A cell constantly gains or loses small molecules to its environment because the small molecules are able to diffuse through the cell membrane. We will build a model for this process. Suppose a molecule is present in the cell at a concentration $C(t),$ and present in its environment at a concentration $C_{}($you may assume $C_{}$ is a constant$).$ One model for the diffusion of molecules across the cell membrane is that the rate at which molecules travel through the membrane is proportional to the difference in concentration between the cell and its surroundings. That is$,$ Rate at which molecules flow out of cell $=k(C-C_{}).$ The constant $k$ is known as the permeability of the membrane; $k>0,$ and $k$ depends on the surface area of the cell and the chemistry of the membrane, as well as the type of molecule. Complete parts $($a$)$ through $($d$).$

$($a$)$ Starting with a word equation for the amount of small molecules in the cell, show, if the cell volume is $V,$ then $\frac{dC}{dt}=-\frac{k}{V}(C-C)$

The rate at which molecules flow out is $\frac{dC}{dt}.$ Solving for $\frac{dC}{dt},$ verifies this differential equation.