As shown in the figure, a hollow cast iron sphere with inner radius $r_1$ and outer radius $r_2.$ Suppose that a gas stayed within the interior of the sphere and the concentration of a gas at the interior of the sphere keeps at $c_{A0}.$ The concentration of the gas around the outside environment is $c_{As.}$ Assume that the diffusion coefficient of the gas through the cast iron is kept at a constant value of $D_A.$

$($a$)$ Starting from the mole balance of the gas, find the concentration profile between $r_1$ and $r_2$ at steady state. $(17\%)$

$($b$)$ Find the mass transfer rate.

$(8\%)$