We showed that one of the most often used forms of Fick's Law for multi-component systems could be written as:

\[\overrightarrow{\mathbf{J}}_{i}=-\underbrace{D_{i j o}\left(1+\frac{\partial \ln \gamma_{i}}{\partial \ln x_{i}}\right)}_{D_{i j}} \vec{abla} c_{i}=c_{i}\left(\overrightarrow{\mathbf{v}}_{i}-\overrightarrow{\mathbf{v}}^{c}\right)\]

Kinetic theory derivations of the flux equation lead to an expression for the gradient in chemical potential, $abla \mu_{i}$, of the form:

\[\vec{abla} \mu_{i}^{c}=\frac{R T x_{j}}{D_{i j o}}\left(\overrightarrow{\mathbf{v}}_{j}-\overrightarrow{\mathbf{v}}_{i}\right)\]

Show that these two forms are equivalent representations for the binary case with species $i$ and $j$. Remember that the chemical potential for species $i$ is given by:

\[\mu_{i}^{c}=\mu_{i o}^{c}+R T \ln \left(\gamma_{i} x_{i}\right) \text { Use expressions for } J_{i} \text { to solve for } v_{i} \text { and } v_{j}\]