Let the volume average velocity $(v)$ in a $Z-$component mixture be defined as follows:

$v^{*}={\displaystyle \underset{i=1}{\overset{Z}{}}}{}_i(\frac{?}{\text{b}\text{a}\text{r}}\frac{(V{)}_i}{M_i})v_i={\displaystyle \underset{i=1}{\overset{Z}{}}}\frac{c_i}{b}ar(V{)}_i{v}_i$

in which $\frac{?}{\text{b}\text{a}\text{r}}(V{)}_i$ and $M_i$ denote, respectively, the partial molar volume and the molecular weight

of component $i.$ Then define the mass flux with respect to the volume average velocity $(j_i^{**})$

as

$j_i^{*}={}_i(v_i-v^{*})$

$($a$)$ Show that for a binary system of A and $B,$

$j_A^M=(\frac{?}{\text{b}\text{a}\text{r}}\frac{(V{)}_B}{M_B})j_A$

To do this, you will need to use the identity $\frac{c_A}{b}ar(V{)}_A+\frac{c_B}{b}ar(V{)}_B=1.$ Where does this come from?

$($b$)$ Show that the Fick's $($first$)$ law of diffusion then assumes the form

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

To verify this, you will need the relation $\frac{?}{\text{b}\text{a}\text{r}}(V{)}_{A}grad{c}_A\frac{+}{b}ar(V{)}_{B}grad{c}_B=0.$ What is the origin of this?