Theory of thermal unimolecular reactions at low pressures. I. Solutions of the master equation

Abstract

The master equation for a thermal unimolecular reaction in gases at low pressures is formulated. Steady-state solutions are derived in analytical form with an exponential model of collisional transition probabilities, (i) for vibrational energy transfer in molecules with variable densities of states, and (ii) for combined rotational and vibrational energy transfer in molecules with variable heights of the centrifugal barriers. Other models of transition probabilities are treated numerically. The diffusion limit of energy transfer is discussed. In all cases, the nonequilibrium populations of excited states and the weak collision efficiency factors βc are calculated.

On collisional energy transfer in recombination and dissociation reactions: A Wiener–Hopf problem and the effect of a near elastic peak

“…[1][2][3][4][5] Our interest in the subject was prompted by studies of ozone whose formation and isotopic effects have been of much recent interest. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] In general, the formation of a molecule AB is described by A + B AB ‫ء‬ , ͑1͒…”

“…[26][27][28][29][30][31][32][33][34][35][36] Analytical treatments of vibrational energy transfer have been given for particular cases. 5,[34][35][36][37][38] In particular, a detailed discussion of the original master equation and of its steady-state approximation is given by Penner and Forst,38 who expressed the solution in terms of hypergeometric functions.…”

“…͑2͒ is usually obtained from the pressure dependence of the reaction rate of the overall reactions ͑1͒ and ͑2͒, using the solution of the collisional master or steadystate equation to fit these experimental reaction rate versus pressure data. 3,5 To this end, a functional form for the collision energy transfer probability, denoted here by Z͑EЈ , E͒, is typically assumed, and its parameters are calculated from the fit. The functional forms used for this purpose are usually the exponential model introduced by Rabinovitch, used in Sec.…”

Average collisional vibrational energy transfer quantities. The exponential model

“…9 Comparison with previous work. The present results may be compared to the 7 functions presented by Troe 9 and BC. As pointed out by BC, Troe's function (eq.…”

“…The general and correct interconversion between these quantities is not facile. A relation was given by Tr6e 8 ' 9 for the EXP model as,…”