Fick's First Law is still valid even when steady-state conditions do not exist. However, it is not very convenient to use it in this form for solving transient diffusion problems, for which the concentration at any position depends on both time and position. It is therefore more convenient to derive a second version of this equation, Fick's Second Law, which contains an explicit dependence on time. In this chapter we will first derive the equation; a second-order partial differential equation. We will then learn how solutions to this equation can be obtained for a large variety of situations depending on the problem geometry, the initial and boundary conditions and the time-frame over which we need a valid solution.

Fick's Second Law

We consider the situation illustrated in Fig. 3.1. At two positions along the y-axis at y1 and y2, a distance Δy apart, the concentration of the solute is C1 and C2 respectively. We can use Fick's First Law to determine the amount of solute per unit time which enters this element at y1 and which leaves it at y2. Because matter must be conserved, the difference between these must be equal to the rate at which solute accumulates within the element.