Geoscience Reference
In-Depth Information
However, we know from Maxwell's thermodynamic relations that
∂
T
=−
∂
S
∂
P
V
∂
T
(7.87)
P
where
V
is volume. Thus, Eq. (7.86) becomes
∂
T
∂
P
S
=
∂
T
∂
S
∂
V
∂
T
(7.88)
P
P
The definition of
α
, the coefficient of thermal expansion, is
∂
V
∂
T
1
V
α
=
(7.89)
P
The definition of specific heat at constant pressure
c
P
is
mc
P
=
T
∂
S
∂
T
(7.90)
P
where
m
is the mass of the material.
Using Eqs. (7.89) and (7.90), we can finally write Eq. (7.88)as
∂
T
∂
S
=
T
α
V
mc
P
(7.91)
P
Since
m
/
v
=
ρ
, density, Eq. (7.91) further simplifies to
∂
T
∂
P
S
=
T
α
ρ
c
P
(7.92)
For the Earth, we can write
d
P
d
r
=−
g
ρ
(7.93)
where
g
is the acceleration due to gravity. The change in temperature with radius
r
is therefore given by
∂
T
∂
r
S
=
∂
T
∂
P
d
P
d
r
S
T
α
ρ
c
P
g
=−
ρ
T
g
c
P
α
=−
(7.94)
For the uppermost mantle, the adiabatic temperature gradient given by
Eq. (7.94)isabout 4
10
−
4
◦
Cm
−
1
(0.4
◦
Ckm
−
1
) assuming the following values:
×
T
, 1700 K (1427
◦
C);
10
3
Jkg
◦
C
−
1
.
At greater depths in the mantle, where the coefficient of thermal expansion
is somewhat less, the adiabatic gradient is reduced to about 3
10
−
5
◦
C
−
1
;
g
, 9.8ms
−
2
; and
c
P
, 1.25
α
,3
×
×
10
−
4
◦
Cm
−
1
(0.3
◦
Ckm
−
1
). Figure 7.16 illustrates a range of possible models for the tempera-
ture through the mantle. Many estimates of the increase of temperature with depth
in the mantle have been made: all agree that the temperature gradient though the
upper mantle will be approximately adiabatic. If the upper and lower mantle are
separately convecting systems, the temperature will increase by several hundred
degrees on passing through the boundary layer at 670 km. In the outer part of
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