . 0 26 f 8 . 0
Max P I E O . ~ sec-1 X 10-8
6.4 f 1.0 6.4 f1.0 12 3 t 2 . 0 26 f 3 . 0
Reference 6a. Data obtained:& 77” K in 1-butanol-isopentane glass (3:7). e~ were recalculated using the more accurate T F ~and values in this table. Estimated errors by comparison with data in ref 3a, c, and 6b are 2075, except for triphenylene which may J. D. Laposa, E. C. Lim, and R. E. Kellogg, J . Chem. Phys., 42, 3025 (1965), in EPA a t 77°K. Estimated error have a 30% error. Data required a t Reference 5a, obtained in deoxygenated cyclohexane at room temperature. Estimated error is 5%. is 10%. 77°K are assumed to be same as those obtained a t room temperature. This probably introduces a maximum error of 15%. e Reference See A. A. Lamola and G. S. Hammond, J . Chem. Phys., 43, 2129 (1965), for references and discussion of the 2, in EPA a t 77°K. From eq 9. Estimated errors propagated from other reported values in EPA a t 77°K. These values are probably reliable to f0.02. parametric errors are listed. * From eq 4. Estimated errors are 15% except for triphenylene which is 11%. ’ This agrees with the average of data reported in ref 3a and 6b. a
+F
’
The experimentally determined value of ~ ‘ I S C from eq 9 is equal to the rate of intersystem crossing from 81 to T1 when only one triplet state is accessible from SI. I n the case where two triplets are accessible from the lowest singlet state, the experimentally determined ~ ‘ I s c is ~ ‘ I S ~ C=
kbc
+ JC2~~c(kZ1~c7pe”) (10)
~ ‘ I S Cis the intersystem rate constant from SI to Ti, k211C is the rate constant for internal conversion from T2to T1, and 7pe” is the experimental lifetime of Tz. If PIC = 1/7pe”, as is usually assumed
where
k’ISC
=
k’IS0
+
(11)
k21SC
For the case where three triplet states are accessible from the lowest singlet state ~‘ISC
=
+ k 2 ~ ~ ~ ( k 2 1+~ ~ ~ ~ e ” ) k ~ I s C ( k ~ ~ I C ~ p e ” k+ ~ ~ ~ ~ ~ p e ”) ’ (12)
E’ISC
IC10~~7p~”’
Here 7pe”’ is thLe experimental lifetime of T3, and V’IC are internal conversion rate constants from T3 to Ti. If k211c = 1/7pe” and (k311c I C ~ ~ I C=) l / 7 p e ” ’
+
~‘IBC
=
~‘ISC
+ k21so + Ic31sc
(13)
Of the existing data on triplet-triplet absorption of aromatic compounds, only that of Keller and Hadley6a appears to satisfy the assumptions made in the derivation of eq 9. A.s an example of the proposed method, these data with values of ~ ’ I S C are collected in Table I along with maximum values of JC’ISC obtained from eq 4. We have recalculated the Q(X) values using more accurate +F and. T F ~data listed in Table I. Because 7~~ data at 77°K are not available for most of the compounds, we have assumed that the values obtained at room temperature in cyclohexane are identical with 77°K values in EPA. This introduces an estimated error of 15%.
These limited and not highly precise data allow some interesting conclusions to be made. First, agreement (to within experimental error) between values calculated from 9 and 4 shows that the S1 to So internal conversion rate constant is negligible compared to those for fluorescence and intersystem crossing, confirming a previous statement to this effecta2 The data for naphthalene and deuterated naphthalene show no isotope dependence (to within experimental error) for kalsc in agreement with previous indirect evidence based on +F values of phenanthrene, triphenylene, and naphthalenea2 Since the isotope effect on ~ ‘ I S C is expected to be largest for naphthalene7 (with the largest Tl-Sl energy difference), there should be no isotope effect on k*16~for triphenylene and phenanthrene, both of which have smaller T1-S1 energy differences. The values of JC’ISC are lo6 to lo7 sec-l and show an inverse dependence upon the T& energy gap as predicted by Kasha.8 The higher values of Jc’Isc for triphenylene and phenanthrene may be due to one or more triplet states lying between SI and TI,since the two compounds are predicted to have three and two triplet states, respectively, below The precision of ~ ‘ I S C could be markedly improved by precise T F data ~ at 77°K which is not now available. The proposed method of obtaining kaIsc should prove invaluable in further studies of the effect of temperature and intramolecular factors on this rate constant. We hope that experimentalists involved in energy transfer studies will use this method to investigate this important parameter. Hopefully, more precise methods of
(7) M. R. Wright, R. P. Frosch, and G. W. Robinson, J. Chem. Phys., 33, 934 (1960). (8) M.Kasha, Radiation Res. Suppl., 2 , 243 (1960). (9) D.R. Kearns, J . Chem. Phys., 36, 1608 (1962); (assignments I or 11).
vohme 78, Number 9 September 1968
COMMUNICATIONS TO THE EDITOR
3356 obtaining BT will be developed thereby improving the utility of this method. MATERIALS SCIENCES LABORATORY LOCKHEED-GEORGIA COMPANY MARIETTA, GEORQIA30060
DONALD R. SCOTT OTTO hlALTENIEKS
RECEIVED MAY10, 1968
concluded that the error was about 1% for air in water and 5% for carbon dioxide in water. If one assumes a constant partial specific volume ( VA) for the solute, the diffusion equation including the convective terms is
E-[.r* dt*
where C* = (C - Cm)/(Ca - C,) and a = ~ V A . The problem is completely described by also specifying the relation between daldt and the flux at the interface (see Readey and Cooper’)
A Note on the Disvolving of Stationary Spheres, Especially Gas Bubbles
Sir: Krieger, Mulholland, and Dickey1 have reported in this journal an elegant experimental method of studying the dissolving of spherical gas bubbles in a liquid. A particularly attractive feature is that the conditions approach very closely the spherical symmet,ry which is a required condition for obtaining any reasonably straightforward theoretical interpretation. By analogy with the equivalent problem of heat transfer to a sphere with constant surface temperature, the rate of change of radius (a) with time (t) is3
where D is the effective diffusivity, p the density of the sphere (of pure solute) ; C, is the dissolved concentration of solute in equilibrium at the surface of the sphere; and C, is the original uniform concentration of the solute in the solution. Introducing t’he dimensionless parameters a* = a/ao, t* = Dt/ao2, and PO = (C, C,)/p, where a. is the initial radius, eq 1 becomes da* dt*
1
Several authors have assumed that l/a*, which leads to a*2
= 1-
2&&*
l / ( ~ t * ) ” *
