koff - LEPA

Γ /Γ eq
1.0
0.8
0.6
0.4
0.2
0.0
0
2000
4000
Kinetic-Steady state diffusion
Diffusion control
Kinetic-Time dependent
diffusion layer thickness
t /s
6000
8000
10000
Fig. 8. Comparison between stirred and unstirred diffusion controlled reactions . ( K=10 9 M –1 ,
kon=10 6 M –1 ·s –1 , c bulk = 0.1 pM, D=10 –10 m 2 ·s –1 , δ = 100 µm and ! max " 10 #9 mol·m #2 ).
This figure shows that the perturbation theory gives a rather good approximation to the exact
solution eq.(38).
3.4 Steady-state approximation for kinetic-diffusion control - General case
Eq.(26) can be more generally written as
d!(t)
dt
= konc surf ! max ( 1"! ) " koff! max! = " J = D
or in terms of surface coverage
d!(t)
dt
= konc surf ( 1!! ) ! koff! =
D
!" max
( ) (40)
! cbulk " c surf
c bulk ! c surf ( ) (41)
from which we can as before obtain an expression for the surface concentration
c surf
( )
( )
= cbulk + Da ! / K
1+ Da 1!!
The differential equation (41) now reads
or
d!(t)
dt
=
( )
konc bulk ( + Dakoff ! ) ( 1!! )
( )
1+ Da 1!!
" 1+ Da 1!! %
$
'd! =
# ! !" ( ! +1)
&
dt
td ( )
! koff! = koncbulk !! konc bulk + koff 1+ Da 1!!
( )
We can integrate this expression explicitly assuming no initial coverage to get
Da! ! 1+ Da " % " " " +1%
%
#
$ " +1&
' ln 1!!
#
$
#
$ " &
'
&
'
( )
t " +1
=
td (42)
(43)
(44)
12