thermodynamics - Confusion about Fick's first law

Consider a binary system of mass transport (A, B). Some of mass transfer books (Skelland and Welty) say that the relation $$J_A= -C D_{AB} \frac{dx_A}{dz} \tag{I}$$ is more general than $$J_A= -D_{AB} \frac{dc_A}{dz} \tag{II}$$ Where $C$ is the total concentration of system.

Now we know that $J_A+J_B=0$. Regarding this equality, I say that the first equation $(I)$ always gives $D_{AB}=D_{BA}$ since $dx_A=-dx_B$.

Second equation $(II)$ gives $D_{AB}=D_{BA}$ under the condition that $C$ is constant.

1. 
   

   So Fick says that if $C$ is constant then $D_{AB}=D_{BA}$.

   
   


2. 
   

   Skelland says that under specific operational conditions for a binary system $D_{AB}=D_{BA}$ and so Fick's relation is true if $C$ is constant.

   
   

Which statement (1) or (2) is correct? What is Bird's opinion on this matter?

Answer

If $J_A$ is the molar flux of $A$ defined with respect to the molar-averaged velocity of the mixture $A+B$ then relation (I) is the most general case as it requires no assumptions to use. Furthermore, this relation can be derived from kinetic theory.

You are correct in stating that relation (II) requires the assumption of the total concentration being constant. In gaseous system, where the pressure is relatively constant, this is a good assumption and the two relations are equivalent. Likely (although don't take my word for it), when Fick did his experiments in the distant past he formulated his law of mass transfer in systems where the assumption of total constant concentration was constant.

Note that when defined with respect to the volume-averaged mixture velocity, relation (II) is actually the correct phenomological description of the flux.

This is explained in more detail in the first chapter of Analysis of Transport Phenomena by Deen. I suggest you take a look at it.