American Mineralogist, Volume 67, pages 470482, 1982

Heat capacities and entropies of Mg2SiOa, Mn2SiOa, and Co2SiOabetween 5 and 380 K

Rrcnnno A. RonE. Bnuce S. HpvrrNcwey

U.S. Geological SurveyReston, Virginia 22092

eNo Hurvrrutro Terer

Research Institute for lron, Steel and other MetalsTohoku University, Sendai, Japan

Abstract

The heat capacities of synthetic single crystats of Mg2Sio4 (forsterite), Mn2Sioa(tephroite), and co2Sioa (cobalt olivine), were measured between 5 and 3g0 K using anadiabatically shielded calorimeter. Mg2SiOa is diamagnetic, and its heat capacity follows anormal sigmoidal curve at low temperatures. co2sioa shows a single sharp )r-typetransition at 49.85+0102 K associated with the antiferromagnetic ordering of the magneticmoments of the Co2+ ions into a collinear spin arrangement below 49.8 K. In contrast toco2sioa, Mn2Sioa has two transitions in ci, a sharp L-type transition at 47.3g10.05 K anda smaller "shoulder" in Ci centered near 12 K. The upper transition corresponds to theparamagnetic (disordered) to collinear antiferromagnetic ordering of the Mn2+ moments,whereas the shoulder near 12 K corresponds to the change from the collinear to a cantedspin structure. Our calorimetric values for the antiferromagnetic-paramagnetic transitiontemperature (N6el Temperature) are in excellent agreement with those obtained by powdermagnetic susceptibility measurements, 49t2 K and 5015 K for co2Sioa and Mn2Sioarespectively.

The thermal Debye temperature, 0$, of Mg2Sioa calculated from our c$ measurementsbetween 6.3 and 13.8 K is 768+15 K and agrees well with the elastic vaiue ofi of 758 Kbased on the mean sound velocity calculated from the room temperature elastic stiffnessconstants (ci:) of Graham and Barsch.

At 298.15 K (25"c) the molar heat capacities are 118.6, 128.7 , and 133.4 J/(mol . K) andthe molar entropies are 94.11-r0.10, 155.910.4 and, 142.6-+0.2 Jl(mol . K) respectively forMg2SiO4, Mn2SiOa, and Co2SiOa.

Introduction

The olivine group minerals are one of the moreimportant constituents of igneous rocks and arebelieved to be a dominant phase in the upper mantleofthe earth. They are also significant as the precur-sor minerals from which certain commercial talcand asbestos deposits have formed. Accurate val-ues for their thermodynamic properties over a widetemperature range are thus necessary for the quan-titative solution of both geochemical and geophysi-cal problems.

The olivines have the general formula M2(SiO4)where M is Mg, Fe, Mn, Ni, Ca, and Co. Theycrystallize in the orthorhombic space group pbnm(62) with 4 formulas per unit cell. yCa2SiOa (calcio-0003-0G)v82l050G0470$02.00 470

olivine) occurs very rarely in nature and Co2SiOadoes not occur as a discrete mineral. However.studies by Ross and others (1954) indicate that Coand Ca are common, though minor components inmany olivines, and fCazSiO+ is important in thephase chemistry of Portland cement clinker.

The crystal structure of the olivine form ofMg2SiOa has been refined by Hazen (1976) and thatof Co2SiOa by Morimoto et al. (1974). Fajino et al.(1981) have refined the structures of synthetic for-sterite and tephroite prepared by one of us (H.T.).

The olivine structure is based on a slightly dis-torted hexagonal close-packed array of oxygen at-oms in which one-eighth of the tetrahedral intersti-ces are filled by silicon and one-half of theoctahedral interstices are occupied by the divalent

ROBIE ET AL.: ENTROPIES OF MgzSiOa,, Mn2SiOa, and Co2SiOa 471

cation M. The divalent cations occupy two crystal-lographically distinct sites normally called M(1) andM(2). The M(1) site has the point symmetry 1whereas M(2) has point symmetry m.

Powder neutron diffraction studies by Nomura e/al. (1964) indicate that in CozSiO+ the spins associ-ated with the Co2* ions are collinearly orderedparallel to [001] at temperatures below 49 K. Similarstudies by Santoro et al. (1966) show that tephroitehas a canted antiferromagnetic spin structure below13 K, and has the same collinearly ordered spinstructure as CozSiO+ between 13 and 47.4 K.

Previous studies of the low-temperature heatcapacities of Mg2SiOa were made by Kelley (1943)over the temperature range 53.2-295.0 K. King andWeller (reported by Mah, 1960) measured C[ ofMnzSiOr between 52.3 and 296.1K. Orr (1953) hasmeasured the heat content of MgzSiO+ between 298and 1800 K and Christiansen (reported by Mah,1960) has determined lft-Iffx up to 1795 K forcrystals and liquid Mn2SiOa. We are unaware of anyprevious measurements on the heat capacity or heatcontent of cobalt olivine.

MaterialsMg2SiOa

The forsterite sample was a single crystal boulegrown by the Czochralski method using an iridiumcrucible. The starting materials were 99.9 percentpurity MgO and SiO2 which were blended in the 2: 1molar proportions and sintered in a resistance fur-nace for 6 hours at 1300'C. The crystal was pulledfrom the molten MgzSiO+ (:1900'C) at 3 to 10 mm/hour with a rotation rate of 20 to 30 revolutions/minunder a nitrogen atmosphere (flow rate 1-3 liter/min). Details of the apparatus and growth proce-dures are given by Takei and Kobayashi (1974) whoalso provide chemical analyses, unit cell parametersand densities of several other boules grown underthe same conditions. The boule initially was 2.2 cmin diameter with a length of 5.6 cm. In order to fitinto our calorimeter it was sliced into quarters, witha diamond saw, parallel to the long axis.

Electron microprobe analyses of our sample byL. B. Wiggins (U.S. Geological Survey) are given incolumns 2 and 3 of Table 1. The tabulated valuesare the averages of 7 and 6 separate analysesrespectively.

Mn2SiOa

The tephroite sample was also a single crystalboule, initially 1.7 cm maximum diameter by 5.7 cm

in length. The starting materials were 99.9 percentMnCO3 and SiOz reagents mixed in the 2:1 molarratio and fired at 1000'C under a nitrogen atmo-sphere for several hours. The boule was grown fromthe melt (-1350"C) in an iridium crucible under aprotective atmosphere of nitrogen plus hydrogen(500:1 by volume) using the Czochralski technique.The pulling rate was 3 to 8 mm/hour with a rotationrate of 10 to 15 revolutions per minute. Additionaldetails of the growth procedure and chemical analy-ses, X-ray cell parameters, and optical absorptionspectra for crystals grown under the same condi-tions with the same apparatus are given by Takei(1976). Microprobe analyses of the heat capacitysample are listed in columns 5 and 6 of Table 1 andare the averages of 8 and 3 separate analysesrespectively.

Co2SiO4

The cobalt orthosilicate sample was composed offragments of 3 separate boules. The fragmentsweighed between 3 and 10 grams. The crystals weregrown by the floating zone method using the appa-ratus and techniques described by Takei (1978)'Electron probe analyses of the cobalt orthosilicatesample are listed in columns 8 and 9 of Table l.

Experimental measurements

The heat capacity measurements were made us-ing the intermittent heating method under quasi-adiabatic conditions using the cryostat described by

Table l. Chemical analyses of MgzSiOr, Mn2SiOa and Co2SiOasingle crystals used for heat capacity measurements.

4 5 6 7 8 9

2 9 . 7 5 2 9 . 3 0 2 e . 4 1 ? . 8 . 6 2 2 7 . 3 2 7 . 4

1 2 3

Si02 42 .71 42 .11 42 .26

t4s0 51 .29 58 .31 57 .05

l'1n 0

Co0

70.25 69 .A6 70 ,44

7 t . 3 8 7 t . 8

1 0 0 . 0 0 1 0 0 . 4 2 9 9 . 3 0 1 0 0 . 0 0 s 9 . 1 6 q 9 . 8 4 1 0 0 . 0 0 c s . l 9 9 . 5

I

4

5 , 6

7

Mg2Si04

t ' 4 i c r o p n o b e a n a l y s i s h y L . B l , l i g g i n s , U ' s . G - " o l o g i c a lSurvey

l,4n,5i 0,

M i c r o p r o b e a n a l y s t s h y L . B - l , l i g q i n s , U . s . G e o l o g i c a lSurvey , Res ton , VA

Co 25 i 04

l l i c r o p r o b e a n a l y s i s b y L . B . l i i g g i n s . t J . S . G e o l o g i c a lS u f v e y , R e s t o n , V A

472 ROBIE ET AL.: ENTROPIES OF MgzSiOq Mn2SiOa, and Co2SiOa

Robie and Hemingway (1972); together with thecalorimeter and semiautomatic data acquisition sys-tem described by Robie et al. (1976). Temperatureswere determined with a Minco model S 1059-lplatinum resistance thermometer (Ro : 100.02ohms) which had been calibrated by the Tempera-ture Measurements Section of the National Bureauof Standards on the International Practical Tem-perature Scale of 1968 (IPTS-68) between 13.8 and505 K, and the provisional temperature scale usedin this laboratory between 4.2 and 13.8 K as dis-cussed by Robie, et al. (1978). The samples weresealed in the calorimeter under pure helium gas at apressure of 5 kPa, (approximately 4 x l0-5 mol ofHe).

The formula weights used were based on the 1975values for the atomic weights and are 140.694,201.960, and 209.950 g mol-r for Mg2SiOa,Mn2SiOa, and Co2SiOa respectively. The sampleweights, corrected for buoyancy, used in our mea-surements were 46.537 ,41.200 and 46.012 grams forMg2SiOa, Mn2SiOa, and Co2SiOa respectively.

For the measurements on Mg2SiOr the calorime-ter contributed 88 percent of the total measuredheat capacity at 10 K, 60 percent at 100 K and 43percent at 300 K and above. For Mn2SiO4 thecalorimeter contributed approximately 53 percentto the measured heat capacity between 60 and 380K, dropping to 28 percent at 25 and 5 percent at l0K. For the measurements on Co2SiOo the calorime-ter was 60 percent at l0 K, 38 percent at 50 K andaveraged 55 percent between 50 and 380 K. Thecorrections for deviations from true adiabatic con-ditions were less than 0.2 percent above 30 K andless than 0.1 percent between 35 and 250 K exceptfor the measurements made with small temperatureincrements (A?<0.5 K) in the neighborhood of theantiferromagnetic transitions.

Mg2SiOa is diamagnetic and its heat capacityfollows a normal sigmoidal curve at low tempera-tures. Co2SiOa and Mn2SiOa are both paramagneticand as was anticipated show a sharp \-type transi-tion in Q, arising from the antiferromagnetic order-ing of the Co2+ and Mn2* spins at low tempera-tures. In addition, MnzSiOr exhibits a second,rounded, maxima (shoulder) in C! near 12 K which,on the basis of the neutron diffraction studies ofSantoro et al. (1966), we believe is related to thechange from a collinearly ordered antiferromagneticspin arrangement to a canted arrangement of thespins with decreasing temperature.

Our experimental results are listed in Tables 2-4

Table 2. Experimental molar heat capacity measurements onsynthetic single crystals of MguSiOa (forsterite).

r e n p ' " " 1 ; : i . ,

r e n p . " , 1 : : 1 . ,

r e n p .

K J / ( s ' K ) K J / ( c . K ) K

I l e a tc a p 6 c l t y

J / ( s . K )

S e r i e s 1

2 9 9 . 4 5 0 . 8 4 5 13 0 4 . 8 0 0 . 8 5 5 23 1 0 . 6 1 0 . 8 6 5 13 r 5 . 3 7 0 . 8 7 5 r3 2 2 . 0 9 0 , 8 8 3 93 2 7 . 7 7 0 . 8 9 3 53 3 3 . 5 6 0 . 9 0 2 4

s e r l e s 2

3 3 5 , 4 4 0 . 9 0 5 33 4 1 . 3 1 0 . 9 1 3 43 4 6 . 9 7 0 . 9 2 2 4

S e r l e s 3

3 5 I . 0 6 0 , 9 2 5 93 5 5 . 2 0 0 . 9 3 2 43 5 0 . 9 5 0 . 9 4 0 73 6 7 . I 3 0 . 9 4 9 43 7 3 . 7 5 0 . 9 s 7 53 8 0 . 3 r 0 , 9 6 5 4

S e r l e s 4

5 2 . 8 7 0 . 0 4 9 I 55 8 . s 3 0 . 0 6 5 1 16 3 . 5 4 0 . 0 8 r 1 46 8 . 8 2 0 . 0 9 9 7 67 4 , 2 1 0 , I 2 0 37 9 . 6 3 0 . r 4 2 I8 5 . 1 6 0 , L 6 5 29 0 . 7 7 0 . 1 8 9 29 6 . 4 I O . 2 t 3 7

r 0 r , 9 9 0 . 2 3 8 01 0 7 . 5 0 0 , 2 6 2 21 1 2 . 8 3 0 . 2 8 5 4I I 8 . 0 2 0 . 3 0 8 01 2 3 . 1 0 0 . 3 2 9 8t 2 8 . 2 5 0 . 3 5 r 6

S e r l e s 5

I 3 3 . 5 4 0 . 3 7 4 11 3 9 . 1 7 0 . 3 9 6 7t 4 4 . 7 3 0 . 4 1 8 9I 5 0 . 2 8 0 . 4 4 0 51 5 5 . 6 5 0 . 4 6 0 91 6 1 , 0 0 0 . 4 8 0 81 6 6 . 5 5 0 . 5 0 0 8t 7 2 . 1 5 0 . 5 2 0 7r 7 7 . 6 3 0 . 5 3 9 5

S e r l e s 6

r 7 5 . 3 1 0 . 5 3 r 51 8 0 . 7 9 0 . 5 4 9 91 8 6 . 4 1 0 . 5 5 8 5I 9 2 . 0 2 0 . 5 8 6 2L 9 7 . 7 3 0 . 6 0 3 92 O 3 . I + 6 O . 6 2 1 12 0 9 . 0 8 0 . 6 3 7 5

S e r l e s 7

2 0 7 , 8 6 0 . 5 3 3 82 L 3 . 4 2 0 . 6 4 9 52 r 9 . r l 0 . 6 6 5 52 2 4 . 9 2 0 . 6 8 r I2 3 0 . 6 2 0 . 6 9 6 I2 3 6 . 4 0 0 . 7 1 0 72 4 2 . 2 5 0 . 7 2 4 92 4 A . O t 0 . 7 3 8 92 5 3 . 8 4 0 . 7 5 2 72 5 9 . 7 4 O , 7 6 6 22 6 5 . 5 6 0 . 7 7 8 3

s e r l e 6 8

2 7 0 . 3 4 0 . 7 8 8 52 7 6 . 2 0 0 . 8 0 0 52 8 2 . 0 8 0 . 8 r 2 42 8 7 . 7 t 0 . 8 2 3 42 9 3 . 4 6 0 . 8 3 4 32 9 9 . t 4 0 . 8 4 4 4

S e r i e s 9

7 . 6 6 0 . 0 0 0 0 r 89 . t 4 0 . 0 0 0 r 4 9

I 0 . 1 0 0 . 0 0 0 2 1 5I l . l l 0 . 0 0 0 3 0 5t 2 , 3 5 0 . 0 0 0 4 0 1r 3 , 8 2 0 . 0 0 0 5 7 8L 5 . 4 7 0 . 0 0 0 8 3 0L 7 . 2 2 0 . 0 0 1 0 0 41 9 . I 8 0 . 0 0 1 6 9 82 1 . 4 6 0 . 0 0 2 4 6 32 3 , 7 | 0 . 0 0 3 4 0 22 5 . 8 5 0 . O O 4 5 2 4

S e r i e s 1 O

6 . 2 3 0 . 0 0 0 0 3 71 . 5 6 0 . 0 0 0 0 8 68 . 4 4 0 . 0 0 0 r 1 7

S e r l e s 1 1

2 6 , O 4 0 . 0 0 4 6 2 12 8 . 5 8 0 . 0 0 6 4 2 53 1 , 4 3 0 . 0 0 8 8 4 53 4 . 7 0 0 , O 1 2 4 43 8 . 2 5 0 . 0 r 7 3 24 1 . 8 5 0 . 0 2 3 3 04 5 . 8 8 0 . 0 3 1 3 35 0 . 1 2 0 . 0 4 1 5 8

in their chronological order of measurement. Theheat capacities of Mg2SiOa and Co2SiOa are shownin Figure I and those of Mn2SiOa in Figure 2.

For Mn2SiOa and CozSiOr we made replicatemeasurements in the region of the C! anomaliesusing temperature increments of 0.3-0.18 K inorder to delineate the exact form ofthe heat capaci-ty in the transition regions. These are shown inFigures 3 and 4 for Co2SiOa and Mn2SiO4 respec-tively.

Thermodynamic properties of Mg2SiOa, Mn2SiOa,and Co2SiOa

Our experimental heat capacities were graphical-ly extrapolated to 0 K using a plot of Cf,lT versus Pas shown in Figure 5.

Above 30 K the heat capacity data were analyti-cally smoothed using an orthogonal polynomial

ROBIE ET AL.: ENTROPIES OF MgzSiOt,, Mn2SiOa, and CozSiOq 473

Table 3. Experimental molar heat capacity measurements on Table 4. Experimental molar heat capacity measurements on

synthetic single crystals of MnzSiOa (tephroite). single crystals of CozSiO+ (olivine structure).

Ternp H e a t

C a p a c i t y

Teqp H e a tC a p a c i t y

H e a !

C a p a c i t y

Ie 'np HeatC a P a c i t Y

m o l . K m o l . K J / ( u o l . K ) K J / ( n p l . K )

S " r i e s I3 0 3 . 5 3 l 2 C . B3 0 c . 6 3 I 3 l . l3 1 5 . 3 6 1 3 2 . 2 -3 2 1 . 0 5 1 3 3 . 33 2 6 . 6 9 I 3 4 . 4

S e . i e s 1 l3 2 6 - 4 3 1 3 4 . 43 3 2 . 3 4 1 3 5 . 73 3 3 . 5 7 1 3 6 . 83 4 4 . 1 9 1 3 7 . 63 5 n , 9 4 I 3 8 . 73 5 / . 1 6 1 3 9 . 73 6 3 . 4 3 I 4 0 . 7i 6 9 . 5 € ' I 4 l . 73 1 5 . 1 \ 4 1 4 2 . 63 B l . C q I 4 3 . ' , 1

S e r r e s I I I5 ) . 1 2 2 e . 8 35 2 . q 9 3 q . q q6 0 . 0 4 i 2 . 6 t l

S , o n i e s I V

7 0 . 0 67 4 . 9 7

3 4 . 6 93 7 . l tl n . ? 1

S e r i e s X2 1 9 . a ] 1 0 8 . 42 2 _ 5 . 5 3 1 1 0 . 22 3 1 . 4 2 ] l 2 . 0? - 3 7 , 3 2 1 1 3 . 72 4 3 . 2 4 | 5 . 3

S e r i e s \ I? - 4 9 . 1 2 _ l l 5 . 42 5 4 . 9 6 I I 5 . 8

S e r i e s X I2 5 0 - l I 1 1 1 , ?? 5 6 . q 3 l l s . B? 6 1 . 9 7 I 2 D . 32 - 6 7 , 9 2 1 ? l . 82 7 i . 8 A 1 2 3 . 32 . 7 9 . 8 5 1 2 4 . 12 8 5 . 7 a 1 2 , 6 . O2 9 1 . 6 e ) 2 t . 5

S e r i e s X I I I4 . 6 5 1 . q 9 75 . 5 1 2 . 6 2 56 . 5 0 3 . t 0 2L 5 0 4 . 6 0 1q . 5 ? 5 . 4 3 09 . 6 8 6 . 8 0 0

1 0 . 7 0 1 . 1 4 91 1 . 7 0 1 . 1 e 61 2 . 8 9 7 . 2 q 11 4 . 2 ) 7 . 5 i Jt 5 . 6 5 1 . 9 ? 6l 1 . 2 ) 8 . 6 1 31 8 . 9 1 1 . 4 6 62 0 . 8 6 I q . 5 5

2 ? - . q \ I I . 8 92 \ . 2 4 I 3 . 4 1? - 1 . 1 5 I 5 . 2 33 0 . 7 q 1 1 . 7 43 4 . 2 5 2 i , q 43 8 . 1 0 2 5 . ? 6t r 2 . ? 3 3 0 4 q4 5 , 5 5 4 3 , 5 q *

S - r i e s X I V4 2 . 7 2 i l . 6 24 3 . 6 1 i 3 . ? C4 4 . 3 r ) 1 4 . 7 14 5 , 0 q i 6 . 2 34 5 . 5 1 3 5 . 3 84 5 . B 5 3 A . l 94 6 . 1 4 3 q . 3 54 6 , 4 2 3 q . 9 24 6 . 6 ^ 4 l , n 04 6 . 3 1 ' 1 1 , 6 04 1 . a n 4 2 . 1 54 1 . 1 8 4 i . 5 ^1 1 , 3 9 4 F , , O 64 1 . 6 8 t l . 9 24 4 . n 4 3 0 . 7 14 q . 4 2 2 9 . 5 84 8 . 3 1 2 9 . 4 2

Ser i es I3 0 9 . 1 5 1 3 5 . 83 t 5 . 0 0 I 3 7 . l3 2 A . 8 2 I 3 8 . 33 2 5 . 0 9 I 3 9 . 5332.55 I 40 .73 3 8 . 4 8 I 4 l . S

Ser i es I I3 4 4 . 3 2 1 4 3 . 1350.22 144.23 5 6 . 1 9 1 4 5 . 3362.24 1 ,15 .43 6 8 . 3 / 1 4 1 . 43 1 4 . 5 7 I 4 8 . 53 8 0 , 8 C I 4 9 , 5

S e . i e s I I I5 0 . 1 2 3 2 . 1 75 1 . 0 2 2 / . 6 15 2 . I 0 2 6 . 5 25 3 . 1 7 2 - 6 . 2 15 4 . 2 3 2 6 . 3 055.25 26 .515 3 . l 3 2 1 . 6 56 2 . 9 7 3 A . 2 16 7 . 9 0 3 3 . 3 91 2 . 9 2 3 6 . 7 71 I . 9 7 4 0 . 2 78 3 . 0 5 4 3 . 8 3B B . 2 ) 4 1 , 3 7

S e r i e s I V3 8 . 6 3 4 1 . 6 79 4 . 1 0 5 1 . 3 49 9 . 3 6 5 1 . 8 1

1 0 4 . 6 0 5 8 . 1 6I 0 9 . 9 5 6 l . 5 0I r 5 . 5 4 5 1 . 8 9121.25 68 ,26t 2 6 . 8 t 7 1 . 4 5

S e r i c s V) 3 2 . a 2 t 4 . 5 1I 3 8 . 2 9 1 7 . 6 41 4 3 . 3 2 9 1 . 4 31 4 9 . 4 0 8 3 . 1 5I 5 5 . 0 2 8 5 . i 1 5150. t t3 88 .531 6 6 . 6 6 9 1 . 1 5

S e r i e s V Il t 2 _ . 3 7 9 3 . 6 21 1 7 . 9 5 9 5 . 9 1 11 8 3 , 5 1 9 8 . 3 Ct 8 9 . 3 5 1 0 r ) . 6

S e r i e s V I I1 9 4 . 7 1 1 0 2 . 62 0 0 . 3 8 1 0 4 . 72 0 6 . 0 1 I 0 6 . 8211 .76 I 08 . B2 1 7 . 4 1 1 1 0 . 72 2 3 . 2 1 1 1 2 , 5

S e r i e s V l I I2 2 8 . 8 2 1 I 4 . 5234.50 115.22 4 A . 2 6 I I 8 . 02 4 6 . 0 1 I I 9 . 7

S e r i e s I X2 5 1 . 2 0 1 2 1 . 2251.18 123.0263.04 1?4.6264.85 125.22 7 4 . 5 9 1 2 1 . 7?n0.?-5 129,2285.90 I 30 .62 9 1 . 5 0 1 3 2 , 02 9 7 . 0 6 I 3 3 . I

Se . i es x1 . 5 3 , ^ 2 3 59 . 5 7 . O B ? 5

1 0 . 6 1 . ) 2 9 5I 1 . 6 4 , 1 9 0 212.89 .29331 4 , 3 8 . 4 B l 31 5 . 1 3 . i 9 8 91 8 . 0 6 1 . 3 0 220.01 2 .4232 2 . 4 1 3 . 0 8 824.94 4 ,5562 t . 4 5 6 . 3 5 93 0 . 0 8 8 . 6 2 13 2 . 9 1 I I . 5 83 5 . 9 5 I 5 . 3 939.41 24 .5143.23 28 .55

S e r i e s X I5 . 1 5 . 0 0 8 16 . 1 6 . A 1 a ?7 . 5 1 . 0 2 8 38 . 4 6 . 0 5 1 3s .65 .oBBn

1 0 . 7 3 . 1 3 8 0

S e . i e s X I I16 .05 36 ,484 1 . 6 9 4 3 . 4 048.03 45 .7 11 8 . 3 2 4 7 . ? 14 8 . 5 4 4 3 . 9 248.16 50 .844 i1 .97 53 . I 54 9 . 1 1 5 5 . 8 04 9 , i 7 5 9 . 3 14 9 . 5 5 6 4 , 0 549,73 64 .04*49.95 39 .4 /

S - . r i e s V] N ) 2 4 2 t 4r q 3 . 4 2 4 5 . 4 .3 , 1 . 6 7 4 r l . 5 r l9 4 . ? 2 5 t . q 6

S i : r i e s V I. 3 1 i 4 4 5 . 5 4 1, 8 , 7 q 4 9 , 5 9q 3 , 9 6 5 1 . q 2q ! . 6 r 5 5 - I t

S : r i e s V I II n 0 . 3 9I 0 6 . t 6i l 1 . 5 31 1 7 . 1 91 2 2 , o 21 2 _ ' \ . . ?I 3 4 . n 2I l q . 3 tI 4 1 . 7 3I 5 0 - l 7t 5 5 . 6 /

5 5 . 5 65 q . 9 76 2 . O t i6 5 , 2 46 P ' . 3 17 ) . i 27 4 , 1 it 5 . t 51 9 . 3 19 t . 9 18 4 . 3 3

5 . r i e s V I I i, 5 1 . 2 9t 6 1 . 1 1| 7 2 . 9 8l l a . t \ 1

8 6 . 9 4q 9 . r 29 t . 7 79 4 . l 7

I 8 4 . 8 3 9 5 . 4 4I 9 0 . 7 5 q 8 . 6 4

S e r i . s I X1 9 5 . 5 2 I C 0 . 72 A 2 . 3 2 1 0 2 . 12 0 l l , l q 1 0 4 . 7? 1 4 . a 3 1 0 6 . 6

previous (unpublished) data on the CalorimetryConference Copper sample (Osborne, Flotow, and

curve fitting routine (except in the transition re- Schreiner, 1967) between 4.5 and 20 K, we believegions). The analytically and graphically smoothed that our smoothed Ci values are accurate to approx-parts were joined smoothly and corrected for curva- imately 15 percent between 5 and 15 K, increasingture. Our measurements for Mg2SiOa have an aver- in accuracy to 10.1 percent for temperatures aboveage deviation from the smoothed curve of 10.06 20 K except in the immediate region of T1q wherepercent between 25 and 380 K and maximum devi- the small temperature increments reduced the accu-ations of 0.5 percent. racy to 11.0 percent.

From replicate measurements on all three sam- Smooth values for the thermodynamic functionsples in the range 5 to 10 K together with our C;,S$-.Sfi, Gft-m)ff,and(Gt-I13)/TforMg2SiOa,

ROBIE ET AL.: ENTROPIES OF MgzSiOa, Mn2SiOa, and Co2SiOa

Y

=

z>.

Uo

UF

UT

Nf

^t

^ fr .{

"A-ry

F

F.l

. T

dJ-

,ftr,s

r rtr

t,t

dtrl

no4

C

f-{T

S A.P ! M92SiO4 USGS

o Mg25iOo

(KELTEY r943)

a Co2SiOo

)T

TEMPERATURE, IN KETVINSFig. 1. Experimental molar heat capacity of synthetic single

crystal Mg2SiO4 (forsterite), formula weight 140.694 g mol-r andsynthetic single crystal Co2SiOa formula weight 209.950 g mol 1.

Earlier measurements of Kelley (1943) on powdered Mg2SiOabetween 53.2 and 295.0 K are shown for comparison.

Mn2SiOa, and Co2SiO4 are listed at integral tem-peratures in Tables 5, 6, and 7 up to 380 K.

At 298.15 K the entropy changes, Sies-Sfi, forMg2SiOa, Mn2SiOa, and CozSiO+ 94.11-{-0.10,1 55.9 -f 0.4, and 142.6-10.2 Ji(mol.K) respectively.The zero point entropy, Sfi, is presumably zero foreach of these three olivines. Accordingly, the S|-Sfivalues are equal to ,Si and are the correct values foruse in thermodynamic calculations.

Mg2SiOa

Kelley (1943) measured the heat capacity ofMg2SiOa prepared from "a mixture of oxides con-taining an 0.82Vo excess of magnesium oxide overthe theoretical 2:l molar ratio was rotated in thechamber of a porcelain ball mill for nine hours toassure intimate mixing. The mixture was moist-ened, tamped into a nickel cartridge, dried at 100',and finally heated in a tube furnace at 1150-+20' forfour days under a vacuum of l0-4 mm. After thistreatment, the product contained 3.67o uncombinedmagnesium oxide. It was reground and similarlyheated for five additional davs at 1180-+10o. Analv-

JXX

Ja?"

?.X

X

x MnzSiOr

(TEPHROTTEx '

r00 1 5 0 350 400

TEMPERATURE, IN KEI -VINS

Fig. 2. Experimental molar heat capacities of synthetic singlecrystal Mn2SiO4 (tephroite), formula weight : 201.960 g mol-r.

1 5 0

1 4 0

1 3 0

't20

l r 0

r00

l 0

o

t 4 0

1 3 0

r20

'n0

. 100o= e oz- 8 0>.

u 6 0

I

l 0

Y

o=

zF

L)

4

U

u-t' \"/

I

,( a Co2SiOa

(Co-OtlVlNE)

3d

20 30 40 50

TEMPERATURE, IN KEIVINS

Fig. 3. Molar heat capacity of Co2SiOa in the region of theparamagnetic-antiferromagnetic transition, TN = 49.7610.02K.

55

50

10

35

30

25

20

l o

5

0

I

*

ilII

) a i

X

fo'""

x Mn2SiO,

(TEPHROTTE)

ROBIE ET AL.: ENTROPIES OF MezSiO+, Mn2SiOa, and Co2SiOa 475

Table 5. Molar thermodynamic properties of MgzSiO+(forsterite). Formula weight = 140.694 g mol t.

50

T E I ] I ' . i l E A TC A P A C I T Y

T C *

X D L I I I :

f , I : T R O P Y E N T I I A L P Y C I B I S T N E R G Y

F U I t C T I O i t F L r l t C T I 0 l i

r s i - s i r r r i - H i r r r - < c i - r i r z r

J / ( t r o 1 ' K )

v 3 5

o: ^ ^

zF a s

U

l) zv

Ur t 5

5I O1 52 A2 53 03 54 04 55 0

6 07 08 09 0

1 0 01 1 01 2 01 3 0I 4 01 5 0

1 6 0I 7 0l 8 c1 9 02 A O2 1 02 2 42 3 02 4 A2 5 0

2 6 A2 7 r )2 P o2 -ea3 0 03 1 03 2 03 3 C

3 5 0

0 . 0 0 3 2 c . 0 0 I 00 . 0 2 8 4 0 . 0 0 8 e0 . 1 0 6 0 . 0 3 2 7a . 2 7 5 0 . 0 8 3 40 . 5 7 0 0 . 1 7 3 61 . 0 5 9 0 . 3 1 6 91 . 8 0 a 0 . 5 3 32 . P 4 3 C . 8 3 94 . r 7 1 . r . 2 4 75 . 7 A 7 1 . 7 6 A

9 . 8 0 4 3 . 1 6 3r a . 6 9 5 . 0 3 22 A . 2 0 7 . 3 4 72 6 . 1 2 1 0 . 0 73 2 . 2 6 1 3 . 1 33 8 . 4 4 1 6 . 5 04 4 . 5 5 2 0 . 1 I5 0 . 5 1 2 3 . 9 rs 6 . 2 7 2 7 . 8 66 1 . 8 1 3 1 . . 9 4

6 7 . L 21 2 . r 97 7 . 0 2P 1 . 6 08 5 . 9 59 0 . 0 79 3 . 9 89 7 . 7 0

1 0 1 . 31 0 4 , 6

3 6 . 1 04 0 . 3 24 4 . 5 84 8 . 8 75 3 . L 75 7 . 4 66 l . 7 46 6 . 0 01 0 . 2 4] 1 . 4 4

0 . 0 0 0 70 . 0 0 6 6o . o 2 4 90 . 0 6 3 70 . 1 _ 1 2 9a . 2 4 2 90 , 4 1 C0 . 6 4 60 . 9 6 IL 3 6 1

2 . 4 1 93 . 8 1 45 . 5 1 37 . 4 7 19 . 6 4 2

t 4 . 4 41 6 . 9 9r 9 . 5 92 2 . 2 2

2 4 . P 62 1 . 5 03 0 . 1 23 2 . 7 t3 5 . 2 63 7 . 7 14 A - 2 44 2 . 6 64 5 . 0 34 7 . 3 4

4 9 . 6 15 1 . 8 25 3 . q P5 6 . 0 95 B . 1 46 0 . 1 56 2 . 1 16 4 . 0 16 5 . 8 76 7 . 6 8

6 9 . 4 51 1 . r 1

5 2 . 5 r5 1 . 1 1

0 . o 0 0 30 . 0 0 2 20 . 0 0 7 80 . 0 1 9 7a . 0 4 a 10 , 0 7 4 00 . I 2 30 . 1 9 30 . 2 8 60 . 4 0 7

0 . 7 4 4r . 2 1 71 . 8 3 52 . 5 9 53 . 4 9 24 . 5 2 0

6 . 9 2 3P , . 2 7 69 . 7 r 7

r 1 . 2 4t 2 . e 21 4 . 4 7r 6 . t 71 7 . 9 rr 9 . 6 92 I . 5 02 3 . 3 52 5 - 2 r2 7 . r C

2 t . 0 03 0 . 9 13 2 . P 43 4 - 7 13 C . 7 03 8 . 6 44 C . 5 84 2 . 5 24 4 . L 64 ( , . 4 4

4 8 . 3 35 0 . 2 65 2 . 1 { r3 r . 5 23 6 . 3 4

l 0

0 r0

o08

007

o v v v

E

- o05

F

o q 0 0 4U

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

TEMPERATURE, IN KELVINS

Fig. 4. Molar heat capacity of MnzSiO+ in the region of the lowtemperature spin ordering transition, 47.38+0.05K.

TEMPERATURE. IN KETVINS5 1 0 t 5

Mg, SiOo

SERIES 9 ^

S E R I E S I O T

Co2 S iOo a

0 r00 200 300

TEMPERATURE2 IN K2Fig. 5. Cp/T versus 72 extrapolation of heat capacity data on

MgzSiOr and Co2SiO4 to 0 K.

sis now gave 98.6Vo magnesium orthosilicate,O.SVIuncombined magnesium oxide, no free silica, andno magnesium metasilicate". His Ci results (53.2 to295.0 K) differ frorn our measurements by a maxi-mum of 0.5 percent over the common temperaturerange. Our value for Stes, 94.11!.10 Ji(mol'K) is 1.1percent less than that obtained by Kelley (1943).Almost all of this difference arises from the empiri-cal extrapolation of his data from 50 to 0 K. Overthat part of the temperature range for which heactually measured Ci his value for,SisrS3o and oursagree to 0.3 percent.

We have also combined our heat capacity resultsfor the range 300-380 K with the heat content dataof Orr (1953) between 398.1 and 1807.6 K to gener-ate values for the molar heat capacity at higher

1 0 7 . P 1 P . ( , 11 1 0 . 9 8 2 . 7 31 1 3 . 7 8 6 . ? , 21 1 6 . 4 9 0 , 8 51 1 9 . 1 9 4 . 8 5r 2 t . 6 s 8 . 7 9t 2 4 , A 1 C 2 . 71 2 6 . 2 1 0 6 . 51 2 8 . 3 1 I 0 . 31 3 c . 2 1 1 4 . 1

3 6 0 1 3 2 . 1 I I 7 . 83 7 A 1 3 4 . 1 l 2 l . A3 { ' C 1 3 5 . { 1 1 2 5 . 02 7 3 . r 5 1 l I . t ? 4 - O 22 f r 8 . 1 5 I 1 8 . 6 9 4 l I

476

Table 6. Molar thermodynamic properties of Mn2SiOa(tephroite). Formula weight : 2019ffi g mol-r.

T E } 1 P . I I E A T N } ' T I I O P YC A P A C I T Y

t a ,

T E L V I I ]

( s r - s o )

. i / ( D o 1 ' k )

ROBIE ET AL.: ENTROPIES OF MgzSiOt, Mn2SiOa, and Co2SiOa

Table 7. Molar thermodynamic properties of Co2SiOa. Formulaweight : 209.950 g mol-r.

f , l l T l t l L P Y c l x t s E t L i l i c YF L r t i C T I 0 l i F U I : C T l 0 l .

i l : _ . - r ; ^ ) / T - ( c - - t i ^ \ / iI r r I u _

i . I I I P . I I ' A T E I ' T R o P YC A P A C I T Y

T C -

r u L v l l i

u t i T l A L t Y c I D E S l l i r R C lF U I I C T I O N } ' U N C T I O I ]

<s i - s i l t r ; i - r r i l r r -<c i - r r i > r r

5 2 . 2 r rl 0 6 . 8 9 3t 5 7 . 7 6 02 0 1 0 . 0 82 5 1 3 . 2 13 0 1 7 . 0 53 5 2 r . 1 64 0 2 7 . 1 14 5 3 6 . 0 e5 0 2 9 . 9 r

5 5 3 0 . 7 16 0 3 2 . L . )7 0 3 7 . 2 98 0 4 3 , 2 99 0 4 9 . 4 2

1 0 0 5 5 . 4 r

o , 0 0 5 0 0 . 0 0 r 40 . 1 0 r 0 . 0 2 3o . 5 7 4 0 . 1 3 01 . 9 7 5 0 . 4 5 84 . ( t O 1 1 . I 5 98 . 5 5 5 2 . 3 2 6

1 4 . 1 3 4 . 0 3 92 t . 7 1 6 . 3 9 33 3 . 1 8 9 . 5 5 33 8 . 5 0 1 4 . 2 22 6 . 4 1 1 6 . 6 0

r l o1 2 01 3 01 4 0t 5 01 6 01 7 01 P 0l a 02 0 0

2 r02242 3 42 L A2 5 02 6 02 7 02 8 C2 9 A3 0 0

l 0

2 02 53 0J J

4 0

5 0

1 6 01 7 01 0 01 9 02 0 02 1 02 2 02 3 02 4 42 5 0

2 6 02 7 02 8 02 9 03 0 03 1 03 2 03 3 03 4 03 5 0

1 2 3 . 8t 2 6 . 5r 2 9 . 11 3 1 . 61 3 3 . 91 3 6 . 1t 3 a . 21 4 0 . 2r 4 2 . 2r 4 4 . 1

1 2 4 . 91 2 9 . 7r 3 4 . 31 3 8 . 91 4 3 . 4t 4 7 . 8t 5 2 . 21 5 6 . 4I 6 0 . 7I 6 4 . 8

I 6 P . 9r 7 2 . 9t 7 6 . 9l 3 r . lr 4 2 . 6

1 . 2 6 74 . 2 6 37 . 2 1 79 . 7 7 3

r 2 . 3 t1 5 . 0 4l 8 , 0 12 t . 2 92 4 . 9 92 8 . 8 5

3 I . 7 13 4 . 4 63 9 . 8 04 5 . 1 75 0 . 6 25 6 . 1 4

6 1 . 7 0

7 2 . 8 27 4 . 3 58 3 . 8 3a 9 . 2 50 4 , 6 1

9 9 . 9 11 0 5 . 1 21 1 0 . 3

1 1 5 . 31 2 0 , 3t 2 5 . 2I 3 0 . 01 3 4 . 71 3 9 . 41 4 3 . 9r 4 a . 41 5 2 . 8r 5 7 . 2

t 6 r . 5r 6 5 . 71 6 9 . 81 7 3 . 8t 7 7 . 8I t I . 8I 8 5 . 6I 8 9 . 4t 4 5 . 41 5 6 . 4

0 . 8 0 e2 . 1 1 24 . 2 4 45 . 3 8 46 . ( , 2 7

8 . 0 1 09 . 6 4 3

r 1 . 5 1t 3 . 7 41 6 . 0 2

r 7 . 2 91 8 . 4 82 0 . 8 02 3 . 2 32 5 . 8 02 t , . 4 7

3 r . r 83 3 . 9 23 6 . 6 53 9 . 3 64 2 . O 44 4 . 6 64 7 . 2 44 9 . 7 65 2 . 2 25 4 . 6 2

5 6 . 9 55 9 . 2 26 r . 4 35 3 . 5 86 5 . 6 76 7 . 1 16 9 . 6 87 | . 6 t7 3 . t t 87 5 . 3 0

7 7 . O 77 8 . 7 9P O . 4 68 2 . 1 08 3 . 6 98 5 . 2 48 6 . 7 48 8 . 2 17 0 . 3 07 4 . 9 6

0 . 4 5 9r . 5 5 r2 . 9 7 34 . 3 4 95 . 6 8 27 . 0 1 14 . 3 6 79 . 7 1 4

1 1 . 2 51 2 . 8 3

1 4 . 4 21 5 . 9 81 9 . 0 02 1 . 9 42 4 . A 22 7 . 6 A

3 0 . 5 23 3 . 3 53 6 . r 73 8 . 9 84 r . 7 94 4 . 5 94 7 , 3 75 0 . 1 55 2 . 9 05 5 , 6 4

5 6 . 3 66 1 . 0 76 3 . 7 56 6 . 4 16 9 . 0 57 I . 6 67 4 . 2 57 6 . 8 27 9 . 3 78 1 . 8 9

8 4 . 3 98 6 . 8 68 9 . 3 1- q t . 7 49 4 . 1 49 6 . 5 29 e . e 8

I C r . 2 11 5 . 0 78 1 . 4 3

2 8 . 1 03 4 . 7 54 r . 6 44 8 . 5 95 5 . 2 76 1 . 5 86 7 . 5 37 3 . t 47 8 . 4 48 3 . 4 6

1 9 . 1 92 4 . O 52 9 . r 43 4 . 4 53 9 . 9 24 5 . 4 8

5 6 . 1 36 2 . 3 46 7 . 9 3

0 . 0 0 1 o0 . 0 1 8 60 . r c 6o . 3 7 20 . 9 3 51 . 8 5 63 . I 8 85 . 0 0 31 . 4 4 A

1 1 . 1 51 2 . 6 0

1 3 . 8 4r 6 . 3 81 9 . 1 02 t . 9 , o2 4 . 9 92 8 . 0 33 I . O B3 4 . 1 03 7 . 0 84 0 . 0 1

4 2 . 8 74 5 . 6 14 8 . 4 05 1 . 0 55 3 . 6 45 6 . 1 55 8 . 5 96 0 . 9 66 3 . 2 76 5 . 5 2

6 7 . 7 16 9 , 8 37 1 . 9 L7 3 . 9 27 5 . 8 87 7 . 7 97 9 . 6 4t r 1 . 4 58 3 . 2 08 4 . 9 2

6 6 , 5 98 4 . 2 28 9 . 8 07 0 . 4 9

0 . 0 0 0 40 . 0 0 4 5c . 0 2 4c . 0 8 6o . 2 2 40 . 4 7 00 . 8 5 11 , 3 9 02 . 1 1 33 . 0 6 94 . 2 0

5 . 3 5 51 . 6 7 7

1 0 . 0 4t 2 . 4 5r 4 . 9 31 7 . 4 52 0 . o 22 2 . 6 32 5 . 2 62 7 . 9 2

3 0 . 6 03 3 . 2 B3 5 . 9 13 8 , 6 64 t . 3 44 4 . 0 24 6 . 6 94 9 . 3 45 I . 9 95 4 . 6 2

5 7 . 2 35 9 . 8 26 2 . 4 06 4 . 9 66 7 , 5 07 0 , 0 27 2 . 5 27 5 . O 07 7 . 4 57 9 . 8 9

8 2 . 3 r8 4 . 7 08 7 . 0 76 0 . 6 46 7 . O 3

6 r . 2 36 6 . 8 07 2 . A 87 7 , C 78 1 , 8 18 6 . 3 09 0 . 5 69 4 . 5 79 8 . 3 6

1 0 1 . 9

1 0 5 . 3I 0 8 . 51 1 1 . 61 1 4 . 5I L ? . 21 1 9 . 81 2 2 . 3t 2 4 . 7t 2 7 . O1 2 9 . r

6 07 08 09 0

1 0 01 1 01 2 01 3 01 4 01 5 0

8 8 . 1 9 7 3 . 4 79 2 , ( , 5 7 8 . 9 59 6 , 8 6 ? , 4 . 3 6

r 0 0 , 8 e . 9 . 1 rt o t t . 6 9 4 . 9 81 0 8 . 1 1 0 0 . 21 1 1 . 5 1 0 5 . 31 1 4 . 8 1 1 0 . 31 1 7 . 9 1 1 5 . 3r 2 0 . 9 1 2 0 . 1

3 1 0 t 3 r . 23 2 0 r 3 3 . 13 3 0 1 3 5 . 13 4 0 t 3 6 . 93 5 0 1 3 8 . 63 6 0 1 4 0 . 23 7 0 1 4 1 . P3 E O 1 4 3 . 02 7 3 . t 5 1 2 3 . 12 9 8 . r 5 1 2 8 . 7

3 6 0 1 4 6 . 03 7 0 1 4 7 . 73 8 0 t 4 9 . 32 7 3 , r 5 1 2 7 . 42 9 8 . t 5 L 3 3 . 4

temperatures;

Cp : 87.36 + 0.08717r - 3.699x1067-2+ 843.610'5 - 2.237x10-sT1

(298-1800K). This equation fitted the 23 datapointswith an average deviation of 0.2 percent and joinsthe low temperature Cfi curve smoothly.

The enthalpy change for the reaction

2MgO + SiO2: Mg2SiOa (1)

has been measured by Torgeson and Sahama(1948), and King et al. (1967) by HF(aq) solutioncalorimetry. These authors' results were correctedby Hemingway and Robie (1977) for SiO2 particlesize effect. Navrotsky (1971) obtained -62.8!4.2kJ for LIfios (1) by molten salt calorimetry. Her

value is however based on an estimated value forAH.or,, of MgO in the molten 2PbO'B2O3 used todissolve Mg2SiOa and SiO2. This estimated vahrefor A.EI".6 of MgO in the 2PbO.B2O3 melt has theopposite sign from that measured at a later time byNavrotsky and Coons (1976) at a slightly differenttemperature and we therefore did not include thisresult in our analysis. Charlu et al. (1975) obtained-62.6-1 1.1 kJ at 970 K and Kiseleva et al. (1979)obtained -58.211.4 kJ for Llff"./t for this reactionboth by molten salt solution calorimetry. Kiselevaet al. (1979) value corrected to 298 K using the heatcapacity data of Robie et al. (1979) yields A,ffies :-56.5 kJ, and is in excellent agreement with thatadopted by Robie et al. (1979, p. 364),-56.7-r.7 kJ/mol which we believe is

LIffes :the best

ROBIE ET AL.: ENTROPIES OF MgzSiO+ MnzSiOa,

Table 8. Thermodynamic properties of Mg2SiOa at high G = 261.28 -temp€rature and Alf and AGe for the reaction 2MgO + SiO2 :

MgzSiO+.

and CozSiOt

1.378x 1,0-zT- 2217.7T-o'5 (Zgg-t0oo r)

for Mn2SiOa.Jeffes et al. (1954) obtained LIfrsE: -49.2+2.9

kJ for the reaction

2MnO + SiO2 eMn2SiO4

manganosite o,-quartz tephroite

by HF(aq) solution calorimetry (at 25'C). Heming-way and Robie (1977) adjusted this result to.-47.1-*'3.0

kJ to account for the very fine particlesize of the quartz used in the AI1"o1 measurements.Navrotsky and Coons Gnq obtained L,Iffeg :-48.5-'-1.4 kJ for this reaction using molten2PbO'B2O3 as the solvent. Schwerdtfeger and Muan(1966) give -43.5-r 1.7 kJ for the Gibbs free energychange, A6f,, for this reaction at 1423 K based onmeasurements of the equilibrium gas ratios of CO/CO2 coexisting with phases in the "FeO"-MnO-SiO2 system. Biggers and Muan (1967) obtainedACfrszt : -42.3!3.4 kJ from equilibrium measure-ments of the exchange reaction

CoSiO3 + 1/2 Mn2SiO4 e MnSiOr+ 1/2 Co2SiOa (3)

combined with the AGF values for CozSiO+ fromLebedev et al. (1962) and Aukrust and Muan (1963).

Using A6f, from Schwerdtfeger and Muan, andBiggers and Muan, and our heat capacity (andentropy) data together with those for MnO and SiOztabulated by Robie et al. (1979) we can also obtain avalue for Llfrslfor reaction (2) as follows: from theheat capacity data, we obtain ASi.r+zr and from therelation

L I i " : A d + T A S "

we get ffi1ay and using Kirchoff's relation calcu-late A1{.2es : -53.7t2.1 kJ from Schwerdtfegerand Muan and -53.5-+4.4 kJ, from Biggers andMuan respectively. Schreiber (1963) studied thereaction

2MnCOr + SiO2 = Mn2SiO4 + 2CO2 (4)

between 738 and 813 K. From his equation forlogfco,as a function of temperature and our entropydata, and using unpublished measurements of Robieand Hemingway on the entropy of MnCOr, we getAG?oo : -44.2 kJ/mol for the formation of Mn2SiOafrom the oxides, and -49.3 kJ/mol for AI4m (2).

The agreement between the 5 independent valuesis only fair and we therefore simply adopted a

477

TEMI . (HT-Hze8) /T s i - (cT- r12e. ) /T c i ENTHALev

K J / r n o l . K J / m o l . K J / n o l . K J / n o l . K k J l o c l

G I IJBSF R I E

ENERG Y

2 9 8 . 1 5 0 . 0 0 9 4 , i l1 . 1 0

400 32 .92 1 3 t .975 0 0 5 5 . 0 1 1 6 3 . 9 26 0 0 t l . 2 t I 9 r , 6 5/ 0 0 8 3 . 7 3 2 1 6 . 1 2300 93 .81 238.06

9 4 . 1 1 1 1 8 . 6 0 - 5 6 . 6 91 . 1 0 I . 6 9

9 9 . 0 s 1 3 7 . 7 1 - 5 6 . 7 3108.91 148.28 -56 .84I 2 0 . 4 4 1 5 5 . 1 t - 5 7 . 1 21 32_.39 I 6 l .75 -57 .531 4 4 . 2 5 1 6 6 . 8 3 - 5 8 . 0 6

I 55 .80 171.25 -59 .001 6 6 . 9 4 1 7 5 . 1 4 - 5 8 . 7 11 7 7 . 6 5 1 7 8 . s 6 - 5 8 . 2 81 8 7 . 9 3 t 8 t . 5 4 - 5 1 , 7 41 9 7 . 7 9 l r J 4 , 0 9 - 5 7 . 1 0201 .26 186. )2 -56 .39216.35 187.93 -55 .52225.08 189.2? -54 .84233,49 I 90 .09 -54 .052 4 1 , 5 9 I 9 C . 5 4 - 5 3 . 2 c

- 5 6 . i ?

- 5 5 . 2 0- 5 6 . 0 5- 5 5 . 8 7- 5 5 . 6 2- 5 5 . 3 3

(2)

900I 000l l 0 0I 200r 300I 400I 500I 600I 700I 800

- 5 4 . 9 1-54.46- 5 4 . 0 7

- 5 3 . 4 I- 5 3 . I 3- 5 2 . 9 3- 5 2 , 1 8-52 .F ,6- 5 2 , 6 2

value. Combining this value with the enthalpies offormation of periclase and quartz from Robie et a/.,we obtain Ll4,zgs : -2170.4+ 1.4 kJ/mol. Using theentropies of the elements adopted by Robie er a/.and the entropy offorsterite obtained in the presentwork we get AGi,2es : -2051.0-r1.4 kJ/mol.

The heat capacity equation for forsterite wascombined with the value of ffi,2es adopted aboveand the high temperature thermodynamic functionsfor periclase and quartz tabulated by Robie et al.(1979) to obtain the values of A/lt and AGi listed inTable 8. Our values for AGi differ significantly fromthose obtained by the e.m.f. measurements of Roget al. (1974) between 1373 and 1573 K.

MnzSiOa

King and Weller's measurements on Mn2SiO4between 52.3 and 296.1 K (Mah, 1960) were madeon "crystalline manganese orthosilicate (teph-roite)" with no further description of the prepara-tion of the material. Their measured values aresystematically larger than ours, being about 2 per-cent greater at 50 K and decreasing to +0.7 percentat 300 K. Their entropy value at 298K includes anempirical extrapolation below 51.0 K of 24.9 Jl(mol'K) and was apparently made without anyknowledge of the two low-temperature magnetictransitions. In the same way as described forMg2SiOa we combined our low temperature Q, datawith the heat content measurements of Christensen(Mah, 1960) to generate

102,1 7 251.97I 09 .28 276.22I I 5 .43 293.08I 20 . Br 308. 741 2 5 . 5 9 3 2 3 . 3 8129.84 337. I 0I 3 3 . 6 6 3 5 0 . 0 1r 3 7 . 1 0 3 6 2 . 1 8I 40 . I 9 37 3 .681 42 .98 384.56

478 ROBIE ET AL.: ENTROPIES OF MgzSiOq, Mn2SiOa, and Co2SiOa

rounded average for ffi,2es (2) of -50.8-r2.5 kJlmol. We combined this value with ancillary datafrom Robie et al. (1979) to obtain -1732.0t 2.9 and-1631.5-f 3.0 kJ/mol for AI/i,2e3 and AGF,zrs respec-tively. In Table 9 we list the high temperaturethermodynamic properties for tephroite togetherwith AIIt and AGg for reaction (2).

CozSiot

There are apparently no high temperature heatcapacity data for Co2SiOa. At 298.15 K AS' is-4.8+0.4 J/K for reaction (5) based on our entropyfor Co2SiOa and those tabulated by Robie andothers (1979) for CoO and SiO2. In order to use thehigh temperature equilibrium data we have estimat-ed the heat capacity of Co2SiOa between 400 and1500 K by using the sum ofthe heat capacities of2moles of CoO and 1 of SiO2, joining these estimatessmoothly with our measured values for Co2SiOabetween 300 and 380 K, making proper allowance;that is, by smoothing thru the temperature region ofanomalous heat capacities caused by the a-B transi-tion in qwrtz. Our estimated heat capacities forCo2SiOa are represented by the equation

q: 343.9 - 0 .1006? + 3 .814x l | -sP- 31757-o's (289-1500 K)

This equation was used together with our measuredentropy at298.15 K to calculate Si and lli-Ifrgsforuse in a thirdlaw treatment of the high temperatureequilibrium studies on Co2SiOa.

Navrotsky (1971) and Navrotsky er al. (1979)obta ined LH\as -21 .6-12 .1 , and A!s6 :-23.0-f 0.8 kJ respectively, for the reaction

2CoO + SiO2: Co2SiOa (5)

by solution calorimetry using 2PbO.B2O3 as thesolvent. The Gibbs energy change for this reactionhas been determined by Lebedev et al. (1963)between 1073 and 1423K, and by Aukrust and Muan(1963) (1273-1573 K) by CO reduction equilibria,and by Kozlowska-Rog and Rog (1979) by solidstate emf studies.

We have used the experimental data of Lebedevet al. (1962) and of Aukrust and Muan (1963) for theequilibrium

Co2SiOa + zCO : ZCo + SiO2 + ZCO2 (6)

over the temperature range 1083 to 1573 K. To theexperimental A@ data of Lebedev et al. for (5) weadded AGfr for the reaction

Table 9. Thermodynamic properties of MnzSiO+ at highertemperatures and A-El" and Ad for the reaction 2MnO + SiOr :

Mn2SiOa.

G I B B S( t l f - i l ) 9 s ) / r s t - ( G t - H 2 e B ) / r c D E N T H A L P Y F R E t E N E R G T

J / m o l ' K J / m o l . K J / m o l ' K J / , i o l ' K k J / n o l k J / n o l

TEIIP.

400500600700iJ00

900I 000

0 . 0 0 1 5 5 . 9 0r0 .40

3 5 . 0 0 1 9 6 . 1 75 8 . 0 6 2 2 9 . 6 114.88 254.6487.79 284.109 8 . 0 5 3 0 6 . 7 8

106.42 327.21iI I 3 .40 345.75

I I 9 . 3 0 3 6 2 . 1 61 2 4 , 3 6 3 7 8 . 4 1128.14 392.9?132.56 405,431 3 5 . 9 2 4 1 9 . 0 61 3 8 . 9 0 4 3 0 . 9 0I 3 9 . 4 5 4 3 5 . 2 1

I 79 .21 -53 ,53 -43 .371 8 0 . 6 8 - 5 3 . 7 8 - 4 2 . 4 21 8 1 . 8 1 - 5 4 . 0 5 - 4 t . 4 71 8 2 . 6 7 - 5 4 . 4 6 - 4 0 . 4 91 8 3 . 3 0 - 5 5 . 0 0 - 3 9 . 4 6ta3 . t4 -55 .12 -38 .411 8 3 . 3 1 - 5 5 , 7 4 - 4 1 . 0 9

1 5 5 . 9 C 1 2 8 . 5 0 - 5 0 . 8 4 - 4 9 . 3 5t 0 . 4 0 1 0 , 1 5 ! 2 . 5 0 ! 2 . 6 0

l 6 l . l 7 1 4 4 . 8 4 - 5 1 . 1 3 - 4 8 . 7 9l 7 l , 6 1 1 5 5 . 1 7 - 5 1 . 3 2 - 4 8 . 1 91 3 3 , 7 5 r 6 2 . 4 3 - 5 1 . 5 4 - 4 7 . 5 41 9 6 , 3 1 1 6 7 , 1 1 - 5 r . 9 0 - 4 6 . 8 6208.13 111.80 -52 ,47 -45 .09

2 2 0 . 7 3 r 7 4 . 9 r - 5 3 . 5 5 - 4 5 , 2 42 3 2 . 3 5 1 7 7 . 3 3 - 5 3 . 5 5 - 4 4 . 3 0

i l00I 200I 300r 400I 500I 500r 620

243.46251.05264,182 1 3 . 8 7283. I 4292. ( )0295,15

ZCo + 2CO2:ZCoO + ZCO (7)

using data from Robie et al. (1979) for SiO2 and theJANAF Thermochemical Tables for CoO to obtainAG+ (1083-1573 K) for reaction (6). We did not usethe equation given by Lebedev et al. for reaction (5)because in deriving it from their experimental data,they had used values for AGi,1 for CO taken fromRichardson and Jeffes (194q which are incorrect.Aukrust and Muan (1963) obtained their AGfr valuesfor (5) by combining their equilibrium constants for(6) with their own measurements for (7). A@ forreaction (5) from the CO reduction equilibriumstudies are in reasonable agreement but are -3 KJmore negative than the AG" obtained from the emfmeasurements. A third-law treatment of these dataleads to LIffss for reaction (5) of -23.0-1 1.5 kJ.Combining our values with ffi,2es for CoO andSiO2 from rANAF (1971) or Robie et al. (1979), weobtain AGitsa : -1305.8-13.0 kJ/mol and A.Ef,2es :-14ll.6-12.9 kJ/mol for Co2SiOa (from the ele-ments).

At 975 K, the mean temperature of the moltensalt calorimetric determination of Alf for reaction(5), our treatment yields -27.9 kJ. This is approxi-mately 5.6 kJ more negative than the value given byNavrotsky et al. (1979).

Magnetic entropies and heat capacities of Mn2SiOaand Co2SiOa

In order to separate the magnetic contributions tothe heat capacity from those arising from the latticevibrations we have adopted the corresponding

ROBIE ET AL.: ENTROPIES OF MezSiO',, MnzSiOe, and CozSiOq 479

oE

F

./_./ l r l

,/(56org 6o-2Rln2 i - r l

,,

,/,iJ

Scars oo

/.

I

o r:0"."",",","1

i" k"t ' i : :Fig. 6. Plot of the entropy of lCa2SiOa and the quantity

Scorsioo - 2Rln2.

states model suggested by Stout and Catalano(1955). We used both our Ci data for Mg2SiOa andthose of King (1957) for yCa2SiO4 with the corre-sponding states model to estimate the lattice entro-pies of MnzSiOa and Co2SiOa.r

We proceeded as follows: first, from a large scaleplot of our smoothed values for the molar heatcapacity of Mg2SiOa and those of King (1957) for yCa2SiOa, which cover only the range 52 to 295 K,we calculated the ratio TIT' where ?' is the tempera-ture at which Ci of 7-Ca2SiOa is the same as C[ forMg2SiOa at the temperature ?. The ratio TIT' wasthen plotted as a function of temperature and extro-polated to zero kelvin. From the extrapolated val-ues of TIT' and our measured value of Ci (I) ofMg2SiOa we generated values of Ci for yCa2SiOafor temperatures between 5 and 52 K. These esti-mated values for Ci of yCa2SiO4 wer€ combinedwith King's measured values above 52 K to calcu-late ^Sfr for yCa2SiO+ at 5 K intervals. Next weplotted Si for yCa2SiOa and the quantity (S1,"." -2R1n6) for Mn2SiOa where 2R1n6 is the limitingvalue for the magnetic entropy at high temperature,and derived the ratio TIT' wherc again T' is the

rAlthough neither Mg2SiOa or yCa2SiOa are ideal models forcalculating the lattice heat capacity or entropies of the transitionmetal olivines, we use them because Zn2SiOa, which would havebeen a more satisfactory model, does not crystallize in theolivine structure---€ven up to 150 kilobars (15 MPa) pressure,(Syono et al. l97l).

temperature at which the quanity (Si - 2Rln6) forMn2SiOa has the same value as ,S'(Ca2SiOa) at thetemperature ?, see Figure 6. This ratio was linearlyextrapolated to 0 K and used to estimate the latticeentropies for MnzSiOa. This estimated lattice entro-py was then subtracted from our smoothed valuesfor the total entropy of Mn2SiOa, listed in Table 5,to obtain the entropy associated with the orderingof the magnetic moments of the Mn2+ ions. Theresultant magnetic entropy is shown in Figure 7.Our measurements for Co2SiOa were treated in asimilar fashion. For Co2+ the spin quantum number^S is 3/2. However, at low temperature Coz* fre-quently behaves as though S were 1/2, that is theeffective spin S' : ll2 (e.9., Carlin and van Duyne-veldt, 1977, p. g:71) and accordingly for ourcorresponding states estimation of the lattice entro-py we used the quantity (,St - 2Rln2) for CozSiO+.We therefore expect the magnetic entropy ofCozSiO+ to approach 2Rln2 for ? D Trq and this is infact what we observed.

From Figure 7 we see that for Mn2SiOa themagnetic entropy is only about 75 percent of 2Rln6(29.79 J/mol.K) at the N6el temperature, and that ofCozSiOr is only about 71 percent of 2Rln2 (11.53 J/

0 1 . 0 2 . 0

T/T N6"t

Fig. 7. Magnetic entropies of CozSiOa and Mn2SiOa.

20Y

oE

o;o

4E0 ROBIE ET AL.: ENTROPIES OF MezSiOa,, Mn2SiOa, and Co2SiOa

mol'K) at fN. This indicates that appreciable shortrange order is retained by the magnetic spin sys-tems at temperatures well above Zry and in fact thetotal expected magnetic entropies are not fullyexcited until temperatures of the order of twice theN6el temperature have been reached.

The lattice heat capacities of Mn2SiOa andCozSiOa were derived from our estimated latticeentropies utilizing the thermodynamic relation

Cl.t : (dsht/d7)

Values for C61 so obtained were subtracted fromo'rr measured total molar heat capacities (Figs. 4and 5) to obtain the magnetic heat capacities ofMn2SiOa and Co2SiOa listed in Tables 10 and 11.

The Debye temperature of Mg2SiOa

The Debye temperature, 0o is the most usefulsingle parameter for characterizing the thermalproperties of a solid. The Debye temperature maybe calculated from low temperature heat capacitymeasurements from equation (8)

6: Ie2ls)qrraRf lcufit3 (s)

where q is the number of atoms in the "molecule"and R is the gas constant (8.3143 J/(mol.K)). Thisrelation is strictly rigorous only for temperatures ofthe order of T < fu/50, Blackman (1955). From our

Table 10. Magnetic heat capacity of Mn2SiOa (tephroite) in theneighborhood of the N6el temperature, 47.38 K.

*liT?,, 37[;:i'l] J/?;rt.K) ,,?ffiy.,r

Table 11. Magnetic heat capacity of Co2SiOa (cobalt olivine) inthe neighborhood of the N€el temperatwe, 49.76 K.

Co ( to ta l )J / (mo l K) ,r?#t'*r ,r?ffiY'*l

3 9 . 4 143.2846. 0547.6948. 0848.3248. 5448.7648.9749.1749.3749. 554 9 . 9 55 0 . 1 25 1 . 0 252.105 3 . l 754 .235 5 . 2 5

20.6728. 553 6 . 4 84 3 . 4 045.7 |47 .2148.9250.845 3 . 1 55 5 . 8 05 9 . 3 16 4 . 0 539.4732.772 7 . 6 726.6226.2126.3026.5r

8 . 8 51 1 . 3 813.221 4 . 3 114.5714.731 4 . 8 81 5 . 0 41 5 . 1 8_t c . JJt5.471 5 . 6 01 5 . 8 81 6 . 0 rt6 .67t 7 . 4 618.261 9 . l 01 9 . 7 8

1 l . 8 2L 7 . 1 72 3 . 2 629.093 1 . l 432 .4834. 043 s . 8 03 7 . 9 740.4743 .844 8 . 4 523.5916.761 1 . 0 09 . 1 6

7 . 2 06 . 7 3

heat capacity results for T < 15 K, Table l, weo b t a i n 6 : 7 6 8 1 t 5 K .

The Debye temperature is also related to themean velocity of sound in a solid by relation (9)

6: (trlt<)tfqNp/4zMlr/3v- (9)

where h is Planck's constant (6.62618 x 10-34J'sec), k is Boltzman's constant (1.38066 x 10-23J'K-l), q is the number of atoms in the "molecule",N is Avogadro's number (6.02209 x 1023 mol-r), pis the density in g ' cf,-3, and M the formula weightin grams. Robie and Edwards (1966) have shownhow the mean sound velocity, vm, may be calculat-ed for a crystal of any symmetry from the singlecrystal elastic constants and the density by meansof numerical integration with a computer.

We have calculated the Debye temperature ofsynthetic forsterite from the room temperature elas-tic stiffness constants (c;i) of Graham and Barsch(1969), and of Sumino et al. (1977) using a modified(nonrneN-IV) version of the computer programmentioned by Robie and Edwards (1966). Thesecalculations yield v- : 5.553 and 5.532 km/sec fromthe data of Graham and Barsch and from Sumino eta/. respectively. The corresponding ff298; are757.9 and 755.9 K. Sumino et al. (1977) also mea-sured the elastic constants at 83 K from which weobtained v- : 5.604 km/sec and 6s (83) : 766.6 K.From results of Sumino et al. we estimate 6tOt :

TempKel v i ns

34.253 8 . 1 04 2 . ? 342.724 3 . 6 144.394 5 . 0 945.574 5 . B 546.1446.4246.644 6 . 8 147.004 7 . t 847.6848. 0448.424 8 . 8 151.7252.99

20.9425.263 0 . 4 93t.623 3 . 2 034.7 I7 A t 1

36.8838. 19? O ? E

39.924 1 . 0 04 1 . 6 042.75+ J . 5 041.923 0 . 7 129. 5829.0?29.833 0 . 9 9

5 . 6 8

9 . 9 410.271 0 . 8 111.2911.7 472.041 2 . 1 51 ? . 4 0t 2 . 5 8L 2 . 7 312.8412.961 3 . 0 81 3 . 4 11 3 . 6 51 ? 0 1

14.171 6 . 1 41 7 . 0 4

15.261 7 . 5 82 0 . 5 521.352 2 . 3 923.4224.4924.8426.0426.95t t . 5 4

28.2728.7 629.7 930.482 8 . 5 1l 7 . 0 615.671 4 . 8 51 3 . 6 9L J . v )

ROBIE ET AL.: ENTROPIES OF MezSiOe, Mn2SiOa, and CozSiOa 4El

768.6 K which is in excellent agreement with ourcalorimetric value.

'At the lowest temperatures of our measurements,the heat capacity of MnzSiO+ and Co2SiOa arecomposed of a lattice (Debye) term plus a contribu-tion arising from the spin waves (magnons) (see forexample, Gopal, 1966). Both of these terms vary asT3 and accordingly we cannot evaluate their Debyetemperatures (6) from our measurements. Onemay however make a first order estimate of thecoefficient of the spin wave term as follows: fromthe room temperature elastic constant values ofSumino (1979) for Mn2SiOa and Co2SiO4 one calcu-lates 6 of 533 and 548 K respectively. Assumingthese (elastic) Debye temperatures to be indepen-dent of temperature, the lattice heat capacity maybe calculated from (10)

Cy: |215 rraqn7dltdo (10)

for T ( enl50. This calculated lattice heat capacityis then substracted from the measured (total) heatcapacity to obtain that arising from the spin waves.At 10 K for Co2SiOa the spin wave heat capacity isapproximately 18 percent of the measured molarq.

AcknowledgmentsThis work was in part supported by the U.S. Department of

Energy through the Los Alamos Scientific Laboratory, contractX69-9915F-3. We thank our U.S. Geological Survey colleagues,H.T. Haselton, Jr., and Susan Werner Kieffer for their manyuseful suggestions for improving the manuscript.

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ROBIE ET AL.: ENTROPIES OF MgzSiOa,, Mn2SiOa, and Co2SiOa

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Manuscript received, July 20, 1981 ;acceptedfor publication, January 20, 1982.