Dependence of entropy on volume for silicate and oxide minerals:
A review and a predictive model TrvrorHv J. B. Hor-r.aNn
Department 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 may be used to predict the magnitude of the dependence of entropy on volume for silicate minerals. The origins for the volume effect as well as the effect of variation in coordination state of cation polyhedra on the thirdlaw entropy are explored and rationalized within the framework of simple lattice-vibration theory. It is shown that Einstein and Debye theories for solids predict a value for (05/0V)r^ of about 1.0 J'K-''cm-3, precisely the value 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 multiple regression of this body of measured data. The entropy-volume-coordination model fits these data better than any previously published scheme and works remarkably well even for transition metal-bearing phases when allowance for magnetic disordering is made.
Phases such as magnetite and hematite that undergo magnetic disorder at temperatures above 298 K can be accommodated within the model by correcting for the partial disorder at 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 uncertainties typically 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, so the need for reliable estimates of entropy for mineral phases increases accordingly. While recent years have seen a 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 still many phases remaining that require methods of estima- tion. The reasons for this need, which will never entirely disappear, are (l) calorimetric determinations of entropy require 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 quantity for measurement; or (3) the end-members to be deter- mined are fictive in the sense that they are not stable for the chosen structure or composition.
In principle at least, entropies (and all other thermo- dynamic functions) may be calculated from spectroscopic determination of the lattice-vibration spectrum for solid crystalline materials. Rigorous determinations have rare- ly been attempted (but see Salje and Werneke, 1982, for andalusite and sillimanite) although simplifications in modeling the phonon density of states function, as in Kief- fer (1980), have produced encouraging results. Apart from the fact that this approach is even more labor-intensive
0003-004x/89/0102-0005$02.00 5
than direct calorimetry and therefore unlikely to become commonly used, there still remains the problem of un- availability of the required material in pure form or in the relevant structural state. The astonishing success of Price et al. (1987) in accurately predicting the heat ca- pacity, entropy, and compressibility of forsterite from a set of independently derived interionic potentials holds obvious promise for the future-if this technique be- comes widely used (and extended to a larger system than Mg-Si-O). However, until such time, simpler, quicker methods must be found that aford reliable estimates of mineral entropies and that can be based on a minimum of measured properties.
Past efforts to find estimates of entropy have been re- markably successful at the 5-l5o/o accuracy level, and all involve some modification of the Newmann-Kopp rule, which is based on the observation that heat capacities (and therefore entropies) of complex compounds may be estimated by summing, in stoichiometric proportions, the heat capacities or entropies of simpler chemical entities.
Latimer (1951,1952) and Fyfe et al. (1958) used entro- pies of the elements and of oxides respectively to estimate entropies of more complex compounds. Fyfe et al. noted the positive correlation between molar volume and en- tropy and incorporated a simple volume correction factor in their estimation scheme:
Sr: )n,S, + k(Vj - 2n,V,),
where ,$ is the entropy (to be determined) of phase j, V, is the molar volume of phase 7, whereas S, and V, arc entropies and volumes of the n, oxide components I re- quired to make up phaseT, and k is an arbitrary consmnr determined from measurements.
Helgeson et al. (1978) improved matters somewhat by taking structurally analogous mineral phases as compo- nents instead of oxides; this has the advantage that dif- ferences in coordination state between the components and the phase "being built" are minimized. It has long been recognized that the coordination state affects the entropy; the difference between Al in octahedral and tet- rahedral sites may be readily evaluated and has been used in simple calculations (e.g., Holland and Richardson, 1979) on mineral stability. Recognizing the importance of coordination, Robinson and Haas (1983) used multi- ple regression to derive a set of fictive oxide components from the available measured mineral entropies and heat capacities. The result was a set of oxide components, in varying coordination states, which could be summed di- rectly to estimate the entropy of any desired oxide or silicate. While having the advantage over the structural analogue approach of path independence, the neglect in Robinson and Haas's model of the volume correlation, which was so successful in earlier approaches, makes their method 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 predict accurate entropies for common pyroxene and amphibole components that led to the model proposed here. Some of the more notable entropy discrepancies in the Robin- son and Haas method (see column RH in Table 2) are hercynite (+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 of which would cause unacceptable errors in phase-equilib- rium calculations. The volume-corrected methods of Fyfe et al. (1958) and Helgeson et al. (1978) generally work quite well, and there are sound theoretical reasons for a positive correlation between volume and entropy that warrant a brief review before presenting the revised es- timation method and results.
TrrB nNrnopy-voluME RELATroNsHrp REvtEwED Although 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 that mass 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 expansion to compressibility, is generally a small positive number, this equation does not provide a satisfying explanation of the effect at an atomistic or structural level. The reason
for a positive correlation of entropy with volume can be seen most easily by considering the role of lattice vibra- tions in determining the heat capacities and entropies of crystalline solids. The simplest quantum model for lattice heat capacity was devised by Einstein to explain the fall off 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 heat capcity for I mol of a phase containing n atoms in its formula unit is given by
Cr: 3nRlu'e"/(e" - l)'1,
where R is the gas constant, and u : hu/kT, .vv.rth h and k 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 not match the behavior of real crystals perfectly at low tem- peratures, it does simulate the trend remarkably well for a 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 whose atoms vibrate at a characteristic frequency has the effect of moving apart the component atoms, thus reducing their bond stiffnesses and lowering their vibrational frequency (in proportion to (Vo/tr)'/'). Figure I illustrates the general dependence ofheat capacity on vibrational frequency, and from 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 temperatures for solids.
The magnitude of the volume effect on the entropy of an 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 expression above for the entropy, one finds by substitution that
( d S / d V ) 2 , 8 : n R u ' / l V ( e ' - l ) ( l - e fl.
In this expression, Z is the molar volume of the mineral concerned. Evaluating this expression at 298 K for all phases listed in Table l, the mean value of (dS/|V)r* is found 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-
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, with only one adjustable parameter, the heat capacity and en- tropy of crystals, and it should give a reliable estimate of the volume dependence of the entropy. The relationship of normal mode frequencies to entropy will be explored further later, but in passing it should be noted that the effect of volume on heat capacity will lessen at higher temperature where thermal agitation is more pronounced and will tend to mask the smaller volume-related effect.
It is for this reason that high-temperature heat capacities may be modeled quite reasonably by a simple additivity of oxides approach without consideration of the volume coffection (as done by Berman and Brown, 1985).
It is interesting to inquire whether an increase in the sophistication of the assumptions used alters this result, and so the above exercise was repeated using the Debye model for heat capacities. In place of a single frequency to characterize the vibrational spectrum, Debye assumed a 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 that taken above, using the Debye (D) expression for the en- tropy with 0r: hv^ */k expressed in parameterized form, glves
S r r r . o : n ( l 3 l 4 l / 0 " - 3 . 8 1 )
where the constants were determined from a least-squares fit to the Debye entropy function at 298 K and entropy is in J.K-' mol '; hence,
( A S / A V ) , , s : ( , S + 3 . 8 1 n ) / 3 V ,
which on evaluation (Table 1) gives a slightly lower value of (dS/dZ)rn, : 0.93 + 0.10 J'K ''cm 3. Given that the Debye approximation is often used best in the low-tem- perature limit and the Einstein model often works well for approximating the high-temperature behavior, an av- erage might be appropriate, and a value of unity for the proportionality constant k is accepted, as predicted from simple lattice-vibration models.
Trrs nnLlrroNSHIp BETwEEN ENTROPY AND
COORDINATION
Having examined the entropy-volume relationship, I now turn to look briefly at the role ofcoordination state.
As a simple example, I will take the CarSiOo minerals larnite and calcium olivine that have entropy values of 12'7.6 J'K' and 120.5 J'K ', respectively, despite larnite having the smaller volume. Clearly larnite owes its higher entropy to the distorted and much larger site for one of its two Ca ions (see Table 1), while presumably it man- ages to keep its volume reduced by a suitable packing arrangement of alternating large (M2) and small (Ml) Ca sites. Thus the dominant control of the entropy difference between Ca olivine and larnite lies in the coordination change from 6 to 8 one of its Ca sites, as would be pre-
20
c,
1 5 J atom-ll 0
5
0
Debye Heat Capacities
2 . . 3
O Forsterite Cp
1 0 = 5 0 0 K 2 0 = 8 0 0 K 3 e = 1000K
0 100 ZO0 300 400 500
T K
Fig. 1. Heat capacities calculated from the Debye theory us- ing three values for the Debye temperature 0o to show the effect of varying the vibrational frequency (0o : hv/k). Filled circles are 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 the polyhedra. 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, and in 4-fold tetrahedra in sillimanite. The entropies at 298 K a r e 8 2 . 3 . 9 1 . 4 . a n d 9 5 . 8 J ' K ' ' m o l - ' , r e s p e c t i v e l y ; t h u s it is usually argued that the entropy increase in these min- erals occurs in response to reduction in the coordination state. However, in these polymorphs the entropies in- crease in the reverse order from that predicted by the frequencies ofthe internal vibrational modes; the average polyhedral Al-O bond is shortest and stiffest in the tet- rahedral sites (sillimanite) and longest in the octahedral sites (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 internal vibrational modes of the characteristic Al polyhedra in these silicates that dominate the sequence of lowest optic
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
( _ 2 6 . 8 ) f 4 4 9 8 1 1 5 . 6 1
(_2e.8)+
6 7 . 1 8 1 5 9 . 0 1
(_2s.8)+
6 4 5 9 1 5 5 . 7 2
(_ 14.s)+
6 0 . 4 0 1 3 3 5 1
31.69 95.5 l, 3 Merwinite (merw)
44.53 126.2 1
(- 13.4)+
39.78 a0.6 1
1 8 . 8 2 5 0 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, 2lalsio2 Microcline (micr)
0.5r"rKro, 0.5r4lAlro3, 3r4rsio, Kaliophilite (kal)
0 5r"lKrO, 0.5l4lAlro3, I4lSi02 Leucite (lc)
0.srbrK2o, 0.5r4rAlro3, 2r4rsio, Albile (ab)
0 srelNaro, 0.st4rAlroo, 34rSi02
3telcaO, 16rFerO3, 3SiO,
3 9 9 3 8 1 . 7 1 0 6 3 . 5 6 1 3 5 . 3 1 1 6 6 . 1 9 1 4 2 . 7 1 0 6 2 . 6 8 1 3 2 . 5 1 0 3 5 . 1 6 8 7 . 6 1
(_ 14.9)+
272.70 549.1 1 0 0 7 9 1 9 9 . 3
98.47 253.1 1
108.72 2't4 2 1 5 9 . 8 9 1 3 3 . 3 1 8 8 . 3 9 1 8 4 . 3 1
100 04 207.4 1
13.22 44 I
( 14.e)+
1 6 . 7 6 3 8 . 1 1 1 . 2 5 2 7 . 0 2 5 . 5 8 5 0 . 9
40.75 92.9
( - 13.4)+
6 7 . 8 8 1 6 0 . 8
(_ 13.4)+
6 5 . 9 2 1 6 2 5
(_26.8)+
44.09 82.3
50 03 95.8
5 9 . 1 1 1 2 0 . 5 5 1 . 6 1 2 7 . 6 90.24 198 6 92 54 209.2 5 1 .4 8 1 0 8 1 s 5 . 6 5 1 2 9 . 3 46.30 124 2
(_26.8)+
43.66 94.1
2 3 3 2 2 4 0 7 . 1 4 8 . 6 1 1 3 3 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 1
0.5rrrNaro, 0 5rrAlro3, r4tsio,
Muscovite (mu) 140.83 287.7 1
0.srotKro, r6tAlro3, 0 StotAlzoo, 34rsior, Hro(a)
Phlogopite (ph) 149.64 315.9 12
0 slblKro, 3FrMgO, 0.5t1lAl2oo, 3lolsior, HrO(b)
Talc (ta) 136.25 260.8 1
3rrMgo, 4rrsior, Hro(b)
Pyrophyllite (pph) 127.61 299.4 1
16141203, 4r4rsi02, Hro(a)
Anthophyllite (anth) 265.4 537.0 10
2r4MgO, 516rMgO, 8r41SiOr, H,O(b)
Clinochlore (clin) 210.9 397.6 13
516rMgO, 0.5t6tAlzOs, 0 SrntAlzos, 3rrsior, 4Hro(b)
Margarite (ma) 129.60 263.6 't4
telcao, t6lAl2oo, I4rAlrOo, 2l4iSior, Hro(a)
Paragonite (pa) 132.11 277.1 12
0.srerNa2o, 16rA1103, 0.5r4iAlro3, 3r4rsior, Hro(a)
Diaspore (dia) 17 76 35.3 1
0.516141203, 0.5Hro(a)
Gibbsite (gib) 32.03 68.4 1
0.5r6tAlro3, 1 sH,O(a)
Prehnite (pre) 140.26 292.8 14
2lerCaO, 0.516rAlrOs, 0 storAl2o3, 3SiOr, HrO(b)
Chrysotile (chy) 107.46 221 3 1
316rMgO, 2r4tsiOr, 2HrO(a)
zoisite (zo) 135.88 295.9 14
2rrrcao, 1.516rA1203, 3r4isior, 0.5Hro(b)
A l m a n d i n e ( a l m ) 1 1 5 . 1 1 2 9 9 . 6 1 5
3rsrFeo, 16rA1103, 3sio, (_40.1)+
P y r o p e ( p y ) 1 1 3 . 1 8 2 6 6 . 3 j 6 3rsrMgo, r6iAlro3, 3sio,
Grossular (gr) 125 35 254.7 17
3tslcao, 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 H e m i n g w a y , 1 9 8 4 a ; ( 7 ) S h a r p e t a l . , 1 9 8 6 ; (8 ) Robie et al., 1982a; (9) Robie et at., 19.82b; ( 1 0 ) K r u p k a e t a t . , i9 B 5 ; ( 1 1 ) H a s e t t o n e t d t . , ts e + ; 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; i17i P e r k i n s e t a l . , 1 9 7 7 ; ( 1 8 ) R o b i e e t a t . , 1 9 8 7 .
modes, and hence entropies, but some unspecified exter- nal modes associated with the linking of the polyhedral units. We must therefore expect the common occurrence of phases in which the positive entropic effect of smaller and more rigid polyhedra is offset by the lower rigidity
of the interconnecting framework. Given that the entropy and other thermodynamic properties are functions of the average vibrational spectrum, we might expect that the molar volume should reflect the overall bonding, and hence vibrational, state of the crystal. In the aluminosil-
icate example, the entropies of the three polymorphs are indeed propoftional to their molar volumes.
From the above discussion it is clear that the effects of cation coordination as well as the volume contribution must be incorporated into the additivity models for min- eral entropy estimation, although in many complex sili- cates it may be impossible to ascribe anomalous entropy to any one particular structural feature. Any model based solely on the coordination state and molar volume can form only a crude guide to the vibrational spectrum, but as shown below may be sufficiently accurate in predicting entropies of silicates from a minimum of information.
Some non-lattice-vibrational contributions to entropy Contributions to the entropy arising from phenomena other than lattice-vibration effects have been discussed in considerable detail in the literature, and the interested reader may turn to the review article by Ulbrich and Waldbaum (1976) for more detail. For the present pur- poses, all such contributions to the 298-K entropy must be removed before looking at the relationship of entropy with volume, and so terms arising from magnetic as well as site disordering and possible electronic efects arising from Jahn-Teller site distortions and other crystal-field effects must be taken into account.
The site<onfigurational entropy terms involved in, for example, Al-Si order-disorder on tetrahedral sites, have been removed from the tabulated entropies, and only the calorimetric entropies have been used in the following analysis. Care must be taken when using tabulated entro- pies; in the tables of Robie et al. (1979), certain phases have 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 this is always justified, and in the case of muscovite, the ex- perimental phase relations are consistent only with a largely ordered state. Strong short-range order can reduce the entropy contribution to very small values, and recent work suggests that (Al,Si) in muscovite is ordered on a Iocal 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-capacity anomalies (I peaks) that contribute to the entropy at 298 K. 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), whereas for Fe3* and Mn2*, s is 5/2 and so the entropy contribu- tion becomes R ln 6 (14.9 J.K '.atom '). In the analysis to follow, these ideal entropies for magnetic disorder have been subtracted from the phases that have a low-temper- ature magrretic transformation. The minerals hematite and magnetite 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 the entropy 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.:
S u , o . , : S - " * [ l - ( 1 - T / 7 . ) ]
where S-.. is the maximum entropy expected for the transformation. i.e..
*"^: J" 7 o,
and Cp is the excess heat capacity relative to the fully ordered phase. It is conventional, in Landau theory, to take excess properties relative to the disordered, high- symmetry, phase, whereas the above expressions have been rearranged according to the more usual petrological convention. For hematite (7. : 955 K, ,S."* : 2R ln 6), the entropy contribution at 298 K, from the equation above, 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 is not negligible, and the value of 5.1 J'K r.mol-r should be subtracted from the calorimetric entropy of hematite if one wishes to determine the lattice-vibrational portion of the entropy. A similar argument for magnetite (2. : 848 K) suggests that the entropy of magnetite should be decremented by 8.4 J.K-' mol-'to allow for the \ tail effect. 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) that occurs at 1 15 K.
A further problem is the variable entropy contribution from transition-metal ions (in this study in FeO, FerOr, TiOr) due to polyhedral site distortions. The subject has been discussed by Wood (1981) who showed that crystal- field effects can be large and important for these cations under some circumstances. The effects discussed by Wood should be maximal only at very low temperatures and for extreme differences in the shape of a distorted and un- distorted polyhedron; the success of the simple model to be discussed in the next section implies that at 298 K these 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 both volume and coordination and the fact that both of these are only an approximate guide to the vibrational contri- bution to the specific heat and entropy, one opts for the simplest possible model-that of Fyfe et al. (1958)-but allows some components to be represented in several dif-
s v
Phase. hat meas.
Tlele 2. Results of regression for S - yof phases ferent coordination states: The Fyfe et al. expresslon 1s recalled 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 the entropy and volume of the phase j. A certain amount of experience with measured data has shown that the value of k is approximately 1.0 if the units of entropy are in J.
K '.mol 'and volumes are in cm3.mol '. A petrological consequence ofthis is that solid-solid reactions involving no change in coordination state should have dP/dT: AS/
AZof approximately l0 bar.K-'. An example of such a reaction might be tremolite + 2 diopside * talc. Unpub- lished experiments (Jenkins and Holland, in prep.) are in good agreement with this estimate, which should be rea- sonable for reactions, such as the one above, that involve little or no change in coordination. In contrast, reactions such as jadeite + quartz + albite are driven by larger entropies, in part due to the increase in coordination state ofNa 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 the above equation to the form
S,: kV, + 2n,(5, - kV,),
where the second term on the right is a constant for each oxide component. The values of the (St - kV,) and k were determined by least squares from measured entropies of oxides 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 the constant 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 by regression. 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 deviation being 4. I J.K '.mol ' (for tremolite). Table 2 shows the calculated results and the residuals in entropies for the phases used, and Table 3 lists the values for the entropies associated with each oxide component and their uncer- tainties (lo).
The assignment of the components chosen requires brief comment. 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 orthopyroxenes and the M4 site in orthoamphiboles. Similarly, tsrQa6 refers to M2 clinopyroxene and M4 amphibole sites as well as the Ca in larnite and sphene. I8-'OlCaO represents Ca in much enlarged sites (e.g., in margarite) andlor those difficult to define precisely in terms of coordination num- s v
calc. resid RH ko kE
m t 0 . 3 1 4 hem 0.283 Timt 0.364 CaFt 0.249 DCFI 0.358
acm 0.351
j d 0 . 7 1 2
ilm 0.237
s p 0 7 3 7
ru 0 379
trid 0 037 mang 0.229 l i m e 0 1 6 1 per 0 071 c o r 0 1 7 6 h e r c 0 . 1 5 9 h e d 0 . 1 3 7 f s 0 . 3 1 3
ky 0.353
sill 0.230 caol 0.400 larn 0.379
geh 0.208
ak 0.409
m o n t 0 . 1 2 3
sph 0.298
fa 0.294
fo 0.346
crd 0 502 teph 0.621 w o 0 . 1 7 0 c a t s 0 . 1 5 4 d i 0 2 3 4
en 0 533
r h o 0 1 6 0
It 0 417
an 0 259
merw 0.478 micr 0.632
kal 0.425
l c 0 . 3 1 5
ab 0.481
ne 0.323
mu 0 530
phl 0.394
t a 0 . 1 8 9
pph 0.348
anth 0.533 clin 0.879
ma 0.240
pa 0.350
dia 0.039
gib 0.258
pre 0.304
chy 0.551
zo 0.377
alm py g r andr
8 1 .7 0 8 1 . 0 2 0 6 8 52.03 50.24 1.79 9 5 . 2 9 9 4 1 8 1 .1 1 7 0 . 5 8 7 2 1 8 1 . 6 0 9 1 . 8 1 9 4 1 3 2 . 3 2 9 1 .0 8 8 8 . 1 8 2 . 9 0 7 3 . 0 7 7 4 3 6 1 . 2 9 63.83 63.41 0.42 4 0 . 8 0 4 1 . 3 6 - 0 5 6 3 1 . 4 3 3 2 6 3 - 1 20 1 7 . 3 7 1 7 4 5 - 0 0 8 3 1 . 5 9 3 3 4 1 1 . 8 2 21.34 21.94 -0 60 1 5 . 7 0 1 5 7 5 - 0 0 5 25 35 22 60 2.75 5 2 1 4 5 3 3 7 1 . 2 3 92.92 93.0s -0 13 96.54 96 45 0.09 3 8 . 2 1 4 0 . 0 4 - 1 . 8 3 45.76 43.19 2.57 6 1 3 9 6 1 . 3 4 0 . 0 5 7 6 . 0 0 7 2 . 1 9 3 . 8 1 1 0 8 3 6 1 0 8 . 0 8 0 2 8 1 1 6 . 6 7 1 1 5 . 4 1 1 . 2 6
5 6 . 6 2 5 5 . 1 4 1 4 8 77.65 77 .45 0 20 7 7 . 9 4 7 9 . 0 0 - 1 . 0 6 50.45 48.94 1 51 1 7 3 . 9 0 1 7 6 . 5 1 - 2 . 6 1 84.80 84 28 0.52 41.76 39.39 2.37 7 1 .7 4 7 0 . 5 6 1 .1 8 7 6 . 5 1 7 8 . 0 2 - 1 . 5 1 6 9 . 8 6 7 1 .7 1 - 1 8 5 52.44 50 86 1.58 276.40 280 52 -4 12 9 8 . s 1 9 8 . 1 5 0 . 3 6 1 5 4 6 0 1 5 3 7 5 0 . 8 5 1 0 5 4 8 1 0 6 . 5 7 1 .0 9 7 3 3 7 7 1 .6 7 1 . 7 0 95.93 93 32 2.61 1 0 7 . 3 6 1 0 6 . 5 3 0 . 8 3 7 0 . 1 9 7 1 . 6 4 - 1 4 5 1 4 6 . 8 7 1 4 9 0 8 - 2 . 2 1 1 6 6 3 0 1 6 5 4 5 0 . 8 5 124 54 124.48 0.06 1 1 1 7 9 1 0 8 . 1 1 3 . 6 8 27't 60 267 90 3.70 1 8 6 7 0 1 8 6 . 5 8 0 j 2 1 3 4 0 0 1 3 6 . 4 6 2 . 4 6 1 4 5 0 0 1 4 4 . 8 4 0 . 1 6 1 7 . 5 5 1 9 . 1 5 - 1 . 6 0 36.48 34.87 1 61 1 5 2 . 5 0 1 5 4 . 2 7 - 1 . 7 7 1 1 3 . 8 0 1 1 3 . 5 7 0 . 2 3 1 6 0 . 0 0 1 5 8 . 7 0 1 . 3 0 184.45 184.45 0 1 5 3 . 1 2 1 5 3 1 2 0 129.35 1 28.53 0.82 1 5 4 . 5 6 1 5 6 1 7 - 1 .6 1
1 1 4 1 . 2 8 1 1 2 1 2 6 1 . 2 0 1 . 3 3 1 .0 5 1 .1 9 0 . 9 6 1 . 0 8 0 . 9 2 1 . 0 6 - 1 5 . 7 0 . 9 5 1 . 0 9 1 . 2 0 1 . 3 4 - 0 . 8 0 . 9 0 1 . 0 s 1 .0 9 1 . 2 3 1 . 0 0 7 8 0 . 8 9 1 .0 5 1 . 4 5 - 1 .5 0.91 1 .01 4 . 2 1 . 0 2 1 .1 8 8 . 5 0 9 1 1 . 0 8 1 1 . 2 1 . 0 0 1 . 1 3 - 6 . s 0 . 9 1 1 . 0 4 - 2 . 0 1 . 0 1 1 . 1 5 3 . 0 0 . 8 5 1 . 0 1 0.9 0.84 0.98 -6 7 0.83 0.93 3 . 0 0 . 9 9 1 . 1 2 6 . 2 0 . 9 0 1 . 0 2 0 . 1 0 . 9 2 1 . 0 3 2.9 0.87 0.99 0 . 9 6 1 . 0 8 2 . 7 1 . 0 9 1 2 2 5 . 7 0 . 9 2 1 0 6 1 6 . 9 0 . 7 4 0 8 5 1 . 1 0 1 2 2 - 0 . 8 0 . 8 4 0 . 9 5 - 3 . 5 0 . 9 1 1 . 0 5 - 8 . 0 0 . 9 1 1 . 0 4 1 2 0 . 9 1 1 . 0 5 1 .0 1 1 . 1 4 - 2 1 . 0 0 . 8 6 0 . 9 9
0 0 . 8 2 0 . 9 4 - 1 0 1 0 4 1 . 1 6 - 0 8 0 8 1 0 . 9 1 4 . 0 0 . 8 9 0 . 9 9 12.1 0.84 0.94 0 0 . 8 6 0 . 9 7 2 . 7 0 . 9 3 1 . O 4 1 . 4 0 . 8 7 1 . 0 0 4 . 4 0 8 8 1 . 0 1 - 7 I 0 . 8 3 0 . 9 7 - 3 4 0.82 0.96 5 . 8 0 . 8 7 1 . 0 1 - 1 4 2 0.85 0.99 - 0 I 0 . 8 8 1 . 0 3 2 . 4 0 . 9 0 1 . 0 4 - 0 . 3 0 . 9 5 1 . 1 5 3 . 9 0 . 9 9 1 .1 8 - 2 . 8 0 . 8 9 1 0 2 9 . 5 0 . 9 0 1 0 5 4 I 0 . 9 3 1 . 0 7 1 0 5 1 . 0 9 1 . 2 4 - 1 3 . 7 1 . 0 1 1 . 1 6 4 2 9 0 . 8 8 1 . O 2 0 . 9 2 1 . 0 5
Nofei Columns (S - y).""" and (S - y)* are measured and catcutated S - y; resid is the residual in calculated entropy; RH is the equivalent residual calculated from the tables of Robinson and Haas (1983); ko and kE 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 the influence of each observation on the least-squares solution. with hat : 0 denoting no influence and hat : 1 denoting extreme influence (forcing the fit through that datum).
. Abbreviations are given in Table 1.
ber. KrO was partitioned intot"lKrO framework sites such as in feldspars and feldspathoids, and IolKzO sites such as in interlayer mica positions (and the large cavity of leu- cite); thus size and degree ofvibrational freedom seem to categorize KrO ditrerences. NarO is partitioned into small t8lNazO as in pyroxene (M2) and amphibole (M4) sites and large Ie-r2lNa2O as in framework silicates and micas.
HrO was the most difficult parameter to define; in the end, it was decided to split HrO up into only two cate- gories for simplicity. Although the nature of the bonding ofthe proton in hydrous silicates is very variable, it was found convenient to split mineral structures into a high- entropy HrO(a) and a low-entropy HrO(b) group. The micas fall naturally into high-entropy dioctahedral and low-entropy trioctahedral groups that may be rational- ized on the basis ofthe distortion ofthe proton position in dioctahedral micas away from the normal to the mica sheets; in trioctahedral micas, the three full octahedral sites repel the proton equally (Bailey, 1984). In diocta- hedral micas, the vacant M2 site causes the proton to be deflected largely into the vacant space, allowing for a greater degree of vibrational freedom. Chlorites are trioc- tahedral, with talc-like and brucite-like layers in which there is strong hydrogen bonding of the brucite hydroxyl groups to the talc layer oxygens, and so chlorites are at- tributed to the HrO(b) group, as are talc, phlogopite, and the amphiboles. Diaspore, gibbsite, and serpentine are allocated to the HrO(a) group because of the larger degree offreedom ofthe OH groups that are not strongly bonded to tetrahedral layers as in micas or, in the case ofserpen- tines, because the layer mismatch between the octahedral and 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 separate evaluation; the entropy contribution from Ca in garnet re,t(QaO) is less than expected whereas the entropy con- tributions of iron r",t6eO) and Mg te,r(MgO) are larger. For Mg, a small ion in a very large and somewhat distorted cage, the extra entropy is readily rationalized, but the low entropy for Ca seems anomalous.
To see the efect of ignoring the volume dependence of the entropy, a regression of the same data was performed without the volume terms. as done bv Robinson and Haas
( 1 e 8 3 ) ,
{: >n,s" (2)
and the results are given in Table 3. The standard devia- tion of the residuals and the mean absolute deviation of the 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 that with 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 C o m p o n e n t S - Y o s , v S O g
trrSiOz 16lAlzOs torAlzOg 16rMgO tolMgo rtrMgo rs,rMgO rorCaO rorcaO rerolcao IdlCaO rsrFeO rdlFeO {6rMnO 16lTiOz t"rFezOg IolNarO tg;2tNazO htKrO rorK20 H,O(a) H,O(b)
1 7 . 4 5 22.60 28.89 1 5 . 7 5 18.77 2 1 . 0 6 z o . u o 21.94 2 7 . 3 7 34.37 1 7 . 8 6 3 0 . 7 8 3 6 . 5 0 33 41 32 63 50.24
3 0 . J 2
79.49 79.55 87.96 1 5 . 7 1
7 . 4 4
0 . 3 8 0.84 1 . 0 6 0 5 3 1 7 7 1 3 6 0 8 8 0 . 8 0 0 . 8 4 0.70 0.67 0.83 0.67 1 . 2 0 1 . 5 4 1 . 6 0
J . O /
3 . 1 2 3.03 0 9 1 o.87
40.30 0.39
43.78 0.84
7 2 . 0 7 1 . 0 5
26.67 0.54
38.30 1 77
27.74 1 .35
33.87 0.88
3 9 . 5 9 0 . 8 0
38.73 0 81
48.25 0 71
29.47 0.99
43.24 0.83
44.96 0.68
46.28 1.20
5 1 . 9 4 1 . 5 4 8 0 . 5 1 1 . 6 1
65.86 4.58
9 7 . 2 8 3 1 3 1 1 4 . 3 5 3 . 7 7
120.37 3.37
30.03 0.91
20.74 1.03
Nofe. The first two columns of data, S y and its uncertainty, refer to the entropy-volume model, Eq. 1 in the text; the last two columns, S and its 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 model also fits the measured entropies rather better than the data of Robinson and Haas (1983) and may be useful in estimating entropies of mineral end-members for which no reliable volume data arc available.
Appr-rca.rroNs
One of the motivating reasons behind this study was the need to estimate the entropies for minerals in a more general project to derive a reliable thermodynamic data set for petrological calculations. In earlier works (Powell and Holland, 1985; Holland and Powell, 1985), a prelim- inary thermodynamic data set was generated and used to obtain more powerful methods of characterizing meta- morphic conditions, particularly pressure (Powell and Holland, 1988). The project is now at an advanced stage, involving many more mineral end-members and phase- equilibrium constraints, and has required the estimation ofentropies ofseveral phases where they were not known.
It is also useful to have reliable entropy estimates for minerals even if complex thermodynamic calculations are not required, for instance when determining approximate slopes ofunivariant reactions from the Clausius-Clapey- ron equation. The entropies for a number of rock-form- ing mineral end-members of interest are presented (Table 4), using the methods outlined above. Because it appears that 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 entropy terms have been omitted, as the degree of order in most silicates is at present poorly constrained. It is left to the
TABLE 4. Predicted entropies for unmeasured mineral end-members Y S t o "
(cm3 mol ,) (J K ' mol t) (J K 1.mol 1)
Johannsenite
Mg-Tschermak's pyroxene Tschermakite
Endenite Pargasite Cummingtonite Grunerite Ferropargasite Ferroactinolite Glaucophane Ferroglaucophane Riebeckite
Magnesioiriebeckite
Mg-celadonite.
Eastonite Annite Siderophyllite Manganophyllite Na-phlogopite- Amesite Daphnite Mn-clinochlore.
Tschermak's talc.
Minnesotaite Mn-talc.
Mg-chloritoid.
Fe-chloritoid.
Mn-chloritoid.
Mg-staurolite' Fe-staurolite.
Mn-staurolite.
Mg-carpholite- Fe-carpholite.
Mn-carpholite.
Sapphirine Mg-pumpellyite- Vesuvianite Fe-cordierite.
Mn-cordierite.
Deerite
Chain silicates 179 123
R e r
588 582 533 734 703 705 534 624 6 9 1 602 Sheet silicates
288 306 405
J O C
421 306 388 542 568 250 358 373 132 162 167 Others
885 993 1 0 1 3
194 223 229 790 629 2008 470 483 1 531
CaMnSir06 MgAl(SiAl)O6
CarM 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)s FesAl(AlSia)Olo(OH)s MnsAl(AlSi3)Oio(OH)s Mg,Al(Si3Al)O,o(OH), Fe3(Si4)Oio(OH), Mn3(Si4)O1o(OH), MgAlrSiOs(OH), FeAl,SiOs(OH), MnAl,Si05(OH),
Mg4Al,sSi750€H4 FelAlrsSiT 5O4oH4 MniAlloSiz s046H4 MgAl,Si,O6(OH)4 FeAl,Si,O6(OH)4 MnAlrSirO6(OH)4 Mg6(M gAl)AlB(Al,Sia)O@
CalAl5MgSi60rl(OHD CaleMgrAlll SirsO6r(OH)e Fe2Al4SisO1,
MnrAl4SisOlo Fe?t Fe8+ Sil,O4o(OH)10 68 1
5 8 . 9 265.3 270.9 272.4 264.7 278.0 279.4 282.8 2 6 0 . 5 265 I 274 I 2 7 1 . 3
1 3 9 . 7 1 4 7 . 5 1 5 4 . 3 1 5 0 . 5 1 5 7 I 144.5 209.2 213.4 2 1 9 . 0 1 3 2 . 9 1 4 7 . 9 1 5 0 5
6 8 . 8 6 9 8 7 1 0
442.6 448.8 452.2 1 0 5 . 9 1 0 6 . 9 1 0 8 . 2 395.7 295.5 852.0 2 3 7 . 1 241.2
5 5 9 . 5
4 4 7 5
b 5 o o o
2 2
2
3 4 3 4
o
2 3
2
2 2 2 '11
9 1 0
2
1 6 4
1 2
Notei Entropies calculated from the entropy-volume model with no provision for Mg-Al or Si-Al disorder; however, the ideal magnetic entropy contribution 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 thermodynamic calculations.
user's judgement (or prejudice!) to add an appropriate entropy for disorder.
As a worked example, an estimation is made of the entropy of carpholite, MnAl,SirOu(OH)", for which the volume is 108.2 cm3.mol-r. All the Al is in octahedral coordination, and it is assumed that the HrO is nonnal, i.e., like the trioctahedral micas. Thus, from Table 3, and Equation (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 for Mn2*, of R ln 6, to yield S :229 + , r." ,.-o1-t, the value given in Table 4.
CoNcr,usroNs
The old concept of additivity of oxide components with a 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 in mineral structures. The proportionality constant in the entropy expression to allow for the volume efect has been found to be 1.0 if units of J, K, cm3, and mol are used, and this value is in excellent agreement with predictions from the behavior of simple Einstein and Debye solids.
The entropies of silicate and multiple oxide phases may be estimated with uncertainties of about +2 to +3 J.K-'.mol-' for most materials, but with slightly larger uncertainties (+3 to +6 J'K-''mol-t) for phases contain- ing transition-metal ions. These uncertainties are lower than for any other simple method of estimation and in
many cases are not much larger than the experimental uncertainties themselves. In conclusion, molar volume should be viewed as a reliable monitor of the average bonding and vibrational state of minerals, a quality that makes molar volume useful in estimating entropies.
AcrNowr,nocMENTS
The helpful comments of Roger Powell at an early stage of this project are much appreciated, as are reviews by Bob Newton and Sue Kieffer that led to improvements in the discussion.
RnrnnBNcps crrno
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M,c.NuscnrFr REcEIVED Mencs 2, 1988 MeNuscnrpr AccEPTED Seprer''rsen 6, 1988
