Higher-order large-time asymptotics for a reaction of the formnA+mB→C

Abstract

This study examines the large time asymptotic behavior for the family of reactions of the form nA+mB-->C when the reactants A and B are initially separated. Once the reactants are brought into contact they are assumed to react with a kinetic rate proportional to A;{n}B;{m} . A planar reaction front forms and usually moves away from its initial position to invade one of the reactant solutions. The position of the reaction front relative to the initial position where the reactants were put in contact x_{f} for large times t , is found theoretically to satisfy the expansion x_{f}=2sqrt[t][alpha+alpha_{2}t;{-2sigma}+alpha_{3}t;{-3sigma}+O(t;{-4sigma})] , where sigma=1(n+m+1) and alpha , alpha_{2} , and alpha_{3} are constants. This expansion is valid provided that n and m are positive constants less than or equal to 3. The implication of this is that when n+m not equal3 , x_{f} either tends to zero or infinity, while if n+m=3 then there exists the possibility of x_{f} tending to a finite nonzero constant. For fractional order kinetics, n and m are arbitrary positive constants, however, for simple reactions n and m are positive integers. Hence, the reaction A+2B-->C is the only reaction of the form nA+mB-->C with n and m being positive integers less than 4 in an infinite domain that can lead to a reaction front approaching at a finite but nonzero distance from the position at which the two liquids first met.