page_title: Catalytic resonance theory
section_title: Theory
hierarchical_section_title: Catalytic resonance theory / Theory
caption_attribution_description: English: Sinusoidal Resonance Response
context_page_description: In chemistry, catalytic resonance theory was developed to describe the kinetics of reaction acceleration using dynamic catalyst surfaces. Catalytic reactions occurring on surfaces that undergo variation in surface binding energy and/or entropy exhibit overall increase in reaction rate when the surface binding energy frequencies are comparable to the natural frequencies of the surface reaction, adsorption, and desorption.
context_section_description: The basis of catalytic resonance theory utilizes the transient behavior of adsorption, surface reactions, and desorption as surface binding energy and surface transition states oscillate with time. The binding energy of a single species, i, is described via a temporal functional including square or sinusoidal waves of frequency, fᵢ, and amplitude, dUᵢ:
  Other surface chemical species, j, are related to the oscillating species, i, by the constant linear parameter, gamma γᵢ₋ⱼ:
  The two surface species also share the common enthalpy of adsorption, delta δᵢ₋ⱼ. Specification of the oscillation frequency and amplitude of species i and relating γᵢ₋ⱼ and δᵢ₋ⱼ for all other surface species j permits determination of all chemical surface species adsorption enthalpy with time.
  The transition state energy of a surface reaction between any two species i and j is predicted by the linear scaling relationship of the Bell–Evans–Polanyi principle which relates to the surface reaction enthalpy, ΔHᵢ₋ⱼ, to the transition state energy, Eₐ, by parameters α and β with the following relationship:
  The oscillating surface and transition state energies of chemical species alter the kinetic rate constants associated with surface reaction, adsorption, and desorption. The surface reaction rate constant of species i converting to surface species j includes the dynamic activation energy:
  The resulting surface chemistry kinetics are then described via a surface reaction rate expression containing dynamic kinetic parameters responding to the oscillation in surface binding energy:
  ,
  with k reactions with dynamic activation energy. The desorption rate constant also varies with oscillating surface binding energy by:
  .
  Implementation of dynamic surface binding energy of a reversible A-to-B reaction on a heterogeneous catalyst in a continuous flow stirred tank reactor operating at 1% conversion of A produces a sinusoidal binding energy in species B as shown.  In the transition between surface binding energy amplitude endpoints, the instantaneous reaction rate (i.e., turnover frequency) oscillates over an order of magnitude as a limit cycle solution.
caption: Response of an A-to-B reversible unimolecular catalytic reaction to a sinusoidal oscillation in surface binding energy.
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