6308
J. Phys. Chem. 1992,96,6308-6313
Correlations of Values of Average Energy Transfer from Highly Excited Polyatomic Molecules with Heats of Vaporization and Boiling Temperatures Izback Oref Department of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel (Received: February 5, 1992; In Final Form: March 27, 1992)
Correlations of the average energy transferred in a highly excited polyatomic molecule-bath gas binary collision with the heat of vaporization and boiling point of the bath gas are presented. It is shown that a unique correlation exists between the logarithms of the average energy transferred in a collision and the values of the two quantities reported above. Examples are given for excited toluene, azulene, and substituted cycloheptatriene collisions with mono- and polyatomic bath gases,
1. Introduction
Thermal unimolecular reactions, chemical activation, energy removal from highly excited molecules in the ground or excited states and photoquenching are some of the processes which depend intimately on the average energy transferred in a nonelastic collision, (ALQBu.l4 For example, the shape of the rate coefficient vs pressure falloff curve of unimolecular reactions depends on the intermolecular energy transfer probability function, which determines the value of (hE),11 and the average energy transferred per down collision ( h E ) d . At low levels of excitation the energy transfer process is well understood theoretically and experimenThe repulsive Landau-Teller potential plays a major role plots predicted by the SSH-T theoryS and log probability vs give good agreement with experimental results. At high levels of excitation, however, little is known about the nature of the intermolecular potential and how it effects the values of ( The complexity of the system defies development of theoretical models and, therefore, general correlations of ( AQd with physical quantities can be of great help to the experimentalist and point the way for future theoretical developments. Parmenter and co-workers4s6have proposed propensity rules for collision-induced cross sections uM for state changes within the vibrational manifold of electronically excited molecules in collision with a bath molecule, A** M A* M. They propose a simple relation between uM and the well depth of the Lennard-Jones potential of the neat bath gas molecules eMM
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unit efficiency. Nevertheless, very interesting correlations are found. One correlation is with the polarizability of the colliding molecules via the London dispersion potential (1-3) where I is the ionization potential, a is the polarizability, and C is a constant dependent on the bath gas.g For many bath gas molecules colliding with a given reactant, e.g. NOC17 or CH3NC,8 the quantities denoted by the subscript A are constants for a given reactant, and since the ionization potentials are almost constant for many large bath gases, the value of Vdis is proportional to the value of the polarizability of the bath gas. Indeed, the collisional efficiency shows a clear correlation with the value of the polarizability of the bath gas.* The force of the collision' is a quantity which includes the parameters of the potential. For a Lennard-Jones potential it measures the steepness of the repulsive part (1-4)
A correlation is obtained in the case of NOCl thermal decomposition in a limited number of bath gases. No correlation is obtained in the case of CH3NC,8which is explained by the fact that the attractive and not the repulsive force dominates the energy transfer process. In addition, values o f t and u are not well known, In UM = In c 8 ( € M M / k ) ' / 2 (1-1) and the scatter in the available data is so large as to render a correlation invalid. where C is a constant and 8 is related to the Lennard-Jones well Dipole moments as prime correlation parameters do not appear depth of the A*-A* interaction. For a series of M colliding with promising. There is no clear-cut correlation and many large A*, 0 is a constant, and so the details of the A*-A* interactions molecules with small dipole moments show high efficiency. The need not be known in detail since they are constant for a given depth of the Lennard-Jones potential, e, was found to be a good A* colliding with a series of bath gases. Equation 1-1 is reported correlation parameter of the dissociation energy of 1,-M complex to hold provided the interactions leading to the change of state where M is He, Hz, D2,Ne, and k.'"The linear correlation holds are dominated by intermolecular attractive forces, Le., the cross for this limited set of bath gases, but it was not tested with a large sections are of the order of magnitude of kinetic theory hard sphere sample of bath gases and is not expected to hold in all cases.I0 cross sections or larger. An additional constraint on the appliHowever, a plot of collisional efficiency as a function of e shows cability of eq 1-1 is that resonance energy transfer is absent in poor correlation, which may be attributed in large part to unthe energy transfer process. certainty in the values of e.* Propensity rules for collisional efficiency in unimolecular reThe major difficulty in finding a unique correlation between actions were attempted by Volpe and Johnston7and Rabinovitch collision efficiency or cross section and a single molecular paand co-workers.'v8 The collisional efficiency 8, on a per collision rameter such as a,e, u, or p rests in the fact that the values derived basis is defined as for these quantities are interrelated.'V8 As e changes, so do the collision integral Q('*')* and ULJ, and as p increases, so does the polarizability a. These multiple dependences led Russel and SionslI to use successfully an empirical correlation of icdine atom recombination rates with the boiling points of bath gases. Volpe and Johnston7 correlated the collisional efficiencies of a limited where k is the low-pressure rate coefficient, 1is the reduced mas, and uu is the Lennard-Jones collision diameter &-' = d%(2,2)*, number of bath gases in the thermal unimolecular decomposition of NOCl with the boiling temperature of the bath gas. Chan et where uHS is the hard sphere diameter and Q(2,2)* is the Lena1.8 made the same correlation with over 100 gases using as a probe nard-Jones collision integral. The collision efficiency is a quantity the unimolecular decomposition of CH,NC. The underlying of limited usefulness since most large polyatomic colliders have
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0022-3654/92/2096-6308$03.00/0
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6309
Energy Transfer from Polyatomic Molecules
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Figure 1. (AE),I1as a function of the heat of vaporization of the mon' ~ 700 cm-l) and atomic gases for hot toluene13 (52 OOO cm-'), a ~ u l e n e (16 ethylcycl~heptatriene'~ (41 600 cm-'). Lines are polynomial best fit to the data.
philosophy is that the boiling temperature is a readily available physical quantity which imbedswithin itself all the various factors which affect intermolecular interactions, namely size, weight, polarizability, cross section, and perhaps more. This correlation, interesting as it is, is hardly useful for large molecules since they all have unit efficiency. A quantity of much more basic importance is the average energy transferred per collision ( AE)an,which can be used for comparison with the first moment of the collision energy transfer probability distribution function, used in the solution of the master equation to obtain, for example, the pressure falloff behavior or the temporal decay rate of internal energy. The average energy transferred in a collision is also related to the second moment of the distribution'* (AE),, which provides useful information on the nature of the distribution function. In the following sections we develop some relations which lead to good and simple correlations between energy transfer and heat of vaporization and boiling temperature.
Results and Discussion The long range interactions which govern the intermolecular energy transfer process cannot be correlated directly with the two-parameter Lennard-Jones potential. This spherically symmetric potential does not take in consideration the mutual orientational effects of the reactant and the bath molecules, and the average parameters of this potential, t and u, are no substitute for the detailed potential of interaction. The heat of vaporization does include all the various interactions which also affect the energy transfer process, namely, the size, polarizability, and number of internal modes as well as the parameters of the spherical potential such as t and the classical turning point of the potential.
3000
Figure 2. (AE),IIfor hot ethylcycloheptatrieneI5 (41 600 cm-I) as a function of the heat of vaporization of polyatomic bath gases. Line is a best fit to the data.
In the limiting case where the Lennard-Jones potential is assumed to apply, AHvapassumes a simple form
AHvap= constant Z t
ETHYL-C-HEPTATRIENE GROUP 1
2.24
2000 , cm-'
(2-1)
where Z is the number of interacting neighbors. To show the correlation between heat of vaporization and average energy transferred in a collision ( AE)aIIand for easy handling of the data, the bath molecules were divided into groups. Group 1 represents monatomics, group 2 diatomics, group 3 triatomics, and group 4 polyatomics. The dependence of ( on the heat of vaporization of the bath molecule is given in Figure 1 for hot toluene,13azulene,14and ethyl~ycloheptatriene'~ colliding with monatomics (group 1) where the lines are the best fit through the data. The saturation type curve is typical and appears in systems such as thermal isomerization and chemical activation. Figure 2 shows ( AE)allfor ethylcycloheptatriene15colliding with polyatomic bath gas molecules, group 4. There is a clear correlation between the two quantities. Correlation between AH, and Tb.In its simplistic form AHv and the boiling temperature Tb are related through the entropy of vaporization, which is almost constant for many normal liquids (Trouton's rule). Deviations from Trouton's rule were addressed by providing empirical expressions to correlate A H v with Tb. Hildebrand and Scott @vel6for the heat of vaporization (cal/mol) at 298 K and 1 atm AHv = -2950
+ 23.1Tb + 0.02Tb2
(2-2)
Kistiakow~ky'~ gives a different relation between the two variables AHv = 8.15Tb
+ 4.576Tb log Tb
(2-3)
Figure 3, taken from ref 16, shows the correlation between heat of vaporization and boiling temperature for many substances. Figure 4 shows the same correlation for the monatomic and diatomic gases. The slope of the line is 18.5 cal/K, close to Trouton's rule value. The outstanding feature is the almost linear relation between the two quantities for all molecules and the perfect linearity for the inert monatomic gases. So, whatever the interactions, a point we return to later, the ratio of the two quantities, the entropy of evaporization,is almost constant. We are justified therefore in replacing AHvby TbrHildebrand's rule notwithstanding,in future correlations and will do so since Tbis the more easily obtainable physical quantity. Correlation between (AE).,, and Tb. The preceding sections lead us naturally to question whether there is a correlation between the average energy transferred per collision (AE), and the boiling temperature of the BATH gas molecule. Figures 5 and 6 show that this is true for monatomic gases. Regardless of the energy content or the nature of the excited molecule, saturation curves, with He the least efficient collider, are observed. This should be contrasted with the fact that t/u vs Tbis linear, indicating that
6310 The Journal of Physical Chemistry, Vol. 96, No. 15. 1992
Oref
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