4952
J. Phys. Chem. 1984, 88, 4952-4955 39(exptl) - x[39(dimer)] - (1 - x)[39(monomer)] = 0
A simultaneous equation fit was then used to test the simulation. The 157-ys time delay spectra evidencing the highest dimer concentration were used for this fit. Mass 39 is the largest peak in both the experimental cluster mixture and the monomer spectrum, so it was employed as the fitted datum. The simulation spectra are generated as a series, with the 6.0-mJ total ion current set as 1.O. The lower power spectra are found to decline (in total ion current) in an essentially quadratic dependence, as expected. The 6.0-, 5.0-, and 4.0-mJ values for mass 39 in the monomer, dimer and 157-ys delay experimental spectra were inserted into the equation
(2)
where x is the experimental dimer fraction at that time within the molecular pulse. A perfect fit over all three laser powers with a single x is impossible, but the monomer and dimer were varied to minimize the variation from zero in eq 2. This procedure generated the “dimer simulations” shown in Figure 8. These represent the best present estimates of the MPI mass spectra of the pure dimer species under the specified conditions. Registry No. Toluene, 108-88-3.
The Nonstatistical Multiphoton Ionizatioh-Dissociation of the van der Waals Toluene Dimer J. Silberstein, N. Ohmichi, and R. D. Levine* The Fritz Haber Molecular Dynamics Research Center, The Hebrew University, Jerusalem 91 904, Israel (Received: April 5, 1984)
The persistence of van der Waals bimer ions in the fragmentation pattern of the toluene dimer ion can be accounted for by using a simple, physically motivated constraint. The computations predict that many of these bimers are in predissociative states. Comparison is made with the observed patterns of Squire and Bernstein (preceding paper). Uncertainties in the dissociation energies of the bimer ions preclude however a quantitative check.
Introduction The experimental study’ of the fragmentation, following multiphoton ionization (MPI), of van der Waals dimers provides a critical test of the maximal entropy computationZof the mass spectrum. The considerable frequency mismatch between the vibrations of the monomer and that of the van der Waals bond suggests that the dissociation of such dimers will not be well described by a simple statistical limk3 Indeed we find that the observed’ fragmentation pattern of the toluene dimer is not consistent with the predictions of the extreme statistical limit as previously introduced.2 In particular, there is a qualitative manifestation of this failure. In a given observed fragmentation pattern, where many small (and hence energy expensive) ions are present, one still detects nonnegligible quantities of large ions. This is unexpected in the extreme statistical limit for in that limit the pattern is simply determined by a balance between energetic and entropic considerations. At low levels of excitation, energy wins. One sees mostly the more stable and larger ions. At high levels of excitation, entropy wins and the fragmentation is extensive. The maximum entropy formalism is not limited to computing the fragmentation pattern in the extreme statistical limit. The latter is a special case where the distribution of species (whether ionic or neutral) is constrained only by the universal conservation conditions (energy, elements, charge) and is otherwise of maximal entropy. One can however impose additional constraints. The central result of this paper is that imposing one such constraint whose interpretation is clear-cut suffices to remove the qualitative discrepancy between experiment and the extreme statistical limit. We do not report a quantitative comparison with experimental results since we have too much latitude at the moment for achieving a good fit to experiment. The reason is that input to the computation is the partition function of the dimer ion (and (1) D. W. Squire and R. B. Bernstein, preceding paper in this issue. (2) (a) J. Silberstein and R. D. Levine, Chem. Phys. Lett., 74, 6 (1980); (b) J. Chem. Phys., 75, 5735 (1981). (3) J. Jortner and R. D. Levine, Adu. Chem. Phys., 47, 1 (1981).
0022-3654/84/2088-4952$01.50/0
of any other ionic or neutral potential fragment). Since neither the dissociation energy nor the geometry and certainly not the frequencies of the dimer ion are known, we have too many parameters at our d i ~ p o s a l . ~ A contributing factor to the success of the simple deviance from statistics description proposed below is, of course, that the MPI-fragmentation of toluene monomer is well described by the extreme statistical limit. This is confirmed both by a computation-free test5 and by computational comparisons2b,6with experiment.
The Extreme Statistical Limit The entropy of the distribution of fragments is2 S = -kxX,[ln X, I
+
In x , ~- In x]
(1)
1
k is the gas constant (per molecule). Xj is the number of molecules of species j (whether neutral or charged), and summation is over all potential fragments. X (which is not a priori known) is the total number of fragments
x = cx, J
(2)
xlj is the fraction of molecules of species j in the quantum state i. In the extreme statistical limit the entropy is maximized subject to the following ubiquitous constraints: (a) conservation of probability EXij = 1 i
(b) conservation of energy (4) Any other theory which requires additional input, e.g. the parameters of the transition state for the dissociation of the dimer ion, will have an even wider latitude. ( 5 ) D. A. Lichtin, R. B. Bernstein, and K. R. Newton, J . Chem. Phys., 75, 5728 (1981). (6) N. Ohmichi, J. Silberstein, and R. D. Levine, Isr. J. Chem., in press.
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4953
van der Waals Toluene Dimer for all species, (c) conservation of matter
where akjis the number of atoms of type k in a molecule of species
j , ck is the number of atoms of type k in X , molecules of the parent
(Xois arbitrary, and one can show6 that, at a given ( E ) ,X j (and X , cf. (2)) is proportional to X,.), and K is the number of distinct elements in a molecule of the parent, and (d) conservation of charge For k = K + 1, akj is the charge on species j and Ck is the charge on X , molecules of the parent. One can start the computation with the neutral parent provided the free electron is included as a possible fragment7 In doing so, it is necessary however to take into account the discreteness of the photons. Typically and here too we shall start from the parent ion. Subject only to these constraints the resulting fragmentation pattern is analytically found to be2 K+ 1
Xj = XQj exp(-
C
k- 1
Ykakj)
(4)
+
Here, the K 1 parameters Yk are the Lagrange multipliers for the K + 1 constraints (3c) and (3d). Qj is the partition function Qj
= CexP(-peij)
(5)
I
where we reiterate that all energies refer to a common zero. /3 is the Lagrange multiplier for the constraint (3b). ( E ) is the energy absorbed from the laser by Xo molecules of the parent. Current experiments cannot measure ( E ) . Hence, here and elsewhere we compute the fragmentation pattern as a function of ( E ) . In practice, this is done by varying p and using (3b) to compute the value of ( E ) for each value of p. At each value of ( E ) there are thus K 2 unknowns in (4), the K + 1 yL)s and X . They are (uniquely) determined by the K + 2 equations (3c), (3d), and (2). Two factors govern the fraction of molecules of species j in the extreme statistical limit. One is purely thermochemical, namely the partition function. It favors the more stable and largers species. The other is the exponential factor. Positive Lagrange multipliers favor the smaller species (since akjis always nonnegative and is smaller for smaller species). As ( E ) is increased, so do the Lagrange mltipliers. (See the many computational examples in ref 2b.) Smaller fragments (with a high heat of formation and hence low Q ) can only appear when the Lagrange multipliers are already positive so that the exponential factor outweighs the partition function. But once the Lagrange multipliers are positive, the exponential factor in (4) strongly discriminates against the larger (many atom containing) species. That is the dilemma faced in interpreting the observed' fragmentation pattern of the toluene dimer. Both the large ions and the small ions are observed in the same pattern. The large ions seem to have fewer atoms than suggested by their stoichiometric formula and/or to have a larger partition function than they should rightly have.
+
Constraints Two-color multiphoton ionization of isotopically mixed benzene dimers9 indicates that at least for very soft ionization the excitation is substantially localized on one of the monomers and that each half retains its isotopic composition. These findings are clearly at variance with the assumptions inherent in the extreme statistical limit (equipartitioning of the energy and equivalence of all atoms of the same mass number. On the other hand, the energies we (7) N. Ohmichi, J. Silberstein, and R. D. Levine, J. Phys. Chem., 85, 3369 (1981). ( 8 ) The partition function is an effective volume in phase space. (See,e.g., J. E. Mayer and M. G. Mayer, "Statistical Mechanics", Wiley, New York, 1977, Section 7p.) Other things being equal, larger species will have larger partition functions. (9) K. H. Fung, H. L. Selzle, and E. W. Schlag, J. Phys. Chem., 87,5 113 (1983).
are concerned with are significantly higher than those employed in the very soft ionization experiments.) Our first approach was therefore to completely decouple the two halves of the dimer. That is, we write the parent molecular ion as a bimer ion (C7H8-.T)+ where T has the same energy levels as a toluene monomer but its excitation and the excitation of the van der Waals bond (indicated by the dots) are independent of the excitation of the toluene half. Furthermore, since no atom scrambling takes place, the bimer ion (C7H8-.T)+ has seven carbon atoms and eight hydrogen atoms (and one positive charge). It differs from a toluene monomer ion, C7H8+,in that the dimer contains the conserved "atom" T. In other words, we regard the initially unexcited half of the dimer as a separate distinct entity. In addition to the dimer we also included such species as (C7H7--T)+and, of course, the neutral species T. Computationally, we simply take K = 3 for the conservation of elements constraints where k = 1 is carbon, k = 2 is hydrogen, and k = 3 is T. Since we have essentially no information on heats of formation of the bimers with T, only the (C7HpT)+and (C7H7-T)+ T-containing ions were included in the computation. Among the neutral species only T was included.1° It is also worthwhile to examine the physical significance of the constraint in terms of the Lagrange multiplier. The Lagrange multiplier Yk (k = 1, ..., K ) can be regarded as the (Planck) chemical potential of element k. The chemical potential in the more familiar (energy) units is k k = yk/@ The chemical potential for the j t h species is y j = @kj = CakjYk k
(6)
In other words, eq 4 can be written in the thermodynamic-like form Xj = XQj exP(-kj/kT)
(7)
In the extreme statistical limit the chemical potentials of each element have the same value for all the species. The constraint introduced in this paper is that the chemical potentials of the two halves of the dimer ion are not equal. The carbon and hydrogen atoms in the T half are not scrambled with their counterparts in the C7H8half. Computations Restricting the energy exchange between the C7H8and the T halves of the parent dimer ion was not found essential. Presumably at the higher energies required for fragmentation the resonance transfer of vibrational energy between the two halves is more efficient than near the ionization threshold. It may also be that quantitative agreement with experimental results does require a restricted energy exchange, but at the present level of uncertainty about the thermochemistry of these bimers such speculations cannot be resolved. There is one place where the notion of a restricted energy exchange does survive in the results shown below. This has to do with computing realistic partition functions. At the high mean excitation energies which are relevant in the present problem, it is essential to recognize that molecules have only a finite number of bound states (Le. they can dissociate). The summation in (5) contains therefore a large (but finite) number of terms. Correcting for the possibility of dissociation depends on the question of over how many degrees of freedom the energy is distributed. The dissociation energy of the bimer ions is presumably low, and hence the cutoff correction is quite important. We are unable to account for the observed fragmentation pattern if we assume that vibrational excitation in the C7H8or in the T halves of the (C7H8--T)+ bimer is available to dissociate the van der Waals bond. It is essential not to misinterpret this point. We are not saying that energy flow into the van der Waals bond is prohibited. We are talking of state counting. A state of high vibrational excitation in the C7Hs and/or the T parts of (C7H8-T)+ and low vibrational (10) In principle, (C,H8-T)+ could dissociate say to C3H3+ and (C,H,.-T) neutral van der Waals bimer. Since such species are expected to have very low dissociation energies, they were not included.
4954
Silberstein et al.
The Journal of Physical Chemistry, Vol. 88, No. 21. 1984 30-
( E ) = 2 5.8ev
ul
c
0
5 20-
-
+
F
0
+-
c
0
0
8
33.3-
‘0-
0 100
150
250
200
300
350
< E > (eV) Figure 1. Fragmentation pattern of the toluene van der Waals dimer ion vs. ( E ) , the mean energy (in eV) per parent molecule. Shown are the fractions of C7H7+ions, (C7H7-.toluene)t ions, C4+ions (primarily masses 52-50; cf. Figure Z), CJ’ ions, and Cz+ ions.
excitation of the van der Waals bond is counted as discrete. In principle, such a state is degenerate with and hence coupled to an unbound state where the vibrational excitation of C7H8and/or T is lower and more than the dissociation energy is present in the van der Waals bond. Strictly speaking, therefore, the former state is a predissociating one and lies in the continuum. We count it as discrete. The details of computing the partition function are given in the Appendix. In summary, we regard the predissociative states of the bimer as stable. Other than the way of state counting for (C7H8-.T)+ and (C7H7--T)+the computation runs exactly as in the extreme statistical lirnkZbv6The only difference is that some species contain another “atom” T. We reiterate that the internal energy levels of T are those of toluene. Results A typical breakdown curve for toluene dimers is shown in Figure 1. The result is “typical” in that reasonable variations in the input thermochemical data of the two bimer ions do not alter the qualitative features of the curve at low energies. At the higher energies where the fraction of the bimer ions becomes negligible, the curve is, of course, independent of such input. The most important thermochemical inputs” are the dissociation energies. (27 and 24 kcal/mol for (C7H8--T)+and (C7H7*-T)+,respectively, are used in Figure 1. The qualitative features are quite insensitive to the first one. The second could be lowered by 5 kcal/mol without significant changes in the curve.) As long as the vibrational frequency of the van der Waals bond is below I/(hp) (8’ = 2200 K in Figure 2), its precise value matters not. The following special features of Figure 1 deserve comment: (a) As in the observed’ patterns, there is practically no parent dimer (C7Hs.-T)+. Only the parent minus H, (C7H7-T)+, is present to any extent, this despite our having allowed the parent to have a higher dissociation energy. (b) As in the observed patterns, there are bimer ions ((C7H7-T)+ present together with the smaller fragments. These are the predissociating states discussed in the previous section. (c) As the energy ( E ) is increased, one notes a decline followed by a secondary appearance of the (C7H7--T)+ ions. The original decline is that due to dissociation of (C7H7--T)+ to C7H7+and T. As ( E ) is further increased, the theory begins to populate the higher energy predissociative states of the bimer, hence their secondary appearance. A similar increase of the fraction of (C7H7--T)+ions with increasing laser power can also be seen in the experimental results (Figure 10 of ref 1). It is suggestive to correlate these experimental and theoretical observations. In so far that the secondary hump in the fraction ~
~~
~
~~~~
(1 1) The input thermochemical data set is very extensive. For each species (whether ionic or neutral) we require the heat of formation of the electronic ground state, its dissociation energy, the vibrational frequencies, and moment(s) of inertia (likewise for any low-lying electronic excited state). For those wishing to repeat the computation we have available (upon request) a computer printout providing both a complete data set and a sample output.
masslcharge Figure 2. Computed mass spectral fragmentation pattern of the toluene dimer ion at ( E ) = 25.8 eV. Note the near absence of the parent (C7Hs-toluene)+ ion, the high fraction of C4 and C3 ions (as seen in the experiment’), and the presence of C2ions. The value of B corresponding to ( E ) = 25.8 eV is B’ = 2200 K. The m / e scale is one unit,
of (C7H7-T)+ ions vs. ( E ) plot is very typical of our results, such a correlation may not be entirely unreasonable. (d) We compute a nonnegligible fraction of C6 ions which are not seen in the experiment. (e) Among the smaller ions present in the energy range where bimer ions are significantly present, masses 51 (C4H3+),39 (C3H3+),52 (C4H4+),and 27 (C2H3+)predominate (Figure 2). (f) The fragmentation pattern below mass 91 (C7H7+) is similar to that of toluene monomer ionZbJ’at lower energies. The reason is quite simple. Since we do not constrain the energy exchange between the two rings in the dimer, such dimer ions which are highly excited dissociate. Those fragmentation patterns which show the presence of a nonnegligible fraction of bimer ions correspond (in our results) to comparatively low mean energy uptake, ( E ) ,per dimer molecule (cf. Figure 2). Since the fraction of bimer ions is not very high, the equations that need be solved for the Lagrange multipliers are almost the same as those for toluene monomer. At higher laser powers (not reported in ref 1) where no intact bimer ions survive, the fragmentation pattern would be essentially identical with that of toluene monomer. On the whole, our interpretation of the mass spectrum is closer to the second explanation of Squire and Bernstein. Concluding Remarks The main features in the observed fragmentation pattern of the toluene dimer ion can be accounted for by assuming that there is no atom exchange between the two halves of the dimer. It is not necessary to restrict the energy transfer between the two halves of the dimer. It is however necessary to restrict the energy transfer between the relative motion of the two monomeric units and their internal motion. Such restriction implies that dimer states with high internal excitation of the monomeric units are stable. Strictly speaking, such states are predissociative. The (assumed) large frequency mismatch between the internal vibrations and the relative stretch and twist motions implies that such predissociative states will have a very long lifetime,3 and we thus take them to be stable. It is a robust conclusion of the present computation that such predissociative states make the major contribution to the fraction of bimer ions at the higher excitation energies. Acknowledgment. We thank Prof. R. B. Bernstein for an early communication of the results reported in ref 1. This work was partly supported by the Office of Naval Research and the U. S.-Israel Binational Science Foundation. The Fritz Haber Research Center is supported by the Minerva Gesellschaft fur die Forschung, mbH, Miinchen, BRD. Appendix. The Vibrational Partition Function at High Excitation Levels The partition function defined by ( 5 ) is the usual thermodynamic partition function at the temperature T, p = l / k T (except that the energy levels of all species refer to a common zero). At low temperatures, the sum in ( 5 ) is rapidly convergent so that it is unimportant to exclude the states past the dissociation threshold.
4955
J . Phys. Chem. 1984,88, 4955-4959 At higher laser powers, the energy ( E ) per dissociating parent molecule is high, corresponding to a low value of @. It is then essential to limit the summation in ( 5 ) to levels below dissociation. For the temperatures (1500-3000 K) of interest in the present study this affects primarily the vibrational contribution to the partition function. In the rigid rotor-harmonic oscillator approximation6 the partition function factorizes. In the classical limit it is given by Qvib
= Jm&b(E)
exP(-E/kT) d E
(A. 1)
where pvib(E) = E'r'/(S,-l)!IIhvjj i
(-4.2)
sj is the number of vibrational modes of species j , and vij is the frequency of the ith mode (i = 1, ...,sj) in species j . We use the
truncated harmonic oscillator approximation where the upper limit of integration in (A.l) is replaced by the dissociation energy. To evaluate the integral, we write it as ovib E
JDPvidE) exP(-E/kT) d E = Qvib(1 - F)
('4.3)
where F is the Hinshelwood12 fraction
si- 1
C [(D/kT)'/i!]
exp(-D/kT)
(A.4)
i=O
For a given dissociation energy D and a t a given temperature T, the higher sj is the higher the vibrational energy content and hence the higher the fraction F of molecules that have dissociated and are not to be counted. For the special case of a van der Waals bimer we write the vibrational partition function as a product of three terms: two partition functions, one for each monomer, and a partition function for the van der Waals bond. In the latter, si in (A.2) and hence in (A.4) is restricted to a small integer. (Its precise value matters not. We used 6). This is equivalent to counting the predissociative states of the bimers as stable. Had we computed one vibrational partition function for the entire bimer by using (A.3), then at the low value of D (51 eV) for the van der Waals bond the high value 1 at all temperatures of of si for the bimer would lead to F interest. Registry No. Toluene, 108-88-3.
-
(12) C. N. Hinshelwd, Proc. R. SOC.London, Ser. A, 113,230 (1927).
Methyl Orange as a Probe for Photooxidation Reactions of Colloidal TiOp Graham T. Brown and James R. Darwent* Department of Chemistry, Birkbeck College, University of London, London WCl E 7HX. England (Received: April 12, 1984)
Colloids containing TiOz supported by PVA can only sensitize the reduction of Methyl Orange. In contrast unsupported Ti02 colloids catalyze the photooxidation of Methyl Orange and concomitant reduction of 02.H202inhibits oxidation of Methyl Orange in a manner analogous to noncompetitive enzyme inhibition. This suggests the Hz02intercepts a precursor to the species responsible for dye oxidation. A kinetic analysis shows that lo4 M H2O2can intercept 50% of photogenerated h+ before recombination with e-, whereas Methyl Orange reacts with surface radicals, (TiO.)s. Only 1 in 450 photogenerated h+ lead to (TiO.)s and in the absence of H202charge recombination is the major reaction pathway. Cationic surfactants (CTAC) and cationic polymers (Polydmeama and Merquat 100) increase the rate of Methyl Orange oxidation. This results from up to a fivefold increase in surface oxidation compared to charge recombination.
Introduction The interface between an electrolyte and a semiconductor is a unique environment in which to catalyze and control chemical reactions. A potential gradient is formed at a semiconductor/liquid junction.' This electrostatic field can drive mobile holes and electrons to different regions of the semiconductor material and hence provide oxidizing and reducing species. In addition, the surface may function as a conventional heterogeneous catalyst by adsorbing reactants and providing catalytic acid/base groups. Recently, several research groups, including ours, have shown that colloids containing semiconductor particles provide an excellent medium in which to investigate electron transfer reactions across a semiconductor/liquid i n t e r f a ~ e . ~ Using -~ colloidal TiOz (1) Heller, A. Acc. Chem. Res. 1981, 14, 154. (2) Henglein, A. Ber. Bunsenges. Phys. Chem. 1982,86, 241. (3) Duonghong, D.; Ramsden, J.; Gratzel, M. J . Am. Chem. SOC.1982, 104, 2977. (4) Kuczynski, J. P.; Milosavijevic, B. H.; Thomas, J. K. J. Phys. Chem. 1983, 87, 3368. ( 5 ) Rossetti, R.; Brus, L. J. Phys. Chem. 1982, 86, 4470. (6) Moser, J.; Gratzel, M. J. Am. Chem. SOC.1983, 105, 6547.
0022-3654/84/2088-4955$01 .50/0
particles stabilized by polyvinyl alcohol (PVA), we studied the photosensitized reduction of Methyl Orange ((CH3)2NC6H4N= NH4C6S03-Na+).9 In this system, PVA and water are rapidly oxidized by photogenerated holes (h+), so that photogenerated conduction band electrons were effectively trapped on the particles and their subsequent transfer to Methyl Orange was monitored by flash photolysis. Significantly, this work provided quantitative information about the effect of pH on the rate of interfacial electron transfer from TiOz to Methyl Orange and oxygen. We would now like to report further work on colloidal TiOz, but in this case using naked (unsupported) particles and colloids containing cationic surfactants and polyelectrolytes which unlike PVA are not readily oxidized. Consequently, the photogenerated holes can now oxidize probe molecules such as Methyl Orange, so that the colloids behave quite differently from thosd stabilized by PVA. As a result, we were able to investigate the rates of (7) Fox, M. A,; Lindig, B.; Chen, C. C. J. Am. Chem. SOC.1982, 104, 5828. (8) Darwent, J. R. J. Chem. SOC., Faraday Trans. 1, 1984,80, 183. (9) Brown, G. T.; Darwent, J. R . J. Chem. SOC.,Faraday Trans. 1, accepted for publication.
0 1984 American Chemical Society
