Problem $1$

Living cells that are homogeneously distributed $($immobilized$)$ within

an agarose gel require glucose to survive. An important aspect of

the biochemical system design is the evaluation of the effective dif$-$

fusion coefficient for glucose $($species A$)$ into the cell$-$immobilized

gel. Consider the experiment shown in the margin where a slab of

the cell$-$immobilized gel of $=1\mathrm{.}0cm$ thickness is placed within

a well$-$mixed aqueous solution of glucose maintained at a concen$-$

tration of $c_{As}=50$mmo$\frac{l}{L}.$ The glucose consumption within the

cell$-$immobilized gel proceeds by a zero$-$order process given by

$R_A=-k_0=-0\mathrm{.}05$mmo$\frac{l}{L}\frac{?}{\text{m}\text{i}\text{n}}$

The solubilities of the glucose in both water and the gel are the

same; that is$,$ the concentration of glucose on the water side of the

water$-$gel interface is equal to the concentration of glucose on the gel

side of the water$-$gel interface. A syringe mounted at the center of

the gel carefully excises a tiny sample of the gel at various points for

glucose analysis. Develop a model in final integrated form to predict

the concentration profile of glucose within the gel. State clearly all

assumptions used. How can the glucose profile be used to evaluate

a diffusion coefficient? This system is analogous to the example de$-$

scribed in class for oxygen diffusion in a spherical coordinate system

$($TPIIlec$16.$pdf$).$