Douce A.P. Thermodynamics of the Earth and Planets
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238 Chemical equilibrium
boiling point. This agrees with the fact that melting requires absorption of heat in order to
break atomic bonds in the solid, so that the entropy of melting is always a positive quantity
(Worked Example 4.1).
On the isothermal projection, Fig. 5.4, the slope of the G surface for the gas is also steeper
than that for the liquid. This is the geometric expression of the higher molar volume, or
lower density, of the gas relative to the liquid (equation (4.133)). The curvature of the gas
surface on the isothermal projection is also greater than that of the liquid. From (4.136)we
see that this reflects the greater compressibility of the gas (a consequence of the molecules
being further apart and of the weakness of intermolecular forces).
We can make an important generalization: the high entropy phase is on the high temper-
ature side of a phase transition, and the high density (= low molar volume) phase is on
the high pressure side of a phase transition. We can also derive an algebraic expression for
the slope of the phase transition. For a system of constant chemical composition, the total
differential of the Gibbs free energy of reaction d(
r
G), is given by:
d
(
r
G
)
=
∂
(
r
G
)
∂T
P
dT +
∂
(
r
G
)
∂P
T
dP. (5.2)
From the linearity of differentiation (the derivative of a sum is the sum of the derivatives)
and equations (4.132) and (4.133) it is simple to show that:
∂
(
r
G
)
∂T
P
=−
r
S (5.3)
and:
∂
(
r
G
)
∂P
T
=
r
V , (5.4)
where
r
S and
r
V , the entropy and volume of reaction, are the differences in entropy
and volume across the phase transition. Substituting in (5.2) we get:
d
(
r
G
)
=−
r
SdT+
r
VdP. (5.5)
Along a phase transitionthe Gibbs free energy of reactionis identically zero, i.e. d(
r
G)=0.
It follows from (5.5) that the slope of the phase transition is given by:
dP
dT
=
r
S
r
V
. (5.6)
This equation is known as the Clapeyron equation, and the slope of a phase transition is
often called the Clapeyron slope. It is important to keep in mind that both
r
S and
r
V
are generally functions of P and T, so that the Clapeyron slope is not constant.
5.2 Equilibrium among pure chemical species
In the preceding section we used a one-component system to derive the thermodynamic
condition of chemical equilibrium and some of the properties of phase transitions. All
of these results generalize to equilibrium in multicomponent systems. In multicomponent
systems more than one phase is stable on at least one of the sides of a phase boundary
(Chapter 6). Properly speaking a phase boundary in a multicomponent system is not a
239 5.2 Equilibrium among pure chemical species
phase transition, but the use of the phrase “phase transition” to refer to multicomponent
phase boundaries is common in the planetary sciences, for instance, as in the “spinel–garnet
phase transition” or the “perovksite phase transition”. In any event, because Gibbs free
energy is an extensive property (i.e. it is additive) phase boundaries between multi-phase
assemblages in multicomponent systems have the same geometric properties as first order
phase transitions. The goal of the rest of this chapter is to build the mathematical framework
needed to calculate chemical equilibrium in systems composed of an arbitrary number of
components and in which phase compositions may be variable. There are several ways of
doing this. In the most general case, one starts with a list of chemical components and a set
of possible phases, and seeks the subset of phases (and perhaps their compositions, if they
can vary) that minimizes Gibbs free energy for every combination of intensive variables
and for a constant bulk composition of the system. A way to visualize this procedure, and
indeed to implement it, is to calculate the Gibbs free energy surface for each possible phase
and then seek the combination of surfaces that minimizes G subject to the constant bulk
composition constraint. Phase boundaries are then defined by intersections between these
surfaces, in a manner analogous to what we did in Fig. 5.2. We examine this approach to
phase equilibrium in the context of species distribution in homogeneous fluids (Chapters 9
and 14). In this chapter we follow a different route which, if not as general, is pedagogically
clearer. We will assume that we know beforehand the phase boundary that we want to locate.
The location of the phase boundary will then be found by solving for the set of intensive
variable combinations that makes
r
G =0. In other words, we will calculate the intersection
between Gibbs free energy surfaces directly, without calculating the surfaces themselves.
This is procedurally simpler than finding Gibbs free energy minima, but requires more prior
qualitative knowledge of the system of interest. Specifically, we need to know what is the
exact phase assemblage or phase boundary that applies to our problem. In most instances this
is either known from observations or can be inferred with reasonable certainty (Chapter 6).
We begin by considering chemical equilibrium among chemical species in their standard
state. The precise meaning of standard state will become clear later on, but for now we
note that a pure chemical species at a specified temperature and pressure and in a well-
defined physical state is one possible definition of standard state. For example, at a given
temperature and pressure, pure liquid H
2
O, or pure solid H
2
O, or pure H
2
O gas are all
possible (though different) standard states for the chemical species H
2
O.
Consider a balanced chemical reaction involving an arbitrary number of chemical
species, each of them in its standard state, where ν
i
is the stoichiometric coefficient
of species i (ν
i
< 0 for reactants, ν
i
> 0 for products). Chemical equilibrium among
the species occurs at all coordinate combinations for which
r
G = 0. The coordi-
nates can be any intensive variables, for example, pressure and temperature, but other
choices of intensive variables are also possible. Finding the equilibrium position con-
sists simply of solving the equation
r
G = 0 for the intensive variables of interest.If
the chemical reaction takes place among chemical species in their standard states, then
we symbolize the Gibbs free energy change of the reaction at pressure P and temperature
T by
r
G
0
P ,T ,
where the superscript
0
means standard state. For every thermodynamic
variable Z, we define the difference operator for a balanced chemical reaction,
r
,
as follows:
r
Z =
i
ν
i
Z
i
, (5.7)
240 Chemical equilibrium
where Z
i
is the molar value of Z for species i. This is a generalization of equation (1.86).
The equilibrium condition for a reaction among chemical species in their standard state is
thus given by:
r
G
0
P ,T
=
i
ν
i
G
0,i
P ,T
=0, (5.8)
where G
0,i
P ,T
is the standard state Gibbs free energy of species i at P and T.
The linearity of differentiation allows us to combine G and any of its derivatives (S,V , Cp,
etc.) using (5.7). Therefore, using equation (5.1.7) we write the standard state Gibbs free
energy change for a chemical reaction at P and T as follows:
r
G
0
P ,T
=
r
f
G
0
1,298
−
r
S
0
298
(
T −298
)
−T
T
298
r
C
P
T
dT +
T
298
r
C
P
dT +
r
P
1
V(P,T)dP
. (5.9)
In this equation we have placed the difference operators for the heat capacity terms under
the integral signs. This implies that heat capacities for all of the species can be combined
linearly, which is possible if they are all given by the same polynomial function. This is
almost always the case (see Box 5.1 and Software Box 1.1), but if it is not then the integral
for each species must be performed separately. The pressure integral generally cannot be
treated linearly, so equation (5.9) will not be made explicit in pressure for now.
From equation (5.1.2), the difference in Gibbs free energy of formation among chemical
species is:
r
f
G
0
1,298
=
r
f
H
0
1,298
−298
r
S
0
298
. (5.10)
Now, (S
0
298
), the “entropy of formation” of a species, is the difference between the Third
Law entropy of the species and those of its constituent elements, given by equation (4.70).
Chemical stoichiometry requires that the sum of chemical elements be the same for both
sides of a chemical reaction (this is what balancing the reaction is all about!), so that
r
for the entropies of the constituent elements must always vanish. This means that:
r
S
0
298
=
r
S
0
298
, (5.11)
where
r
S
0
298
is simply the difference in reference state Third Law entropies between
products and reactants, as given by equation (5.7). Following equation (1.98) we also write:
r
f
H
0
1,298
=
r
H
0
1,298
, (5.12)
where
r
H
0
1,298
is the difference in reference state enthalpies of formation between prod-
ucts and reactants. Substituting in (5.9) and simplifying we get the following equilibrium
equation:
r
G
0
P ,T
=
r
H
0
1,298
−T
r
S
0
298
−T
T
298
r
C
P
T
dT +
T
298
r
C
P
dT +
r
P
1
V(P,T)dP
, (5.13)
241 5.2 Equilibrium among pure chemical species
which we can also write as follows, for the sake of keeping track of where the various
contributions to the Gibbs free energy of reaction come from:
r
G
0
P ,T
=
r
H
0
1,T
−T
r
S
0
1,T
+
r
P
1
V(P,T)dP
=
r
G
0
1,T
+
r
P
1
V(P,T)dP
. (5.14)
Equation (5.13) (or (5.14)) is the starting point for all our calculations of chemical equilib-
rium. Setting
r
G
0
P ,T
=0 (equation (5.8)) allows us to calculate heterogeneous equilibrium
among pure phases of fixed composition. The set of intensive variable combinations that
satisfy the equation is the phase boundary. This is what we did in Section 5.1. The appli-
cation of equation (5.13) goes beyond calculating phase diagrams among species in their
standard states, however. We will see that calculating heterogeneous equilibrium among
phases of variable composition, and calculating homogeneous chemical equilibrium within
a phase, are simple extensions of (5.13).
Worked Example 5.1 Calculation of a phase boundary: the spinel–garnet transition in planetary
mantles, part (i)
The mineral assemblage of the Earth’s upper mantle, and almost certainly of the upper
mantles of the other terrestrial planets, consists of olivine, orthopyroxene, clinopyroxene
and an aluminous phase. The aluminous phase changes with increasing pressure, from
plagioclase to spinel to garnet. Although the modal abundance of the aluminous phase is
generally subordinate to those of the other upper mantle phases, its identity exerts a powerful
control on the melting relationships of mantle lherzolite, such as melting temperature, melt
productivity and major element melt compositions. Each aluminous phase imparts distinct
trace element signatures to mafic magmas formed in its stabilityfield, and they can also affect
the physical properties of mantle rocks, such as density and elastic parameters. It is thus of
interest to know the pressure–temperature conditions under which the plagioclase–spinel
and spinel–garnet transitions occur in planetary mantles. Here we focus on the spinel–
garnet transition. Plagioclase peridotites are restricted to shallow and high-temperature
environments in the suboceanic mantle. It is left as an exercise to the reader to find out the
limits of plagioclase stability in planetary mantles.
All of the mantle phases involved in the spinel–garnet transition are Fe–Mg solid
solutions. In addition, both pyroxenes dissolve significant amounts of Al, Cr is a major com-
ponent of spinel and also enters pyroxenes and garnet, and Ca is an important component in
garnet. All of these compositional characteristics result in considerable complications when
trying to determine the conditions under which the spinel to garnet transition takes place,
which have been the subject of many published studies (Green & Ringwood, 1967; Asimow
et al., 1995; Robinson & Wood, 1998; Klemme & O’Neill, 2000a; Klemme, 2004). We begin
with the simplest possible model for the transition, which is the Mg end-member reaction:
MgAl
2
O
4
+2Mg
2
Si
2
O
6
Mg
2
SiO
4
+Mg
3
Al
2
Si
3
O
12
. (5.15)
In this simple model the spinel–garnet transition is the phase boundary defined by equili-
brium (5.15). Calculation of the phase boundary consists of finding the set of all P –T
combinations for which
r
G = 0. We need an explicit function for the pressure integral
in equation (5.13), and the simplest one comes from assuming that crystalline solids are
242 Chemical equilibrium
incompressible and undergo no thermal expansion. The following equation for the phase
boundary then follows directly from equations (5.13) and (5.1.10):
r
H
0
1,298
+
T
298
r
C
P
dT −T
r
S
0
298
+
T
298
r
C
P
T
dT
+
(
P −1
)
r
V
0
solids
=0,
(5.16)
where
r
V
0
solids
is the volume change of the reaction calculated from the molar volumes of
the solid phases at the reference state, 1 bar and 298.15 K. We shall relax the constant volume
assumption in Chapter 8 – consider equation (5.16) as an “interim rough approximation”
only, the goodness of which needs to be determined.
We have one equation, (5.16), in two unknowns, P and T. We say that this is a system
with one degree of freedom, meaning that we can freely specify one of the two variables,
and solve the equation for the other one. The equation is linear in P but not in T, and the
heat capacity integrals are messy polynomial functions that do not have analytical roots.
Solving for P by hand would be simple, if computationally intensive. Trying to come up
with an approximate solution for T by hand is much harder. Either solution, however, is
very easy to implement in a symbolic computation system such as Maple, as explained in
Software Box 5.2.
Software Box 5.2 Calculation of a phase boundary among pure phases. Spinel–garnet
equilibrium, part (i)
The file th_template_3.mw contains Maple procedures that solve for a phase
boundary among pure solid phases, assuming constant volume. The procedures in
this worksheet solve equation (5.16). Calculation of the Gibbs free energy change of
reaction is placed in its own procedure, named dGPT. This procedure is called as a
function by Maple’s equation solver, which makes solving for a phase boundary very
straightforward. There are two different procedures that do this.
Pbound: solves for pressure along the phase boundary, at a specified temperature,
by making dGPT =0. The temperature range and increment are specified in the
procedure call, as are the name of the table containing the reaction stoichiometry
and the name of the file where output is to be sent. Since the
r
G =0 equation is
linear in P the solution is very fast.
Tbound: solves for temperature along the phase boundary, at a specified pressure, by
making dGPT =0. The pressure range and increment are specified in the procedure
call, as are the name of the table containing the reaction stoichiometry and the name
of the file where output is to be sent. In this case the
r
G = 0 equation is non-
linear (and messy) because of the heat capacity integral. An initial temperature
guess must therefore be supplied in the procedure call, and this guess is updated at
each iteration of the solver (for a different pressure) with the solution for the last
iteration. If the procedure fails to find a solution it is almost certainly because the
initial temperature guess is in an unfeasible solution region. Enter a different initial
guess and try again.
The data for the spinel–garnet phase boundary are stored in tab-delimited format in a
file named spgrt.
243 5.2 Equilibrium among pure chemical species
1000 1100 1200 1300 1400 150
0
10
15
20
25
Temperature (°C)
Pressure (kbar)
forsterite
+
pyrope
spinel +
2
enstatite
Garnet lherzolite
Spinel lherzolite
MOR adiabat
Mantle solidus
Fig. 5.5 Phase boundary between spinel lherzolites and garnet lherzolites calculated with equation (5.16), assuming
(incorrectly) that no Al dissolves in orthopyroxene. The MOR adiabat is a possible adiabatic thermal gradient beneath
Earth’s mid-ocean ridges, calculated from equation 3.32: P = P
0
+(C
p
/αV)ln(T /T
0
) , assuming T
0
=1623 K
(1350
◦
C) at P
0
=3 kbar. The solidus is a possible lherzolite solidus (from McKenzie & Bickle, 1988). The calculated
phase boundary predicts that initial melting under mid ocean ridges should take place in the stability field of garnet
lherzolites, which is thought not to be the case (see also Chapter 10).
The results of the calculations are shown in Fig. 5.5. The figure suggests that, if the
terrestrial mantle were composed of the elements Mg, Al, Si and O only, and under the
incorrect assumption that all species remain in their standard states, garnet would become
stable under mid-ocean ridges at P ∼17 kbar. They also suggest that mid-ocean ridge
basalts (MORB) should form entirely in the stability field of garnet lherzolites. We will
come back to these results and see the extent to which our conclusion will have to be
modified once we account for some of the changes in phase compositions that actually
take place in the mantle.
The geometric properties of a multicomponent phase boundary are the same as those
of a one-component phase transition, but in this case the assemblage with higher entropy
is the one that is on the high-temperature side of the phase boundary, and the assemblage
with lower molar volume is on the high-pressure side. The Clapeyron slope of the phase
boundary is the ratio of the change in entropy to the change in volume between reactant
and product assemblages.
5.2.1 An important digression: uncertainties in calculated equilibrium conditions
There is a question that we need to address. This is: how sensitive is a calculated phase
boundary, such as the one shown in Fig. 5.5, to uncertainties in the values of reference state
thermodynamic properties? We begin our analysis of this question with Fig. 5.6, which
244 Chemical equilibrium
1000 1100 1200 1300 1400 150
0
10
15
20
25
Temperature (°C)
Pressure (kbar)
Garnet lherzolite
Spinel lherzolite
Fig. 5.6
Intersection between the Gibbs free energy surfaces for the assemblages spinel +2 enstatite (solid contour lines) and
garnet +forsterite (dashed contour lines). Dots show four intersection points between the surfaces. Phase boundary,
the same one as in Fig. 5.5, is shown by thick curve. Note the very shallow intersection between the Gibbs free energy
surfaces of the two condensed assemblages.
shows contours on the Gibbs free energy surfaces for the assemblages spinel + 2 enstatite
(solid lines) and forsterite + pyrope (dashed lines), together with the phase boundary for
reaction (5.15) (this diagram is equivalent to Fig. 5.2). There is a crucial message here,
the importance of which I cannot overemphasize: the two Gibbs free energy surfaces are
almost exactly parallel to one another. Clearly, even a small relative displacement of the
G surfaces can have a large effect on their intersection, which is what defines the phase
boundary. This is in stark contrast to the phase diagram for H
2
O (Figure 5.2), in which the
intersection between the G surfaces for liquid and gas is sharp. It is important to understand
the reason for the difference. The local slope of the Gibbs free energy surface is given
by the entropy and volume of the assemblage (see Fig. 4.13). Entropy and volume are
significantly different for a liquid and a gas of the same chemical composition, leading to
a sharp intersection such as that in Fig. 5.2. In contrast, in a reaction in which only solid
phases participate the entropies and volumes of the reactant and product assemblages are
typically so close that the two surfaces meet almost tangentially.
We can attach some numbers to this conclusion. From equations (4.132) and (4.133) and
the linearity of differentiation the temperature and pressure derivatives of
r
G are
r
S
and
r
V , respectively. The values of these functions for reaction 5.15 at a characteristic
temperature of 1400
◦
C are ∼6.9JK
−1
mol
−1
and 0.8 J bar
−1
mol
−1
. This means that a
relative displacement of the G surfaces of, say, 1 kJ corresponds to a displacement in their
intersection of ∼150
◦
Cor∼1.25 kbar. These are certainly not negligible values, but their
significance becomes even more crucial when they are compared to the absolute Gibbs
245 5.3 Phases of variable composition
free energies of the assemblages, i.e. to the actual values of the contours in Fig. 5.6. For
reaction (5.15), the Gibbs free energies of the assemblages spinel + 2 enstatite or forsterite
+ pyrope at 1400
◦
C are ∼8800 kJ mol
−1
. A displacement of 1 kJ mol
−1
corresponds to
approximately 0.01% of this value.
Although this analysis is based on a specific example, the conclusion is general: even
small errors in the determination of reference thermodynamic properties can have enormous
effects on calculated phase diagrams. This is true despite the fact that enthalpies and Gibbs
free energies are referenced to an arbitrary zero, because their measurement nevertheless
entails measuring energy transfers of the order of thousands of kilojoules. And, in any case,
entropy values are absolute values, by the Third Law. A description of the experimental and
mathematical procedures used to determine thermodynamic properties is beyond the scope
of this book; see, for example, Anderson, 2005; Berman, 1988; Holloway and Wood, 1988;
Holland and Powell, 1998; Anderson, 1995.
In addition to errors in reference state thermodynamic properties, ignoring the compress-
ibility and thermal expansion of the solid phase can have important energetic implications,
with large effects on the positions of calculated phase boundaries. We discuss this in
Chapter 8, where we will see that the constant volume assumption is not generally an
acceptable approximation, except for surface and near-surface conditions. Uncertainties in
solution properties (discussed later in this chapter) and in the energetics of higher-order
phase transitions (discussed in Chapter 7) must be considered too.
5.3 Phases of variable composition: chemical potential revisited
5.3.1 Equilibrium among chemical species in an arbitrary state
Most planetary materials, whether solids, liquids or gases, are phases of variable composi-
tion. We refer to such phases as solutions. The chemical species that make up a solution,
such that the amount of any of them can be varied independently of all the other species,
are called solution components or phase components. We will often refer to them simply
as components, but they must not be confused with the system components defined in
Section 5.1. For example, consider a system made up of olivine (forsterite–fayalite) and
orthopyroxene (enstatite–ferrosilite) solid solutions. This system can be described with the
three system components FeO, MgO and SiO
2
, as they constitute a linearly independent
set that spans the composition of the system. None of these is, however, a phase compo-
nent, as their amounts cannot be varied independently of the others while preserving the
integrity of the phases olivine and orthopyroxene. The appropriate phase components in
this case are Mg
2
Si
2
O
6
,Fe
2
Si
2
O
6
,Mg
2
SiO
4
and Fe
2
SiO
4
. This set does not constitute a
set of system components, as they are not linearly independent. It is generally clear from
the context whether one is referring to system components or phase components, but the
type of component will be specified if there is any possibility of confusion. We shall return
to this topic in Chapter 6.
We now seek an equation that describes chemical equilibrium among phase components
in solutions of variable composition, or, equivalently, among chemical species that are not
necessarily in their standard states. Consider a system made up of h phases, in which there
are k phase components among which it is possible to write a balanced chemical reaction.
We make no claims as to the relative values of h and k, nor as to whether or not other
246 Chemical equilibrium
phase components that are not part of the chemical reaction between the k components are
also present. For instance, if we consider a system made up of the solid solutions spinel,
orthopyroxene, olivine and garnet (h = 4), we can write a balanced chemical reaction
among the phase components MgAl
2
O
4,
Mg
2
Si
2
O
6
,Mg
2
SiO
4
and Mg
3
Al
2
Si
3
O
12
(k =4),
regardless of whether or not there are other phase components that do not participate of
this chemical reaction. If we consider a homogeneous gas phase composed of the system
components C, H and O (h = 1) we can write a balanced chemical reaction among the
phase components CH
4
,H
2
O, CO
2
and H
2
(k = 4), even if other phase components are
present (e.g. O
2
, CO, C
2
H
6
, etc.). Other chemical reactions involving these components
are of course possible.
For now we will identify each phase by a number j (0 <j ≤ h) and each component
by another number i(0 <i≤ k). We arrange the is so that all components in each phase
are identified by consecutive numbers. Thus, phase 1 consists of components 1 through φ
1
,
phase 2 of components φ
1
+1toφ
2
, and so on where φ
i
is the identity of the last of the
phase components of interest that is present in phase i. We study the behavior of the system
when infinitesimal amounts of matter are transferred among phase components. Let dn
i
be an infinitesimal change in the amount of phase component i, and ν
i
the stoichiometric
coefficient of component i in the balanced chemical reaction among the k phase components.
If we choose to make ν
1
= 1, which we can do without loss of generality, then we have
dn
i
=ν
i
dn
1
for all k phase components.
We use equation (4.129) to write the change in Gibbs free energy for each phase, dG
j
,
when matter is exchanged at constant temperature and pressure. For phase 1 we have:
dG
1
=
i=φ
1
i=1
µ
i
dn
i
, (5.17)
where µ
i
is the chemical potential of component i. Similarly,
dG
2
=
i=φ
2
i=φ
1
+1
µ
i
dn
i
(5.18)
and so on until we get to the last phase:
dG
h
=
i=φ
h
i=φ
h−1
+1
µ
i
dn
i
, (5.19)
where obviously it must be φ
h
=k. The total change in the Gibbs free energy of the system
when matter is exchanged at constant pressure and temperature is given by:
dG =
j=h
j=1
dG
j
=
i=φ
1
i=1
µ
i
dn
i
+
i=φ
2
i=φ
1
+1
µ
i
dn
i
+···+
i=φ
h
i=φ
h−1
+1
µ
i
dn
i
=
i=k
i=1
µ
i
dn
i
.
(5.20)
247 5.3 Phases of variable composition
Applying the stoichiometry constraint (dn
i
=ν
i
dn
1
) this equation becomes:
dG = dn
1
i=k
i=1
ν
i
µ
i
. (5.21)
At equilibrium it is dG=0 (equation (4.130)), and because we have specified that transfer of
matter does take place, dn
1
=0. We then arrive at the following fundamental equation that
expresses the condition of isothermal and isobaric chemical equilibrium among chemical
species in any state related by a balanced chemical reaction:
i
ν
i
µ
i
=0. (5.22)
This equation is formally identical to (5.8), except that standard state Gibbs free energy
has been replaced by chemical potential. All explicit references to specific phases have
disappeared from equation (5.22), so that it is valid for heterogeneous equilibrium among
solutions, homogeneous equilibrium within a solution, or a combination of these situations.
I chose to identify phases and components with numbers because it simplifies the notation
in the algebraic steps that lead to the compact equation (5.22).This notation is not convenient
in practice, however, as it is better to state explicitly the identity of each chemical species and
of the phase in which it is present. In explicit applications I shall therefore replace µ
i
by µ
c
p
,
where c is the component and p is the phase, and similarly, if needed, ν
i
by ν
p
c
. For example,
if we consider equilibrium among the Mg-bearing species in spinel, orthopyroxene, garnet
and olivine solid solution, then equation (5.22) becomes:
µ
MgAl
2
O
4
spinel
+2µ
Mg
2
Si
2
O
6
opx
=µ
Mg
3
Al
2
Si
3
O
12
garnet
+µ
Mg
2
SiO
4
olivine
. (5.23)
Even though (5.22) (or a specific application such as (5.23)) is the rigorous thermodynamic
definition of chemical equilibrium among chemical species in any arbitrary state, it does
not by itself allow us to calculate an equilibrium condition. In order to do so we need to find
a function that relates chemical potential to phase composition and standard state Gibbs
free energy. We tackle this problem in subsequent sections, but before that we explore some
other important properties of the chemical potential.
5.3.2 Chemical potential as a driving force for mass transfer
Consider two solutions, call them A and B, and a chemical species σ that is a component
in both phases and that can be transferred between them. Simple examples could be the
species NaAlSi
3
O
8
, which can be a component of the phases plagioclase, alkali feldspar
and silicate melt, or the species H
2
O contained in the phases seawater, moist air and ice.
By equation (5.22) equilibrium between the phases requires that the following condition
be met:
µ
σ
A
=µ
σ
B
. (5.24)
Equilibrium requires that the chemical potential of a component be the same in all phases
in which the component is present. The importance of this conclusion is that it is always
true, and in particular it is independent of how one chooses to define the standard state. For
example, in order for sanidine to coexist at equilibrium with a silicate melt the chemical