On the Velocity Distribution in a Chemically Reacting Gas

Abstract

The departure of the velocity distribution function from the Maxwellian form and the corresponding correction to the reaction-rate formula have been calculated for a gas undergoing a slow bimolecular reaction. Kinetic theory methods based on the Boltzmann transport equation have been used. This work differs from a previous calculation by Prigogine and Xhrouet in two respects: (1) the energy dependence of the reaction cross section is taken in a form that is consistent with the conventional simple collision theory (in the previous work this was not the case and the results are sensitive to this choice), and (2) a simplified derivation of the important formulas is given, in which we follow the original methods of Chapman based on the equations of change instead of the integral-equation techniques of Enskog. The calculations take into account the effect of the reaction in depleting the tail of the Maxwell distribution, but do not take into account the effects of the heat of reaction. It is found that when k T is one-fifth the activation energy, the reaction rate is 8% less than that predicted by the conventional simple collision theory.

Nonequilibrium Effects in a Bimolecular Chemical Reaction in a Dilute Gas

“…Introducing an additional term in the perturbation solution of the Boltzmann equation we have shown [23] that the nonequilibrium effect on the rate constant of chemical reaction can be much larger than in Refs. [5][6][7][8] and that similarly as in the Lorentz gas [10][11] the results are accurate for slow reactions only. These results [23] have been also confirmed by the molecular dynamics simulation method developed by Gorecki [24][25][26].…”

“…We will analyze the differences between the nonequilibrium values of kA and the equilibrium values of kA(0). These rate constants can be calculated as and We adapt the line-of-centers model due to Present [5] where k is the unit vector in the line-of-centers of colliding molecules A modelled as hard spheres with a diameter dA, sF is the steric factor accounting for the fact that in order to have the reactive collisions it is necessary to have a peculiar orientation of the colliding molecules, g* is the threshold relative velocity connected with the threshold energy of the reaction (1) by the relation where mA is the molecular mass. We introduce the reduced dimensionless threshold energy ε* Introducing for σ Ar the line-of-centers model (see Eq.…”

“…Prigogine and coworkers [1,2] have used the perturbation Chapman-Enskog [3,4] method of solution of the Boltzmann equation and they have shown that this effect diminishes the rate constant of chemical reaction. Present [5,6] has analyzed this effect for the reactive crosS-section described by the line-of-centers model, which is very suitable for such an analysis. In analytical results for the rate constant of chemical reaction, described by this model with total neglecting of products, the nonequilibrium effects do not exceed 8%, whereas for the Maxwell-Boltzmann distribution function this model leads to the Arrhenius expression [5,6].…”

“…Present [5,6] has analyzed this effect for the reactive crosS-section described by the line-of-centers model, which is very suitable for such an analysis. In analytical results for the rate constant of chemical reaction, described by this model with total neglecting of products, the nonequilibrium effects do not exceed 8%, whereas for the Maxwell-Boltzmann distribution function this model leads to the Arrhenius expression [5,6]. Shizgal and Karplus [7] have confirmed the results of Present.…”

Contributions of steady heat conduction to the rate of chemical reaction

“…[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] Under nonequilibrium situations such as gases under the heat conduction and the shear flow, their nonequilibrium effects on the rate of chemical reaction have attracted attention among researchers. [12,13,14,15,16,17,18,22,23] However, to derive the effect of steady heat flux on the rate of chemical reaction in the line-of-centers model, we need the explicit velocity distribution function of the steady-state Boltzmann equation for hard-sphere molecules to second order in density and temperature gradient.…”