Geology Reference
In-Depth Information
5.1 Thermal expansion
We will start with density, and its inverse, specific volume. Density,
ρ
, is the mass
of a unit volume of material. Specific volume,
V
, is the volume occupied by a unit
mass of material. Upper-mantle rocks have a density of about 3300 kg/m
3
.The
specific volume of upper-mantle rock is then
1
/
3300 m
3
/kg
10
−
4
m
3
/kg
.
V
=
1
/ρ
=
=
3
×
(5.1)
We more commonly work with density, but volume may be easier to visualise for
the moment.
Thermal expansion occurs when material warms up. Suppose it starts with a
volume
V
0
and a temperature
T
0
, then it warms up to temperature
T
and attains
a volume
V
. The way thermal expansion works is that the change in volume is
proportional to the initial volume and to the change in temperature. We can write
this as
V
−
V
0
=
αV
0
(
T
−
T
0
)
.
(5.2a)
We can also write it as
V
V
0
=
αT,
(5.2b)
where
V
T
0
. Thus
V
is the change in volume, and
V/V
0
is the
fractional
change in volume. The quantity
α
in Eqs (5.2) is called
the
coefficient of thermal expansion
. It is different for every material and thus it
is called a material property. So, from Eq. (5.2b), we can say that the fractional
change in volume is proportional to the change in temperature, and the constant of
proportionality is the coefficient of thermal expansion.
Thermal expansion can also be expressed in terms of changes in density. Thus,
from Eq. (5.2a)
=
V
−
V
0
and
T
=
T
−
1
ρ
−
1
ρ
0
=
ρ
0
−
ρ
α
ρ
0
T ,
=
ρρ
0
which can be rearranged into
ρ
0
−
ρ
ρ
ρ
ρ
=−
=
αT.
This is close to the form of Eq. (5.2b), but there is a minus sign and, more
significantly, the denominator is the
final
density, rather than the initial density.
The minus sign arises because we choose to define the change in density as the
final density minus the initial density. The second difference, the final density
in the denominator, would complicate the algebra when using the relationship.
Fortunately, the density changes in the mantle context (and in most convection)
