Consider the precipitation of a spherical B$-$rich phase $($bet phase$)$ from a dilute solution $(\backslash$alpha phase$)$ of B in A$.$ Suppose the original concentration of B in the solid solution is C$\_0=5$ x $10^21$ atoms$/$cm$^3,$ the diffusion coefficient of B is D $=2$ x$10^(-10)$ cm$^2/$sec$,$ and the interface transfer parameter of B is M$=2$x$10^(-6)$ cm$/$sec$.$ The equilibrium concentration of B in the $\backslash$alpha and the $\backslash$beta phases $($C$\backslash$alpha and C$\backslash$beta $)$ are $1\mathrm{.}625$x$10^21$ atoms$/$cm$^3$ and $3\mathrm{.}75$x $10^22$ atoms$/$ cm$^3,$ respectively. In a quasi$-$steady state, the averaged concentration of B in the bulk $($C$\_$t $)$ remains approximately the same as C$\_0.$ When the radius of the beta particle is r $=800$ nm$,$

$($a$)$ what is the concentration of B next to the a$/$b interface, C$\_$r $?$

$($b$)$ and what is the b particle growth rate $($ dr$/$dt$)?$

Use the diffusion flux via Fick$`$s Law $($ Johnson$-$Mehl$-$Avrami equations$)$