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( 4 0 pts ) Let's analyze the dissolution of an aspirin tablet. Assume that the dissolution process is controlled by diffusion of the drug

(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 r0(constant) and that the drug concentration in the liquid right next to the pill (i.e., at the pill's
surface, {:r=r0) is always saturated (i.e.,c(r0,t)=cs 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)=ur,tr 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, =rg(t)
will not go to zero at the pill surface ). What would happen if we defined =r-r0g(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 cs=1molmm3,D=1mm2s, and r0=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?
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