Note that 



2 fb 



f b e - s2 ds = — f b e~ sl ds-~ [ a e~ s2 rfs = erf (b) -erf (a) 



The error function is defined for a positive upper limit of integration. Here the 

 upper limit may be negative. Note that 



v^ 



2 r-a^ 



ds = 1 — erfa., a > 



Thus 



u(z,t) = 



C 



r .,'z—h+2ni\ „[z—l+2nl\ 



exi[—=— )-erf — =^- ) 



%kt' / \\/4ftF / 



I 



f .z+l+2nl\ Jz+h+2nl 

 r 4kT J W4fe! / 



^2nK 



.(21) 



where the sign of the term is changed if the argument is negative. 



As an example of the special case, consider a layer of solvent of thickness 

 h such that h/l = 0.2. Let this layer initially lie on a layer of solution with con- 

 centration C . The subsequent distribution of concentration C divided by C 

 (that is, C/C ) is plotted as a function of z/l for the following values of the non- 

 dimensional parameter kt/l 2 : 0.01, 0.0004, and 0.0001 (fig. AD. 



The time required to reach the above indicated distributions from the initial 

 state is given in table Al. Here / = 50 cm, solutions are aqueous, concentration is 

 in gm-molecules per liter, and k is given in cmVday. 



TABLE Al. EXAMPLES OF DIFFUSION TIME 



Material 



Concentration 



Temperature, °C 



Diffusion 

 Coefficient (k) 



kt/l 2 



Time (t), days 



NaCl 



0.1 tq 1 



15 



0.94 



0.01 



0.0004 



0.0001 



26.6 

 1.06 

 0.27 



Sugar 



1.0 



12 



0.254 



0.01 



0.0004 



0.0001 



98.4 

 3.94 

 0.984 



CdS0 4 



2.0 



11.04 



0.246 



0.01 



0.0004 



0.0001 



102. 

 4.06 

 1.02 



It can be shown that the gradient at h/l = 0.2 is (for— not too large) 



proportional to l/vkt and directly proportional to I. For a given gradient, the time 

 required for a sharp interface to deteriorate to that gradient is inversely proportional 

 to k. 



20 



