In class, we discussed the problem of mass$-$transfer through a stagnant gas film and obtained expressions for the molar flux of A i$.$e$.N_{Ay}$ as well as for the concentration profile i$.$e$.x_A.$

$($a$)$ Show that the two expressions can be combined to get

$\frac{1-x_A}{1-x_{A1}}=$exp$(\frac{N_{Ay}y}{c{D}_{AB}})$

$($b$)$ Obtain the same result by directly integrating the equation relating $N_{Ay}$ with the concentration $x_A,$ using the fact that $N_{Ay}$ is a constant

$($c$)$ Note what happens when the mass$-$transfer rate becomes very small. Expand the equation in $($a$)$ in a Taylor series and keep two terms only, as is appropriate for small $N_{Ay}$ is small. What happens to the curvature of the concentration profile when $N_{Ay}$ is very small? Show schematically by plotting the concentration profile.