Theory of the Earth
Don L. Anderson
Chapter 5. Thermodynamics and Equations of State
Boston: Blackwell Scientific Publications, c1989
Copyright transferred to the author September 2, 1998.
You are granted permission for indi
vidual, educational, research and
noncommercial reproduction, distribution, display and performance of this work
in any format.
Recommended citation:
Anderson, Don L. Theory of the Earth.
Boston: Blackwell Scientific Publications,
1989.
http://resolver.calt
ech.edu/CaltechBOOK:1989.001
A scanned image of the entire book may
be found at the following persistent
URL:
http://resolver.caltech.edu/CaltechBook:1989.001
Abstract:
The total energy contained in a system, such as a mineral, is called the internal
energy, U, and includes the kinematic and potential energy of all the atoms. It
depends on temperature, pressure and position in the field of gravity.
hermodynamics
Equations
of
State
The
art
of
narrative consists
in
concealing from your audience
every
-
thing
it wants
to
know until
after
you
expose
your
favorite
opinions on
topics foreign
to
the
subject.
I
will now begin,
ifyou
please,
with
a
horoscope located
in
the
Cherokee Nation;
and end
with a
moral
tune
on
the
phonograph.
-0.
HENRY,
"
C
A
BB
A
G
E
S
AND
K
I
N
G
S
"
THERMODYNAMICS
The total energy contained
in
a system,
such
as
a mineral,
is called the
internal energy,
U,
and
includes the kinematic
and
potential
energy
of
all
the
atoms.
It depends
on
tem
-
perature, pressure
and
position
in
the
field
of
gravity.
For
an
infinitesimal change of
the system, the
law
of
conserva
-
tion
of
energy,
or
thejrst
law
of
thermodynamics,
is
dU
=
dQ
-
dW
where
Q
is the heat
flow
and
W
is the mechanical work, for
example the change of
volume
acting against a hydrostatic
pressure
dW
=
PdV
The
enthalpy
or heat content
of
a system
is
H=U+PV
dH=dU
+PdV+
VdP
The energy contents cannot
be
determined
in
absolute
terms;
they
are
only
known as differences. The usual, but
arbitrary, zero point is
known
as
the
standard state
and
is
denoted
A
H
O
.
The
heat
capacity
or
specijic
heat
is the heat required
to
raise a
unit
mass
of the
material
by
one degree. This can
be
done at constant
volume
or at constant pressure
and
the
corresponding symbols are
Cv
and
C,,
For
minerals,
A
certain fraction
of
the heat
entering a system,
dQIT,
is not available
for mechanical work.
The
integral of
this
is
the entropy,
S,
defined
from
giving the
second law
of
thermodynamics,
TdS
=
dU
+
dW
which
applies
to
reversible processes, processes
that
do
not
lose energy to the environment.
In
irreversible processes,
Entropy
is
a measure
of
the energy associated
with
the
random arrangement
and
thermal
motion
of
the
atoms
and
that is therefore
unavailable
for external
work.
At
absolute zero temperature a perfectly
ordered
crystal
has
zero entropy;
with increasing
temperature a certain dis
-
order or randomness
is introduced. The entropy at tempera
-
ture
T
is
At
high
temperature,
When
a mineral
undergoes
a change
of
phase'at temperature
T
involving
a change
in
enthalpy
or latent heat of
transfor-
mation
AH,
there is a discontinuous change
of
entropy:
AS
=
AHIT
The
mechanical part
of
the free energy
U
is the Helm
-
holtz
free energy
F:
F=U-TS
dF
=
dU
-
TdS
-
SdT
=
-P
dV
-
S
dT
giving
P
=
-(dFIdV)T
When using
P and
T as independent variables,
instead
of
V
and
T,
it is
convenient to use
the
Gibbs
free energy,
G:
G=H-TS=U+PV-TS=F+PV
For
a reversible process,
dG=
VdP
-SdT
TABLE
5
-
1
Differentials
of
Thermodynamic Parameters
If
W
is any
thermodynamic function, the
volume
and
pressure derivatives at constant temperature
may
be
related
by
writing
(aw/ev),
=
(aw~ap),
(apiav),
We
can also write
(dWldT)"
=
(dWIdT)P
+
aKT(dW/dP),
where
a
is the
volume
expansion
coefficient.
Thermodynamic
Identities
There are a
variety
of
relations
between
the partial differ
-
entials
of
the standard thermodynamic parameters. Some
of
the standard forms are:
Differential
Constant
element
T
P
V
S
-
PV
KSV
PV
-
yTS
-S+yrnC,
K,V-
yTS
Stacey
(1977).
U
Internal energy
*
V
Volume*
H
Enthalpy
*
y
Gmneisen
parameter
F
Helmholtz
free
energy
*
a
Volume expansion coefficient
G
Gibbs
free
energy
*
p
Density
S
Entropy
*
rn
Mass
of
material
*
T
Absolute temperature
K
Bulk modulus
=
incompressibility
P
Pressure
C
Specific heat
Subscripts signify parameters held constant.
*Parameters proportional to
mass.
THERMODYNAMICS
81
du
=
(auias),
ds
+
(auiav),
dv
=
T
ds
-
P
dv
dH
=
TdS
+
VdP
dF=
-SdT-PdV
dG
=
-SdT
+
VdP
The Maxwell relations are:
(dTlaV),
=
-
(aP/dS),
=
-
yTIV
(dSl~3V)~
=
(dP/aT)v
=
ypCv
=
aKT
(dTIaP),
=
(~3VldS)~
=
yTIKs
-
(dSIdP),
=
(dVIdT),
=
aV
Table
5
-
1
represents all possible partial differentials
of
the
standard parameters. The individual entries are
to
be
taken
in
pairs. Thus
(dTIdP),
is
aT
at constant
S
(that
is,
yT)
divided
by
dP
at constant
S
(that
is,
K,)
giving
(dTIdP)s
=
yTIKs
The
following
partial differentials are
of
particular
interest:
(dPldT)v
=
?PC,
is
the differential form
of
the
Mie
-
Griineisen
equation
and
gives
the variation
in
pressure
in
heating
at
constant
volume.
(daldP),
=
(lIK+)
(dKT/dT),
connects the pressure dependence
of
the
coefficient
of
ther
-
mal
expansion
with
the temperature dependence
of
the
bulk
modulus. The relation
T(daIdT)v
=
-
p(8CvIdP)T
is useful in the high
-
temperature limit
where
C,
=
3R
(R
being
the gas constant)
and
a
is independent
of
T at
con
-
stant
V
and nearly
independent
of
T
at constant
P.
Table
5
-
2
gives thermodynamic data for
a few
minerals.
The combination
aK,
occurs
in
many
thermodynamic
relationships. The
following second
derivative
thermody
-
namic
identities are therefore useful:
Chemical Equilibria
The fact that
a mineral
assemblage changes
into
a different
assemblage means that the
new
association
has
a lower
free
energy
than
the old.
At
equilibrium
both assemblages
have
the same free
energy.
The stable phase
has the lowest
free
energy,
at the
given
pressure
and
temperature
and
mineral
association,
of
all alternative phases.
In
general, the denser
phases are favored at
high
pressure
and
low
temperature.
The
partial
molal free
energy
or
chemical
potential
per
mole
of
species
i
is
F,
,
where
a,
is the
activity
of
a chemical species,
and
Fp
is the
free energy
in
a standard state. The total
energy
is
G
=
n,F,.
where
G is
Gibbs free energy
and
n,
is the
number
of
moles
of
species
i.
At
constant temperature
and
pressure,
dG
=
F,
dn,
F,
=
Mi/..&,
TABLE
5-2
Thermodynamic Properties
of
Minerals
Mineral
CP
cv
by
Ks
Y
0
erg/g/K
(lo6)
(K
x
10
-6)
(kbar)
(K)
MgO
CaO
'41203
Mg,Si04
MgSiO,
MgA1204
SiO,
Garnet
Garnet
and
Mi
is the molecular weight and
pi
is
the
chemical
po
-
tential
per
gram.
The
change
of
activity with
pressure is
The
total Gibbs
free
energy
of
a system
of
C
compo
-
nents and
p
phases
is
where
n{
is the
number
of
moles
of
the
component
i
in
phase
j
and
p{
is its chemical potential in phase
j.
The
equilibrium
assemblage,
at
a given pressure and
temperature, is found
by minimizing
G.
Taking
the
standard
state
of
i
to
be
pure
i
in
phase
j
at
the
pressure and
temperature
of
interest,
where
Hi,,,
Si,,
and
V$
are
the enthalpy and
entropy of
pure
i
in
phase
j
at
P
=
0
and
T
and
volume
at
T.
With
the
chosen
standard
state
for
the
activity,
a$
in
phase
j
contain
-
ing pure
i
is one.
The
activity
of
pure liquids
or
pure
solids
is unity.
In
an
ideal solution
ai
is equal
to
the mole fraction
of
component
i.
At equilibrium
the
standard
-
state free
-
energy
change
at
the pressure and temperature
of
interest is
AG:,,
=
-
RT
In
K
where K is the equilibrium
constant. Consider the hypo
-
thetical reaction
where two molecules
of
A
react to
form
one molecule each
of
C
and
D.
The rate
of
reaction is
proportional
to
the
col-
lisional probability between any two
molecules,
which is
related to the product
of
the
concentrations.
For
equilibrium
the rates
of
the two reversible reactions
are the same,
and
where
(A),
(C)
and
(D)
are
the concentrations
or
activities.
For
the
general
reaction,
then
The
equation
for
equilibrium
yields
dP
AS
AH
-
-
-
-
dT
AV
TAV
which
is the
Clausius
-
Clapeyron
equation.
For
a system in
equilibrium the
following
relation
holds between
the
number
of
coexisting phases
p,
compo
-
nents
c,
and
degrees
of
freedom
f:
This
is
the
phase rule
of
J.
Willard
Gibbs.
The
phases
are
the parts
of
the
system
that can be mechanically separated,
for example, the
minerals and any
coexisting
liquid and
gas.
The
components
are the
smallest
number
of
chemical
species
necessary to make
up
all
the phases.
The
degrees
of
freedom
are generally the temperature, pressure
and
composition.
The
components distribute themselves
over all
the
phases
of
the
system. No
phase can be
without
some contri
-
bution from
all
components since the chemical potential
or
activity
of
each component
must be the
same
in
all phases
of
the system. The
phase
rule places a limitation
on
the
number
of
minerals that
can occur
in
equilibrium
in
a
given
rock. The
maximum
number
of
phases can be
at
-
tained only in an invariant
system,
one
with
P
and
T
fixed.
If
both
P
and
T
vary
during
the
process
of
formation
of
a
rock,
then
which
is the
mineralogical phase rule
of
Goldschmidt.
Because
of
the
phenomenon
of
solid
solution, the number
of
different
minerals in a
rock
is less than the number
of
components.
Table
5
-
3
is a compilation
of
the
terms
and
relations
introduced
in
this section.
THEORETICAL
EQUATIONS
OF
STATE
The
equation
of
state
of
a substance
gives
the pressure
P
as
a function
of
volume
V
and
temperature
T:
The general
expression
for
the
free
energy
of
a crystal can
be written
in
terms
of
three
functions
where
X
=
VJV
=
plp,
is the dimensionless
volume rela
-
tive
to
the
volume
at
normal conditions
and
8
is a charac
-
teristic temperature,
such as
the
Debye
or
Einstein tempera
-
ture.
U(X)
is
the
potential
part
of
the free
energy, which
depends only
on
the
volume.
The
second
term is
the
phonon
term
and is usually calculated
from
the
Debye
or
Einstein
theory.
The
third
term
represents
high
-
temperature correc
-
tions
to the
equation
of
state. This
term,
which
is generally
TABLE
5
-
3
Notation and Basic Relationships
V
=
Specific volume
V,,=VatP=O
p
=
Density
=
M
I
V
p,
=patP=O
T
=
Absolute temperature
P
=
Pressure
S
=
Entropy
a=---
=
Volume
thermal expansion
V
dT
K
=-v
()
--
=
p
(E)
=
Adiabatic bulk modulus
as=Kl
sP-
-V2--V2=
p
s
(z)
=
elastic
ratio
y
=
aKTIpCv
=
crKs/pCp
=
Griineisen ratio
C,
=
Specific heat at constant volume
C,
=
Specific heat at constant pressure
V,,
V,
=
Velocity
of
compressional
and
shear
waves
0
=
Characteristic temperature
=
Mean atomic weight
Bulk modulus
=
Incompressibility
=
l/Compressibility.
6
=
[alnKlalnp],
=
Second Griineisen
ratio
small, is
due
to anharmonic lattice oscillations, formation
of
point defects
and
thermal excitation
of
conduction elec
-
trons in metals.
For most
geophysical problems
U(X)
is the
dominant term.
The potential energy
of
a crystal can
be
written as the
sum
of
an
attractive potential,
which holds
the atoms to
-
gether,
and
a repulsive potential,
which
keeps the crystal
from collapsing:
where
r
is the interatomic spacing and
A,
B,
m
and
n are
constants, different from those
in
the last section. The func
-
tional form
of
the repulsive potential is uncertain,
and
an
exponential form is also often used.
The pressure
is obtained
by
differentiation:
The isothermal
bulk
modulus,
KT,
is
The
bulk modulus
is
also called the incompressibility.
At
P
=
0,
V
=
V,
and
KT
=
KT(0).
The
PV
equation
of
state
can therefore
be
written
as
and
the
bulk modulus
as
The
pressure derivative
of
KT
at
P
=
0
is
K$(O)
is approximately
4
for
many
substances. Since
the
repulsive potential is
a stronger function
of
r
than the
at
-
tractive potential,
n
>
m
and
3
<
n
<
6
for
K$(O)
=
4.
THE
G
R
U
N
E
I
S
E
N
RELATIONS
Griineisen
(1912)
introduced the concept
of
a
"
thermal
pressure
"
derived from the pressure
of
a collection
of
atoms
vibrating under the excitation
of
the
energy associated
with
nonzero temperature.
A
crystalline
solid
composed
of
N
at
-
oms has
3N
degrees
of
freedom,
and
the solid can
be
viewed
as
a collection
of
harmonic oscillators. The
energy
levels
of
a harmonic oscillator are nhv,
where
n are succes
-
sive integers
and
h is Planck's
constant.
In
thermal equilib
-
rium
a given
energy level
is populated with the probability
exp(
-
nhvlkT),
where
k is
Boltzmann's
constant. The in
-
dividual oscillators
have
a
frequency
vi,
and
these are
con
-
sidered
to
be
independent
of
temperature
but
dependent
on
the volume,
V.
The quantity
d
log
vi
Yi
=
-
-
d log
V
is
involved
in
calculations
of
the thermal pressure
and
is
known
as the
Griineisen
ratio.
If it is
assumed that
all
the
yi
are the same,
then
Vi
=
v-Y
The
Griineisen
equation
of
state
is
where
Po
is the pressure at absolute zero
and
UD
is the in
-
ternal energy
of
the oscillators in a
volume
V
due
to
the
elevated
temperature:
Differentiating P
with
respect to temperature gives
where
C,
is the specific heat
at
constant
V.
From the ther
-
modynamic relations
where
a
is the volume coefficient
of
thermal expansion, the
following relation can
be
derived:
which
is called the
Griineisen
relation.
The thermal energy
of
a crystal is equal to
the
sum
over all oscillators
and,
therefore,
over
all pertinent fre
-
quencies.
In
the
Debye
theory the
sum
is replaced
by
an
integral,
and
it is
assumed
that all frequencies
of
vibration
are
bounded
by
some
maximum
value
vi
<
v,.
In
an elastic solid three modes
of
wave
motion
are per
-
mitted, one compressional mode
and two
shear
modes hav
-
ing orthogonal particle motions. The total thermal energy is
therefore
where
No
is the number
of
atoms
per
unit volume.
The
maximum
oscillation frequency
is
related to the
volume
available to
the
oscillator
and
the
velocity
of
elastic
waves. In
the
Debye
theory a
mean
sound velocity is im
-
plied,
and
thus
where
N,
is
Avogadro's
number,
p
is the density,
and
is
the
mean
atomic
weight
(molecular
weight
divided
by
the
number
of
atoms
in
the
moleculej.
In
the
Debye
theory it is
assumed that
velocity
is isotropic
and
nondispersive, that
is,
independent
of
direction
and
frequency.
The
Debye
temperature is
defined
as
and
therefore
where
V,
is mean
velocity:
The
Debye
temperature can
be
estimated from the
ve
-
locities
of
elastic
waves
and,
therefore, can
be
estimated for
the mantle from seismic data.
In
principle,
the
velocities
should
be measured
at frequencies
near
v,
(
-
lOI3
Hz) since
there is some dispersion. Ignoring dispersion,
however,
is
consistent
with
Debye's
original assumption. There are also
optical modes,
as well as
acoustic modes,
and
these are ig
-
nored
in
the simple theories.
By
differentiation
of
U
we
obtain for the thermal pressure
U,
d log 0
p*
=
-
-
V
d log
V
and,
therefore,
-
d log 0
-
d log
v,
-
-
=
d log V
d log
V
At
high
temperature,
01T
<<
1,
At
very
low
temperature,
At
intermediate temperature,
where
5
=
hvlkT.
The thermal pressure
in
the mantle is estimated
to be
between 10
and
200 kilobars, increasing
with
depth. The
Debye
temperature increases
by
about a factor
of
2
through
the mantle,
and
the
Griineisen
parameter probably remains
close to
1.
The specific heat
can
be
written
At
T
>>
0
we
have the
classic high
-
temperature limit,
Silicates
show
a close approach to the
"
classical
"
values
at
temperatures greater
than about
1000°C.
Under
these con
-
ditions
C,
approaches
6
calI0C
g atom for each particle
of
the chemical formula. The
mean
atomic
weight
for
most
rock
-
forming minerals is close to
20,
so
the specific heat at
high
temperatures is close
to 0.3
calIoC
g.
The variation
of
specific heat
with
pressure is
The
specific heat probably only decreases
about
10
percent
at the highest mantle pressures,
and
its variation is therefore
small relative to the changes expected for bulk
modulus
and
thermal expansion.
Most
of
the interior
of
the Earth is hot,
well above
the
Debye
temperature. This
means
that the
Earth's
interior
probably can be treated with classical solid
-
state physics
concepts.
I
say
"
probably
"
because the interior
of
the Earth
is at simultaneous
high
temperature
and high
pressure
and
these are competing effects. The quantization
of
lattice
vi
-
brations
and
the departures from classical behavior that are
of
interest
to
quantum and low
-
temperature physicists are
not relevant except, in some cases,
when
extrapolating from
laboratory measurements to the high temperatures in
the
in
-
terior. The close relationship
between
y
and
the elastic con
-
stants
and
their pressure derivatives means
that
y
can be
estimated from seismology.
The
thermal pressure,
P,
can
be viewed
as the radia
-
tion pressure exerted on the solid
by
completely
diffuse
elastic waves, that
is,
-
-
up
ae,
us
ae,
2
-
-
8,
av
0,
av
where
the
Urn,
V,
and
6,
are
the
thermal energies, elastic
wave
velocities
and
characteristic temperatures associated
with the longitudinal
(P)
and
transverse
(S)
waves.
At
high
temperature
we
have
The
thermal pressure can
be
written
in
a form analogous to
the perfect gas equation:
Q
P*
=
-
RT,
Q
=
y,
+
2ys
v
where
Q
is of
the order
of
5
or
6
for
many
elements
and
is
near
4
for
MgO
and
A1203.
EFFECT
OF
TEMPERATURE
ON
BULK
MODULI
The pressure
and
the isothermal
bulk modulus
are
volume
derivatives,
at
constant temperature,
of
the free energy
F(V,T).
The corresponding adiabatic quantities are
vol
-
ume
derivatives
of
the internal energy
U
(V,S)
at constant
entropy. The equation
of
state
of
simple solids subjected
to hydrostatic pressure can
be
written
in
two
alternative
forms. The
vibrational
formulation splits the free
energy
of
the solid into the
lattice
energy,
U,(V),
which
is the energy
of
a static solid
of
volume
V
in
its
electronic ground state,
and
a
vibrational
energy
U*(KT).
The
thermal
formula
-
tion splits the free energy into a nonthermal
cohesive
energy
U,(V)
of
the solid
of
volume
V
at
0
K
and
a
thermal
energy
U*(V,T).
Note
that
the lattice
and
cohesive energies depend
only
on volume
and
the terms
with
asterisks depend, in
gen
-
eral, on
both volume and
temperature;
in
the Hildebrand
approximation the thermal
and
vibrational energies are
taken
to
be
a function
of
temperature alone,
this being
a
good
approximation at
high
temperatures
where
the heat
capacity at constant
volume has
attained its classical value.
The cohesive energy is
the
free
energy required
to assemble
the atoms from
infinity to
form
the
rigid lattice;
it includes
both
static lattice
and
zero
-
point energy contributions. The
total vibrational
energy of
the solid
is the sum
over all
the
modes
of
lattice vibration
of
all
the
particles.
The
vibra
-
tional energy
Um(V,T)
consists
of
the zero
-
point vibrational
energy,
U*(V,O),
of
the normal
modes at
T
=
0
K
plus the
energy required to heat the lattice at constant volume,
V,
from
0
K,
to
TK;
that is,
The
Helmholtz
free
energy,
in
the Hildebrand approxi
-
mation, can be written, for example,
F(V,T)
=
U,(V)
+
U*T
-
TS(V,T)
Since
P
=
-
(aFIdV),
and
we
have
and
The quantity
is of
the order
of
-
1.
The quantity
aKT
is of
the order
of
10 to 100
bar/K
for elements
and
is between
about 30
and
70 for compounds
of
interest in the deeper mantle. The
quantity
is
of
the order
of
-
50
bar/K,
and
a temperature rise
of
some 2000
K changes the bulk modulus
by
about
100 kbar,
which
is about
10 percent
of
estimated
values
for
the bulk
modulus
in
the mantle.
The
following
relations are
useful
and
serve
to define
the
second Griineisen
parameter,
6:
The elastic
moduli
of
a solid are
affected by
tempera
-
ture both implicitly, through the
volume,
and
explicitly.
Thus, for example,
and
The
measured
variation
of
KT
with
temperature
is,
then,
where
(a
In
KT/dT),
is the intrinsic temperature dependence
of
KT.
(a
In
KS/dT),
is positive. There is a general tendency
for
(d
In
KT/dT),
to
be
smaller at high
Tie.
Experiments
show
and
all
of
which
are useful
when
trying to estimate the
effects
of
pressure, volume,
and
temperature on the adiabatic
bulk
modulus.
Note
that these are all experimental
and
thermo
-
dynamic inequalities and are independent
of
the equation
of
state.
We
also
note that
6,
<
K'
<
6,
8~
-
6s
=
Y
The seismic parameter
@
is simply
@
=
Kslp
so that
The pressure
in
the
mantle
rises
to about
1500 kbar,
which, for
(dKldP),
=
4,
corresponds
to
a 6000
-
kbar in
-
crease
in
the
bulk
modulus. Temperature can therefore
be
treated
as
a small perturbation on the general trend
of bulk
modulus,
or
@,
with
depth,
at least in the deeper part
of
the
mantle.
THERMAL
EXPANSION
AND ANHARMONICITY
Because the attractive
and
repulsive potentials
have
a dif
-
ferent dependence
on
the separation
of
atoms, the thermal
oscillation
of
atoms
in
their (asymmetric) potential
well is
anharmonic or nonsinusoidal. Thermal oscillation
of
an
atom causes the
mean
position
to be
displaced,
and
thermal
expansion results.
(In
a symmetric,
or
parabolic, potential
well
the mean
positions are unchanged, atomic vibrations
are harmonic,
and
no thermal expansion results.) The
Debye
model
is
restricted
to
assemblages
of
harmonic oscillators
and,
strictly speaking, cannot
be
used
to discuss
anhar-
monic
effects such
as
thermal expansion. Anharmonicity
causes atoms to take
up new
average positions
of
equilib
-
rium, dependent on
the
amplitude
of
the vibrations and
hence on the temperature,
but
the
new
positions
of
dynamic
equilibrium
remain nearly
harmonic.
At
any given
volume
the harmonic approximation
can
be
made
so that the char
-
acteristic temperature,
6,
and
frequency are
not
explicit
functions
of
temperature. This
is called the quasi
-
harmonic
approximation.
If
it is
assumed
that a change
in
volume
can
be
adequately
described
by
a change
in
8,
then
the
frequency
of
each normal
mode of
vibration is changed in
simple proportion
as
the volume
is changed. The Griineisen
parameter