Determination of 222 Rn Diffusion Coefficient in Japanese Soils

P-1b-48
Compartment 2
V2 C2
Sample soil
Co(z) Po
Compartment 1
V1 C1
Rn source
z
z= Ho
z=0
Figure 1 The schematic cross section of diffusion chamber.
The time dependent equation describing the concentration, C 0 , is given by Fick’s Law:
dC
dt
0
D
0
d
2
0
2
dz
C
+ P − λC
= (1)
where λ =the decay constant for 222 Rn.
dC
dt
0 =
0
In steady state 0 and Eq.1 transforms into:
d
2
0
2
dz
C
C
P
0
0
− λ 0 = −
(2)
D0
D0
The boundary conditions at equilibrium are:
C 0 (z=0)=C 1 and C 0 (z=H 0 )=C 2 (3)
Eq (2) becomes
d
2
0
2
dz
with
C
C
P
0 0
− = −
(4)
2
l0
D0
2 D0
l
0 = ,
0
λ
l =diffusion length.
The solution of Eq.4 is given as:
⎛ z ⎞ ⎛ z ⎞ P
C0 () z Asinh
⎜ B cosh +
l
⎟
⎜
0
l
⎟
⎝ ⎠ ⎝ 0 ⎠ λ
0
= +
(5)
From the boundary conditions in Eq.3 it follows that:
C
=
C cosh β − P0 (1
sinh β
2 − 1
−
cosh β ) / λ
A (6a)
B
C
1
P0
−
λ
= with
H
=
l
0
β (6b)
0
2