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Thermal effects in a modified law of mass action
Abstract
Non-isothermal effects in activated processes are considered by means of mesoscopic non-equilibrium thermodynamics. Charge
transfer through electrode surfaces is used as a model problem. It is shown that as a generalization of classical non-equilibrium thermodynamics,
the theory is capable of incorporating thermal effects intoa non-linear description of activated processes. This results
in a modified law of mass action accounting for non-isothermal conditions. Generalized versions of Nernst and Butler–Volmer
equations allowing for thermal gradients are presented as a consequence of the modified rate law. Some distinct advantages of
the formalism over its classical counterpart are discussed.Introduction Activated processes are ubiquitous in nature and constitute
a basic mechanism in the evolution of many systems.
Chemical reactions, nucleation, surface growth,
elastoplasticity, and crossing of energy gaps in semiconductors
are, to name a few, examples of activated
processes.
The traditional treatment of Arrhenius-type laws for
activated processes [1,2] has limited itselfto the description
of systems in isothermal heat baths, thereby
neglecting the possible existence of thermal gradients
hindering or enhancing the rate of the process. The presence
of such temperature gradients may lead to significant
effects as in the case of evaporation and
condensation [3,4], nucleation [5], growth [6,7], polymer
crystallization [8] and chemical vapordeposition [9].
A classical non-equilibrium thermodynamic [10]
treatment of activated processes is not possible due to
their intrinsically non-linear nature. When these processes
are analyzed at shorter time scales as done by
mesoscopic non-equilibrium thermodynamics (MNET)
[11,12] it is possible to obtain a description for the
non-linear kinetics. MNET has been successfully usedto describe a wide variety of isothermal activation phenomena
including nucleation [13], aggregation and
agglomeration [14,15], evaporation and condensation
[16] and charge transfer through an electrode surface
[17].
As a generalization of classical linear non-equilibrium
thermodynamics, mesoscopic non-equilibrium thermodynamics
should be capable of incorporating all thedifferent
thermodynamic forces into its non-linear
description. The present work intends to extend the
treatment of activated processes to non-isothermal regimes.
For this purpose the framework of MNET is
used to analyze charge transfer through electrode surfaces
beyond the work or Rubi and Kjelstrup [17]
who, assuming isothermal conditions, showed that
MNET can be used toobtain the law of mass action
and the Butler–Volmer equation for an anodic electrochemical
reaction.
For the sake of generality the anode and the cathode
are given separate consideration at an initial stage. The
electrode surfaces are described using Gibbs_ excess variables.
The central assumption of local equilibrium is taken
along an internal mesoscopic space describing theadvancement of the chemical reaction. It is not only
the chemical potential of the reacting species that is assumed
to vary along this internal space; the electric potential
and the temperature are also argued to obey
some functionality with respect to this coordinate. The
resulting activated expression for the current density is
a modified law of mass action and its directoff-spring
a modified Butler–Volmer equation.
The organization of the present paper is as follows. In
Section 2 the system under consideration is described
and the relation between different reacting species is
identified. In Section 3 the entropy production for the
polarized electrode surface is presented, the reaction
coordinate is introduced and the resulting flux–force...
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