Consider the unidirectional, steady state diffusion process in a dilute solution of a solute A dissolved in solvent B$.$ Diffusion occurs along the $z$ axis of an $(x,y,z)$ Cartesian coordinate frame of reference.

a$.$ Starting with the general equation of mass transfer and using Fick's law, show that the concentration $C_A$ of species $A$ is governed by the following differential equation:

$\frac{d^2{C}_A}{d{z}^2}=0.$

b$.$ Solve the above equation subject to the boundary conditions $C_A(z_1)=C_{A1}$ and $C_A(z_2)=C_{A2}.$

c$.$ Provide a qualitative sketch of the concentration profile.

d$.$ Derive expressions for the flux $N_A,$ mass transfer rate $M^{}$ across an area of cross section $S$ and total mass transfer $M$ achieved in a time interval $.$