Rotational equilibrium assumption

The rotational equilibrium assumption implies substantial conceptual and technical simplifications of the kinetic description of chemical lasers. In this section we shall consider the validity and consequences of this assumption and compare the very different lasing mechanisms and efficiencies implied by fast and slow rotational relaxation. 

The physical content of the rotational equilibrium assumption is that rotational (like translational) relaxation is practically instantaneous on the time scale of the other rate processes in the laser cavity including, in... 

In addition to simplifying the population rate equations the rotational equilibrium assumption also leads to fewer photon equations (2). Explicitly, instantaneous rotational relaxation implies that in each vibrational band, 1, there is only one active transition o,1-mj—1, 7 at... 

The rotational equilibrium assumption will be valid when X l, where... 

For a normal machine, this assumption is valid after the rotating element has achieved equilibrium. When the forces associated with rotation are in balance, the rotating element will center the shaft within the bearing. However, several problems directly affect this self-centering operation. First, the machine-train must be at designed operating speed and load to achieve equilibrium. Second, any imbalance or abnormal operation limits the machine-train s ability to center itself within the bearing. 

The model [39] was developed using three assumptions the conformers are in thermodynamic equilibrium, the peak intensities of the T-shaped and linear features are proportional to the populations of the T-shaped and linear ground-state conformers, and the internal energy of the complexes is adequately represented by the monomer rotational temperature. By using these assumptions, the temperature dependence of the ratio of the intensities of the features were equated to the ratio of the quantum mechanical partition functions for the T-shaped and linear conformers (Eq. (7) of Ref. [39]). The ratio of the He l Cl T-shaped linear intensity ratios were observed to decay single exponentially. Fits of the decays yielded an approximate ground-state binding... 

As for the theoretical treatment, we could only try to include the eleetrostatie solute-solvent interaetions and, in faet, we corrected the electronic potential energies for the solvation effeets by simply adding as calculated according to the solvaton model [eq. (2)]. The resulting potential curves are to be seen as effective potentials at equilibrium, i.e. refleeting orientational equilibrium distributions of the solvent dipoles around the eharged atoms of the solute molecule. In principle, the use of potentials thus corrected involves the assumption that solvent equilibration is more rapid than internal rotation of the solute molecule. Fig. 4 points out the effects produced on the potential... 

The overall OD vibrational distribution from the HOD photodissociation resembles that from the D2O photodissociation. Similarly, the OH vibrational distribution from the HOD photodissociation is similar to that from the H2O photodissociation. There are, however, notable differences for the OD products from HOD and D2O, similarly for the OH products from HOD and H2O. It is also clear that rotational temperatures are all quite cold for all OH (OD) products. From the above experimental results, the branching ratio of the H and D product channels from the HOD photodissociation can be estimated, since the mixed sample of H2O and D2O with 1 1 ratio can quickly reach equilibrium with the exact ratios of H2O, HOD and D2O known to be 1 2 1. Because the absorption spectrum of H2O at 157nm is a broadband transition, we can reasonably assume that the absorption cross-sections are the same for the three water isotopomer molecules. It is also quite obvious that the quantum yield of these molecules at 157 nm excitation should be unity since the A1B surface is purely repulsive and is not coupled to any other electronic surfaces. From the above measurement of the H-atom products from the mixed sample, the ratio of the H-atom products from HOD and H2O is determined to be 1.27. If we assume the quantum yield for H2O at 157 is unity, the quantum yield for the H production should be 0.64 (i.e. 1.27 divided by 2) since the HOD concentration is twice that of H2O in the mixed sample. Similarly, from the above measurement of the D-atom product from the mixed sample, we can actually determine the ratio of the D-atom products from HOD and D2O to be 0.52. Using the same assumption that the quantum yield of the D2O photodissociation at 157 nm is unity, the quantum yield of the D-atom production from the HOD photodissociation at 157 nm is determined to be 0.26. Therefore the total quantum yield for the H and D products from HOD is 0.64 + 0.26 = 0.90. This is a little bit smaller ( 10%) than 1 since the total quantum yield of the H and D productions from the HOD photodissociation should be unity because no other dissociation channel is present for the HOD photodissociation other than the H and D atom elimination processes. There are a couple of sources of error, however, in this estimation (a) the assumption that the absorption cross-sections of all three water isotopomers at 157 nm are exactly the same, and (b) the accuracy of the volume mixture in the... 

Let us now apply the technique in some specific cases. The existence of optical activity of several compounds in solution has been explained due to the presence of several active forms of the compound in equilibrium with each other and various assumptions about the forms were also put forward. The equilibrium between the different forms depended on external conditions. But a definite explanation was put forward in 1930 about tartaric acid and it was said that the molecule exists in the following three conformations and each of which makes a certain contribution to the rotation observed. 

Basilevsky et al. [1982] proposed a mechanism of ionic polymerization in crystalline formaldehyde that was based on Semenov s assumption [Semenov, 1960] that solid-state chain reactions are possible only when the products of each chain step prepare a configuration of reactants that is suitable for the next step. Monomer crystals for which low-temperature polymerization has been observed fulfill this condition. In the initial equilibrium state the monomer molecules are located in lattice sites and the creation of a chemical bond requires surmounting a high barrier. However, upon creation of the primary cation (protonated formaldehyde), the active center shifts toward another monomer, and the barrier for addition of the next link diminishes. Likewise, subsequent polymerization steps involve motion of the cationic end of the polymer toward a neighboring monomer, which results in a low barrier to formation of the next C-0 bond. Since the covalent bond lengths in the polymer are much shorter than the van der Waals distances of the monomer crystal, this polymerization process cannot take place in a strictly linear fashion. It is believed that this difference is made up at least in part by rotation of each CH20 link as it is incorporated into the chain. 

The Forster cycle calculations on the dissociation of 9-anthroic acid (Vander Donckt and Porter, 1968a) have been criticized (Werner and Hercules, 1969) on the grounds that excited state rotation of the carboxyl group invalidates the equal entropy assumption of the Forster cycle. Pace and Schulman (1972) found that this dissociation equilibrium was not established in the excited-state but obtained a p/ (Sj )-value from fluorescence titrations for protonation of the carboxyl group. As expected its basicity was increased in the Sj state a Apisf-value of 6 units was recorded. 

Extending the theory to interpret or predict the rovibrational state distribution of the products of the unimolecular dissociation, requires some postulate about the nature of the motion after the unimolecularly dissociating system leaves the TS on its way to form products. For systems with no potential energy maximum in the exit channel, the higher frequency vibrations will tend to remain in the same vibrational quantum state after leaving the TS. That is, the reaction is expected to be vibrationally adiabatic for those coordinates in the exit channel (we return to vibrational adiabaticity in Section 1.2.9). The hindered rotations and the translation along the reaction coordinate were assumed to be in statistical equilibrium in the exit channel after leaving the TS until an outer TS, the PST TS , is reached. With these assumptions, the products quantum state distribution was calculated. (After the system leaves the PST TS, there can be no further dynamical interactions, by definition.)... 

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