Treanor distribution function

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 ... 

The W-exchange rate coefficient can be expressed as vv(-E) exp(-5vv ) (see Section 2.6.5), where 5w is an adiabatic relaxation parameter. Solution of the linear kinetic equation j E) = 0 with flux (3-126) gives the Treanor distribution function (3-124). 

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. 

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). 

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

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. ... 

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 ... 

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... 

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

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