The Gross View of Nonsteady-State Diffusion

What has been done so far is to consider steady-state diffusion in which neither the flux nor the concentration of diffusing particles in various regions changes with time. In other words, the whole transport process is time independent. What happens if a concentration gradient is suddenly produced in an electrolyte initially in a time-invariant equilibrium condition Diffusion starts of course, but it will not immediately reach a steady state that does not change with time. For example, the distance variation of concentration, which is zero at equilibrium, will not instantaneously hit the final steady-state pattern. How does the concentration vary with time  

Consider a parallelepiped (Fig. 4.15) of unit area and length dx. Ions are diffusing in through the left face of the parallelepiped and out through the right face. Let the concentration of the diffusing ions be a continuous function of x. If c is the concentration of ions at the left face, the concentration at the right force is 

Pick s law [Eq. (4.16)] is used to express the flux into and out of the parallelepiped. Thus the flux into the left face is 

the net outflow of ions per unit volume per unit time is D d c/dx ). But this net outflow of ions per unit volume per unit time from the parallepiped is in fact the sought-for variation of concentration with time, i.e., dcldt. One obtains partial differentials because the concentration depends both on time and distance, but the subscripts X and r are generally omitted because it is, for example, obvious that the time variation is at a fixed region of space, i.e., constant x Hence, 

This partial differential equation is known as Pick s second law. It is the basis for the treatment of most time-dependent diffusion problems in electrochemistry. 

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