Geoscience Reference
In-Depth Information
r
2
u
d
t
of the cylinder swept during the time d
t
the volume 4
π
×
the number density
ρ
of
molecules in the unit volume:
3
RT
M
1
2
4
r
2
r
2
d
N
=
π
ρ
u
d
t
=
2
π
ρ
(12.23)
where the factor 1/2 accounts for the fact that a collision involves two molecules. The mean
distance between collisions (mean free path)
is equal to the distance
v
d
t
traveled by a
molecule during the time d
t
divided by the number of collisions d
N
:
u
d
t
1
=
u
d
t
=
(12.24)
2
π
r
2
ρ
2
π
r
2
ρ
At 1 bar (10
5
Pa) and 25
◦
C, the number density of nitrogen molecules (
r
10
−
9
m,
=
0.19
×
M
=
28) in a pure N
2
atmosphere is:
6.022
10
23
10
5
×
V
=
N
1
P
RT
=
10
25
m
−
3
ρ
=
=
2.4
×
(12.25)
8.31
×
293
The mean speed is
√
3
RT
=
√
3
515 m s
−
1
. The mean free
/
M
×
8.31
×
298
/
0.028
=
10
−
9
2
2.4
10
25
]
−
1
10
−
7
m.
Such a very small distance between collisions shows that no gas molecule under the
pressure conditions of the Earth's surface should ever leave the atmosphere. This is the
standard tropospheric regime. It is only under the conditions of the upper atmosphere,
where the gas molecules are rarefied, that gas loss may start becoming significant. Let us
try to evaluate how high in the atmosphere we should go before we see this happen. We
show in
Appendix H
that the variation of temperature in an adiabatic atmosphere in which
conduction and radiation can be neglected is given by:
path
is [
(
2
)(
3.14
×
×
=
1.8
×
T
0
1
z
Mg
C
P
T
0
T
=
−
(12.26)
where
z
stands for the altitude above the ground,
M
for the gas mean molar weight,
g
for
the acceleration of gravity,
C
P
for the specific heat at constant pressure, and subscript 0
refers to conditions at the surface of the planet. The density is given by:
ρ
=
ρ
0
1
C
P
T
0
z
C
V
R
Mg
−
(12.27)
where
C
V
stands for the specific heat at constant volume. For a diatomic gas such as
N
2
or O
2
,
C
V
=
5
7
2
R
, so that the density of the atmosphere decreases
with the elevation even faster than its temperature. The length scale (
C
P
/
2
R
and
C
P
=
=
Mg
(
×
) / (
×
)
≈
100 m) is characteristic of a 1 degree drop in temperature.
We now understand that when the pressure drops enough to reduce the collisions, tem-
perature has also decreased to the point where most molecules do not travel fast enough
anymore to leave the gravity field! Heating by solar ultraviolet radiation, however, drasti-
cally affects the situation provided the appropriate gases are present at the right elevation.
We have seen that diatomic molecules such as N
2
,O
2
, and H
2
do not absorb radiation. But
if enough H
2
O, CO
2
,CH
4
, and O
3
is present in the atmosphere and in particular at high
altitude, the planet starts losing the high-velocity fraction of different gases. Absorption
3.5
8.31
0.028
9.81
