420 Diffusion, Permeability, and Active Transport /23 : I 



of biological cells, and on the still smaller scale of reacting molecules, 

 diffusion becomes very rapid. To diffuse throughout a cell requires 

 only milliseconds. 



From the point of view of molecules, the diffusing solute may be 

 thought of as jumping from one quasi-equilibrium site to the next, 

 perhaps an Angstrom unit away. There will be greater probability of 

 a molecule jumping from a region of higher concentration than vice 

 versa. Thus, diffusion will lead toward an equalizing of concentration. 

 When the concentration is equal throughout the container, the partial 

 molal free energy G will also be equal throughout. Diffusion leads 

 toward the equilibrium condition : 



dG = 



At boundaries separated by membranes (for example, cell membranes), 

 the rate of diffusion may be markedly slowed. Although not really a 

 different type of phenomenon, the rate of diffusion through a limiting 

 membrane is called the permeability. Most membranes are permeable 

 only to certain substances. If the concentrations of substances to which 

 the membrane is impermeable are different on the two sides of the 

 membrane, the solvent will tend to move toward the greater concentra- 

 tion. This is described by assigning an osmotic pressure to the solution. 



When some of the molecules which cannot pass the boundary are 

 charged, an electrical potential may be developed across the membrane. 

 This is called a Donnan potential. In this case, the equilibrium concen- 

 trations of the ionic species may be different on the two sides of the 

 membrane. 



In Chapter 4, on the conduction of nerve impulses, it was pointed out 

 that the ion concentrations inside nerve axons are different from those 

 outside. These differences cannot be explained by passive diffusion 

 through a membrane subject only to osmotic and electrical forces. 

 Rather, certain ions are actively transported at the ultimate expense 

 of metabolic energy. Active transport is not restricted to membranes 

 of axons. Forced diffusion in a direction different than indicated by 

 electrical and concentration gradients probably is a common occurrence 

 in all cells. 



Many experiments have been carried out to measure diffusion rates 

 and permeabilities, as well as to demonstrate the role of active transport 

 against electrochemical gradients. Behind each of these experiments, 

 indeed as an essential part of each, is the mathematical theory of diffusion 

 and permeability. Without it, the experiments would be meaningless. 

 In this chapter, the basic mathematical development is presented. It is 

 hoped that the reader will not be misled into feeling that the experi- 

 ments are less important than the theory, for this is surely not the case. 



