Onsager Reciprocity Conditions

We begin here the study of thermodynamics in the proper sense of the word, by exploring a variety of physical situations in a system where one or more intensive variables are rendered nonuniform. So long as the variations in T, P, /x or other intensive quantities are small relative to their average values, one can still apply the machinery of equilibrium thermodynamics in a manner discussed later. It will be seen that the identification of conjugate forces and fluxes, the Onsager reciprocity conditions, and the rate of entropy production play a central role in the analysis provided later in the chapter. 

THE LINEAR PHENOMENOLOGICAL EQUATIONS, AND THE ONSAGER RECIPROCITY CONDITIONS... 

Reference to the preceding section, and to Eq. (6.4.4), shows that if steady state conditions are applied and if there are no constraints placed on the various forces all currents vanish thus, the first term drops out. This is also the condition for which the rate of entropy conduction is a minimum. Thus, considering 8 to be an implicit function of the fluxes, the left hand side must then vanish. The Onsager reciprocity condition is thereby established Lik - l i One should note that if the reciprocity condition were to fail the various XA would be interrelated, contrary to the assumption that they are independent. Furthermore, the assumption of the Onsager relations leads directly to the minimization of 8 under steady state conditions. 

Prove that the phenomenological coefficients L12 and L21 in Eq. (6.8.1) satisfy the Onsager Reciprocity Condition. 

This set of equations obeying the Onsager reciprocity conditions obtains only near equilibrium. It is generally the case that the linear approximation for the reactions is valid over a far more limited range than is the linear approximation for the types of processes discussed in Sections 6.7-6.11. 

In conjunction with Eq. (6.3.6) we introduce the Onsager reciprocity conditions (OCR) which will be derived below under a restricted set of circumstances. The OCR read as follows ... 

We provide here a simplified derivation of the Onsager Reciprocity Conditions (ORC) that is applicable only when steady state conditions hold. A full derivation, which is based on the machinery of statistical mechanics, is beyond our purview the derivation, based on Tykodi s work, is satisfactory for present purposes. In following the derivation the reader is advised to write out the steps in... 

Chemical potentials entering these equations are relative to the vacancy chemical potential, h ai = Hai - fiv and = hzt - t v- We use to express diffusion fluxes the Onsager reciprocity condition, Laizt = LztM-... 

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