Effective number of oscillators

Unfortunately, as with all oversimplified theories, there are limitations for the application of the latter equation to ions close to the dissociation threshold, hi these cases, the number of degrees of freedom has to be replaced by an effective number of oscillators which is obtained by use of an arbitrary correction factor. [7] However, as long as we are dealing with ions having internal energies considerably above the dissociation threshold, i.e., where E - Eo)/E = 1, the relationship is valid and can even be simplified to give the quasi-exponential expression... 

In this high-pressure or equilibrium limit these models cannot be distinguished and the justification of a choice between them has to be made on other grounds. They will differ under conditions in which the internal energy distributions P E) and P Eij. . . , E ) are no longer the equilibrium values. But then the differences will be not in form but in the effective number of oscillators assignable to the decomposition. 

The other parameter was again modified to take account of the effective number of oscillators and the zero point energy, using the Whitten-Rabinovitch formula... 

The primary step is similar to that of azomethane. One interesting distinction from axomethane is the shift of the fall-off region to lower pressures and the increase in the effective number of oscillators (from 12 to 23). This is due to the higher efficiency of the C-F as compared to the C-H vibrations to exchange energy with the critical C-N stretching motion. 

Numerical evaluation of the Kassel integral permitted a comparison between theoretical and experimental fall-off behaviour . With an average molecular diameter of 5.5 A the calculated rate coefficient-azoethane pressure curve showed the best agreement with experiment at an effective number of oscillators of 18, somewhat less than half of the maximum 2N— 6. Because of the complexity of the reaction the experimental curve is probably in error, rendering comparison unreliable. Similar calculations for azomethane using the earlier uninhibited kinetic data showed best agreement with experiments at a molecular diameter of 4.7 A and an effective number of oscillators of 12, one half of the total normal modes of vibrations. 

The theoretical curve defined by the reciprocal of Eq. (4-14) is calculated as follows. is obtained from a plot of In/oo versus 1/T the high-pressure rate constant 4 0 is measured directly or obtained from extrapolation of the plot of l//exp versus 1/P the effective number of oscillators n/2 is obtained by locating the pressure at which begins to fall off it is assumed that at this pressure the rate of activation is equal to the first-order rate of reaction, that is, ac W = > exp 9 relation which will yield a value of n/2 after insertion of the experimental value of and a reasonable value for the elastic-hard-sphere diameter d. 

Since only few electronic oscillators contribute significantly to the source in the expansion (Cl), the summation can be truncated at some effective number of oscillators M Mwithout sacrificing accuracy. 

Oscillator Strength. Another useful quantity is the oscillator strength/ or "/-number." It is defined as the "effective number of electrons that can oscillate." In classical electromagnetic theory the intensity of absorption is given by... 

Oscillator strength (f Number) A measure of the intensity of a spectral band a classical concept (giving the effective number of electrons taking part in a certain transition) adapted to wave mechanics. For a transition between state i and state j,... 

Fig.4.4-5a demonstrates the effect of p on the transient response of C2 and Cn, i.e. the first and the last perfectly mixed reactors in the upper plug flow reactor in Fig.4.4-5. The tracer was introduced into reactor 2. It is obsereved that by increasing p from 80 to 800, the number of oscillations for reaching the steady state concentration 1/22 in the system is increased. It should also be noted that for p = 800, the distance between two successive peaks corresponding to C2 and Cn is...

Treindl and Fabian (1980) studied the effect of oxygen on parameters of the B-Z reaction in the presence of Ce(IV)/Ce(III) redox catalyst and malonic acid, citric add or 2,4-pentanedione as substrates. They concluded that the effect of oxygen was in its catalytic influence on the oxidation of the substrate with Ce(IV) ion. Under this influence, the number of oscillations as well as the induction period and the first oscillation period diminished. 

D Alba and Serravalle (1981) compared the effect of various catalysts on the B-Z reaction by using an electrochemical method, [see Botre et al. (1981)]. The catalysts compared are Ce(S04)2, ferroin and FeS04 and their various combinations. The number of oscillations, the average frequencies and the duration of oscillations were measured and tabulated for different cases. 

Fig. 16. Product relative translational energy distribution72 P(E T) points are experimental solid line RRKM calculations using an effective S( 16-9,151 and 13-9, respectively) dashed line RRKM calculations using full number of oscillators (S = 27).

Z°eff is the effective number of valence electrons in the molecule, df/dE is the differential form of the oscillator strength fn from Equation 1. 

In addition to an optimization of the set of examined frequencies and the number of oscillations, beneficial effects of a separate determination of the heat exchange with environment were identified. Either by an additional heating measurement or by an evaluation of direct signal components, a precise identification of the heat exchange coefficient is possible. This reduces markedly the required range of frequencies for the accurate identification of heat capacity and thermal conductivity. 

FIGURE 54 The energy-dependent oscillator strength or the effective number of valence electrons per atom contributing to interband transitions in amorphous diamond, crystalline diamond, and graphite [132]. 

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