422 Diffusion, Permeability, and Active Transport /23 : 2 



next. At any given cross section, c will also vary in time. These varia- 

 tions can be described by a single partial differential equation. This 

 equation will now be developed. 



Consider two planes at x t and x 2i spaced as shown in Figure 1 . In a 

 homogeneous liquid, the probability of a molecule jumping either in the 

 + x or —x direction is equal. The mass per unit time of molecules 

 jumping from plane x± to plane x 2 will be proportional to the concentra- 

 tion c x at x x . Likewise, the mass per unit time going from x 2 to x x will 

 be proportional to c 2 . These masses will also be proportional to the 

 cross-sectional area A. Analytically, one may express this as 



^ = pA( Cl - c 2 ) = -flAAc (1) 



where Am is the net mass transfer in the + x direction across the surface 

 at x 2> and fi is a probability parameter. 



It seems clear that if the two planes x x and x 2 are far enough apart, 

 the probability constant ft must be very low, whereas if they are close, ft 

 should be large. The exact dependence of /? on the separation can be 

 approximated by 



'-^-£ 



where D is a constant. Then Equation 1 may be rewritten 



Am _ . Ac 



-r- = -DA — 



At Ax 



Taking the limits as At and Ax go to zero reduces this to 



£--*"* (3) 



Equation 3 can be derived starting from thermodynamics as well as 

 from other points of view. Although several of these add refinements, 

 they all contain the basic assumption made in writing Equation 2. This 

 assumption can be justified empirically because experiments with a wide 

 variety of gases and solutes in different solvents confirm Equation 3. 

 For most purposes, it is correct; however, it is meaningless to apply 

 Equation 3 to distances of the order of a few angstrom units or times 

 comparable to the period that a molecule remains in a quasi-equilibrium 

 state (10 -14 sec). The cases discussed in the remainder of this chapter 

 have been restricted to those to which Equation 3 can be applied. 



Equation 3 is all that is necessary to mathematically analyze the 

 diffusion problems encountered in this text. The constant D is called 

 the diffusion constant or sometimes the Fick diffusion constant. Equation 3 



