Chapter 7.1, Problem 49E

Concentration of a solute According to the Fick’s law, the diffusion of a solute across a cell membrane is given by

$c^{\prime }\left(t\right)=\frac{kA}{V}\left[C-c\left(t\right)\right],\left(1\right)$

Where A is the area of the cell membrane, V is the volume of the cell, c(t) is the concentration inside the cell at time t, C is the concentration outside the cell, and k is constant. If c_o represents the concentration of the solute inside the cell when t = 0, then it can be shown that

$c\left(t\right)=\left(c_0-C\right)e^{-kAt/V}+C.\left(2\right)$

a. Use the last result to find $c^{\prime }\left(t\right)$.

b. Substitute back into Equation (1) to show that (2) is indeed the correct antiderivative of (1).