Quaternion-Based Kinetic Operator for Modeling Competitive and Incompatible Chemical Reactions Incorporating Angular Error Probabilities

We propose a novel kinetic operator based on quaternion probability formalism combined with angular error metrics to model chemical reaction systems involving multiple competing and mutually incompatible pathways. The method introduces a quaternion representation of reaction probabilities, where each component corresponds to an independent yet non-coexisting reaction pathway. The angular error is derived from experimentally measurable quantities by treating the theoretical and experimental results as orthogonal components of a right triangle, allowing a geometric interpretation of the deviation. The integration of this angular error into the quaternionic probability framework enables the identification of the dominant reaction under competition, even in the absence of equilibrium. The approach is extended to account for fractional-order kinetics and a modified Arrhenius operator, allowing for complex temperature-dependent behaviors. Two case studies are presented: (1) competition between three reactants producing two mutually exclusive products, and (2) competition between amino acids for a catalyst. Numerical simulations demonstrate the feasibility and potential of the model in predicting reaction dominance under noisy experimental conditions. This framework opens new avenues for integrating algebraic operator theory, fractional calculus, and statistical geometry into reaction kinetics.