RATE OF DIFFUSION OF IODINE IN KI 491 



t)U . _ !,„!, 



7— = — n ■ sm nx ■ e , 

 ox 



diiferentiating again with respect to x we have, 



OH „ _ i.„2f 



—— = — n^ ■ COS nx ■ e *" '. 

 ox^ 



Substituting in equation (I) we have an identity; hence, u = 

 COS nx ■ e~ ''"'', is a solution. In exactly the same way we can show that 

 u = sin nx ■ e"*"'' is a solution; and hence, 



. n 



(II) u = ^^ (a„ cos nx + b„ sin nx) ■ e 



is a solution. If this is to satisfy our conditions we must have for 

 t = 0, u = f (x) known i.e. Mq, and when a; = • or a; = r 



— - = 0, at all times! 

 ox 



(Here r is the total length of the column, i.e. both layers.) 

 Differentiating equation II, and putting a; = 0, we have: 



dx~ -^n = Q 



n (a„ sin nx — 6„ cos nx) ■ e *"'' = 0. 



For this case sin nx is identical with zero; but as e"*""'' can not be zero, 

 6„ must be zero for all values of n. If now we put x = r the above 

 equation becomes, 



(III) ^- = / — n • a„ sin ?ir = 



ox ^—1 



Hence, nr must be some multiple of ir, i.e., the values of nr must be 



0, It, 2ir, Stt, 47r, 

 Or n must be, 



0, x/r, 27r/r, 37r/r, 47r/r, • • • 



Equation (II) now becomes, 



-■n'-U -iir'kt 



u = ao cos • e + a, cos Tx/r ■ e r^ + a2 cos 2ira;/r • e r^ -f 



-9irM -16ir.%i _1_ . . . 



03 COS 3xa;/r • e ^^ +a4 cos 4xa;/r • e r^ i" ' ' " , 

 It now remains to evaluate the coefficients a^, ai, 02, ■ • , • 



