CHAPTER 5 THERMODYNAMICS OF SOLUTIONS (43 p)

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1
CHAPTER 5
THERMODYNAMICS OF SOLUTIONS
(43 p)
At the outset it may be remarked that the chief difficulty in the study of solutions is that thermodynamics
provides no detailed information concerning the dependence of the chemical potential (or other
thermodynamic functions) on the composition. (Denbigh, 1971
, p. 215
)
The relations we developed in the previous chapter apply only to closed systems (i.e.,
systems of constant bulk composition) consisting of phases of constant composition. In
the natural world, however, most geologic materials are solutions of variable
composition, a
solution
being defined as a homogeneous phase formed by dissolving one
or more substances (solid, liquid, or gas) in another substance (solid, liquid, or gas). A
solution may be a solid (e.g., a solid solution, such as olivine or plagioclase), a liquid
(e.g., an aqueous solution resulting from dissolution of NaCl crystals or CO
2
in water, or
a silicate melt generated by partial melting of rocks), or a gas (e.g., a mixture of O
2
and
CO
2
). A solution may be ideal or non-ideal, and in many cases the value of a
thermodynamic function at the same pressure and temperature may be significantly
different, depending on whether the solution is treated as ideal or non-ideal. In this
chapter we will learn how to evaluate quantitatively the thermodynamic properties of
solutions, excluding aqueous electrolyte solutions, which will be discussed in
Chapter

7
.
We will see that gas mixtures are treated somewhat differently than liquid and solid
solutions. For non-ideal gas mixtures the strategy commonly employed is to modify the
ideal gas equation to fit the available
P-V-T
data. In the absence of equations of state
comparable to the ideal gas equation, non-ideality in liquid and solid solutions commonly
are described as deviation from ideality through excess functions, which are then used to
derive
P-V-T
equations of state (Anderson and Crerar, 1993,
p. 417
).
To describe the composition of a solution, we need to know what are the substances
present in the solution and in what quantities. The substances in a solution may be
specified in terms of “components” in the sense of the Gibbs phase rule, as defined in
Section 4.2.3
, but many reactions cannot be represented in terms of “components” in the
true sense. We will use the less restrictive term
constituents
as defined by Anderson and
Crerar (1993, p. 242), a constituent of a phase or a system being “any combination of
elements in the system in any stoichiometry.” Constituents may be “components”, actual
chemical species present in a solution (e.g.,
€
HCO
3

in a H
2
O-CO
2
solution) or even
hypothetical species, or end-members of a solid solution series (e.g., NaAlSi
3
O
8
and
CaAl
2
Si
2
O
8
for plagioclase). The quantity of any constituent
i
in a solution is expressed
either as number of moles or
mole number
(
n
i
) or as
mole fraction
(
X
i
).
2
5.1 CHEMICAL POTENTIAL
5.1.1 Partial molar properties
In general, the thermodynamic properties of a solution cannot be calculated by regarding
it as a mechanical mixture of its constituents. For example, if we add 1 mole of NaCl to
1 kg of water, the dissolved NaCl will dissociate into Na
+
and Cl

, and the volume of the
solution will not be equal to the sum of the two volumes; actually it will be slightly less
because of electrostatic interactions between the ions and H
2
O molecules. Similarly, the
entropy and the free energy of a plagioclase solid solution will not be the sum of the
entropies and free energies of its end-members, NaAlSi
3
O
8
and CaAl
2
Si
2
O
8
because of the
entropy and free energy associated with the mixing of the two end-members.
In order to address the effect of composition of a system on its extensive
thermodynamic properties we introduce the concept of partial molar properties (also
referred to as partial molal properties by some authors). Consider an extensive property
Y
of a homogeneous phase
α
that consists of
k
different constituents. Let the amounts
(moles) of the constituents in the whole of the phase be denoted as
€
n
1
α
,

n
2
α
,
...,

n
i
α
,
...,

n
j
α
,
...
n
k
α

. Mathematically, the corresponding
partial molar property
of
the
i
th constituent in the phase
α
,

€
y
i
α
, is defined as a partial derivative of
€
Y
α
taken at
constant temperature, pressure, and
€
n
j
α
:
y
i
α

=


Y
α

n
i
α








P
,

T
,

n
j
α
(5.1)
where
€
n
j
α

refers to all constituents in the phase
α
other than
i
(i.e.,
j

i
). In other words,
€
y
i
α
represents the change in
€
Y
α
resulting from an infinitesimal addition of the
i
th
constituent (so as not to change its overall composition) when the pressure, temperature,
and the amounts (i.e., mole numbers) of all other constituents in the phase are held
constant. The partial molar quantity
€
y
i
α
is an intensive property.
As
Y
is a function of temperature, pressure, and composition, the total differential of
Y
,
dY,
is
€
dY
α

=


Y
α

T






P
,

n
i
α
,

n
j
α
dT

+


Y
α

P






T
,

n
i
α
,

n
j
α
dP

+


Y
α

n
i
α






P
,

T
,

n
j
α

dn
i
α
i

1
i

k


(5.2)
which can be integrated at constant temperature and pressure to obtain (see, for example,
Fletcher, 1993, p. 108)
€
Y
α

=

n
i
α


Y
α

n
i
α






i

P
,

T
,

n
j
α
=

n
i
α

y
i
α
i


(5.3)
3
Thus, at constant temperature and pressure, an extensive property of a multi-constituent
phase is the sum of the corresponding partial molar property of all its constituents
multiplied by their respective mole numbers.
Dividing both sides of
Equation 5.3
by the total number of moles,
€
n
i
α
i

, we obtain
Y
α

=

X
i
α

y
i
α
i


(5.4)
where
€
Y
α
is the corresponding molar quantity for the phase
α
and
€
X
i
α

is the mole fraction
of the
i
th constituent in the phase. Note that for a pure phase (i.e., a phase composed of
one constituent only so that
i
= 1 and
€
X
i
α

=
1
), the partial molar quantity is the same as
the corresponding molar quantity
€
(
Y
i
α
)
:
€
Y
α

=

X
i
α

y
i
α

=

y
i
(
X
i
=
1)
α

=

Y
i
(
X
i
=
1)
α

i


(5.5)
If a system consists of several multi-constituent phases (
α
,
β
,
γ
, ….), we can write an
equation similar to
5.3
for each phase and add them to obtain the value of
Y
for the
system:
€
Y
system

=

n
i
α


Y
α

n
i
α






i

P
,
T
,
n
j
α
α


=

n
i
α

y
i
α
i

α


(5.6)
_______________________________________________________________________
Example: Estimation of partial molar quantities of dissolved constituents in a solution
Let us consider a simple binary solution of
n
1
moles of NaCl (solute) in
n
2
moles of H
2
O
(solvent). The volume of the
€
NaCl

H
2
O
solution, according to
Equation 5.3
, is
€
V
solution

=

n
1
v
NaCl
solution

+

n
2
v
H
2
O
solution
(5.7)
To express
Equation 5.7
in terms of mole fractions of the constituents,
X
NaCl

=

n
1
/ (
n
1
+
n
2
)
and
X
H
2
O

=

n
2
/ (
n
1
+
n
2
)
=
1

X
NaCl
, we divide each term in
the equation by (
n
1
+
n
2
):
V
solution

=

X
NaCl

v
NaCl
solution

+

X
H
2
O

v
H
2
O
solution
(5.8)
where
V
solution
is the molar volume of the solution. We can determine graphically
€
v
NaCl
and
€
v
H
2
O
corresponding to the given composition of the
€
NaCl

H
2
O
solution if we know
from experimental data how
V
solution
varies with composition (
Fig. 5.1
).
4
5.1.2 Definition of chemical potential
In order to extend the application of the auxiliary thermodynamic functions (
Equations
4.36 to 4.38
) to systems that may undergo a change in composition, Gibbs in 1876–1878
introduced the term
chemical potential
(

). The chemical potential of the
i
th constituent
in phase
α
,
€

i
α
, can be defined mathematically as a partial derivative of
U
α
(at constant
S
,
V
,
n
j
),
H
α
(at constant
S
,
P
,
n
j
),
A
α
(at constant
T
,
V
,
n
j
), or
G
α
(at constant
P
,
T
,
n
j
), but the
most useful definition is with respect to
G
α
:
€

i
α

=


G
α

n
i
α






P
,
T
,
n
j
α

(5.9)
This is because
€

i
α
in
Equation 5.9
is defined at constant temperature and pressure, and
thus represents
partial molar free energy
. Thus,
the chemical potential of a constituent i
is its partial molar free energy
. Note that for a pure phase, the partial molar free energy
is equal to the molar free energy (i.e.,
G
i
α

=


i
α
) at constant temperature and pressure.
Thus, for a reaction, such as
4.92
, involving only pure phases, the change in chemical
potential,
Δ

r
, is the same as the change in free energy,
Δ
G
r
at constant temperature and
pressure:
5
€
Δ

r

=
2

MgSiO
3
clinoenstatite



Mg
2
SiO
4
forsterite




SiO
2
α
-
quartz

=
2
G
(clinoenstatite)


G
(forsterite)


G
(
α
-
quartz)
=

Δ
G
r

Now let us see how we can express the free energy of a multi-constituent phase
α
in
terms of the chemical potentials of its constituents. Expressing
G
α
as
a function of
P
,
T
,
and composition,

the total differential

€
dG
α
can be written as
dG
α

=


G
α

T








P
,
n
i
α
,

n
j
α
dT

+


G
α

P








T
,

n
i
α
,
n
j
α
dP

+



G
α

n
i
α








P
,

T
,

n
j
α
dn
i
α
i


(5.10)
Substituting for the temperature and pressure partial derivatives (
Equations 4.55
and
4.56)
,
dG
α

=


S
α

dT

+

V
α

dP

+


i
α

dn
i
α
i


(5.11)
At constant pressure and temperature (
dP
= 0,
dT
= 0),
Equation 5.11
reduces to
dG
α

=


i
α

dn
i
α
i

(5.12)
which can be integrated to yield
G
α

=


i
α

n
i
α
i

(5.13)
Note that for one mole of a pure phase,
G
=

. For a system composed of several multi-
constituent phases (
α
,
β
,
γ
, ….), at constant
P
and
T
,
G
system

=


i
α

n
i
α
i

α

(5.14)
Equations 5.10 to 5.14
are valid for both closed systems involving composition
changes due to reactive combination of existing constituents within the system and for
open systems in which composition changes may occur also due to mass transfer between
the system and surroundings. Note that these equations are consistent with those derived
for closed systems of constant composition in Chapter 4. For example,
Equation 5.11
reduces to
Equation 4.54
when the term accounting for compositional variation vanishes
in the case of constant composition (i.e.,
€
dn
i
α

=
0
).
Like all partial molar properties,
€

i
α
varies with temperature, pressure, and
composition (see
Sections 5.2 and 5.3
).
6
5.1.3 Criteria for equilibrium and spontaneous change among phases of variable
composition
Consider a hypothetical system composed of two phases,
α
and
β
, with a common
chemical constituent
i
. If
dn
i
moles of
i
are transferred from
α
to
β
at constant
temperature and pressure, then conservation of mass requires that

dn
i
α

=

dn
i
β
The change in the free energy of the system is
dG

=


i
α

dn
i
α

+


i
β

dn
i
β

=


i
α

dn
i
α

+


i
β
(–
dn
i
α
)
=
(

i
α


i
β
)
dn
i
α

For a reversible process at constant temperature and pressure,
dG
= 0 (
Equation 4.57
)
and, since
€
dn
i
α


0
,

i
α

=


i
β
Extending the above argument to pairs of phases in a system containing several phases
(
α
,
β
,
γ
, …), it can be shown that at equilibrium (constant temperature and pressure),

i
α

=


i
β

=


i
γ

=
........
(5.15)
Thus, in a system at equilibrium (constant temperature and pressure), the chemical
potential of a constituent must be the same in all phases in the system among which this
constituent can freely pass. If two phases are separated by a semi-permeable membrane
that is permeable to some constituents but not to others,
Equation 5.15
would apply only
to those constituents that can pass freely through the membrane.
A spontaneous reaction should occur (at constant temperature and pressure) if either
dG
< 0 or
dG
> 0. If
dG
< 0,
€
(

i
α


i
β
)
dn
i
α

<
0
, and since
€
dn
i
α


0
,

i
α

>


i
β
(5.16)
Similarly, if
dG
> 0,

i
β

>


i
α
(5.17)
In either case, to minimize the free energy of the system, spontaneous transfer of
i
would
occur from a phase in which its chemical potential is higher to another phase in which its
chemical potential is lower, until the chemical potentials become equal. This is why

i
is
called a “potential”. Chemical potential gradients due to differences in chemical
potentials are the cause of all processes of diffusion. The direction of decreasing
chemical potential usually, but not always, coincides with the direction of decreasing
concentration.
7
_______________________________________________________________________
Example: Equilibrium and mass transfer between calcite and aragonite
Let us take another look at the calcite – aragonite equilibrium (
Fig. 4.12
):
CaCO
3
(calcite) = CaCO
3
(aragonite)
(4.100)
which we discussed in the previous chapter and see how chemical potentials of the
constituent CaCO
3
in these phases are related to equilibrium or mass transfer between
them (
Fig. 5.2
).
Consider a system consisting of the polymorphs calcite and aragonite at pressure and
temperature defined by point X. At this condition of pressure and temperature, which lies
in the stability field of aragonite,

CaCO
3
aragonite

<


CaCO
3
calcite

, and material should flow from the
calcite phase to the aragonite phase until all the calcite disappears. The opposite should
happen if the system is placed in the stability field of calcite, such as the point Y. In this
case,
€

CaCO
3
calcite

<


CaCO
3
aragonite

, and material should flow from the aragonite phase to the
calcite phase until all the aragonite disappears. At any point, such as Z, on the reaction
boundary, calcite and aragonite are in equilibrium,

CaCO
3
calcite
=


CaCO
3
aragonite

, and there should be
no net flow of material between the two phases once equilibrium is established.
Figure 5.2
Pressure-temperature phase diagram for the calcite-aragonite system
illustrating conditions for equilibrium and spontaneous reaction in terms of the chemical potentials of
CaCO
3
in the two polymorphs. At the pressure and temperature represented by point X, calcite should
spontaneously change to aragonite; at point Y, aragonite should spontaneously change to calcite; and at
point Z, calcite and aragonite should coexist in equilibrium
.
8
5.1.4 Criteria for equilibrium and spontaneous change for a reaction
To find the condition of equilibrium at constant temperature and pressure for a reaction in
terms of chemical potentials, we define a quantity called
the extent of reaction
(
ξ
) as
d
ξ

=

dn
i
ν
i
(5.18)
where
ν
i
is the stoichiometric coefficient of the
i
th constituent in the reaction.
Substituting for
dn
i
in
Equation 5.12,
at constant temperature and pressure,
(
dG
)
T
,
P

=

(
ν
i

i


i
)
d
ξ
i

G
∂ξ








T
,
P

=

ν
i
i



i
(5.19)
The convention is that
ν
i
is positive if it refers to a product and negative if it refers to a
reactant.

At equilibrium,
G
must be a minimum with respect to any displacement of the
system. So,

G
∂ξ








T
,
P

=
0
(5.20)
Thus, the most general condition of equilibrium for a reaction at constant temperature and
pressure, irrespective of composition and physical state (solid, liquid or gas) of the
substances involved, is
ν
i
i



i

=
0
(5.21)
For example, the breakdown of plagioclase (plag) to garnet (gt), sillimanite (sil), and
quartz (qz) during metamorphism of rocks can be represented in terms of phase-
components as
€
3CaAl
2
Si
2
O
8
anorthite
plag

=

Ca
3
Al
2
Si
3
O
12

+

grossular
gt
2Al
2
SiO
5
sillimanite
sil

+

SiO
2
quartz
qz
(5.22)
The condition of equilibrium for this reaction at the
P
and
T
of interest is (taking
ν

i

as
positive for products and negative for reactants):
€
Δ

r,

T
P

=

3

CaAl
2
Si
3
O
12
gt

+
2

Al
2
SiO
5
sil

+


SiO
2
qz



3

CaAl
2
Si
2
O
8
plag

=
0
(5.23)
The evaluation of chemical potentials will be discussed in
Section 5.3
.
9
5.1.4 The Gibbs-Duhem equation
Differentiating
Equation 5.13
, we get
dG
α

=


i
α

dn
i
α

+

n
i
α

d

i
α
i

i

(5.24)
Subtracting the above equation from
Equation 5.11
, we obtain what is known as the
Gibbs-Duhem equation
for a single phase:

S
α

dT

+

V
α

dP



n
i
α

d

i
α
i


=
0
(5.25)
The Gibbs-Duhem equation describes the interdependence among the intensive variables
(pressure, temperature, and chemical potentials) in a single phase at equilibrium. In a
multiphase system, a Gibbs-Duhem equation can be written for each phase. The
equations impose the restriction that in a closed system at equilibrium, net changes in
chemical potential will occur only as a result of changes in temperature or pressure or
both.
The Gibbs-Duhem equation is particularly useful for closed systems at constant
pressure and temperature, in which case
Equation 5.25
reduces to
n
i
α

d

i
α
i


=
0
(5.26)
Equation 5.26
tells us that there can be no net change in chemical potential at
equilibrium if pressure and temperature

are held constant. In other words, if the phase
α
contains
k
constituents, the chemical potential of only (
k
– 1) constituents can vary
independently. To illustrate the significance of this statement, let us consider the
NaCl–H
2
O solution we mentioned earlier. For this phase in equilibrium at constant
pressure and temperature,
n
NaCl
solution

d

NaCl
solution

+

n
H
2
O
solution

d

H
2
O
solution

=
0
In this closed binary system, we can change independently the chemical potential of
either H
2
O or NaCl, but for the equilibrium to be maintained, the corresponding change
in the chemical potential of the other constituent must satisfy the above relation. This is
an important constraint for binary systems as it is useful in the calculation of phase
equilibria and in the design of phase equilibria experiments.
Box 5–1. Derivation of the Gibbs phase rule
Consider a heterogeneous system in equilibrium, consisting of
c
components and
p
phases as defined in
Section 4.2
. In order to derive the phase rule, we need to count the
number of independent variables for the system and the number of equations relating
10
these variables; the difference between these two numbers is the variance of the system
(
f
).
Each component adds a variable in the form of its chemical potential, but for
c
components only (
c
– 1) of these are independent because of the restriction imposed by
Equation 5.26
for each phase, giving
p
(
c
– 1) independent compositional variables for
the system. Including
P
and
T
, the total number of independent variables for the system
is
p
(
c
– 1) + 2.
At equilibrium (constant temperature and pressure), the chemical potential of each
component must conform to
Equation 5.15
, which gives (
p
– 1) equations for
p
phases.
Thus, the total number of equations relating the
p
phases and
c
components is
c
(
p
– 1).
We can now compute the variance of the system as
f
= [
p
(
c
– 1) + 2] – [
c
(
p
– 1)] =
c

p
+ 2
This is the Gibbs phase rule (
Equation 4.20
).
5.2 VARIATION OF
€

i
α
WITH TEMPERATURE, PRESSURE, AND
COMPOSITION
5.2.1

Temperature dependence of chemical potential
From
Equation 5.10
, at constant pressure,
€
dG
α

=


G
α

T






P
,

n
i
α
,

n
j
α
dT

+



G
α

n
i
α






P
,

T
,

n
j
α
dn
i
α
i


Since
G
is a state function,
dG
is an exact differential (
see Box 4.3
):


T


G
α

n
i
α








P
,

T
,
n
j
α










n
i
α

=



n
i
α


G
α

T








P
,
n
i
α
,

n
j
α










T
Using the definition of
€

i
α
(
Equation 5.9
) and
Equation 4.55
for the partial derivative of
G
with respect to temperature, we obtain

the variation of
€

i
α
with temperature:


i
α

T








P
,

n
i
α
,
n
j
α

=


S
α

n
i
α








P
,
T
,

n
j
α

=


s
i
α
(5.27)
5.2.2 Pressure dependence of chemical potential
11
An analogous derivation, using
Equation 4.56
for the partial derivative of
G
with respect
to pressure, yields the variation of
€

i
α
with pressure:


i
α

P








T
,

n
i
α
,
n
j
α

=


V
α

n
i
α








P
,

T
,

n
j
α

=

v
i
α
(5.28)
In view of the definition of

i
, it is not surprising that the above two equations are
analogous to
Equations 4.55
and
4.56
, with the molar property replaced by the
corresponding partial molar property.
5.2.3 Compositional dependence of chemical potential
We have shown that the free energy of a multi-constituent phase at the pressure and
temperature of interest can be calculated using
Equation 5.13
and that of a multi-phase
system using
Equation 5.14
. In order to perform such calculations, we must first learn
how to calculate the chemical potential of a given constituent in a particular phase,
€

i
α
,

at
the temperature and pressure of interest.
We employ the following strategy for calculation of

€

i
α
at constant temperature and
pressure. We first determine the chemical potential of the pure phase composed entirely
of pure
i
at some
standard state
pressure and temperature
(
P
0
,

T
0
)
, and then add a
composition-dependent term to account for the deviation from the standard state. The
general
algebraic
form of the resulting equation, as will become evident in course of
subsequent discussions, is

i
α
(
P
,
T
)
=


i
α
(pure)
(
P
0
,

T
0
)
+

RT
ln
a
i
α
(5.29)
where
R
is the Gas Constant and
€
a
i
α
is the
activity
of
i
in phase
α
. The activity of a
constituent in a solution is related to, but generally not equal to, its concentration; it may
be viewed as the “thermodynamic concentration” or “effective concentration” of a
constituent in a solution. The term
€

i
α
(pure)
(
P
0
,

T
0
)
represents the chemical potential of
pure
i
at the chosen standard state
(
P
0
,

T
0
)
and is called the
standard state chemical
potential
of

i
; it is independent of the composition of
α
.
The standard state serves the same purpose as the
reference state
we used in
Chapter 4
to calculate the thermodynamic parameters at
P
and
T
, but it is a more useful
concept. A standard state has four attributes: temperature, pressure, composition, and
physical state (solid, liquid, solution, etc.). The choice of the standard state is a matter of
convenience in a given case, and it may be different for different substances in a single
reaction
(Powell, p. 54)
(whereas the reference state temperature and pressure are the
same for all substances in a reaction). The standard state may even be a hypothetical
state. The only requirement is that we know or can determine the values of the
thermodynamic parameters of a substance in the standard state. The values of

i
α
(
P
0
,

T
0
)

and

a
i
α

depend on the standard

state

chosen, but the value of

€

i
α
(
P
,
T
)

is
independent of the standard state chosen. Note that

a
i
α

has no unique value; its value
12
depends on the chosen standard state (see
Box 5. 2
). It is

always a function of
composition and it may or may not be a function temperature or pressure, depending on
the chosen standard state temperature and pressure.
For the sake of brevity, we often omit the superscript denoting the phase to which the
i
th constituent refers and the standard state information, and write
Equation 5.29
as

i

=


i
0

+

RT
ln
a
i
(5.30)
in which the standard state chemical potential

is identified by the subscript “0”. For
almost all applications it is logical to choose the temperature of interest (
T
) as the
standard state temperature (i.e.,
€
T
0
=

T
), in which case

i
0
becomes a function of
temperature and
€
a
i
a function of pressure and composition. This is called a variable
temperature standard state. For such a choice we will have a series of standard states
corresponding to a series of equilibrium states at different temperatures, one for each
temperature. In principle, we could choose a fixed temperature standard state (e.g.,
€
T
0
=

298.15
K
), but this may render the computations unnecessarily more cumbersome.
We will include the standard state temperature and pressure information in the equations
where doing so would be a helpful reminder.
Equation 5.30
is the basis for calculation of chemical potentials of constituents in all
multi-constituent solutions — solids, liquids, or gases. Assuming that the standard
chemical potential of a constituent is known, the critical factor in this calculation is the
value of its activity. As will be elaborated in the following sections, the deviation of
€
a
i
from
€
X
i
depends on whether the solution is ideal or non-ideal and, if non-ideal, the
assumed model of non-ideality.
Box 5–2. Activities and chemical potentials based on two commonly chosen
standard states
Let us consider the simple case of a pure phase consisting of a single constituent
i
, so that
the molar quantities are numerically equal to their corresponding partial molar quantities
at the pressure (
P
) and temperature (
T
) of interest.
(a) Standard state: pure substance,
P
(bar),
T
(K)

€
G
T
P

=


i
(
P
,

T
)
=


i
0
(
P
,

T
)
(5.31)
Comparison with
Equation 5.29
shows that

RT
ln
a
i

=
0 or
a
i

=
1
(5.32)
This is the reason why in some petrologic problems it is advantageous to choose pure
constituents at
P
and
T
as the standard state. For 1 mole of a substance, we can write
Equation 4.78
in terms of molar values as
13

€
G
T
P

=

G
T
1

+

V

dP
1
P

(4.61)
For a reaction,

€
Δ
G
r,

T
P
=

Δ

r
(
P
,

T
)
=
Δ
G
r,

T
1

+

Δ
V
r

dP

1
P


=

Δ
H
r,

T
1



T

Δ
S
r,

T
1

+

Δ
V
r

dP

1
P

(5.33)
(b) Standard state: pure substance, 1 (bar),
T
(K)

G
T
P

=


i
(
P
,

T
)
=


i
0
(
1
,

T
)
+

RT
ln
a
i

=

G
T
o

+

RT

ln
a
i
(5.34)
Thus, the activity corresponding to this standard state is given by

€
RT
ln
a
i

=

V
i

dP
1
P

(5.35)
For a reaction at
P
and
T
,

€
Δ
G
r,

T
P
=

Δ

r
(
P
,

T
)
=

Δ
G
r,

T
1

+

RT

ln
a
i

i

(products)


ln
a
j
(reactants)
j











=

Δ
H
r,

T
1



T

Δ
S
r,

T
1

+

Δ
V
r

dP

1
P

(5.36)
Evidently, both the standard states will yield identical values for
€
Δ
G
r,

T
P
.
It is permissible to choose different standard states for the different substances
involved in a reaction. In fact, for a reaction involving both solid and gas phases, it is a
common practice to choose pure solids at
P
and
T
of interest as the standard state for
solid phases and pure, ideal gas at 1 bar and
T
as the standard state for the gas phase.
5.3 RELATIONSHIP BETWEEN GIBBS FREE ENERGY CHANGE AND
EQUILIBRIUM CONSTANT
Since the chemical potential of a substance is related to its free energy and the
equilibrium constant of a reaction is defined as a ratio of the activities of substances
taking part in the reaction, there must be a relationship between the free energy change
and equilibrium constant of the reaction. For a reaction in equilibrium at
P
and
T
, it can
be shown that (
Box 5–3
)
€
Δ
G
r
0

=



RT
ln
K
eq
(5.37)
14
where
€
Δ
G
r
0
is the
standard Gibbs free energy change
for the reaction, with each reactant
and product in its standard state, and
K
eq
is the equilibrium constant of the reaction. This
relation is universally valid and one of the most useful relations in chemical
thermodynamics, because it allows us to compute equilibrium constants from free energy
data and
vice versa
. The relation, however, is a peculiar one in the sense that the two
sides of the equation refer to completely different physical situations. The left-hand side
refers to free energy change of a reaction involving pure substances at chosen standard
state temperature(s) and pressure(s), without any connotation of an equilibrium state; the
right-hand side refers to a system in equilibrium and involves calculation of activities of
each constituent at
P
and
T
relative to the chosen standard state(s).
(White, 2000, p. 92)
If we choose a fixed–temperature standard state (e.g., 298.15 K),
Equation 5.37
is valid
only at that standard state temperature; if we choose a variable–temperature standard
state,
Equation 5.37
is valid for all temperatures, but
€
Δ
G
r
0
is then a function of
temperature. The same consideration applies to pressure.
Box 5–3. Derivation of the relation

€
Δ
G
r
0

=



RT
ln
K
eq
Let us consider the following balanced reaction in equilibrium at (
P
,
T
):
x X + y
Y +…
=
c C + d D + …
reactants
products
Following
Equation 5.30
for the chemical potential of a constituent at
P
and
T
,
€

X
=

X
0
+
RT
ln
a
X
;
€

Y
=

Y
0
+
RT
ln
a
Y
;
€

C
=

C
0
+
RT
ln
a
C
;
€

D
=

D
0
+
RT
ln
a
D
The change in chemical potential for the reaction,
€
Δ

r
, can be calculated as
€
Δ

r

=
(c

C

+
d

D
)

(x

X

+
y

Y
)

=
c (

C
0

+

RT
ln
a
C
)
+
d (

D
0

+

RT
ln
a
D
)


x (

X
0

+

RT
ln
a
X
)

y (

Y
0

+

RT
ln
a
Y
)

=
(c

C
0

+
d

D
0


x

X
0


y

Y
0
)
+

RT
(ln
a
C

+
ln
a
D


ln
a
X


ln
a
Y
)

=

Δ

r
0

+

RT
ln
a
C
c

a
D
d
a
X
x

a
Y
y







=

Δ
G
r
,

T
P
The general equation for
€
Δ

r
can be written as

€
Δ

r

=

Δ

r
0

+

RT
ln
a
i
ν
i

=

Δ
G
r,
T
P
i


(5.38)
15
where
ν
i
is the stoichiometric coefficient of the
i
th constituent in the reaction, positive if
it refers to a product and negative if it refers to a reactant, and
€
Δ
G
r,
T
P
is given by
Equation

4.91
. If the
standard Gibbs free energy change
for the reaction (i.e., the free
energy change with each of the reactants and products in its standard state) is denoted by
€
Δ
G
r
0
, then
€
Δ

r
0

=

Δ
G
r
0
. Since at equilibrium,
€
a
i
ν
i

=

K
eq
i

and
€
Δ
G
r,
T
P

=
0
,

€
Δ
G
r
0

=



RT
ln
K
eq
(5.37)
5.4 GASES
5.4.1 Pure ideal gases and ideal gas mixtures
An ideal gas is conceptualized as consisting of vanishingly small particles that do not
interact in any way with each other, i.e., there are no forces or energies of attraction or
repulsion among the particles
(Anderson and Crerar, 1989, p. 229).
We start with pure
ideal gases, which are the easiest to model, because the equation of state for ideal gases is
well established (
Equation 4.7)
. For a pure ideal gas, the molar volume (
Equation 4.22
)
is the same as the partial molar volume (
Equation 5.5
). Thus, at constant temperature
T
,
€
d


=

v
i

dP

=

V
i

dP

=

RT
P

dP
(5.39)
Integrating between the limits of an arbitrarily chosen standard pressure
€
P
0
and the
pressure of interest
P
, we obtain


=


0
(
P
0
,

T
)
+

RT
ln
P
P
0

(5.40)
Comparison with
Equation 5.30
shows that the activity of a pure ideal gas is
P
/
P
0
and it
is a dimensionless number. If the standard state is conveniently chosen as the pure ideal
gas at 1 bar pressure, the equation reduces to


=


0
(1,

T
)
+

RT
ln
P
(5.41)
A mixture of two ideal gases behaves as an ideal gas as the particles of each
constituent gas is considered to have no interaction with any other particle in the mixture;
i.e., there is no change in enthalpy or volume due to mixing. (Ideality in a liquid solution,
on the other hand, assumes a complete uniformity of intermolecular forces irrespective of
composition.) For a constituent
i
in a mixture of ideal gases, the chemical potential
€

i
at
P
and
T
may be expressed in terms of its partial pressure
P
i
:

i

=


i
0
(
P
0
,

T
)
+

RT
ln
P
i
P
0
(5.42)
16
Comparison with
Equation 5.30
shows that the activity in this case is
€
P
i
P
0

and it is a
dimensionless number. Taking the standard state as pure
i
at 1 bar and
T
, we get
€

i

=


i
0
(1,

T
)
+

RT
ln
P
i
(5.43)
According to Dalton’s law of partial pressures (
Box 4–1
),
P
i
=
P X
i
where
X
i
is the mole
fraction of
i
and
P
is the total pressure.
________________________________________________________________________
Example: Partial pressures of constituents in a gas mixture
A gas mixture consisting of CO, CO
2
, and O
2
is at equilibrium at 600 K (327
o
C). What
are the partial pressures of CO
2
and CO if the total pressure is 1 bar and the partial
pressure of O
2
is 0.8 bar? Assume that each gas behaves ideally. (After Richardson and
McSween, 1989 p. 60). The standard free energies of formation at 1 bar and 600 K are (in
kJ.mol
–1
):
€
Δ
G
f
1
(CO
2
)
=
–395.2
;
€
Δ
G
f
1
(CO
)
=
–164.2
; and
€
Δ
G
f
1
(O
2
)
=
0.0
(Robie and
Hemingway, 1995).
The first step is to recognize that the partial pressures of the constituents in the gas
mixture is controlled by the reaction
CO + 1/2 O
2


CO
2
(5.44)
Taking the standard state for the gases as 1 bar and 600 K, and
a
i
=
P
i

(ideal gases),
Equation 5.37
gives
€
Δ
G
r
,600
1
=

RT
ln
K
eq
where
€
Δ
G
r
,
600
1
=
Δ
G
f
,
600
1
(CO
2
)

Δ
G
f
,
600
1
(
CO
)

0.5
Δ
G
f
,
600
1
(
O
2
)

=
–395.2

(–
164.2)

0.0
=
– 231.0 kJ
=
– 231,
000 J
and
€
K
eq
=
a
CO
2
gas
a
CO
gas
(
a
O
2
gas
)
0.5
=
P
CO
2
P
CO

P
O
2
0.5
Substituting,
€
K
eq
=
exp

Δ
G
r, 600
1
RT






=
exp
231,
000
(8.314) (600)






=
1.29 x 10
20
Since
€
P
CO
2
+
P
CO
+
P
O
2
=
P
total
=
1 bar
,
€

P
CO
2
=

P
CO

P
O
2
0.5

K
eq

, and
€

P
O
2
=
0.8 bar
,
17
€
P
CO
=
1
-
P
O
2
1
+
K
eq
(
P
O
2
)
0.5
=
1.734 x 10
– 21
bar
€

P
CO
2
=

P
CO

P
O
2
0.5

K
eq
=
(1.734 x10
– 21
) (0.8)
0.5
(1.29 x 10
20
)
=
2.000 x 10
– 1
=
0.2 bar
Since
€
P
i
=
P
total
X
i
(
Equation 4.9
), at equilibrium, the gas would be a mixture essentially
of CO
2
and O
2
, with only a trace of CO.
________________________________________________________________________
5.4.2 Pure non-ideal gases
In the high temperature–high pressure environments encountered in igneous and
metamorphic petrology, the gas phase commonly behaves as a non-ideal (or
real
) gas.
An equation of the same algebraic form as
Equation 4.22
can be applied to a pure non-
ideal gas by replacing pressure (
P
) with a new variable called
fugacity
(
f
), which is
related to pressure by a
fugacity coefficient
(
χ
) defined as
χ

=

f
P
such that
f
P


1 as
P


0
(5.45)
That is, as pressure
P
tends to zero, the value of
χ

approaches unity and the gas
approaches ideality. Now we can derive a relation for a non-ideal gas analogous to
Equation 5.40
in which the pressure is replaced by fugacity:


=


0
(
P
0
,

T
)
+

RT
ln
f
f
0

(5.46)
where
f
is the fugacity corresponding to pressure
P
of interest and
f
0
the fugacity
corresponding to the chosen standard state pressure
P
0
. Comparison with
Equation 5.30
shows that the activity in this case is
f
f
0
, again a dimensionless number. For
convenience of computation, we choose a standard state for which
f
0
=
1
at all
temperatures. The only substance for which this is true is an ideal gas (i.e.,
χ

= 1) at a
pressure of 1 bar (i.e.,
P
0
=
1
). With this choice of standard state (
Fig. 5.3
),
Equation
5.46
reduces to


=


0
(
1
,

T
)
+

RT
ln
f

=


0
(
1
,

T
)
+

RT
ln (
P

χ
)
(5.47)
Fugacity coefficient is a function of both temperature and pressure. The classical
method of obtaining the fugacity and fugacity coefficient of a pure, real gas at specified
P
and
T
is by experimental measurement of its molar volume
(
V
real
)
at several pressures up
to
P
(at constant
T
) followed by calculation using the equation (see
Box 5–4
):
18
€
ln
χ

=
ln
f
P

=

1
RT

V
ideal



V
real


0
P


dP
=



1
RT

RT
P



V
real






0
P


dP

(5.48)
Box 5.4 Calculation of fugacity of a real gas at pressure and temperature of interest
The fugacity of a pure, real gas at constant temperature
T
can be calculated from its
measured volumes at various pressures up to the pressure of interest (
P
) and its
theoretical volume at the same pressures if it behaved as an ideal gas. To derive an
appropriate mathematical relation for this calculation (Nordstrom and Munoz, 1994), let
us define a quantity
φ
at the
P
and
T
of interest as:
φ

=

V
ideal gas



V
real gas

=

RT
P



V
real
Rearranging and integrating between the limits
P
0
(standard state pressure) and
P
(pressure of interest), we obtain
€
V
real

dP
P
o
P


=

RT
P



φ







dP
P
0
P


=

RT
ln
P
P
0


φ

dP
P
0
P

19
Since for a pure gas,

molar volume is the same as partial molar volume, using
Equations
5.28
and
5.46
, we get
V
real

dP

P
0
P

=

d

P
0
P


=


(
P
,

T
)



0
(
P
0
,

T
)
=

RT
ln
f
f
0
We can now write
RT
ln
f
f
0

=

RT
ln
P
P
0



φ

dP
P
0
P

which on rearrangement gives
RT
ln
f
P

=

RT
ln
f
0
P
0



φ

dP
P
0
P

Considering
Equation 5.45
in terms of
€
P
0
and
€
f
0
,
when
P
0


0,

f
0
P
0


1,
so that
RT
ln
f
0
P
0


0
Substituting for
φ
, we get the following equation for calculation of the fugacity of a gas:

€
ln
χ

=
ln
f
P

=

1
RT

V
ideal



V
real


0
P


dP
=



1
RT

RT
P



V
real






0
P


dP

(5.48)
which on rearrangement gives

€
RT
ln
f

=

RT
ln
P

+

V
real


V
ideal


0
P


dP
(5.49)
As illustrated in
Fig. 5.4
for water as an example, the integrand in
Equation 5.48
can
be evaluated graphically. The
P–V–T
method, however, is difficult, expensive and time
consuming when experiments are conducted above one kbar pressure and 500
o
C. As
such, very little experimental
P-V-T
data are available for gases of geologic interest
beyond about 10 kbar and 1,000
o
C (Holloway, 1977).
For a pure gas, thermodynamic properties, including fugacities, within and beyond
the range of
P–V-T
measurements are commonly obtained by formulating equations that
conform to the available
P–V–T
data for the gas. One approach involves generating a
polynomial, without the constraint of an equation of state, for the sole purpose of
obtaining the best fit (i.e., lowest residuals) with the experimental data. This was the
approach used by Burnham et al. (1969b) for calculating fugacity coefficients of H
2
O to
20
1,000
o
C and 10,000 bars and by Holland and Powell (1990) for calculating fugacities of
H
2
O and CO
2
. over the range 1–50 kbar and 100–1600
o
C. Another approach is to derive
an appropriate equation of state, based on molecular thermodynamics, that can not only
account for the existing
P–V–T
data but also is likely to yield satisfactory results on
extrapolation. For example, Kerrich and Jacobs (1981) used a modified version of the
Redlich–Kwong (MRK) equation of state (
Equation 4.24)
to obtain fugacity coefficients
for H
2
O and CO
2
gases at a series of temperatures and pressures up to 1,000
o
C and 10
kbar, whereas Holland and Powell (1998) used what they called Compensated-Redlich-
Kwong (CORK) equations to calculate fugacities of H
2
O and CO
2
over the range 0.5–120
kbar and 200–1,400
o
C (
Appendix , from Holland & Powell, 1998 – necessary?
).
Mäder and Berman (1991) used a van der Waals type of equation and claimed that it
yielded thermodynamic properties for mineral equilibrium calculations reliably in the
range 400-1800 K and 1 bar-42 kbar with good extrapolation properties to higher
pressure and temperature.
Figure 5.4
A plot of the difference between the ideal and real volumes of H
2
O at 500
o
C as a function of
pressure, using data from Burnham et al. (1969).

The shaded area under the curve is equivalent to the
integrand in
Equation

5.49
with P = 2000 bars, and can be used to solve for the fugacity of H
2
O at 500
o
C
and 2000 bars. The data for the plot are from Burnham et al. (1969), who give the calculated value of the
fugacity as 721 bars, which translates into a fugacity coefficient of 0.361 (
Equation 5.45
).
5.4.3 Non-ideal gas mixtures
For a constituent
i
in a mixture of non-ideal (or real) gases, the chemical potential
€

i
at
P
and
T
may be expressed in terms of its corresponding fugacity
f
i
by an equation
analogous to
Equation 5.46
:
21
€

i

=


i
0
(
P
0
,

T
)
+

RT
ln
f
i
f
i
0

(5.50)
where
€
f
i


=

χ
i

P
i
,
such that
f
i
P
i


1 as
P


0
(5.51)
and
f
i
0
is the fugacity of pure
i
at the standard-state pressure
P
0
. In this case, activity of
i
, a dimensionless number, is given by
a
i

=

f
i
f
i
0
(5.52)
As in the case of pure non-ideal gases, if the standard state is chosen as the state in which
the pure gas has unit fugacity (i.e.,
f
i
0
=
1
) and behaves as an ideal gas
(
i.e.,
χ
i

=

1), then
P
0

1
. This is because the only substance for which the fugacity is 1.0 at all
temperatures is an ideal gas at unity pressure. With this standard state,
Equation 5.50
reduces to

i

=


i
0
(1
,

T
)
+

RT
ln
f
i

(5.53)
If the mixing of the gases is assumed to be ideal, according to the Lewis fugacity
rule discussed in the next section (
Equation 5.67
),
f
i

can be calculated as
f
i

=

f
i
pure

X
i
regardless of the proportion of other constituents in the gas mixture. For example, the
fugacity coefficient of pure H
2
O gas at 800
o
C and 4 kbar is 0.843 (Kerrick and Jacobs,
1981), which translates to
€
f
H
2
O
pure
=
4.0 x 0.843
=
3.372 kbar
. In an ideal gas mixture
with
€
X
H
2
O
=
0.2
,
€
f
H
2
O
gas mixture
=
0.2 x 3.372
=
0.674 kbar
at the pressure and temperature
of interest, irrespective of other constituents in the gas mixture
(see N & M, p 155)
. The
rule works best at high temperatures, at low pressures, and for dilute solutions of
i.
In principle, the ideal mixing model, should not be applicable to real gas mixtures,
such as H
2
–H
2
O, CO
2
–H
2
O, and CO
2
–H
2
O–CH
4
, composed of unlike molecules. This is
because interactions between unlike molecules generally result in volumes, and hence
fugacities, that are not the same as obtained by a linear combination of the end-member
volumes. CO
2
-H
2
O fluids are of particular importance in geochemistry, because they
play vital roles in crystallization of magmas and metamorphism of rocks. A number of
mixing models have been proposed for non-ideal CO
2
-H
2
O fluids (for example, Holloway
1981; Kerrick and Jacobs, 1981; Saxena and Fei, 1988; Duan et al., 1996, Aranovich and
Newton, 1999). A commonly used model is by Kerrick and Jacobs (1981) that uses a
modified Redlich-Kwong (MRK) equation of state for the pure gases and simple mixing
rules for binary CO
2
-H
2
O fluids. Aranovich and Newton (1999) proposed equations for
the activity coefficients of CO
2
and H
2
O (
Box 5–5
) that are simpler in analytical form and
not tightly linked to any particular equation of state for the end-member gases.
22
Calculated activities according to both models show significant positive non-ideality of
mixing that increases with decreasing temperature at constant pressure and with
increasing pressure at constant temperature (
Fig. 5.5
).
Figure
5.5
.
Activity-composition
relationship
in
CO
2
-H
2
O
fluid
mixture
at
600
o
C
and
1000
o
C
at
pressures
ranging
from
1
kb
to14
kb
as
calculated
by
Aranovich
and
Newton
(1999).
The
mixture
becomes
closer
to
an
ideal
mixture
with
decreasing
pressure
at
constant
temperature
and
with
increasing
temperature
at
constant pressure.

[Am. Mineral., 84, Fig. 8, 1330].
Box 5–5. Activity coefficients of H
2
O and CO
2
in binary H
2
O–CO
2
fluids
The equations given by Aranovich and Newton (1991) for the activity coefficients (
€
χ
i
) of
CO
2
and H
2
O in a non-ideal CO
2
–H
2
O fluid in the range 600–1000
o
C and 6–14 kbar are:

€
RT
ln
χ
H
2
O
=
(
X
CO
2
)
2
W
V
H
2
O
o
(
V
CO
2
o
)
2
(
V
H
2
O
o
+
V
CO
2
o
)(
X
H
2
O
V
H
2
O
o
+
X
CO
2
V
CO
2
o
)
2








(5.54)

€
RT
ln
χ
CO
2
=
(
X
H
2
O
)
2
W
V
CO
2
o
(
V
H
2
O
o
)
2
(
V
H
2
O
o
+
V
CO
2
o
)(
X
H
2
O
V
H
2
O
o
+
X
CO
2
V
CO
2
o
)
2








(5.55)
where
€
V
H
2
O
o
and
€
V
CO
2
o
are the specific volumes of pure H
2
O and CO
2
at a given
P
(in
kbar) and
T
(in K),
€
X
i
the mole fraction, and
€
W
=
(A
+
B
T
[1

exp(–20
P
)]
+
C
PT
. The
experimentally determined best-fit values of the constants are: A = 12893 J, B= – 6.501 J.
K
–1
, and C = 1.0112 J.K
–1
.kbar
–1
. Aranovich and Newton (1991) took the volumes of H
2
O
and CO
2
from the CORK equations of Holland and Powell (1991). The experiments were
conducted in the range 600–1000
o
C and 6–14 kbar, but calculations down to one kbar or
below at temperatures down to 500
o
C are likely to be reasonably accurate with these
equations.
23
Equation 5.52
is commonly used as the general definition of activity, because the
activities of all the other ideal and non-ideal gas systems discussed above are special
cases that can be derived from this definition. Moreover, this definition of activity
applies equally well to liquids and solids (condensed phases), because all constituents of
a phase have a fugacity, wh