where 



Pi->2 = rate of movement of traced substance 

 from compartment 1 to compartment 2. 



S, = amount of traced substance in compart- 

 m ent 1 . 



R, = absolute amount of tracer in compart- 

 ment 1 . 



a, = specific activity of compartment 1, 



^ = water, 



s = sediment. 



t = time. 



If a tracer is introduced into the water, its 

 movement from the water can be described: 



d R ^, = -rate of movement of tracer from 

 dt water to sediment 



+ rate of return of tracer from sedi- 

 ment to water. 



At any instant the rate of movement of tracer 

 fronn water to sediment = the rate of movement 

 of traced substance from water to sediment 

 times the fraction labeled =pa„, and similarly 

 the rate of movement of tracer from sediment 

 to water = p a^ 



and since 



or d R, 



dt 



paw + P^s 



and since many compartments may be present 

 in the sediment 



d R„ 

 ~dF 



+ (Pwj^s, awj +Pw2^S2 aw2 ■ 

 + (Pw^^S, ^Wn + PWn-^2 ^Wp 

 ■•■ (PS|-*w, as, +Ps,->W2 as, ■ 



- 



+ (Ps2^w, as2 +PS2-W2 asj ■ ■ 



+ (PSn,->-W, aSn, +PS„^W, ^j^ 



• + Pw2->-Sn,aw2 ) • • • 



■ + Ps,->Wn as,) 

 + Ps2-»-Wn as^) . . 



For determining the net exchange rate 

 across the sediment-water interface, it should 

 be possible to describe the water-sediment 

 exchange system by equations for only two 

 exchanging compartments, water and sediment, 

 during a short interval after the introduction 

 of tracer (principle of lumping), giving: 



d R„ 



- P(a.s 



.) 



and near t 



dt 

 0, aj =g 0, therefore 



d R, 



dt 



■P aw 



R, 



\, 



d R, 



dt 



p^w 



and 



d R, 



Rv 



dt 



Assuming p and S^ constant and integrating 

 over linnits 



to 



dt 



In R, 



In R, 



In R = In R, 



S 



^t 



and 



2.303 log R s 2.303 log R - P . 



& w => w t 







Sw 



log R^ s log R^ 



2.303 S„ 



Therefore, lumping of the water and sediment 

 compartments into only two compartments and 

 assuming that a^ = 0,are permissible as long 

 as a plot of log Rw vs. t gives a straight line. 

 This is shown to be true for the first 90 min- 

 utes in figure 13. 



Now since 



d R 



dt 



a "paw during the first 90 minutes 



then near t = 



d R, 

 dt 



26 



