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given by the flow of entropy in and out of the volume element and by the entropy production inside the volume element:

(4.1)

      Here, s(x, t) is the entropy density, Js(x, t) is the entropy flux in the laboratory frame of reference and σ(x, t) is the entropy production. As the entropy density and flux depend on both position and time, we use partial derivatives. We consider only one-dimensional transport problems.

      Below a more explicit expression for σ will be found by combining

      •mass balances;

      •the first law of thermodynamics;

      •the local form of the Gibbs equation.

      We shall do this and compare the resulting expression for the time rate of change of the entropy density with Eq. (4.1). This will make it possible to identify the entropy flux as well as the entropy production. We shall find that σ can be written as the product sum of the conjugate fluxes and forces in the system, see Sec. 1.1. In the derivation, we follow Refs. [23, 30]. Electroneutral, polarizable, non-equilibrium systems, with and without chemical reactions are of interest. These are the systems that we encounter most often in nature, and also in industry. We shall model transport of heat, mass and charge in systems where reactions occur. We recommend to use the symbol list for check of dimensions in the equations. An introduction to non-equilibrium thermodynamics for homogeneous systems was given by Kjelstrup, Bedeaux, Johannessen and Gross [2, 3].

      Consider a volume element V which is in local equilibrium in a polarizable, electroneutral bulk phase. The volume element does not move with respect to the walls (the laboratory frame of reference), see Fig. 4.1. All fluxes are with respect to the laboratory frame of reference.

      The volume element has a sufficient number of particles to give a statistical basis for thermodynamic calculations. Its state is given by the temperature T(x, t), chemical potentials μj(x, t) for all the n neutral components, pressure p(x, t) and the equilibrium electric field E_eq(x, t) = D_eq(t)/ε(x, ^t). We shall take ε constant in the homogeneous phases. Both equilibrium electric fields are then independent of the position:

      Figure 4.1A volume element in a bulk phase with transport where j′ indicates one of the fluxes.

(4.2)

      The number of positive particles equals the number of negative particles, but their distribution within the volume element can lead to a polarization density.

4.1Balance equations

      The balance equation for component j is

(4.3)

      where Jj(x, t) are the component fluxes in the laboratory frame of reference, all directed along the x-axis. Furthermore, νj is the stoichiometric constant of component j in a chemical reaction, which is positive if j is a product and negative if j is a reactant, and r(x, t) is its rate in the volume element. For simplicity, we consider only one reaction. It is easy to add more reactions [23]. The reaction Gibbs energy is

(4.4)

      In our description of reactions at surfaces, we shall also need the contribution to the reaction Gibbs energy due to the neutral species

(4.5)

      where the sum in Eq. (4.5) is only over the neutral components. The difference between Δ_nG and the usual expression Δ_rG is due to the chemical potentials of ions and electrons. We explain why we need Δ_nG in Secs. 10.6 and 10.7.

      Exercise 4.1.1.Derive Eq. (4.3) for r = 0 by considering changes in a volume element that is fixed with respect to the walls.

      •Solution: The change in the number of moles of a component Nj(t) in a small volume V is due to the flux of the component in and out of the volume. We have

      where Ω is the cross-sectional area of the volume orthogonal to the flux direction (the fluxes are flows per unit of area). The area is equal to the volume divided by dx. In the limit of small dx, we have

      By dividing this equation left and right by the (constant) volume, one obtains Eq. (4.3).

      The conservation equation for charge is

(4.6)

      where z(x, t) is the charge density. The systems that we consider can all be described as electroneutral. It follows that ∂j/∂x = 0 so that the electric current density j is constant. The electric current is due to the relative motion of the charge carriers in the system, like for instance of electrons relative to the metal ion lattice. This relative motion is such that it preserves charge neutrality.

      According to the first law of thermodynamics, the change in internal energy density per unit of time is equal to:

(4.7)

      where Jq(x, t) is the total heat flux in the laboratory frame of reference. The product [∂ϕ(x, t)/∂x]j is the electrical work done to the volume element. The time derivative of the polarization density is the displacement current, j_displ(x, t) ≡ ∂P(x, t)/∂t. The product E(x, t)j_displ(x, t) gives the work per unit of volume by the displacement current. This is a capacitive contribution which, for instance, is important when there is an oscillatory electric field (see Chapter 21). In the appendix at the end of this chapter, a discussion is given for the relation between Eq. (4.7), the first law, the definition of the total heat flux, and the relation between ∂ϕ(x, t)/∂x and E(x, t). An important assumption

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