23 : 2/ Diffusion, Permeability, and Active Transport 



421 



The mathematical theory can never be completely divorced from 

 experiment any more than the opposite is possible. 



The development both of the individual sections and of the over-all 

 chapter illustrates the approach of the mathematical physicist as well as 

 of the biophysicist. One feature of this approach is to start with simple 

 idealized situations which can be described exactly by mathematical 

 formulae. These are then gradually expanded, improved, made to 

 correspond more exactly to nature. In the process, the theories may 

 become mathematically cumbersome, but throughout the entire develop- 

 ment, the intuitive picture is colored and influenced by the simple, 

 idealized approximation which can be exactly solved. 



2. Diffusion Equations 



The example of diffusion which is easiest to describe in mathematical 

 terms is that in which diffusion occurs in one dimension (direction) only. 

 This mathematical development is presented in this section, the more 



Ax 



CM 



Figure I. One-dimensional diffusion. In this figure, oxygen 

 is considered to be diffusing down a long pipe, such that the 

 concentration is constant all over the plane at x x . The con- 

 centration value at a second plane at .v 2 however may be 

 different from that at the plane at x 1 . 



general case of three dimensions being left to the reader. The uni- 

 dimensional situation can be visualized as some substance, for example 

 2 , diffusing through a liquid which fills a long pipe of constant cross- 

 sectional area A. This is illustrated in Figure 1. With suitable pre- 

 cautions, the concentration c of the 2 will be constant throughout any 

 given cross section. In contrast, c will vary from one cross section to the 



