23 : 2/ Diffusion, Permeability, and Active Transport 



423 



is often called the Fick diffusion equation or Fick's law. It can be expressed 

 in an alternate form which is easier to handle, although conceptually 

 identical. To derive this alternative form, consider again the pipe of 

 constant cross section A, and so on, redrawn in Figure 2. The mass 



1 dm a n dC 



A dt ~ U dt 



o 2 C 



c__> 



Ax 



1 $!I!± _n dCb 



B dt ~ ° dx 



\AI/=/4Ax 



x b 



Figure 2. One-dimensional diffusion and continuity. This 

 figure is used to illustrate the relationship of the change of 

 concentration within A V to the 2 diffusing in at x a and out 

 at x b . 



change per unit time in the volume A V between planes x a and x b is the 

 difference between that entering at x a and that leaving at x b . That is 



^m- 



dx 



dx 



or dividing both sides by Ax and taking the limit as Ax goes to zero, 



dc cPc 



81 ~ dx 2 



(4) 



This is an alternative form of Equation 3, valid for the case in which no 

 oxygen is generated or destroyed. 



Equation 4 can be readily generalized to the three-dimensional 

 diffusion equation as 



d 2 c c 2 c 

 Ty 



dt 



~fa 2 " dH 2 " ~dz 9 - 



(5) 



Both Equations 3 and 5 may be written in the vector notation. By 

 defining the mass current J as 



J 



t\ dmY 

 = \A~di) 



(6) 



where Tz is a unit vector normal to A, Equation 3 may be rewritten as 



J = DVc (7) 



