A substance is diffusing out of a spherical liquid droplet suspended in water. The concentration $c$ of the substance in the water at the surface of the droplet is determined by the partition coefficient $K$ from equilibrium thermodynamics, while the concentration far away from the droplet is zero.

The steady$-$state concentration profile can be modeled by the boundary value problem

$D\frac{1}{r^2}\frac{d}{dr}(r^2\frac{dc}{(d)r})=0$

$c(R)=\frac{K}{M}$

$c()=0,$

where $D$ is the diffusion coefficient, $R$ is the radius of the droplet, $$ is the density of the substance in the droplet, and $M$ is the molar mass of the substance.

Assuming that the concentration of the substance in water reaches steady state quickly compared to the time it takes the droplet to dissolve, the mass balance for the droplet is

$\frac{dm}{(d)t}=4R^{2}DM{c}^{\text{'}}(R),$

where $m$ is the mass of the droplet.

$($a$)$ Solve for $c(r).$

$($b$)$ Starting from the mass balance, formulate an initial value problem for the radius of the droplet $R$ if the droplet radius is $R_0$ when $t=0,$ then solve for $R(t).$

$($c$)$ How long does it take for the droplet to dissolve completely?

$2.$ A continuous cryopreservation process incorporates a solute from aqueous solution into frozen ice. The solute concentration is that of bulk solution, $c_{},$ a distance $$ behind the solidification front. The ice is removed at a continuous velocity $U.$ The concentration $c$ of the solution as a function of position $x$ relative to the solidification front at $x=0$ can be modeled by the boundary value problem

$D\frac{d^{2}c}{(d)x^2}-U\frac{dc}{(d)x}=0$

$c(-)=c_{}$

$D{c}^{\text{'}}(0)=U(1-K)c(0),$

where $D$ is the diffusion coefficient of the solute and $K$ is the partition coefficient of the solute in ice. Solve for $c(x).$