Support for the form of the generalized version of Fick's law introduced in this chapter can be found in the thermodynamics of irreversible processes. Here we introduce a brief description of this field. More detailed descriptions can be found in the books by Hase (1968) and De Groot and Mazur (1962). The starting point is the rate of entropy production described in the following subsection.

Entropy production due to heat transport was discussed.

We recall the expression here:

\[g_{s}=-\frac{1}{T^{2}} \boldsymbol{q} \cdot abla \boldsymbol{T}\]

The starting point was the definition of the change in internal energy,

\[d \hat{u}=T d \hat{S}-p d \hat{V}\]

For a multicomponent system the change in internal energy is given by

\[d \hat{u}=T d \hat{S}-p d \hat{V}+\sum_{i} \mu_{i} d \mathcal{M}_{i}\]

where \(\mathcal{M}_{i}\) is the number of moles of component \(i\) in the mixture and \(\mu_{i}\) is its chemical potential.

Show that, in the presence of a diffusion flux, the entropy production due to diffusion contributes the following additional term:

\[g_{s}(\text { diffusion })=\sum_{i}^{n} J_{i} \cdot abla\left(\mu_{i} / T\right)\]

If the term is split into a flux and a driving force, we obtain a phenomenological model for diffusion. Follow up this line of thought and show that the irreversible thermodynamics provides a platform for development of the constitutive models for diffusion. Note that there is an additional contribution to entropy production due to chemical reaction, which is not included in the transport models.