$(40$ pts$)$ Let's analyze the dissolution of an aspirin tablet. Assume that the dissolution process is

controlled by diffusion of the drug into the surrounding stomach fluid $($assumed to be static$).$ In other

words, what is the concentration profile of drug outside the pill? Assume that the pill is spherical with

radius $r_0($constant$)$ and that the drug concentration in the liquid right next to the pill $($i$.$e$.,$ at the pill's

surface, $\{$:$r=r_0)$ is always saturated $($i$.$e$.,c(r_0,t)=c_s$ always$).$

a$.$ What differential equation do you have to solve? Look up the spherical coordinate Laplacian and

use symmetry arguments to reduce the number of independent variables.

b$.$ What are the initial $/$ boundary conditions? Think carefully...the pill is very small compared to the

size of your stomach....

c$.$ Transform your PDE into something you can solve by substituting $c(r,t)=u\frac{r,t}{r}$ into the PDE

and then simplifying it$.$ Make sure to transform the initial and boundary conditions as well.

d$.$ After applying the transform in part $($c$),$ you should get a simple PDE for $u(r,t).$ How did we solve

this kind of problem in class? Examine the boundary condition at the pill's surface carefully; recall

that we had to have $0$ as one of the BCs to work this kind of problem. Unfortunately, $=\frac{r}{g(t)}$

will not go to zero at the pill surface $).$ What would happen if we defined $=\frac{r-r_0}{g(t)}?$ Work

the problem through using this new definition for $.$ Transform the initial and boundary conditions;

you should be able to solve for $u(r,t)$ just like we did in class.

What is the concentration profile $c(r,t)?$

e$.$ What is the molar flux of drug leaving the pill surface as a function of time?

f$.$ What is the functional form of the steady$-$state profile?

g$.$ Show how the diffusion process evolves by plotting $c(r,t)$ for $t=1,5,50,$ and $1000sec$ on the

same graph. Assume that $c_s=1mo\frac{l}{m}{m}^3,D=1m\frac{m^2}{s},$ and $r_0=3mm.$

h$.$ What is the steady$-$state concentration at $r=20mm?$ How long does it take the concentration at

this point to reach $90\%$ of its steady state value?