4037
RATEEQUATION FOR REVERSIBLE PHOTOISOMERIZATION splitting constants of y protons of alkyl substituents is much more difficult. I n the present time, the changes observed can only be considered to be a further proof of an important role played by the hyperconjugation type mechanism in the interaction mechanism of the unpaired electron with the y protons.
Acknowledgments. I wish to thank Dr. J. PospfBil for the incentive to complete the present work and also for the derivatives of pyrocatechol which he kindly
supplied. My thanks are also due to Dr. J. Petrinek for the preparation of the benzoquinone derivatives and for his valuable comments on the manuscript and discussion, and to Dr. I. Buben, Dr. 0. Ryba, and Dr. K. Ulbert for their aid and discussions. The program for the calculation of the spin densities was supplied by courtesy of the coworkers of Dr. R. Zahradnik, to whom I am much indebted for his consultation in quantum chemistry. Technical assistance was kindly provided by Mrs. J. VaiikovA and Miss V. Hartingerovit.
An Exact Solution to the Rate Equation for Reversible Photoisomerization
by Joseph Blanc RCA Laboratories, Princeton, New Jersey
08640
(Receiued July 6,1970)
An exact solution to the differential equation proposed by Zimmerman, et al., for reversible photoisomerination is presented. The kinetic equation is given by a rapidly convergent expansion m
c0In 161 + 6-1 where 6 made.
=
C,P =
- { ( F + w T t ) / y m 1 + constant
D m - D and the C i s are explicit analytic functions of
Research in these laboratories has been reported’J on reversible photoisomerization of indigoid dyes. Because of our own work and because of relatively large discrepancies in quantum yields which have been reporteda in the central case of stilbene photoisomerization, we have been concerned with obtaining an accurate solution to the kinetic differential equation for reversible photoisomerization which would be amenable to systematic error analysis. It is the principal purpose of this article to exhibit such a solution. hv
For the photochemical isomerization T C, let D be the optical density at time t, y the fraction of isomer C present at time t, ym the fraction of C a t photostationary state, ET and EO molar extinction coefficients of isomers T and C, respectively, a t frequency Y , [A] total isomeric concentration (moles/ unit volume), F the number of incident einsteins per unit time and unit area, 1 the optical path length, and @T the quantum yield for the process T -+ C. E’S and D’sare to the base e . I n their classic paper, Zimmerman, Chow, and Paik4 showed, defining a variable z =ym - y, that the solution to the photochemical rate equation (in the absence of thermal isomerization) is formally
Dm.
A brief discussion of errors is also
D -= dx 1 - exp(-D) z
F@TET~ {+y constant }(1)
The slope of I vs. t therefore gives @T multiplied by experimentally determinable parameters. Zimmerman, et al., by an expansion of the integrand in terms of D, obtained an approximation of the form I = s In 1x1 rx ux2. They did not give explicit expressions for s, r, or u,which are in fact cumbersome functions of the experimental parameters 1, ET, EC, ym,and [AI. A much more transparent approximation can be obtained by rewriting I entirely in terms of D: D = 1[A] { ET ( E C - ET)^ ). Defining, in analogy to x , 6 E Dm- D, eq 1 can be rewritten
+
+
+
(1) J. Blanc and D. L. Ross, J. Phys: Chem., 7 2 , 2817 (1968). (2) D.L. Ross, J. Blanc, and F.J. Matticoli, J . Amer. Chem. Soc., 92, 5750 (1970). (3) For a summary of stilbene quantum yields, see D. L. Ross and J. Blanc in “Photochromism,” G. H. Brown, Ed., Wiley-Interscience, New York, N. Y., 1971,Chapter6, in press. (4) G. Zimmerman, L-Y. Chow, and U. J. Paik, J. Amer. Chem. Soc., 80, 3258 (1958). The Journal of Physical Chemistry, Vol. 74, No. 83,1970
JOSEPHBLANC
4038 I
Using the same expansion as Zimmerman, carried to the same order in D
The coefficients in this expression appear to be power series in D,, with an uncertain radius of convergence. This approximation suggests, however, that an exact solution of the form
I
=
In 161
CO
+
m
CnGn n-1
where the (7s,’ are analytic functions of D,, can be found. If the expansion parameter is taken to be 6, which is the “natural” parameter by analogy with thermally reversible first-order reactions, such an exact expansion can be found with the (7,s’ explicit functions of D,. Before proceeding, define y = exp( -D m ) (therefore 0 5 y 5 1); definef(0) = (1 - y)-I = 1 y y2
+ . . . (y2 < 1) and f(n)
1) = ydf(n)/dy for n in terms of 6, y, and D,
-d6 ‘1 - y exp(6)
> 0.
m
nZ-1
+ + mnymand note f ( n +
Now rewrite eq 2 entirely
+
It is interesting to note that for the limiting case D, + 0, I equals 11, exactly, and therefore Iz can be viewed physically as a correction for photochemical “backreaction.” 11 is a standard integral which equals In (1 - y exp (6)) - 6. We are interested, however, in expressing 11 as a power series in 6 for combination with It. For - 1 < y exp (6) < 1, the logarithm can be expanded and 11 =
-
6
-
m
(y exp(6))m/m; now nZ-1
expand exp(m6) in powers of 6 and collect terms in 6”. The result is, omitting a constant term m
11
=
- n-1
f(n - 1)6”/n!
(5) m
To obtain
- y exp (S))-l as k - 0 (y exp < y exp (6) < l), expand exp (k6)
Iz,expand
(1
( 6 ) k ) (valid for -1 once again, collecting terms in 6”, then integrate term by term
I2 = D,f(O) In 161
+ D,
eo
Cf(n)P/n.n!
(6)
n=l
Thus, as desired
I
=
I1
+ I2 = Co In 161 +
m
CnP
(7)
n=1
-
with CO = Dmf(0)and C n = (D,f(n) - nf(n 1))/ n!-n. The first three C i s are explicitly Co = D,/ The Journal of Physical Chemistry, Vol. 74, No. $8, 1970
- y);
+
[r(l Dm) - 11/(1
-
rI2,and Cz = Y)~. Further coefficients can be obtained recursively if needed: they are not listed here as they are unlikely to be useful in practice since all higher coefficients vanish for large D, and are small for small D,. Equation 7 can be shown to converge for all physically meaningful (nonnegative) values of D and D,. I note h y only that the proper limits are obtained in the limits D, D,, 6 -t 0 and D, D, -+ 03 ; I = (In 161) and (DmIn 161 - a), respectively. The latter result clearly cannot be obtained from the approximate eq 3. For D , = 2, the approximate (eq 3) us. the exact (eq 7) coefficients are Co = 2.33 us. 2.37 (2% error); CI = - 0.833 us. - 0.796 (5% error) and Cz = 0.042 us. 0.028 (50% error). The approximation gets rapidly worse as D , exceeds 2 and rapidly better in the opposite direction. Little appears to be gained by using the approximation, although it is quite adequate for many experimental situations. I now give a brief error analysis, discussing only the logarithmic term which almost always dominates the slope of I us. t. For D , + 0, the fractional error in CO,Le., ACo/Co,is AD,/2, where AD, is the uncertainty in D,; this error decreases slowly with increasing D,. If D , is a measured optical density (rather than a calculated one; see below), an upper limit to AD, can be taken to be =0.010 (=0.004 in density to the base 10 units), and uncertainties in quantum yields due to this source should not exceed 0.5%. Previous workers have presumably used the approximation for I in the form suggested by Zimmerman, et al., L e . , in terms of concentrations and absorption coefficients. Equation 3 or 7 can be recast in terms of these parameters; from the point of view of this paper, this amounts to calculating D, in terms of the other parameters. Error estimates for the CnJscan be made, but I make no such numerical estimates here as, except for the original work of Zimmerman, et al., insufficient details of experimental methods and data reduction have been given to make such estimates reliably. It should be pointed out, however, that errors of 10% or more could plausibly arise even with standard laboratory practices, unless particular care is taken to eliminate them. For example, inaccuracies in 9, and (ET - e)of 1% would generate errors of more than 3% each in calculated Co with parameters appropriate to stilbene photoisomerization. Finally, I note that the “method of initial slopes” proposed by Lippert and Liiderb and previously used by us1i2 is quite correct in principle, but may be difficult to apply in practice without ambiguity. I n short, for conversions small enough that “back reaction” is negligible, and if the photochemical system is initially completely in one isomeric form, the method proposes that the quantum yield be determined from (1
y [ D , (y
(6)
CI =
+ 1) + 2y - 2]/4(1
-
E.Lippert and W. Luder, J . Phys. Chepn.,
66, 2430 (1962).
PHOTOCHEMISTRY OF PEROXODIPHOSPEIATES
4039 results from eq 7 and from the method of “initial slopes.”
independent of D,. However, in the face of scatter of experimental data, a criterion is needed for judging over what tim.e interval dD/dt is in fact constant. Such a criterion is clearly available by comparison of
Acknowledgments. This work is the result of a collaborative effort on photochromic indigoids with D. L. Ross. I am grateful to J. P. Wittke for a useful critique of the manuscript.
The Photochemistry of Peroxodiphosphates. The Oxidation of Water and Two Alcohols by Roger J. Lussier, William M. Risen, Jr., and John 0. Edwards* Metcalf Chemical Laboratories, Brown University, Providence, Rhode Island
OB912 (Received May 8, 1970)
The photolytic reactions of peroxodiphosphatespecies in aqueous solution, in aqueous ethanol, and in aqueous 2-propanol produce phosphate plus oxygen, acetaldehyde, and acetone, respectively. Variables, which have been investigated for their influence on stoichiometry and quantum yield, are pH, reactant concentrations, and light intensity. In the water oxidation, the results are comparable to those found in the photolytic oxidation of water by peroxodisulfate; nonchain radical mechanisms apparently are involved in both cases but there are differences in detail. The photolytic oxidations of the alcohols by peroxodiphosphate also proceed by mechanisms involving radical intermediates with chain lengths dependent on alcohol nature and pH. Mechanisms which explain the results are presented.
Introduction I n the course of our studies of inorganic peroxides, important differences between the mechanistic chemistries of peroxodiphosphate’ and peroxodisulfate2 became apparent. One of these differences was the lack of facility of the peroxodiphosphates to undergo homolytic scission. ~
~
0
+ ~ 4 2- ~ 0 2 -
Thus, peroxodiphosphates do not noticeably oxidize water at 80” or below, although the oxidation of water by peroxodisulfate to form oxygen is a radical process with conveniently measured rates a t 60’. Estimates’” indicate that homolytic scission of SZOs2-is about two powers of ten more rapid than scission of P20&. Furthermore, alcohols are oxidized by peroxodisulfate via radical chain mechanisms at 40 to 60” SzOs2-
+ RzCHOH +R&O + 2S04*- + 2H+
No comparable oxidation of alcohols by peroxodiphosphate occurs at convenient temperatures. Battaglia,a in a survey of ultraviolet spectra of peroxides, has shown that the spectra of peroxodiphosphate ion (P~OB~--) and its protonated forms (H4-PzOan-> are very similar to those of other peroxides.
This similarity of spectra led us to predict that peroxodiphosphates would be activated to radical reactions comparable to those observed with hydrogen peroxide4 and peroxodisulfates at 253.7 nm. We report here the results of an investigation of the oxidation of water and of aqueous alcohols by photo-
* To whom correspondence should be addressed. (1) (a) M. M. Crutchfield in “Peroxide Reaction Mechanisms,”
J. 0. Edwards, Ed., Wiley-Interscience, New York, N. Y., 1962, p 46 ff; (b) M.M. Crutchfield and J. 0. Edwards, J. Amer. Chem. Soc., 82, 3533 (1960); (c) Sr. A. A. Green, J. 0. Edwards, and P. Jones, Inoro. Chem., 5 , 1858 (1966); (d) M. Andersen, J. 0. Edwards, Sr. A. A. Green, and Sr. M. D. Wiswell, Inorg. Chim. Acta, 3, 655 (1969). (2) For reviews on peroxydisulfate mechanisms, see: (a) A. Haim and W. K. Wilmarth in “Peroxide Reaction Mechanisms,” edited by J. 0. Edwards, Wiley-Interscience, New York, N. Y.,1962, p 176 ff; (b) D.A. House, Chem. Rev., 62, 185 (1962); (0) E. J. Behrman and J. E. McIsaac, Jr., in “Mechanisms of Reactions of Sulfur Compounds,” Vol. 2, N. Kharaach, Ed., 1968,pp 193-218. (3) C. J. Battaglia, Ph.D. Thesis a t Brown University, 1962. (4) C’,, M. C. R. Symons in “Peroxide Reaction Mechanisms,” J. 0. Edwards, Ed., Wiley-Interscience, New York, N. Y., 1962, p 137 ff. (6) (a) Reference 2a; (b) M. Tsao and W. K. Wilmarth, J. Phys. Ch.em., 63, 346 (1969); (c) C. R. Giuliano, N. Schwartz, and W. K. Wdmarth, tbuE., 63, 353 (1969); (d) L. Dogliotti and E. Hayon, ibU., 71, 2511 (1967); (e) E. Heckel, A. Henglein, and G. Beck, Ber. Bunsenoes. Phys. Chem., 70, 149 (1966); (f) M. H. B. Mariano in “Radiation Chemistry,” Vol. I, Advances in Chemistry Series No. 81,American Chemical Society, Washington, D. C., 1968,p 182. The Journal of Phyeical Chembtry, Vol. 74, No. 23,1970
