Irreversible behavior in small systems – Solvent effect and calculation of a chemical rate constant

In this paper the approach of irreversible chemical reactions presented in J. Electroanal. Chem. 660 (2011) 332 is extended to take into account the presence of a solvent. Here we investigate the evolution of a small system enclosed in a box in contact with a solvent represented by an assembly of independent oscillators. Under very general conditions concerning the thermal equilibrium of the solvent and some time scales a Smoluchovski equation is obtained for the small system. The solvent creates a transition from a quantum to a classical regime. To the mass of the particle we must add an effective mass and the potential existing in the box is replaced by a quantity containing this potential and dynamical properties. To be illustrative we calculate the chemical rate constant in the case of a particle inserted in a fixed double well potential. The interest of this model system is that it can be solved exactly. For the chemical rate constant we discuss the meaning of the activation energy and the expression of the so-called prefactor. A comparison with other theories is shortly presented. New applications are suggested.

► A quantum Smoluchovski equation is derived from a path integral formalism.
► In presence of a solvent model a transition towards a classical regime is obtained.
► The approach is used to investigate a simple irreversible chemical reaction.
► The concept of activation energy is discussed and the pre-factor is determined.
► A comparison with standard theories is presented.