Vibrational distribution Treanor

The Gibbs distribution (3-35), together with (3-34) and (3-36), leads to a nonequilibrium vibrational distribution of diatomic molecules known as the Treanor distribution (Treanor, Rich, Rehm, 1968) ... 

Figure 3-3. Comparison of Treanor and Boltzmann vibrational distribution functions.

One solution of the non-linear kinetic equation Jy iE) = 0 with flux (3-128) is again the Treanor distribution function (3-124), which is, however, not the only solution of the equation another solution, which is a plateau-like vibrational distribution, will be discussed in some detail in the next section. 

The vibrational distribution is determined by the Treanor function (3-124) at lower energies ... 

Vibrational distribution similar to that presented in Fig. 3-7 also takes place in the regime of intermediate excitation, which occurs when < Syy, but the Treanor effect is... 

The population of vibrationally excited states at the Treanor minimum E = Exr) is large in this regime, and the non-linear resonant W exchange dominates and provides a plateau at > i Tr even though < 5yy. At low levels E < E-y ), the linear non-resonant W exchange dominates over the non-linear one. It does not change the vibrational distribution function, however, because both non-resonant and resonant components of the W exchange result in the same Treanor distribution at < Ej. ... 

The vibrational distribution in N2 at Ty = 3000 K and different translational temperatures is shown in Fig. 3-8. Comparison of different theoretieal vibrational distribution fime-tions for the same eonditions is presented in Fig. 3-9 (dashed lines designate (E) = 1). In addition to the differenee between Treanor and Boltzmann distributions. Fig. 3-9 also shows... 

Numerically, E (15-20) x quasi continuum actually takes place at very high levels of excitation of the asymmetric vibrational mode, close to the dissociation energy (see Fig. 5-14). Thus, most of the vibrational distribution function relevant to CO2 dissociation in this case, in contrast to the one-temperature approach, is not continuous but discrete. The discrete distribution function /(Va, Vs) over vibrational energies (5-16) can be presented analytically according to Licalter (1975a,b, 1976) in the Treanor form ... 

Treanor Effect for Low Discrete Leveis of CO2 Symmetric Osciiiations. Explain the vibrational distribution function related to low discrete levels of CO2 symmetric modes, presented in Fig. 5-15. Why does this vibrational distribution clearly combine two exponential functions with two separate temperatures - a vibrational one and a rotational-translational one ... 

This kind of process has first been described by Tre-anor. Rich, and Rehm [1,24] and has been used successfully to explain the vibrational distribution in diatomic gases. It is called "enharmonic VV-pumping" and is now very often referred to as "Treanor pumping". For quantitative treatment we refer to the literature [3,6]. However, we have to add a few more points to the discussion of the CO-plasma. 

The expression (83) yields the non-equilibrium quasi-stationary Treanor distribution Treanor et al. (1968) generalized for a multi-component reacting gas mixture taking into account anharmonic molecular vibrations and rapid exchange of vibrational quanta. 

When the system of vme is solved by retaining v-v terms only, under an initial disequilibrium between the ground and the first vibrational level, defined by a vibrational temperature 0, = Ei0/kln (Nq/N, one obtains a Treanor distribution in the form10 ... 

Inclusion in the vme of v-t terms only produces a Boltzmann distribution at the gas temperature Tg. The inclusion of e-v, v-v and v-t terms produces the distributions of Fig. 8 e-v processes tend to establish the non equilibrium vibrational temperature 8 j > Tg. v-v processes, which have rates several orders of magnitude larger than v-t ones, at low vibrational quantum numbers, tend to create a Treanor distribution up to approximately v = v. Above a given level, v-t processes dominate the v-v ones and determine a Boltzmann tail characterized by a temperature approaching Tg. The plateau, which extends from approximately v = vi up to the onset of the Boltzmann tail, is connected to the near resonant v-v terms. The dependence of kd on pressure (Fig. 9) can be interpreted along these lines by examining the normalized distributions of Fig. 10. The same arguments apply to the data of Fig. 11. 

Figure 28 shows the N distribution at t = tmax and for times t > tmax. In Fig. 28a we have also reported the instantaneous Treanor s distributions at the relevant 0i(t) s values. These distributions, which are analytical solutions of the system of vibrational master equations including v-v rates only, should represent upper limits to the actual Nv distributions, which include all v-t deactivating processes. 

One appreciates that the low lying vibrational levels only satisfay Treanor s distributions, while from approximately v - 10 on a plateau is present in the N distributions, the population of which is much higher than the corresponding Treanor s values. Concentrations belonging to the plateau grow with a N0 law, while the Nv/Np+i ratio approximately fits Eq. (14). This plateau should therefore be attributed to the recombination process, as outlined above. 

At t = tmax the vibrational temperature 01 reaches a value of 1650 K, the system being able to sustain appreciable vibrational populations of higher levels by means of v-v exchanges. The N distribution (see Fig. 29) is, at t = tmax, very close to the corresponding Treanor s distribution which in this case overestimates the populations. As time evolves, the concentration of deactivating oxygen atoms increases and the... 

Marrone-Treanor Model (Marrone Treanor, 1963). This model assumes an exponential distribution (with parameter U) of probabihties of dissociation from different vibrational levels. If t/ c , the probabilities of dissociation from all vibrational levels are equal. The non-equilibrium dissociation factor Z can be found within the framework of the model... 

Here Xe is the coefficient of anharmonicity and B is the normalizing factor. Comparison of the parabolic-exponential Treanor distribution with the linear-exponential Boltzmann distribution is illustrated in Fig. 3-3. A population of highly vibrationally excited levels at TV > To can be many orders of magnitude higher than that predicted by the Boltzmaim distribution even at vibrational temperature. The Treanor distribntion resnlts in very high rates and energy efficiencies of chemical reactions stimulated by vibrational excitation in plasma. 

The Treanor distribution function (see Section 3.1.8) makes the W flux (3-122) equal to zero. Thus, the Treanor distribution is a steady-state solution of the Fokker-Planck kinetic equation (3-116), if W exchange is a dominating process and the vibrational temperature Tv exceeds the translational temperature Tq ... 

The exponentially parabolic Treanor distribution function, which provides a significant overpopulation of the highly vibrationally excited states, was illustrated in Fig. 3-3. To analyze the quite complicated W flux (3-122), it can be divided into linear and non-linear components ... 

Here F is a constant proportional to quantum flux along the vibrational spectrum. In addition to the Treanor distribution, solving kinetic equation (3-132) also gives the hyperbolic plateau distribution ... 

Distributions (3-t67) for the mixture of two isotopes are illustrated in Fig. 3-14. The population of lower vibrational levels is larger for the heavier isotope (1, usually small additive), which corresponds to the Treanor effect and relation (3-165). The situation is opposite at higher levels of excitation, where the vibrational population of a relatively light isotope exceeds that of a heavier one. This phenomenon is known as the reverse isotopic effect (Macheret et al., t980a,b). 

Non-Equilibrium Statistical Treanor Distribution for Vibrationally Excited Molecules. Based on the non-equilibrium Treanor distribution function, find the average value of vibrational energy taking into account only relatively low vibrational levels. Find an application criterion for the result (most of the molecules should be located in the vibrational levels lower than the Treanor minimum). 

The situation changes considerably when the rate of the VV energy exchange in collisions of relaxing molecules exceeds that of VT processes. Then, at the first stage, the quasi-resonant W exchange results in a quasi-stationary distribution with the total number of vibrational quanta equal to Ef/hco. This quasi-stationary distribution function, sometimes called Treanor distribution [485], is of the form... 

Figure 9.5 The relative population of vibrational states of CO in an O2—CS2—He flame plotted vs. the vibrational quantum number vfor different times after ignition. The COM molecules are mainly produced by the very exoergic, see Figure 5.17, O + CS reaction. The Treanor distribution, that neglects the V—T relaxation at high vs, is shown as a dashed line [adapted from S. Tsuchiya, N. Nielsen, and S. H. Bauer, J. Phys. Chem. 77, 2455 (1973)].

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