Considering the case of a pure component i diffusing through a membrane of polymer $p,$ and restricting

the flux of the polymer to be zero, the following flux relation is obtained:

$N_i=-\frac{{}_i}{l{k}_{B}T}\frac{D_{i,p}}{x_p}\frac{d{}_i}{d}$

where ${}_{}$Di,p is the MS diffusivity between component i and the glassy polymer and $xp$ is the polymer mole fraction. As the membrane swells in response to the sorption of component $i,$ the increase of the membrane thickness $l$ will act to decrease flux, while the decrease in polymer mole fraction will act to increase flux.

Show detailed mathematical steps and provide appropriate thermodynamic assumptions to derive the above equation.