Drug diffusion coefficients across membranes, $D,$ of model drug, $1,$ such as lidocaine and dihydroquercetin from hydrogels were

measured using a Franz diffusion cell thermostated at $37C.$ The cell consists of two well$-$stirred volumes separated by a cellulose

acetate membrane or a rat skin. To measure a diffusion coefficient with this cell, both the upper $($or donor$)$ and lower $($or receptor$)$

compartments are originally filled with distilled water. At time $t=0,$ a certain amount of $1$ is placed in the donor compartment,

$C_{1,\text{d}\text{o}\text{n}\text{o}\text{r}\text{.}}^0$ After a known time, both donor and receptor compartments are sampled and their concentrations, $C_{1,\text{d}\text{o}\text{n}\text{o}\text{r}}$ and $C_{1,\text{r}\text{e}\text{c}\text{e}\text{p}\text{t}\text{o}\text{r}},$

are measured respectively by high performance liquid chromatography. Assuming the flux across the membrane quickly reaches a

steady state value, $A$ is the area available for diffusion, $$ is the membrane thickness, $V_{\text{d}\text{o}\text{n}\text{o}\text{r}}$ and $V_{\text{r}\text{e}\text{c}\text{e}\text{p}\text{t}\text{o}\text{r}}$ are volumes of the donor and

receptor compartments respectively:

$(1)$ calculate the flux across the membrane

$(2)$ write an overall mass balance on the two adjacent compartments

$(3)$ obtain the differential equation by combining $(1)$ and $(2)$ and integrate to achieve the desired result.

$(4)$ explain how to use the result to determine $D?$