Anerica* Min*atrogistVol. 57, pp. 524-553 (1972)

GIBBS FREtr ENERGY. ENTHALPY. AND ENTROPY OF TENROCK-FORMING MINERAI^S: CALCULATIONS,

DISCREPANCIES, IMPLICATIONS1

E-aN Znn, U. S. Geolo,gical Suruey, Washi,ngton, D. C . 20242.

Ansrnacr

The standard 298'K, 1 bar Gibbs free energy of formation and enthalpy offormation of the following minerals were calculated from reversed hydrothermalequilibrium data: Diaspore, tremolite, zoisite, prehnite, laumontite, wairakite,clinochlore, muscovite, paragonite, pyrophyllite. Calculation was by a procedurepreviously described by the author, and uncertainties, due to uncertainties inthe hydrothermal dafa and in the thermochemical data input, are included.Where entropy d,a+"a did not exist, they were calculated from pairs of hydro-thermal p-? brackets. The zeolites have anomalously high entropy values, re-flecting the highly mobile state of molecular IIzO in the structure. The calculatedGibbs free energy of muscovite is about l0 kcal more negative than calorimetricmeasurements; however, much of this discrepancy could be caused by a grosserror in the basic thermochernical pararneters lor the aluminum silicates (comn-dum, kyanite, andalusite, sillimanite, kaolinite) suggesting error in the thermo-chemistry of corundum. The inconsistency of about 34 kcal/gram atom of Alis implicit in the availa.ble therrnochemical values of aluminum-bearing minerals,but there is at present no a prinri means to decide which sets of data are wrong.Until this point is resolved, the validity of all therrnochemical calculations formineralogical phase reactions involving aluminum is suspect.

INrnooucrrorq

Thermodynamic parameters of minerals can be computed from ex-perimental phase equilibrium data when the data define one or morereliably reversed p-T brackets. Various methods of calculation havebeen proposed in the past (see, for instance, Orville and Greenwood,1965; Robie, 1965; Weisbrod, 1968; Anderson, 1970) ; all those citedexcept Robie used rectilinear graphical fitting of several experimentalpoints. The writer (1969, 1971) previously pointed out that graphicalfitting of data has disadvantages, and advocated a method that usesthe reversed p,-7 brackets as such without any assumption as to howthese brackets might be related to one another. The major drawbackof the method is the need to know most of the thermodynamic param-eters of individual phases (including the volatiles when these are in-volved in the reaction) before the calculations can be made. However,with the steady accumulation of thermodynamic data for rock-formingphases, this method is becoming more widely applicable. One major

l Publication authorized by the Director, U. S. Geological Survey

,Zt:

THERMODYNAMIC PNOPENTIES OF MINERALS 525

advantage of the method is that it allows direct estimation of thermo-chemical uncertainties; another is that any mutual inconsistency ofthe data points is quickly revealed.

Efforts to derive thermodynamic parameters from equilibrium hy-drothermal data received a major boost with the recent publicationof precise p-T-V measurements of HzO over wide p-7 ranges (Burn-ham etol., 1969). The data of Burnham et aI. were converted by Fisherand Zen (1971) to the same reference state used by Robie and Wald-baum (1968) for HzO and for other minerals, so they can be useddirectly to derive further thermochemical parameters. A test of thereliability of the method was made by Fisher and Zen (1971) whocalculated the standard Gibbs free energy of formation of brucite,for which high-precision calorimetric data are also available. The cal-culations used a single hydrothermal p-T bracket, and took into ac-count the uncertainties involved in each step. The resulting Gibbs freeenergy is in excellent agreement with the calorimetric value and showsa comparable determinative uncertainty. This comparison is deemedjustification of extension of the method of calculation.

Uxrts, Svltnols, eNo AeenEvrATroNS

G,o Standard (298 K, 1 bar) Gibbs free energy of formation of aphase from the elements

H,o Standard (298 K, 1 bar) enthalpy of formation of a phasefrom the elementsStandard (298 K, 1 bar) entropy ("Third Law") of a phaseStandard (298 K, 1 bar) entropy of formation of a phasefrom the elements

ASr," Sum of entropy of formation of product solids minus the sumof entropy of formation of reactant solids at T and p

A^9r,"o The corresponding sums referring to the standard state(298 K, 1 bar)

AV " Sum of total volumes of product solids minus the sum oftotal volumes of reactant solids

gibbs/gf Unit of entropy, 1 Gibbs : I calorie per degree, abbreviatedGb (The Gibbs/gf is commonly denoted by the unitlessunit e.u.) For original definition, see Giauque et al., 1960

calfbar Unit of volume, l.caI/bar : 41.842 ccT Temperature in Kelvins (K)p Pressure in bars (b) or kilobars (kbar)T " Temperature at which an univariant reaction is at equilibriumP" The corresponding pressuregf Unit gram-formula weight

sosro

'Xt E-AN ZEN

' THsnuocHEMrcAL Ce;,cur,erroNs

Details of the prbcedure of thermochemical calculations from hy-drothermal data are given in Zen, 7971 and Fisher and Zen, IgZt.The sources of information in each set of calculations are given hereunder each appropriate section and also in Tables 1 and 2, or thedata are taken from Robie and Waldbaum (1968). The data of Robieet al. (1967) fdr volumes were used insofar as possible to insureinternal consistency.

Where sufficient numbers of reversed p-? brackets exist for a givenreaction, it becomes possible to make several estimates of the thermo-chemical parameters. For instance, if there are ?? separate brackets,(re-l) independent evaluations of the entropy of a phase are possible,using the relations in %en,1971, eq. 12. In practice, all n points areused in the (z-1) calculations, but pairs of points are selected so thatthey are remote from each other in order to minimize errors causedby small differences. If the discrepancies between separate estimatesare not excessive, they can commonly be resolved. The method ofresolution is described in the section on "Errors".

What corstitutes an acceptable value of the entropy for a phaseis of course a matter of judgment where no independent measure-rnents are available. Occasionally, despite any permissible adjustmentthe calculated entropy may turn out to be near zero or even negative;such results are prima facie evidence of poor experimental data andthe calculations are abandoned. In practice, entropy values sornewherenear the value estimated by the "oxide sum" method (Fyfe et al., 19'58,p.25), using a value o,f 10 Gb/gf for H2O', are considered reasonable(ibi.d., p. 117). For phases of high density, values less than this summay be expected; for zeolites having molecular HzO, values consider-ably greater than this sum are reasonable.

When the entropy values of all phases of a given reaction are ob-tained, it is a simple matter to calculate the standard Gibbs freeenergy of formation of the phases from the elements, the standardstate being at 298 K and 1 bar, using the relationship (Fisher andZen,1977, eq.8):

AG(TE,P') : 0 : AGr,"o(298, 1)

- ['" Asr," d? + ['" av.dp * Gu,o*(Tr, pr)J zsA .11

The standard enthalpy of formation is obtained from the definition forGibbs free energy,

G = H - T S .

5nTHERMODYNAMIC PROPERTIES OF MINERALS

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530 E-AN ZEN

The differences between the sum of entropies of formation of theproduct solid phases and that of the reactant solid phases, ASy,", is avery slowly varying function of temperature and also of pressure(Fisher arrd Zen, 1971, p. 299). Therefore, good estimates of Gro for aphase can be obtained by approximating the integral JA&," d,T by thequantity ASr,,o A?. The difference between the two, where data allowestimation, is not more than about I kcal even in the worst case. Theapproximation will be called the "constant-entropy" method. Wherehigh-temperature heat, capacity data do exist, for instance from thetabulation of Robie and Waldbaum (1968), a more accurate calculationis possible. Then the integral JA&," dT can be replaced by a sum,Ior aSr- A? in which the mean value, AS* is obtained from entropy-of-formation values for successive even-hundred degree entries. This willbe called the "summation" method.

To facilitate calculation, the Third Law entropy values for commonrock-forming minerals, as listed in Robie and Waldbaum (1968) foreven hundred degrees K, are converted into entropy-of-formationvalues. The results are given in Table 1, in units of Gibbs/gf. Theanthophyllite data are from Mel'nik and Onopriyenko (f969). Themineral formulae are those of Robie and Waldbaum; for anthophyllitethe formula is based on 24 oxygens.

The pressure-volume contribution to the calculations of the energyand entropy-of-formation is based on the assumption that the integralttV"dp can be replaced by the term AZ'AP. The approximation isreasonable because the volume term itself is always a small contribu-tion, and the difference between the volumes of the solid product,phases and the solid reactant phases is largely independent of tem-perature and pressure. For example, an unusually large value of A7"might be I cal/bar; for a pressure range of 3 kbar this would leadto a contribution of 3 kcal. Even as much as a l0 percent error in thevolume data would lead only to a Gibbs free energy error of 300 cal.If a temperature difference of 50 deg is associated with the 3 kbarpressure difference, the corresponding contribution to the entropywould be 6O gb/gf, and a 10 percent error in volume would lead to6 sb/sf of etror, which is significant but not fatal. The compressibilityof solids being on the order of 10-G per bar, it seems improbable thatthe pressure correction for the differences of volumes of reactants andproducts could amount to 10 percent of a7s; the effect of elevatedtemperature on ,AZs tends to offset the pressure effect.

To be rigorous, one should carry out the temperature and pressureintegrations in sequence, e.g., frorn 298 K, I bar to 7u, I bar, thenfrom 1 bar to prbar at ?u. Our assumption that the volume integrand

THENMODYNAMIC PROPENTIES OF MINERALS 531

can be replaced by a constant term amounts to stating that the resultof the calculation is independent of the sequence of integration.

In all the calculations, the fluid phase is assumed to be pure HrO sothat GH"s : lrs"o. The various paths of salculation are summarizedin Figure 1.

Ennons

Estimating the uncertainties associated with each calculated ther-mochemical value is a complex problem, involving sources of errorof different kinds. One property of the p-T brackets is that it is astep function: Within the bracket, there is equal chance that the trueequilibrium p, T value lies anywhere, but outside the bracket thechance is, by the definition of limits of the brackets, zero. Thus thedata do not represent conventional distribution function, and standardstatistical procedures do not apply.

The experimental results are commonly reported in terms of anlower temperature, 71, which is the lowest observed temperature atsome pressure at which the endothermal reaction proceeded. Thereis a corresponding upper temperature, 7,.r. To each temperature is as-sociated an uncertainty in measurement, 8?1 and 87p; the true bracketlies within the range of. T1- 8?1 and Tz + 8?2.

Thus, to each p-T bracket is associated flrst a thermochemical un-certainty because the true location of p", Tu values is unknown. ForGibbs free energy Gyo, this uncertainty is readily calculated by com-paring the Gyo at the limits of the bracket with the calculated value.Second, there is the uncertainty in G10 associated with the input valuesfor the free energy, volume, and entropy of the individual phases.This uncertainty is readily calculated (Fisher and Zen, 1971), buthow to combine the two types of uncertainty is not obvious becausethe first source does not involve normal distribution. I have simplyadded the two sources together.

In this way, when the Gibbs free energy of formation is calculated forthe same phase from different brackets, there results an uncertainty foreach calculated value. The "best" value as well as the associateduncertainty are computed as follows. Let the uncertainty associatedwith a particularvalue Go be g;, and letw : l/go".Then the "best"value of Go, or G, is dn : Eo=r* Gnwr/Z, w; and, tnte "best" estimateof the uncertainty, 0, : (/Er-ru ,,)"'(see Mandel, !964, p. 132 fr;I am much indebted to G. M. Anderson for suggesting using thisapproach). I have adopted the two-a convention in reporting uncer-tainties (Robie and Waldbaum, 1968). Calculation of the uncertaintyassociated with the standa,rd enthalpy of formation from the Gibbs

Experimentalp, T data

Use swation

subs t i tu te in toequation fore a c h , p . Tbracket

Use constantentropy nethod,subs t i tu te . in toequation foreach p, T bracket

Use pairs ofbrackets todetemine { , "

Check againstoxido sm forreasonableness

Find "consistentrlp, T values to usein Gt calculations

632 E-AN ZEN

Fro. 1. Scheme of calculation of thermodynamic parameters for minerals fromhydrothermal equilibrium data.

THERMODYNAMIC PROPERTIES OF MINERALS 533

free energy and entropy data simply follows the conventional procedurefor combining sums; see Fisher andZen, 1971.

Estimating the uncertainty in entropy calculations is more difficultbecause each calculation involves pairs of. p-T brackets. Use ofbrackets that are not adjacent to each other is preferred, as suchpairs tend to reduce the uncertainty. For each pair, a maximum valueand a minimum value of the entropy of formation of a phase canbe computed, and to each extremal value can be attached an errordue to uncertainties in the data input. In practice, for each pair ofp-T brackets, one gets a mean value of the entropy directly, and thespread of values gives an estimate of the uncertainty. From thevarious pairs of p-? values, then, the "best" entropy-of-formationvalue and the "best" uncertainty can be calculated by the formulaegiven above. From the value of entropy of formation thus derived,the "Third Law" entropy value, So, is obtained. The uncertainty dueto the uncertainties of the entropy data input, is obtained from theroot-mean-square of the uncertainties of Lhe Third-Lau entropiesof the solid phases plus those of hydrogen and oxygen gas, takinginto account the stoichiom,etric coefficients of the reaction.

The determination of a "best" value of the entropy of formationof a solid phase allows estimation of an internally consistent set ofp and T values for the univariant equilibrium. This is done by plotting,for each pair of brackets, the entropy values against p and T, andreading off the plot the values of p and 7 corresponding to the "best"entropy value. Any remaining discrepancies arising from the use ofdifferent pairs of brackets are resolved by taking the mean values.These adjusted p and ? values are automatically within the experi-mental p-? brackets and are used in the G7o calculations.

Entropy values that do not permit such treatment, because the

"best" value falls outside the range obtained from one or more pairs

of p:7 brackets, are handled as individual problems and discussed inthe appropriate sections.

Rnsur,rs oF CAr,cuLATroNS

Diaspore

Haas and Holdaway (1970) studied the dehydration of diasporeaccording to the reaction

2 H A l O r : A l , O s + H , Odiespore ooruDdum v&por

The p-T brackets are: (1) 398 :t 5"C, 1.75 kbar; (2) 409 t 5oC,

5M E-AN ZEN

2.4 kbar; (3) 420 :t soc, 3.5 kbar; (4\ 428 -+ 8oC, 4.8 kbar (thefirst and last points as well as the uncertainties were given by Haasduring oral presentation of his paper). Using the mean temperatures,the data of Tables 1 and2, those in Robie and Waldbaum (1968) andthose in Fisher and Zen (1971) , G1o of diaspore was obtained fromeach of the brackets; the constant-entropy approximation was used.The mean value is -2lg.g3 + .38 kcal/gf diaspore; the uncertaintyincludes about 50 cal for the temperature brackets. The actual valuesof the four brackets are respectively -219.98, -220.01, -219.91, and-219.81 kcal. The H10 of diaspore calculated from the Gibbs freeenergy value is -238.69 :t 0.38 kcal. The G10 value may be comparedwith the values given by Wagman et aI,. (1968) and by Fyfe andHollander (1964), -220 kcal.

Trem,ol;ite

The upper thermal stability of magnesian tremolite was studied byBoyd (f 959) . The reaction is:

Ca,MguS_ieO,,(OID, : 2CaMgSi,O. + t$.*.?,*r + $19:

+ H,O

The univariant, curve passes through the following po'ints: (1) 810'C,0.6 kbar; (2) 835'C, 1 kbar; (3) 855'C, 1.5 kbar; and (4) 870"C,2kbar. These values were obtained by interpolation from the p-T dia-gram of Boyd; I estimate the temperature uncertainty to be 10'C.

Usingthe data of Tables l and 2, of Robie and Waldbaum (1968),and of Fisher and,Zen (1971), and using the summation method forthe entropy term, the Gyo of tremolite has a mean value of -2775.20! 2.62 kcal; the actual values are respectively -2775.46, -2775.22,-2775.W, and -2775.03 kcal. To the uncertainty associated with thedata input has been added a maxirnum of 0.2 kcal for the temperaturebracket. The large uncertainty of data input reflects the large un-certainty in the Gibbs free energy of diopside, 2.2 kcal per gram-formula. The Ilyo of tremolite at 298 K and I bar is -2949.00 'r 2.64kcal.

The Gibbs free energ' and enthalpy values are to be comparedwith the data given by Robie and Waldbaum (1968) , resp. -2,779)137:t 4150 cal and -2,952,985 L 4l4A cal. These values and my resultsshow acceptable overlap. In the present, calculations, the Gibbs freeenergy and entropy values of clinoenstatite have been used, eventhough in the experimental work enstatite was the phase encounteredlthe approximation was necessary because no thermochemical dataexist for enstatite. One might expect the Gibbs free energy for ensta-

THENMODYNAMIC PROPENTIES OF MINERALS 535

tite to be more negative than for clinoerrstatite at row temperatureand pressure, so the calculated tremolite values are apt to be toopositive; the deviation is in the right direction for the discrepancybetween the present calculated value and the direct thermochemicardetermination given by Robie and Waldbaum.

The Gibbs free energy value for diopside in Robie and Waldbaum(1968) was derived from the arithmetic mean of two independentvalues for the heat of solution of this phase. one, determined on aniron-bearing natural sample (Robie and Waldbaum, 1g68, ref. 116),leads to a Gyo of -724 534 cal/g|; the other, on an artifi.cial sampleof unspecified composition (Robie and Waldbaum, 1g68, ref. gb), leadsto a G1o of. -727 044 cal/gf .If the more negative value for diopsideis used in the preceding computations, a Gf for tremolite of. -2ZZZ.Zkcal/gf would have resulted, in much better accord with the varue inRobie and waldbaum. ,However, in view of the compositionar uncer-tainties associated with both diopside sampres, the agreement isprobably illusory. This feeling is reinforced by the fact that thetremolite used by weeks (1950) in his enthalpy determination showedlarge compositional departure from the ideal formula. Therefore, Ihave simply used the rather artiflcial mean Glo value for diopside,and also retained the large uncertainty, as given by Robie and Wald-baum.

ZoisiteThe stability relations of zoisite were studied by Newton (1966).

The reaction I analysed is the upper stability of zoisite in the pres-ence of quarlz:

nc*Al:l*9',oH * ll9: : uc*llP,io'+ "?.'.*P.','g',

* ?*.?.oNewton used different types of experimental apparatus to achieveequilibrium under different sets of p.-T conditions, but applied pres-sure correction for the results, so r have presumed that his reportedp-? values are reduced to the same datum. The experiments werecarried out with all solid reactant and product phases present, so therelative growth or diminution of phases is a good measure of thedirection of the reversible reaction. The p-T values cited below aretaken from Newton, 1966, tables 2 *ndB: (1) 610-640oC,4.6 kbar;(2) 650'C, 5.G5.9 kbar; (3), 650-700'C, 6 kbar; (4), ZZA"C, Z.a-Z.Bkbar; (5), 750-770oC,8 kbar. The final p-? values adopted, usingthe method of deriving the mean entropy values explained in the"Errors" section, are, respectively, dl8oO, 4.6 kbar; 650oC, b.6 kbar;670oC, 6 kbar; 720"C,7.2kbar;765,oC, 8 kbar.

536 E-AN ZEN

With five p-? brackets, four independent estimates of the entropyof zoisite were'made. The pairs of values selected were': 1, 4; l, 5;2,4; and.3,5 of the above list. The S1o has a mean value of -319'54

Gb/gl, the mean uncertainty is 3'4 Gb/gf. The entropy obtainedis 67.9 -+ 3.9 Gb/gf, taking into account the uncertainties of the

thermochemical data input. The value may be compared with the

"oxide-sum" estimate, which is 72.3 Gb without applying any volume

correction. The entropy value of HzO used in the estimate was 10

Gb/gf (Fyfe et aI., !958, p. 117). The agreement is good consideringthat zoisite is a fairly dense phase so its entropy should be less than

the oxide sum.From the S1o of zoisite, the G1o of zoisite is calculated, using the data

of Table 2 and assuming that ASlp of the solids is a constant. The re-

sult is -1552.66:L 1.4 kcal per gram formula of zoisite. The actualyalues for the five p-T brackets are respectively -1552.72, -1552'57,-1552.62, -1552.62, and -1552.8O kcal. The Hf of zoisite from the

elements is -1647.93 :t 1.8 kcal. No uncertainty due to the T or p

brackets is attached because this is already taken into account in the

range of entropy values.

Prehnite

The thermal stability of prehnite was studied by Liou (1971a) '

The reaction is:

Ca,AlzSieO,o(OID, : CaALSi,Oe * CaSiO' + H'Op r e h D i t e e n o r t b i t e w o l l a s t o n i t o v o D o r

:L 0.003 calfbar.Four independent estimates of the entropy of prehnite are possible

with the five sets of. p-T data. I used the combinations o'f points

of 10 Gb/gf.using the mean entropy of formation value given above, the G10

of prehnite aL 298 K, 1 bar from the elements is next calculated to

THERMODYNAMIC PROPERTIES OF MINERALS i)lJ /

be -1389.82 ! I.71kcal. The actual values are respectively -1389.83,-1389.79, -1389.80, -1389.81, and -1389.88 kcal. The H10 ol prehniteis -1479.99 :L 1.86 kial. As for zoisite, no uncertainty due t"o lhe p-Tbrackets is given because the uncertainty due to the entropy term isa better estimate of the same phenomenon.

Wairaldte

The upper stability limit of the zeolite mineral, wairakite, wasrecently studied by Liou (1970). The reaction is:

CaAlzS"LO','.2H,O: Cr,*.t;P,i:O. + ,"P.t"|,, + ?*9Liou gave the following reversed p-T brackets: (1) 330:L 5oC,0.5kbar; (2) 348 :t 5,oC, 1' kbar; (3) 372 -+ 5,"C, 2 kbar; and (4) 3S5-f 5"C, 3 kbar. For reasons discussed bblow, the mean temperaturesare used in my calculations and no adjustment for internal con-sistency has been made.

Liou (1970) also gave numerous unit cell parameters for the waira-kite prepared in his experimental work and from natural occurrences;these pararneters lead to closely agreeing cell volumes. From these,an avera,ge value of 4.558 r- O.l cal/bar-gf was obtained (Table 2).

From the data cited above and given in Table 2, three independentestimates of the entropy of wairakite are possible. From the pairsof points 1, 31 1, 4; and 2,4, Lhe respective ranges are -336.38 to-336.90; -335.78 to -335.80; and -335.30 to -339.85. The valuesdo not all overlap, but the discrepancy is not significantly large. Theadopted mean value of Sio is -335.79 -+ A.2 Gb/Cf ; the narrow un-certainty range is deceptive and results from the values of the pairof points 1,4. The entropy of wairakite aL298 K and 1 bar is f10.86+ 1.0 Gb/gf, where an arbitiary but more realistic estimate of theuncertainty has been given.' The entropy value is considerably greater than the "oxide surii4estimate of 81.2 Gb,/gf when the entropy contribution of HzO is takento be 10 Gb/gf . Part of the unusually high entropy of wairakite canbe attributed to the low density of the phase (see Fyf.e.et aI.,1958,p. ll7 for discussion). However, the bulk of the anomaly mrist beassociated with the locisely attached zeolitic water in the structure.If we assign all the "excess" entropy to such a cause without cor-recting for density, we get 15 Gb/gf of H2O, so that the HzO con-tribution to the wairakite entropy would be about 25 Gb/gf , a valuegreater than that of liquid water, 16.7 Gb/gf,

The eomparison may seem speculative because of uneertanties in

538 E-AN ZEN

the validity of the computed entropy value itself, or in the possibilitythat the wairakite equilibrated in Liou's experiments was not stoichio-metric, especially in its HzO content. Howevet, the same effect is foundin the entropy of analcime, to which wairakite is structurally analogousexcept for slight distortions that reduce the symmetry from cubic tomonoclinic. The entropy of analcime, NaAlSizOo' H2O, calorimetricallyobtained by Kelley and King (1961; see Robie and Waldbaum, 1968),is 56.03 Gb/gf at 298 K and 1 bar. The oxide sum Yalue' using 10GVSf for H2O, is 44.8 Gb/gf, and the deviation is 11 Gb/gf, which re-sembles the 15 Gb/gf excess computed for wairakite, and the HzOcontribution to entropy likewise is higher than the entropy of liquidwater. A dehydrated wairakite could be expected to show anomalouslylow entropy instead. It appears that in both wairakite and analcimethe molecular HzO exists in the structure in a highly mobile state,having fewer or looser bonds to other HzO molecules than in theshort-range structured liquid water.

Because of the lack of overlap of entropy values, no internallyconsistent set of adjusted temperatures is possible. The mean tem-peratures reported by Liou (1970) therefore were used directly forGibbs free energy calculations. The deviations from this cause arenegligible. From the mean entropy value, the Gyo of wairakite iscomputed to be -1,477.29 !.1.54 kcal; the individual values of thef.ov p-T brackets are -1477.30, -1477.28, -1477.26, and -1477'32

kcal. The Hyo of. wairakite is -1577.41:t 1.63 kcal.An independent check of the thermochemical parameters for waira-

kite can be obtained because another reaction involving this phasehas been studied:

CaAl, S"inq,, ..2H,o : ca-A.t Si,.gl"qgP, . H,o + r"P"i9,,

This reaction, involving only solids, has been studied by Liou (1971b)who gave two reversed points: 305oC,3.4 kbar, and 39OoC, 4.4kbar.These points give an entropy value of 90 Gb/gf for wairakite, which ismuch too low compared with the value of 110.9 Gb/gf obtained fromthe wairakite-anorthite reaction. Because the control on the pointsis inferior to that on the wairakite-anorthite reaction this value isrejected, even though it is still reasonable compared with the 81Gb/gf estimate from the oxide sums.

If we used the value 110.9 Gb/gf for wairakite, the Gyo of waira-kite can be calculated from the wairakite-lawsonite reaction. The meanresult is -7479.3 kcal, which is about 2 kcal too negative comparedwith the data obtained from the wairakite-anorthite reaction. Using

THERMODYNAMIC PROPERTIES OF MINERALS 539

the entropy of wairakite derived from the wairakite-lawsonite re-action would make this value about O kcal more negative.

Lau'm"ontite

Laumontite has been experimentally studied in recent years byCrawford and Fyfe (1965), Nitsch (1968), Thompson (1970a), andLiou (1971b). All these studies pertain to the reversible reaction:

CaAlzSinO,r.4HrO + Ca,Al,Si,O?(OH),.H,O + 2SiO, + 2H,Oquartr vapor

Because of large volume differences between laumontite and law-sonite, Lhe p-T projection of this univariant reaction is nearly parallelto the temperature axis. The results of Thompson agree with thoseof Nitsch, but the results of Liou lie at a slightly higher pressure.The curve of Crawford and Fyfe (1965) is discordant with all theothers and will not be considered.

A second reaction involving laumontite was also studied by Liou:It is the breakdown of laumontite to form wairakite:

CaAl,SinO,r.4H,O : CaALSiO,,.2H"O + 2H"OlsuEont i tg w e l r s k i t e vapor

We will examine these two reactions to deduce thermochemical pa-rameters for laumontite, and then compare the results from Liou'sdata with those to be obtained from Nitsch's and Thompson's data.

tr'or the laumontite-lawsonite reaction, Liou (1971b) gave lwo p-ybrackets, at 210 t 5oC, 3 kbar, and 250 -f 5oC, 3.2 kbarl. Usingthese values, the poorly-controlled Syo of laumontite becomes -440.6Gb/gf, corresponding to an entropy of 118 Gb. The Gyo of laumontiteis then -1600.1 kcal, and lhe H1o, -1731.7 kcal. No meaningfulestimate of uncertainties is possible.

Liou (1971b) also gave five reversed p-T brackets for the dehy-dration of laumontite to form wairakite. Using the data previouslyderived for wairakite, an independent set of values for laumontitecan be derived. The p-T brackets are: (1) 235 t 5'C, 0.5 kbar; (2)255 t 5oC, l kbar; (3) 282 -+ 5oC,2 kbar; (4) mZ L 5"C,8 kbar;and (5) 327 ! soC, 6 kbar2. A problem arose when pairs of bracketswere used to calculate the entropy of laumontite. The entropy rangesfor different pairs are wide, and different ranges do not overlap; the

lNote that the ?-value for the 3.2 kbar point given in the abstract (Liou,1971b) is apparently in error.

'Again, note that the yT values in the abstract (Liou, lg7fb) are not thesame as thoge in the main text.

I &umont i te l e w a o a i t o

540 E-AN ZEN

entropies calculated seem to become more negative as the p-? valuesof the brackets increase. This fact suggests some systematic experi-mental error; as the calculation of wairakite did not lead to such atrend, the suspicion is that the laumontite compositions may havediffered systematically in Liou's experiments. Using the pairs ofbrackets 1, 3; 1, 4; l, 5; and 2,5, however, the calculated S70 is-439.0:t 4.6 Gb/gf; the uncertainty is formal and obviously toolow because the individual values range from -424 lo -a57 Gb/1f..It is interesting to,note, though, that the mean value is nearly sameas the value obtained frorn the laumontite-lawsonite equilibrium. Theentropy of laumontite at 298 K, 1 bar is 119.3 ! 4.6 Gb/gf..

The lack of a consistent entropy made it necessary to calculatethe G10 of laumontite using the mean p-7 values. The G10 values are-1596.46, -1596.68, -1596.97, -1596.82, and -1595.98 kcal; thearlerage value is -1596.6:t 1.8 kcal, and Lhe H1o -1727.5 L 2.2kcal. It must be stressed again that the data above are not "con-sistent" in the sense used in this paper. Ifowever, the values derivedfrom the two reactions by Liou do overlap within the uncertainties.

Thompson (1970a) determined a p-T bracket for the reactionlaumontite : lawsonite * quarbz * vapor by using the weight-lossmethod, at"25A"C,2.75 ! 0.25 kbar. If we accept an entropy of 119.3Gb/gf for laumontite from Liou's study of the same reaction (an as-sumption that seems reasonable because the slopes of Liou's curve andthe one preferred by Thompson (1970a, p.271) are parallell note thatThompson's slope was obtained by circuitous extrapolation frorn Craw-ford and Fyfe's (1965) data), we obtained the G70 of laumontite of-1599.5 kcal, in fair agreement with the values obtained from Liou'sdata, despite the fact that Thompson's p,-7 value is obviously atlower pressure than Liou's curve for the same reaction.

A second univariant reaction studied by Thompson (1970a) wasthe direct dehydration of laumontite to anorthite:

CaAl,SinO',.4H,O : CaALSLO8 + 2SiO, + 4H,Olaumont i t € anor th i te qu&rlz vapor

for which f.our p-T brackets were given: (1) 310 -+ 10oC, l kbar;(2) 317 I 10oC, 2 kbar; (3) 338 -+ 10oC, 4 kbar; and (4) 347 t10oC,6 kbar. Using the pairs 1,3; 1,4; and 2,4,I obtained a straightariihmetic mean S1o of laumontite of -454.9 i 6.3 Gb/1f., and arange from -444.3 to -629 Gb,/gf. The wide range reflects in partthe narrow temperature difference between brackets and the formaluncertainty. is meaningless. However, the mean value of entropy im-plies an entropy of 103 Gb/gf, which is low in view of the fact thatthe oxide-sum estimate, using 10 gb for H2O, is 104 Gb, and a zeolite

THERMODYNAMIC PROPERTIES OF MINERALS 541

should certainly have an entropy exceeding the oxide sum. The valueof -444 Gb/gf would give an entropy in close agreement with thatderived from Liou's data. Using this last value gives a G1o of. laumon-tite of -1599.5 kcal. The Gibbs free energy is not sensitive to theentropy (hence, p-? slope) values and the closely agreeing valuesshould be fairly reliable. The recommended value and uncertaintyare given in Table 2.

The entropy of 119.3 Gb/gf for laumontite from Liou's data leadsto an "excess" entropy of HzO of 4 Gb/gf, much smaller than that inwairakite or analcime. The magnitude of the "excess" agrees with thatof leonhardite (Robie and Waldbaum, 1968), which is equivalent toa laumontite that has lost 1/8 o! the molecular H:O: for leonharditethe anomaly is 3.3 Gb/gf H2O.

This discussion underscores the problem of hydrothermal phase

equilibrium studies involving zeolites which could have variableamounts of HzO: How would one ascertain the amount of HzO in

the zeolite along a particular "univariant" reaction curve, and how

would one ascertain that the amount remains uniform for all experi-ments that defi.ne the curve? As far as I know this problem has notbeen tackled, and to that extent all existing hydrothermal data involv-ing zeolites are of ambiguous significance. On the basis of the preced-

ing discussion, one might conclude that the consistency of Liou's datausing laumonitie-lawsonite (not a zeolite) relations' and using lau-montite-wairakite relations, and the consistency of his data usinglaumontite-wairakite relations and using wairakite-anorbhite relations,suggest that he indeed dealt with the same zeolites having the samestates of hydration. Comparison of the anomalous entropy values withthe oxide-sum estimate and with the values for leonhardite suggestsfurther that Liou probably dealt with the fully hydrated laumontiteas well as wairakite, whereas Thompson may not have.

Nitsch (1968) did not publish his p-7 data for the breakdown oflaumontite to lawsonite, and his figure showing the experimentaldata does not allow aecurate estimate. As his data agree closely withthose of Thompson (1970a, p.273), no further calculation seems war-ranted.

Cl;inochlore

The thermodynamic parameters for the low-Al, pure Mg-end mem-ber of chlorite, clinochlore, have been estimated from the breakdowncurve reported by Fawcett and Yoder (1966) according to the reac-t i on :

Mg"ALSIO,.(OH), : Mg,SiOo l2MgSiOs * MgAl,On + 4H'O

542 E-AN ZEN

Unfortunately there is only one good reversed point, at 831 :L OoCand 10 kbar, and a questionably reversedl point aI 785:t 15oC, 5kbar. Accepting these Lwo p-T brackets at their face values, one couldget an essentially uncontrolled estimate of the entropy of clinochlore.Using the mean temperatures, we obtain a S1o of -|aT Gb/gf., or a Sovalue of 85 Gb/gf. This value seems low because the oxide-sum esti-mate is about 114 Gb/gf. Next, I adjusted the b kbar temperatureto that of the lower limit, at 770"C, and the 10 kbar value to theupper limit at 837'C. These adjustments result in a value of Syo of-523 Gb/gf, corresponding to a So value of 109 Gb/gf, which is morereasonable. The corresponding G1o of. clinochlore is -1974 kcal, andthe,Fllo is -2130 kcal.

These values are very rough, as their derivation pushed the meagreexperimental data beyond warranted limits. Ilowever, consistent withthe experimental data, the entropy is a maximum measure, and theGibbs free energy and enthalpy, correspondingly, the most positivevalue. Therefore, it is of interest to compare the Gibbs free energyvalue with the only other published estimate (Helgeson, 1969, p. 784)for Mg-chlorite derived from solubility measurements made by Mac-kenzie and Garrels (1965). The value of Third Law entropy was esti-mated by Helgeson to be 112 Gb/gf, and the G1o was -1954.8 kcal,nearly 20 kcal less negative than my results. The difference coulddrastically affect predictions of course of phase reactions involvingchlorite. Because in Mackenzie and Garrels' work (1g65, fig. 1) equi-librium was approached only from the supersaturation direction,Helgeson's Gibbs free energy value is, if anything, too positive, andthe value of -1974 kcal mav be a better estimate of the true value.

Muscouite

The Gibbs free energy of formation of muscovite was determinedby Barany (1964), using conventional calorimetry, and by Reesmanand Keller (1965), using solubility measurements. The standard valuesat 298 K and 1 bar are, respectively, -1330.1 kcal and -1328.7 kcal.Routine calculations using reversed hydrothermal equilibrium data(Velde, 1966; Day, 1970), however, revealed unexpected inconsist-encies.

r The question is prompted by two considerations. First, oxide mixtures ratherthan the product/reactant phases were used in some runs, thus delineating syn-thesis rather than stability fields; second, the plotted points in Fawcett andYoder's Figure 2 do not all correspond to the runs listed in their Table 2. Thediscrepancy is not explained, and includes the questionable 5-kbar point.

THERMODYNAMIC PROPERTIES OF MINENALS 543

Velde (1966) studied the equilibrium:

*t'S::91:5orDz : S"lP,ilo- +"*l:k+ rroHe gave frve p-T brackets forthe equilibrium: (1) 560 i 10"C, 1 bar;(2) 620 -+ 10oC, 160 bars; (3) 660 -t- 10oC, 1 kbar; (4) 688i 13'C,2 kbar; and (5) 730 i 10oC, 8 kbar. As the entropy of muscovite hasbeen calorimetrically determined to high temperatures, no separatedetermination of the entropy was attempted. Rather, the entropy termwas calculated by the summation method. Using these values, those ofTables 1 and 2, and those of Fisher and Zen (1971)., the G10 of mus-covite turns out to be, in order of. p-T listing above, -1348.1 kcal,-1341.9 kcal, -1340.2 kcal, -1340.0 kcal, and -1338.1 kcal. Thespread is 10 kcal, much greater than expected.

In corresponding with Velde, it was brought out that the point at 1bar is quite unreliable, and so it is disregarded in the ensuing dis-cussion. The four remaining Gibbs free energy values are in a muchtighter grouping, averaging at -1340.1 :t 1.5 kcal, including the un-certainty due to the temperature brackets, less than 0.2 kcal. Usingthe mean value, a Ilo is computed to be -1431.1 + 1.5 kcal.

Anderson (1970) also discussed the problem of deriving consistentthermodynamic parameters of muscovite from hydrothermal data,using the work of Velde as an example. Anderson derived an internallyconsistent univariant curve, and showed that one such curve passesthrough the following four p-7 points, all within Velde's stated uncer-tainties: 620oC,0.16 kbar; 655oC, I kbar; 680oC, 2kbar; and 729oC,8 kbar. Using these values, the corresponding G7o of muscovite are-f341.9 kcal, -1340.4 kcal, -f339.8 kcal, and -1338.1 kcal, with amean of -1340.0 kcal.

There is no sensible difference between these values and those ob-tained by using the means of Velde's temperature brackets, againshowing the general insensitivity of calculated Gibbs free energies tothe precise location ol lhe p-T points. The persistence of a 3 kcal spreadin the Gibbs free energy value in using Anderson's adjusted data, how-ever, remains a puzzle. Polymorphism of the feldspar or muscovite atdifferent p-? brackets could be a cause of the data scatter, but this isentirely speculative, and its seems doubtful that polymorphism couldaccount for 3 kcal of discrepancy. Use of the assumption of constantA7" is not the cause, for even at a pressureamounts to only about 1 kcal.

Next, I used the data of Day (1970) forcovite in the presence of quartz:

of 8 kbar the pI/ term

the breakdown of mus-

544 E-AN ZEN

zut'il:9ll5oH)' * Sl9; : *li3o' +"*,t*l,o* + H'o

Day gave lhree p-T brackets at 595 -+ 15oC, I kbar; 649 :t 10'C,2 kbar; and 662 + 6oC, 3 kbar; he did not otherwise specify experi-mental methods or other details. Nonetheless, using the same methodapplied to Velde's data, the G.o of. muscovite are computed to be,respectively, -1339.8 kcal, -1340.1 kcal, and -1339.9 kcal, with amean at -1339.9 * 1.7 kcal. The 1110 of muscovite is -1431.0:t 1.7kcal. These values are in excellent agreement with those derived fromVelde's data. The uneertainty due to the temperature bracket is atmost 0.4 kcal, which is to be added to the uncertainty of 1.5 kcal fromthe thermochemical data input.

Both Velde's and Day's data, point to a Gibbs free energy (and anenthalpy value) for muscovite which disagrees with that determinedby Barany, -1330.1 kcal, and by Reesman and Keller, -1328.7 kcal.One's first thought is that the entropy value for muscovite, determinedby Weller and King (1963), might be questioned because the chemicalanalysis of the sample they used showed 3.2 percent FezOa, 0.5 percentFeO, and 0.8 percent MgO, which were not corrected for in their cal-culations. However, this cannot be the major source of discerpancy be-cause the same entropy value was used in my calculations. The nextpossibility is that despite its unlikelihood, microcline was the realequilibrium phase in the hydrothermal work. To check this, I recal-culated Day's results by assuming equilibrium with microcline. Theresulting G1o of muscovite is -1337.4 -r 1.4 kcal, and the I/1o is-1428.5 * 1.5 kcal. The smaller uncertainty reflects the fact that theentropy of microcline was measured whereas that of sanidine wascalculated (Waldbaum, 1968). The values are somewhat closer to thecalorimetric value, but the gap is still wide. We will come back to thisproblem in a later section.

Paragonite

The upper thermal stability limit of paragonite was recentlystudied by Chatterjee (1970). The reaction is:

NaAl3S.iso.o(oH), : *r;4.,,l$l'o' +"*."kPJ + II,o

Chatterjee reported six reversed p-T brackehs as follows: (1) 540 :t10oC, 1 kbar; (2) 565 i 10oC,2 kbar; (3) 590 i 10oC,3 kbar; (4)633 i 8oC, 5 kbar; (5) 635 i 15oC, 6 kbar; and (6) 660 =L 10oC,7 kbar. It is obvious, from points (4) and (5), that the mean tempera-tures are not mutually consistent. Using the mean for the entropy, thefinal adjusted tempertures adopted are, respectively, 541oC, 567"C,

THERMODYNAMIC PROPERTIES OF MINERALS 545

587"C, 630'C, 641oC, and 661oC. These points still show some incon-sistency (mainly point 5) but no further refinement was attempted.

The entropy of formation of paragonite was next calculated, usingthe pairs of points 1,6; 1,5; 2,4;2,6; and 3,6. The calculation assumedthat high albite was the equilibrium phase. Chatterjee (1970) gaveno information on this point, but subsequent examination (N. D.Chatterjee, 1971, oral communication) showed that the cell parametersof the product albite ale nearly those of high albite. The mean valueof S1o of paragonite turns out to be -305.08 :L 4.6 Gb/gf and a spreadfrom -298.5 to -314.7 Gb/gf. The implied entropy 80 of paragoniteis 66.1 t SGb/gf, which uncertainty includes those due to data input.The oxide sum, using 10 Gb/gf for HzO is 67.6 Gb/gf.

Using the mean entropy value and again the data for high albite,the G10 of paragonite was next computed. The mean value is -1328.37i 1.9 kcal, and the individual values are respectively -1328.32,-1328.34, -1328.39, -1328.47, -1328.34, and -1328.42 kcal. TheH1o of paragonite is -1419.33 :t 2.1 kcal. No uncertainty due to thep-7 brackets is given because this is included in the uncertainty forentropy.

The above thermochemical parameters for paragonite may becompared with the values obtained by Chatterjee (1970), based onthe same experimental data, but calculated by a procedure modifiedfrom Weisbrod (1968), involving a direct linear extrapolation of thedata points rather than on a point-by-point calculation as is donehere. Chatterjee's values are: For entropy,67.8:L 3.9 Gb; for Gibbsfree energy, -1327.4 + 4.0 kcal; for enthalpy, -l4l79 ! 2.7 kcal'-The agreement is very good. This comparison makes clear that ifenough good experimental brackets are available, reliable thermo-chemical parameters can be obtained by any sound method. However,unless there are enough good points to fix a straight line, extrapolatoryprocedure can lead to large thermochemical errors, and point-by-pointcalculations are preferable.

Chatterjee (in press) recently determined the equilibrium relationsof the reaction

paragonite * quartz : high albite * andalusite + vapourand deduced thermochemical data for paragonite. The values are ingood agreement with those given above.

Pyrophyll;ite

The value of Gyo of pyrophyllite from hydrotherrnal data was c&l:culated by Zen (1969). Since then, new data have appeared that neces-sitate revision of previous values and some new conclusions.

Three reactions were considered (Zen. 1969\:

E.AN ZEN

AI,Si,O'(OH)4 + 2SiO, : Al,SiO,o(OID, * H,Oq u o r t ! D y r o p h y l l i t o v l p o r

Al,SinO'.(OIDz : AlgSiOs + 3SiO, + H,ODyrophyl l i te stdalqsi to qu4at, vspor

and

ALSi,O,r(OH), + 3Al,O3 : 4Al,SiOu * HrOpyrophy l l i t€ coruDduh kyeai to vapor

For the calculations, I used an oxide-sum estimate of the Third Lawentropy value of pyrophyllite of 63.6 Gb/gf., given by Fonarev (1967).The Gibbs free energy value of pyrophyllite based on the kaolinitereaction turned out to be about 7 kcal more positive than given bythe andalusite reaction, the value within each group is tightly clus-tered regardless of the source of data, and the value based on thecorundum-kyanite reaction is similar to that based the andalusitereaction. The Gibbs free energy value based on solubility measure-ments (ReeSman and Keller, 1968) is comparable to that from theandalusite reaction. Despite these facts, the value based on the kao-linite reaction cannot be ruled out as invalid because this value, infact, when combined with the Gibbs free energy value for muscoviteof Barany (1964), predicts mineral assemblages in accord with petro-graphic experience, whereas the value based on the andalusite reactionclearly does not.

The Third-Law entropy of pyrophyllite was recently measuredby King and Weller (1970). The pyrophyllite was a natural sample,its source was not stated, but its chemical analysis is comparable withthe theoretical value for pyrophyllite. The slightly low value of SiOzand high value of HzO might suggest some kaolinitic contamination,but the amount is small and probably can be neglected, as was doneby the authors.

The Third law entropy is 56.6 + 0.5 Gb/gf. This is 6.7 Gb/gf lessthan the oxide-sum estimate, comparable with the discrepancy of 7.2Gb/gf between the measured entropy of talc and the oxide sum. Theoxide sum is the same as that obtained by adding together the entropiesof 2 boehmite and 4 qtaft,z (62.7 Gb/gf), whereas the measured valueis same as that obtained by adding together the entropies of 2 diasporeand 4 qtartrz (56.a Gb/gf ).

The revised entropy value of pyrophyllite means that the Gibbsfree energy also will be more negative. The new results for Gyo forpyrophyllite are given in Table 3, which uses also the new Gibbs freeenergy data for HzO (Fisher and Zen, 1971) and new experimentaldeterminations of the stability of pyrophyllite (Thompson, lg70b).

THENMODYNAMIC PROPERTIES OF MINENALS 547

The point previously attributed to Hemley for the kaolinite-pyrophyl-lite reaction is replaced by the better and more recent data point of300'C i 10oC, 1 kbar (Reed and Hemley, 1966).

Table 3 shows that use of the new entropy value for pyrophylliteleads to nearly uniform changes of the G1o values, so that the differ-ence between the mean of the kaolinite * quartz reaction and themean of the andalusite * quartz reaction remains al 7 kcal (compareZen, 1969), and the result of the corundum-kyanite reaction is morenegative than the result of either of the other two reactions. The morenegative values are close to the measurements of Reesman and Keller(1968) of -1258.7 kcal. Reesman, on the basis of additional experi-ments, now prefers a value of about -1263.5 kcal (A. L. Reesman,letter, September, 1969) .

bar) Gibbs flee energy of fomtion of pyrophyllite flon €lements

290

3 0 0 ! r 0

325 t 20

345 r l0

3?5 i 15

390 ! L0

4 0 5 r 5

Pt = PH2O, bars ci , kcal kference

r. Kaol ini te + 2 quart i = pyrophyl l i te + H20

I000

1 0 0 0

2 000

4 0 0 0

2000

7000

Average

-L253.49 t r .27*

- 1 2 5 3 . 4 1 j I . 3 7

- 1 2 5 3 . 1 8 I r , 4 8

- 1 2 5 4 . 0 9 I 1 . 3 8

- L 2 5 2 . 8 6 r ! . 2 0

-L252,62 ' L.39

- L 2 5 3 . 6 0 r 1 . 3 3

- ] 2 5 3 . 2 r I ' 3 k c a l

R. o. Fownier, 1969,oral comuicationReeil and Hemley, l-966

Thompson' .1970b

Atthaus. 1966

Avelage stanalaral (298 K. I bar) enthafpy of folmation from the elements -1342'7 1 1'3 kcal

400 r 15

410 r 15

4 3 0 r 1 5

4 9 0 r 5

5 2 5 ! 5

II . pyrophyt l i te = sdalusi te + 3 q@rtz + H20

1OOO -1260.14 ! I .55 genley' 1967

18OO -1259,97 t I .52 Kerr ick, 1968

3900 -1259.52 t L.52 sMe

2 o o o - 1 2 6 1 . 2 0 t 1 . 3 8 A l t h a u s ' 1 9 6 6

? 0 0 0 - 1 2 6 0 , 0 4 + 1 . 3 7 s 8 e

Average -1260.2 r 1.5 kcal

Avelage stdalard (298 Kr L bar) enthalpy of foffition flom the elenents -1349'7 t l'5 kcal

I I I . pyrophyl l i te + 3 coluds= 4 kyi l i te + H20

- 1 2 6 2 , A 3 r 2 . 4 5 Matsushina et aI . , 19675 2 0 r 5 7 0 0 0

standald (29a K, 1 bar) enthalpy of fomation from the elerents -1352.3 r 2 '5 kcal

*No bracket ibfofrtion avail.abl.e"

E-AN ZEN

D is ans sian : M usao uite, P ar,ag,onite, P yr op'hy ltite D ata

The inconsistency in the value of the Gibbs free energy of pyro-phyllite could be resolved by assigning a 7 kcal error in the Gibbsfree energy for the aluminum silicate phases, as previously discussed(Zen, 1969). The known phase relations of kyanite, andalusite, andsillimanite indicate that their relative free energies cannot be in errorby more than a very few hundred calories, so if one Gibbs free ener.gyvalue is off by 7 kcal, all the polymorphs will be off by as much. The'concordant Gibbs free energy value of pyrophyllite, calculated fromthe reaction pyrophyllite * 3 corundum,: 4 kyanite * vapour thensuggests that the value for corundum may be off by a similar amount.This conclusion, that the corundum value and the values for thealuminum silicate polymorphs are internally consistent, is borne outby the consistent Gibbs free energy of both muscovite and paragonitecalculated from the "quarLz present" reaction (Day, 1970; Chatterjee,in press) and from the "quartz absent" reaction (Velde, 1966; Chat-terjee, in press also discussed the same point).

Thus if it is indeed true that the sillimanite value and corundumvalue are off by 7 kcal, then the discrepant Gibbs free energy of mus-covite, based on hydrothermal data and on calorimetry, can be recon-ciled. As 7 keal per two gram atoms of Al (as in andalusite andcorundum) means 10 kcal per three gram atoms of Al (as in musco-vite), the change would eliminate all of the present discrepancy of10 kcal .

The Gibbs free energy of formation of K-feldspar is more negativethan that of the Na-feldspar in the same structural state by about gkcal. Because the difference in the framework internal energy of aK-feldspar and a Na-feldspar is not expected to amount to more thana few hundred calories, the bulk of the g kcal should be associatedwith the Na-K substitution. The structures of muscovite and ofparagonite are basically the same (Burnham and Radoslovich, 1964;Radoslovich, 1960), so the bulk of the free energy difference betweenparagonite and muscovite again could be associated with the Na-Ksubstitution. One might expect this difference to be on the order of8-10 kcal. The calculated difference is about 12 kcal when the high-temperature feldspar polymorphs were used. Thus one might expectthat if the muscovite value is nearer -1830 than -1340 kcal, theparagonite value should be nearer -1320 than -lBB0 kcal. This wouldin fact be the case because the paragonite value also implied the samepossible discrepancy in the corundum data and would be revised byprecisely the same amount.

THERMODYNAMIC PROPERTIES OF MINERALS ilg

In the earlier study (Zen, 1969) ii was suggested that in order todecide which of the two Gibbs free energy values of pyrophyllite wasmore nearly correct, two petrographic criteria could be applied. Onewas the breakdown of kaolinite to pyrophyllite in the presence ofquartz, and the other was the reaction of pyrophyllite * microcline(a virtually unknown assemblage) to form muscovite * qtarLz (a

ubiquitous assemblage). My discussion was partly faulty because Itacitly assumed that the chain of Iogical arguments can be decoupledby supposirg that hydrothermal studies of the kaolinite reaction(leading to the more positive Gibbs free energy of pyrophyllite) wereerroneous. It seems a bad supposition because the reactions are re-ported to have been reversed, and a 7 kcal error would entail a tem-perature error of many hundreds of degrees. The second petrographictest involving muscovite is now vitiated by the adjustment in pyro-phyllite data and new uncertainties in the muscovite value itself. It

appears that we could use the consistent values of -1260 kcal forpyrophyllite and -1340 kcal for muscovite, and predict that muscovite* quartz is more stable by 6 kcal, or use the value of' -7253 kcal forpyrophyllite and -1330 kcal for muscovite and predict a 3 kcal

margin in favor of muscovite * quartz. The combination of -1260

kcal for pyrophyllite and -1330 kcal for muscovite would imply thatpyrophyllite * microcline is a more stable assemblage, contradictingall petrographic data, so these two values cannot be both correct.

One could run through a parallel series of calculations for the reac-

tion of paragonite * quartz versus albite * pyrophyllite. The formeris a common assemblage in metamorphic rocks of all grades (Zen and

Albee, 1964), the latter found only rarely (Tobschall, 1969) and in

rocks that include other strange assemblages such as pyrophyllite *biotite (instead of muscovite + chlorite), and thus may be out ofphase equilibrium. Using values of Gibbs free energy of - 1260 kcal

for pyrophyllite and -1327 kcal for paragonite, the reaction at near-

surface conditions is a few kcal in favour of paragonite * quartz'

Analogous to the kaolinite * quartz - pyrophyllite reaction, however,

changing the value of pyrophyllite to -7253 kcal would require, for

consistency, changing the paragonite value to -1318 kcal, and the

same conclusion on the stability of paragonite * quartz would be

arrived at.It appears that because of interdependence of the thermochemical

data, no resolution of the dilemma is available short of direct calori-metric determination of the Gibbs free energy of paragonite andpyrophyllite, and redetermination of those of corundum, the aluminumsilicate polymorphs, muscovite and kaolinite. This is an urgent task

550 E-AN ZEN

Table 4.--T'wo sets of internalLy consistent Gibbs free energy of

formation values for phases. Ttre two sets are mutually incompatible.

AndlalueiteSiLlinaniteKyaniteMuscoviteParagonitePyrophyl.liteKaoliniteDiasporeCorundun

set 1

-584.13 kcal-583.60 kcal-584.00 kcal

-1340. kcal-1328. kcal-!260. kcal-910. kcal-219.9 kcal-378.08 kcal

Set 2

-577. kcal-5?6. kcaL-a I t . Kcar

-1330. kcal-1318. kcal-1253. kcal

-902.87 kcal

-2L6. kcal-37I. kcal

because so much of the available thermochemical data on rock-formingminerals ultimately hinge on the correct knowledge of the thermo-chemical parameters of corundum. Table 4 is an example of possiblesets of internally consistent data on some important minerals; one serassumes the calorimetric corundum data to be correct, the other setassumes the calorimetric muscovite data to be corect.

Acr.rowr,oncunNrs

I thank G. M. Anderson, N. D. Chatterjee, T. M. Gordon, IL C. Helgeson,J. S, Huebner, R. A. Robie, and II. R. Shaw for reviewing the paper and makingsuggestions for its clarification. J. R. Fisher, A. L. Reesman, Priestley ToulminIII, and D. R. Waldbaum contributed by numerous discussions of problems intherrrochemistry and io procedures of calculation. To all these friends, myhearty thanks.

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Manuscript receiued,, Septem.ber p4, 1971; accepted, lor pwblic<ttion, October 29,lstl.