TY - JOUR
T1 - Higher-order large-time asymptotics for a reaction of the form nA+mB→C
AU - Trevelyan, Philip Michael Joseph
PY - 2009/1/9
Y1 - 2009/1/9
N2 - 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^nB^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 xf for large times t, is found theoretically to satisfy the expansion xf = 2vt [a + a2t^(-2s) + a3t^(-3s) + O(t^(-4s))], where s = 1 / (n+m+1) and a, a2, and a3 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=3, xf either tends to zero or infinity, while if n+m=3 then there exists the possibility of xf 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.
AB - 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^nB^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 xf for large times t, is found theoretically to satisfy the expansion xf = 2vt [a + a2t^(-2s) + a3t^(-3s) + O(t^(-4s))], where s = 1 / (n+m+1) and a, a2, and a3 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=3, xf either tends to zero or infinity, while if n+m=3 then there exists the possibility of xf 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.
UR - https://journals.aps.org/pre/abstract/10.1103/PhysRevE.79.016105
UR - https://pure.southwales.ac.uk/en/publications/higherorder-largetime-asymptotics-for-a-reaction-of-the-form-nambc(e16ea4b8-2f8b-4bd0-9795-cd6aa5653089).html
U2 - 10.1103/PhysRevE.79.016105
DO - 10.1103/PhysRevE.79.016105
M3 - Article
C2 - 19257105
SN - 1539-3755
VL - 79
JO - Physical Review E
JF - Physical Review E
M1 - 016105
ER -