Entropy Production and Diffusion Matrix

In irreversible thermod3mamics, the second law of thermodynamics dictates that entropy of an isolated system can only increase. From the second law of thermodynamics, entropy production in a system must be positive. When this is applied to diffusion, it means that binary diffusivities as well as eigenvalues of diffusion matrix are real and positive if the phase is stable. This section shows the derivation (De Groot and Mazur, 1962). 

Unlike mass, which is conserved, leading to dp/dt= —V ], entropy is not conserved. That is, in addition to entropy flux that would lead to variation in entropy in a given volume, entropy is also produced from nowhere. Hence, entropy balance is written as 

Replacing ds/dt by the entropy balance equation, du/dt by the energy conservation equation, and dpkidt by the mass balance equation, we have 

Since AGy = the Gibbs free energy change for the reaction / = J2 kjV-k we finally have 

That is, the entropy production in the volume consists of three terms, each of which is due to an irreversible process. The first term is the heat conduction term, the second is the mass diffusion term, and the third is the chemical reaction term. The above equation is known as the entropy production equation. 

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