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 $\frac{?}{{?}_{\frac{?}{?}}{C}_0},C_{},C_0,$ 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 $x(t).$