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294 4. Adsorption and Ion Exchange

ate
Absolute r, r and equilibrium ate elative r ,
,
For any reactor the conservation principle can be represented by the following relation-
ship:

rate of i into volume element – rate of i out of volume element rate of production of i
within the volume element = rate of accumulation of i within the volume element

In the ideal batch-stirred tank reactor (BSTR), the fluid concentration is uniform and there
are no feed or exit streams. Thus, only the last two terms in the previous equation e xist.
For a volume element V , the mass balance becomes (Smith, 1981)

CV d i
rV i (4.109)
t d

where C i is the concentration of species i at any time. If the volume of the reaction mix-
ture is constant, the aboe equation becomes v

C d

i
r
i (4.110)
t d
where r is the intrinsic rate of the reaction. F it is xchange systems, or adsorption and ion-e
i
more appropriate to call this rate the “absolute (uptake) rate.” Note that this is also the def-
inition of a homogeneous reaction rate. The sameness is due to the fulfillment of the
requirements of uniform concentration, temperature, and constant volume in an ideal
BSTR.
The absolute rate, i.e. the amount adsorbed per unit time, is a function of the dif fusion
coefficient, i.e. the relatie rate expressed by v U ( t ), as well as of equilibrium expressed by
v the aailable or actie sites for adsorption or ion exchange. Since for v t > t 1
2
qt () 1
q t ( )
2
Ct () 1 (4.111)
Ct ( )
2
then

V V
qt () C t C t C C t C C t
qt ( )
2 1 2 1 o 2 o 1 (4.112)
m m
where

C o i U t C o (4.113)
C
C t ()
i