For diffusion in binary mixtures, the Fick's Law of Diffusion may be expressed in the

following form:

$j_A=-D_{AB}$grad${}_A$

Starting from the above, prove the following:

$($a$)j_A=-(M_A\frac{M_B}{M^2})D_{AB}grad{x}_A$

$($b$)grad{x}_A=\frac{1}{c{D}_{AB}}(x_A{N}_B-x_B{N}_A)$

$($c$)v_A-v_B=-D_{AB}ln(\frac{x_A}{x_B})$