Myron G. Best. Igneous and metamorphic 2003 Blackwell Science

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Although discussions in later chapters will explore

speciﬁc implications of Rb-Sr isotopic compositions to

magma genesis, it is pertinent here to present general-

ized models of Sr-isotope evolution in the Earth. In a

simple model that assumes an initial meteoritic compo-

sition and no fractionation of Rb and Sr during the en-

tire history of the Earth, all rocks would have the same

Sr/

Sr ratio today after its 4.5-Gy history. This uni-

formity in

Sr/

Sr ratio is decidedly incorrect. In-

stead, a single-stage model can be assumed (Figure

2.26a) in which a partial melting event occurred in the

mantle at, say, 1.5 Ga that generated magma of greater

Rb/Sr ratio than the mantle source because of the dif-

fering incompatibility of these two elements. Solidiﬁca-

tion of this magma created crust whose

Sr/

Sr ratio

evolved at a greater rate along a steeper path because

of the higher Rb/Sr ratio than in mantle peridotite.

However, this single-stage model is still not valid be-

cause crustal rocks have a range of

Sr/

Sr ratios, to

as high as about 0.74, and mantle rocks and magmas

derived from them have ratios of about 0.702–0.711. A

multistage model must, therefore, apply in which Rb-

enriched magmas have been repeatedly extracted from

the mantle throughout Earth history (Figure 2.26b).

The granitic continental crust has been derived, prob-

ably indirectly, from the mantle over most of the history

of the Earth; older crustal segments now have the high-

est

Sr/

Sr ratios.

Samarium-Neodymium Systematics. There are many

isotopes of these two light REEs, but the one of most

relevance to geochronology and petrology is

147

Sm,

which decays by alpha emission to

143

Nd with a half-

life of 106 Gy. The abundance of the radiogenic daugh-

ter product is referred to by a ratio with the stable

144

Nd isotope, that is,

143

Nd/

144

Nd. The Sm-Nd iso-

topic system is similar to the Rb-Sr system just de-

scribed and is handled in much the same manner.

However, one important difference between the two

isotopic systems is the much greater half-life for

147

decay, which limits the usefulness in dating to rocks

more than 1 Ga or so in age.

The initial ratio (

143

Nd/

144

Nd)

serves as a valuable

petrogenetic tracer. Rocks with high Sm/Nd ratios

(i.e., light REE-depleted) develop higher isotope ratios

with the passage of time, whereas rocks with low

Sm/Nd ratios (light REE-enriched) develop lower iso-

tope ratios (Figure 2.27). This behavior means that en-

riched sources like the continental crust have lower

(

143

Nd/

144

Nd)

but higher (

Sr/

Sr)

. Another im-

portant contrast with Sr isotopes is the very small dif-

ference in the

143

Nd/

144

Nd ratios (0.510 to 0.514,

or 1%), which stems from the small variations in

Sm/Nd ratios of 0.1 to 2.0 in rocks, in contrast to

Rb/Sr ratios, which vary from 0.005 to over 100. The

small differences in Sm and Nd are related to their sim-

ilar crystal chemical behavior. Because of this similarity,

there are no reservoirs on Earth especially rich in Sm or

Nd like the Sr-rich oceanic waters that produce Sr-rich

marine carbonates and the Rb-rich continental crust.

Sm-Nd isotopes offer signiﬁcant advantages over

other systems such as Rb-Sr and U-Pb because Sm and

Nd occur in major minerals such as pyroxenes and pla-

gioclase and are not easily mobilized. Comparison of

Nd and Sr isotopic ratios can constrain models of mag-

matic evolution, for example, mantle partial melting

and crustal contamination, to a better degree than use

of Sr isotopes alone.

As a ﬁnal note regarding Sr and Nd isotopic ratios,

143

Nd/

144

Nd and

Sr/

Sr in the remainder of this

textbook will always refer to the initial ratio.

2.6.3 Cosmogenic Isotopes: Beryllium

Cosmogenic isotopes are produced when high-energy

cosmic rays interact with nuclei of atoms in the atmo-

sphere or on the surface of the Earth to produce new

isotopes. Many of these cosmogenic isotopes are unsta-

ble and decay to other isotopes, including

C—which

is the basis of a dating method for material less than

about 40,000 years old. Isotopes of H (tritium), Al, Cl,

and Be are also formed; only the latter is discussed

here.

Composition and Classiﬁcation of Magmatic Rocks

246810

Isochron at time t

(

2.25 Schematic Rb-Sr isochron diagram for cogenetic rocks or

minerals decaying through time. Three samples of rocks or

minerals (solid circles) having identical initial (

Sr/

Sr)

ratios crystallized at the same time, t

, from cogenetic rocks or

magma, after which the isotopic system remained closed. The

initial

Rb/

Sr ratios of the three samples differed because of

different Rb/Sr ratios; muscovite, for example, would have a

greater ratio than plagioclase. Subsequent to time t

, individ-

ual rocks or minerals track along straight lines having a slope

of 1 due to decay of

Rb to

Sr. At any one time, such as t

and so on, points representing the analyzed samples deﬁne a

straight line isochron whose positive slope is dictated by the

age of the system since crystallization occurred to close it.

48 Igneous and Metamorphic Petrology

2.26 Schematic strontium isotope evolution diagrams for the Earth. Filled circle represents an assumed parental chondritic meteorite whose initial

(

Sr/

Sr)

ratio at 4.5 Ga, when the Earth accreted from primordial material, was 0.69897  0.00003. Note that growth lines are slightly

curved because the amount of

Rb decaying to

Sr decreases through time. (a) Single-stage model with hypothetical partial melting event

at 1.5 Ga. Thereafter, mantle growth is at a slightly lower rate because of extracted Rb from a depleted mantle. (b) Multistage model in which

the mantle has been partially melted many times throughout its history to create a growing, evolving crust. Metasomatic processes create an

incompatible-element-enriched mantle (Section 11.2.2); other parts of mantle are relatively depleted through partial melting processes.

0.5135

0.5130

0.51264

0.5125

0.5120

0.702 0.704 0.706

0.7052

0.708 0.710

Continental

crust

(−)

(+)

(−)

high Rb/Sr

high Sm/Nd

w Rb/Sr

low Sm/Nd

high Rb/Sr

low Sm/Nd

Enriched sources

Depleted sources

low Rb/Sr

high Sm/Nd

MANTLE

Mantle

array

OΙB

MORB

Bulk silicate

Earth

143

144

2.27 Sr and Nd isotope ratio correlations. All terrestrial rocks are derived from a primordial bulk silicate Earth. The mantle array is deﬁned by

relatively depleted, mantle-derived, mid-ocean ridge basalt (MORB) and more enriched ocean island basalt (OIB), as well as fragments

(xenoliths) of the suboceanic mantle (labeled MANTLE) brought up in erupted OIBs. Continental rocks plot well off the diagram to the

lower right. The 

and 

notation refers to the difference between the measured

Sr/

Sr and

143

Nd/

144

Nd ratios in rock samples and

the reference bulk Earth ratio (for further details see Rollinson, 1993). (Redrawn from Rollinson, 1993.)

Of the three isotopes of beryllium, only

B is stable,

whereas

Be and

Be are radioactive isotopes con-

stantly produced as cosmic rays fragment stable iso-

topes of oxygen and nitrogen in the atmosphere.

decays to stable

Li with a half-life of only 53 days. The

more geologically useful

Be transforms by beta decay

to stable

B with a half-life of 1.5 million years. After

production, the cosmogenic Be isotopes are rapidly re-

moved from the atmosphere by precipitation of rain

and snow. Because Be is relatively insoluble in water, it

is then absorbed or otherwise incorporated into or-

ganic and inorganic sediment particles that settle to the

bottom of the oceans. As the oceanic crust is sub-

ducted and partially melted, the

Be isotopes are par-

titioned into ascending and erupting magmas, provid-

ing unequivocal evidence that oceanic sediment

supplied at least some of the mass of subduction zone

magma. Moreover, if detected,

Be isotopes provide

insights into the time involved in subduction, magma

generation, and ascent. If no sediment were subducted,

Be would be found in any associated volcanic

rocks, but, in fact, several investigators have found

traces of

Be in rocks erupted above subduction

zones. Of course, the absence of

Be does not prove

that sediment was not involved because

Be has a very

short half-life.

SUMMARY

Magmatic rocks have a broad continuous spectrum

of chemical, mineralogical, and modal compositions

that are produced by a virtually inﬁnite variety of

conditions in the magma source and subsequent evolu-

tionary processes. Variation diagrams facilitate presen-

tation of compositional data so that meaningful differ-

ences or similarities and evolutionary patterns can be

discerned.

Rock classiﬁcations attempt to systematize these

compositional continua, as well as recognizing a wide

range of fabrics and ﬁeld relations, in order to under-

stand better the origin of rocks. One rock can be given

several labels depending on the intent of the applied

scheme of classiﬁcation.

Magmatic rocks are subdivided on the basis of ﬁeld

relations into plutonic and volcanic, whose fabric is

chieﬂy phaneritic, on the one hand, and aphanitic,

glassy, or volcaniclastic, on the other. Mineralogical

mnemonics are used to convey modal attributes. Spe-

ciﬁc rock-type names in plutonic/phaneritic rocks are

based on the modal proportion of alkali feldspar, pla-

gioclase, quartz, and feldspathoids for felsic rocks and

the proportion of plagioclase, clinopyroxene, orthopy-

roxene, and olivine for maﬁc and ultramaﬁc rocks.

Other major rock-forming minerals nonessential to

their classiﬁcation include amphibole, biotite, and Fe-

Ti oxides. Rock-type names for volcaniclastic rocks

depend on clast size. Names for volcanic/aphanitic

(glassy) rock-types depend mostly on the relative

amounts of (Na

O  K

O) and SiO

The CIPW normative composition of magmatic

rocks, together with actual mineral constituents, facili-

tates their classiﬁcation according to degrees of silica

and alumina saturation. Other coherent comagmatic

kindreds and trends that can be discerned on variation

diagrams include the alkaline rock suite, whose mem-

ber rock types are enriched in alkalies relative to silica,

and the subalkaline suite. This widespread suite of

rocks, in which alkali concentrations are modest so that

rocks are usually silica saturated to oversaturated, is

further subdivided into the tholeiitic subsuite showing

relative Fe enrichment and the calc-alkaline subsuite in

which evolved rocks are less Fe-enriched and more fel-

sic. Still other comagmatic kinships can be discerned

with respect to variations in SiO

and K

O. A particu-

lar rock type such as basalt can occur in different rock

suites and can be found in different global tectonic set-

tings. However, speciﬁc rock suites tend to be associ-

ated with speciﬁc tectonic settings, forming petrotec-

tonic associations. Thus, tholeiitic basalt makes up

most of the oceanic crust produced at spreading ridges;

tholeiitic basalt and tholeiitic andesite (and locally

dacite) compose island arcs at convergent ocean-ocean

plate junctures; calc-alkaline basalt, andesite, dacite,

and rhyolite (and their phaneritic equivalents) consti-

tute medium- to high-K magmatic rocks at continental

margin subduction zones; and midplate oceanic islands

are of tholeiitic and alkaline basaltic rocks. Highly al-

kaline rocks occur in relatively stable continental cra-

tons and in continental rifts where basalt and rhyolite

also form a bimodal association.

Contrasting sources, global tectonic settings, and

evolutionary processes affecting magmas are recorded

in trace element and isotopic-tracers in rocks. In con-

trast to major elements, whose distribution between

crystals and melt in a magma is controlled by phase

equilibria, trace elements have more widely ranging

concentrations, which reﬂect partitioning according to

compatibility constraints between different major and

accessory minerals and coexisting silicate melt. Trace

element partitioning patterns can indicate speciﬁc min-

erals with which partial melts equilibrated at their

source and which minerals might have been subse-

quently fractionated from the magma. Radiogenic iso-

tope ratios can be used to establish the chronology of

magmatic processes and place constraints on magma

sources and interactions between a magma and its sur-

roundings. The stable isotopes of oxygen serve as pet-

rogenetic tracers as well as geothermometers of mineral

growth.

Composition and Classiﬁcation of Magmatic Rocks

CRITICAL THINKING QUESTIONS

2.1 What are the hindrances, rationale, and justiﬁca-

tion for classiﬁcation of magmatic rocks? Is clas-

siﬁcation just meaningless busywork?

2.2 Brieﬂy indicate the basis for each of the several

classiﬁcations of magmatic rocks described in

this chapter.

2.3 Contrast rock types with rock suites.

2.4 What are three fabric heteromorphs of granite?

2.5 From Figures 2.8 and 2.12, describe systematic

variations in mode and silica among granite,

granodiorite, and diorite. From Table 2.2, de-

scribe and explain systematic changes in major

oxides.

2.6 Explain in detail how it is possible for a quartz-

normative basalt to contain no real quartz in its

mode, only plagioclase, pyroxene, olivine, and

Fe-Ti oxides. A nepheline normative basalt of

the same mode containing no actual nepheline.

What is the role of pyroxene?

2.7 Discuss silica saturation in a simple model

magma consisting of O, Si, Al, and K, which can

potentially crystallize quartz, K-feldspar, leucite

(KAlSi

), and kalsilite (KAlSiO

). Create a

diagram for this hypothetical system like Figure

2.14.

2.8 Discuss differences in the major and trace ele-

ment compositions of basalt and relate these dif-

ferences to global tectonic setting.

2.9 What is the relation between crustal thickness

and K

O at a given SiO

content for magmatic

rocks in subduction zones?

2.10 What are the bases for distinguishing between

compatible and incompatible trace elements?

Give examples of each in rhyolitic and in

basaltic magmas.

2.11 What are the rationale and the goal of investiga-

tions of trace elements in magmatic rocks? Of

isotopes?

2.12 What factors control variations in the ratios of

stable isotopes? Radiogenic isotopes?

2.13 Contrast the behavior and implications of Sm-

Nd and Rb-Sr isotope systems.

PROBLEMS

2.1 Classify the garnet and spinel peridotites from

Table 11.1 using Figure 2.10b.

2.2 Plot average granite and granodiorite in Table

2.2 in an enlarged photocopy of Figure 2.8 us-

ing the amounts of normative Q, Or, and Ab in

lieu of modal Q, A, and P.

2.3 Calculate the normative composition of a mag-

matic biotite and hornblende from Appendix A.

Classify these minerals, as if they were rocks, as

to degree of silica saturation. Discuss how crys-

tallization of these minerals in magmas controls

the availability of silica for potential crystalliza-

tion of quartz.

2.4 From problem 2.3 classify the selected biotite

and amphibole as to degree of alumina satura-

tion.

2.5 Calculate the normative compositions of the

pantellerite in Table 13.10, the phlogopite-rich

leucite lamproite in Table 13.11, and the

shoshonite in Table 13.6. Classify these rocks in

the total alkali-silica diagram (Figure 2.12). Dis-

cuss whether these rocks belong to the alkaline

rock suite.

2.6 Calculate the bulk partition coefﬁcients for Ni

and for Rb in the garnet peridotite in Table 11.1

using partition coefﬁcients in Table 2.5.

2.7 Construct a chondrite-normalized diagram for

the spinel peridotite in Table 11.1. Describe this

diagram. What rock in Figure 2.23 does the

pattern of the spinel peridotite most closely re-

semble?

50 Igneous and Metamorphic Petrology

UNDAMENTAL

UESTIONS

ONSIDERED IN

HIS

HAPTER

1. What are the basic concepts of thermodynamics

and why is thermodynamics important in

petrology?

2. How can the ﬂows of matter and energy in

changing rock-forming systems be predicted and

interpreted using thermodynamic models?

3. How do phase diagrams reﬂect the thermo-

dynamic stability of different states of matter?

4. What is kinetics, why is it important in petrology,

and in what ways do kinetic factors limit the

application of thermodynamics to rock-forming

systems?

INTRODUCTION

Unlike sedimentary systems, in which many ﬂows of

energy and matter are readily observed and studied in

modern environments, natural magmatic and meta-

morphic systems are usually not amenable to direct

investigation. Only a few nonexplosively extruded

magmas can be examined at close range. Accordingly,

igneous petrologists have turned to controlled labor-

atory experiments and theoretical models to enhance

their understanding of the behavior of rock-forming

systems, supplementing inferences from rocks. Theory

and experiment can often demonstrate what is impos-

sible and what is possible in a geologic system. At a

minimum, they provide fruitful insights.

However, despite their advantages, such investiga-

tions of natural geologic systems have inherent weak-

nesses. Laboratory studies can never duplicate the

large dimensions and long time scales of geologic sys-

tems—as large as the entire Earth and commonly mil-

lions of years. Natural systems are always more com-

plex than the experiment or model used to simulate

them, necessitating simplifying assumptions that may

deviate from the real world to varying degrees. The ul-

timate test of the validity of any petrologic interpreta-

tion lies in the fabric, composition, and ﬁeld relations

of real rocks. Nature does not lie, but our interpreta-

tions of it can.

With these caveats in mind, we consider in this

chapter one of the foundation stones of petrogenetic

theory—thermodynamics—and the kinetic factors that

limit its application to real petrologic systems.

3.1 WHY IS THERMODYNAMICS

IMPORTANT?

In Section 1.3, the concepts of stability and equilibrium

were introduced, showing that rock-forming processes

in changing geologic systems tend toward a state of sta-

ble equilibrium in which the energy is the lowest possi-

ble. Metastable states having less than the minimal pos-

sible energy can persist indeﬁnitely because of an

activation energy barrier impeding the change. Stabil-

ity and equilibrium were illustrated by hypothetical ex-

amples of three boulders that have differing amounts

of gravitational potential energy because of their differ-

ent position on hilly terrain and of hot lava ﬂowing

downhill and losing heat, thus reducing its gravita-

tional potential and thermal energy.

CHAPTER

Thermodynamics

and Kinetics:

An Introduction

The melting of rock to generate magma sufﬁces to il-

lustrate the inadequacy of minimization of thermal and

gravitational potential energies as a universal indicator

of how all systems spontaneously change. During the

spontaneous melting of mantle rock in an ascending

plume beneath the island of Hawaii, the plume system

has moved to a state of higher gravitational potential

energy while simultaneously losing thermal energy by

adiabatic cooling (discussed later). Is the “rule” for at-

tainment of stable equilibrium cited in Section 1.3 in

error, or is there another form of energy that is mini-

mized in these melting systems, and in natural

processes in general?

It turns out that a state of stable equilibrium in rock-

forming systems does indeed have the lowest possible

energy, but of a new form. This new form of energy, or

energy function, incorporates potential, kinetic, ther-

mal, chemical, and mechanical (work) energies in such

a way as to be generally applicable to all natural sys-

tems. Evaluation of this new energy function allows

one to predict the direction of changes that naturally

occur in rock systems as their T, P, and chemical com-

position change. J. W. Gibbs formulated this new en-

ergy function about a century ago. Called the Gibbs

free energy, this new energy function can be considered

as a thermodynamic potential energy by analogy with

gravitational potential energy in systems, such as the

hypothetical boulders in Figure 1.6, whose stability is

governed by gravity.

To begin to appreciate this new Gibbs free energy

and how it can be applied to a rock-forming system,

some elementary concepts of thermodynamics must be

introduced. Thermodynamics is a set of mathematical

models and concepts that describe the way changes in

T, P, and chemical composition affect states of stable

equilibrium in rock-forming systems. Thermodynamics

is a powerful petrologic tool if an appropriate model

that accurately mimics the real rock system can be con-

structed to enhance understanding of the system’s

working. The goals of thermodynamics in petrology are

essentially twofold:

1. Predict how rock-forming systems respond to

changes in P, T, and chemical composition

2. Interpret the nature of P, T, and chemical composi-

tion in ancient rock-forming systems from the

chemical compositions of the constituent minerals

and glass in the magmatic rock

The plan of the following pages is, ﬁrst, to pre-

sent some “nuts and bolts” of thermodynamic mod-

els and concepts, including types of simple ideal

systems, properties of state that characterize sys-

tems, ideal processes of change, and energy changes;

and, second, to introduce the concept of entropy as

a necessary ingredient in formulating the Gibbs free

energy.

3.2 ELEMENTARY CONCEPTS

OF THERMODYNAMICS

3.2.1 Thermodynamic States, Processes,

and State Variables

Geologic systems—parts of the universe set aside in

our mind for investigation—are commonly more com-

plex than, for example, the systems of a laboratory

chemist. Geologic systems can be as large as the entire

Earth and may endure for millions of years; they tend

to be poorly deﬁnable and ever changing. Ideal end-

member systems (Figure 3.1) are as follows:

1. Isolated system: No matter or energy can be trans-

ferred across the boundary of the system and no

work can be done on or by the system. Considering

the span of time over which most identiﬁable geo-

logic systems operate, no part of the Earth, nor

even all of it, can be considered perfectly isolated

because transfers of energy and movement of mat-

ter typify the dynamic Earth.

2. Open system: The opposite of an isolated system.

Matter and energy can ﬂow across the boundary

and work can be done on and by the system. Most

geologic systems are open, at least in the context of

their long lifetimes.

3. Closed system: Energy, such as heat, can ﬂow

across the boundary of the system, but matter can-

not. The composition of the system remains exactly

constant. Because movement of matter is slow

across boundaries of, for example, a rapidly cooling

thin dike, it can be considered to be virtually

closed. On the other hand, a large intrusion of

magma that remains molten for tens of thousands

52 Igneous and Metamorphic Petrology

ISOLATED

SYSTEM

Energy

(including

work)

Energy

(including

work)

Matter

OPEN

SYSTEM

CLOSED

SYSTEM

ADIABATIC

SYSTEM

Matter

Work

3.1 End-member thermodynamic systems. Classiﬁcation is accord-

ing to ﬂows of matter and energy across their boundary with

the surroundings.

of years while various ﬂuids move across the wall-

rock contact is not a closed system.

4. Adiabatic system: A special category of an isolated

system is one in which no heat can be exchanged

between the system and the surroundings. It is

thermally insulated, but energy can be transferred

across the system boundary through work done on

or by it. Thus, an ascending, decompressing mantle

plume or magma body cools as it expands against

and does PV work on the surroundings, into which

little heat is conducted because the rate of conduc-

tion is so slow.

Any thermodynamic state of a system can be char-

acterized by its state properties, or state variables.

Some state properties are extensive in that they depend

on the amount of material, such as the mass and vol-

ume of some chemicals constituting a system. Other

state properties are intensive and are independent of

the amount of material present; they have a deﬁnite

value at each point within the system, such as T, P, and

concentration of a particular chemical species. Density

(mass/volume) is also an intensive property. For exam-

ple, at T  25°C and P  1 atm, the density of stable

graphite anywhere within a crystal is 2.23 g/cm

whereas that of diamond is 3.51 g/cm

P and T are extremely important intensive variables

whose change in rock-forming systems is responsible

for changes in their state; for example, increasing T

causes solid rock to melt. Values of the geobaric and

geothermal gradients in the Earth can be used to ap-

proximate P and T at a particular depth, assuming

these gradients to be uniform (Sections 1.1.3 and 1.2).

Otherwise uniform gradients can locally be perturbed,

such as the geothermal gradient around a shallow

crustal magma intrusion.

Rock-forming geologic processes can be thought of

as thermodynamic processes: those that affect a ther-

modynamic system as it changes from one state to an-

other. What happens in transit—the time-dependent

kinetic path—between the initial and ﬁnal states is not

important at this point, but the nature of kinetic paths

will be considered later in Section 3.6. In a thermody-

namic process there could be all kinds of work done on

or by the system, ﬂows of heat in or out, chemical re-

actions and movement of matter, tortuous paths taken,

and so forth. However, two ideal, end-member ther-

modynamic processes can be recognized: irreversible

and reversible. In an irreversible thermodynamic

process the initial state is metastable, and the sponta-

neous change in the system leads to a more stable,

lower-energy ﬁnal state. The conversion of metastable

volcanic glass into more stable crystals—a process

called devitriﬁcation—under near-atmospheric condi-

tions is one example of an irreversible process. The en-

ergy trough in which metastable glass is stuck has to be

surmounted, overcoming the activation energy, before

it can move to a lower energy state. The atomic bonds

in the glass must be broken or reformed into more sta-

ble crystalline structures. Devitriﬁcation of glass oc-

curs spontaneously in the direction of diminishing en-

ergy, never in reverse (Figure 3.2).

In a reversible process, both initial and ﬁnal states

are stable equilibrium states and the path between

them is a continuous sequence of equilibrium states. A

reversible process is never actually realized in nature; it

is a hypothetical concept that is used to make the math-

ematical models of thermodynamics work, and that’s

all that can be said without an extended discussion.

3.2.2 First Law of Thermodynamics

Suppose that a change in the internal energy, dE

of a

rock system, such as a mineral grain, is produced by

adding some amount of heat to it, dq. As a result of ab-

sorbing heat, the mineral grain expands by an incre-

ment in volume, dV, doing an increment of PV work,

dw  PdV, on the surrounding mineral grains. Ac-

cording to the ﬁrst law of thermodynamics, or law of

conservation of energy, the increase in internal energy

due to heat absorbed is diminished by the amount of

work done on the surroundings, or

3.1 dE

 dq  dw  dq  PdV

In the “scientiﬁc” convention, heat added to (or work

done on) a system, as in this formulation, is positive

and work done by (or heat withdrawn from) the system

on its surroundings is negative.

3.2.3 Enthalpy

A new state property, the enthalpy, H, is deﬁned here as

3.2 H  E

 PV

Upon differentiation, this equation becomes dH 

 PdV  VdP. Combining with equation 3.1 gives

3.3 dH  dq  VdP

Thermodynamics and Kinetics: An Introduction

Energy

Final

stable

state

Initial

metastable

state

Spontaneous

process

Never!

3.2 Spontaneous, irreversible thermodynamic process. A system in

an initial higher-energy metastable state moves spontaneously

to a lower-energy, stable state. The reverse never happens

spontaneously.

For isobaric (constant-pressure) changes, dP  0,

equation 3.3 becomes

3.4 dH

 dq

Although H may appear to be simply another and un-

necessary label for q, the enthalpy is signiﬁcant in pro-

viding a measure at constant P of how much heat is

gained or lost from a system. Three possible changes in

a system are accompanied by a gain or loss of heat, as

follows:

1. Chemical reactions, such as the reaction of quartz

plus calcite creating wollastonite plus CO

2. A change in state, such as crystals melting to liquid,

that occurs at a ﬁxed T once that T is reached in

heating a system.

3. A change in T of the system where no change in

state occurs, such as simply heating crystals below

their melting T. Combining equations 1.4 (in differ-

ential form) and 3.4 gives dq

 C

dT  dH

3.5 C

 (dH/dT)

The last two changes can be illustrated by a plot of

enthalpy versus T for the diopside CaMgSi

system

as it is heated at constant P (Figure 3.3). As diopside

crystals absorb heat up to their melting point, (3.,

above), T increases proportionally with the heat capac-

ity, C

. The slope of this line is (dH/dT)

 C

. At the

melting T absorbed heat does not increase T, but is

consumed in breaking the atomic bonds of the crys-

talline structure to produce the more random liquid ar-

ray. This relatively large amount of absorbed heat at the

constant T of melting, (2., above), is the latent heat of

melting, or enthalpy of melting, H

. Once melting is

complete, additional input of heat into the system

raises the T of the melt proportional to its heat capac-

ity. The slopes of the T-H lines above and below the

melting point are the heat capacities of the melt and the

crystals, respectively.

Note in Figure 3.3 that the latent heat involved in

changing the state of the system from liquid to crystal,

or vice versa, is similar to the heat absorbed in chang-

ing the T of the crystals or liquid by hundreds of de-

grees. Thus, melting of rock in the Earth to generate

magma absorbs a vast amount of thermal energy, which

moderates changes in T in the system.

Chemical reactions, (1., above), either release or ab-

sorb heat: They are either exothermic or endothermic,

respectively. If catalyzed by a spark, the reaction be-

tween hydrogen and oxygen releases a burst of heat—

a rapid exothermic reaction, or explosion. Solidiﬁca-

tion of magma exothermically releases the latent heat

of crystallization. In contrast, dissolution of potassium

nitrate in water endothermically absorbs heat so that

the container becomes cold. Thus, whether heat is re-

leased or absorbed in a chemical reaction provides no

consistent clue as to the direction a reaction moves

spontaneously.

3.2.4 Entropy and the Second and Third Laws

of Thermodynamics

Another way to look at spontaneity is in changes in the

distribution or concentration of energy. Spontaneous

thermal processes lead to a more even concentration of

heat. A bowl of hot soup on the table eventually reaches

the same T as the room. Heat ﬂows spontaneously from

a hot body to a cold, eliminating the difference in T.

Without an uneven concentration of thermal energy, or

the opportunity for heat ﬂow, no work can be done. As

heat ﬂows from an intrusion of hotter magma into the

cooler wall rocks, PV work of volumetric expansion is

performed on them. The heat also drives endothermic

chemical reactions of wall rock metamorphism. Water

in an enclosed lake on a high plateau has gravitational

potential energy relative to sea level, but so long as it is

isolated from sea level and cannot ﬂow, the concentra-

tion of energy is uniform and no work can be done.

However, in the natural course of events, a river drains

the lake into the sea, forming a process path along the

potential energy gradient between the high- and low-

potential-energy levels. Work can then be done, driving

turbines to generate electricity, eroding the river chan-

nel, transporting sediment, and so on.

One statement of the second law of thermodynam-

ics is that spontaneous natural processes tend to even

out the concentration of some form of energy, smooth-

ing the energy gradient. A hot lava ﬂow extruded from

a lofty volcano cools to atmospheric T as it descends

down slope, thereby reducing differences in thermal

54 Igneous and Metamorphic Petrology

Constant P

Crystals

0 400 800 1200 1600

∆H

144 kJ/mol

Melt

Temperature

of melting

(dH/dT)

= C

= 0.28 kJ/mol°C

T (°C)

600

400

200

H (kJ/mol)

3.3 Enthalpy-T relations for CaMgSi

at 1 atm. (From Weill DF,

Hon R, and Navrotsky A, The igneous system CaMgSi

CaAl

-NaAlSi

: Variations on a classic theme by

Bowen. In: Hargraves RB, ed., Physics of Magmatic Processes.

with permission of Princeton University Press.)

and gravitational potential energy between initial and

ﬁnal states in accordance with the second law.

Eventually, billions of years from now, all of the

thermal energy in the Earth will be consumed in tec-

tonism, volcanism, and other processes and dispersed

into outer space. No mountains or volcanoes will be

erected and erosion in the solar-powered hydrologic

system will wear everything down to some common

level (assuming the Sun does not run out of nuclear en-

ergy!). Without differences in the concentration of

thermal and gravitational potential energy no geologic

work can be accomplished and the planet will be geo-

logically dead!

The measure of the uniformity in concentration of

energy in a system is called the entropy, S. The more

uniform the concentration of some form of energy, the

greater the entropy. The geologically dead planet will

have maximal entropy.

Another, more useful, way to deﬁne entropy is to re-

late it to the internal disorder in the system. This pro-

vides an alternate statement of the second law: In any

spontaneous process in an isolated system there is an

increase in entropy, that is, an increase in disorder. The

law in this form is illustrated by Figure 3.4, where

white and black balls in the boxes represent molecules

of two different gases. The spontaneous mixing of the

two gas molecules results in an increase in “mixed-

upness,” disorder, randomness, or entropy. Note that

there is no accompanying change in energy in this mix-

ing process. Thus, another driving “force” for a spon-

taneous process is an increase in entropy, even though

there may be no change in the energy.

At decreasing T, crystals become increasingly or-

dered, less atomic substitution is possible, and their en-

tropy decreases. The third law of thermodynamics

states that at absolute zero, where the Kelvin tempera-

ture is zero (0K 273.15°C), crystals are perfectly

ordered and all atoms are ﬁxed in space so that the en-

tropy is zero.

A convenient way to think of relative entropies is

that a gas made of high-speed molecules in random tra-

jectories has a greater entropy than the compositionally

equivalent liquid array, which, though still somewhat

disordered, has linked atoms. The compositionally

equivalent crystalline solid has still lower entropy, be-

cause its atoms form an ordered array. As an example,

for water,

steam

 S

liquid water

 S

ice

3.2.5 Gibbs Free Energy

The boulder-on-the-hill example (Figure 1.6) has two

major ﬂaws as an analogy for the way natural systems,

in general, change spontaneously from a higher to a

lower energy state. First, it isn’t always gravitational po-

tential energy that is minimized. Second, the analogy

does not take into account the fact that in an isolated

system a process can proceed spontaneously without

any change in energy, but it does proceed with increas-

ing entropy (Figure 3.4). To overcome these two ﬂaws,

a new extensive property of a system is deﬁned in such

a way as to serve as a universal directionality pointer for

spontaneous reactions. This new property is called the

Gibbs free energy, G, and is deﬁned by the expression

G  H  TS. Combining with equation 3.2

3.6 G  E

 PV  TS

In differential form this becomes

3.7 dG  dE

 PdV  VdP  TdS  SdT

Remembering the work-pressure-volume relation (dW 

PdV), we can write a parallel expression for the heat-

temperature-entropy relation

3.8 dq  TdS

Equation 3.7 can be simpliﬁed by substituting this

equality and also by making a substitution from equa-

tion 3.2 to obtain

3.9 dG  VdP  SdT

Thermodynamics and Kinetics: An Introduction

Final state

Final

state

Initial state

initial

final

∆S >O

+ S

∆S

= S

final

− S

initial

Initial

state

3.4 Entropy increases in a spontaneous, irreversible process in an

isolated system. Bottom left, a hypothetical isolated system—a

box ﬁlled with atoms of two gases (black and white balls) sep-

arated by an impermeable wall. Bottom right, the wall has

been removed in the box, and the atoms of the two gases have

mixed spontaneously and irreversibly as a result of their mo-

tion (kinetic energy). An increase in disorder or randomness of

the atoms in the system and an increase in entropy, S  0. No

change in energy has occurred. (Redrawn from Anderson,

1996.)

Equation 3.9 is a useful thermodynamic expression

that allows us to make powerful statements regarding

the direction of changes in geologic systems as the in-

dependent intensive variables of state, T and P, change.

The extensive entropy and volume properties of the

system are also relevant factors.

If P and T remain the same through any sponta-

neous change in state, that is, dP  dT  0, then, from

equation 3.9

3.10 dG

P, T

 0

This is simply the condition for a minimum (or maxi-

mum) in G in P-T space where the slope of the tangent

to the energy function is zero, or horizontal. Figure 3.5

shows a system that has moved to a state of minimal

energy and stable equilibrium from a higher-energy,

metastable state at constant P and T in a closed (con-

stant composition) system. Note that the energy change,

G

P, T

, between the initial metastable state and the ﬁnal

stable state is negative: G

P, T

 0.

Without further justifying details (see Anderson,

1996) a summary statement can be made as follows: In

some spontaneously changing systems, increasing en-

tropy is the dominant factor, whereas in others, de-

creasing energy is the dominant factor. The Gibbs free

energy is formulated in such a way that it always de-

creases in a spontaneous change in the state of a system.

In other words, the Gibbs free energy of the ﬁnal sta-

ble state is lower than that of the initial metastable state

(Figure 3.5). The analogy with the boulders in the val-

ley (Figure 1.6) is now complete. The Gibbs free en-

ergy is a thermodynamic potential energy that, like

gravitational potential energy for the hypothetical

boulders, is the lowest possible in a state of stable equi-

librium for a changing system.

3.3 STABILITY (PHASE) DIAGRAMS

The foregoing concepts of stability couched in equa-

tion 3.9 can be portrayed graphically. For a system of

constant chemical composition, the two intensive vari-

ables that govern the state of the system, whether it is

stable or unstable, are P and T. On a diagram where P

is usually represented on the y-axis and T on the x-axis,

it would be expected that over a particular range of P

and T one state of the system is more stable and over

another range in P and T another state is more stable,

with some sort of boundary between these two states.

Across the boundary, the difference in the free energy

of the two states, G, changes sign, as the G of the

more stable state in its stability ﬁeld is exceeded by the

G of the less stable state. A stability diagram shows

which state is more stable as a function of P, T, or other

variables.

In rock-forming systems, different states are mani-

fested by the existence of different phases or assem-

blages of phases. A phase diagram shows which phase,

or assemblage of two or more phases, is more stable

as a function of P, T, or other variables. A phase is

deﬁned as a part of a system that is chemically and

physically homogeneous, bounded by a distinct inter-

face with adjacent phases, and physically separable

from other phases. Phases may be gaseous, liquid, or

solid. Minerals in rocks are solid crystalline phases,

whereas glass is a solid amorphous phase that has no

ordered atomic structure. Crystalline grains of olivine

and plagioclase in a basalt are two separate and dis-

tinct solid phases. Even though made of albite

(NaAlSi

) and anorthite (CaAl

) end mem-

bers, a homogeneous plagioclase solid solution crystal

is a single phase because the end members cannot be

physically separated.

Figure 3.6 is a phase diagram that portrays stability

relations of a hypothetical crystalline phase (crystals)

and its compositionally equivalent melted liquid phase

separated by the melting (freezing or crystallization)

curve. Melting and crystallization are fundamental

changes in state that are caused by changes in P and T.

Obviously, such changes are the very core of dynamic

magma systems and deserve special consideration.

Other phase diagrams that portray stability relations of

only crystalline phases are of no less importance in

petrology.

Over the range of P and T where crystals are more

stable in Figure 3.6, the Gibbs free energy of crystals is

less than that of liquid, G

 G

. On the other hand, in

the P-T region where liquid is more stable, the reverse

is true, or G

 G

. For any value of P and T exactly on

the melting (crystallization) curve that is the boundary

line of the two regions, crystals and liquid coexist to-

gether in equilibrium and their free energies are equal,

 G

56 Igneous and Metamorphic Petrology

 G

Metastable

diamond

G decreases

to a minimum

graphite

stable

graphite

 G

diamond

 ∆G

P,T

 0

3.5 Gibbs free energy decreases in a spontaneous change in a

closed system where the initial and ﬁnal states are at the same

P and T. In this example, the energy of diamond is greater than

that of graphite at the same P and T, or G

diamond

 G

graphite

so the change in energy, G

P, T

, in the spontaneous process is

negative, or G

diamond

 G

graphite

G

P, T

 0. Note the acti-

vation energy barrier, E

, that must be surmounted in order

for the change to occur.