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  • Author: D. Bedeaux x
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It is shown how one can derive the impedance of a polarized electrode surface from irreversible thermodynamics. The oxygen electrode is studied as an example. A Nyquist diagram is constructed for the case that the electrode conducts by electrons only, and the electrolyte conducts by oxygen ions or vacancies. The electrode surface contributes to the diagram with two semi-circles. One semi-circle is due to production of dipoles from adsorbed oxygen atoms. The other is due to the dipole moment of the ion-electron hole pair.


The current description of a formation cell using nonequilibrium thermodynamics gives an overall picture. It does not consider the behaviour of the temperature, chemical potential and electric potential locally. Thus it is unclear where temperature and potential change in the cell. Are these quantities discontinuous at the electrode surfaces, and if yes: how this should be described. Using methods developed in our earlier work we present such a local description. A differential equation is derived and solved for the transference coefficient of the salt in the electrolyte between the electrodes. Boundary conditions are essential for this purpose.

We derive the impedance for the hydrogen electrode in the polymer membrane fuel cell from irreversible thermodynamics. The results predict a surface contribution to the cell impedance that can give two semi-circles in the Nyquist diagram. The equivalent circuit of the impedance is shown. The high-frequency contribution is connected to the oscillation of dipoles consisting of free charges in the surface, while the low-frequency contribution is connected to the electrochemical reaction. This can be explained by a slowly relaxing proton conducting polymer network at the reaction site.


We examine the problem of energy-efficient production in an industrial process. By energy-efficient we mean minimum entropy production. We use the possibility to redistribute the production in different times or parts of the system for a given total production, and show that a distribution, that equipartitions the derivative of the local entropy production rate with respect to the local production, minimizes the entropy production. Equipartition in time implies stationary state production. Equipartition in space implies production for a given position independent force. The same constant derivative of the local entropy production rate is found if ones optimizes the production for a given total entropy production.

Close of equilibrium the equipartition condition is found to reduce to the isoforce principle. Further from equilibrium, this reduction is extended to a whole class of nonlinear flux-force relations. We show that, when one increases the total production, the entropy production per unit produced starts to increase linearly, as a function of this total production.

It is shown which process conditions give an optimum path with an equipartition of the entropy production rate. How this relates to the isoforce principle is discussed. In general constraints on process conditions restrict the freedom to optimize, and therefore make it impossible to realise the most favorable conditions. The importance of the Onsager relations for the systematic description of the optimization is discussed.


This paper treats the simple example of heating a system from a given initial to a given final temperature with minimum entropy production. The allowed control for the process is the selection of K temperatures for intermediate heat baths. The problem is sufficiently simple to allow analytic approaches and we compare the optimal solution with the solution prescribed by equal thermodynamic distance (ETD). We find that ETD coincides with the optimum if the heat capacity is constant. For a temperaturedependent heat capacity, ETD deviates from the exact optimum. ETD however matches the optimal solution to second order in 1/K.