COMMUNICATIONS TO THE EDITOR
4174 dyes used in fluorescence microscopy and histochemistry, provides a valid model for energy transfer under biological conditions. The extremely facile intermolecular energy transfer between aggregates, in particular, raises interesting possibilities about such processes being operative in the connective tissue matrix, where the polyanions are in similar association with a range of polycations. It has also been found possible to render dye-polyanion systems resistant to fading under conditions of illumination. The small amounts of methylene blue or thionine added do not affect the absorption spectra, but seem to decrease the photodecomposition or photoxidation reactions of acridine orange in the presence of air. The decrease in fading brought about by methylene blue is much more marked than the quenching of dimer fluorescence. This useful finding is consistent with very rapid energy transfer involving a number of excited states. The high efficiency of transfer makes it possible for energy transfer from higher excited states to occur. Acknowledgment. This work was supported in part by a grant from the Medical Research Council and the United States Department of Agriculture PL 480 Grant FG-UK- 147.
E ( { v))
Z({ej)niePp*ei
=
(1)
le1
where v, is the chemical potential of the i t h atomic species and p is the statistical temperature. The function E ( { v is essentially the Laplace transform of Inverting the transform, we have Z({ e
1)
1).
where v f f can be conveniently chosen (within the convergence limit) on the real axis of vj. We can evaluate X by making use of the fact that the only important contributions to X come from energy levels which result from the atoms being associated jn stable species. Let vii (i = 1 to n) be the number of molecules of the ith stable compound and write the stoichiometric relations for the decomposition of molecules to atoms in the form i=l
niail = el ( j = 1 to m )
where aij are stoichiometric coefficients. Then
* To whom correspondence should be addressed. DEPARTMENT OF CHEMISTRY THEUNIVERSITY, NOTTINGHAM, ENGLAND
canonical partition function for the set of atomic components { e ). The atomic composition of the system is, of course, unchanged by chemical reaction. The grand partition function X ( [ v )) is then given by
R. B. CUNDALL C.LEWIS
~ ( v= ) CZ({e})ir,epuiei = le1
C ~ ( { n } ) r ~ ~(3)( ~ i
in1
where At = irjePyJasJis the activity of the ith moleaular DEPARTMENT OF CHEMISTRY AND P. J. LLEWELLYN APPLIEDCHEMISTRY G. 0. PHILLIPS* species. In the last equality we have made use of the fact that only atomic configurations corresponding to THEUNIVERSITY SALFORD, M5 4WT, ENGLAND RECEIVED JULY 25, 1970
Chemical Equilibrium and the Anti-Helmholtz Function.
A Statistical Interpretation
Sir: Recently Duffin and Zener have discussed the problem of chemical equilibrium as formulated in the language of geometric programming. They establish that the usual method of minimizing the Helmholtz free energy with respect to concentration of products and reactants is equivalent to maximizing an “anti-Helmholtz” function with respect to the activities of the atomic species present in the system. They also give a statistical mechanical interpretation of this approach in terms of the method of Darwin and Fowler. The well known relation between the “selector variables” of the Darwin-Fowler method and chemical activities suggests an alternate statistical interpretation of their results which clearly reveals the relationship between the atomic activities and the anti-Helmholtz function. Let e, (i = 1 to m) be the number of atoms of the ith atomic component in the system and Z({e]) be the T h e Journal of Physical Chemistry, Vol. 74, N o . $9,1970
stable compounds give an important contribution to the sum. If we substitute eq 3 into eq 2 and evaluate the resulting integrals by the method of steepest descents, we obtain an expression for Z in terms of the minimum of the function Z*({ v)) = E / r t e P ~ with ~ e ~ respect to the atomic chemical potentials { v) , This is equivalent to the maxiniization of the anti-Helmholtz function F* = - 1 / p In Z* = Ziv,ei - PV where PV is the pressurevolume product. As an example, in the case considered by Duffin and Zener, when all the species are ideal gases, then
E
= C i r i ( x i ~ J n z / n= a ! exp(Ctx,njeP~~a~~) in!
where x i is the molecular partition function of the ith stable species. From eq 3 we have
which when computed by the method of steepest descents yields the result of Duffin and Zener
(1) R. J. Duffin and C. Zener, J.Physc. Chem., 74,2419 (1970).
COMMUNICATIONS TO THE EDITOR
4175 original site of activation or by rupture of the original ring (eq 2b).4 Equal amounts of decomposition of the two rings is direct evidence for internal energy randomization prior to decomposition. Variation of pressure D. K. HOFFMAN of the system and the rate of collisional stabilization w of the hot HBC molecule (eq 3)
This is the statistical analog of the maximization of the anti-Helmholtz function for the equilibrium between reacting ideal gases. INSTITUTE FOR ATOMICRESEARCH AND DEPARTMENT OF CHEMISTRY IOWA STATEUNIVERSITY AMES,IOWA50010 RECEIVED AUGUST24, 1970
CF2-CF-CF-CF2*
\/' CHz Intramolecular Energy Relaxation.
Sir: Working theories of unimolecular decomposition2 assume that internal energy relaxation occurs on a time scale that is short relative to decomposition; effective randomness of the internal energy distribution in excited molecules leads to the prediction of random incidence of decomposition events. Considerable evidence of various kinds and of varying degrees of cogency has now been accumulated that supports the general valid~ there can be no doubt ity of this p o s t ~ l a t e ,although that failure must occur for sufficiently short-lived species. We have for some time been attempting to make an even more direct and quantitative experimental test of intramolecular energy relaxation than has hitherto existed. To this end we have applied the technique of chemical activation in order simultaneously to symmetrize and activate a substrate molecule of interest. The substrate finally chosen for study was hexafluorovinylcyclopropane (HVC). Addition of CD2 to HVC produces symmetrical, vibrationally excited hexafluorobicyclopropyl (HBC)
\/
+ lCD2
--j
CHz CF2-CF-CF-CF2"
\/
\/
CH2
CD2
v CH2
\/
CDz
+CF2--CF-CF=CD2
C D2
\/ CHz
\/
CD2
+ ni (3)
provides a time scale against which the rate of internal relaxation can be measured. Obviously one wishes to shorten this time scale by pushing the system to higher and higher pressures; this kind of experiment has the advantage over others that have been made recently at high p r e ~ s u r ein~ ~that ~ three-body effects which can vitiate the lattere as valid tests of the energy-randomization postulate will be self-canceling by the present test. In these experimeonts, HVC was prepared by the photolysis at 2800 A of a ketene-perfluorobutadiene mixture. This fluorocarbon system is preferable to its hydrocarbon analog, which we first investigated, since the reactions of methylene are cleaner with the fluorocarbon; also, decomposition of the hot product proceeds uniquely by split-off of a CF2 m ~ i e t yrather ,~ than by a variety of ring isomerization reactions that ensue for the hydrocarbon analog. The HVC was purified and photolyzed again with ketene-& in the presence of excess CO bath gas which contained 0.5% 02. These diatomics inhibited complicating reaction of any %H2 formed.7i8 After careful and complete gas chromatographic separation of the tetrafluorovinylcyclopropane products from all other components of the reaction mixture, analysis was made with a 3IS9 mass spectrometer. Rupture of products I and I1 by electron impact produces cyclopropyl and vinyl ions and results
(1)
where the asterisk signifies vibrational excitation. Datoms were used as a tracer to distinguish the original and product rings and otherwise differ unimportantly from H atoms. The excited HBC may decompose by disruption of the product ring (eq 2a) which was the CFz-CF-CF-CF2"
\/
CF~--CF-CF-CF~
A Novel
and Direct Test of the RRK-RRKM Postulate la
CF-2CF-CF=CFz
+ M A
+ CFz
(1) (a) Work supported by the National Science Foundation; (b) Standard Oil predoctoral fellow. (2) (a) L. Kassel, "Kinetics of Homogeneous Gas Reactions," Reinhold, New York, N. Y. 1932; (b) R. A. Marcus and 0. K. Rice, J. Phys.Colloid Chem., 55,894 (1951). (3) The first striking evidence was given by J. N. Butler and G. B. Kistiakowsky, J . Amer. Chem. SOC., 8 2 , 759 (1959) ; for a review see L. D. Spicer and B. S. Rabinovitch, Ann. Rev. Phys. Chem., 21, 349 (1970). (4) N . C. Craig, T. Hu, and P. H. Martyn, J . Phys. Chem., 7 2 , 2234 (1968). (5) D. W. Placzek, B. 9. Rabinovitch, and F. H. Dorer, J . Chem. Phys.,44,279 (1966)). (6) J. Aspden, N. A . Kawaja, J. Reardon, and D. J. Wilson, J . Amer. Chem. Soc., 91,7580 (1969). (7) B. A. DeGraff and G. B. Kistiakowsky, ibid., 71, 1553 (1967); 71,3984 (1967). (8) J. W. Simons and B. S. Rabinovitch, J. Phys. Chern., 6 8 , 1322 (1964).
The Journal of Physical Chemistry, Vol. 74,No. $3, 1970
