$($Welty $26\mathrm{.}26)$ Consider a spherical gel bead $6\mathrm{.}0$ mm in diameter containing a biocatalyst

uniformly distributed within the gel. Within the gel bead, a homogenous, first$-$order reaction:

$1$

$->,$ with k$1$ is $0\mathrm{.}019$ s$-1$ is promoted by the biocatalyst. The gel bead is suspended within

water containing a known, constant, dilute concentration of solute A $($cA$\backslash$infty $=0\mathrm{.}02\backslash$mu mol$/$cm$3).$

a$)$ Develop a differential equation in terms of the concentration of A $($cA$)$ that models this

system. State all assumptions and boundary conditions necessary to completely specify the

differential equation.

b$)$ Show that the analytical solution for the concentration profile is given by:

$()=\backslash$infty

$$

$$

sinh$(1/)$

sinh$(1/)$

c$)$ What is the total consumption rate of solute A by one single bead in units of $\backslash$mu mol A per hour?

The diffusion coefficient of solute A within the gel is $2\backslash$times $106$ cm$2/$s$.$