American Mineralogist, Volume 74, pages 5-13, 1989

Dependence of entropy on volume for silicate and oxide minerals:A review and a predictive model

TrvrorHv J. B. Hor-r.aNnDepartment of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, England

AssrRAcr

Simple quantum models (Einstein and Debye) of lattice heat capacity and entropy maybe used to predict the magnitude of the dependence of entropy on volume for silicateminerals. The origins for the volume effect as well as the effect of variation in coordinationstate of cation polyhedra on the thirdlaw entropy are explored and rationalized withinthe framework of simple lattice-vibration theory. It is shown that Einstein and Debyetheories for solids predict a value for (05/0V)r^ of about 1.0 J'K-''cm-3, precisely thevalue found from regression of a set of 60 experimentally measured entropies and volumes

of silicates and oxides. An additive model for estimating the entropies of mineral end-members at 298 K, based upon the scheme advocated by Fyfe, Turner, and Verhoogen(1958), but allowing for coordination changes, is developed and evaluated by multipleregression of this body of measured data. The entropy-volume-coordination model fits

these data better than any previously published scheme and works remarkably well evenfor transition metal-bearing phases when allowance for magnetic disordering is made.Phases such as magnetite and hematite that undergo magnetic disorder at temperaturesabove 298 K can be accommodated within the model by correcting for the partial disorderat 298 K using simple Landau theory. The model predicts silicate and oxide entropies in

the system K.,O-NarO-CaO-MgO-FeO-FerO.-MnO-TiOr-AlrO3-SiO, with uncertaintiestypically in the range of 0-2o/o, even for the Fe-, Mn-, and Ti-bearing phases.

INlnotucrroN

As increasing reliance is being placed on thermody-namic modeling to interpolate and extrapolate experi-mentally determined mineral equilibria in petrology, sothe need for reliable estimates of entropy for mineralphases increases accordingly. While recent years have seena remarkable number of new precise experimental deter-minations of the heat capacities and, by direct integra-tion, the entropies of mineral end-members, there are stillmany phases remaining that require methods of estima-tion. The reasons for this need, which will never entirelydisappear, are (l) calorimetric determinations of entropyrequire considerable effcrt and are time-consuming; (2)often a chosen mineral end-member cannot easily be ob-tained either in pure enough form or in sufficient quantityfor measurement; or (3) the end-members to be deter-mined are fictive in the sense that they are not stable forthe chosen structure or composition.

In principle at least, entropies (and all other thermo-dynamic functions) may be calculated from spectroscopicdetermination of the lattice-vibration spectrum for solidcrystalline materials. Rigorous determinations have rare-ly been attempted (but see Salje and Werneke, 1982, forandalusite and sillimanite) although simplifications inmodeling the phonon density of states function, as in Kief-fer (1980), have produced encouraging results. Apart fromthe fact that this approach is even more labor-intensive

0003-004x/89/0102-0005$02.00 5

than direct calorimetry and therefore unlikely to becomecommonly used, there still remains the problem of un-availability of the required material in pure form or inthe relevant structural state. The astonishing success ofPrice et al. (1987) in accurately predicting the heat ca-pacity, entropy, and compressibility of forsterite from aset of independently derived interionic potentials holdsobvious promise for the future-if this technique be-comes widely used (and extended to a larger system thanMg-Si-O). However, until such time, simpler, quicker

methods must be found that aford reliable estimates ofmineral entropies and that can be based on a minimumof measured properties.

Past efforts to find estimates of entropy have been re-markably successful at the 5-l5o/o accuracy level, and allinvolve some modification of the Newmann-Kopp rule,which is based on the observation that heat capacities(and therefore entropies) of complex compounds may beestimated by summing, in stoichiometric proportions, theheat capacities or entropies of simpler chemical entities.Latimer (1951,1952) and Fyfe et al. (1958) used entro-pies of the elements and of oxides respectively to estimateentropies of more complex compounds. Fyfe et al. notedthe positive correlation between molar volume and en-tropy and incorporated a simple volume correction factorin their estimation scheme:

Sr: )n,S, + k(Vj - 2n,V,),

HOLLAND: REVIEW OF ENTROPY-VOLUME DEPENDENCE

where ,$ is the entropy (to be determined) of phase j, V,is the molar volume of phase 7, whereas S, and V, arcentropies and volumes of the n, oxide components I re-quired to make up phaseT, and k is an arbitrary consmnrdetermined from measurements.

Helgeson et al. (1978) improved matters somewhat bytaking structurally analogous mineral phases as compo-nents instead of oxides; this has the advantage that dif-ferences in coordination state between the componentsand the phase "being built" are minimized. It has longbeen recognized that the coordination state affects theentropy; the difference between Al in octahedral and tet-rahedral sites may be readily evaluated and has been usedin simple calculations (e.g., Holland and Richardson,1979) on mineral stability. Recognizing the importanceof coordination, Robinson and Haas (1983) used multi-ple regression to derive a set of fictive oxide componentsfrom the available measured mineral entropies and heatcapacities. The result was a set of oxide components, invarying coordination states, which could be summed di-rectly to estimate the entropy of any desired oxide orsilicate. While having the advantage over the structuralanalogue approach of path independence, the neglect inRobinson and Haas's model of the volume correlation,which was so successful in earlier approaches, makes theirmethod less powerful than it could have been.

It was the lack of TiOr, MnO, and Fe2O3 in the Rob-inson and Haas scheme as well as its inability to predictaccurate entropies for common pyroxene and amphibolecomponents that led to the model proposed here. Someof the more notable entropy discrepancies in the Robin-son and Haas method (see column RH in Table 2) arehercynite (+ll y.tq-';, cordierite (+17 J.K '), jadeite(- 16 J.f- ';, tremolite (-21 J.K-,;, leucite (+ l2 J.K r),pyrope (-14 J.K ,) and grossular (+43 J.K'), all ofwhich would cause unacceptable errors in phase-equilib-rium calculations. The volume-corrected methods of Fyfeet al. (1958) and Helgeson et al. (1978) generally workquite well, and there are sound theoretical reasons for apositive correlation between volume and entropy thatwarrant a brief review before presenting the revised es-timation method and results.

TrrB nNrnopy-voluME RELATroNsHrp REvtEwEDAlthough the relationship of entropy to mass is well

known and has been discussed and used by Latimer ( I 95 I,1952), in the methods to be discussed below, fictive com-ponents are summed to "build" mineral entropy so thatmass is conserved. In their discussions, Fyfe et al. (1958)called upon the relation

@S/dV)r: a/0 : (dP/dT)v

to explain the positive correlation of entropy with vol-ume and to justify the positive sign of the correlation.While it is true that u/8, the ratio of thermal expansionto compressibility, is generally a small positive number,this equation does not provide a satisfying explanation ofthe effect at an atomistic or structural level. The reason

for a positive correlation of entropy with volume can beseen most easily by considering the role of lattice vibra-tions in determining the heat capacities and entropies ofcrystalline solids. The simplest quantum model for latticeheat capacity was devised by Einstein to explain the falloff of heat capacity to zero as the absolute zero of tem-perature is approached. In this model the crystal is as-sumed to be composed of 3N independent one-dimen-sional harmonic oscillators vibrating with frequency z,where Nis Avogadro's number. The Einstein model heatcapcity for I mol of a phase containing n atoms in itsformula unit is given by

Cr: 3nRlu'e"/(e" - l) '1,

where R is the gas constant, and u : hu/kT, .vv.rth h andk being Planck's constant and Boltzmann's constant, re-spectively. The entropy according to this model is

S: 3nR{lu/(e, - l) l - ln(l - e-")}.

While the heat capacity of an Einstein crystal does notmatch the behavior of real crystals perfectly at low tem-peratures, it does simulate the trend remarkably well fora simple one-parameter model.

Next to be considered is the role of molar volume,whose importance lies in its relationship to the term el :hv/kT. EnlarySng the cell volume of a simple crystal whoseatoms vibrate at a characteristic frequency has the effectof moving apart the component atoms, thus reducing theirbond stiffnesses and lowering their vibrational frequency(in proportion to (Vo/tr)'/'). Figure I illustrates the generaldependence ofheat capacity on vibrational frequency, andfrom the definition ofthe third-law entropy,

s,: J' ? or.

one can see clearly that lowering the frequency by length-ening and weakening the bonds causes an increase in en-tropy. To be more rigorous, C" must be converted to C,with the relation

Cr: C, + TVu2/8,

although the difference is negligible at low temperaturesfor solids.

The magnitude of the volume effect on the entropy ofan Einstein solid is given by

(A S / A V),,8 : @ u/ 0 V)(0 S / |u),n"

which, from u : uo(Vo/V)v, and the Einstein expressionabove for the entropy, one finds by substitution that

(dS/dV)2,8: nRu' / lV(e ' - l ) ( l - e f l .

In this expression, Z is the molar volume of the mineralconcerned. Evaluating this expression at 298 K for allphases listed in Table l, the mean value of (dS/|V)r* isfound to be 1.07 + 0. 1l J.K-'.cm :. This value is a nu-merical measure of the constant k in the Fyfe et al. en-

HOLLAND: REVIEW OF ENTROPY-VOLUME DEPENDENCE

tropy-estimation scheme discussed above and is in ex-cellent agreement with the value found empirically below.

Although the above model is simple, the Einstein ap-proximation is nevertheless remarkable in describing, withonly one adjustable parameter, the heat capacity and en-tropy of crystals, and it should give a reliable estimate ofthe volume dependence of the entropy. The relationshipof normal mode frequencies to entropy will be exploredfurther later, but in passing it should be noted that theeffect of volume on heat capacity will lessen at highertemperature where thermal agitation is more pronouncedand will tend to mask the smaller volume-related effect.It is for this reason that high-temperature heat capacitiesmay be modeled quite reasonably by a simple additivityof oxides approach without consideration of the volumecoffection (as done by Berman and Brown, 1985).

It is interesting to inquire whether an increase in thesophistication of the assumptions used alters this result,and so the above exercise was repeated using the Debyemodel for heat capacities. In place of a single frequencyto characterize the vibrational spectrum, Debye assumeda quadratic density ofstates g(v): av2, which is the cor-rect behavior in the low-frequency limit for a continuum,with a cut-off value at /-"*. A similar approach to thattaken above, using the Debye (D) expression for the en-tropy with 0r: hv^ */k expressed in parameterized form,glves

Srrr .o: n( l3 l4 l /0" - 3 .81)

where the constants were determined from a least-squaresfit to the Debye entropy function at 298 K and entropyis in J.K-' mol '; hence,

(AS/AV), ,s : ( ,S + 3.81n) /3V,

which on evaluation (Table 1) gives a slightly lower valueof (dS/dZ)rn, : 0.93 + 0.10 J'K ' 'cm 3. Given that theDebye approximation is often used best in the low-tem-perature limit and the Einstein model often works wellfor approximating the high-temperature behavior, an av-erage might be appropriate, and a value of unity for theproportionality constant k is accepted, as predicted fromsimple lattice-vibration models.

Trrs nnLlrroNSHIp BETwEEN ENTROPY ANDCOORDINATION

Having examined the entropy-volume relationship, Inow turn to look briefly at the role ofcoordination state.As a simple example, I will take the CarSiOo mineralslarnite and calcium olivine that have entropy values of12'7.6 J'K' and 120.5 J'K ', respectively, despite larnitehaving the smaller volume. Clearly larnite owes its higherentropy to the distorted and much larger site for one ofits two Ca ions (see Table 1), while presumably it man-ages to keep its volume reduced by a suitable packingarrangement of alternating large (M2) and small (Ml) Casites. Thus the dominant control of the entropy differencebetween Ca olivine and larnite lies in the coordinationchange from 6 to 8 one of its Ca sites, as would be pre-

20

1 5c,

J atom-ll0

5

0

Debye Heat Capacities

2. . 3

O Forsterite Cp

1 0 = 5 0 0 K2 0 = 8 0 0 K3 e = 1000K

0 100 ZO0 300 400 500TK

Fig. 1. Heat capacities calculated from the Debye theory us-ing three values for the Debye temperature 0o to show the effectof varying the vibrational frequency (0o : hv/k). Filled circlesare experimentally measured C, data for forsterite'

dicted by the simple vibrational models discussed above.Kiefer (1982) has drawn attention to the fact that the

entropy is sensitive to the position of the lowest-frequen-cy optic modes, which are those usually associated with

internal vibrational modes of cation polyhedra in sili-cates, and one should thus expect, as a general rule, that

increasing coordination state will lead to an increase in

entropy.Exceptions to this simple notion are numerous, and it

is often impossible to apply it straightforwardly to com-plex silicates because in many instances it is not the in-

ternal modes associated with a coordination polyhedron

that dominate the low-temperature heat capacity, but the

external modes associated with the linkages between thepolyhedra. A classic example is the case of the three alu-

minosilicite polymorphs kyanite, andalusite, and silli-manite. All three minerals contain SiOo tetrahedra and

chains ofedge-connected AlOu octahedra, but an extra Al

in the formula occurs in 6-fold octahedra in kyanite, in

irregular 5-fold coordinated polyhedra in andalusite, andin 4-fold tetrahedra in sillimanite. The entropies at 298

Kare 82.3. 91.4. and 95.8 J 'K ' 'mol- ' , respect ive ly ; thusit is usually argued that the entropy increase in these min-

erals occurs in response to reduction in the coordinationstate. However, in these polymorphs the entropies in-

crease in the reverse order from that predicted by the

frequencies ofthe internal vibrational modes; the averagepolyhedral Al-O bond is shortest and stiffest in the tet-rahedral sites (sillimanite) and longest in the octahedralsites (as in kyanite), yet Kiefer (1982) noted that the low-

est-frequency optic mode occurs at I l5 cm-r in silliman-ite, at 156 cm-' in andalusite, and at 237 cm ' in kyanite.

Thus it is reasonable to infer that it is not the internalvibrational modes of the characteristic Al polyhedra in

these silicates that dominate the sequence of lowest optic

HOLLAND: REVIEW OF ENTROPY-VOLUME DEPENDENCE

TABLE 1' Entropy, volume, and composition data for phases used in the regression

Phase. St Ref $ Phase- tr- q Ref g

Magnetite (mt)FerO", FeO

Hematite (hem)Fe203

Titanomagnetite (Timt)2FeO, t6rTiOz

Calcium ferrite (CaFt)I6rCaO, FerO3

Dicalcium ferrite (DCFt)2t6loao, FerO3

Acmite (acm)0 5r8rNaro, 0.5FerO3, 2rorSiO,

Jadeite (id)0.5l8lNaro, 0 5t6lAlro3, 2l4rSiO,

llmenite (ilm)FeO, t6tTiOe

Spinel (sp)rrMgO, tor4;,9.

Rutile (ru)16lTiOz

Tridymite (trid)tatSiOz

Manganosite (mang)rorMnO

Lime (lime)16rCaO

Periclase (per)r6tMgo

Corundum (cor)t6tAlzOg

Hercynite (herc)FeO, tel41r9.

Hedenbergite (hed)clcao, Feo, 2r1tsio,

Ferrosilite (ts)2FeO, t.tSiOz

Kyanite (ky)16rAlro3, r4tsio,

Sillimanite (sill)0 5t6lAlros, 0.5r4lAlro3, r4lsio'

Calcium olivine (caol)2l6loao, Ia)SiO,

Larnite (larn)r6rcao, {8tcao, r4lsio,

Gehlenite (geh)telcao, IslCaO, {4lAlro3, t4lsi02

Akermanite (ak)IsrCaO, prcao, rarMgo, 2rarsio,

Monticellite (mont)r6icao, r6lMgo, r4lsio,

Sphene (sph)rorcao, I6lTiOr, t4lSiO,

Fayalite (fa)2FeO, t rSiOz

Forsterite (fo)2t61MgO, rarsi02

Cordierite (crd)216rMgO, 2r4tAlro3, 5r4lsio,

Tephroite (teph)216lMnO, tarSiO,

30.27( 1e.e)f

82.3 1(-s .1)+

46.82 142.1 1(_26.8) f

44 98 1 15.6 1(_2e.8)+

67 .18 159 .0 1(_2s.8)+

64 59 155.7 2(_ 14.s)+

60.40 133 5 1

31.69 95.5 l, 3 Merwinite (merw)

44.53 126.2 1

(- 13.4)+39.78 a0.6 1

18.82 50 3 1

26.s3 43.9 1

Wollastonite (wo)I6rCaO, SiO,

Calcium Tschermak's pyroxene (cats)IsrCaO, 0.5t6lAlro", 0.StolAl.O", tnls19,

Diopside (di)rErcao, 16rMgo, 2r4sio,

Enstatite (en)lTiMgO, 16rMgO, 2tarsiO,

Rhodonite (rho)16rMno, r.rsio,

Tremolite (tr)2l8loaO, 516rMgO, 8t4rsior, Hro(b)

Anorthite (an)telcaO, IalAlrO3, 2ta1siO,

3telcao, r6lMgo, 2lalsio2Microcline (micr)

0.5r"rKro, 0.5r4lAlro3, 3r4rsio,Kaliophilite (kal)

0 5r"lKrO, 0.5l4lAlro3, I4lSi02Leucite (lc)

0.srbrK2o, 0.5r4rAlro3, 2r4rsio,Albile (ab)

0 srelNaro, 0.st4rAlroo, 34rSi02

3telcaO, 16rFerO3, 3SiO,

39 93 81.7 10

63.56 135.3 1 1

66.19 142.7 10

62.68 132.5 1 0

35.16 87.6 1(_ 14.9)+

272.70 549.1

100 79 199.3

98.47 253.1 1

108.72 2't4 2 1

59.89 133.3 1

88.39 184.3 1

100 04 207.4 113.22 44 I( 14.e)+

16 .76 38 .1

11.25 27.0

25.58 50.9

40.75 92.9( - 13.4)+

67.88 160.8(_ 13.4)+

65.92 162 5(_26.8)+

44.09 82.3

50 03 95.8

59.1 1 120.5

51 .6 127 .6

90.24 198 6

92 54 209.2

51 .48 108 1

s5.65 129.3

46.30 124 2(_26.8)+

43.66 94.1

23322 407.1

48.61 133 4( 2e 8)+

'I

1

1

1

1

4

5

o

o

'I

1

1

'I

7

1

I

I

1

I

Nepheline (ne) 54.16 124.4 10.5rrrNaro, 0 5rrAlro3, r4tsio,

Muscovite (mu) 140.83 287.7 10.srotKro, r6tAlro3, 0 StotAlzoo, 34rsior, Hro(a)

Phlogopite (ph) 149.64 315.9 120 slblKro, 3FrMgO, 0.5t1lAl2oo, 3lolsior, HrO(b)

Talc (ta) 136.25 260.8 13rrMgo, 4rrsior, Hro(b)

Pyrophyllite (pph) 127.61 299.4 116141203, 4r4rsi02, Hro(a)

Anthophyllite (anth) 265.4 537.0 102r4MgO, 516rMgO, 8r41SiOr, H,O(b)

Clinochlore (clin) 210.9 397.6 13516rMgO, 0.5t6tAlzOs, 0 SrntAlzos, 3rrsior, 4Hro(b)

Margarite (ma) 129.60 263.6 't4telcao, t6lAl2oo, I4rAlrOo, 2l4iSior, Hro(a)

Paragonite (pa) 132.11 277.1 120.srerNa2o, 16rA1103, 0.5r4iAlro3, 3r4rsior, Hro(a)

Diaspore (dia) 17 76 35.3 10.516141203, 0.5Hro(a)

Gibbsite (gib) 32.03 68.4 10.5r6tAlro3, 1 sH,O(a)

Prehnite (pre) 140.26 292.8 142lerCaO, 0.516rAlrOs, 0 storAl2o3, 3SiOr, HrO(b)

Chrysotile (chy) 107.46 221 3 1316rMgO, 2r4tsiOr, 2HrO(a)

zoisite (zo) 135.88 295.9 142rrrcao, 1.516rA1203, 3r4isior, 0.5Hro(b)

Almandine(alm) 115.11 299.6 153rsrFeo, 16rA1103, 3sio, (_40.1)+

Pyrope (py) 113.18 266.3 j63rsrMgo, r6iAlro3, 3sio,

Grossular (gr) 125 35 254.7 173tslcao, 16rA1103, 3SiO,

Andradite (andr) 't32.04 286.6 18(-2e.8)+

. Phase name, abbreviation, and composition. Numbers in brackets represent coordination state (see text)- - Y, volume, in cm3.mol ' .

1 S, entropy, in J K 1 mol-,.+ Entropy corrected by this amount for magnetic and/or other disorder (see text).$ References: (1) Robie et al., 1979; (2) f_o 9t al., 1977; (3) Anovitz et al., t985; (a) Hasetron et at., 1987; (5) Bohten er at., 1983; (6) Robie and

Hemingway, 1984a; (7) Sharp et a l . , 1986; (8) Robie et a l . , 1982a; (9) Robie et at . , 19.82b; (10) Krupka et at . , i9B5; (11) Hasetton et dt . , tse+; 1tz;Robie and Hemingway, 1984b; (13)Hen-d-erson et al., 1983; (14) perkins et at., 1980; (15) Bohb; et;t., 1986; itol Has'en6n and Westrum, 1980; i17iPerkins et a l . ,1977; (18) Robie et at . , 1987.

modes, and hence entropies, but some unspecified exter-nal modes associated with the linking of the polyhedralunits. We must therefore expect the common occurrenceof phases in which the positive entropic effect of smallerand more rigid polyhedra is offset by the lower rigidity

of the interconnecting framework. Given that the entropyand other thermodynamic properties are functions of theaverage vibrational spectrum, we might expect that themolar volume should reflect the overall bonding, andhence vibrational, state of the crystal. In the aluminosil-

HOLLAND: REVIEW OF ENTROPY-VOLUME DEPENDENCE

icate example, the entropies of the three polymorphs areindeed propoftional to their molar volumes.

From the above discussion it is clear that the effects ofcation coordination as well as the volume contributionmust be incorporated into the additivity models for min-eral entropy estimation, although in many complex sili-cates it may be impossible to ascribe anomalous entropyto any one particular structural feature. Any model basedsolely on the coordination state and molar volume canform only a crude guide to the vibrational spectrum, butas shown below may be sufficiently accurate in predictingentropies of silicates from a minimum of information.

Some non-lattice-vibrational contributions to entropy

Contributions to the entropy arising from phenomenaother than lattice-vibration effects have been discussed inconsiderable detail in the literature, and the interestedreader may turn to the review article by Ulbrich andWaldbaum (1976) for more detail. For the present pur-poses, all such contributions to the 298-K entropy mustbe removed before looking at the relationship of entropywith volume, and so terms arising from magnetic as wellas site disordering and possible electronic efects arisingfrom Jahn-Teller site distortions and other crystal-fieldeffects must be taken into account.

The site<onfigurational entropy terms involved in, forexample, Al-Si order-disorder on tetrahedral sites, havebeen removed from the tabulated entropies, and only thecalorimetric entropies have been used in the followinganalysis. Care must be taken when using tabulated entro-pies; in the tables of Robie et al. (1979), certain phaseshave had an arbitrary configurational term added, an ex-ample being muscovite for which the full -4R[0.75

ln(0.75) + 0.251n(0.25)1, amounting to 18.7 J.K '.mol-',

has been added. It remains a debatable point whether thisis always justified, and in the case of muscovite, the ex-perimental phase relations are consistent only with alargely ordered state. Strong short-range order can reducethe entropy contribution to very small values, and recentwork suggests that (Al,Si) in muscovite is ordered on aIocal basis (Herrero et al., 1987).

Magnetic order-disorder transformations at low tem-peratures are quite common in minerals containing tran-sition metals and give rise to substantial heat-capacityanomalies (I peaks) that contribute to the entropy at 298K. Although the tr anomaly occurs at different tempera-tures and varies in size in different minerals, the contri-bution to the entropy is ideally given by S : R ln(2s +1), where s is the spin quantum number. Thus, for Fe2t,s is 2 and contributesR ln 5 (13.4 J.K I 'atom t), whereasfor Fe3* and Mn2*, s is 5/2 and so the entropy contribu-tion becomes R ln 6 (14.9 J.K '.atom '). In the analysisto follow, these ideal entropies for magnetic disorder havebeen subtracted from the phases that have a low-temper-ature magrretic transformation. The minerals hematite andmagnetite require special comment as they are character-ized by having their magnetic tr transitions at tempera-tures well above 298 K, but have long tails to their \anomalies extending down to temperatures below 298 K;

thus there is likely to be some small contribution to theentropy arising from the incipient disorder of the mag-netic spins below 298 K. One way of estimating the mag-nitude of this contribution is to apply simple Landau the-ory (for a good mineralogical review ofthe basic concepts,see Carpenter, 1988) to these magnetic transformations.Landau theory for tricritical behavior leads to the follow-ing useful expression for the entropy as a function of tem-perature below I.:

Su,o. , : S-"* [ l - (1 - T/7. ) ]

where S-.. is the maximum entropy expected for thetransformation. i.e..

*"^: J" 7 o,and Cp is the excess heat capacity relative to the fullyordered phase. It is conventional, in Landau theory, totake excess properties relative to the disordered, high-symmetry, phase, whereas the above expressions havebeen rearranged according to the more usual petrologicalconvention. For hematite (7. : 955 K, ,S."* : 2R ln 6),the entropy contribution at 298 K, from the equationabove, is 5.1 J.K r.mol-r, amounting to l7o/o of the max-imum entropy that would be gained only at Z: 955 K.The conclusion is that even though 298 K is well below?"., the effect of the tail in the tr heat-capacity anomaly isnot negligible, and the value of 5.1 J'K r.mol-r shouldbe subtracted from the calorimetric entropy of hematiteif one wishes to determine the lattice-vibrational portionof the entropy. A similar argument for magnetite (2. :

848 K) suggests that the entropy of magnetite should bedecremented by 8.4 J.K-' mol-'to allow for the \ taileffect. Magnetite still requires a further adjustment of ap-proximately -2R ln 2 J.K '.mol-' to allow for the dis-ordering transformation (normal to inverse spinel) thatoccurs at 1 15 K.

A further problem is the variable entropy contributionfrom transition-metal ions (in this study in FeO, FerOr,TiOr) due to polyhedral site distortions. The subject hasbeen discussed by Wood (1981) who showed that crystal-field effects can be large and important for these cationsunder some circumstances. The effects discussed by Woodshould be maximal only at very low temperatures and forextreme differences in the shape of a distorted and un-distorted polyhedron; the success of the simple model tobe discussed in the next section implies that at 298 Kthese crystal-field terms probably make only a small con-tribution to the entropy.

Pnoposno ESTTMATToN METHoD AND REsuLTs

Realizing the need to incorporate the efects of bothvolume and coordination and the fact that both of theseare only an approximate guide to the vibrational contri-bution to the specific heat and entropy, one opts for thesimplest possible model-that of Fyfe et al. (1958)-butallows some components to be represented in several dif-

1 0 HOLLAND: REVIEW OF ENTROPY-VOLUME DEPENDENCE

s vPhase. hat meas.

Tlele 2. Results of regression for S - yof phases ferent coordination states: The Fyfe et al. expresslon 1srecalled as

f : > n J + k ( v j - Z n , v , ) ,

where S, and V, are entropies and volumes of oxide com-ponents i, present in amount n,, and,!, and V, are theentropy and volume of the phase j. A certain amount ofexperience with measured data has shown that the valueof k is approximately 1.0 if the units of entropy are in J.K '.mol 'and volumes are in cm3.mol '. A petrologicalconsequence ofthis is that solid-solid reactions involvingno change in coordination state should have dP/dT: AS/AZof approximately l0 bar.K-'. An example of such areaction might be tremolite + 2 diopside * talc. Unpub-lished experiments (Jenkins and Holland, in prep.) are ingood agreement with this estimate, which should be rea-sonable for reactions, such as the one above, that involvelittle or no change in coordination. In contrast, reactionssuch as jadeite + quartz + albite are driven by largerentropies, in part due to the increase in coordination stateofNa and the change from octahedral to tetrahedral co-ordination of Al, and have somewhat larger dP/dT slopes(around 20 bars.K-').

The method used here involves rearrangement of theabove equation to the form

S,: kV, + 2n,(5, - kV,),

where the second term on the right is a constant for eachoxide component. The values of the (St - kV,) and k weredetermined by least squares from measured entropies ofoxides and silicates, with multiple regression returning k: 1.00 within error, in pleasing agreement with the sim-ple harmonic oscillator model. Thus we may drop theconstant k and fit the simpler model, taking S,' to rep-resent (,S, - 4),

Si: Vt I 2n,5,' ( l)

to the experimentally measured entropies in Table I byregression. The resulting standard deviation ofthe resid-uals was 1.77 J'Kr.mol r, and the average absolute de-viation was l.4l J.K-t.msl-t, with the worst deviationbeing 4. I J.K '.mol ' (for tremolite). Table 2 shows thecalculated results and the residuals in entropies for thephases used, and Table 3 lists the values for the entropiesassociated with each oxide component and their uncer-tainties (lo).

The assignment of the components chosen requires briefcomment. The 6-fold coordination of cations in the ox-ides AlrOr, MgO, CaO, FeO, TiOr, and MnO are straight-forward as are tetrahedral coordination for SiOr, AlrO3,and MgO. In addition it was necessary to consider trrMgO

to represent the slightly larger M2 site in orthopyroxenesand the M4 site in orthoamphiboles. Similarly, tsrQa6refers to M2 clinopyroxene and M4 amphibole sites aswell as the Ca in larnite and sphene. I8-'OlCaO representsCa in much enlarged sites (e.g., in margarite) andlor thosedifficult to define precisely in terms of coordination num-

s vcalc. resid RH ko kE

mt 0.314hem 0.283Timt 0.364CaFt 0.249DCFI 0.358acm 0.351jd 0.712ilm 0.237sp 0737ru 0 379trid 0 037mang 0.229l i m e 0 1 6 1per 0 071cor 0 176herc 0.159hed 0.137fs 0.313ky 0.353sill 0.230caol 0.400larn 0.379geh 0.208ak 0.409mont 0.123sph 0.298fa 0.294fo 0.346crd 0 502teph 0.621wo 0 .170cats 0.154di 0 234en 0 533rho 0 160It 0 417an 0 259merw 0.478micr 0.632kal 0.425lc 0.315ab 0.481ne 0.323mu 0 530phl 0.394ta 0.189pph 0.348anth 0.533clin 0.879ma 0.240pa 0.350dia 0.039gib 0.258pre 0.304chy 0.551zo 0.377almpygrandr

81 .70 81.02 0 6852.03 50.24 1.7995.29 94 18 1 .1 170.58 72 18 1.6091 .81 94 13 2 .3291 .08 88.18 2.9073.07 74 36 1.2963.83 63.41 0.4240.80 41.36 -0 5631.43 32 63 -1 2017 .37 17 45 -0 0831 .59 33 41 1 .8221.34 21.94 -0 6015 .70 1575 -0 0525 35 22 60 2.755214 53 37 1.2392.92 93.0s -0 1396.54 96 45 0.0938.21 40.04 -1.8345.76 43.19 2.5761 39 61.34 0.0576.00 72.19 3.81

108 36 108.08 0 281 16 .67 1 15 .41 1 .2656.62 55.14 1 4877.65 77 .45 0 2077.94 79.00 -1.0650.45 48.94 1 51

173.90 1 76.51 -2.6184.80 84 28 0.5241.76 39.39 2.3771 .74 70.56 1 .1876.51 78.02 -1.5169.86 71 .71 - 1 8552.44 50 86 1.58

276.40 280 52 -4 1298.s1 98.15 0.36

154 60 153 75 0.85105 48 106.57 1 .0973 37 71 .67 1.7095.93 93 32 2.61

107.36 106.53 0.8370 .19 71 .64 -1 45

146.87 149 08 -2.21166 30 165 45 0.85124 54 124.48 0.06111 79 1 08.1 1 3.6827't 60 267 90 3.70186 70 186 .58 0 j 2134 00 136.46 2.46145 00 144.84 0.1617 .55 19 .15 -1 .6036.48 34.87 1 61

152.50 154.27 -1.77113 .80 113 .57 0 .23160.00 158.70 1.30184.45 184.45 0153 .12 153 12 0129.35 1 28.53 0.82154 .56 156 17 - 1 . 61

1 1 4 1 . 2 81 1 2 1 2 61 .20 1 .331 . 05 1 . 190.96 1.080.92 1.06

-15.7 0.95 1.091 .20 1 .34

-0.8 0.90 1.0s1 .09 1.23

1.0 0 78 0.891 .05 1.45

- 1 .5 0.91 1 .014 .2 1 .02 1 . 188.5 0 91 1.08

1 1 . 2 1 . 0 0 1 . 1 3-6 . s 0 .91 1 .04- 2 . 0 1 . 0 1 1 . 1 5

3.0 0.85 1.010.9 0.84 0.98

-6 7 0.83 0.933 .0 0 .99 1 .126.2 0.90 1.020.1 0.92 1.032.9 0.87 0.99

0.96 1.082.7 1.09 1 225.7 0.92 1 06

16.9 0.74 0 851 . 1 0 1 2 2

-0.8 0.84 0.95-3.5 0.91 1.05-8.0 0.91 1.04

1 2 0.91 1.051 . 0 1 1 . 1 4

-21.0 0.86 0.990 0.82 0.94

- 1 0 1 0 4 1 . 1 6- 0 8 0 8 1 0 . 9 1

4.0 0.89 0.9912.1 0.84 0.940 0.86 0.972.7 0.93 1.O41 .4 0 .87 1 .004.4 0 88 1.01

-7 I 0.83 0.97-3 4 0.82 0.96

5.8 0.87 1.01-14 2 0.85 0.99-0 I 0.88 1.03

2.4 0.90 1.04-0 .3 0 .95 1 .15

3.9 0.99 1 .18-2.8 0.89 102

9.5 0.90 1 054 I 0 .93 1 .07

1 0 5 1 . 0 9 1 . 2 4-13 .7 1 .01 1 .16

429 0.88 1.O20.92 1.05

Nofei Columns (S - y).""" and (S - y)* are measured and catcutatedS - y; resid is the residual in calculated entropy; RH is the equivalentresidual calculated from the tables of Robinson and Haas (1983); ko andkE are the entropy/volume proportionality constants, defined in the text,for the Debye and Einstein theories; hat is the diagonal term from the hat(least-squares projection) matrix (Belsley et al, 1980) and indicates theinfluence of each observation on the least-squares solution. with hat : 0denoting no influence and hat : 1 denoting extreme influence (forcing thefit through that datum).

. Abbreviations are given in Table 1.

HOLLAND: REVIEW OF ENTROPY-VOLUME DEPENDENCE

ber. KrO was partitioned intot"lKrO framework sites suchas in feldspars and feldspathoids, and IolKzO sites such asin interlayer mica positions (and the large cavity of leu-cite); thus size and degree ofvibrational freedom seem tocategorize KrO ditrerences. NarO is partitioned into smallt8lNazO as in pyroxene (M2) and amphibole (M4) sitesand large Ie-r2lNa2O as in framework silicates and micas.HrO was the most difficult parameter to define; in theend, it was decided to split HrO up into only two cate-gories for simplicity. Although the nature of the bondingofthe proton in hydrous silicates is very variable, it wasfound convenient to split mineral structures into a high-entropy HrO(a) and a low-entropy HrO(b) group. Themicas fall naturally into high-entropy dioctahedral andlow-entropy trioctahedral groups that may be rational-ized on the basis ofthe distortion ofthe proton positionin dioctahedral micas away from the normal to the micasheets; in trioctahedral micas, the three full octahedralsites repel the proton equally (Bailey, 1984). In diocta-hedral micas, the vacant M2 site causes the proton to bedeflected largely into the vacant space, allowing for agreater degree of vibrational freedom. Chlorites are trioc-tahedral, with talc-like and brucite-like layers in whichthere is strong hydrogen bonding of the brucite hydroxylgroups to the talc layer oxygens, and so chlorites are at-tributed to the HrO(b) group, as are talc, phlogopite, andthe amphiboles. Diaspore, gibbsite, and serpentine areallocated to the HrO(a) group because of the larger degreeoffreedom ofthe OH groups that are not strongly bondedto tetrahedral layers as in micas or, in the case ofserpen-tines, because the layer mismatch between the octahedraland tetrahedral sheets allows more flexibility in the vi-brational freedom ofthe hydroxyl groups.

It was also found that the large, S-fold coordinated (dis-torted cube geometry), sites in garnet required separateevaluation; the entropy contribution from Ca in garnetre,t(QaO) is less than expected whereas the entropy con-tributions of iron r",t6eO) and Mg te,r(MgO) are larger. ForMg, a small ion in a very large and somewhat distortedcage, the extra entropy is readily rationalized, but the lowentropy for Ca seems anomalous.

To see the efect of ignoring the volume dependence ofthe entropy, a regression of the same data was performedwithout the volume terms. as done bv Robinson and Haas(1e83) ,

{: >n,s" (2)

and the results are given in Table 3. The standard devia-tion of the residuals and the mean absolute deviation ofthe residuals were 3.26 and 2.50 J. K'. mol', respective-ly, about twice the values of the volume-corrected model;however, the worst deviations were 9.85 J.K-'.mol-'(larnite), 8.1 J.K-'.mol-' (cordierite), and 7.5 J.K '.mol '

(leucite). Inspection of the results for AlrO, reveals thatwith no volume correction, the difference between octa-hedral and tetrahedral coordination is 28.3 J.K-'.mol-t,whereas for the volume-corrected model, the difference

Trele 3, Values for use with the entropy models

Component S- Y os,v S Og

trrSiOz16lAlzOstorAlzOg16rMgOtolMgortrMgors,rMgOrorCaOrorcaOrerolcaoIdlCaOrsrFeOrdlFeO{6rMnO16lTiOzt"rFezOgIolNarOtg;2tNazOhtKrOrorK20

H,O(a)H,O(b)

17.4522.6028.8915 .7518.7721 .06zo.uo21.9427.3734.3717 .8630.7836.5033 4132 6350.2430.J2

79.4979.5587.9615.717.44

0.380.841 .060 5 31 7 71 3 60 8 80.800.840.700.670.830.671 .201 .541 .60J . O /

3 . 1 2

3.030 9 1o.87

40.30 0.3943.78 0.8472.07 1.0526.67 0.5438.30 1 7727.74 1 .3533.87 0.8839.59 0.8038.73 0 8148.25 0 7129.47 0.9943.24 0.8344.96 0.6846.28 1.2051 .94 1.5480.51 1.6165.86 4.5897.28 3 13

1 14.35 3.77120.37 3.3730.03 0.9120.74 1.03

Nofe. The first two columns of data, S y and its uncertainty, refer tothe entropy-volume model, Eq. 1 in the text; the last two columns, S andits uncertainty, refer to the simple additivity model with no volume correc-tion. Eo. 2 in the text.

is only 6.3 J'K r'mol r. The volume-independent modelalso fits the measured entropies rather better than thedata of Robinson and Haas (1983) and may be useful inestimating entropies of mineral end-members for whichno reliable volume data arc available.

Appr-rca.rroNs

One of the motivating reasons behind this study wasthe need to estimate the entropies for minerals in a moregeneral project to derive a reliable thermodynamic dataset for petrological calculations. In earlier works (Powelland Holland, 1985; Holland and Powell, 1985), a prelim-inary thermodynamic data set was generated and used toobtain more powerful methods of characterizing meta-morphic conditions, particularly pressure (Powell andHolland, 1988). The project is now at an advanced stage,involving many more mineral end-members and phase-equilibrium constraints, and has required the estimationofentropies ofseveral phases where they were not known.It is also useful to have reliable entropy estimates forminerals even if complex thermodynamic calculations arenot required, for instance when determining approximateslopes ofunivariant reactions from the Clausius-Clapey-ron equation. The entropies for a number of rock-form-ing mineral end-members of interest are presented (Table4), using the methods outlined above. Because it appearsthat magnetic transitions in silicate minerals tend to oc-cur at very low temperatures, the tabulated entropies in-clude the ideal magnetic contributions. It should be re-emphasized, however, that site<onfigurational entropyterms have been omitted, as the degree of order in mostsilicates is at present poorly constrained. It is left to the

t 2 HOLLAND: REVIEW OF ENTROPY-VOLUME DEPENDENCE

TABLE 4. Predicted entropies for unmeasured mineral end-members

Y S t o "(cm3 mol ,) (J K ' mol t) (J K 1.mol 1)

JohannseniteMg-Tschermak's pyroxeneTschermakiteEndenitePargasiteCummingtoniteGruneriteFerropargasiteFerroactinoliteGlaucophaneFerroglaucophaneRiebeckiteMagnesioiriebeckite

Mg-celadonite.EastoniteAnniteSiderophylliteManganophylliteNa-phlogopite-AmesiteDaphniteMn-clinochlore.Tschermak's talc.MinnesotaiteMn-talc.Mg-chloritoid.Fe-chloritoid.Mn-chloritoid.

Mg-staurolite'Fe-staurolite.Mn-staurolite.Mg-carpholite-Fe-carpholite.Mn-carpholite.SapphirineMg-pumpellyite-VesuvianiteFe-cordierite.Mn-cordierite.Deerite

Chain silicates179123R e r

588582533734703705534624691602

Sheet silicates288306405JOC

421306388542568250358373132162167

Others885993

1 0 1 3194223229790629

2008470483

1 531

CaMnSir06MgAl(SiAl)O6CarM g3Alr(Si6Alr)Orr(OH),NaCarM95(Si?Al)Orr(OH),NaCarM g4Al(Si6Alr)Orr(OH),MgTSi.Orr(OH),Fe?SiBOrr(OH),NaCarFe4Al(Si6Al,)Oz(OH),CarFesSisOrlOH),NarMg3AlrSisOrr(OH),NarFeoAlrSisOrlOH),NarFeoFerSioOrr(OH),NarMgsFerSi6Oz(OH),

KMgAI(Sil)O,o(OH),KM grAl(SirAlr)Oro(OH),KFe3(Si3Al)O,o(OH),KFe,Al(Si,Al,)Oa(OH),KMns(SioAl)O,o(OH),NaM93(Si3Al)O,o(OH),M glAlr(AlrSir)O,o(OH)sFesAl(AlSia)Olo(OH)sMnsAl(AlSi3)Oio(OH)sMg,Al(Si3Al)O,o(OH),Fe3(Si4)Oio(OH),Mn3(Si4)O1o(OH),MgAlrSiOs(OH),FeAl,SiOs(OH),MnAl,Si05(OH),

Mg4Al,sSi750€H4FelAlrsSiT 5O4oH4MniAlloSiz s046H4MgAl,Si,O6(OH)4FeAl,Si,O6(OH)4MnAlrSirO6(OH)4Mg6(M gAl)AlB(Al,Sia)O@CalAl5MgSi60rl(OHDCaleMgrAlll SirsO6r(OH)eFe2Al4SisO1,MnrAl4SisOloFe?t Fe8+ Sil,O4o(OH)10

68 158.9

265.3270.9272.4264.7278.0279.4282.8260.5265 I274 I271 .3

139.7147.5154 .3150 .5157 I144.5209.2213.4219 .0132.9147 .9150 568.86 9 871 0

442.6448.8452.2105 .9106 .9108 .2395.7295.5852.0237.1241.2559.5

4

4

75b

5ooo

222

3434o

232

222

'119

1 02

1 6

41 2

Notei Entropies calculated from the entropy-volume model with no provision for Mg-Al or Si-Al disorder; however, the ideal magnetic entropycontribution has been added, on the assumption that transition metal-bearing silicates undergo tow-temperature magnetic transitions.. Names with elemental prefixes are not valid mineral names but represent idealized end-member (standard state) compositions in thermodynamiccalculations.

user's judgement (or prejudice!) to add an appropriateentropy for disorder.

As a worked example, an estimation is made of theentropy of carpholite, MnAl,SirOu(OH)", for which thevolume is 108.2 cm3.mol-r. All the Al is in octahedralcoordination, and it is assumed that the HrO is nonnal,i.e., like the trioctahedral micas. Thus, from Table 3, andEquation (l),

Scaryr,orit" : V.u,phorit + (,S - V)*"o * (S - Z)tuto,ro,+ 2(,S - V)",o, t 2(,S - Z)H,o(b): 108.9 + 33.4t + 22.60 + 2( t7.45)+ 2(7.44):214.7 * magnelic term.

Finally, one must add the ideal magnetic contribution forMn2*, of R ln 6, to yield S :229 + , r." ,.-o1-t, thevalue given in Table 4.

CoNcr,usroNs

The old concept of additivity of oxide components witha volume correction to yield estimates of mineral entro-pies is a useful one and has been improved upon by con-sideration of variable coordination states of cations inmineral structures. The proportionality constant in theentropy expression to allow for the volume efect has beenfound to be 1.0 if units of J, K, cm3, and mol are used,and this value is in excellent agreement with predictionsfrom the behavior of simple Einstein and Debye solids.

The entropies of silicate and multiple oxide phases maybe estimated with uncertainties of about +2 to +3J.K-'.mol-' for most materials, but with slightly largeruncertainties (+3 to +6 J'K-''mol-t) for phases contain-ing transition-metal ions. These uncertainties are lowerthan for any other simple method of estimation and in

HOLLAND: REVIEW OF ENTROPY-VOLUME DEPENDENCE l 3

many cases are not much larger than the experimentaluncertainties themselves. In conclusion, molar volumeshould be viewed as a reliable monitor of the averagebonding and vibrational state of minerals, a quality thatmakes molar volume useful in estimating entropies.

AcrNowr,nocMENTS

The helpful comments of Roger Powell at an early stage of this project

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M,c.NuscnrFr REcEIVED Mencs 2, 1988MeNuscnrpr AccEPTED Seprer''rsen 6, 1988