Average vibrational energy

Next we come to the case of vibrational energy. At this point the size of the vibrational quanta is too large to use the continuous integration method and we must use a discrete summation. 

In addition, we remind ourselves that we are counting the energy starting above Eq = —. The discrete summation of the terms in Q is essentially the same as Planck used for the treatment of 

Make note that this is the average energy above the zero-point energy. In this case, Cy is not easy to calculate. Define x = assume one mole of oscillators so that (Nji y k = R. 

FIGURE 1 S.1 The vibrational heat capacity of a quantized oscillator as a function of x = —. Note that the high temperature hmit will occur at x = 0. 

FIGURE 15.2 A graph of the heat capacity, Cy, for HCl using a fundamental of 2990.2 cm 

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Figure 8. Photoelectron spectrum (PES) and Penning ionization electron spectrum (PIES) of nitric oxide radical. Average vibrational energy spacing of the first band amounts to 285 and 284 cm", respectively (104).

By applying Eqs. (4) or (36) we can calculate the average vibrational energy of a molecule with N vibrational modes, and we do this with respect to the zero of the vibrational potentials, implying that we will include all zero-point energies ... 

Table I lists some characteristic wave lengths from the work of Gregory (9). The calculations of f shown in Figure 2 are taken from the work of Clayfield and Lumb.(lO) By using these calculations one can determine the attractive energy per pair of particles at various separation distances, and determine for any particular value gf Aj2i> Xj, and radius (a) the critical value of H that makes U j 21= kT, where k is the Boltzmann constant and kT is the average vibrational energy of a pair of particles flocculated at separation distance H. If Uj2l is greater than -kT the particles will nearly always bounce apart on collision, but if it is less than -kT the particles tend to flocculate.

The distribution at f = 0 is divided by a factor of 3. The positions of the distributions on the vertical axis are determined by the average vibrational energy for that state. (From Ref. 25.)... 

Both the UV excitation and IRF methods depend critically upon calibration curves to determine the average vibrational energy < > within the molecule. Despite much theoretical and experimental effort by both sets of... 

The photolysis of CH2CO at 193 nm has also been studied by Fujimoto et al. (50). A CO laser was used to probe the vibrational excitation of the CO produced. The product CO contains an average vibrational energy of 6.4 kcal/mol. The vibrational population distribution corresponds to about 4000 K in good agreement with Sonobe and Rosenfeld (49,50). 

The results indicate that the fraction of the radicals formed in higher vibrational levels increases as the available energy increases but ratio of the average vibrational energy to the available energy remains constant. The rotational excitation also remains fairly constant and the rotational distribution can be described by a Boltzmann distribution. The latter obser-... 

The unimolecular 1,2-HF and 2,3-HF elimination reactions of CF3CHFCH3 have been characterized using the chemical activation technique for an average vibrational energy of 97 kcalmol-1.36 The transition state for 1,2-HF elimination has a two-fold larger pre-exponential factor than that for 2,3-HF elimination, because three F atoms attached to carbon atoms of the four-membered ring have lower frequencies than those in a CF3 group. 

In the (classical) high-temperature limit, when hi/s -C kBT, the exponential can be expanded to first order and the average vibrational energy is kBT. The approximation hvs -C kis l is, however, not well satisfied for typical molecular vibrational frequencies, except at temperatures that exceed several thousand degrees. The average rotational energy of a rigid rotor is... 

We assume that the rate constant is calculated according to transition-state theory. Calculate the barrier height Eq (in the unit kJ/mol), using Ea and the exact average vibrational energies. 

The model we are to consider is that of an adsorbed molecule capable of free translation parallel to the surface and vibrating normal to the surface. The average vibrational energy aEyih is now not equated to zero we write instead... 

Further evidence of the formation of transient anions in the condensed phase Is provided by resolving the vibrational structure of the state of N2 In the v=l, 2, and 3 decay channels In the solid. From this result, shown In Fig. 5b, we can estimate (j4) the average vibrational energy (0.29 eV) and the lifetime (a 3x10" 5 sec.) of the resonance. These values, are close to those found in the gas phase (0.27 eV and 3.5xl0"l5 sec (30), respectively). Furthermore, the shift in the energy of the oscillatory structure... 

Average vibrational energies, dissociation probabilities and effectiveness of vibrational energy for methane with a fixed kinetic energy. 

Figure 11 Variation in average vibrational energy (cm1) vs. time (ps) for relaxation from the CH (v = 3) overtone. The energy lost from the initially excited CH stretch, and the energy gain for the other local modes are shown. Total energies are shown for 6 CH stretch, 6 CC stretch, 6 CCH wag, and 3 CCC bend modes.

Average Vibrational Energy of the CO Formed in Some 0( P) +Alkyne/Allene and Some Photodissociation Reactions... 

Experiments were also carried out using isotopically labeled 0 atoms. Under these conditions, the C 0 laser probe cannot detect the C 0 produced by the attack on the center C atom, the entire CO absorption signal is due to the C 0 formed from the already present in the CjOj molecule. The C 0 vibrational distribution thus obtained is also shown in Fig. 20, and is noticeably colder than that produced by the unlabeled O atoms. This impUes that vibrationally hotter CO is produced from the center C atom. The deconvolution of these data showed that the average vibrational energy carried by CO from the center C atom amounted to about 15 kcal/mole, compared with only about 4 kcal/mole for the two CO s derived from the C = 0 bonds initially present in the C3O2. 

Figure 5.5 The calculated Boltzmann ro-vibrational energy distribution for isobutane at the two indicated temperatures. At 80 K, the average vibrational energy is reduced to just 17 cm- , while the rotational energy is 83 cm- . The vibrational frequencies were taken from Weitzel etal. (1991).

A more direct method for determining the average vibrational energy in molecules... 

Consider losses of average vibrational energy, Sv = J Ef E)dE. Multiplication of the Fokker-Planck equation (3-116) with the following integration leads to the energy balance equation ... 

The left-hand side of the equation presents the change in average vibrational energy per molecnle, which in one mode approximation can be expressed by the Planck formnla ... 

The balance of average vibrational energy, located on an asymmetric mode of a CO2 molecitle is mostly determined by eV and W -relaxation processes as well as dissociation itself ... 

The expression for the average vibrational energy (5-47) includes strmmation over all CO2 vibrational modes (r) with statistical weights g, when necessary. When the specific... 

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