Variational Solution of the 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 $\mathrm{\psi \hspace{0.17em}=\hspace{0.17em}A\; [}\exp {\mathrm{(\gamma c}}^2{\mathrm{)\hspace{0.17em}+\hspace{0.17em}ac}}^2\mathrm{\hspace{0.17em}+\hspace{0.17em}b]}$ where $a$ and $b$ are fixed by the auxiliary conditions and $A$ by the equation of constraint and $\gamma$ 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 $\text{ε*\hspace{0.17em}/\hspace{0.17em}kT\hspace{0.17em}=\hspace{0.17em}20\hspace{0.17em}→\hspace{0.17em}30}$ where $\mathrm{\epsilon *}$ is the activation energy; however, the correction is still negligibly small. Small improvements are obtained for fast activated reactions and for free‐radical recombinations.