# 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?

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.