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The Canadian MineralogistVol. 38, pp. 1313-1328 (2000)

“THERMODYNAMICS OF A MAGMATIC GAS PHASE” 50 YEARS LATER:COMMENTS ON A PAPER BY JOHN VERHOOGEN (1949)

JAMES NICHOLLS§

Department of Geology and Geophysics, University of Calgary, 2500 University Drive NW,Calgary, Alberta T2N 1N4, Canada

ABSTRACT

In 1949 John Verhoogen published a paper entitled Thermodynamics of a Magmatic Gas Phase in The University of Califor-nia Publications in the Geological Sciences. This paper documented the first sophisticated application of thermodynamics to aproblem in igneous petrology. In that paper, Verhoogen examined many aspects of the behavior of a magmatic gas phase: devel-opment and composition of a gas phase, deposition from a gas phase, variation of vapor pressure over a cooling magma. He alsodiscussed two other possible processes in magmas: distillation and crystallization of indifferent states. Distillation occurs whena constituent has a higher concentration in the vapor than in the melt. The volatile constituents of magmas, H2O, CO2, and H2S,obviously fit the definition of constituents that undergo distillation. Verhoogen did not discuss the reverse process, partitioning ofa constituent into the melt rather than the gas phase. HF is a constituent that undergoes reverse distillation; its concentration ishigher in a melt than in a gas phase. Volatile constituents that undergo reverse distillation cause magmas to vesiculate at shallowerdepths than those that undergo distillation. Indifferent states occur during processes that change the masses of the phases of asystem but leave their compositions unchanged. Univariant systems and azeotropes are familiar examples of indifferent states. Atthe time Verhoogen wrote his paper, there were limited amounts of experimental data on NaAlSi3O8–H2O and not much else.Thermodynamic databases constructed in the last decade provide the tools needed to explore processes described by Verhoogen50 years ago.

Keywords: thermodynamics, igneous petrology, magmatic gas phase, distillation, indifferent states.

SOMMAIRE

En 1949, Jean Verhoogen publia un article intitulé Thermodynamics of a Magmatic Gas Phase dans la revue The Universityof California Publications in the Geological Sciences. Dans cet article, il documenta la première application sophistiquée del’approche thermodynamique à un problème en pétrologie ignée. Il examina plusieurs aspects du comportement d’une phasevapeur: développement et composition de la phase vapeur, précipitation à partir de celle-ci, variation en pression exercée parcelle-ci en contact avec un magma au cours de son refroidissement. De plus, il discuta deux autres processus possibles dans unmagma: distillation et cristallisation selon les états indifférents. La distillation a lieu dans les cas où un composant possède uneconcentration plus élevée dans la phase gazeuse que dans le bain fondu. Les composants volatils du magma, par exemple H2O,CO2, et H2S, répondent clairement à la définition des composants sujets à une distillation. Verhoogen n’a pas traité du processuscontraire, le partage d’un composant dans le bain fondu plutôt que dans la phase gazeuse, par exemple le composant HF, dont lecomportement est en fait une distillation inverse: sa concentration est plus élevée dans le bain fondu que dans la phase gazeuse.

* Presidential address, Mineralogical Association of Canada, delivered on May 31st, 2000 in Calgary, Alberta.§ E-mail address: nicholls@geo.ucalgary.ca

Volume 38 December 2000 Part 6

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1314 THE CANADIAN MINERALOGIST

INTRODUCTION

In 1949, John Verhoogen published a paper entitledThermodynamics of a Magmatic Gas Phase in The Uni-versity of California Publications in the Geological Sci-ences (Fig. 1). The paper, 44 pages long, contained 90numbered equations, more that were not numbered, andno figures. This paper was the first sophisticated appli-cation of thermodynamics to a problem in igneous pe-trology.

Verhoogen (Fig. 2) was born in Brussels, Belgiumin 1912. He made major contributions to igneous andmetamorphic petrology, to our understanding of conti-nental drift and plate motion, to our understanding ofthe heat budget of the Earth, and to the study of rockmagnetism. Verhoogen died in 1993.

As a graduate student at the University of Liège,Verhoogen learned thermodynamics from Professors

Prigogine and Defay. These professors wrote a treatise,Thermodynamique chimique conformément aux métho-des de Gibbs et de Donder. D.H. Everett translated the1950 edition into English (Prigogine & Defay 1954).Much of Thermodynamics of a Magmatic Gas Phase isbased on the work of Prigogine & Defay, whichVerhoogen acknowledged in his paper. Several of theideas and concepts expressed by Verhoogen in Ther-modynamics of a Magmatic Gas Phase are more com-pletely developed by Prigogine & Defay (1954).

In Thermodynamics of a Magmatic Gas Phase,Verhoogen examined several aspects of the behavior ofa magmatic gas phase. Table 1, which shows the tableof contents of his paper, lists the topics he discussed. Inthis paper, I will look at a few topics introduced by

FIG. 1. Facsimile of the cover page of Verhoogen’s 1949paper.

FIG. 2. John Verhoogen in 1958 as the recipient of the ArthurL. Day Medal from the Geological Society of America.

Les composants volatils qui ont cette propriété provoquent une saturation du bain fondu en phase gazeuse à plus faible profondeurque dans les cas sujets à une distillation. Les états indifférents apparaissent où il y a changement des masses des phases d’unsystème sans changement en composition. Les systèmes univariants et ceux qui contiennent un azéotrope sont des exemplesfamiliers d’états indifférents. A l’époque de Verhoogen, il y avait un peu d’information expérimentale à propos du systèmeNaAlSi3O8–H2O, et très peu sur les autres systèmes importants. Les banques de données thermodynamiques devenues disponiblesau cours de la dernière décennie fournissent les outils nécessaires pour explorer à fond les processus décrits il y a cinquante anspar Verhoogen.

(Traduit par la Rédaction)

Mots-clés: thermodynamique, pétrologie ignée, phase gazeuse magmatique, distillation, états indifférents.

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THERMODYNAMICS OF A MAGMATIC GAS PHASE 1315

Verhoogen into igneous petrology. Some have beenstudied thoroughly, whereas some have been ignored.

To describe gas phase phenomena, calculations weremade with a modified version of the thermodynamicdatabase constructed by Ghiorso & Sack (1995), a solu-bility model for CO2 in a silicate melt developed byPapale (1999), and some speculations on the behaviorof volatile substances in silicate melts on my part.

FUNDAMENTAL RELATIONS

The first section of Thermodynamics of a MagmaticGas Phase provides a complete coverage of the funda-mental relations of thermodynamics. In eight pages,Verhoogen developed the relationships between Gibbsenergy and chemical potentials, entropy, and volume.Partial molar quantities are defined, and their place inthermodynamic theory explained. Finally, he derivedversions of the Gibbs phase rule and Duhem’s theorem.

Definition of equilibrium

There is really nothing new in fundamental relationsin classical thermodynamics since Gibbs. However,petrologists have become more precise about the rela-tionship between thermodynamic calculations and realprocesses. Anderson (1996) and Greenwood (1989), inparticular, have emphasized that thermodynamics pro-vides a model to compare with reality. I would like toextend this idea to the concept of equilibrium. I suggestthat the concept of equilibrium belongs to the thermo-dynamic model rather than to reality.

Thermodynamic systems can be described in twodifferent ways. There is, first, the real collection of at-oms, molecules, phases, and energy that constitutes thereal world. Then there is the mathematical structure thatrepresents the real world. The literature contains sev-eral definitions of equilibrium in thermodynamic sys-tems. Some definitions are phrased in terms of thebehavior of the real collection. Other definitions aremathematical statements. Prigogine & Defay (1954,p. 69, Eq. 6.23) use the value of a linear combination ofchemical potentials to describe the equilibrium condi-tion:

i

i iv∑ =µ 0 (1)

where �i is a stoichiometric coefficient and �i is thechemical potential of the chemical entity i. Verhoogen(1949, Eq. I.14) also described this criterion as the equi-librium criterion. Equation (1) represents the equilib-rium for a chemical transformation that can besymbolized:

i

i iv x∑ = 0 (2)

where xi is a chemical formula, and Equation (2) repre-sents a balanced chemical transformation. This criterionfor equilibrium will hereafter be called the First State-ment.

Denbigh (1981) described thermal equilibrium withhis zeroth law: “.... if bodies A and B are each in ther-mal equilibrium with a third body, they are also in ther-mal equilibrium with each other.” In other words, bodiesin equilibrium, at least bodies in thermal equilibrium,have the same temperature. Later in his text, Denbigh(1981, p. 263) stated that equality of chemical poten-tials for a component is the only proper criterion forequilibrium with respect to diffusion. This is the samestatement as that of Verhoogen (1949). The criterion isa particular application of the first statement.

Nordstrom & Munoz (1985) stated that equilibriumoccurs when the equality holds in the Clausius equa-tion:

dQ – TdS ≥ 0 (3)

where Q is the heat gained by the system, T is the tem-perature in Kelvin, and S is entropy. Equality holds forreversible processes.

Anderson & Crerar (1993) defined equilibrium withtwo statements: 1) “A system at equilibrium has noneof its properties changing with time, no matter how longit is observed.” 2) “A system at equilibrium will returnto that state after being disturbed, that is, after havingone or more of its parameters slightly changed, [they

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1316 THE CANADIAN MINERALOGIST

will] then change back to the original values.” Otherstatements of equilibrium postulate a minimum for ther-modynamic potentials that depend on a particular set ofvariables. For example, equilibrium obtains if the Gibbsenergy for a system is a minimum at some fixed valuesof pressure and temperature.

All of these statements are correct in that, if the cri-teria hold, then the system is at equilibrium. However,except for the definition used by Prigogine & Defay(1954) and Verhoogen (1949), the statements are suffi-cient rather than necessary criteria for equilibrium inchemical systems. In some systems, the other statementswill be incorrect, whereas the statement by Prigogine &Defay (1954) and Verhoogen (1949) (the First State-ment) will still be correct. If the other statements ob-tain, then so does the First Statement. The converse isnot true. The First Statement can obtain although one ormore of the others do not obtain. Appendix A containscounter examples of the sufficient statements.

In summary, the sufficient statements are:1. Equilibrium systems do not change, no matter

how long they are observed.2. At equilibrium, the rate of the forward reaction

equals the rate of the reverse reaction.3. When infinitesimally perturbed, equilibrium sys-

tems return to their original equilibrium state.4. A minimum in the value of a thermodynamic

potential occurs at equilibrium.5. Equilibrium systems are at constant temperature.Usually, the systems that contain one phase are not

described with chemical potentials. Consequently, theFirst Statement is not often applied to such systems. Thefirst statement applies, however, even in the trivial caseof one phase. Suppose two grains of kyanite, a and b.Then,

�aAl2SiO5 – �b

Al2SiO5 = 0 (4)

To quote Verhoogen (1949, p. 94) “At equilibrium, thechemical potential of all constituents must be the samein all phases.”

Representations of equilibrium by linear combina-tions of chemical potentials equal to zero do, in fact,provide convenient methods for defining equilibriumsystems. Equilibrium systems are those parts of realitythat can be adequately and appropriately represented bysuch linear combinations. The non-equilibrium partsmust be represented by linear combinations of chemi-cal potentials different from zero. Describing systemsin terms of equality and inequality constraints puts theinitial focus on whether or not that part of reality can bedescribed in terms of equilibrium thermodynamics orwhether the description must be a dynamic one. Thecauses for the differences can then be interpreted interms of geometrical walls, membranes and barriers tothe transfer of matter and energy.

Although many equilibrium thermodynamic systemscannot be realized in nature, they often provide adequate

representations of reality. If even part of a natural sys-tem can be adequately approximated by an equilibriumsystem, then significant simplification over a dynamicmodel usually follows. Consequently, a precise defini-tion of equilibrium leads to precise descriptions of rep-resentations of reality and a better understanding ofprocesses that affect chemical systems.

Equilibrium systems are in many cases equated withparts of reality characterized by reversible reactions orchanges. Although systems containing reversible pro-cesses are systems at equilibrium, the converse need notbe true. Some systems containing irreversible processescan be analyzed and described with equilibrium state-ments. One such set of systems is the set that containsisenthalpic changes. Examples include expansion ofgases into a vacuum (the Joule–Thompson effect; seeLewis & Randall 1961, p. 48), adiabatic decompressionduring uplift (Waldbaum 1971), and rapid ascent ofnephelinites (Nicholls 1990, Trupia & Nicholls 1996).Such systems should be distinguished from systemsdescribed by what are called irreversible thermodynam-ics (e.g., the Soret effect).

Duhem’s Theorem

Verhoogen (1949, p. 98) introduced this remarkabletheorem to petrology. Duhem’s Theorem is anotherphase rule. The familiar Gibbs Phase Rule is:

FG = c – � + 2

where FG is usually called the degrees of freedom of thethermodynamic model, c is the number of components,and � is the number of phases in the model. Degrees offreedom often cause confusion. Actually, the degrees offreedom express the number of variables one must fixor set before all the other variables describing the modelcan be calculated. The remarkable feature of Duhem’sTheorem is that the degrees of freedom are 2 if the com-position of the thermodynamic model is known. Thismeans that one can often specify temperature and pres-sure and then calculate all other properties of the model,including phase compositions and relative abundancesof those phases. Thermodynamic models thus have alarge number of independent properties that can be com-pared with nature. If the calculated properties of themodel adequately describe the properties of the realsystem, one gains insight into processes of rock forma-tion.

Both Verhoogen (1949) and Prigogine & Defay(1954) derived Duhem’s Theorem in terms of massesof the components forming a system. Their derivationshad the unfortunate result of preventing the significanceof Duhem’s Theorem from being appreciated. Findingor knowing the masses of components composing a rockbody, for example, can be fraught with difficulty. Con-sequently, petrologists looked to the Gibbs Phase Rulefor insight because it invokes only intensive variables,

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THERMODYNAMICS OF A MAGMATIC GAS PHASE 1317

such as concentrations; extensive variables, such asmasses of the components, need not be considered. Therequirement that the masses of the components beknown is too stringent. The weaker requirement that thecomposition of the system be known is sufficient.Whether the system is the size of a batholith or a labo-ratory crucible, if the system is at equilibrium, then theproperties of the phases are independent of size. Thesize of the system can be arbitrarily set and all the prop-erties calculated except the two that are required byDuhem’s Theorem (Nicholls 1990).

As an example of the guidance Duhem’s Theoremcan give, consider the problem of calculating the satu-ration conditions of a phase in a melt. Duhem’s Theo-rem states that one need fix only two variables to

calculate the state of the model system if the composi-tion of the system is known. The state of the system isknown when the temperature, pressure, compositions ofall the phases, and the relative amounts of all the phasesare known. One variable to fix is the pressure. The satu-ration state at a fixed pressure can then be calculated.However, temperature cannot be arbitrarily fixed be-cause an arbitrary temperature may not be on the satu-ration surface at a fixed pressure. Consequently, thetemperature must be a calculated quantity. Also, thecomposition of the saturating phase must be a calculatedquantity because, again, an arbitrary composition maynot be the composition of the saturating phase at a fixedpressure. Constraining the system to a saturation stateimplicitly fixes the amount of the saturating phase at

FIG. 3. P–T diagram for the system NaAlSi3O8–H2O, which serves as a model for a mag-matic system that can coexist with a gas phase.

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1318 THE CANADIAN MINERALOGIST

zero, which fixes the second variable required byDuhem’s Theorem. The melt is forced to have the samecomposition as the system, which is a composition thatis known by hypothesis. A solution composed of n com-ponents has n – 1 independent compositional variablesbecause the mole fractions sum to one. For each of then components, one can write a form of the First State-ment. This procedure provides a system of n equations,which can be solved for the n – 1 independent composi-tional variables and the saturation temperature. In sum-mary, Duhem’s Theorem guarantees that no matter whatthe chemical complexity of the solid solution or the melt,one can calculate the composition of the saturating phaseand the temperature of saturation at a fixed pressure.

DEVELOPMENT OF A GAS PHASE

The system NaAlSi3O8–H2O as a model

Several petrologists have used the systemNaAlSi3O8–H2O as a model system for the generationof a magmatic gas phase (Goranson 1938, Burnham1974, Nicholls 1980). A calculated P–T diagram isshown on Figure 3. At a fixed H2O concentration, thediagram can be divided into four areas: melt (red), meltplus gas phase (yellow), melt plus solid (tan), and solidplus gas phase (blue). The nearly vertical lines, labeledwith wt% H2O contents in the melt, are the coordinateswhere solid albite crystallizes from a melt containingdissolved H2O. The heavy line labeled 8 separating thered and tan regions is the liquidus curve for an H2O-undersaturated melt that is in equilibrium with albite.The line separating the P–T region of melt (red) frommelt plus gas phase (yellow) marks the P–T coordinatesof vesiculation in the model system on release of pres-sure. The solidus, the line separating the blue regionfrom the others, where melt, gas, and solid coexist, ap-pears as a line in a P–T section, but is an area in a T–Xsection. This univariant line represents a set of indiffer-ent states, a topic for later discussion.

If a melt is originally at a temperature greater thanthe solid–melt isopleth, then on cooling, albite will crys-tallize when the P–T coordinates coincide with thesolid–melt isopleth. If the magma cools at constant pres-sure, the H2O concentration in the melt will increase,and the concentration of albite (condensed component)will decrease. For P–T coordinates on the univariant,three-phase curve, gas, melt, and solid coexist. As thetemperature or pressure drops, the amount of gas andsolid will increase at the expense of melt. The P–T co-ordinates remain on the univariant curve until all themelt has crystallized and evaporated. At any instant, amagma will contain a fixed amount of volatile speciesdissolved in the melt.

In order for a gas phase to form, pressure–tempera-ture conditions will have to coincide with the appropri-ate vesiculation isopleth (Fig. 3). As the P–T conditionsdrop below the isopleth, the melt loses H2O to the gas

phase, and the concentration of H2O in the melt de-creases. If the pressure drops a small amount, then theamount of gas phase will be small. In order to producea large amount of gas from an undersaturated magma,the pressure drop must be large. Most large plinian erup-tions, such as occurred at Mount St. Helens, probablyare initiated by a large change in pressure. The MountSt. Helens eruption is considered to have resulted fromthe pressure release caused by the landslide that imme-diately preceded the eruption. Volcanic activity atMount St. Helens was the subject of Verhoogen’s Ph.D.thesis at Stanford University, which was later publishedin The University of California Publications in the Geo-logical Sciences (Verhoogen 1937).

The isopleths carry into multivariate systems. Theliquidus and vesiculation curves, in particular, have ana-logues in multicomponent systems. The positions of theliquidus and vesiculation curves depend on the concen-trations of the condensed components and the concen-trations of the volatile components. The latter can havea significant effect on the location of the vesiculationcurves. If the vesiculation curves are displaced to lowerpressures, then magmas must move up in the Earth (orother planets) to form a gas phase. If the vesiculationcurves are displaced to higher pressures, then a gasphase will separate at greater depths. The saturation orvapor pressure of a gas phase depends, among otherfactors, on the volatile species dissolved in a magma.Some volatile species will displace the vesiculationcurves to higher pressures, and others species will dis-place them to lower pressures. CO2 and HF are examplesthat cause the contrasting effects relative to the H2Ovesiculation curve (Fig. 4).

FIG. 4. P–T diagram showing the vesiculation curves for 0.5wt% volatile species dissolved in a melt with the composi-tion of an alkali olivine basalt from Hawai’i.

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THERMODYNAMICS OF A MAGMATIC GAS PHASE 1319

In the section on Development and Composition ofa Gas Phase, Verhoogen (1949) provided general rela-tions between the thermodynamic properties of the meltand gas phases. He pointed out that the properties of thepure components are not reliable indicators of the prop-erties of the components in the melt or gas phase solu-tions.

DISTILLATION

Verhoogen (1949) defined distillation from a mag-matic gas phase as a process that occurs when the con-centration of a constituent increases in the gas phase anddecreases in the silicate melt. H2O and CO2 will undergodistillation where both are dissolved in the melt. Rhy-olitic melts in hydrothermal experiments show dispro-portionately larger amounts of vesiculation if CO2 isadded to melts saturated with H2O (Ian Carmichael,pers. commun.).

The opposite effect, increasing concentration in themelt and decreasing concentration in the gas phase, canalso occur. Most constituents display this behavior (e.g.SiO2, TiO2, etc.). One unexpected constituent that un-dergoes reverse distillation is HF. Appendix B containsthe details of the calculation.

Figure 4 shows the vesiculation pressures of an HF-bearing melt with the composition of an alkali olivinebasalt from Hawai’i. An HF-bearing melt will vesicu-late at pressures approximately one-tenth the pressuresof vesiculation of the equivalent H2O-bearing melt. HFincreases the solubility of H2O in silicate melts andraises the saturation temperatures of (OH, F)-bearingsolids. This latter effect is shown for apatite on Figure 5.

Zoning of apatite is a consequence of the partition-ing of HF into the melt and its retention by the meltduring crystallization. Fluorapatite typically is brown,whereas hydroxyapatite is clear (Fig. 6A). Apatitemicrophenocrysts in intermediate to felsic volcanicrocks commonly have a clear center and a brown rimthat results from the increasing HF concentrationbrought about by fractional crystallization.

Even a small concentration of HF can affect the erup-tion of intermediate to felsic rocks. The Craters of theMoon lava field, in Idaho, comprises minor basalt andlarger numbers of intermediate and felsic lava flows oftrachybasalt, tristanite, and trachyte (Stout et al. 1994).The rocks have SiO2 concentrations as high as 63% andfeldspar contents as high as 80% by volume (Stout etal. 1994, Fig. 6). Melts with these concentrations shouldhave viscosities similar to those of rhyolitic melts, yetthe morphology of the flows suggests a viscosity simi-lar to that of basaltic melts (Fig. 7). For example, basaltswith a SiO2 content near 45% and a liquidus tempera-ture near 1250°C have a viscosity of approximately 3 �102 Pa s (Shaw 1972). Rhyolites with a SiO2 contentnear 65% have a viscosity near 3 � 106 Pa s at theirliquidus temperatures (1050°C). Trachytes from theCraters of the Moon, Idaho, with a SiO2 content near

63%, should have a viscosity near 5 � 106 Pas at theirliquidus temperature (1050°C).

INDIFFERENT STATES

In part V of his 1949 paper, Closed Systems,Verhoogen introduced the concept of indifferent statesto petrology. A chemical system is in an indifferent stateif the concentrations of the phases in the system changebut the compositions of the phases do not. Examples ofsystems in an indifferent state include systems with twoor more polymorphs. Two polymorphs coexist along aunivariant line. In crossing from one side of the line tothe other, the relative proportion of the two phaseschanges but, being polymorphs, their compositions donot change. In the writings on indifferent states, pres-sure and temperature have been considered constants.In this example, they do not have to be constant. How-ever, in the general case, pressure and temperature haveto be constrained to constant values to satisfy the defi-nition of an indifferent state. An example is providedby a system with two solid solutions, a pure solid, and amelt. There is one degree of freedom under the GibbsPhase Rule, and the compositions of the solids changealong the univariant curve. Another common exampleof a system with an indifferent state is one with an azeo-trope (Fig. 8). Azeotropes can occur at minimum ormaximum temperatures at constant pressure. Likeunivariant curves with solutions, an azeotrope is in anindifferent state if the pressure and temperature are con-stant because the composition of the azeotrope changeswith pressure and temperature (e.g., Denbigh 1981, p.221).

FIG. 5. P–T diagram for apatite saturation as a function ofH2O and HF concentration in a melt with the compositionof an alkali olivine basalt from Hawai’i.

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1320 THE CANADIAN MINERALOGIST

FIG. 6. A. Photomicrograph ofan apatite microphenocrystin a hawaiite from Kohalavolcano, Hawai’i. B. Olivinephenocryst in glassy picrite(oceanite) from Kılauea vol-cano, Hawai’i. C. Glass in-clusion from olivine pheno-cryst in Figure 6B. Spinel(black) and augite (?) inclu-sions in the glass.

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THERMODYNAMICS OF A MAGMATIC GAS PHASE 1321

FIG. 7. View of contrasting morphology of flows in rhyolite domes (Mono Craters, California) and trachybasalt flows (Cratersof the Moon, Idaho).

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The concept of general conditions for indifferentstates in petrological systems has not been exploredsince Verhoogen (1949). Although typical indifferentstates occur in simple systems (Fig. 8), I do not knowwhether analogous features occur in complex systems.I also do not know whether processes of melting andcrystallization will bring the system to a maximum or

minimum in temperature (Fig. 8). Consequently, whatfollows is speculation on my part.

Indifferent states may be significant in partial melt-ing of mantles of rocky planets. The primary melts sup-plied to Kılauea volcano, Hawai’i are, at least in someinstances, picritic (Nicholls & Stout 1988, Wright 1971).Within the glassy picrites that were erupted fromKılauea volcano in 1968 are olivine phenocrysts withglass inclusions (Fig. 6B). Included within the glass in-clusions are inclusions of spinel and augite (?) (Fig. 6C).On a plot of (Fe + Mg)/K versus Si/K, the rock compo-sitions define a fractionation path controlled by olivinefrom picrite to olivine tholeiite (Fig. 9). Joining the frac-tionation path controlled by olivine is a fractionationpath controlled by olivine, plagioclase, and augite fromolivine tholeiite to tholeiite (Fig. 9). Data points repre-senting the compositions of the glass included in oliv-ine fall below the extension of the olivine – plagioclase– augite fractionation path (dashed line, Fig. 9). Afterbeing trapped, the glass (melt) inclusion was affectedby plating of olivine on the wall of the inclusion and bycrystallization of the minerals in the inclusion. Conse-quently, the closest point on the fractionation path wherethe inclusion could have been trapped is directly abovethe data point. For the extreme glass composition, themelt at the time of trapping contain approximately 18%MgO. This is a minimum estimate, and the primary meltmay have had a higher concentration of Mg. One cancalculate a composition from a mixture of olivine,orthopyroxene, and clinopyroxene that has the samecomposition as a picrite with 18% MgO. If spinel is in-cluded in the assemblage, the MgO concentration mustbe near 22%. Consequently, the mass-balance criterionfor an indifferent state in the source region beneathHawai’i is satisfied. Whether the temperature and pres-sure at which this indifferent state could exist is a sub-ject for future research, as is whether the actual mantlecomposition is close to the requirement for an indiffer-ent state.

If an indifferent state exists in the upper mantle, itcould play a part in the formation of large igneous prov-inces. Large igneous provinces are massive emplace-ments of mafic extrusive and intrusive rocks in the crust.Examples are the Columbia River basalts, the KerguelenPlateau, and the Deccan Traps (Coffin & Eldholm1994). One of the largest is the approximately 120 MaOntong–Java province. The volume of magma extrudedto form this province was 55 ± 20 � 106 km3 (Coffin &Eldholm 1994). This volume is equivalent to a spherewith a diameter between 405 and 470 km. If the mag-mas of the Ontong–Java province formed by partialmelting of the mantle, then a large part of the low-velocity zone must have been affected. If the primarymelts were basaltic, then they would have formed bysmall amounts of partial melting, and the volume of low-velocity zone affected would be large. At larger amountsof partial melting, a smaller volume would be affected,but the primary melts would be picritic or approach a

FIG. 8. Binary systems with azeotropes. A system with thecomposition of the azeotrope is in an indifferent state at thepressure and temperature of the azeotrope. A system in anindifferent state can change from completely melted to allsolid without changing pressure and temperature. The azeo-trope temperature can be either a maximum or a minimumat constant pressure. Maximum temperatures are rare inchemical systems relevant to the formation of rock-form-ing minerals.

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picritic composition, and these would have to fraction-ate to a basaltic composition before eruption. If thesource region for the Ontong–Java magmas was in anindifferent state, the melting would be 100%, and thevolume affected by the melting process would be mini-mal. I would like to stress, again, that the idea of anindifferent state in the mantle is complete speculation.

An indifferent state may also play a role in the forma-tion of amphibolites in a regional metamorphic terrane.Basaltic melts can crystallize directly to amphiboles.Nephelinites from Volcano Mountain, Yukon Territory,have chemical characteristics that suggest they formedfrom an amphibole source-rock (Francis & Ludden1995) or from source rocks that isochemically recrys-

FIG. 9. (Fe + Mg)/K versus Si/K plot depicting the formation of basaltic melts from picriticmelts by fractionation of olivine at Kilauea Volcano, Hawaii. The compositions of theglass inclusions are consistent with melts with a minimum of 18% MgO being precur-sors to the melts that erupted at the surface with 15% MgO.

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1324 THE CANADIAN MINERALOGIST

FIG. 10. A Thompson space diagram (Thompson 1982a, b, Nicholls et al. 1991) for anephelinite from Volcano Mountain, Yukon Territory. The rock composition can berecalculated as mixtures of olivine, clinopyroxene, and amphibole that fall in the grayarea. The point where lines representing Di = 0, Fo = 0, and mc = 0 meet has the coor-dinates in Thompson space where the rock composition can be recast as an amphibole.

tallized from amphibole-rich assemblages. That thesenephelinites could exist as pure amphibole is illustratedon Figure 10. In addition, there is experimental evidencethat basaltic melts can crystallize as amphibole. Yoder& Tilley (1962, p. 442) reported an experimental run ona high-aluminum basalt at 1150°C and 0.5 GPa H2Opressure that quenched to rare magnetite and amphib-ole. Consequently, there is both analytical and experi-mental evidence that basaltic melts with the composi-tions of amphiboles exist and that they can crystallizeto amphiboles at pressures expected in the crust.

ACKNOWLEDGEMENTS

I thank the members of the Mineralogical Associa-tion of Canada for allowing me to be president of theAssociation. I thank the members of the MAC Execu-tive and Council for their support. I thank R.F. Martinfor his suggestions on improving the manuscript.

REFERENCES

ANDERSON, G.M. (1996): Thermodynamics of Natural Systems.John Wiley & Sons, Inc., Toronto, Ontario.

________ & CRERAR, D.A. (1993): Thermodynamics inGeochemistry, The Equilibrium Model. Oxford UniversityPress, Oxford, U.K.

BURNHAM, C.W. (1974): NaAlSi3O8–H2O solutions: a thermo-dynamic model for hydrous magmas. Bull. Soc. fr. Minéral.Cristallogr. 97, 223-230.

COFFIN, M.F. & ELDHOLM, O. (1994): Large igneous provinces:crustal structure, dimensions, and external consequences.Rev. Geophys. 32, 1-36.

DENBIGH, K.G. (1981): The Principles of Chemical Equilib-rium. Cambridge University Press, Cambridge, U.K.

FRANCIS, D. & LUDDEN, J. (1995): The signature of amphibolein mafic alkaline lavas, a study in the northern CanadianCordillera. J. Petrol. 36, 1171-1191.

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THERMODYNAMICS OF A MAGMATIC GAS PHASE 1325

GHIORSO, M.S. (1990): Application of the Darken equation tomineral solid solutions with variable degrees of order–dis-order. Am. Mineral. 75, 539-543.

________ & SACK, R.O. (1995): Chemical mass transfer inmagmatic processes. IV. A revised and internally consist-ent thermodynamic model for the interpolation and ex-trapolation of liquid–solid equilibria in magmatic systemsat elevated temperatures and pressures. Contrib. Mineral.Petrol. 119, 197-212.

GORANSON, R.W. (1937): Silicate–water systems: the “osmoticpressure” of silicate melts. Am. Mineral. 22, 485-490.

________ (1938): Silicate–water systems: phase equilibria inthe NaAlSi3O8–H2O and KAlSi3O8–H2O systems at hightemperatures and pressures. Am. J. Sci. 235A, 71-91.

GREENWOOD, H.J. (1989): On models and modeling. Can. Min-eral. 27, 1-14.

LEWIS, G.N. & RANDALL, M. (1961): Thermodynamics.McGraw-Hill Book Co., New York, N.Y.

LIANG, Y. (2000): Dissolution in molten silicates: effects ofsolid solution. Geochim. Cosmochim. Acta 64, 1617-1627.

NICHOLLS, J. (1980): A simple thermodynamic model for esti-mating the solubility of H2O in magmas. Contrib. Mineral.Petrol. 74, 211-220.

________ (1990): Principles of thermodynamic modeling ofigneous processes. In Modern Methods of Igneous Petrol-ogy – Understanding Magmatic Processes (J. Nicholls &J.K. Russell, eds.). Rev. Mineral. 24, 1-23.

________ & STOUT, M.Z. (1988): Picritic melts in Kilauea –evidence from the 1967–1968 Halemaumau and Hiiakaeruptions. J. Petrol. 29, 1031-1057.

________, ________ & GHENT, E.D. (1991): Characterizationof partly-open-system chemical variations in clinopyroxeneamphibolite boudins, Three Valley Gap, British Columbia,using Thompson space calculations. Can. Mineral. 29, 105-124.

NORDSTROM, D.K. & MUNOZ, J.L. (1985): Geochemical Ther-modynamics. The Benjamin/Cummings Publishing Co.,Inc., Menlo Park, California.

PAPALE, P. (1999): Modeling of the H2O + CO2 fluid in silicateliquids. Am. Mineral. 84, 477-492.

PRIGOGINE, I. & DEFAY, R. (1954): Chemical Thermodynam-ics. Longmans, Green and Co., Ltd., London, U.K.

SACK, R.O. (1980): Some constraints on the thermodynamicmixing properties of Fe–Mg orthopyroxenes and olivines.Contrib. Mineral. Petrol. 71, 257-269.

SHAW, H.R. (1972): Viscosities of magmatic silicate liquids:an empirical method of prediction. Am. J. Sci. 272, 870-893.

STOUT, M.Z., NICHOLLS, J. & KUNTZ, M.A. (1994): Petrologi-cal and mineralogical variations in 2,500–2,000 yr B.P. lavaflows, Craters of the Moon Lava Field, Idaho. J. Petrol. 35,1681-1715.

THOMPSON, J.B. (1982a): Composition space: an algebraic andgeometric approach. In Characterization of Metamorphismthrough Mineral Equilibria (J.M. Ferry, ed.). Rev. Mineral.10, 1-32.

________ (1982b): Reaction space: an algebraic and geomet-ric approach. In Characterization of Metamorphismthrough Mineral Equilibria (J.M. Ferry, ed.). Rev. Mineral.10, 33-52.

TRUPIA, S. & NICHOLLS, J. (1996): Petrology of Recent lavaflows, Volcano Mountain, Yukon Territory, Canada. Lithos37, 61-78.

VERHOOGEN, J. (1937): Mount St. Helens, a Recent Cascadevolcano. Univ. Calif. Publ. Geol. Sci. 24, 263-302.

________ (1949): Thermodynamics of a magmatic gas phase.Univ. Calif. Publ. Geol. Sci. 28, 91-135.

WALDBAUM, D.R. (1971): Temperature changes associatedwith adiabatic decompression in geological processes. Na-ture 232, 545-547.

WRIGHT, T.L. (1971): Chemistry of Kilauea and Mauna Loa inspace and time. U.S. Geol. Surv., Prof. Pap. 735, 1-40.

YODER, H.S., JR. & TILLEY, C.E. (1962): Origin of basalt mag-mas: an experimental study of natural and synthetic rocksystems. J. Petrol. 3, 342-532.

Received July 1, 2000.

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1326 THE CANADIAN MINERALOGIST

Equilibrium systems do not change,no matter how long they are observed

As a counter example to this definition of equilib-rium, consider a glass of water and ice in an environ-ment at room temperature. The temperature in the glassis 0°C. In fact, the glass contains an assemblage ofphases that defines a point on the thermodynamic tem-perature scale. Yet, obviously the system is changingwith time as the ice melts. Purists will protest that thetemperature is not really 0°C, but slightly greater thanthat. However, as a representation of reality, the math-ematical representation of reality is adequate. A tem-perature scale based on phase assemblages out ofequilibrium would be inconvenient at best and unusablenormally. Consequently, for the practical purpose of thetemperature scale, a mixture of ice and water is an equi-librium system, and the appropriate and adequate ex-pression is:

�IceH2O – �Water

H2O = 0 (5)

an example of the First Statement.

At equilibrium, the rate of the forward reactionequals the rate of the reverse reaction

Again, the mixture of ice and water example pro-vides a counter example. The rate of melting is fasterthen freezing, yet, as shown above, the system can berepresented as an equilibrium system.

When infinitesimally perturbed, equilibrium systemsreturn to their original equilibrium state

Consider a large volume of water and ice into whicha drop of hot water is added. The temperature quicklyreturns to 0°C, but the small amount of ice that meltedto cool the drop does not freeze again. Unless the equi-librium state is defined by temperature alone, then thesystem of ice and water does not return to its originalstate when infinitesimally perturbed.

A minimum in the value of a thermodynamic potentialoccurs at equilibrium

A counter example to this condition is more difficultto contrive than the others. Consider a solid solution thatcan have a varying degree of order, for example, an il-menite–hematite solid solution. Ghiorso (1990) derivedan expression for the chemical potential of the hematitecomponent:

µ ∂∂

∂∂Hm

s yG y G

y s Gs=

– – (6)

where y is the mole fraction of Fe2O3 in solid solution,and s describes the degree of order of Fe and Ti on themetal sites of the structure. Disordered FeTiO3 is char-acterized by s equal to zero, whereas fully orderedFeTiO3 has a value of s equal to one. G is the Gibbspotential for the solution. In general, s is evaluated bysetting the partial derivative of G with respect to s equalto zero (see, for example, Sack 1980). This exercise isequivalent to minimizing the Gibbs potential for thesolid solution with respect to diffusion. However, it mayhappen that a solid solution encounters an environmentwhere it will react with another phase under conditionswhere rates of internal diffusion in the solid solution aretoo slow to alter the state of order (e.g., Liang 2000).Under such conditions, the chemical potential of Fe2O3still exists (Eq. 6) and could be used to describe an equi-librium change in a geochemical system with an expres-sion of the form of Equation (1). In such a system, theGibbs potential for the system will not be a minimumbecause the Gibbs potential for the solid solution willnot be a minimum.

Equilibrium systems are at constant temperature

A counter example to this statement can be visual-ized by recourse to that magical construct, an adiabaticsemipermeable membrane. Suppose a solution on oneside of the wall and a vapor consisting of the pure sol-vent on the other. The wall is assumed permeable to thesolvent but impermeable to the solute. As an example,suppose that the solvent is water and the solution isideal. The chemical potential of H2O in the solution is:

�WaterH2O = –68317 – 16.712TW + RTW lnx. (7)

R is the gas constant, and TW is the temperature of thesolution. The chemical potential of H2O as steam, as-suming ideal gas behavior, is:

�SteamH2O = –5795 – 45.11TS + RTS lnp. (8)

TS is the temperature of the vapor. If the chemical po-tentials in Equations (7) and (8) are set equal, an equa-tion in four unknowns results, TW, TS, x, and p. It remainsto show that the equation can be satisfied by values ofthe unknowns that are physically realizable:

0 < TW < ∞TW < TS < ∞ (9)0 < p < ∞0 < x < 1

Figure A1 shows a diagram drawn for a set of param-eters that satisfy Equations (7) and (8), and the inequali-ties in (9).

APPENDIX A. COUNTER EXAMPLES TO THE OTHER DEFINITIONS OF EQUILIBRIUM

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THERMODYNAMICS OF A MAGMATIC GAS PHASE 1327

In Part V of his paper, Verhoogen discussed the pos-sible role of osmotic pressure in igneous processes.Goranson (1937) discussed the osmotic pressure in vol-canic processes using NaAlSi3O8–H2O as a model. BothGoranson (1937) and Verhoogen (1949) suggested thatthe walls of a magma chamber could act as a semiper-meable membrane open to the gas phase but closed tothe silicate melt. However, unless the walls of magmachambers are also insulating, there can be a differencein temperatures across the chamber wall. Consequently,both temperature and pressure differences may obtain,and the composition of the gas phase outside the cham-ber may be different than if osmotic conditions did notobtain. The equations in this section describe differencesin osmotic temperature and are analogous to the equa-tions that describe differences in osmotic pressure. Ageneralized discussion of osmotic conditions that in-clude pressure, temperature and compositional effectshas not been formulated, at least to my knowledge.

FIG. A1. Diagram showing the equilibrium conditions be-tween an ideal aqueous solution and H2O vapor through animpermeable membrane. Pressure, p, is plotted on the hori-zontal axis, and the mole fraction of H2O in the solution, x,is plotted on the vertical axis. The solution and vapor are atdifferent temperatures.

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1328 THE CANADIAN MINERALOGIST

One assumes equilibrium between melt and vapor,and asks how the saturation of a particular componentchanges with the change in concentration of anothercomponent. In particular, one wishes to calculate theeffect of HF (component 2) on the solubility of H2O(component 1).

At equilibrium, one has:

�1gas = �H2O

melt (10)

Finding the differential of each side of equation 10gives:

– –

,

s dT v dP s dT

v dP dnN

N

1 1

1

1

2

2 2

gas gasH Omelt

H Omelt

iH Omelt i

ii

j

n

+ =

+ +=

=

∑∑

µ(11)

where µ H Omelt

j2 , is the modified derivative:

∂µ∂

H Omelt

iP T n

melt

i

N

in n

j

2

1

=∑

, ,

.

If HF and H2O are the only two components thatleave or enter the melt, equation 11 simplifies to:

– –

,

,

s dT v dP s dT v dP

dn n

dn n

gas gasH Omelt

H Omelt

H O H Omelt

H O

i

N

i

H O HFmelt

HF

i

N

i

1 1

1

1

2 2

2 2 2

2

+ = +

+

+

=

=

∑

∑

µ

µ(12)

The final step in the derivation is to convert terms withdni into terms with yi and dyi. The mole fraction of com-

ponent i, yi, is defined as y n ni i

i

N

k==∑

1

. The de-

rivatives with respect to ni and nj are:

∂∂

∂∂

yn y n

yn y n

i

i P T ni

i

N

k

i

j P T ni

i

N

k

k

k

= ( )

=

=

=

∑

∑, ,

, ,

–

–

11

1

(13)

Equation 14 contains four variables as differentials, P,T, nH2O and nHF. The mutual variations in two variablescan be calculated by holding the other two variablesconstant. For example, the variation of the pressure ofsaturation with change in concentration of HF at con-stant T and moles of H2O is given by:

∂∂

µ

ln –

–

, ,

,

1

2

2 2

yP

v v

HF

T n n

H O igas

H O HFmelt

k H O

( )

= ( )(14)

The relations in equation 14 are used to arrive at equa-tion 15.

At constant P and T, one can calculate the change inconcentration of H2O at saturation with change in HFconcentration. The result is:

∂∂

µµ

yy

y

y

H O

HF P T n

H O

HF

H O HFmelt

H O H Omelt

k

2

2 2

2 2

1

1

= ( )( )

, ,

,

,–

–

–(15)

If there is a known relationship between the gain andloss of H2O and the gain or loss of HF, then this rela-tionship eliminates one of the dni, and one of the othervariables can be left in the equation. For example, ifdnH2O = – dnHF, then:

∂∂

ln –1 yP

H O

T,n ,dn –dn

H Ogas

H O, H Omelt

H O, HFmelt

2

k H2O HF

2

2 2 2

v – v

–

( )

= ( )µ µ( )

=

(16)

APPENDIX B. CALCULATING THE EFFECT OF A SECOND COMPONENT ON THE SOLUBILITY OF A FIRST

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