Brief Communication

Journal of Mathematical Chemistry

, Volume 49, Issue 2, pp 328-334

First online:

Alternate solutions for two particular third order kinetic rate laws

  • Brian Wesley WilliamsAffiliated withDepartment of Chemistry, Bucknell University Email author 

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Alternate solutions to third order rate law expressions which are first order in one reactant and second order in another for the generalized reaction \({{aA}+{bB}\mathop \rightarrow \limits^{k_A }{P}}\) with a limiting reactant and for simple cubic autocatalysis \({{A}+{ 2B}\mathop \rightarrow \limits^{k_A }{3B}}\) are presented and discussed. These solutions are expressed in terms of the Lambert function W[x] with argument x(t) = α exp(αβ t), where α and β are empirical parameters. These expressions permit the concentrations of the reactant species to be explicitly represented as dependent functions of time. For the generalized reaction, the solution involves considering exactly which species is limiting, so that the correct branch of the Lambert function can be determined. For simple cubic autocatalysis, the solution is shown to be consistent with the characteristics associated with clock reactions. An exact expression for the “induction time” associated with a clock reaction described by this mechanism is also derived.


Third order rate law Kinetics Lambert W Cubic autocatalysis Clock reaction