Chapter03.pdf

W. M. White

Geochemistry

Chapter 3: Solutions

Chapter 3: Solutions and Thermodynamics of

Multicomponent Systems

3.1 Introduction

n the previous chapter, we introduced thermodynamic tools that allow us to predict the equilibrium

mineral assemblage under a given set of conditions. For example, having specified temperature, we

were able to determine the pressure at which the assemblage anorthite+forsterite is in equilibrium

with the assemblage diopside+spinel+enstatite. In that reaction the minerals had unique and invariant

compositions. In the Earth, things are not quite so simple: these minerals are present as solid solu-

tions*, with substitutions of Fe 2+ for Mg, Na for Ca, and Cr and Fe 3+ for Al, among others. Indeed, most

natural substances are solutions; that is, their compositions vary. Water, which is certainly the most in-

teresting substance at the surface of the Earth and perhaps the most important, inevitably has a variety

of substances dissolved in it. These dissolved substances are themselves often of primary geochemical

interest. More to the point, they affect the chemical behavior of water. For example, the freezing tem-

perature of an aqueous NaCl solution is lower than that of pure water. You may have taken advantage

of this phenomenon by spreading salt to de-ice sidewalks and roads, or adding salt to ice to make ice

cream.

In a similar way, the equilibrium temperature and pressure of the plagioclase+olivine ® clinopy-

roxene+spinel+orthopyroxene reaction depends on the composition of these minerals. To deal with

this compositional dependence, we need to develop some additional thermodynamic tools, which is the

objective of this chapter. This may seem burdensome at first: if it were not for the variable composition

of substances, we would already know most of the thermodynamics we need. However, as we will see

in Chapter 4, we can use this compositional dependence to advantage in reconstructing conditions un-

der which a mineral assemblage or a hydrothermal fluid formed.

A final “difficulty” is that the valance state of many elements may vary. Iron, for example, may

change from its Fe 2+ state to Fe 3+ when an igneous rock weathers. The two forms of iron have very dif-

ferent chemical properties; for example Fe 2+ is considerably more soluble in water than is Fe 3+ . Another

example of this kind of reaction is photosynthesis, the process by which CO 2 is converted to organic

carbon. These kinds of reactions are called “oxidation–reduction”, or “redox” reactions. The energy

your brain uses to process the information you are now reading comes from oxidation of organic car-

bon — carbon originally reduced by photosynthesis in plants. To fully specify the state of a system, we

must specify its “redox” state. We treat redox reactions in the final section of this chapter.

Though Chapter 4 will add a few more tools to our geochemical toolbox, and treat a number of ad-

vanced topics in thermodynamics, it is designed to be optional. With completion of this chapter, you

will have a sufficient thermodynamic background to deal with a wide range of phenomena in the

Earth, and most of the topics in the remainder of this book.

3.2 Phase Equilibria

3.2.1 Some Definitions

3.2.1.1 Phase

Phases are real substances that are homogeneous, physically distinct, and (in principle) mechanically

separable. For example, the phases in a rock are the minerals present. Amorphous substances are also

phases, so glass or opal would be phases. The sugar that won't dissolve in your ice tea is a distinct

phase from the tea, but the dissolved sugar is not. Phase is not synonymous with compound . Phases

*The naturally occurring minerals of varying composition are referred to as plagioclase rather than anorthite, olivine

rather than forsterite, clinopyroxene rather than diopside, and orthopyroxene rather than enstatite.

October 1, 2009

W. M. White

Geochemistry

Chapter 3: Solutions

need not be chemically distinct: a glass of ice water has two distinct phases: water and ice. Many solid

compounds can exist as more than one phase. Nor need they be compositionally unique: plagioclase,

clinopyroxene, olivine, etc., are all phases even though their composition can vary. Thus a fossil in

which the aragonite (CaCO 3 ) is partially retrograded into calcite (also CaCO 3 ) consists of 2 phases. Sys-

tems, and reactions occurring within them, consisting of a single phase are referred to as homogenous ;

those systems consisting of multiple phases, and the reactions occurring within them, are referred to as

heterogeneous .

3.2.1.2 Species

Species is somewhat more difficult to define than either phase or component . A species is a chemical

entity, generally an element or compound (which may or may not be ionized). The term is most useful

in the context of gases and liquids. A single liquid phase, such as an aqueous solution, may contain a

number of species. For example, H 2 O, H 2 CO 3 , HCO – , CO 2 + , H + , and OH – are all species commonly

present in natural waters. The term species is generally reserved for an entity that actually exists, such

as a molecule, ion, or solid on a microscopic scale. This is not necessarily the case with components, as

we shall see. The term species is less useful for solids, although it is sometimes applied to the pure end-

members of solid solutions and to pure minerals.

3.2.1.3 Component

In contrast to a species, a component need not be a real chemical entity, rather it is simply an algebraic

term in a chemical reaction. The minimum number of components * of a system is rigidly defined as the

minimum number of independently variable entities necessary to describe the composition of each and every phase

of a system. Unlike species and phases, components may be defined in any convenient manner: what

the components of your system are and how many there are depend on your interest and on the level of

complexity you will be dealing with. Consider our aragonite-calcite fossil. If the only reaction occur-

ring in our system (the fossil) is the transformation of aragonite to calcite, one component, CaCO 3 , is

adequate to describe the composition of both phases. If, however, we are also interested in the precipi-

tation of calcium carbonate from water, we might have to consider CaCO 3 as consisting of 2 compo-

nents: Ca 2+ and CO 2 .

There is a rule to determine the minimum number of components in a system once you decide what

your interest in the system is; the hard part is often determining your interest. The rule is:

c = n - r 3.1

where n is the number of species , and r is the number of independent chemical reactions possible between these

species . Essentially, this equation simply states that if a chemical species can be expressed as the alge-

braic sum of other components, we need not include that species among out minimum set of compo-

nents. Let’s try the rule on the species we listed above for water. We have 6 species: H 2 O, H 2 CO 3 ,

HCO – , CO 2 + , H + , and OH – . We can write 3 reactions relating them:

HCO – = H + + CO 2 –

H 2 CO 3 = H + + HCO –

H 2 O = H + + OH –

Equation 3.1 tells us we need 3 = 6 – 3 components to describe this system: CO 2 + , H + , and OH – . Put

another way, we see that carbonic acid, bicarbonate, and water can all be expressed as algebraic sums

the hydrogen, hydroxyl, and carbonate ions, so they need not be among our minimum set of compo-

nents.

In igneous and metamorphic petrology, components are often the major oxides (though we may often

chose to consider only a subset of these). On the other hand, if we were concerned with the isotopic

* Caution: some books use the term number of components as synonymous with minimum number of components .

October 1, 2009

W. M. White

Geochemistry

Chapter 3: Solutions

AlO(OH)

Al 2 O 3

Al(OH) 3

H 2 O

Figure 3.1. Graphical Representation of the System Al 2 O 3 -H 2 O.

equilibration of minerals with a hydrothermal fluid, 18 O would be considered as a different component

than 16 O.

Perhaps the most straightforward way of determining the number of components is a graphical ap-

proach. If all phases can be represented on a one-dimensional diagram (that is, a straight line repre-

senting composition), we are dealing with a two component system. For example, consider the hydra-

tion of Al 2 O 3 (corundum) to form boehmite (AlO(OH)) or gibbsite Al(OH) 3 . Such a system would con-

tain 4 phases (corundum, boehmite, gibbsite, water), but is nevertheless a two component system be-

cause all phases may be represented in one-dimension of composition space, as is shown in Figure 3.1.

Because there are two polymorphs of gibbsite, one of boehmite, and two other possible phases of water,

there are 9 phases possible phases in this two-component system. Clearly, a system may have many

more phases than components.

Similarly, if a system may be represented in 2 dimensions, it is a three-component system. Figure 3.2

is a ternary diagram illustrating the sys-

tem Al 2 O 3 –H 2 O–SiO 2 . The graphical rep-

resentation approach reaches it practical

limit in a four component system because

of the difficulty of representing more than

3 dimensions on paper. A four compo-

nent system is a quaternary one, and can

be represented with a three-dimensional

quaternary diagram.

It is important to understand that a

component may or may not have chemical

reality. For example in the exchange reac-

tion:

NaAlSi 3 O 8 + K + = KAlSi 3 O 8 + Na +

we could alternatively define the exchange

operator KNa -1 (where Na -1 is -1 mol of Na

ion) and write the equation as:

NaAlSi 3 O 8 + KNa –1 = KAlSi 3 O 8

In addition, we can also write the reaction:

K – Na = KNa -1

Here we have 4 species and 2 reactions

and thus a minimum of only 2 compo-

nents. You can see that a component is

merely an algebraic term.

There is generally some freedom in

choosing components. For example, in

the ternary (i.e., 3 component) system

SiO 2 – Mg 2 SiO 4 – MgCaSi 2 O 6 , we could

choose our components to be quartz,

diopside, and forsterite, or we could

choose them to be SiO 2 , MgO, and CaO.

Either way, we are dealing with a ternary

H 2 O

g,by,n

ka,ha,di,na

d,bo

Al 2 O 3

SiO 2

a, k ,s

Figure 3.2. Phase diagram for the system Al 2 O 3 –H 2 O–

SiO 2 . The lines are called joins because they join phases.

In addition to the end-members, or components, phases

represented are g : gibbsite, by : bayerite, n : norstrandite

(all polymorphs of Al(OH) 3 ), d : diaspore, bo : boehmite

(polymorphs of AlO(OH)), a : andalusite, k : kyanite, s :

sillimanite (all polymorphs of Al 2 SiO 5 ), ka : kaolinite, ha :

halloysite, di : dickite, na : nacrite (all polymorphs of

Al 2 Si 2 O 5 (OH) 4 ), and p : pyrophyllite (Al 2 Si 4 O 10 (OH) 2 ).

There

are

also

polymorphs

quartz

(coesite,

stishovite,

tridymite,

cristobalite, α -quartz,

and β -

quartz).

October 1, 2009

W. M. White

Geochemistry

Chapter 3: Solutions

system (which contains MgSiO 3 as well as the three other phases).

3.2.1.4 Degrees of Freedom

The number of degrees of freedom in a system is equal to the sum of the number of independent in-

tensive variables (generally T & P) and independent concentrations (or activities or chemical potentials)

of components in phases that must be fixed to define uniquely the state of the system. A system that

has no degrees of freedom (i.e., is uniquely fixed) is said to be invariant, one that has one degree of

freedom is univariant, etc. Thus in an univariant system, for example, we need specify only the value

of one variable, for example, temperature or the concentration of one component in one phase, and the

value of pressure and all other concentrations are then fixed, i.e., they can be calculated (assuming the

system is at equilibrium).

3.2.2 The Gibbs Phase Rule

The Gibbs ‡ Phase Rule is a rule for determining the degrees of freedom , or variance , of a system at equi-

librium . The rule is:

ƒ = c −φ+ 2 3.2

where ƒ is the degrees of freedom, c is the number of components, and φ is the number of phases. The

mathematical analogy is that the degrees of freedom are equal to the number of variables less the num-

ber of equations relating those variables. For example, in a system consisting of just H 2 O, if two phases

of coexist, for example, water and steam, then the system in univariant. Three phases coexist at the tri-

ple point of water, so the system is said to be invariant, and T and P are uniquely fixed: there is only

one temperature and one pressure at which the three phases of water can coexist (273.15 K and 0.006

bar). If only one phase is present, for example just liquid water, then we need to specify 2 variables to

describe completely the system. It doesn’t matter which two we pick. We could specify molar volume

and temperature and from that we could deduce pressure. Alternatively, we could specify pressure

and temperature. There is only 1 possible value for the molar volume if temperature and pressure are

fixed. It is important to remember this applies to intensive parameters. To know volume, an extensive

parameter, we would have to fix one additional extensive variable (such as mass or number of moles).

And again, we emphasize that all this applies only to systems at equilibrium.

Now consider the hydration of corundum to form gibbsite. There are 3 phases, but there need be

only two components. If these 3 phases (water, corundum, gibbsite) are at equilibrium, we have only 1

degree of freedom, i.e., if we know the temperature at which these 3 phases are in equilibrium, the pres-

sure is also fixed.

Rearranging equation 3.2, we also can determine the maximum number of phases that can coexist at

equilibrium in any system. The degrees of freedom cannot be less than zero, so for an invariant, one

component system, a maximum of three phases can coexist at equilibrium. In a univariant one-

component system, only 2 phases can coexist. Thus sillimanite and kyanite can coexist over a range of

temperatures, as can kyanite and andalusite. But the three phases of Al 2 SiO 5 coexist only at one unique

temperature and pressure.

Let's consider the example of the three-component system Al 2 O 3 –H 2 O–SiO 2 in Figure 3.2. Although

many phases are possible in this system, for any given composition of the system only three phases can

coexist at equilibrium over a range of temperature and pressure. Four phases, e.g., a, k, s and p, can co-

exist only along a one-dimensional line or curve in P-T space. Such points are called univariant lines

(or curves). Five phases can coexist at invariant points at which both temperature and pressure are

uniquely fixed. Turning this around, if we found a metamorphic rock whose composition fell within

‡ J. Williard Gibbs (1839-1903) is viewed by many as the father of thermodynamics. He received the first doctorate in

engineering granted in the U. S., from Yale in 1858. He was Professor of Mathematical Physics at Yale from 1871 until

his death. He also helped to found statistical mechanics. The importance of his work was not widely recognized by

his American colleagues, though it was in Europe, until well after his death.

October 1, 2009

W. M. White

Geochemistry

Chapter 3: Solutions

the Al 2 O 3 –H 2 O–SiO 2 system, and if the rock contained 5 phases, it would be possible to determine

uniquely the temperature and pressure at which the rock equilibrated.

3.2.3 The Clapeyron Equation

A common problem in geochemistry is to know how a phase boundary varies in P-T space, e.g., how

a melting temperature will vary with pressure. At a phase boundary, two phases must be in equilib-

rium, i.e., ∆ G must be 0 for the reaction Phase 1 ® Phase 2. The phase boundary therefore describes the

condition:

d(∆G r ) = ∆V r dP - ∆S r dT = 0.

Thus the slope of a phase boundary on a temperature-pressure diagram is:

dP = Δ V r

3.3

Δ S r

where ∆V r and ∆S r are the volume and entropy changes associated with the reaction. Equation 3.3 is

known as the Clausius-Clapeyron Equation , or simply the Clapeyron Equation . Because ∆V r and ∆S r are

functions of temperature and pressure, this, of course, is only an instantaneous slope. For many re-

actions, however, particularly those involving only solids, the temperature and pressure dependencies

of ∆V r and ∆S r will be small and the Clapeyron slope will be relatively constant over a large T and P

range.

Because ∆ S = ∆ H/T , the Clapeyron equation may be equivalently written as:

dP = T Δ V r

3.4

Δ H r

Slopes of phase boundaries in P-T space are generally positive, implying that the phases with the

largest volumes also generally have the largest entropies (for reasons that become clear from a statis-

tical mechanical treatment). This is particularly true of solid-liquid phase boundaries, although there is

one very important exception: water. How do we determine the pressure and temperature dependence

of ∆V r and why is ∆V r relatively T and P independent in solids?

We should emphasize that application of the Clapeyron equation is not limited to reactions between

two phases in a one-component system, but may be applied to any univariant reaction.

3.3 Solutions

Solutions are defined as homogeneous phases produced by dissolving one or more substances in another sub-

stance . In geochemistry we are often confronted by solutions: as gases, liquids, and solids. Free energy

depends not only on T and P, but also on composition. In thermodynamics it is generally most con-

venient to express compositions in terms of mole fractions, X i , the number of moles of i divided by the

total moles in the substance (moles are weight divided by atomic or molecular weight). The sum of all

the X i must, of course, total to 1.

Solutions are distinct from purely mechanical mixtures. For example, salad dressing (oil and vine-

gar) is not a solution. Similarly, we can grind anorthite (CaAl 2 Si 2 O 8 ) and albite (NaAlSi 3 O 8 ) crystals into

a fine powder and mix them, but the result is not a plagioclase solid solution. The Gibbs Free Energy of

mechanical mixtures is simply the sum of the free energy of the components. If, however, we heated

the anorthite-albite mixture to a sufficiently high temperature that the kinetic barriers were overcome,

there would be a reordering of atoms and the creation of a true solution. Because this reordering is a

spontaneous chemical reaction, there must be a decrease in the Gibbs Free Energy associated with it.

This solution would be stable at 1 atm and 25°C. Thus we can conclude that the solution has a lower

Gibbs Free Energy than the mechanical mixture. On the other hand, vinegar will never dissolve in oil

at 1 atm and 25°C because the Gibbs Free Energy of that solution is greater than that of the mechanical

mixture.

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