Geoscience Reference
In-Depth Information
1000
500
200
100
0
T
(
°
C)
−
5
Na
+
Ca
2+
(Basalt)
−
10
Na
+
(Water)
Fe
2+
(Olivine)
−
15
Ar
(K-feldspar)
Yb
3+
(Garnet)
−
20
O
(Quartz)
−
25
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1000/
T
(K)
Figure 5.3
Arrhenius plot of the diffusion coefficients of various elements in different media.
Semi-logarithmic graph. Notice temperatures are in K at the bottom and
◦
Catthetop.
ensures that the flux is directed toward the part of the system with a lower
i
concentration.
Equation
(5.10)
isFick'slaw.
For example, we can calculate the diagenetic diffusive flux of manganese (
i
Mn) in
sedimentary pore water to ocean bottom water. Let us assume that, because of its reducing
character, interstitial water 10 cm beneath the surface has a manganese concentration of
10
=
mol l
−
1
. Seawater, being highly oxidized, has a manganese concentration that is virtu-
ally nil. The diffusivity of manganese in seawater is
D
Mn
μ
10
−
9
m
2
s
−
1
. The local
diffusive flux of manganese to the ocean in absolute terms is therefore:
=
×
0.7
10
−
6
10
−
3
10
−
9
10
×
×
−
0
J
Mn
10
−
16
mol m
−
2
s
−
1
=
0.7
×
=
0.7
×
0.1
Clearly, as we considered the probability of a forward or backward jump as equal (Brow-
nian motion), the mean displacement of an atom is zero. Nonetheless, we can calculate a
measure of the dispersion of a population of atoms initially in position
x
through another
simple statistical concept, the standard deviation of displacement, i.e. the mean quadratic
displacement. During time
t
, the number of jumps by an atom is equal to
Pt
and the
quadratic distance traveled by an atom is
Ptl
2
,or2
D
i
t
. The mean quadratic deviation,
i.e. the distance
δ
over which two-
thirds
of the atoms will be found on either side of their
initial position, is therefore
√
2
D
i
t
. This distance is a measure of the migration char-
acteristic of the atoms under the effect of diffusion and is generally short; it is typically a
few centimeters per year for an ion in seawater, meters per million years for the Fe-Mg
exchange of olivine in the mantle, or a micrometer per million years for oxygen isotope
exchange in metamorphic quartz.
Diffusion is not therefore a process for long-distance transport, but it usually does allow
minerals to achieve equilibrium with each other or with solutions percolating through the
δ
=
