VARIATIONAL SOLUTION OF CHEMICAL-KINETIC BOLTZMANN EQUATION

Variational principles are formulated for the linearized molecular Boltzmann equation and it is shown that if a perturbation of the Maxwell distribution maximizes the nonequilibrium correction to the reaction rate, subject to an equation of constraint, it is a solution of the integral equation describing a gas-phase reaction. A corollary is that the Chapman-Enskog-Burnett subapproximations generate lower bounds for the nonequilibrium correction to the reaction rate. Since it has been shown previously that the Chapman-Enskog method is useless in the treatment of slow activated reactions because the Sonine polynomial expansion diverges, the variational method is applied to the same problem using a trial function that is not in polynomial form. We use the trial perturbation function ψ=A [exp(γc2)+ac2+b] where a and b are fixed by the auxiliary conditions and A by the equation of constraint and γ is an adjustable variation parameter. An order-of-magnitude improvement and increase in the nonequilibrium correction over the previous two-polynomial approximation is obtained for ε*/kT=20→30 where ε* is the activation energy; however, the correction is still negligibly small. Small improvements are obtained for fast activated reactions and for free-radical recombinations.