The equation governing the accumulation of an extensive

quantity $($e$.$g$.,$ concentration, $c),$ as a consequence of mass balance is:

$\frac{\text{d}\text{e}\text{l}\text{c}}{\text{d}\text{e}\text{l}\text{t}}=-$grad$*vec(J)$

a$.$ How would this equation change if there was some rate of accumulation ${}^{}$of the density of material in in differential volume element $dV?$

b$.$ Such an accumulation would correspond to a "source" in the material.

Can you think of any physical examples of sources?

c$.$ Now let ${}^{}=0$ for the sake of simplicity. By definition, Fick's first law relates vec$(J)$ to the gradient of $c$ as

vec$(J)=-$Dgradc

Using this information, derive the heat $($or diffusion$)$ equation, which is

otherwise known as Fick's second law. Assume diffusivity $D$ is constant.