Sergey Macheret

The formulas[4] are used for estimation of non-equilibrium rate coefficients of the simple-exchange endothermic reaction: ${\displaystyle XY(v)+Z\rightarrow X+YZ}$. Such reactions appear in low-temperature nonequilibrium plasmas where substantial fraction of energy input goes into vibrational excitation of molecules. The formulas are obtained by applying classical mechanics methods for high temperatures and semi-classical approximation for moderate temperatures.

Assumptions:

1. Translational-vibrational non equilibrium: ${\displaystyle T>T_{v}}$ or ${\displaystyle T<T_{v}}$.

2. Collisions are collinear and rotational energy effects are negligible.

3. The duration of collision is much faster than the period of molecule vibration.

4. The vibrational energy obeys Boltzmann distribution.

Formulas and relations:

${\displaystyle k(T,T_{v})=Z(T,T_{v})k^{0}(T)}$

Nonequilibrium factor in the high temperature case (${\displaystyle T\geq \theta }$):

${\displaystyle Z(T,T_{v})=\exp \left[-{\frac {E_{a}-W}{\alpha _{M}T_{v}+(1-\alpha _{M})T}}-{\frac {W}{T}}+{\frac {E_{a}}{T}}\right]}$

${\displaystyle \alpha _{M}={\frac {m_{Y}(m_{X}+m_{Y}+m_{Z})}{(m_{X}+m_{Y})(m_{Y}+m_{Z})}},\quad W=E_{a}(1-{\frac {f_{v}}{\alpha _{M}}})}$

Nonequilibrium factor in the low temperature case (${\displaystyle T<\theta }$):

${\displaystyle Z(T,T_{v})=\left[f_{v}\exp(-{\frac {T_{v}}{\theta }})+(1-f_{v})\exp(-{\frac {T}{\theta }})\right]^{E_{a}/\theta }\exp \left({\frac {E_{a}}{T}}\right)}$