Treanor distribution

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. 

In molecular nitrogen large e-v and v-v rates are associated with unusually low v-t rates. This represents a crossing of conditions favourable to the PVM, as discussed in 2.3.1. This situation is immediately reflected in the N distributions of Fig. 19. These distributions consist of a Treanor distribution followed by a long plateau extending up to the dissociation limit. The Boltzmann tail has practically disappeared, as consequence of the low v-t rates, and the influence of Tg on the distribution is limited. The behaviour of the NP distributions is reflected on the k values reported... 

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) ... 

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 ... 

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. 

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 ... 

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. ... 

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 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. 

For the generalized Treanor distribution (83), the multi-temp>erature rate coefficients of exchange reactions occurring as a result of collisions of two molecules take the form... 

Using the Treanor distribution (83) for n,-, the factor Z is given by the relation... 

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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