Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

11.2 Diffusion-controlled reactions 232
(3) With =4 2 as the surface of the sphere with radius around , weobtain
= × 4 2 × []
(11.16)
(4) To determine the concentration profile []
, we integrate over []
:
Z []∞ Z ∞
[]
=
[]
4 2
(11.17)
y
∞
[]| ∞
= −
4
¯¯¯¯
(11.18)
y
[] − []
= − 0 (11.19)
4
(5) is found from the boundary conditions that []
=0at = :
=4 × [] (11.20)
(6) The concentration gradient of [] is obtained by inserting the above result for
into Eq. 11.19:
[] − []
= []
(11.21)
y
³
[]
=[] 1 − ´
(11.22)
This expression is plotted in Fig. 11.3.
(7) To determine , we write the rate of the diffusion controlled reaction as
[ ]
= [] (11.23)
Inserting the expression for from Eq. 11.20, we obtain
[ ]
=4 × [][] (11.24)
The diffusion-limited rate constant is thus (in molecular units)
=4 (11.25)
(8) Nernst-Einstein relation between diffusion constant and viscosity for spherical particles
with radius :
=
(11.26)
6
y
∝
(11.27)
(9) Keepinmindthatdiffusion is an activated process:
= 0 −
Typical value for H 2 O: ≈ 15 kJ mol −1 .
(11.28)