show that $G=kT\frac{}{P}$ and hence the growth rate is given by vprop$\frac{}{P}.$

Clarence Zener $(1905-1993)$ is

widely recognized for his

contributions to materials kinetics.

For instance, he is known for

originating the idea of linearizing

the concentration gradient in the

problem of diffusion$-$controlled

growth. His simplified profile is

given at right, where $"x"$ indicates

the thickness of solid $()$ phase.

a$.$ The conservation of mass means that the two shaded areas in the figure

are equal. From this information, what is the concentration gradient

$d\frac{C}{d}x$ in terms of the quantities $C_0,C_{},C_0,$ and $x?$

b$.$ Based on your solution to part $($a$),$ show that the velocity of the solid, $V=$

$d\frac{x}{d}t,$ depends on particle thickness $x$ as $vprop{x}^{-1}$ and thus $xprop{t}^{\frac{1}{2}}.$

Hint: note from the mass balance condition discussed in class that

Vpropd$\frac{C}{d}x.$ Also, you will need to integrate $V(t)$ to find $x(t).$

PLEASE ACTUALLY DO IT