Two gases are initially separated into two bulbs connected by a narrow diameter capillary. Pure $N_2$ flows from right to left, as shown below. We assume that the system has achieved steady state and no reactions occur between $N_2$ and $O_2.$ We also assume that the velocity vector has only an axial component and there is diffusion only in the axial direction.

The governing equation is given by below:

$v\frac{dC}{dz}=D\frac{d^{2}C}{d{z}^2}$

where $v$ and $D$ are constants. The boundary conditions are:

At $z=L,C=C_0$

At $z=0,C=0$

a$.$ Solve the differential equation to obtain $C(z),$ the concentration profile of $O_2.(20$ pts$)$

b$.$ BONUS: Show that in the limit of very small values of $v\frac{L}{D}$ for $\frac{vL}{D}1),$ the solution becomes linear $)=(\frac{z}{L}$