Computer calculations of the MgO-SiO2-AlO1.5 ternary and higher order phase diagrams through
thermodynamic analysis of phase equilibria
by Matthew James Scanlon
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Chemistry
Montana State University
© Copyright by Matthew James Scanlon (1988)
Abstract:
Published data on the MgO-Sio2-A101.5 ternary system, along with the binaries and single component
phases, are analysed in order to develop a computer model for the ternary system. In doing this a very
useful relationship between the bulk modulus and coefficient of thermal expansion using the
Murnaghan logarithmic equation of state is derived, thus allowing dK/dT to be calculated from thermal
expansion measurements. Also, the alpha-beta transition in quartz was analysed using Pippard's theory
of second order phase transitions, accurate X-ray data and the pressure dependence of the transition
temperature.
Full equations of state for quartz are given up to 1900 K and 4000 MPa. The phase diagram is also
calculated.
From analysis of the phase equilibria in the MgO-SiO2 system and the enthalpy of vitrification of
MgSi03, the enthalpies of fusion of enstatite and forsterite were refined. The final best values for the
heats of fusion were 48.8 ± 4 kJ/mole and 92.9 ± 12 kJ/mole respectively.
Also phase diagrams are calculated at 0.1 and 1000 MPa using Redlich-Kister coefficients.
Methods of dealing with three, four, and five component systems are developed using Redlich-Kister
equations. Portions of the phase diagrams for the MgO-SiO2 A101.5 ternary and the
FeO-Feo1.5-Cao-Sio1.5 system are calculated. S CdL (s>3
COMPUTER CALCULATIONS OF THE MgO- Si O2 - A l O 1 g TERNARY AND
HIGHER ORDER PHASE DIAGRAMS THROUGH THERMODYNAMIC
ANALYSIS OF PHASE EQUI LI BRI A
by
Mat t hew James Sc a n l o n
A t h e s i s submitted in p a r t i a l f u l f i l l m e n t
o f t h e r e q u i r e m e n t s f o r t h e degr ee
of
Doct or
of
Phi l o s o p h y
in
Chemi st ry
MONTANA STATE UNIVERSITY
Bozeman, Mont ana
March
I 988
APPROVAL
of a thesis
submi t t ed
by
Mat t hew James Scanl on
T h i s t h e s i s has been r e a d by each member o f t h e
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b i b l i o g r a p h i c s t y l e , and c o n s i s t e n c y , and i s r e a d y f o r
s u b mi s s i o n t o t h e C o l l e g e o f Gr a d u a t e S t u d i e s .
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ACKNOWLEDGMENT
My. g r a t i t u d e
and t h a n k s t o a l l
s u p p o r t e d me t h r o u g h
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and t h i n
d u r i n g t he course of
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and enc our age men t
during
t h e wor k and t h e w r i t i n g
and E l a i n e Howal d hel ped i n t h e p r o o f
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Pat and Gayl e Cal l i s f o r
friendship.
support
and
V
TABLE OF CONTENTS
INTRODUCTION
The R e d l i c h - K i s t e r
Equati ons
.
.
The Two Component Phase Di agr am.
Ext ended R e d l i c h - K i s t e r
The H i g h e r
Notation
Component Syst ems
...............................................
THE ONE COMPONENT PHASES...................................................................... 14
Equations of
State f o r
Magnesi um Oxi de
........................
14
E q u a t i o n s o f S t a t e f o r S i l i c o n D i o x i d e ........................ 28
The Al pha Qu a r t z t o Bet a Qu a r t z T r a n s i t i o n . . 30
The E q u a t i o n o f S t a t e f o r Bet a Qu a r t z . . . .
42
E q u a t i o n o f S t a t e o f Al pha Qu a r t z ....................... 49
E q u a t i o n o f S t a t e f o r C o e s i t e ................................. 52
The E q u a t i o n o f s t a t e f o r C r i s t o b a l i t e . . . .
60
E q u a t i o n o f S t a t e f o r S i l i c o n D i o x i d e L i q u i d . 66
E q u a t i o n s o f S t a t e f o r Al umi num Oxi de
Al umi num Oxi de ( C, Cor undum) . .
Al u mi n i u m Oxi de ( L i q u i d ) . . . .
The S t o c h i o m e t r i c Phases ........................
F o r s t e r i t e . ...........................................
'
E n s t a t i t e ( Magnesi um S i l i c a t e ) .
S p i n e l ( Magnesi um A l u m i n a t e ) . .
C o r d i e r i t e ...............................................
.
.
.
.
75
75
76
77
. 83
86
THE BINARY SYSTEMS
The M a g n e s i a - S i l i c a B i n a r y .......................................... - - 86
The E n t h a l p y o f F u s i o n o f Magnesi um Oxi de . . 93
The E n t r o p y o f M i x i n g . . ...........................................95
The Heat C a p a c i t y ..........................................
.
.98
The Phase D i a g r a m ........................................................... 102
The A l u m i n a - S i l i c a
Binary
102
vi
THE TERNARY SYSTEM MAGNESIUM OX I DE - SI L I CA-ALUM I NA .
.
. 112
THE FeO- FeO1 ^ - S i O g - A l O 1.-CaO SYSTEM............................................. 124
The FeO- FeO1
gSy s t e m.....................................................................124
SUMMARY........................................................................................................... . 136
REFERENCES C I T E D , ...................................................................................... 138
VI I
LI ST OF TABLES
Ta bl e
1.
Page
The R e d l i c h - K i s t e r c o e f f i c i e n t s f o r a t h r e e
component ' s y s t e m, t h r o u g h s e v e n t h power i n mol e
f r a c t i o n ..............................................
10
2.
Red I i c h - K i s t e r c o e f f i c i e n t s and t h e i r s u b s c r i p t s
c o r r e l a t e d w i t h t h e i r c o r r e s p o n d i n g p o w e r s .......................12
3.
The t h r e e and f o u r component R e d l i c h - K i s t e r
c o e f f i c i e n t s .........................................................................................
4.
C a l c u l a t e d mo l a r vol umes f o r
13
MgO............................................... 22
5.
The e q u a t i o n s o f s t a t e , he at c a p a c i t y e q u a t i o n s and
s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f MgO ( c ) and
MgO ( I ) ...................................................................................................... 23
6.
Val ue s r e p o r t e d f o r t h e e n t h a l p y change bet ween
a l p h a q u a r t z a t 298. 15 k and b e t a q u a r t z a t 1000 K . 29
7.
Ca l c u l a t e d e q u i l i b r i u m c on s t an t s at var i o us
t e m p e r a t u r e s bet ween ou r 1s and Robi e e t a I 1s .
e q u a t i o n o f s t a t e f o r Bet a q u a r t z ............................
29
8.
The f u n c t i o n F (6 ) r e p r e s e n t i n g t h e
d i f f e r e n c e i n vol ume o f a l p h a q u a r t z f r o m a f u l l y
d i s o r d e r e d be t a q u a r t z a t t h e same t e m p e r a t u r e . . •. 39
9.
The e n t r o p y and e n t h a l p y changes f o r
n e a r t h e l ambda p o i n t ............................
alpha quart z
41
10.
The e q u a t i o n s o f s t a t e , h e a t c a p a c i t y e q u a t i o n s and
s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f a l p h a and bet a
q u a r t z . ,...................................................................................................... 46
11.
Compar i son o f t h e t h e r mo d y n a mi c v a l u e s f o r
q u a r t z and c o e s i t e a t h i g h p r e s s u r e s and
t e m p e r a t u r e s .....................................
bet a
50
viii
Tabl e
12.
13.
14.
15.
Page
Compar i son o f t h e p o l y n o m i a l f i t and P i p p a r d
c a l c u l a t i o n s f o r t h e t h e r mo d y n a mi c p r o p r e t i e s
a l p h a and b e t a q u a r t z 60 K bel ow t h e l amda
t r a n s i t i o n ........................................................................
of
55
The e q u a t i o n o f s t a t e , h e a t c a p a c i t y e q u a t i o n s and
s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f c o e s i t e and
h i g h c o e s i t e ..........................................
56
The t h e r mo d y n a mi c p r o p e r t i e s
61
of c r i s t o b a l i t e .
. . .
The e q u a t i o n s o f s t a t e , h e at c a p a c i t y e q u a t i o n s
■ and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f
c r i s t o b a l i t e and l i q u i d q u a r t z ..........................................
65
16.
The e q u a t i o n s o f s t a t e , he at c a p a c i t y e q u a t i o n s
and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f al umi num
o x i d e , cor undum andl i q u i d ............................................................... 70
17.
The e q u a t i o n o f s t a t e , h e a t c a p a c i t y e q u a t i o n
and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f
f o r s t e r i t e ( MggSi O^ ) . . . . . ........................................................
76
18.
The e q u a t i o n s o f s t a t e , he at c a p a c i t y e q u a t i o n s
and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f t h e t h r e e ■
f o r ms o f MgSi O3 ; e n s t a t i t e , p r o t o e n s t a t i t e and
19.
The e q u a t i o n o f s t a t e , he at c a p a c i t y e q u a t i o n and
s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f s p i n e l
( Mg A l 2O4 ) .. . .......................................................................................... 80
20.
AS and AV o f d i s p r o p o r t i o n a t i o n
21.
The e q u a t i o n o f s t a t e , he at c a p a c i t y e q u a t i o n and
s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f c o r d i e r i t e
(Mg2A l 4 S i Oi g ) ......................................................................
for
MgAl 2O4 . . . .
85
85
22.
Publ i shed e s t i mat es of the e n t h a l p y o f f u s i o n of
v a r i o u s compounds i n t h e MgO- Si O2 s y s t e m ................................ 873
2
23.
Excess e n t h a l p i e s o f m i x i n g o f MgO and Si O2 f o r
v a r i o u s model s a t a mol e f r a c t i o n o f 0 . 5 ........................88
ix
Tabl e
24.
Page
Cal cul at ed e q u i l i b r i u m constants f o r the r e a c t i o n :
.ZMg7Si Oad ( I ) - Mg3S i 7O7 + MgO( I )
a t x MgO = 0 . 7 , wher e z i s t n e number o f mol es o f
Mg3S i 2O7 . . . ' ............................................................
95
25.
R e d l i c h - K i s t e r c o e f f i c i e n t s f o r a c i d i c and b a s i c
MgO- Si O2..................................................................................................... 107
26.
R e d l i c h - K i s t e r c o e f f i c i e n t s f o r t he M O . r - Si 0 2
s o l i d and l i q u i d s y s t e m s ................... .... . . * . ................... 108
27.
The m a t r i x MT00P f o r c a l c u l a t i n g t e r n a r y
R e d l i c h - K i s t e r c o e f f i c i e n t s ..................................................... 115
28.
The t r a n s f o r m a t i o n m a t r i x , CNN, f r o m t h e R e d l i c h K i s t e r v e c t o r N^ 23 t o t h e v e c t o r N3 1 2 .............................. 116
29.
C a l c u l a t e d R e d l i c h - K i s t e r c o e f f i c i e n t s f o r excess
e n t h a l p y f r o m t h e Toop- Muggi anu i n t e r p o l a t i o n
a l o n g w i t h ou r f i n a l s e l e c t e d v a l u e s f o r t h e
MgO- Si O2- A l O 1 g t e r n a r y a t 1800 . . . . . . . . . .
30.
C a l c u l a t e d excess e n t h a l p i e s a t 1800
R e d l i c h - K i s t e r c o e f f i c i e n t s from t he
i n t e r p o l a t i o n and our f i n a l s e l e c t e d
t h e MgO- Si O2- A l O 1 g t e r n a r y . . . . .
31.
R e d l i c h - K i s t e r t e r ms t h r o u g h F f o r t h e t e r n a r y
syst em A l O 1 g - S i O2-MgO................................
32.
. 117
K us i n g the
Toop- Muggi ana
values f o r
. . . . . .
. 120
The he at c a p a c i t y e q u a t i o n s and t h e r mo d y n a mi c
p r o p e r t i e s o f v a r i o u s s t o i c h i o m e t r i c compounds i n
. t h e FeO- FeO1 g - S i 0 2 - A 1 0 1 g-CaO syst em . . . . . .
121
. 126
33.
R e d l i c h - K i s t e r c o e f f i c i e n t s f o r t h e FeO- FeO1 c
s o l i d b i n a r y .......................................................................*• • • - 128
34.
Redlich-Kister
liq u id binary
c o e f f i c i e n t s f o r t h e FeO- FeO1 c
...........................................*. . .
. 129
LI ST OF FIGURES
Fi gur e
Page
1.
The c o e f f i c i e n t o f t h e r m a l e x p a n s i o n o f MgO. The S
shaped l i n e w i t h a l t e r n a t i n g l ong and s h o r t dashes
r e p r e s e n t s t h e v a l u e s used i n r e f . 25. The s t r a i g h t
dashed l i n e i s . 0 0 0 0 4 3 5 + 1 . 0 x 1 0
T - 1 0 0 0 ) . The
s o l i d l i n e shows t h e v a l u e s s e l e c t e d i n t h i s w o r k .
Above 300K t h e s o l i d l i n e i s t h a t c a l c u l a t e d f r om
r e f . 16, 17, and 2 0 ........................................................................... 15
2.
The i s o e n t r o p i c b u l k modul us f o r MgO. The s o l i d
l i n e i s c a l c u l a t e d f r o m t h e t h e o r y p r e s e n t e d he r e.
The e x p e r i m e n t a l v a l u e s ar e f r o m S p e t z l e r , b l a c k
c i r c l e s ; Soga and A n d e r s o n , open c i r c l e s ; and f r om
Ander son and. A n d r e a t c h , d i a mo n d s .......................................... 21
3.
C o n t o u r l i n e s f o r t h e Gr u n i e s e n p a r a m e t e r , 6 , f o r
MgO on a P-T f i e l d .
The c o n t o u r i n t e r v a l i s 0 . 0 1 ,
except t h a t at hi gh t e mp e r a t u r e s d o t t e d l i n e s
a t 0 . 0 2 i n t e r v a l s ar e a l s o s hown.......................................... 25
4.
Gr u n e i s e n p a r a m e t e r s , <$, f o r MgO p l o t t e d v e r s u s
vol ume f o r t h e t h r e e p r e s s u r e s 0 . 0 1 , 5000, and
10000 MPa. The c i r c l e s r e p r e s e n t c a l c u l a t e d v a l u e s
a t 300, 400, andn6Q0 K. The d o t t e d l i n e i s f o r
Y = 0. 1 26 52 V1 ‘
............................................................. 26
5.
Heat c a p a c i t y , Cp, f o r a l p h a q u a r t z ne ar t h e l ambda
t e m p e r a t u r e . The s o l i d l i n e r e p r e s e n t s o u r
c a l c u l a t e d v a l u e s . The e x p e r i m e n t a l p o i n t s o f Moser
and S i n e l 1n i k o v ar e shown as open and f i l l e d
c i r c l e s r e s p e c t i v e l y . . . . . ...............................................
6.
C e l l demens i o n s f o r a l p h a and b e t a q u a r t z v e r s u s
t e m p e r a t u r e . Our c a l c u l a t e d f i t s t o t h e v a l u e s o f
Ackerman and S o r r e l ( f i l l e d c i r c l e s ) ar e shown as
s o l i d l i n e s . Ol d e r e x p e r i m e n t a l v a l u e s o f Jay
( 1939) and B e r g e r e t a l . ( 1 9 6 6 ) ar e shown as open
c i r c l e s . . ; . . ............................................... ............................
F i gur e
Page
7.
Vol umes o f be t a q u a r t z a t 0. 1 MPa and a t t h e l ambda
p o i n t p l o t t e d v e r s u s t e m p e r a t u r e . ..................................... 43
8.
Our c a l c u l a t e d c o n t o u r l i n e s f o r t h e vol ume o f bet a
q u a r t z .................................................... ' ................................................. 47
9.
Our c a l c u l a t e d c o n t o u r l i n e s f o r t h e e n t r o p y o f
b e t a q u a r t z .............................................................................................48
10.
( d T / d P ) <- = aVT/ Cp a t 800 K . The s o l i d l i n e i s
ou r c a l c u l a t e d c u r v e . The open c i r c l e s ar e t h e
e x p e r i m e n t a l v a l u e s o f B o e h l e r ...................................................53
11.
( d T / d P ) . = aVT/ Cp a t 1000 K . The s o l i d l i n e i s
ou r c a l c u l a t e d c u r v e . The open c i r c l e s ar e t he
e x p e r i m e n t a l v a l u e s o f B o e h l e r ...................................................54
12.
Heat c a p a c i t y v a l u e s f o r c o e s i t e ( s o l i d l i n e ) and
c r i s t o b a l i t e ( dashed l i n e s ) ........................•........................... 57
.13.
C a l c u l a t e d phase d i a g r a m f o r S i Op . F i l l e d c i r c l e s
ar e f r o m B o e h l e r ( 1 9 8 2 ) , Bohen and B o e t c h e r ( 1 9 8 2 ) ,
M i r w a l d and Massone ( 1 9 8 0 ) , w i t h t h e v a l u e s o f Boyd
and En g l a n d , ( 1960 ) , as open c i r c l e s ....................................59
14.
The t h e r m a l e x p a n s i o n c o e f f i c i e n t o f al umi num o x i d e
v e r s u s t e m p e r a t u r e ........................■.................................................. 72
15.
The a d i a b a t i c b u l k modul us o f A l 0 1 . 5 v e r s u s
t e m p e r a t u r e . The f i l l e d c i r c l e s ar e v a l u e s f r o m
T e f f t ( 1 9 6 6 ) ; t h e open c i r c l e s ar e ou r c a l c u l a t e d
v a l u e s and t h e s o l i d di amonds ar e t h e v a l u e s o f
Soga and Ander son ( I 9 6 7 ) ............................ ■............................... 73
16.
The c a l c u l a t e d phase d i a g r a m f o r MgSi Og. The
e x p e r i m e n t a l p o i n t s shown a r e t h o s e o f Gr o v e r
( 1972 ) and Boyd and Engl and ( 1965 ) ......................................
79
17.
C o n t o u r l i n e s f o r t h e vol ume o f MgAl pO4 , S p i n e l , as
a f u n c t i o n o f t e m p e r a t u r e as r e p o r t e d by Howa I d ,
e t a.I ........................................................................................................8 1*8
1
18.
Co u n t o u r l i n e s showi ng AV f o r t h e r e a c t i o n
MgAl 2O4 (C) = MgO(c) + 2A101 i 5 ( c ) ............................
82
xii
Fi g u r e
Page
19.
E q u i l i b r i u m l i n e f o r the s pi nel d i s p r o p o r t i o n a t i o n '
as c a l c u l a t e d i n t h i s w o r k . .................................................... 84
20.
V a r i o u s c a l c u l a t e d excess e n t h a l p i e s f o r t h e
MgO-Si Og phase d i a g r a m .................................................................. 89
21.
C o r r e c t e d Gi bbs f r e e e n e r g y ( G - 3 8 2 . 1 4 1 ) v e r s u s
mol e f r a c t i o n o f Si Og f o r t h e syst em MgO-Si Og . . .
91
22.
Entropy
in the
filled
mi xi ng
o f m i x i n g v e r s u s t h e mol e f r a c t i o n o f SiOg
syst em MgO- Si Og. C i r c l e s , open s qu ar e s and
s qu ar e s ar e f r o m L i n and P e l t o n 9 i d e a l
and t h i s wor k r e s p e c t i v e l y . ' . . . . . . . . . 9 9
23.
Excess heat c a p a c i t y v e r s u s mol e f r a c t i o n o f Si Og
a t 2123 K f o r t h e MgO-Si Og s y s t e m ............................ • 101
24.
The c a l c u l a t e d phase d i a g r a m a t 0. 1 MPa f o r t h e
MgO-Si Og s y s t e m ................................................................................ 103
25.
A c t i v i t i e s o f S i 0 ? . Open c i r c l e s ar e c a l c u l a t e d
f r o m ou r R e d l i c h - k i s t e r c o e f f i c i e n t s , f i l l e d
di amonds ar e f r o m S. Kambayashi and E. Kat o . . ' .
. 104
26.
A c t i v i t i e s o f MgO. F i l l e d di amonds ar e c a l c u l a t e d
f r o m o u r R e d l i c h - K i s t e r c o e f f i c i e n t s , open c i r c l e s
ar e f r o m S. Kambayash i and E. K a t o ............................ ; . 105
27.
The C a l c u l a t e d phase d i a g r a m a t 1000 MPa. f o r t he
MgO-Si Og s y s t e m ................................................................................. 106
28.
The c a l c u l a t e d c o n t o u r l i n e s f o r t he
Mg O- Si Og - Al O1 5 t e r n a r y s y s t e m .................................................122
29.
The c a l c u l a t e d phase f i e l d s f o r
MgO- Si O2- A l O l i 5 ............................
the t e r n a r y
syst em
123
30.
Co n t o u r l i n e s i n t h e A l 0 1 . 5 - S i Q2 - FeOx syst em
v e r s u s w e i g h t f r a c t i o n c a l c u l a t e d as Fe0 1 . 5 f o r
H2O/ H2 = 1 . 3 .
T e mp e r a t u r e s a r e g i v e n i n 200
de gr e e F a h r e n h e i t i n t e r v a l s ..................................................... 131
31.
C o n t o u r l i n e s i n t h e Al 0 1 . 5 -S i 0 2 -F.e0x syst em
a t 5/o CaO by w e i g h t and HgO/Hg = I • 3 ................................. 132*
32.
C o n t o u r l i n e s i n t h e A l 0 1 . 5 - S i 0 2 ~Fe0 x s yst em
a t 10% CaO by w e i g h t and HgO/Hg = 1 . 3 ..............................133
xii i
Figure
Page
33.
Co n t o u r l i n e s i n t h e A l O i . g - S i Og-FeOx syst em
a t 20% CaO by w e i g h t and AgO/Hg = 1 . 3 ............................ 1344
3
34.
Co n t o u r l i n e s i n t h e A l 0 1 . 5- Si Og-FeOx syst em
a t 20% CaO by w e i g h t and HgO/Hg = 1 . 3 ............................ 135
ABSTRACT
P u b l i s h e d d a t a on t h e Mg O- S i C^ - A l O{ , 5 t e r n a r y
s y s t e m, a l o n g w i t h t h e b i n a r i e s and s i n g l e component
phas es , ar e a n a l y s e d i n o r d e r t o d e v e l o p a c o mp u t e r model
f o r t h e t e r n a r y s y s t e m.
In doi ng t h i s a v er y us e f ul
r e l a t i o n s h i p bet ween t h e b u l k modul us and c o e f f i c i e n t o f
t h e r m a l e x p a n s i o n u s i n g t h e Mur naghan l o g a r i t h m i c e q u a t i o n
o f s t a t e i s d e r i v e d , t h u s a l l o w i n g dK/ dT t o be c a l c u l a t e d
f r o m t h e r m a l e x p a n s i o n meas ur ement s .
Also, the al pha-beta
t r a n s i t i o n i n q u a r t z was a n a l y s e d u s i n g P i p p a r d ' s t h e o r y
o f second o r d e r phase t r a n s i t i o n s , a c c u r a t e X - r a y d a t a and
t h e p r e s s u r e dependence o f t h e t r a n s i t i o n t e m p e r a t u r e .
F u l l e q u a t i o n s o f s t a t e f o r q u a r t z ar e g i v e n up t o 1900 K
and 4000 MPa.
The phase d i a g r a m i s a l s o c a l c u l a t e d .
From a n a l y s i s o f t h e phase e q u i l i b r i a i n t h e MgO-Si O2
sy s t em and t h e e n t h a l p y o f v i t r i f i c a t i o n o f Mg Si O3 , t he
e n t h a l p i e s o f f u s i o n o f e n s t a t i t e and f o r s t e r i t e were
refined.
The f i n a l b e s t v a l u e s f o r t h e h e a t s o f f u s i o n
wer e 4 8 . 8 ± 4 k d / mo l e and 9 2 . 9
± 12 k J / mo l e r e s p e c t i v e l y .
A l s o phase d i a g r a ms ar e c a l c u l a t e d a t 0. 1 and 1000 MPa
using R e d l i c h - K i s t e r c o e f f i c i e n t s .
Met hods o f d e a l i n g w i t h t h r e e , f o u r , and f i v e
component syst ems ar e d e v e l o p e d u s i n g R e d l i c h - K i s t e r
equations.
P o r t i o n s o f t h e phase di a g r a ms f o r t h e MgO-Si Og
A I O1. 5 t e r n a r y and t h e FeO- FeO1 , 5 - CaO- Si Q g - A l 0 1_5 syst em
ar e c a l c u l a t e d .
I
I '
INTRODUCTION
In o r d e r t o
one must t o
all
calculate
a mu l t i c o m p o n e n t
be a b l e t o c a l c u l a t e
the r e a c t i o n s
phase d i a g r a m,
e q u i l i b r i u m constants f o r
bet ween t h e v a r i o u s
phases p r e s e n t .
For
t h e MgO- Si Og- Al gOg t e r n a r y syst em t h e one component
syst ems ar e pu r e MgO, Si Og and Al O^
g.
pu r e s t o i c h i o m e t r i c
MgSi Og,
solids
A l g S i g O j 3 ar e a l s o f or med
MggSi O^,
in t h i s
The a d d i t i o n a l
MgAl gO^ and
phase d i a g r a m .
I n a one component phase di agr am, , v ol u me,
and p r e s s u r e can be mo d e l l e d by f i t t i n g
a power s e r i e s
of these t erms,
so t h a t
species
can be d e t e r m i n e d -.
equation
of
to
lines
for
for
t e r ms
individual
The r e p r e s e n t a t i o n
is
ar e
o f vol ume
known as t h e
a pu r e m a t e r i a l .
capacity,
a power s e r i e s
and p r e s s u r e s .
the
o f t e m p e r a t u r e and p r e s s u r e
state
The h e at
vol umes ar e
these pol ynomial
t h e vol ume c o n t o u r
as a f u n c t i o n
fit
the c a l c u l a t e d
for
The
can h a n d l e up t o t h i r t y - f i v e
a wi de r a nge o f t e m p e r a t u r e s
Once t h e c o e f f i c i e n t s
calculated
t h e vol ume da t a t o
i n b o t h t e m p e r a t u r e and p r e s s u r e .
c o mp u t e r pr ogr am we ar e u s i n g
a c c u r at e over
temperature
Cp , a t c o n s t a n t
p r e s s u r e can be
o f t he f orm
Cp = A + B(T - T0 ) + C(T - T0 ) 2 + D(T - Tq ) 3
+
.
•
Cl )
2
wher e A,
B, C and D ar e c o n s t a n t s , T i s t h e t e m p e r a t u r e
and Tq i s
a standard temperature of
The d e r i v a t i v e
pressure
of the
1000 K o r 29 8. 1 5 K .
he at c a p a c i t y w i t h
respect
to
is
( dCp/ dP ) T = V U d / d T )
- Ta2 V
= - T Xd 2VZdT2 ) p
( 2)
wher e a t h e t h e r m a l
expansi on
Therefore,
c a p a c i t y can be c a l c u l a t e d
pressures
t h e he at
coefficient
f r o m t he. vol ume p o l y n o m i a l s
The t e m p e r a t u r e
enthalpy of
is
l/V(dV/dT)
described
.
at var i ous
earlier.
and p r e s s u r e dependence o f t h e .
a substance
and t h e vol ume t h r o u g h
is re la te d
t o the heat c a p a c i t y
the eq uat i o n
dH = ( d H / d T ) pdT + ( d H/ d P l y d P
( 3)
whi c h becomes
dH = CpdT + (V - TaV)dP
respectively.
particular
Thus,
with
vol ume p o l y n o m i a l
f r o m t h e e l e me n t s
one can c a l c u l a t e
t e m p e r a t u r e and p r e s s u r e .
as. our
a value f o r
t he e n t h a l p y at a
t e m p e r a t u r e and p r e s s u r e r e f e r e n c e d f r o m t he
heats o f f o r m a t i o n
MPa.
( 4)
at
29 8. 1 5 K and t he
t he e n t h a l p y at
We have c ho os er
any
1000 K and .1
s t a n d a r d t e m p e r a t u r e and p r e s s u r e .
The e q u i l i b r i u m
constant
is
Gi bbs f r e e
energies
equilibria
through, the eq uat i on
related
o f t h e component s
AG01- = - R T l n ( K 1 ) .
to the
involved
standard
i n t he
( 5)
3
Si nc e AG = AH - TAS,
the e q u i l i b r i u m constant
can a l s o be
e x p r e s s e d by t h e e q u a t i o n
Ky = e x p ( - AG0ZRT) = e x p ( - A H ° / R T )
ror c o n v e r t i n g
K
to
exp ( AS0ZR)
base 10 and u s i n g J / mo l e y i e l d s
(
.
= 1 0 ( - A H o/ 1 9 . 14464 T ) 1Q( AS° / 19 . 144 64) >
However
in the
literature,
in t he form of P l a n c k ' s
includes
free
function,
Yj = ( G° t
Planck's
tabulated
function
d e f i n e d as
- h°298.15) / T *
The p r e s s u r e dependence o f
f r o m t h e he at
for
capacities
( 8)
Pl ancks f u n c t i o n
pressure
the e q u i l i b r i u m
Wi t h t h e
incorporated
compound a t
information
into
is
dependence.
constant
K = ' i o <- 1H° 2 9 8 . 1 5 / 1 9 . 14464T)
T
equations
Y.
( 7)
t h e t e m p e r a t u r e dependence o f b o t h t h e e n t h a l p y
and t h e e n t r o p y and i s
equation
energy i s o f t e n
6)
d e t e r mi n e d
Thus t h e
can be w r i t t e n
A Y / 1 9 . 14464) _
( g)
d e s c r i b e d above g a t h e r e d and
t h e c o mp u t e r ,
the p r o p e r t i e s
o f any
any t e m p e r a t u r e and p r e s s u r e f o r wh i c h t he
ar e a c c u r a t e can be c a l c u l a t e d .
The. R e d l i c h r K i s t e r . Equations
In or der to
one must model
using
a t wo component phase d i a g r a m,
t h e t h e r mo d y n a mi c p r o p e r t i e s
o f mol e f r a c t i o n
modelling
describe
(x).
Ther e ar e a v a r i e t y
phase d i a g r a ms
known e q u i l i b r i u m
i n t e r ms
constants.
as a f u n c t i o n
o f met hods o f
o f c h e mi c a l
equilibria
Al so s u b l a t t i c e
■
I
4
model s have been pr opo s ed and used s u c e ' s s f u I I y f o r
s p i n e l s . 4 ’ 5 ’ 6,7
calculations
equations
However ,
it
is d i f f i c u l t
o f phase e q u i l i b r i a
ar e b e i n g
use power s e r i e s
used.
t o do
when v a r i o u s
Therefore
it
is
types of
an adv a n t a g e t o
o f a s t a n d a r d f o r m , whi c h ar e a b l e t o f i t
t h e most compl ex s y s t e ms .
The q u a n t i t i e s
component
that
have t o be mo d e l l e d
syst em ar e t h e p a r t i a l
mo I a l
component s and t h e excess q u a n t i t y .
quantity
and t h e excess q u a n t i t y
i n a t wo
quantities
The p a r t i a l
o f bot h
mo I a I
ar e d e f i n e d t h r o u g h t he
f o I I o w i ng e q u a t i o n s
Q1 = ( d Q / d n i ) n 2 ? T i p
Q = n 1Q° +
n 2Q°. + Cn1 + n 2 )Qe
wher e Q1 i s t h e p a r t i a l
is
( 10)
mo l a l
Any power s e r i e s
component
s yst em must
t h e mol e f r a c t i o n
approaches
t h e v a l u e o f t h e mo l a l
Also,
is that
mo l a l
Q
of
a two
conditions.
component ,
quantity,
As
X.,
Q^ , appr oaches
q u a n t i t y o f t h e pur e s u b s t a n c e '
t h e ex c e s s q u a n t i t y ,
The e a r l i e s t
certain
of a p a r t i c u l a r
I the p a r t i a l
I,
and Qe i s t h e excess mo l a l
representation
satisfy
( 11)
Q° i s t h e t h e r mo d y n a mi c
pu r e s u b s t a n c e ,
quantity.
Q°.
q u a n t i t y o f component
t h e t h e r mo d y n a mi c q u a n t i t y ,
q u a n t i t y of the
'
Qe , must a p p r o a c h z e r o .
pr o p o s e d s e t o f power s e r i e s e q u a t i o n s
O.
p r o p o s e d by Ma r g u l e s
i n 1895.
The Ma r g u l e s
equations
ar e
5
Q1 - Q|
= (I
- . X
+ D1X1
1I 2M 1 +
3
B
1X 1
+
C
1X 12
•
.)
Q2 - Q2 = X12 I A 2 + B2X1 + C2X12 + DgX1^ + . • . )
Q6
- x i x2
These e q u a t i o n s
+ Bex I + ^ e x I
s a t i s f y t h e above c o n d i t i o n s ,
each e q u a t i o n
has a d i f f e r e n t
mol e f r a c t i o n
t e r ms w i t h i n
Two t e r ms
ar e o r t h o g a n o I i f
t h e t e r ms o v e r
t e r ms
the
all
space i s
ar e o r t h o g o n a l ,
same f i r s t
three
+ Bex l
three
set
ar e n o t
integral
zero.
If
then t he f o u r
coefficients
^•
however ,
of c o e f f i c i e n t s ,
an e q u a t i o n
the
+ ‘
( 12)
and t h e
orthogonal.
of the product
of
t h e mol e f r a c t i o n
t e r m s e r i e s woul d have
as t h e c o e f f i c i e n t s
in a
term s e r i e s .
Bal e and P e l t o n 9 , 10 have pr o p o s e d t h e f o l l o w i n g
s e t o f e q u a t i o n s t o model t h e t h e r mo d y n a mi c q u a n t i t i e s
? n
(
I
X
1
P
2 a, P,
- QJ =
1
i =0
2 n
2 L 1P1 ( X 1 )
- Qg = x P
1 PO
1 1
n
Pi ( x
= X1Xp s
Qe
• ^ PO
1 1
wher e P . ( x . ) ar e t h e s t a n d a r d Legen dr e p o l y n o m i a l s .
These
I
I
polynomials
ar e c o m p l e t e l y o r t h o g o n a l ; howev er ,
have d i f f e r e n t
coefficients
for
they s t i l l
t h e mol e f r a c t i o n
t er ms
each e q u a t i o n .
0.
Redlich
the f o l l o w i n g
and A.
T . K i s t e r 11, 12
s e t o f power s e r i e s
in
1948 pr opo s ed
equations
in
6
x 2 2 [A
Il
Cr
i
Qi
+
Q2
-
Q2
X
These e q u a t i o n s
the
BfSx1
+
I)
-
D l E x 1
- I)
.
.
.]
:
CfBx1
-
B l f 2X1
.
.]
I ) 2
-
D l E x 1
-
ClBx1
DlBx1
-
Z l f 2X1
+
B l2X1
-
I)
+
D l2X1
-
D 3
.
•
- I.
OJ
X
+
+
+
I)
=C
+
Qe
-
DlBx1
X 1 2 CA
=
B H x 1- -
+
+
-
I ) 2
+
+
.
Cf2X 1
-
A,
D
I 15)
.
D 2
(16)
ar e n o t c o m p l e t e l y o r t h o g o n a l ,
same c o e f f i c i e n t s
-
(14)
however ,
B, and C appear i n a l l
three
equations.
The r e a s o n t h e c o e f f i c i e n t s
mol e f r a c t i o n
t e r ms f o r
f r o m t h e excess p a r t i a l
definition
mo l a l
of the p a r t i a l
expression f o r
equations
-Q^
a r e t h e same i s t h a t
and Qg - Qg ar e d e r i v e d
q u a n t i t y Qe .
mo l a l
t he
From t h e
q u a n t i t y and t h e
t h e t h e r mo d y n a mi c q u a n t i t y g i v e n
10 and 11 t h e f o l l o w i n g
relationship
in
can be
f ou nd
Q1 - QJ + Qe + O i 1 + .I2 H d Q eZ d n 1 )
TjP.
( 17)
Si nc e
CdQeZ d n 1 I t1z = CdQeZ d x 1 I n 2 I d x 1Z d n 1I n 2
118)
d x 1/ d n 1 = X g Z l n 1 + ng)
119)
and
the equat i on
(EO)
Q1 - Q 1 = Qe ' + XgdQe Z d x 1
can be w r i t t e n .
qu antity with
Therefore,
the d e r i v a t i v e
r e s p e c t t o X1 , m u l t i p l i e d
o f t h e excess
by x 2 , p l u s t h e
I
7
ex c e s s q u a n t i t y
Redlich-Kister
For ex a mpl e ,
for
each R e d l i c h - K i s t e r
equation f o r
the p a r t i a l
t h e excess q u a n t i t y f o r
Qp - X1U
- X1 I D U x 1 -
term y i e l d s
mo l a l
t he
quantity.
t he D t erm i s
D 3
( 21)
so t h a t
Qp + X2 M Q eZdx1 ) = Dx2 Ef l
- Zx 1 J U x 1 -
I )3
+ S U x 1 - I J2Z f x 1 - X12 ) ]
( 22)
and
Qp +
Xgf dQeZ d x 1 ) = Dx2 E f l
- Zx1J f Z x 1 -
I )3
+ Q f Z x 1 - D 2 Cx1 - X12 ) ]
+ D E f x 1 - X12 J f Z x 1 Thi s
simplifies
Qe
+
I ) 3]
( 23)
to
Xgf dQe Z d x 1 ) = D f Z x 1 - D 2 ESx12 - Z x 13 - X1
+ I O x 1X2 - I O x 12X2 - x 2 ]
Then u s i n g t h e r e l a t i o n
be r e a r r a n g e d t o
Qe
+
X1 + X2 = I ,
( 24)
t h e e q u a t i o n can
give
Xgf dQe Z d x 1 ) = Xg2D f S x 1 -
D U x1 -
I ) 2-
( 25)
T h e . Two. Component P h a s e . Di a g r a m
To c a l c u l a t e
a t wo component phase d i a g r a m one must
be a b l e t o c a l c u l a t e
the a c t i v i t y . a t
t he t e mp e r a t u r e range o f
coefficients
for
coefficient,
l o g y , can p r o v i d e
activity's
the
interest.
any c o m p o s i t i o n ov e r
Redlich-Kister
l o g a r i t h m of the a c t i v i t y
information
about t he
dependence on C o m p o s i t i o n t h r o u g h t h e e q u a t i o n
8
( 26)
a = Yx
wher e a i s
the a c t i v i t y
and Y i s
At o t h e r t e m p e r a t u r e s t h e
coefficient
Thi s e q uat i o n
if
a c c u r a t e e n t h a l p y and heat
c a p a c i t y d a t a as a f u n c t i o n
can e a s i l y
of temperature
the R e d l i c h - K i s t e r
species
t h e r mo d y n a mi c
are i n e q u i l i b r i u m .
for
l og y
constant
can be c a l c u l a t e d f o r
equilibrium
activities
constant
to the a p p r o p r i a t e
mol e f r a c t i o n
calculated
for
stochiometric
a particular
species.
f r o m t he
power.
Thus,
the
can be
temperature.
suppose one i s c a l c u l a t i n g
curve f o r
The
in the e q u i l i b r i a
a t wh i c h e q u i l i b r i u m e x i s t s
For e x a mp l e ,
equilibrium
reaction.
involved
species
the e q u i l i b r i u m
can a l s o be c a l c u l a t e d
o f each s p e c i e s
if
each o f t h e
equilibrium,
that
a t whi ch
Therefore,
data are a v a i l a b l e f o r
in a p a r t i c u l a r
liquid
coefficients
r e pr es ent the p o i n t
involved
raised
and c o m p o s i t i o n
be c a l c u l a t e d .
The phase d i a g r a m l i n e s
different
( 27)
d e r i v ed from the Gi bbs- Hel mhol t z
Therefore,
ar e a v a i l a b l e ,
from the eq uat i on
= ( H 1 - H?) / I 9 . 1 4 4 6 4 .
is
relationship.
coefficient.
l o g a r i t h m of the a c t i v i t y
can be c a l c u l a t e d
d I o g y / d ( 1/ T)
the a c t i v i t y
the m e l t i n g
The e q u a t i o n
for
of a s o l i d
this
the
into
reaction
two
mi g h t
be
( 28)
The e q u i l i b r i u m
constant
for
this
reaction
can e a s i l y be
9
c a l c u l a t e d f r o m AH0 a t
for
29 8. 1 5 K and t h e
each o f t h e s p e c i e s
using the f o l l o w i n g
at
the t emper at ur e of
activities
interest
equation:
K
= 1 0 ( - A H ° / 1 9 . 14464T)
eq
The e q u i l i b r i u m
Pl a n c k f u n c t i o n
constant
1 0 ( AYj / 19. 14464) , ^
can a l s o be c a l c u l a t e d
( 29)
f r o m t he
o f each s p e c i e s by t h e e q u a t i o n :
Keq = ( a YaZ ) / ( a Y2Z ) ‘
H o we v e r , t h e a c t i v i t y
one so t h a t
this
( 30)
o f the
equation
solid
is
n o r m a l l y equal
to
becomes
^eq = aYaZ'
Substitution
into th is
of the R e d l i c h - K i s t e r
equation,
K
= x..2x „
eq
Y
coefficients
for
l og
we g e t
10( 2 , °9 YV> 10l l O 9 y Zl
( 3 2)
wh i c h becomes
% .2 102 ( x z r [ A
+ B( 4 x Y -
D
( x Y ) [ A + B( 4 x Y - 3)
.]
+
+
.1
xZ 10
Wi t h t h i s
equation
equilibrium
calculated
calculation
point
t h e mol e f r a c t i o n
lies
for
a t whi c h t h e
a particular
by a m i n i m i z a t i o n
process.
t e m p e r a t u r e can be
By r e p e a t i n g t h i s
at v a r i o u s t emper at u r es the e q u i l i b r i u m
can e v e n t u a l l y
be d r a w n .
Extended.Redli c h - K i s t e r . Notation
In d e s c r i b i n g
c ompon ent s ,
R. A.
( 33)
phase d i a g r a ms o f t h r e e
Howal d and I . E l i e z e r
’
or more
’
curve
10
d e v e l o p e d an e x t e n d e d R e d l i c h - K i s t e r
notation
describes
a particular
t h e mol e f r a c t i o n
coefficient.
term f o r
The c o e f f i c i e n t s
F , ar e shown i n Ta b l e
I for
through
a three
whi ch
t he s e v e n t h
component
power ,
sy s t em.
Ta b l e I . The R e d l i c h - K i s t e r c o e f f i c i e n t s f o r a t h r e e
component s yst em t h r o u g h s e v e n t h power i n mol e f r a c t i o n .
A12
A13
CO
CVJ
CO
CO
CO
B 12
A23
B?23
CI 3
C23
C12 3
CO
CVI
_Q - H
O
DI 2 .
D13
°23
D123
D12 3 ‘
E 13
CO
CVJ
LU
E 123
E123
D12 3
O
CI 2
the t e r n a r y .
I represent
by t h e RedI i c h - K i s t e r
number o f
left
coefficient
ar e
The
o v e r ar e t h e t h r e e component t e r m s .
superscripts
t h a n t h e number o f
in
the'mole fr a c ti o n
4
5
^ I 3XI x 3 ^ x I ” x 3 ) 5 F23 x 2 x 3 ( x 2 " x 3^ ^ e t c .
coefficients
t he
b i n a r y s ubsyst ems p r e s e n t
For t h e excess q u a n t i t i e s
t e r ms m u l t i p l i e d
A12 x 1x 2 ’
the t hr ee
CO
CVJ
b i n a r y t e r ms f o r
i n Tabl e
—
H
t h r e e col umns
LU
CVl
LU
The f i r s t
Ed
L I 23
on t h e s e t er ms
subscripts.
subscripts
allowed f o r
a letter
occur w i t h
b i n a r y ter ms.
The
i s a l wa y s t wo l e s s
The mi numum number o f
i s t wo .
Thi s
can o n l y
The maximum a l l o w a b l e number o f
subscripts
i s e q ual
in the al phabet .
is
e q ual
to
one more t h a n t h e
However ,
the actual
number o f
t o t h e number o f component s
ex a mp l e ,
t h e maximum number o f
ho wev er ,
i n a t e r n a r y syst em t h i s
letter's
position
subscripts
i n t h e s y s t e m.
subscripts f o r
is
limited
For
E is
six;
to three
subscripts.
Tabl e 2 i s
superscripts
with
mol e f r a c t i o n
exampl e
Thi s
all
is
useful
their
t e r ms
is
in c o r r e l a t i n g
appropriate
ar e
indicated
a sixth
because E i s
coefficients
power s.
and
The f i r s t
by t h e s u b s c r i p t s .
For
power t e r m i n mol e f r a c t i o n .
the f i f t h
letter
in the a l p hab et .
In
cases one g e t s t h e power by a d d i n g one t o t h e number
representing
position
mol e f r a c t i o n
t e r ms
the s u b s c r i p t s .
in the a l p h a b e t .
ar e x . XgX^ c o r r e s p o n d i n g t o
The n e x t mol e f r a c t i o n
f r o m (V - x . ) m wher e m i s
capital
letter
subscripts,
in t h i s
wh i c h
is
equal
six for
case m = 6 - 3 -
the f i r s t
mol e f r a c t i o n
x 3 , and i
is the f i r s t
subscript
calculated
each s u p e r s c r i p t .
Thus
V i s t h e sum o f each o f
in t h i s
subscript.
case V = X1 + x 2 +
Ther e i s
a l s o a t e r m Zn
each s u p e r s c r i p t ; n i s t h e c o r r e s p o n d i n g
number a s s o c i a t e d w i t h t h e s u p e r s c r i p t
scheme.
each o f
E, mi nus t h e number o f
I = 2.
t er ms
term i s
three
t o t h e number o f t h e
mi nus t h e number f o r
associated with
The f i r s t
Z is defined
from the
as ( x ,
left
- Xr ) ,
and k i s
i n t h e above
wher e j
is the
t he. s u b s c r i p t
second
correlated
12
with
the
the s u p e r s c r i p t . subscript
right.
with
of the R e d l i c h - K i s t e r
For ex a mpl e ,
4 and ( c )
The s u p e r s c r i p t
is
coefficient
t he term £ ^ 3 4
cor relate d with
is c o r r e l a t e d with
has
3.
from the
( a)
correlated
b
Thus f o r
the
c o mp l e t e t e r m i s
h
E I 23x I X2X3 ^ ^ "*
whi c h
2
1
)
is
E123Xl X2X3 ^ X2 + x 3 ^ ^x Z " x 3 ^
Thi s
notation
d i a g r a ms .
can a l s o
be used f o r
h i g h e r component
For exampl e:
E1234X1X2X3X4 ^ X2 - x 4 ^ ^ x 2 _ x 3^
C641X6X4X1 ^x 4 “ x I^
and
D3254 x 3 x 2 x 5 x 4^ x 2
x S^ 1 .
Ta b l e 2. R e d l i c h - K i s t e r c o e f f i c i e n t s and t h e i r
c o r r e l a t e d w i t h t h e i r . c o r r e s p o n d i n g power s
0
a
A
2
e
I
b
B
3
d
C
4
e
D
5
f
E
6
g
subscripts
F
7
h
G
8
i
H
.9
j
T h e . H i g h e r . C o mp o n e n t . S y s t e ms
For a t h r e e
fraction
component
s yst em t h a t
t e r ms up t o t h e s e v e n t h power ,
must be a t
least
fifteen
t wen t y o f the f o u r
t e r ms
present.
component t e r ms f o r
i n c l u d e s mol e
that
is
F, there
Ther e a r e a l s o
t h e excess
13
quantity .
These t e r ms
ar e p r e s e n t e d
order t h a t
t h e y ar e e n t e r e d
into
t h e c omput e r
Ho we v e r , t h e excess q u a n t i t y
Expressions
for
the p a r t i a l
moI a l
is
no t
in the
pr ogr am.
enough.
quantities
ar e al so
T h i s can be a c c o mp l i s h e d as i n t h e t wo component
needed.
case by t a k i n g t h e d e r i v a t i v e
respect
i n Ta b l e 3,
to
o f t h e excess t e r ms w i t h
n , t h e number o f mo l e s .
Ta b l e 3. The t h r e e
coefficients
and f o u r
component
Redlich - K i s t e r
THE.THREE.COMPONENT . TERMS
R®
B123
B123
Cb
L123
Ea
L 123
' Eb
L I 23
E^
h I 23
Fc
r 123
r 123
Fe
r 123
D123
D^
u I 23
Ed
4, 2 3
F 123
Fb
h I 23
THE.FOUR . COMPONENT. TERMS
r aa
L 1234
I-. a a
u 1234
nab
U1234
Dba
u 1234
r aa
" 1234
r ab
" 1234
r ba
L 1234
Fac
L I' 2 3 4
pbb
L 1234
pCa
L 1234
Faa
r 1234 ‘
Fab
r 1234
r-ba
h 1234
Pac
h 1234
pbb
^ 1234
pCa
h 1234
Fad
r 12 34
pbc
h 1234
Pcb
h 1234 ■
pda
r 1234
14
THE ONE COMPONENT PHASES
Equations
of
State
The vol ume as a f u n c t i o n
well
known a t
tabulated
(T -
temperatures
for
Magnesi um Oxi de
of temperature
up t o
o f MgO i s
1700 K, and t h e vol umes
by T o u l o u k i a n 16 ar e e a s i l y f i t
by a q u a r t i c
in
1000 K ):
V = 1 1. 5 64 3 [ I
+ .4.34332 x 1 0 ' 4 (T -
+ 567479 x I O ' 8. ( T -
Figure
I shows v a l u e s o f
coefficient,
tabulated
polynomial
IOOO) 2
. 502119 x I O' 12 ( T -
+ . 821952 x 10- 15
a,
vol umes
(T -
IOOO) 3
I OOO)4 ] .
the t her mal
calculated
1000)
( 34)
expansi on
from t h i s
polynomial,
the
by T o u l o u k i a n , 10 and t h e vol ume
f r o m HowaI d , Moe and Ro y .
25
It
is
clear
that
t h e vol ume dependence upon t e m p e r a t u r e f r o m Ho wa l d , e t
a 1 . 25 has been g i v e n excess c u r v a t u r e
procedure.
given
The s t r a i g h t
line
by t h e
least
squares
bet ween 300 and 1700 K i s
by t h e e q u a t i o n
a = 0 . 0 0 0 0 4 3 5 + 1. 0 x 10" 8 ( T -
The vol umes f r o m t h e t h e r m a l
vol ume p o l y n o m i a l
cm3/ m o l e .
10 00) .
( 35)
e x p a n s i o n e q u a t i o n and t h e
i n t e m p e r a t u r e ag r ee w i t h i n
The l ow t e m p e r a t u r e d a t a
± .001
17,18,19,20
cited
by
a (K)
u
0
500
1000
1500
^uuu
F i g u r e I . The c o e f f i c i e n t o f t h e r m a l e x p a n s i o n o f MgO. The S shaped l i n e
w i t h a l t e r n a t i n g l ong and s h o r t dashes r e p r e s e n t s t h e v a l u e s used i n
r e f . 25. The s t r a i g h t dashed l i n e i s . 0 0 0 0 4 3 5 + 1 . 0 x 1 0 " ° ( T - 1 0 0 0 ) . The s o l i d
l i n e shows t h e v a l u e s s e l e c t e d i n t h i s work . Above 300K t he s o l i d l i n e i s
t h a t c a l c u l a t e d f r o m r e f . 1 6 , 1 7 , and 2 0 .
16
Tou I o u k I a n 16 e x t e n d s down t o 4 K so t h a t
to
sketch
solid
a reasonable
line
in t h i s
c u r v e bet ween 400 and 0 K.
region
of Figure
I gives
it
is
also
extremely
temperature
equation of
and p r e s s u r e .
The e q u a t i o n
accurate
that
with
value f o r
0
to
have
o f MgO as a f u n c t i o n
The Murnaghan
logarithmic
s t a t e 2 1 , 2 2 has been w i d e l y u s e d 2 2 , 2 3 , 2 4 , 2 5 t o
express the pr essur e
dependence o f t h e b u l k modul us
(K).
is
+NP
o
i s t h e b u l k modul us a t
a constant
The
vol umes a t
important
a c c u r a t e d a t a on t h e c o m p r e s s i b i l i t y
wher e K
possible
11 . 1996 and 11 . 2016 cm / mo l e r e s p e c t i v e l y .
Ho we v e r ,
K =K
is
3
and 100 K o f
of
it
( 36)
a standard pressure,
and P i s t h e p r e s s u r e .
and i s
Thi s e q uat i o n
easily extrapolated
a f ew measur ement s
of K at
N is
is
t o hi gh p r e s s u r e s ,
various
so
pressures
a
N can be d e t e r m i n e d .
Ther e have been v a r i o u s
temperature
expressions
for
dependence o f t h e b u l k mo d u l u s .
the
Equat i ons
such as
(37)
K00 " C r
wh e r e ,
K
i s t h e b u l k modul us a t
^^
and p r e s s u r e , h a v e
been used e m p i r i c a l l y
t h e o r e t i c a l l y ; 2 6 ’ 2 7 , 2 8 ho we v e r ,
simple.
a standard temperature
For e x a mp l e ,
the values
MgO by S p e t z l e r 24 ar e 2 7 . 2 t o
24 2 5
’
and d e r i v e d
t h e y ar e p r o b a b l y t oo
for
dK/ dT o b t a i n e d f o r
30. 1 MPa/ K.
These v a l u e s
17
ar e
inconsistent
with
the value c a l c u l a t e d
f r o m S w a l i n 1s
equat i o n ^
dK / dT = ZY2CpZV = 2 0 . 7 MPa/.K
wher e Ks i s
the a d i a b a t i c
Gr u n e i s e n p a r a m e t e r .
( 38)
b u l k modul us and
i s t he
The G r u n e i s e n p a r a me t e r
is
Y = PtVZK1Cv..
( 39)
The d i f f e r e n c e
bet ween t h e a d i a b a t i c
the
b u l k modul us Ky c a n n o t
isothermal
b u l k mo d u l u s ,
account
for
Kg , and
this
discrepancy.
Howa I d ,
logarithmic
Moe and Roy25' have used t h e Mur naghan
equation
V = Vg ( I
but
a constant
of thermal
state
+ NPZKq ) ' 1 / N ,
'
( 40)
value of N r e s u l t s
in negative c o e f f i c i e n t s
expansion
They s o l v e d t h i s
dependence f o r
of
at
hi gh
temperatures
p r o b l e m by i n c l u d i n g
N for
and p r e s s u r e s .
a temperature
MgO, Al Oj i g and MgAl gO^-
However ,
t h e t e m p e r a t u r e dependence o f N i n t h e e x p o n e n t
complicates
a value
obtaining
large
derivatives.
Also,
enough t o a v o i d n e g a t i v e
greatly
i n cr eas ing K to
values
is
O A
inconsistent with
a positive
following
Spetzler' s
value f o r
derivation
dNZdT i s
shows.
d CVdP = ( dKZdT) ZK2
can be e a s i l y
derived
measur eme nt s .
not
Also,
satisfactory,
using
as t h e
The e q u a t i o n .
( 41)
from
d 2 VZdTdP = d 2VZdPdT
( 42)
18
and t h e d e f i n i t i o n s
o f a and K,
( 43)
a = ( I / V) ( d V / d T ) p
and
K = 1/ 3 = - V( d P / d V ) T .
If
t he Mur naghan
logarithmic
the ex p r es s i o n f o r
coefficient
da/d-P = [ ( d K 0 / d T )
domi nant ,
expected.
If
da/ dP becomes
with
is
so t h a t
equation of
the pr essur e
o f thermal
The t e r m d K^ / d T
' ( 44)
becomes by s u b s t i t u t i o n ,
and a t
positive
dN/ dT b e i n g p o s i t i v e ,
as i t
there
of thermal
infinity
any i n t e r a t o m i c
separation
It
is
logarithmic
greater
approaches
form f o r
can n o t
pressure
happen,
t h e vol ume and t he
the nearest
potential
do n o t
zero.
neighbor
Th i s
forces
become i n f i n i t e
at
a
than zero.
entirely
possible
equation of
temperature r a nge.
equation of
However ,
some f i n i t e
Thi s
as i s
pressures
e x p a n s i o n must a p p r o a c h
can be d e r i v e d as sumi ng t h a t
for
higher
s h o u l d be.
is
is
l ow p r e s s u r e s
then at
a t wh i c h da/ dP becomes p o s i t i v e .
coefficient
( 45)
l ow p r e s s u r e s
n e g a t i v e at
negative,
because as P a p p r o a c h e s
used,
+ > ( d N / d T ) ] / ( Kq + NP) 2 .
da/ dP i s
less
is
dependence o f t h e
expansion
negative
dN/ dT i s
state
state
However ,
s t a t e must g i v e
infinity,
that
is
t h e Mur naghan
only val id
any c o r r e c t
hi gh
pressure
vol umes equal
to
z e r o as P
and must r e p r e s e n t
the r e p u l s i v e
over a f i n i t e
t e r ms o f t h e
a mat hemat i cal
interatomic
■
I
19
potentials.
Therefore
any h i g h
s h o u l d be somewhat c o n s i s t e n t
logarithmic
equation
to
hi gh
zero.
If
of state
t h e Mur naghan
Thus,
logarithmic
we wer e
equation
l ed t o
of
state
as
pressures.
Spet z I e r 1s e x p e r i m e n t a l
dN/ dT i s
with
of s t a t e .
c o n s i d e r t h e Mur naghan
bei ng v a l i d
pressure equation
this
is
correct
dependence o f t h e t h e r m a l
24
indicate
that
t he n t h e p r e s s u r e
expansion c o e f f i c i e n t
becomes.
CM
OZ
+
doc/dP = ( dK0 / d T ) / ( K 0
measur ement s
( 46)
can be i n t e g r a t e d , t o g i v e
Thi s
( 47)
cc = - ( d K 0/ d T ) / N ( K 0 + NP).
Thus,
b o t h oc and 3 ap p r o a c h z e r o
t he p r e s s u r e approaches
Rearranging
linearly
in
I / ( K + .NP) as
infinity.
and i n t e g r a t i n g
this
equation
f o r a,
we
ob t ai n.
' k OT
=
wher e T q i s
k
OT0 e x P f - "
a d T )
=
k
OT
0 1v
vt1
V
( 48)
'
‘o
a s t a n d a r d t e m p e r a t u r e a t whi ch t h e
p r e s s u r e b u l k modul us
Wh i l e t h e r e
•and oc a t
/
is
K.
‘o T q
ar e u s u a l l y
hi gh t emper at ues ,
l ow
sufficient
there
is
very
data to
evaluate V
little
data f o r
the e v a l u a t i o n
o f t h e b u l k modul us a t t h e s e h i g h
temperatures.
Therefore,
correct,
it
Ther e
K
will
is
if
this
equation
is
at
all
be e x t r e m e l y u s e f u l .
good a g r e e me n t ^ 5 ^
= 160100 MPa f o r
MgO a t
29 8 . 1 5 K .
^
on t h e v a l u e o f
We f i r s t
used
20
Spetzler's
24
o f dKg/ dT o f
value of
about
N = 3.9;
- 15 MPa/'K.
s t ee p enough t o mat ch e i t h e r
Anderson's
29
data.
MgO r e s p e c t i v e l y .
consistent with
Wi t h
calculate
Ko T =
Most
whi c h
values of
K
at
val ues
not
or Soga and
30
gives
and p o l y c r y s t a l I i n e
i s t a k e n f r o m C a r t e r , et
N and Kq ggg [5 cho s en,
we can
from the r e l a t i o n s h i p
•
met hods g i v e t h e a d i a b a t i c
isothermal
b u l k mo d u l u s ,
bu t
( 49)
b u l k modul us
this
can
be c o r r e c t e d t h r o u g h t h e e q u a t i o n
Ks = K/ ( I - Cf2VKTZCp ) .
This equat i on
gives
( 50)
K5 = 163062 and 150082 MPa a t
300 and
1000 K r e s p e c t i v e l y .
For t e m p e r a t u r e s
for
f r o m t h e c u r v e shown i n F i g u r e
a wer e d e t e r m i n e d
bel ow 300 K, v a l u e s
and Cp v a l u e s wer e t a k e n f r o m t h e JANAF t a b l e s
Bar r on p a p e r ^
full
31
aI .,
o t h e r ' measur ement s.
KoT0 ( VT0/ V T ) *
instead of the
gives
chose a v a l u e o f N = 4. 57
any t e m p e r a t u r e
experimental
easily
24
1968 r e v i e w
crystal
We. f i n a l l y
this
is d e f i n i t e l y
S p e t z l e r 1s
single
our c a l c u l a t i o n s ,
and i s
Th i s
Anderson's
N = 4 . 5 0 and 4 . 5 8 f o r
for
howev er ,
to
and t h e
g e t K = K 5 = 163137 MPa a t 0 K .
s e t o f c a l c u l a t e d Kc v a l u e s
The c a l c u l a t e d
32
curve
is
is
in ex c e lle n t
plotted
I,
The
i n' F i g u r e
agr eement w i t h
2.
the
24
Spetzler
and t h e 296 v a l u e o f
Oft
Anderson and An d r e a t c h .
Anderson and A n d r e a t c h ' s
300 and 800 K v a l u e s o f
n
val ue
a
a t 77 K and a l l
t he v a l u e s of Soga and Anderson
29
21
170000
160000
150000
140000
F i g u r e 2. The i s o e n t r o p i c b u l k modul us f o r MgO. The s o l i d
l i n e i s c a l c u l a t e d f r o m t h e t h e o r y p r e s e n t e d h e r e . The
e x p e r i m e n t a l v a l u e s ar e f r o m S p e t z l e r , b l a c k c i r c l e s ; Soga
and A n d e r s o n , open c i r c l e s ; and f r o m Ander son and
A n d r e a t c h , d i a mo n d s .
22
ar e a b o u t
within
1% h i g h e r t h a t
the experimental
MgO ar e
in f u l l
the v a l i d i t y
state to
pressures wi th
for
MgO f o r
vol umes o f HowaTd, e t
Howal d, et
3 0 , 0 0 0 MPa o u r v a l u e s
Thi s
is
due t o
T a b l e 4.
the
curve,
but t h i s
is
T h e r e f o r e , t he data f o r
agr ee men t w i t h
Tabl e 4 p r o v i d e s
agr ee w i t h
error.
the p r e d i c t i o n s
o f t h e Mur naghan
hi gh
pressures
the c a l c u l a t e d
logarithmic
a constant
assumi ng
equation of
value of
N = 4.57.
c a l c u l a t e d mo l a r vol umes a t
various
c o mp a r i s o n w i t h t h e c a l c u l a t e d
aI . ^
The v a l u e s a t 0 . 1 and 900 MPa
or
o
aI .
w i t h i n ± 0. 01 cm / m o l e .
At
ar e 0 . 0 5 t o
larger
C a l c u l a t e d mo l a r
value of
0 . 0 6 cm / m o l e
larger.
N t h a t we ar e u s i n g .
vol umes f o r
MgO
P1MPa
30000
15000
9000
T, K
.01
300
11 . 24 64
10. 6974
10. 4025
9. 8223
650
11. 3956
10. 8094
10. 4980
9. 8912
1000
1 1. 5643.
10. 9341
. 10. 6035
9. 9665
Once t h e s e c a l c u l a t i o n s
logarithmic
equation
matter to c a l c u la te
pressure.
MgO( I )
of
state
u s i n g t h e Mur naghan
a r e done i t
is
t h e vol ume p o l y n o m i a l ' s
The c o mp l e t e d vol ume p o l y n o m i a l s
ar e g i v e n
i n Tabl e
5,
along wi t h
the
a s i mp l e
dependence on
for
MgO(c)
and
heat c a p a c i t y
23
Ta bl e 5. The e q u a t i o n s o f s t a t e , he at c a p a c i t y e q u a t i o n s
and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f MgO ( c ) and
MgO ( I )
c
__ ~ s
MgO (S) VOLUM E POLYNOMIAL
11.56431
-7.096668E -6
1 .4 0 2 5 7 9 E -1 0
-3.36068E -15
8 .5 9 3206E -20
-2.00574E -24
2 .8 1 9 6 6 1 E-29
4 .3 7 7 3 1 8 1 E-5
-1 .728388E -9
6 .2 1 0 5 0 0 E -f4
-2.15203E -18
7 .1 14327E -23
-1 .96492E -27
3 .0 3 3 2 8 7 E-32
5 .6 7 4 7 9 4 6 E -9
-3.96682E -13
2 .0 4 3 7 7 9 E -1 7
-9.20887E -22
3 .7 0 4 7 8 8 E -2 6
-1.16729E -30
1.941636E -35
-5 .0 2 1 19E-13
-4.12045E -17
4 .5 6 3 1 20E-21
-2.9914.7E-25
1.538755E -29
-5.66303E -34
1.027253E -38
.567479E -08
-.502119E -12
-.412045E -16
.456312E -20
8.219519E -16
-3.69883E -20
2.019368E -24
-1.16580E -28
6.128646E -33
-2.35995E -37
4.424 8 6 5 E -4 2
MgO (LIQUID) VO LUM E POLYNOMIAL
13.993
-.709667E -05
.140258E -09
-.336068E -14
.8 5 9 3 2 1 E -19
-.200574E -23
.281966E -28
.4377 3 2 E -0 4
-.1 72839E -08
.621050E -13
-.2 1 5 2 0 3 E -1 7
.7 1 1433E -22
-.196492E -26
.303329E-31
-.396682E -12
.204378E -16
-.920887E-21
.370479E -25
-.1 16729E -29
.1941 6 4 E -3 4
.821952E -15
-.3 6 9 8 8 3 E -1 9
.201937E -23
-.1 16580E -27
.612865E -32
-.235995E -36
.OOOOOOE+OO
-.29914 7 E -24
.153875E -28
1.566303E-33
.102725E -37
H EA T C A PA C ITY C p
Aa
B
C
E
D
M gO (S)
51.0941
.00310468
-5 .5 6 2 1 8E -07
2 .7 4 7 3 3 0 E -1 0
M gO (L)
53.6 4 8 8
.0 032598
-5 .8 4029E -07
2 .8 8 4 6 9 7 E -1 0
TH E R M O D Y N A M IC
H298
-1.26513E +06
-1.32839E+06
S 298
v IOOO
Y 1000
H 1OOO' H298
PROPERTIES
J M O L '1
J M O L '1
J M O L '1
J M O L '1 K '1
CM3
M gO (S)
M gO (L)
49 .2 7
6 3 .10923
3.2974E +4
34623.
-6.01490E +5
-551278.3
26.9 4
35.
11.248
11.81
a THE HEAT CAPACITY EQUATIONS ARE GIVEN BY Cp
+ D x1 0"9(T-1000) + E x1 07( r 2-1O'6)
x1 0'6(T-1000)
= A + B x1 0*2(T-1000) + C
24
equations
and s e l e c t e d t h e r mo d y n a mi c v a l u e s .
c ap a ci ty equation
for
Howa I d , e t a I . ^
The AH^ug i s
solid
is
fully
equilibrium
MgO( c ) and t h e AH^us a r e t a k e n f r o m
described
constants
The he at
5 7 . 6 5 K J / m o l e.
and can be used t o
for
any r e a c t i o n
Thus t he
calculate
i n wh i c h
it
is
involved.
For MgO l i q u i d
or
he at
changes
capacity.
there
ar e f ew measur ement s o f vol ume
From t h e vol ume and he at
in the f u s i o n
of the a l k a l i
capacity
halides
we have
e s t i m a t e d t h e vol ume and t h e he at c a p a c i t y o f t h e
to
be 21% and 5% g r e a t e r t h a n t h a t
In or der
her e w i t h
to
other
o f the
compar e t h e e q u a t i o n s
pr o p o s e d e q u a t i o n s ,
liquid
solid.
of s t a t e
considered
we have c a l c u l a t e d
Gr unei sen ,
Y =
( 51)
KVACv ,
and t h e A n d e r s o n - G r u n e i sen,
' 6 = ( d Ks / d P ) T par amet er s f o r
pressures.
negligible
( 52)
I-,
MgO o v e r a wi de r ange o f t e m p e r a t u r e s
These p a r a me t e r s show a s ma l l
dependence upon b o t h t h e p r e s s u r e
temperature.
The r e s u l t s
in Figure
3.
F i g u r e 4 shows t h a t
Gr u n e i s e n
p a r a me t e r f o r
of
state
but
of
MgO c a l c u l a t e d
and
lines
of the
f r o m ou r e q u a t i o n
ar e a p p r o x i m a t e l y c o n s i s t e n t w i t h
assumed f o r m
not
y ar e shown as c o n t o u r
the values
and
t h e commonl y
10000
-
5000
1000
1500
TEM PERATURE, K
Figure
3.
Co n t o u r l i n e s f o r t he Gr u n i e s n p a r a m e t e r , y , f o r MgO on a
P-T f i e l d .
The c o n t o u r i n t e r v a l i s 0 . 0 1 , e x c e p t t h a t a t h i g h
t e m p e r a t u r e s d o t t e d l i n e s a t 0 . 0 2 i n t e r v a l s ar e a l s o shown.
26
VOLUME , cm
m ol
F i g u r e 4.
G r u n e i s e n p a r a m e t e r s , y , f o r MgO
p l o t t e d v e r s u s vol ume f o r t h e t h r e e p r e s s u r e s
0 . 0 1 , 5000, and 10000 MPa. The c i r c l e s r e p r e s e n t
c a l c u l a t e d v a l u e s a t 300, 400, and 600 K. The
d o t t e d l i n e i s f o r Y =0 . 12 652 v l - 0 0 5 9 .
27
Y = CVn
with
( 53)
n = 1. 0059 and C = 0. 1 2 6 5 2 b u t
o n l y at p o i n t s
above
600 K .
The A n d e r s o n - G r u n e i s e n
since there
is
a related
parameter,
, is
not p l o t t e d
quantity,
Sj = ( - 1 / a K ) ( - d K / d U p ,
whi c h r e d u c e s t o
( 54)
Gy = N u s i n g
a = - ( dK/ dT) p/NK'.
Thus t h e e q u a t i o n s
equivalent
to
of
( 55)
s t a t e we have p r o p o s e d he r e ar e
choosing a constant
value f o r
Sj.
A n d e r s o n 3 5 , 3 6 , 3 7 , 3 8 ’ 39 has c o n s i d e r e d t h e
approximation
constant.
o f ctK = c o n s t a n t
as an a l t e r n a t i v e
to
Y =
Our e q u a t i o n s g i v e
K = - ( l / N ) ( d K o/dT)
wh i c h
is
fully
recognizes
independent
( 56)
of pressure.
aK c a n n o t be f u l l y
because a a p p r o a c h e s
The q u a n t i t y
( d KQ/ d T )
zero at
As Ander son
i ndependent
of temperature,
0 K as shown i n
is approximately
Figure
i ndependent of
t e m p e r a t u r e o n l y above some mi ni mum t e m p e r a t u r e .
equation ■ KqJ =
Kq 7 q ( Vj q / V j ) N w i l l
values f o r
any t e m p e r a t u r e f o r whi c h
thermal
Kq a t
Our
give reasonable
e x p a n s i o n d a t a ar e a v a i l a b l e .
woul d be b e t t e r
I.
l ow p r e s s u r e
Thi s ap p r o x i ma t i o n
t h a n assumi ng d K^ / d T i s c o n s t a n t .
28
Equations
of
Ther e ar e s e v e r a l
t h e s e ar e a l p h a
oristobalite,
the
to
the
JANAF ( S t u l l
compilations,
significant
the
phase t r a n s i t i o n .
at
primarily
analysis
al pha
and B u l l e t i n
and a d i f f e r e n c e
to
have a w e l l
quartz to
Stull
defined
to
3?
the d i f f e r e n c e
due t o
the f a c t
quartz
transition
changes
that
R o b i e,
is
equilibria
Thus,
is
transition.
aI .
40
i s due
t h e r ma l
as shown by R i c h e t , et
equilibrium
analysis
for
i s" r a t h e r
poor,
t h e a l p h a t o be t a
a l on g p e r i o d o f t i m e ,
have a l a r g e e f f e c t
h e a t c a p a c i t y .of a l p h a q u a r t z .
in the
it
bet ween d i f f e r e n t i a l
in temperature
et
i n t h e s e t wo
and Robi e e t ,
thermal
requires
ar e t h e
bet ween t h e t a b u l a t e d
and d r o p c a l o r i m e t r y ,
The d i f f e r e n t i a l
as
enthalpy of t r a n s i t i o n
beta q u a r t z
and Pr o p h e t
1452,
o f 200 J / m o l e
the v ar i ous
at
literature,
a 380 J / m o l e d i f f e r e n c e
a T.
smal l
the
The t wo s t a n d a r d c o m p i l a t i o n s
in c a l c u l a t i n g
al pha
of
One o f
Ther e ar e v a r i o u s
I OOO K i n t h e
The 380 J / m o l e d i f f e r e n c e
values
dioxide;
and l i q u i d .
phase d i a g r a m as shown i n Ta b l e 7.
important
for
stishovite
and P r o p h e t ) ^
Ther e i s
Si Og
silicon
t h e e n t h a l p y change bet ween a l p h a q u a r t z
i n Tabl e 6 .
aI .
Dioxide
Si Og phase d i a g r a m i s
2 9 8 . 1 5 and b e t a q u a r t z
shown
Silicon
known phases o f
tridymite,
beta q u a r t z
values f o r
for
q u a r t z , beta q u a r t z , c o e s i t e ,
p r o b l e ms w i t h
quartz
State
and
on t h e
29
Ta b l e 6 . Val ues r e p o r t e d f o r t h e e n t h a l p y change bet ween
a l p h a q u a r t z a t 29 8 . 1 5 K and b e t a q u a r t z a t 1000 K .
Rob i e , e t a I . , 1968
K e l l y , 1960
R i c h e t , e t a I . , 1982
Ri c h e t , e t a I . , 1982
*
t h i s wor k
B a r i n and Knacke, 1973
Moser , 19 36.
R o b i e , e t a 1 . 1969
Ghi or so , e t a 1 . 1979
C o m p i l a t i on
Compi I a t i on
Ex p . D. C .
Compi I a t i on
Compi l a t i o n
Compi I a t i on
Compi l a t i o n
Exp Cp
Compilation
Exp DTA
45 6 8 9 . 3
45689. 3
45 6 1 7 . 0
45579.
45452.
45 4 4 4 . 3
45358. 7
45056. 2
44967.
44826. 9
* c a I c u l a t i on u s i n g v a l u e s f o r Si Og g l a s s arid o f ( K r a c e k , e t
a I . , 1953 ) , ( R i c h e t e t a l . , 1982) and ( N a v r o t s k y , e t a l . ,
1980)
T a b l e 7. C a l c u l a t e d e q u i l i b r i u m c o n s t a n t s f o r t h e
h y p o t h e t i c a l r e a c t i o n bet ween t h e t wo e q u a t i o n s o f
f o r Bet a Q u a r t z ; R o b i e , e t a l . and o u r s .
' ( S i O 2 , our = Si O2 , R o b i e , e t a l . )
T (K)
844
800
900
1000
1100
1200
1300
1400
1500
1600
1700
Keq
.9890
. 9928
.9849
.9787
. 9736
. 9694
. 9658
. 9627
. 9598
. 9568
. 9538
state
30
The Al pha Qu a r t z t o
Qu a r t z T r a n s i t i o n
The a l p h a ( a )
first
discovered
order
disorder
to the
quartz
in
to
q u a r t z t r a n s i t i o n was
42
1889 by Le C h a t e l i e r . I t has an
crystal
i n wh i c h t h e r i d g e d
left
is
or t h e r i g h t .
a sixfold
ab out
structure
has a t h r e e f o l d
tetrahedra
scr ew a x i s ,
that
Grimmm and D o r n e r 44 ( 1975)
scr ew
ar e t i l t e d
corresponds
domai ns
either
coexist
d o ma i n s . - The s t r u c t u r e
a v e r a g e o f t h e t wo Dauphi ne t w i n
have e x t e n s i v e l y
746 K and 0. 1 MPa.
The t wo s t r u c t u r e s
ar e known as Dauphi ne t w i n
quartz
beta(B)
l ambda t r a n s i t i o n
The a l p h a q u a r t z
a x i s^3 ,^
Bet a
and
o f beta
t o an
in alpha q u a r t z .
and L i e b a u and Bohm43 ( 1982)
d i s c u s s e d t h e mo t i o n
involved
and t he
domai ns f o r m e d .
Ther e
is
al so evidence f o r
the e x i s t e n c e
o f an
i n c o mme n s u r a t e p h a s e 4 5 ’ 4 6 ’ 4 7 ’ 4 8 ’ 4 9 ’ 5 0 ’ 5 1 ’ 52' I t o
2 K
above t h e a l p h a q u a r t z
a t 846 K .
However ,
to
beta q u a r t z t r a n s i t i o n
s i n c e t h e t e m p e r a t u r e range
i n c o mme n s u r a t e phase i s
phase e q u i l i b r i a
stable
is
negligible
these c a l c u l a t i o n s .
Therefore
phase can be i g n o r e d .
the
literature
transition
transition.
is
is
i n whi c h t h i s
small,
its
effect
compar ed t o t h e e r r o r
for
our p u r p o s e s ,
A high
order
h e at
or
second o r d e r
capacity with
in
this
Ther e has been some d i s c u s s i o n
on w h e t h e r t h e a l p h a q u a r t z t o
a first
on t h e .
in
beta q u ar t z
l ambda
d r o p s o v e r a s ma l l
31
t e m p e r a t u r e range
is
indicative
of
a l ambda t r a n s i t i o n .
The - al pha t o bet a q u a r t z t r a n s i t i o n
capacity
al ong
near 846 K a t
a line
of
has such a heat
l ow p r e s s u r e s
(Figure
i nc reasi ng temperature with
5),
increasing
pressure.
A l s o t h e vol ume change upon t r a n s i t i o n
approaches
z e r o as t h e a l p h a t o
temperature
is
the alpha to
approached,
be t a q u a r t z
mo d e l l e d as f i r s t
Our r e s u l t s
transition
order,
I n a nor mal
of t r a n s i t i o n
the t r a n s i t i o n
first
Richet's^
order
et a I .
However ,
be
41
have done.
as a l ambda
results.
phase t r a n s i t i o n
can be o b t a i n e d
6.
can s t i l l
wh i c h R i c h e t
with
transition
as shown i n F i g u r e
m o d e l l i n g t h. i s t r a n s i t i o n
ag r e e w e l l
equilibrium
beta quar t z
transition
and a l s o
the enthalpy
f r o m t h e vol ume change o f
and t h e p r e s s u r e dependence o f t h e
curve, this
equation
is
dP/ dT = AH/TAV = AS/ AV.
However ,
i n a l ambda t r a n s i t i o n
transition
ar e z e r o ,
possibly
approaching
equation
is
infinity.
very
large,
T h e r e f o r e t h e above
i n e n t r o p y and vol ume upon
t h e e n t r o p y and vol ume can be t r e a t e d
differential.
analogous eq uat i ons
derived.
is
indeterminate.
ar e z e r o ,
as an e x a c t
t h e AV and t h e AH of.
and t h e h e a t c a p a c i t y
S i n c e t h e change
transition
( 57)
By s e t t i n g
Sa = Sg and Va = Vg,
t o t he Cl apeyron eq uat i o n
These e q u a t i o n s
ar e
can be
200
780
8 00
820
840
T1K
F i g u r e 5. Heat c a p a c i t y , Cp, f o r a l p h a q u a r t z ne ar t h e l ambda
t e m p e r a t u r e . The s o l i d l i n e r e p r e s e n t s o u r c a l c u l a t e d v a l u e s , The
e x p e r i m e n t a l p o i n t s o f Moser and S i n e l 1n i k o v ar e shown as open
and f i l l e d c i r c l e s r e s p e c t i v e l y .
33
a - 4.9
#>o O
O
c -5 .4
-IO O
8 4 6 -T , K
F i g u r e 6 . C e l l demens i o n s f o r a l p h a and b e t a q u a r t z v e r s u s
t e m p e r a t u r e . Our c a l c u l a t e d f i t s t o t h e v a l u e s o f Ackerman
and S o r r e l , f i l l e d c i r c l e s , a r e shown as s o l i d l i n e s .
O l d e r e x p e r i m e n t a l v a l u e s o f J a y , ( 1939) and B e r g e r e t
a I . ( 1 9 6 6 ) ar e shown as open c i r c l e s .
34
( d T / d P ) .=■ - ( 1/ VT) ( Aa/ ACp)
■ ( 58)
and
( dT/ dP)
wher e
= - A 3 / Aa.
( 59)
and Sg ar e t h e
vol ume and e n t r o p y o f
Va and Sa ar e t h e vol ume and e n t r o p y o f a l p h a
respectively,
of thermal
I
expansion
compressibility
capacity.
rapidly
Figures
is the temperature,
( 1/ V) ( d V / d P ) j ,
However ,
since
a,
as t h e t r a n s i t i o n
5 and 6 , i t
is
measur ement s o f Aa,
P i p p a r d 53 i n
transition
surface
( 17V ) ( d V / d T ) p ,
cylindrical
quartz
a is the c o e f f i c i e n t
3 is the
and Cp i s t h e
3 and Cp ar e
temperature
very d i f f i c u l t
he at
increasing
is
a p p r o a c h e d , see
t o get
accurate
A 3 and ACp .
1956 d e v i s e d a t h e o r y t o t r e a t
accurately.
bel ow t h e
beta q u a r t z ,
The P i p p a r d t h e o r y t r e a t s
l ambda t r a n s i t i o n
surface.
a l ambda
temperature
Thus t h e e q u a t i o n s
the
as a
he uses ar e
S = S + bT + f 1( T / r •- P)
o
( 60)
V = V0 + aT + f ( T / r
( 61)
and
wher e r
is
-P)
the e q u i l i b r i u m
the e q u i v a l e n t
Sc = S6 *
slope
(dT/dP) .
We ar e u s i n g
expressions,
( 62)
1 / r o f I 01
and
Va = V 1B + W
wh e r e ,
r Q is
( 63)
0 -FI0 ) .
the e q u i l i b r i u m
slope at
0. 1 MPa,
Sa and Sg
35
ar e t h e e n t r o p i e s
respectively,
of
alpha q u a r t z
and b e t a q u a r t z
Vct and Vg ar e t h e vol umes o f a l p h a q u a r t z
and b e t a q u a r t z r e s p e c t i v e l y and 0 =
amount t h e t e m p e r a t u r e
temperature,
i s b e l ow t h e
The i n i t i a l
slope
is
slope
like
Above 1500 MPa t h e
quartz
Pippard e q uat i o n
to
studied.
is
is
straight.
have been a p p l i e d
beta q u a r t z t r a n s i t i o n
So
the
t h e n t h e e n t r o p y can be c a l c u l a t e d .
p e o p l e . 5 5 ’ 5 ^ ’ 57
Dolino,
line
slope
t h e vol ume and f ( 0 ) ar e d e t e r m i n e d f o r
the
Equations
t o t he al pha
by v a r i o u s
The P i p p a r d r e l a t i o n s
ar e used by
e t a 1. 4 7 ’ 49 and Bac hh ei me r and D o l i n o , 46 w h i l e
they c a l l
the t r a n s i t o n
first
order.
The mai n pr obl em
w i t h t hes e approaches
is the experimental
The S i n e l n i k o v 58 he at
c a p a c i t y d a t a ar e a d i s t i n c t
i mpr ov ement
expect
t he
l ambda t r a n s i t i o n
( dT/ dP ) ^has been w e l l
0 . 2 2 7 2 K/MPa and t h e e q u i l i b r i u m
transition
is,
0 . 2 6 5 K/MPa as measur ed by Cohen,
K l e m e n t , and Adams. 54
if
that
T
The e q u i l i b r i u m
that
- I,
o v e r Mo s e r ' s -1936 d a t a . 59
to f in d
temperature
20 mi n u t e s
intervals
However ,
one c a n n o t
c a p a c i t y measur ement s f o r
in a region
or m o r e , ar e r e q u i r e d
temperature.
and r e q u i r e
good he at
d a t a t h e y used.
Fortunately,
smal l
wher e l o n g t i m e p e r i o d s ,
for
equilibrium
a t each
vol ume measur ement s a r e s i m p l e r
o n l y one e q u i l i b r a t i o n
per d a t a p o i n t .
Acker man and S o r r e l 58 have made a c c u r a t e X - r a y
36
measur ement s on powder ed q u a r t z ,
obtaining
d i m e n s i o n s o f b o t h a l p h a and b e t a
Figure
the c e l l
q u a r t z as shown i n
6.
Bet a q u a r t z
referenced
is
h e x ago nal
so t h a t
t o t h e t wo edges o f
the u n i t
These d i m e n s i o n s o f
beta q u a r t z
quardratic
in 0 =
equations
equation
for
(a)
and
a = 4.9978+
cell,
can e a s i l y
- I.
( c ) to
t h e vol ume can be
a and c .
be f i t
We c a l c u l a t e d
by a
t hese
be
0. 3 0 7 6 5 x 1O"^0
( 64)
and
c = 5 . 4 6 0 8 1 2 9 + 0. 2 6 4 7 5 0 x 1O"40
+ 0 . 8 2 2 3 2 8 x 1O" 70 2 .
The c e l l
di mensi ons o f both t he se e q u a t i o n s
angst r oms.
Bot h
(a)
and ( c )
decreasing temperature,
each i s
the
small.
vol ume o f
( c ).
for
should
introduce
6.
is from the e x t r a p o l a t i o n
angst roms,
results
in
(c)
increase wi th
to
150 K bel ow
little
error
in the
Most o f t h e e r r o r
in the
of the c e l l
s h o u l d be w e l l
within
d i me n s i o n
± 0. 005
even o u t t o
i n an e r r o r
The u n i t
cell
very r a p i d l y with
transition
be t a q u a r t z
extrapolating
beta q u a r t z , Fi gur e
The e r r o r
are i n
and t h e t e m p e r a t u r e dependence o f
Therefore,
l ambda t r a n s i t i o n
vol ume
..( 65 )
point
1 5 0 . K bel ow t h e l ambda p o i n t .
3
o f ± 0 . 0 2 1 7 cm / m o l e .
di mensi ons o f
al pha q u a r t z
increasing temperature,
is
approached.
Thus,
Thi s
increase
as t h e
l ambda
a s i m p l e power
37
series
i n 6 = T 1 -T w i l l
power s e r i e s
lies
no t w o r k .
A
with
the
leading
bet ween z er o and one.
l ambda t r a n s i t i o n s
series
We d e c i d e d t o use a
,
term in
where
Ther e ar e enough t h e o r i e s
( Br agg and W i l l i a m s ^ *
1934,
gr oup
t h e o r y by K a n d a n o f f e t a I . , * ^ and Levey e t a l . , * ^ )
0.875,
al most
even up t o
power t o
will
any c h o i c e
use i s
0.95;
0.5.
( a ^ - aa )
b o t h be s i m p l e
power s e r i e s
By d i v i d i n g
by 8 and u s i n g
t er m.
to
o f power f r o m 0 . 2 0 t h r o u g h
however t h e s i m p l e s t
Thus
for
through a
o f d e v e l o p m e n t s , ^ ’ * ^ up t o r e n o r m a l i z a t i o n
justify
(I-A)
in
2
and e a s i e s t
and ( c g - c a )
( 6 ) with
standard
2
no c o n s t a n t
least
s qu ar e s
pr ogr ams we o b t a i n e d t h e e q u a t i o n s
. ( a 3 - a a ) 2 = 0 . 2 2 6 8 0 6 x 1O"30 - 0. 8 6 7 4 0 8 x 1 0 " 76 2
+ 0.259547
x 1 0 " 96 3
- 0.242458
x 1 0 ' 1084
( 66)
and
(c3 -
Figure
C ct )
2 = 0 . 0 1 0 5 9 9 5 x 1 0 " 38 - 0 . 5 7 3 5 4 5 x 1 0 " 76 2
6 shows t h e
di mensi ons o f
+ 0. 3 7 3 4 8 3
x 1 O' 903
- 0. 449171
x 1 0 ' 10O4 .
calculated
curves
for
(67.)
the
cell
b o t h a l p h a and b e t a q u a r t z bet ween 700 and
900 K , and how t h e s e c u r v e s compar e t o t h e measur ed v a l u e s
of
( c ) and ( a ) .
alpha q u a r t z
The s c a t t e r
in the values of
l o o k b a d ; ho wev er ,
from the p l o t t e d
the
largest
smoot h c u r v e ar e ab out
(c)
for
deviations
0 . 0 0 2 5 a n g s t r o ms
38
corresponding
quartz
of
to
about
From t h e
di mensi ons,
an e r r o r
least
squares
intervals
due t o d i s o r d e r
Ghi or so , e t
alpha-to
in quartz
o f 0. 11
is
a l p h a and be t a
intervals
up t o
50.
( 68)
The vol ume
somewhat
larger
bel ow t h e
increase
than
0.60
l ambda p o i n t
dis order in e q u i l i b r i u m q u a r t z .
3
t o 0 . 2 0 5 cm / mo l e r e p o r t e d by
a 1 . 66 and F i l a t o v ,
bel ow t h e
et.
transition
a 1 . 67 f o r
result
curved p o r t i o n s
a first
from o m i t t i n g
of the
last
order
t he
5 to
15
l ambda p o i n t .
The vol umes o f
b o t h a l p h a and b e t a q u a r t z
ar e
from the equation
V = a 2c s i n
120.
We used t h e e x p r e s s i o n s
in 02 .
can be
substantial
beta q u a r t z
calculated
cell
= f (8)
150 de gr e es
v e r y s t e e p and h i g h l y
degrees
vol umes o f bot h
shown i n T a b l e 8 .
cm / mo l e s i n c e even
The AV v a l u e s
al pha q u a r t z
f ( 6 ) a t one degr ee
- V (1,0,1)
still
al pha
from the equat i on
the val ues
is
i n t h e vol ume o f
val ues o f the u n i t
From t h e mo l a r
V (1,0,1)
there
or
0. 046%.
q u a r t z we c a l c u l a t e d
giving
(c)
t h e mo l a r vol umes o f
calculated.
degree
in
So t h a t
8 172 t h r o u g h
i n Tabl e 8 i s
8^ ;
( 6 9)
from
( a)
f (8) logically
h o we v e r ,
and ( c )
should
i n c l u d i n g : t er ms
include
a reasonable f i t
g i v e n by t h e e q u a t i o n
to
t e r ms f r o m
the values
f ( 6 ) = [ Q . 003515080 - 0 . 1 5 8 5 1 4 x 1O"40 2
+ 0. 7 3 3 6 6 0 x I O - 7O3
- 0, 116609, x 1O" 904 ] 1 / 2 .
We j u d g e f r o m F i g u r e
6 that
extrapolation
of
and ( c )
al pha q u a r t z
cell
a n g s t r o ms .
(a)
d i me n s i o n s
Thus, the
values
± 0 . 0 2 2 cm / mo l e even as f a r
l ambda p o i n t .
contributes
The
t o most
cell
any e r r o r s
for
( 70)
in the
be t a q u a r t z
or
i n t he
s h o u l d be l e s s t h a n
o f f ( 0)
out
s h o u l d be w i t h i n
as 150 K bel ow t he
d i me n s i o n
of the e r r o r
0. 005
(c)
of beta q u ar t z
i n f ( 0 ).
T a b l e 8 . The f u n c t i o n F ( 0) r e p r e s e n t i n g t h e d i f f e r e n c e i n
vol ume o f a l p h a q u a r t z f r o m a f u l l y d i s o r d e r e d b e t a q u a r t z
a t t h e same t e m p e r a t u r e .
3
. 10197
K 3
cm / mo I e
0
0.
2
I
. 0159155 . 083456
0
f (0)
4
. 11748
5
. 13105
10
. 18333
15
. 22219
20
. 25398
0
f (0)
25
. 28118
30
. 30510
40
. 34597
50
. 38.044
60
. 41022
0
f (0 )
80
. 46102
100
. 50446
150
. 59937.
' T
u h h
Once t h e
quartz
f u n c t i on f ( 6 ) i s
can be c a l c u l a t e d
Sa = Sg wher e r
o
is
the
k no wn, t he e n t r o p y
of al pha
by use o f t h e e q u a t i o n
( I Z r 0 ) f ( 0 ).
initial
slope of the e q u i l i b r i u m
curve
40
r
bet ween a l p h a and be t a q u a r t z .
Cohen, e t
aI . ^
entropy of
heat
in
O
has been measur ed by
1974 t o be 0 . 265 ± 0. 005 K/MPa.
( S ) can be c a l c u l a t e d
beta q u a r t z
capacity equation f o r
- 0 . 8 1 9 1 6 8 x 1 0 " 5 (T -
IOOO) 2
+ 0 . 5 4 7 3 7 8 x 1 0 " 7 (T -
IOOO) 3
- 0 . 5 8 7 0 7 7 x I 0 " 10 ( T equation
essentially
since
t he heat
no c o n t r i b u t i o n
1000)
( 72)
I OOO)4 .
can be e x t r a p o l a t e d
l ambda p o i n t ,
f r o m t he
quartz
be t a
Cp = 6 9 . 0 3 3 8 + 0. 9 3 0 2 1 5 x 1 0 " 2 (T -
This
The
down t o
150 K bel ow t h e
capacity for
beta q u a r t z
has
f r o m t h e change i n d i s o r d e r .
From t h e e n t r o p y o f a l p h a q u a r t z , S , t h e heat
capacity of
alpha q u a r t z
possible with
( Cp ) can be c a l c u l a t e d .
Th i s
t he. e q u a t i o n
AS = Cp I n ( T 1ZT2 ) .
We have c a l c u l a t e d
I to
( 73)
t he heat c a p a c i t y of al pha q u a r t z
10 d e g r e e i n t e r v a l s
dr awn
in
Figure
beta q u ar t z
5.
resulting
The e r r o r s
as l a r g e
in the experi mental
J / m o l e K,
c a p a ci t y of
Most o f t h i s
beta q u a r t z ,
negligible.
expansion f o r
However ,
in the
in errors
c a p a c i t y of
i n t h e heat
as 3 J / m o l e K.
Cp o f a l p h a q u a r t z
3 J/ mol e K e r r o r
3 J/ mol e K i s
the e r r o r
f (6) fluctuates
over
smoot h c u r v e
i n t h e h e at
and f ( 0 ) c o u l d r e s u l t
c a p a c i t y of alpha q u ar t z
errors
is
is
Bu t ,
ar e o v e r
the
10
f r o m t h e heat
small,
but
not
in the pol ynomi al
in sign.
Thus much o f t h e
41
error
will
cancel
out
upon
integration
The e n t r o p y and e n t h a l p y o f a l p h a
transition
ar e t a b u l a t e d
literature
values
Richet
et
aI .
quartz
i n T a b l e 9,
f r o m Moser 1 9 3 6 , ^
I 982
of the equation.
near t he
al ong w i t h
Robi e
the
1. 978, ^
and
*
T a b l e 9. The e n t r o p y and e n t h a l p y changes f o r
ne ar t h e l ambda p o i n t .
825- 846 K
l ambda
*
al pha quar t z
**
+
AS
AH
2. 6 6 4 2
2230.
1743.
1569. 0
24 43. 5++
1994. '
1925 .
1844. 0
1925. 3
80 0- 8 25 K
' 2.454
775- 825 K
2.369
1866.
1846.
1818. 0
1850. 7
750- 775 K
2. 360
1800.
1800. ■
1791. 7
1802. 0
*
- e x p e r i m e n t a l Cp o f Mos er , 1936
*.* - Robi e , e t a I . , 1978
+ - R i c h e t , e t a I . , 1980 c o m p i l a t i o n d e p en den t on dr op
c a l o r i m e t r i c experi ment s
++ - The i n t e r v a l used i s 826 t o 847 K t o a l l o w f o r t h e .
use o f Tx = 847. A c o r r e c t i o n o f 655 J / m o l e , t r e a t e d as
f i r s t o r d e r t r a n s i t i o n has been added as i n t h e p u b l i s h e d
paper
From o u r e n t h a l p y
values f o r
al pha q u a r t z
r a n g e 750 K t o 846 K combi ned w i t h
capacities
above and bel ow t h i s
can be c a l c u l a t e d
entropy at
to
for
t he
t h e known he at
r a n g e , H1000 - H 298
be 4 5 4 4 4 . 3 ± 70 J / m o l e . ,
1000 K can be c a l c u l a t e d
Al s o t h e
t o be S1000
J / m o l e K by use o f t h e CODATA68 v a l u e f o r
= 116. 215
the entropy of
42
al pha q u a r t z
our AS f o r
at
29 8 . 1 5 K , Sggg ^ 5
t he range
In or der t o
the
literature,
750 t o 846 K .
check our v a l u e o f Hj q 00 - Hggg j 5 w i t h
we needed i t
of t r a n s f o r m a t i o n
experimental
t o make use o f t h e e n t h a l p y
of quartz to
values
glass.
Their
The b e s t
ar e by K r a c e k , e t
T = 29 8. 1 5 K and by N a v r o t s k y , e t
= 985.
= 4 1 . 4 6 J / mp l e K, and
values
aI .
aI .
7 0
69
in
in
1953 f o r
1 980 f o r
T
ar e 9121 ± 250 J / mo l e and 7001 ± 200
J / mo l e r e s p e c t i v e l y .
Thus by t r a n s f o r m i n g t o
29 8. 1 5 K , h e a t i n g t o
985 K , t r a n s f o r m i n g t o
glass
at
beta qu ar t z
at
985 K and t h e n h e a t i n g t o
1000 K , a v a l u e o f H j 000 -
Hggg j 5 can be c a l c u l a t e d
t o be 45452 ± 300 J / m o l e whi ch
is
v e r y c l o s e t o ou r v a l u e o f 45444 ± 70 J / m o l e .
The heat
c ap a ci t y of glass
and b e t a q u a r t z was t a k e n f r o m R i c h e t ,
et
calculation.
aI . ^
for
this
The E q u a t i o n . o f S t a t e . f o r
, Bef a . Quar t z" '
"
Bet a q u a r t z
temperature at
of thermal
unusual
like
have a p o s i t i v e
it
o v e r a 200 K r ange o f
has a n e g a t i v e
as shown i n F i g u r e s
HgO l i q u i d
coefficient
coefficient
6 and 7.
It
is
bel ow 277 K b e t a q u a r t z w i l l
o f t her mal
I n any case t h e b e h a v i o r
expansion c o e f f i c i e n t
(K)
in that
l ow p r e s s u r e s
expansion,
expected t h a t
pressure.
is
expansi on at hi gher
of the t hermal
v e r s u s p r e s s u r e and t h e b u l k modul us
versus t e mp e r a t u r e shoul d not f i t
t h e Mur naghan-
V cm0 mol
23.8
IO O O
F i g u r e 7. Vol umes o f Bet a q u a r t z a t
p o i n t p l o t t e d versus t e m p e r a t u r e .
0. 1 MPa and a t t h e
l ambda
.44
Hildebrand
thesis.
equation of
state
Thus e x p e r i m e n t a l
a function
as d e v e l o p e d e a r l i e r
values
of temperature f o r
mo d u l i
(K)
mo d u l i
for
t h e b u l k modul us as
b e t a q u a r t z ar e needed.
Kr ammer , P a r d u s , and F r i s s e T
elastic
for
in t h i s
be t a q u a r t z
71
have measur ed t h e
f r o m 863 t o
can be used t o c a l c u l a t e
1073 K.
values of the
These
b u l k modul us
from the equat i on
K = (2Sn
+ S33 + Z f S 13 + ZS1 3 ) ) " 1
wher e S r e p r e s e n t s
the value
the e l a s t i c
point
at
compl i ance c o n s t a n t .
V° = Z3 . 70 cm3/ m o I e , K
e t a I . , 71 and e s t i m a t i n g
( 74)
Usi ng
= 73046 MPa f r o m Kammer,
N = 6 , the
vol ume a t
1073 K and 897 MPa can be c a l c u l a t e d
Z3. 4Z cm / mo l e f r o m t h e Mur naghan
logarithmic
the
l ambda
t o be V =
equation of
state
V = V g (I
- NP/K0 ) " 1 / N .
Be l ow IZOO MPa a c u b i c
be used t o d e s c r i b e
transition.
(75)
equation
in pressure
the t emper at ur e of the
The e q u a t i o n we ar e u s i n g
(P)
can
l ambda
is
Tx = 846 + 0. Z65P - 0. 1 13 93ZZ9 x I O- 4 P2
- Z . 4632558 x I O- 9 P3 .
S i n c e t h e vol ume o f b e t a q u a r t z
l ambda t r a n s i t i o n
Figure
7,
it
much h i g h e r
point,
is
( 76)
and t h e vol ume a t t h e
ar e known o n l y up t o
1073 K , as shown
necessary to e x t r a p o l a t e
temperatures.
t hese curves to
For t h e vol ume a t t h e
we have assumed a l i n e a r
relation
in
with
l ambda
temperature
45
to extrapolate
very s l i g h t
for
the
up t o
1700 K , even t ho ug h t h e vol ume has" a
curvature
linear
region
at
l o we r t e m p e r a t u r e s .
The e q u a t i o n
is
Vx = 2 4 . 5 4 9 5 3 - 0 . 0 0 1 0 5 T .
•To e x t r a p o l a t e
t h e vol ume o f be t a q u a r t z
( Vq j ) we have assumed t h a t
mi ni mum and t h a t
positive
( 77)
the
the t hermal
vol ume goes t h r o u g h
expansi on c o e f f i c i e n t
as t h e t e m p e r a t u r e r e a c h e s
polynomial
1600 K .
extrapolations
give
1650 K , wh i c h
1964°® v e r y w e l l
a maximum v a l u e o f K
increasing with
These
= 100, 000 MPa
behavior f o r
a material
i n c r e a s i n g T in the e x p e r i m e n t a l l y
accessible region.
The f u l l
p r e s s u r e p a r a me t e r s
ar e g i v e n
35 vol u me,
The vol ume and e n t r o p y c o n t o u r
ar e shown i n F i g u r e s
and h o r i z o n t a l
( 78)
n i c e l y t o 2000 K.
is reasonable
the beta q u ar t z
1000)
O
O
0
I
I— ■
CO
O
1
bel ow 1373 K and e x t r a p o l a t e s
quartz
The
IOOO) 3
t h e d a t a o f Ackerman and S o r r e l l
with K
becomes
CNJ
I
t—
x I O- 10( T -
+ 0. 144073 x
for
O
O
0
CO
O
- 0. 188601
1
+ 0. 8 5 4 6 3 3 x
at
a
equation
23. 701 3 - 0.. 105873 x 1 0 " 5 (T -
fits
a t .0.1 MPa
t e m p e r a t u r e and
i n Ta b l e 10 f o r
lines
8 and 9.
beta q u a r t z .
f r o m a l p h a and bet a
The c o n t o u r
lines
vol ume and e n t r o p y ar e n e a r l y v e r t i c a l
because t h e s l o p e s o f t h e s e
lines
ar e
l/.aK
46
Ta bl e 10. The e q u a t i o n s o f s t a t e , he at c a p a c i t y e q u a t i o n s
and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f a l p h a and bet a
quartz.
SiO2 (C, BETA QUARTZ)
23.701266
1520712E-4
.2441776 E-8
-.116290E-11
.2140223E-15
-.1058728E-5
.2042034 E-7
-.806657E-11
:3979281 E-14
-.80183 E-18
.854633E-8
-.519449E-10
.1837775E-13
-.368804E-17
. .596339 E-21
-.188601 E-10
.974922 E-13
-.490703E-16
.917195 E-20
-.701148E-24
.1440728E-13
-.714101 E-16
.492830 E-19
-.127623E-22
.1175194E-26
SiO2 (C, ALPHA QUARTZ)
24.340296
-1.218243E-4
1.170823 E-7
-7.99760E-11
3.031544E-14
-5.78773E-18
4.329322E-22
3.238095 E-4
-5.371956E-7
5.474543E-10
-3.38035E-13
1.203374E-16
-2.23346E-20
1.658714E-24
7.855450 E-7
-1.357596E-9
1.376992E-12
-8.22533E-16
2.743039E-19
-4.71042E-23
3.241953E-27
1.018418 E-9
-1.73248E-12
1.886086E-15
-1.14072E-18
3.586603E-22
-5.49837E-26
3.236415E-30
5.083011 E-13
-8.66629E-16
9.986443E-19
-6.00664E-22
1.754802E-25
-2.30557E-29
1.010065E-33
D
E
HEAT CAPACITY (Cp)
A
C
(C, BETA QUARTZ)
69.0387884
9.302148 E-3 .
-.819168 E-5
(C, ALPHA QUARTZ)
98.469291
.175798
THERMODYNAMIC
Y1000
PROPERTIES
J/MOL
(C, ALPHA QUARtTZ) 70.84657
(C1BETA QUARTZ)
76.31535
.31970680E-3
-.587077E-10
.5473777 E-7
-.501216E-10
2.1971990E-7
H1000'H298
J/MOL
H298
J/MOL
S298
J/MOL K
V1000
CM3
47620.06
39899.69
-910700.
-905155.39
41.46
52.63
24.3403
23.701
THE HEAT CAPACITY EQUATIONS ARE GIVEN BY Cp=A+B(T-i 000)+C(T-1000)2+D(T-1000)3
+ E(T-IOOO)4.
1500
*L 1 0 0 0
-p>
500
/
/
/
/
/
/
/
/
1000
2000
3000
4000
Pl MPa
F i g u r e 8 . Our c a l c u l a t e d
quartz.
contour
I i nes f o r
t h e vol ume o f
be t a
16 0
140
120
100
80
50060
40
I
1000
r
I
2000
3000
4000
P1MPa
F i g u r e 9.
Our c a l c u l a t e d
contour
lines
for
t h e e n t r o p y o f bet a q u a r t z
49
and aVT/ Cp r e s p e c t i v e l y ,
coefficient
is
and t h e t h e r m a l
very small .
The a l p h a q u a r t z - b e t a
1643 K and 3400 MPa i s
various
equations
treatment
with
of
of
equations
change
a convenient
of
state
is
to
t h e Mur naghan
logarithmic
a r e compar ed
72
at
compare
earlier.
i n Tabl e
These two
11 al ong, w i t h t h e
beta q u a r t z goi ng
1653 K and 3440 MPa f r o m M i r w a l d and
.
of
S t a t e o f Al pha
For a l p h a q u a r t z ,
as t h e
l ambda p o i n t
t h e b u l k modul us a p p r o a c h e s z e r o
coefficient
According
at
compar e
enough d e t a i l
s t a t e wh i c h we c a l c u l a t e d
state
point
the o n l y o t h e r
i n vol ume and e n t r o p y v a l u e s f o r
Equation
Qu a r t z
entropy
place to
However ,
beta q u a r t z w i t h
of
to coesite
Massone
q u a r t z - c o e s i te t r i p l e
of state.
our e q u a t i o n
equation
expansion
is
approached,
and t h e t h e r m a l
expansi on
and t h e
to.the
slopes
h e a t c a p a c i t y ap p r o a c h i n f i n i t y .
rn
Pippard t h e o r y
t h e vol ume and t h e
ar e bot h
limited
by t h e e q u i l i b r i u m
( d T / d P ) x w h i ch = ( r ) .
T h e r e f o r e , as t h e vol umes
entropy contour
ap p r o a c h t h e
lines
6.
and t he
l ambda t r a n s i t i o n
must bend s h a r p l y as shown i n F i g u r e s
Kl e m e n t 1s F i g u r e
sl ope
8 and 9 and
they
50
T a b l e 11. Compar i son o f t h e t h e r mo d y n a mi c v a l u e s f o r bet a
q u a r t z and c o e s i t e a t h i g h p r e s s u r e s and t e m p e r a t u r e s .
T
P
K
3400
1653
**)
+)
+ +)
ref.
bet a
Qu a r t z
Coesi t e
AQ
*
MPa
1643
*)
Qu a r t z
3440
H
**
+
- 740206
- 739240
- 746483
- 7 482 58
S
**
+
151. 50540
153. 45105
147. 69553
148. 05736
-3.80978
-5.39369
V
**
22 . 9 6 9 6 7
20 . 9 1 7 5 7
-2.05210
H
**
+
++
- 738539
- 737578
- 7 449 44
- 746712
S
**
+
++
151. 08281
153. 93282
148. 12198
148. 49823
-2.96083
-5.43459
-4.2
V
**
+
++
22 . 96209
21 . 6 7 5 4 8
20. 91458
20 . 48 325
-2.04751
-1.19223
-3. I
The change i n t h e H i n ( J / m o l e )
V i n ( c mVmo Te )
T h i s Work
Howa I d , e t a I . , 1985
M i r w a l d and Mas s one,. 1980
The Vo Iume and e n t r o p y o f
t e r ms
i n 8 ° ‘ 5 near t h e
0 = ( J x - T ).
, S in
al pha q u ar t z
l ambda t r a n s i t i o n
a power s e r i e s
as 0 a p p r o a c h e s
zero.
alpha q u a r t z w i t h
- 6405
- 9134
- 6900
( J / m o l e K ) and
bo t h
have
wh e r e ,
These t e r ms a r e v e r y d i f f i c u l t
accurately with
-6277
- 9018
to represent
i n t e m p e r a t u r e and p r e s s u r e
Therefore,
P i p p a r d 1s t h e o r y
we have chosen t o model
50 t o
60 K b e l ow t h e
51
l ambda t r a n s i t i o n .
calculated
The t h e r mo d y n a mi c p r o p e r t i e s
from the equat i ons
^alpha
^beta
3a I p'ha - 3b e t a
r / r Q ( f ( 8 ))
( 79)
1/ r
(
( f (0 ) )
wher e r = ( d T / d P ) ^ t h e e q u i l i b r i u m
0 = (
1/ r
o
-I)
than
and f ( 0)
and r / r
o
slope.
are
allows
is
given
for
The s l o p e d r o p s
slope,
t o 0. 2 27 231
i n vol ume o f t r a n s i t i o n
The c h o i c e o f
in the
for
12000 MPa and r / r Q compensat es " f o r
t h e change
r Q = 0.265,
in Tabl e 8 .
the cur vatur e
equilibrium
a pressure greater
this
instead
t h e change i n e n t r o p y o f t r a n s i t i o n
80)
with
by d e c r e a s i n g
of
increasing
increasing
t e m p e r a t u r e and p r e s s u r e .
The e q u a t i o n
extrapolate
transition
At
alpha q u a r t z
Weaver ,
50 t o
al ong t h r e e
alpha
is
Then w i t h
if
at
these
Fitting
o f t h e edges o f t h e a r e a t o
t h e vol ume and t h e b u l k modul us
quartz,
K q = 37200 MPa,
0 = 55 and 60 g i v e t h e
a
good vol ume d a t a ar e
be f i t .
ar e w e l l
as r e p o r t e d
e t a I . , 7 3 Soga74 and McSki mmn, e t
values at
the
t h e vol ume and
of pressures.
si mpl e
shoul d
60 K b e l o w t h e
can be c a l c u l a t e d
and a s e r i e s
l ow p r e s s u r e s
known f o r
to
beta q u a r t z
from Tabl e 8 , val ues f o r
power s e r i e s
available
for
t e m p e r a t u r e o f 846 K .
temperatures
reliable
state
reasonably well
f ( 0 ) values
entropy of
of
a 1 . 75
The
vol umes and ( dV/ d T)
t h e u p p e r edge f r o m t h e P i p p a r d t h e o r y .
A f ew
by
along
52
intermediate
need t o
be e s t i m a t e d ;
accurate to
of
state
values
at
at
within
800 and
± 0 . 2 cm / m o l e .
in a series
al pha
entropies
in Figures
of
ar e
its
of
State
The q u a r t z
studi ed..
of
AH,
in
slope
t h e vol umes
Tabl e
12,
the e x p e r i me n t a l
and
and
values
10 and 11.
state
has been w i d e l y
provides
a very s t r i n g e n t
s i n c e t h e change
AS,
( AV/ AS) .
ar 6 w e l l
k n o w n . 7^ 57 7
coesite
is
in Tabl e
test
in enthalpy,
ar e s m a l l ,
t hem can cause v e r y
coesite
shown
the equation of
mat ch w i t h
equilibrium
change i n e n t r o p y ,
changes
equilibrium
of
approximations.
Coesite
Thi s e q u i l i b r i u m
and t h e
s ma l l
for
coesite
the eq uat i ons
The c o n t o u r
P i p p a r d t h e o r y shown i n
o f B o e h l e r 76 shown i n F i g u r e s
the entropy
8 and 9 ar e a t h i r d
of the accuracy f o r
how ( d T / d P ) x = aVT/ Cp c o m p a r e s . t o
Equation
The e q u a t i o n s
( d T / d P ) s val ues
successively better
quartz
from the
the
fit
s h o u l d be
1000 K by B o e h l e r . 76
alpha q u a r t z
for
squares
can t h e n be checked a g a i n s t
Two ma j o r t e s t s
state
a good l e a s t
- T = 60 K and a g a i n s t
for
attempt
for
b u t , even e s t i m a t e s
least
obtained
measur ed a t
lines
temperatures
and v e r y
l a r g e changes
in the
The vol ume and b u l k
modul us o f
The e q u a t i o n o f
for
13.
state
.0 4
.O O 1----------- '----------- '----------- '----------1—
0
2000
4000
P1 MPa
F i g u r e 10.
( d T / d P ) ^ = aVT/ Cp a t 800 K. The s o l i d l i n e i s ou r c a l c u l a t e d
c u r v e . The open c i r c l e s ar e t h e e x p e r i m e n t a l v a l u e s o f B o e h I e r .
0 .0 4
CL
O
>
e 0.02
^ e
-
O
0 .0 0 L
0
i
2000
Pt MPa
F i g u r e 11.
( d T / d P ) x = «VT/ Cp a t
c u r v e . The open c i r c l e s ar e t h e
4000
1000 K . The s o l i d l i n e i s our c a l c u l a t e d
exp er i me nt a l values of B o e h l e r .
55
T a b l e 12. Compar i son o f t h e p o l y n o m i a l f i t and P i p p a r d
c a l c u l a t i o n s f o r t h e t h e r mo d y n a mi c p r o p e r t i e s o f a l p h a and
b e t a q u a r t z 60 K bel ow t h e l ambda t r a n s i t i o n . .
P
MPa
T
K
0. I
786
2000
3400
1265
1583
Material
be t a q u a r t z
Pippard t he or y
*
al pha q u a r t z * *
alpha qu ar t z
H
S
V
- 879756
- 1281
99. 91288
-1.548
23. 72101
-.41022
- 881037
- 881052
98 . 3 6 4 9 0
98 . 3 6 4 1 6
23 . 31 228
23. 31228
132. 4389
-1.548
23 . 18649
-.35175
beta q u a r t z
Pippard t h e o r y
*
alpha q u a r t z * *
alpha quar t z
- 800119
- 2023
.
- 802142
- 8 023 56
130. 8909
130. 8132
22.83474
22 . 7 7 6 0 0
beta q u a r t z
Pippard t heory
*
alpha q u a r t z * *
al pha q u ar t z
- 7447 74
- 2515
148. 67349
-1.5487
22 . 9 5 6 1 7
-.35175
- 7 472 89
- 747813
147. 12549
146. 85953
22 . 60 442
22 . 5 4 7 8 4
*)
Val ue s c a l c u l a t e d f r o m t h e b e t a q u a r t z e q u a t i o n o f
s t a t e and t h e P i p p a r d e q u a t i o n s ,
* * ) Val ue s c a l c u l a t e d f r o m t h e a l p h a q u a r t z e q u a t i o n o f
state
In or der t o
bet ween q u a r t z
accurately f i t
and c o e s i t e ,
t h e measur ed e q u i l i b r i a
we had t o
the heat c a p a c i t y equat i on thr ough
its
entropy
coesite
and e n t h a l p y .
listed
in Table
The h e a t
13 i s
successively adjust
successively adjusting
ca p ac i ty equation
for
56
PO
-2
O
O
O
I
CO
O
O
O
-H
-H
I
cn
LD
O
- 0. 7 47 870 x
I
- 0. 4 0 9 4 6 0 x
O
9 . OE ( T/ 800 ) - 1. 23583 + 0. 625 x 10
+ 0. 1 95 243 x 1 0 " 10( T - IOOO) 4
(81)
+ 0 . 4 8 x I O6 ( 1 / T 2 - I x I O6 )
Figure
12 shows t h a t
l ar g e or smal l
t he heat c a p a c i t y
i s not
unreasonably
o v e r t h e t e m p e r a t u r e r ange under
consideration.
T a b l e ' 13. The e q u a t i o n o f s t a t e , heat c a p a c i t y e q u a t i o n s
and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f c o e s i t e and hi gh
coesite.
Sio2 (COESITE)a
0.208005E+02
-.111596E-04
0.420456E-09
-.167317E-13
0.511098E-18
-.929361E-23 .
0.713800E-28
0.300153E-04
-.230112E-08
0.145340E-12
-.672272E-17
0.198214E-21
-.322891E-26
0.218545E-31
0.480077E-09
-.272670E-12
0.420059E-16
-.390897E-20
0.196687E-24
-.478139E-29
0.437280E-34
-.255108E-13
-.783177E-16
0.294235E-19
-.397802E-23
0.234987E-27
-.617087E-32
0.588345E-37
-.689626E-16
0.648684E-19
-.269232E-22
0.378781 E-26
-.227515E-30
0.602315E-35
0.000000E+00
HEAT CAPACITY (Cp)
A
B
C
D
E
Si02 (COESITE)b
-1.235835
THERMODYNAMIC
.00625
-.040946E-4
-.074787E-8
.195243E-10
PROPERTIES
Y1000
JMOL*1
HlOOO*H298
JMOL*1
H298
JMOL*1
S298
V1000
JMOL*1 K*1 CM3
Si02(COESITE)
Si02(HIGH COESITE)
68.77690
71.32204
43080.19
43080.19
-907213.9
-903599.8
40.46715
43.01228
20.8005
20.8005
THE HEAT CAPACITY EQUATIONS ARE GIVEN BY Cp= A+B(T-1000)+C(T-1000)2 +D(T-IOOO)3
+E(T-1 OOO)4. aTHE EQUATION OF STATE FOR COESITE AND HIGH COESITE ARE THE SAME
EXCEPT FORTHE ENTROPY AND PLANCK FUNCTION. bTHE HEAT CAPACITIES FOR COESITE AND
HIGH COESITE HAVE THE ADDITIONAL TERMS .48E6fT2-1 O*6) AND 9.0x E(800/T) WHERE E IS
AN EINSTEIN TERM.
IOO
T1K
F i g u r e 12.
Heat c a p a c i t y v a l u e s f o r
c r i s t o b a I i t e ( dashed l i n e s ) .
coesite
(solid
line)
and
58
The c a l c u l a t e d
298. 15 K i s
wi t h- t h e
Sggg
value repor ted
J/mole.
at
by Hol m,
for
bo t h
value
quartz
± 630 J / m o l e ,
noting that
the e q u i l i b r i u m
ag r e e me n t w i t h
s ma l l
going to
f r o m H o l m , ye t
change
in
Figure
slope t h a t
increase
72
and v a l u e s
i n good
1400 K .
going to
o f the
and t h a t
ar e t o o
79
Si n c e t h e r e must be a
i n t h e e n t r o p y o f a l p h a q u a r t z as i t
l ambda p o i n t ,
ac c omodat e t h e h i g h e r
maki ng a f i r s t
Up t o
ar e
t h e measur ed d a t a o f Boyd and Engl and
substantial
higher
enthalpy
and be t a q u a r t z
ar e t o o n e g a t i v e
and Ma s s o n e .
at
It
13.
AS v a l u e s
and M i r w a l d
coesite
in.the
val ues o f the e q u i l i b r i u m
calculated
approaches t he
AHgyg = 2930
AHgyg = 1339 J / m o l e .
t h e measur ed e q u i l i b r i a
compar ed t o
aI.,
T h i s can
i n t h e s l o p e and ap pea r an c e o f
140 0 K b o t h a l p h a q u a r t z
give
he at
coesite.
a 200 J / m o l e s h i f t
lines
Our c a l c u l a t e d
equilibrium
of
enthalpy value f o r
al ong w i t h the
and f r o m N a v r o t s k y ,
causes a s u b s t a n t i a l
coesite
78
and c o e s i t e g i v e s AHg^Q = 1103
beta q u a r t z
be compar ed t o t h e v a l u e s
Above
T = .
2 9 8 . 1 5 K i s Hggg j g = AH^ = - 9 0 7 2 1 3 . 9 ± 200
± 200 J / m o l e f o r
i s worth
at
Kl eppa and West rum
Our c a l c u l a t e d
Thi s e n t h a l p y
capacities
coesite
= 4 0 . 4 6 7 2 ± 0 . 2 J / mo I e K , whi ch agr ees
$298 15 = 4 0 . 3 7 6 .
coesite
entropy value f o r
slope
t h e o n l y r e a s o n a b l e way t o
is to
temperatures.
order t r a n s i t i o n
i ncr ease the ent r opy of
We a c c o m p l i s h e d t h i s
in coesite with
by
an e n t r o p y
59
2000
befaquartz
coesite
IOOO
alpha
quartz
2000
F i g u r e 13. C a l c u l a t e d phase
c i r c l e s ar e f r om ( B o e h l e r ,
1982; Mi r ma l d and Ma s s one,
and E n g l a n d , 1960, as open
coesite
4000
d i a g r a m f o r Si Og. F i l l e d
1982; Bohen and B o e t c h e r ,
1980) w i t h t h e v a l u e s o f Boyd
circles.
60
of t r a n s i t i o n
equal
t o ASt r a n s
The assumed t r a n s i t i o n
with
the c o n t i n u a t i o n
calculations
= 2. 5451
in c o e s i t e
is
of the quartz
i n c l u d e d as d o t t e d
f
0 . 2 J / mo l e K.
shown i n F i g u r e
to
13,
l ow c o e s i t e
lines.
•The. Equat i on o f . S t a t e . f o r
Cristobalite
Eliezer,
et a l . ^
first
treated
cristobalite
T h i s t r e a t m e n t was based upon t h e JANAF
for
an i n c r e a s e o f
cristobalite
point
for
to
to
0. 11
bring
1175 K .
j/mole
the q u a r t z
cristobalite
goi ng t o
14.
cristobalite
l ow c r i s t o b a l i t e
at
literature
at
et
various
29 8. 1 5 K.
for
a l.^
985 K .
can be
al pha q u a r t z
goi ng t o
The b e s t e n t h a l p y changes
^trans
a 1 . 70 AHt r a n s
yields
Hol m' s
These v a l u e s
( 82)
= 1882. 8 ± 630 J / mo l e a t
Con v er t i n g t hese val ues
1389 ± 250 J / mo l e f o r
as
quartz
= Si O2 C c r i s t o b a l i t e )
u s i n g ou r h e a t c a p a c i t i e s
respectively.
cristobalite
the r e a c t i o n
970 K , and by N a v r o t s k y , e t
J/ mol e at
is reasonable
tempratures
in enthalpy f o r
S i O2 ( b e t a q u a r t z )
ar e by Hol m,
equilibrium
Thus t h e e n t h a l p y changes f o r
c o n v e r t e d t o changes
in the
except
We now have good heat c a p a c i t y v a l u e s
i n t h e changes o f e n t h a l p y f o r
shown i n T a b l e
values,
1978.
K in the en t r o p y of
a l p h a and b e t a q u a r t z and t h e r e
agr eement
32
in
= 90 7 . 9 2 8 ± 250
t o AH283
2356 ± 600 J / m o l e and
and N a v r o t s k y ' s v a l u e s
a l o n g w i t h K r a c e k 1s
69
AH2gg =
61.
2636 ± 290 J/ mo Te f r o m e n t h a l p i e s
29 8. 1 5
K ar e
i n v e r y poor a g r e e me n t .
wor se t h a n t h e e n t h a l p i e s
Thi s
cristobaIite
sampl es r a t h e r
14.
indicates,that
S +++
>298
H298
T lambda
S600
OO
O
O
O
H600"H298
H1000'H298
H1000'H1000 cIua rtz
H298-K29S qUartZ
glass
the probl em i s w i t h
t he
t h a n t h e me t h o d o l o g y .
The t h e r mo d y n a mi c p r o p e r t i e s
this
work
d a t a at
C e r t a i n l y t h e y ar e
from the quart z t o
transition.
Tabl e
of s o l u t i o n
of c r i s t o b a l i t e
**
*
***
+
42.635
43.363
43.363
43.40
43.40
-907916
-907864
-908346
-908346
, ++
-908346
535
525
523
543
535
83.954
84.300
84.60
83.789
83.178
18062
18232
17973
18008
118.216
118.167
118.61
117.80
117.36
117.69
44985
44859
44735
44769
44890
44786
1456
21116
2122
1925
2785
2836
2354
2389
. 18037
83.138
17635
2354
*)
Ri c h e t , e t a I . , 1982
**)
Robi e , e t a I . , 1978
* * * ) S t u l l and P r o p h e t , 1971
+)
Moseman and P i t z e r , 1941
+ +)
EI i z e r , e t a I . , 1978
+++) S. I . u n i t s , 0 / m o l e and J / m o l e K ar e used t h r o u g h o u t
this table
It
better
appears t h a t
the ent ropy of the c r i s t o b a l i t e
known t h a n t h e e n t h a l p y ,
so t h a t
is
the e n t h a l p y of
62
cristobalite
at
29 8 . 1 5 K s h o u l d be a d j u s t e d
known o f t h e e q u i l i b r i u m
instead
of ad justi ng
bet ween q u a r t z
the e n t r o p y .
R i c h e t ' s 41 v a l u e f o r
the en t r op y of
too
l ow a t
the quartz to
Teq = 1108 K .
and changes t o t r i d y m i t e
alkali
silicates.
should c a t a l y z e
c r i stobali t e .
to
Alkali
1141 K .
is
silicates
1141 K .
1160 f o r
H298 15 = - 907915 f o r
in enthalpy f o r
transition
equilibrium
stable
dissolve
the
at
1141
of quartz
to
temperature
The q u a r t z t o
needs
cristo balite
1163 K by H o l m q u i st
the e q u i l i b r i u m ;
l ow c r i s t o b a l i t e .
1163 and
thus,
giving
Then t h e change
al pha q u a r t z t o c r i s t o b a l i t e
2785 f
K r a c e k ' s v a lu .e ° 9 b u t
200 J / m o l e ,
is
wh i c h a g r e e s w e l l
with
h i g h e r than e i t h e r
H o l m ' s 78, or
drop c a l o r i m e t r y
d a t a bet ween
N a v r o t s k y ‘ s 78 v a l u e s .
Us i ng t h e a v a i l a b l e
54 1 . 6 5 and 1834 K , 81, 41
c r i stobalite
was f i t
the
is
Si O2 and
t h e e q u i l i b r i u m must be bet ween
We p i c k e d
of
1150 K i n t h e p r e s e n c e o f
has been o b s e r v e d a t
so t h a t
transition
is
Therefore the t r a n s i t i o n
transformation
(1961),
their
c r i stobalite
the t r a n s f o r m a t i o n
be even h i g h e r t h a n
is
and c r i s t o b a l i t e ,
l ow c r i s t o b a l i t e
Bet a q u a r t z
at
what
We have a c c e p t e d
$298 15 = 4 3 . 3 6 3 J / mo l e K . . However ,
temperature f o r
to f i t
heat c a p a c i t y f o r
to the f o l l o w i n g
polynomial
63
Cp = 9 . 0 E ( T / 8 0 0 ) ' -
I . 49773
+ 0 . 3 2 7 6 1 0 x 1 0 " 3 (T -
1000)
+ 0. 201540 x 1 0 ' 5 ( I
-
IOOO) 2
+ 0 . 6 0 1 1 1 9 x 1 0 ' 8 (T -
IOOO) 3
+ 0. 1 27 126 x I 0 " 11( I
+ 0 . 4 8 x I O6 U Z T 2 wher e E r e p r e s e n t s
keeps t h e h i g h e r
excessively
an E i n s t e i n
polynomial
large w i t h i n
the equat ion
( 83)
t e r m , and t h e
the r egi on
values
for
ar e summar i zed
o f 400 t o
S6 qq - S298
i n Tabl e
4 1 . 2 J / m o l e K , so t h a t
largest
error
c r i stobalite
I x IO '6) ;
I/T
2
t er m
power s f r o m becomi ng
40.3 to
we d e c i d e d t o
IOOO) 4
2000 K wh i c h
represents.
The l i t e r a t u r e
cristobalite
-
treat
with
14 and r a n g e f r om
the e r r o r
in the e n t h a l p y f o r
for
i n AS i s
c r i stobalite.
t h e second o r d e r t r a n s i t i o n
the
Pi ppard r e l a t i o n s .
SI ow( 450 ) = 5 h i g h (4 5 0)
'
1/r
the
Therefore
in
The e q u a t i o n
( Vh i g h ( 4 5 0 )
Vl o w ( 4 5 0 )
The e q u i l i b r i u m
is
( 84)
t h i s t r a n s i t i o n has been
OO
measur ed by Cohen and Kl ement
. t o be ( d T / d P )
= 0. 51
K/ MPa.
slope f o r
The vol ume o f h i g h
described
by t h e p o l y n o m i a l
c r i stobalite
at
l ow p r e s s u r e
is
64
V = 2 7 . 4 3 0 + 0. 605751 x 1 0 ' 5 (T - 0.86013 x 10" 8 ( I
•
-
+ 0. 15697 x I 0 ” 1° ( I
IOOO) 2
-
IOOO) 3
- 0 . 8 8 4 2 9 4 x 1 0 " 14(T fit
1000)
IOOO) 4
( 85)
f r o m t h e d a t a o f Johnson and Andr ews
Skinner
1 9 6 6 , 84 and T o u l o u k i a n
interpolation
450 K o f
results
t h e n AV = 27. 181
change
as c i t e d
l ow c r i s t o b a l i t e
The change
i n vol ume
- 2 6 . 0 4 = 1. 141 cm^ / mol e y i e l d i n g
in entropy of
in
1 9 6 7 . Gr a p h i c a l
i n a vol ume o f
2 6 . 0 4 cm3/ m o I e . 8 3 ’ 88
83
2.237 J/ mol e K.
at
is
a
The l ow c r i s t o b a l i t e
e n t r o p y o f Moseman and P i t z e r 8 * a t 450 K , S^ qw = 63. 443
J / m o l e K can be c o r r e c t e d
better
heat
value f o r
for
high
to
6 4 . 1 9 4 J / mo l e K by u s i n g t h e
c a p a c i t y o f L e a d b e t t e r . and W r i g h t .
S-|
t hen y i e l d s
cristobalite
t he heat c a p a c i t y
an e n t r o p y o f 66. 431
from the
o f hi gh
8 3
J / mo l e K
Pi ppard e q u a t i o n s .
cristobalite
gives
Th i s
Usi ng
Sh i g h ( 6 0 0 )
=
8 3 . 9 5 4 and Sh i g h ( T 0 0 0 ) = H S . 216 J / mo l e K.
The h i g h
pressure
vol umes
for
cristobalite
t h e Mu r n a g h a n - H i d e b r a n d e q u a t i o n o f
MPa and N = 6 .
give
This
value f o r
a reasonable value f o r
cristobalite
equation
of
along w i t h
and i n
state
K
the
t h e Si Og ( T )
speed o f
sound i n
The f u l l
cristobalite
equation
Kq = 14237
i s an e s t i m a t e chosen t o
Si Og l i q u i d .
for
state with
came f r o m
is given
of state.
polynomial
in
Ta b l e
15
Ta b l e 15. T h e ' e q u a t i o n s o f
and s e l e c t e d t h e r mo d y n a mi c
liq u id quartz .
s t a t e , heat c a p a c i t y equat i ons
p r o p e r t i e s o f c r i s t o b a l i t e and
SiO2 (LIQUID)
0.256903E+02
-.621362E-04
0.818605E-08
-.779195E-12
0.407254E-16
-.101736E-20
0.922540E-26
0.882893E-04
-.277794E-07
0.647971 E-11
-.95831OE-15
0.784452E-19
-.314107E-23
0.473867E-28
0.403999E-08
-.622431 E-11 .
0.272915E-14
-.630580E-18
0.682613E-22
-.322660E-26
0.538327E-31
-.115153E-12
-.130836E-14
0.126817E-17
-.421055E-21
0.528631 E-25
-.267420E-29
0.462225E-34
0.130579E-15
-.497927E-18
0.732965E-21
-.291350E-24
0.390861 E-28
-.203681E-32
0.357553E-37
-8.601635E-9
4.231581 E-12
-1.75038E-15
4.879866E-19
-7.45014E-23
5.426761 E-27
-1.45570E-3T
1.569704E-11
-6.83298E-15
2.550793E-18
-6.69296E-22
9.915851 E-26
-7.11521E-30
1.893393E-34
-8.84294E-15
3.541668E-18
-1.21353E-21
3.000796E-25
-4.30461 E-29
3.036980E-33
-8.00699E-38
SiO2 (C, CRISTOBALITE)
27.438
.7023787E-4
1.6423549E-8
-3.58174E-.12
4.927630E-16
-3.42292E-20
8.964361 E-25
6.0575144E-6
-3.158457E-9
1.364669E-12
-3.89561 E-16
6.014507E-20
-4.40513E-24
1.185093E-28
HEAT CAPACITY (Cp)
C
A
SiO2 (L)
B
71.00826
.0184709
(C, CRISTOBALITE) a
3.276100 E-4
-1.497734
THERMODYNAMIC
Y1000
D
E
.601119 E-8
-1.27126E-12
-.532714E-5
2.015400 E-6
PROPERTIES
JM O L'1
H1000"H298
J MOL*1
Si02(L)
(C, CRISTOBALITE)
74.416376
74.27571
44673.87
43940.87
H298
J MOL"1
V1000
S298
J MOL'1 K'1 CM3
-904213.5
-906871.21
46.861
42.677
27.27
27.38
THE HEAT CAPACITY EQUATIONS ARE GIVEN BY Cp=A+B(T-1000)+C(T-1000)2+D(T-1000)3 +
E(T-IOOO)4. aCRISTOBALITE ALSO HAS THE TERMS .4 8 x1 06( r 2-10'6) +9.0 E(T/800),
WHERE E IS AN EINSTEIN TERM.
66
Equation of State
S i li c o n Dioxide
For
liquid
for
Liquid
Si O2 t h e h e a t
c a p a c i t y e q u a t i o n was f i t
t o R i c h e t ' s dr op c a l o r i m e t r y measur ement s . ^ *
capacity equation
The heat
is
Cp = 71 . 00 83 + 0 . 0 1 8 4 7 0 9 ( 1
-
1000)
- 0 . 5 3 2 7 1 4 x 10“ 5 (T - I OOO) 2 .
The e q u a t i o n
is
valid
over
The e n t h a l p y o f f u s i o n
d / mo l e a t
1966 K.
This
measur ement s o f AH f o r
t h e r a nge 400 t o
for
value
R i c h e t 1s v a l u e
is
AHf u s
liquid
is
arrived
the quartz
to
for
value
J/mole.
Thus t h e r e
from the
glass t r a n s f o r m a t i o n
capacity of
of
Our e n t r o p y
Si O2
and l i q u i d .
cristobalite
value f o r
is
overall
S i O2
is
good agr eement
and R i c h e f s.
Ther e ar e a v a r i e t y
properties
of r e por ted
o f Si O2 l i q u i d .
expansion c o e f f i c i e n t
values f o r
Thus,
the
t he t her mal
bet ween 2208 and 2438 K i s
1. 03 x I O" 4 K" 1 as measur ed by Bacon,
W h o l l e y . 86
8 6 2 1 . ± 150
= 17 2. 9484 J / m o l e K and R i c h e f s
17 2. 915 J / m o l e K .
physical
at
the e n t h a l p y o f f u s i o n
S 200O ( I )
bet ween our
is
heat c a p a c i t y o f c r i s t o b a l i t e
=' 8920 ± 1 0 0 0
is
2100 K.
cristobalite
by K r a c e k 69 and N a v r o t s k y , 70 t h e he at
g l a s s '^1 and our
(86)
Has a pi s
a =
and
Bucar o and Dar dy have measur ed t h e
compressibility
o f Si O2 l i q u i d
bet ween 1650 and 2000 K t o
be B = 8 . 5 x I O" 13 MPa, 86 and t h e
ultrasonic
C = 6000 m / s e c 87 bet ween 2060 and 2160 K .
velocity
Us i n g t h e s e
is
67
measur ed v a l u e s
capacity,
and our c a l c u l a t e d
Cp, we c a l c u l a t e
constant
the constant
p r e s s u r e heat
vol ume heat
c a p a c i t y Cy f r o m t h e e q u a t i o n s
Cy = Cp - TVa2K7
( 87)
and
Cv = Cp/ ( I + TMa2c 2ZCp)
wher e V i s
the
vol u me,
i s t he t her mal
c o e f f i c i e n t , K7 i s t h e
ultrasonic
( 87)
is
vol ume h e a t
b u l k mo d u l u s ,
the mol e cul ar wei ght
capacity calculated
incorrect,
and a t
least
o f K7 , a and c must
is too
on t h e o r d e r o f
These v a l u e s
of
large.
5 x 10
For one t h i n g
MPa.
to
beta q u a r t z
is
One woul d e x p e c t t h e b u l k
The v a l u e s
of the thermal
expansion c o e f f i c i e n t
ultrasonic
speed a r e a l s o
in e r r o r
( 88).
values
the bulk
Si O2 l i q u i d
of equation
( 88) is
o f Cy a r e o b v i o u s l y .
The b u l k modul us o f
4
Si Og.
from equat ion
modul us o f
results
c is the
t wo o f t h e t h r e e r e p o r t e d
be wr o n g .
( 8.8 )
expansion
Cy = - 6 1 6 . 3 6 J / mo l e K , and f r o m e q u a t i o n
Cy = 5 5 . 4 6 J / m o l e K .
modul us
isothermal
speed and M i s
The c o n s t a n t
'
be i n t h i s
r a nge
as i s
if
not
smaller.
and t he
shown by t h e
C o n s e q u e n t l y we have de c r e a s e d
4
t h e b u l k modul us a f a c t o r
the thermal
of
100 t o
expansion c o e f f i c i e n t
Wi t h t h e s e changes t h e c o n s t a n t
the
ultrasonic
23 4 3 . 9 m/ s e c .
1. 11525 x 10
MPa and
t o 8. 8 3 8 4 x 10 ^ K * .
vol ume he at
c a p a c i t y and .
speed a r e Cy = 7 9 . 2 6 2 J / mo l e K and c =
Bot h o f t h e s e v a l u e s
ar e now r e a s o n a b l e .
68
Equations
In o r d e r
of
State
oxide d i s so lv e s
use t h e f o r m u l a A l O 1 g.
in oxi de mel t s
t wo A l +^ f o r
The al umi num i o n s
other.
at
very
Al umi num
l ow c o n c e n t r a t i o n s
e v e r y mol e o f al umi num o x i d e .
ar e n o t n e c e s s a r i l y a s s o c i a t e d w i t h
T h e r e f o r e , in order
to give Henry's
one must d e s c r i b e
A l g O3 as A l O 1 g .
Henry's
that
directly
Al umi num Oxi de
t o d e s c r i b e Al gOg i n b i n a r y and h i g h e r o r d e r
phase di agr ams, we need t o
contributing
for
Law s t a t e s
proportional
the a c t i v i t y
of
each
Law b e h a v i o r
a solute
is
t o t h e mol e f r a c t i o n
a = Kx
( 89)
wher e K i s t h e H e n r y ' s
the a c t i v i t y
o f al umi na
mol e f r a c t i o n .
Law c o n s t a n t .
is
described
If
we use Al gO3 t h e n
by t h e
Si n c e t wo A l + ^ ar e d i s s o l v e d
square of t he
p e r A l 3O3
a =YKx2
The a c t i v i t y
( 90)
can a l s o be d e s c r i b e d
a t any c o n c e n t r a t i o n
by
the eq uat i on
a = Yx
thus,
( 91)
in d i l u t e
solutions
Y=Kx.
However ,
( 92)
we ar e u s i n g R. K . c o e f f i c i e n t s
to describe
l og
such t h a t
IogY
=
l og K x . .
T h e r e f o r e , as x a p p r o a c h e s
( 93)
zero
l og Y approaches n e g a t i v e
69
infinity,
formula
wh i c h
for
The h e a t
is
calculated
relative
avoids t h i s
c a p a ci t y equation
16,
valid
temperature
Choosi ng. A l Oj
g as t he
p r o b l e m.
Corundum)
Al Oj
g (c,
c o r u n d u m) ,
was o b t a i n e d t h r o u g h
least
s quar es
o f the t a b u l a t e d
equation
over
for
data of S t u l l
and P r o p h e t .
t h e r a nge 250 t o
2400 K.
dependence o f t h e vol ume o f A l Oj
state
for
first
cor undum i n
1983.
calculated
At t h i s
The
The
and f r o m t h e
vol umes measur ed by Engber g and Zehms.
Moe and Roy
32
g was
f r o m t h e a c c u r a t e NBS X - r a y d a t a ^
Howa I d ,
of
( C,
in Table
fitting
impossible.
al umi num o x i d e
Al umi num Oxi de
listed
is
88
the equation
t i me t h e y
included
a t e m p e r a t u r e dependence o f N i n t h e Mur naghan
logarithmic
4.0 f o r
equation
of
s t a t e . 21,22
cor undum g i v e s d i f f e r e n c e s
Their
o f as much as ± 4%
bet ween t h e measur ed and t h e c a l c u l a t e d
various
half
temperatures,
that
If
and a ( d K s / d T )
v a l u e s o f Ks at
of about
-11 whi ch .is
measur ed by Soga and A n d e r s o n .
the di screpanci es
indicates
a smal l
error
repeated
the c a l c u l a t i o n , w i t h
data.
of temperature
In o r d e r t o
check t h i s
coefficient
it
d i s c r e p a n c y we
o t h e r p a r a me t e r s
We r e c a l c u l a t e d
using
above ar e r e a l
the Murnaghan-Hi l debrand
of
additional
state.
in
described
equation
expansion
value of N =
constraints
and
t h e vol ume as a f u n c t i o n
t o f o r c e t he t hermal
to c o n tin u a lly
increase wi th
Ta b l e 16.. The e q u a t i o n s o f
and s e l e c t e d t h e r mo d y n a mi c
cor undum and l i q u i d .
s t a t e , he at c a p a c i t y e q u a t i o n s
p r o p e r t i e s o f al umi num o x i d e
AIO1 5 (LIQUID)
16.23
-5.879075E-6
1.036895E-10
-2.23345E-15
5.167876E-20
-1.11168E-24
1.482980E-29
0.40000000E-4
-8.81860E-10
2.799532E-14
-8.70313E-19
2.610551E-23
-6.68757E-28
9.853453E-33
4.499999E-10
-6.61392E-14
3.779063E-18
-1.69433E-22
6.561746E-27
-1.98662E-31
3.214079E-36
4.500196E-15
-3.31059E-18
3.412825E-22
-2.21229E-26
1.105066E-30
-3.93645E-35
6.971421 E-40
3.379152E-20
-1.24850E-22
2.324960E-26
-2.17263E-30
1.386229E-34
-.574006E-38
1.103936E-43
AIO1 5 (S,CORUNDUM)
0.130133E+02
-.431438E-05
0.492851 E-IO
-.790190E-15
0.336497E-19
-.209256E-23
0.507648E-28
0.275994E-04
-.627682E-09
0.137846E-13
-.775852E-18
0.145582E-21
-.175490E-25
0.802517E-30
0.397122E-08
-.123187E-12
0.257387E-17
0.872600E-22
0.119748E-25
-.729901E-29
0.567430E-33
-.186255E-11
0.302827E-16
-.129451E-20
0.327643E-24
-.174437E-28
-.638203E-32
0.648833E-36 .
C
D
0.117407E-14
-.288620E-19
0.139498E-23
-.146650E-27
-.426406E-31
' 0.143614E-34
O.OOOOOOE+OO
HEAT CAPACITY (Cp)a
A
B
E
AIO1 5 (LIQUID)
94.69605
.019816
-.123850E-04
AIO1 5(S,CORUNDUM)b
2.47207
.00482352
THEMODYNAMIC
Y1000
QUANTITIES
,545513 E-6
.165569 E-12
-.708193 E-9
JMOL'1
HlOOO"H298
JMOL'1
Hgga
J MOL'1
S298
v TOOO
J MOL"1 K'1 CMj
AIO1 5 (LIQUID)
50.16645
34333.9
-824498.8
69.82366
16.238
AIO1 5 (S,CORUNDUM)
51.11802
38982.328
-837846.
25.44894
13.0133
a THE CAPCITY EQUATION IS GIVEN BY (Cp) = A + B(T-IOOO) + C(T-IOOO)2 + D(T-IOOO) 3
+ E(T-IOOO)4. bAIO1 5(S,CORUDUM) HAS AN ADDITIONAL EINSTEIN TERM 7.5E(T/708).
71
increasing
temperature.
coefficient
included
versus t emper at ur e
the e x t e ns i ve
cor undum c i t e d
K
_
n
single
is
crystal
89
14.
We
measur ement s on
whi c h
included
N = 4 . 2 7 and an e x t e n s i v e
5,290.10
expansi on
shown i n F i g u r e
by Si mmons and Wang
1C = 250800,
o
A graph, o f . t he t h e r m a l
series
o f measur ement s on t h e t e m p e r a t u r e dependence o f K as
published
by T e f f t .
Wi t h
90
N = 4 . 2 7 and K2g 8 / 1 5 = 250800 we wer e a b l e t o
calculate
t h e t e m p e r a t u r e dependence o f Kg .
calculated K
values
the experi mental
H i ldebrand
line
the experi mental
are p l o t t e d
values.
is
still
data.
f 2% o f t h e e x p e r i m e n t a l
giving
a line
Ho wa l d,
with
et a I .
about
for
± 2% i n K
it
is
while
does n o t f i t
t h e Mur naghan-
s m a l l e r t ha n t h a t
However ,
this
values.
An o l d e r
present
line
indicated
is well
by
within
calculation
can be f o u n d
i n the t her mal
cor undum can i n t r o d u c e
in
expansi on
uncertainties
of
through the equation
Ct2 VKT-)]'
not c e r t a i n
values re por ted
Thus,
15 a l o n g w i t h
25
Ks = K [ C p/ ( C v Also,
The s l o p e o f
even more c u r v a t u r e
The u n c e r t a i n t i e s
coefficient
in Figure
The
for
( 94)
whether t he hi gh t emper at ur e
Ks ar e good t o b e t t e r
t h a n ± 2%.
t he Murnaghan- Hi l debr and equat i o n o f
t h e measur ed s l o p e s
(dKs / dT)
state
o f Soga. and
2 .4
CX= CK" 1 x E+5)
2.0
1.8
4-----------h
400
600
800
------1
1000
T (K)
F i g u r e 14. The t h e r m a l
t emper at ure.
expansi on c o e f f i c i e n t
o f al umi num o x i d e ver s us
1200
255000
245000
P (MPa)
235000
225000
-------1
-------------------------------1
-------------------------------1
-----------------------------T—
500
1000
1500
2000
T (K)
Fi gur e 15. The a d i a b a t i c b u l k modul us o f A l Oi #5 v e r s u s t e m p e r a t u r e . The
f i l l e d c i r c l e s ar e v a l u e s f r o m T e f f t ( 1 9 5 6 ) , t h e open c i r c l e s ar e our
c a l c u l a t e d v a l u e s and t h e s o l i d di amonds ar e t h e v a l u e s o f Soga and
Ander son ( 1 9 6 7 ) .
O
—
t
2500
74
A n d e r s o n 29 and T e f f t , 90 i t
experimental. errors
nevertheless
’
to
calculate
vol ume o f cor undum.
by l e a s t
is
squares
Al umi num Oxi de
Ther e i s
enthalpy of
of
fusion
is
the m e l t i n g
decided t o
p r e s s u r e dependence o f t h e
We t h e n
fit
these c a l c u l a t e d
The f u l l
o f Al O^
a t whi ch
g.
it
Thi s
melts,
the d i sc r e p a n c i es
capacity fo r
the
equilibrium
point,
is
substantially
use S h p i ! r a i n ' s
our
. 00296
he at
(T -
the
i s due m a i n l y t o t h e
2327 K.
Shpi T r a i n ' S9 ^
by
so t h e e n t h a l p y
b u t , 50 t o
larger.
150 K above
We have
capacity equation
( 95)
equation.
extrapolate
However ,
well
to
l ow
so we changed t h e e q u a t i o n t o g i v e
reasonable values
at these temperatures.
capacity
is
equation
given
in Tabl e
S h i p l r a i n ' s 91 h e a t o f f u s i o n
Kcal/mole.
values f o r
1000 K )
heat c a p a c i t y
Shpi T r a i n ' s e q u a t i o n does n o t
temperatures,
35 t e r m p o l y n o m i a l
liquid,
at
for
points-
about t he m e l t i n g p o i n t
s ma l l
Cp = 2 7 . 6 6
as a b a s i s
equation of
16.
a hi gh heat
fusion
logarithmic
a l a c k o f good e x p e r i m e n t a l
wor k e x p l a i n s
the
(Liquid)
high t emper at ur e
using
the
procedures.
g i ven i n T a b l e
within
present.
We have used t h e Mur naghan
state
falls
16.
o f A l O ^ g,
The h e a t
We ar e a l s o
AH = 12. 85
usi ng
75
Ther e
liquid.
Ta bl e
is
no e x p e r i m e n t a l
Thus,
16 i s
the equation
calculated
The vol ume o f Al Oj
t han t h e
of
state
for
from e s t i ma t e s
Al Oj
5(I)
of
of
g (c,
cor undum).
be 17 0, 0 00 MPa a t
state
in
o f Vq , Kq , a and N.
Kq and N were
1000 K and 5,
respectively.
Wi t h t h e s e e s t i m a t e s we wer e a b l e t o c a l c u l a t e
equation
g
g ( I ) was e s t i m a t e d t o be 25% l a r g e r
vol ume o f Al Oj
estimated to
d a t a on t h e vol ume o f ' Al O^
the
usi ng the Mur naghan-Hi l debrand equat ion
state.
The S t o i c h i o m e t r i c
Phases
Forsterite
The heat , c a p a c i t y e q u a t i o n
shown i n T a b l e
17 was c a l c u l a t e d
e n t h a l p y d a t a o f R. I .
included
to f i t
K . K . Ke I I y . 9
*^
extrapolates
the
Orr.
Thi s e q uat i o n
well
2100 K .
Forsterite
at
298.15 K i s
hydrofluoric
An E i n s t e i n
is
MggSi O^,
temperature
t e r m was
heat c a p a c i t y d a t a o f
valid
up t o t h e m e l t i n g
at
calculated
92
forstefite,
from the high
l ow t e m p e r a t u r e
Forsterite
is
for
up t o
2000 K and
temperature of
The e n t h a l p y o f f o r m a t i o n
-2168486. 75 JZmol ei
from the en t ha l p y of s o l u t i o n
a c i d , 94 and f r o m t h e e n t h a l p i e s
of
Thi s val ue
with
of formation
o f MgO and c r i s t o b a l i t e .
The t e m p e r a t u r e dependence o f t h e vol ume shown i n
Tabl e
17 i s
calculated
from the t a b ul a t e d
data of
76
Touloukian
vol ume
is
et
aI . ^
The p r e s s u r e
calculated
logarithmic
using
equation
of
t h e Mu r n a g h a n - Hi Tdebr and
state
128134 MPa as e x p e r i m e n t a l l y
Barsch.
dempendence o f t h e
w i t h 1N = 5 . 0 and K^gg ^ g
d e t e r m i n e d by Graham and
95
T a b l e 17. The e q u a t i o n o f s t a t e , h e a t c a p a c i t y e q u a t i o n
and s e l e c t e d t h e r m o d y n a m i c p r o p e r t i e s . o f f o r s t e r i t e
(Mg2 Si O4 ) .
Mg2 SiO4 (C, FORSTERITE)
44.900
-8.767975E-6
2.306197E-10
-7.39389E-15
2.494164E-19
-7.25419E-24
1.170027E-28
8.5512090E-9
-6.45672E-13 ■
4.053964E-17
-2.31366E-21
1.170763E-25
-4.39426E-30
8.138615E-35
0.38494257E-4
-2.026311E-9
9.769012E-14
-4.53815E-18
1.966847E-22
-6.67828E-27
1.170020E-31
1.432073E-13
-9.93583E-17
1.071385E-20
-8.45035E-25
" 5.312951 E-29
-2.27676E-33
0.4533533E-37
-6.64640E-16
1.968706E-20
7.874807E-25
-1.73786E-28
1.530971 E-32
-7.66702E-37
1.645590E-41
HEAT CAPACITY (Cp)a
B
A
.01588391
17.149773
THERMODYNAMIC
Y1000
PROPERTIES
Mg2SiO4 (C, FORSTERITE)
C
D
E
7.14404 E-6
-.324016 E-8
9.39570E+5
H298
J MOL’ 1
S298
’ V1QQO
J MOL’ 1 K’ 1 CM3
172.51588
109559.
-2168486.8 95.1900
43.790
+
O
?
O
O
O
"CO
+
J MOL"1
H1000 "H298
J MOL’ 1
a THE HEAT CAPACITY EQUATION IS GIVEN BY A + B(T-IOOO) + C(T-IOOO)2
E(1/T2 -10’®). b FORSTERITE ALSO HAS AN ADDITIONAL EINSTEIN TERM 10E(T/500).
Enstatite
( Magnesi um S i l i c a t e )
The s t o c h i o m e t r i c
crystal
structures
compound MgSi Og has t h r e e
corresponding
different
to C lin o e n s t a t it e ,
77
orthoenstatite
these t h r e e
and p r o t o e n s t a t i t e .
phases
is
t a k e n f r o m T o u l o u k i an,
T a y l o r ' s * ® t a b u l a t e d thermal
no n me t a l l i e
substances.
wer e t a k e n f r o m C l a r k ,
The vol ume da t a f o r
96
K i r b y and
expansi on data f o r
The b u l k mo d u l i
and v a l u e s
for
t h e s e phases
of N are e s t i m a t e s .
The h e a t c a p a c i t y e q u a t i o n s f o r
the e n s t a t i t e s
Tabl e
and K . K . K e l l y . ^ .
18 a r e f r o m K . K . K e l l y ^
equations
ar e good o v e r t h e r a n g e 298. 15 t o
Figure
16 shows t h e
various
enstatites
experimental
Engl and.
99
calculated
equilibria
u s i n g our e q u a t i o n s
points
ar e t h o s e o f Gr o v e r
of
listed
in
These
1800 K.
bet ween t he
state.The
QR
and Boyd and
'
Spi ne l . ( Ma gne si u m A l u m i nate)2
*
The he at
Ta b l e
values
19 i s
for
c a p a ci t y equation
from B u l l e t i n
Mg- Al
The t e m p e r a t u r e
given
in
Ta b l e
squares f i t t i n g
wor k o f R i g b y ,
vol ume
is
An d e r s o n ,
disorder
1452^
spinel
pressure
spinel,
MgAl gO^,
and t h e n a d j u s t i n g
as d e s c r i b e d by Howal d.,
et
in
these
aI .
25
dependence o f t h e vol ume o f MgAl gO^ i s
19.
T h i s e q u a t i o n was o b t a i n e d
of the values
et
a l . 100
from Schrei ber , ****
Schrieber
by l e a s t
r e p o r t e d by C l a r k
96
from t he
The p r e s s u r e dependence o f t h e
Chang and Bar sch* * * ^ and
and L i e b e r m a n .
Ky =200900 MPa and ( d K/ d P)
of
for
can be c a l c u l a t e d
=N =
at
-10 8
4.19.
The b u l k
Thus,
any t e m p e r a t u r e
modul us
is
t h e vol ume
and
using the Murnaghan-Hi ldebrand eq uat i on of
state.
78
Ta b l e 18. The e q u a t i o n s o f s t a t e , he at c a p a c i t y e q u a t i o n s
and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s . o f t h e t h r e e f or ms
o f Mg Si 0 g ; e n s t a t i t e , p r o t o e n s t a t i t e and o r t h o e n s t a t i t e
MgSiO3 (C1PROTO ENSTATITE)
32.997
2.9510326E-5
-.1034629E-4
-1.668996E-9
2.942106E-10
8.591046E-14
-9.89823E-15
-4.12356E-18
2.742496E-19
1.40521 IE-22
MgSiO3 (C, ENSTATITE)
-2.68003E-10
-8.50053E-14
9.850342E-18
-7.21290E-22
3.087455E-26
-4.12588E-12
1.994086E-1
-8.48925E-21
3.122188E-25
-7.90949E-30
3.813557E-15
-2.10916E-19
1.007556E-23
-4.53786E-28
1.486818E-32
32.1346
4.1944547E-5
-1.155055E-5
-2.778779E-9
3.870193E-10
1.688601E-13
-1.57082E-14
-9.80906E-18
5.162945E-22
6.561695E-19
-2.01031E-26
-2.23521 E-23
3.961003E-28
3.815435E-31
MgSiO3 (C.CLINOENSTATITE)
1.3826728E-8
-1.22745E-12
9.135715E-17
-6.24682E-21
3.714745E-25
-1.56209E-29
3.093022E-34
-3.09543E-14
-2.82446E-16
3.643220E-20
-3.47599E-24
2.574122E-28
-1.23274E-32
2.619277E-37
-1.00521 E-15
-1.78068E-19
2.064981 E-23
-1.99796E-27
1.539655E-31
-7.60931 E-36
1.648066E-40
32.0476
-1.104299E-5
3.351322E-10
-1.19992E-14
3.487301 E-19
5.0511436E-9
-4.60007E-13
3.381681 E-17
-2.13649E-21
8.695828E-26
-3.89573E-12
1.909745E-16
-7.36881 E-21
1.952870E-25
-1.57312E-30
8.830467E-21
2.925781 E-20
-2:90019E-24
1.928241E-28
-7.64917E-33
3.3448084E-5
-2.033051 E-9
1.122201 E-13
-5.75027E-18
2.056T86E-22
HEAT CAPACITY (Cp)
A
B
PROTOENSTATITE
.0168455
123.586
ORTHOENSTATITE
.0168455
123.586
CLINOENSTATITEa
.01195776
55.721425
THERMODYNAMIC
iC
E
D
-.164109E-4
.159434E-7
-.151109E7
-.164109E-4
.159434E-7
-.151109E7
-.9537309E-5
-.229344 E-8
b.397042 E-10
H298
S298
H1000'H298
Y1OOO
J/MOL K
J/MOL
J/MOL
J/MOL
PROPERTIES
-1548467.0
66.2475
117.1442
77235.
PROTOENSTATITE
-1548597.8
67.86
75126.1
116.55215
CLINOENSTATITE
-1547400.
66.7835
77235.
117.98970
PROTOENSTATITE
THE HEAT CAPACITIES ARE GIVEN BYTHE EQUATION (Cp)=A+B(T-1000)+C(T-1000)2
+D(T-1000)3+E(T'2-10'6). a CLINOENSTATITE HAS AN EINSTEIN TERM 8.5(1000/T)
bTHIS TERM IS E(T-IOOO)4
V1OOO
CM3
31.599
31.470
32.384
79
prot oenst at i t e
T (K)
or t hoenst at i t e
cl i noenst at i t e
P (MPa)
F i g u r e 16. The c a l c u l a t e d phase d i a g r a m f o r MgSi Og.
The e x p e r i m e n t a l p o i n t s shown ar e t h o s e o f Gr o v e r
1972, and Boyd and Engl and 1965.
80
T a b l e 19. The e q u a t i o n o f s t a t e , h e a t c a p a c i t y e q u a t i o n
and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f s p i n e l
( Mg Al 204 ).
MgAl2 O4 (C, SPINEL)
.404136E+02
-.535877E-05
.745188E-10
-.124816E-14
.224941E-19
-.389813E-24
.449630E-29
.419510E-08
-.164180E-12
.532159E-17
-.157824E-21
.436263E-26
-.102675E-30
.142237E-35
.285336E-04
-.793759E-09
.199552E-13
-.483529E-18
.113512E-22
-.236766E-27
.304801E-32
-.146062E-11
.260203E-16
-.197197E-21
-.856107E-26
.557514E-30
-.187940E-34
.309217E-39
-.457875E-17
.341202E-20
-,124942E-24
:270666E-29
-.236600E-34
-.770728E-39
.247142E-43
HEAT CAPACITY (Cp)*
THERMODRNAMIC
D
-.574449 E-4
3.52436 E-8
O
iO
O
.0370550
185.173
C
_<
B
A
JMOL'1
H1000"H298
J MOL'1
H298
J MOL'1
S298
J MOL'1 K"1 CM3
MgAI2O4 (C, SPINEL)
153.580
112079.
-2300553.
80.63002
O
O
O
QUANTITIES
40.4136
a THE HEAT CAPACITY IS GIVEN BY (Cp) =A + B(T-IOOO) + C(T-IOOO)2 + D(T-IOOO)3.
Cont our
Figure
17.
lines
for
Di f e r e n c e s
t h e vol ume o f Mg Al 2O4 ar e shown i n
from Figure
Roy ar e due t o t h e a d o p t i o n
equation
of
state.
Spinel
higher temperatures
2.
of
Howa I d ,
Moe and
o f a Murnaghan-Hi l debrand
u n der goe s d i s p r o p o r t i o n a t i o n
according to the r e ac t i o n
Mg Al 2O4 = MgO( c ) + 2A101 >5 ( c ) .
The c o n t o u r
spinel
lines
at
sho wi n g AV f o r
a r e shown i n F i g u r e
the pr es s u r e- t e mp e r a t u r e
18.
plane
( 9 6)
the d i s p r o p o r t i o n a t i o n
The. e q u i l i b r i u m
shows s u b s t a n t i a l
line
in
of
140000
120000
100000
P (MPa)
60000
40000
800
T(K)
1200
Fi g u e 17. C o n t o u r l i n e s f o r t he vol ume o f MgAl gO^,
o f t e m p e r a t u r e as r e p o r t e d by Howal d, e t a I .
1600
spinel
as a f u n c t i on
82
100000
75000
P (MPA)
50000
25000
1800
F i g u r e 18.
Mg Al 2O4 (C)
C o u n t o u r l i n e s sho wi n g AV f o r
= MgO(C) + Z Al O1 5 ( C) .
the r e a c t i o n
83
c u r v a t u r e , as shown i n
Figure
t o t h e AS o f t h e r e a c t i o n
disproportionation-is
entropy of disorder
by t h e
interchange
19.
shown
Thi s
is
due p r i m a r i l y
i n T a b l e 20.
The AS o f
l a r g e and n e g a t i v e due t o t h e
bet ween t h e Mg and Al
of these
i ons w i t h i n
cations,
caused
the s pi nel
crystal
structure.
■Cordierite
It
state
is
for
essential
cordierite,
the t e r n a r y
have a r e a s o n a b l e
Mg2A l ^ S i 5Oj g ,
phase d i a g r a m a t
to calcula te
the equation
t h e vol umes by l e a s t
temperatures:
able.to
to
calculate
of
400,
it
shows up i n
hi gh t e m p e r a t u r e s . In or der
state
s qu ar e s
T = 300,
since
equation of
for
cordierite
at the f o l l o w i n g
900,
t h e vol umes a t
1400 and 1500 K .
estimate
values
respectively,
for
listed
We wer e
these temper at ur es
t h e c o e f f i c i e n t o f t h e r m a l e x p a n s i o n da t a l i s t e d
g5
Memoi r 97.
In o r d e r t o c a l c u l a t e t h e p r e s s u r e
dependence o f t h e vol ume
we f i t
in
from
in
Ta bl e 21 we had t o
o f N and K o f 4 . 8 and 12 0, 000 MPa
since there
is
no p r e s s u r e d a t a a v a i l a b l e
cordierite.
The h e a t
capacity equation
is taken d i r e c t l y
along w i t h
f r om Robi e,
et
for
cordierite
aI .,
Bulletin
i n Ta b l e 21
1452^
t h e s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s .
♦
I 1000
P (MPa)
10000
9000
1600
T (K)
F i g u r e 19. E q u i l i b r i u m
in t h i s work.
line
for
t he s p i n e l
disproportionation
as c a l c u l a t e d
85
Tabl e 20.
AS and AV o f ( d i s p r o p o r t i o n
T
(K)
9000.
9500.
10 0 0 0 .
15000.
MgAl ^O^
AS
( J/ moIe K)
3 A-V
( c n r / mo I e )
P
( MPa)
5 77. 5 '
1097 .
1326.
1 502.
for
■-2.654
-2.620
-2.600
-2.580
-1.3997
-4.6693
-6.6834
-7.9860
’
T a b l e 21. The e q u a t i o n o f s t a t e , heat c a p a c i t y e q u a t i o n
and s e l e c t e d t h e r mo d y n a mi c p r o p e r t i e s o f c o r d i e r i t e
(Mg2A l 4S i O 1 8 )
Mg2Al4Si5O18 (C, CORDIERITE)
0.899349E-05
-.442406E-09
0.193252E-13
-.444596E-18
-.114674E-21
0.225578E-25
-.129101E-29
0.234078E+03
-.848087E-05
0.207740E-09
-.559773E-14
-.856411E-20
0.237592E-22
-.140682E-26
0.451668E-08
-.229776E-1.2
0.105119E-16
-.928989E-21.
0.261934E-24
-.422462E-28
0.233536E-32
-.285238E-11
0.132829E-15
-.553999E-20
0.374617E-24
-.248256E-28
-.429451E-32
0.537906E-36
-.230909E-14
0.109807E-18
-.120853E-23
-.106071E-26
0.170194E-30
-.923538E-35
0.000000E+00
D
E
THE HEAT CAPACITY (Cp)a
A
C
B
698.34
THERMODYNAMIC
.043339
-8.211200E+6
-5.0003E+3
QUANTITIES
Y1000
JMOL'1
H1000"H298
J MOL'1
H298
J MOL'1
V1000
S298
J MOL'1 K'1 CM3
Mg2 AI4 Si5 O 18
693.26
433303.
-9161524.
407.20
a THE HEAT CAPACITY EQUATION IS GIVEN BY Cp = A + B(T-1000) + C(1/T2 - 10"6) +
D(1/(T)1 /2-1/(1000)1/2)
234.078
86
THE BINARY SYSTEMS
In d e s c r i b i n g
equations
of
s u b s y s t e ms ,
inherent
state
and t o
binary
ternary
syst ems
for
the s i n g l e
all
be a b l e t o
s u b s y s t e ms .
phase d i a g r a m i n c l u d e s
it
is
n e c e s s a r y t o have
component
calculate
activities
The MgO- Si C^ - Al O^
the t h r e e
g ternary
b i n a r y s u b s y s t e ms :
MgO- Si O2 , MgO- Al O1 g and Si O2 - A l O 1 g . To c a l c u l a t e
activities
of these
coefficients
for
syst ems one must
the a c t i v i t y
in the
. The M a g n e s i a - S i l i c a
The f i r s t
syst em t h a t
MgO- Si O2 b i n a r y .
important
with
this
This b i n a r y
s y s t e m f o r wh i c h
a c c u r a t e data
system i s t he
the e n t h a l p i e s
If
we had a c c u r a t e
an excess
compositions.
heats
from a n a l y s i s
of fusion
of
discuss
However ,
the best
of these t h r ee
the
an e x t r e m e l y
amount o f
pr obl em
calorimetric
data
MgO, Mg2Si O^ and MgSi Og.
for
enthalpy fo r
is
one ma j o r
of d i r e c t
for
and t he
Binary
sy s t em i s
enthalpies
However ,
of fusion
lack
( a)
introduction.
a substantial
is a v a ila ble.
for
calculate
I shall
the
have R e d l i c h - K i s t e r
coefficient
ex c e s s e n t h a l p y He as d i s c u s s e d
in the
t h e s e compounds we c o u l d
the
liquid
published
a t t wo
values f o r
the
compounds have come m o s t l y
phase e q u i l i b r i a ,
and i t
is
possible th at
87
t he MgO-SiOg system i s as we l l
listed
known as, t he o t h e r
systems
i n Tabl e 22. 10^ l d 9
Ta b l e 22. P u b l i s h e d e s t i m a t e s o f t h e e n t h a l p y o f f u s i o n
v a r i o u s compounds i n t h e MgO-Si Og sy s t em.
T
(K)
3105
AHf
( ku/ moI e )
syst em
MgO
57.65 ± 8
MgO-CaO
104
6 0 . 9 7 ± 16
MgO-Si O2
a
77. 1
MgO-Si Og
105
77 . 4
MgO-ZrOg
32, 106
MgSi O3
107
48.8- ± 3
MgO-Si Og .
a
6 0 . 4 ±. 15
MgO-Si Og
b
61. 5
MgSi O3- Ca S i O3
106, 40
MgSi O3- Ti Og
32,1.08 .
75. 3 f
2156
a)
b)
reference
number
solid
95 . 4
1830
Mg2Si O4
T h i s wor k f i n a l
T h i s wor k f i r s t
of
21
58.6
Mg2Si O4 - FegSi O 4 109
71. 1 ± 21
Mg2Si O4- Ti Og
32, 108
9 2 . 8 8 ± 12
MgO-Si Og
a
118. 5 ± 26
MgO-Si Og
b
value
estimate
.
88
It
is
liquids
Ther e
from the
is
giving
heat s
one r e c e n t l y
The.heat
of
solution
published
capacities
a b o u t 48 k J / m o l e .
the enthalpy, of
o f MgO l e a d s t o
x = 0.5 f o r
silica
of glasses
value
of the
s h o u l d g i v e an e n t h a l p y o f
than t h i s ,
with
t o get e n t h a l p y of mi x i ng
an e n t h a l p y o f v i t r i f i c a t i o n
MgSi Og.
solid
possible
ne ar
liquid,
glass
somewhat
available
57.6 k J / mol e f o r
t he heat
larger
t h e he at o f
val ue
of fusion
- 20 kJ a t
The m i x i n g o f MgO and
s h o u l d be e x o t h e r m i c and v a r i o u s
when t h e model s
and
Co mb i n a t i o n o f t h i s
an excess e n t h a l p y o f m i x i n g o f
even t h o u g h
985 K.
system,* * ^
mode I s * * ^ ’ * ^ ’ ^ ag r ee r o u g h l y on t h e m a g n i t u d e .
true
the
o f 42 k J / m o l e f o r
fusion
t h e MgO-Si Og l i q u i d .
liquids
in t h i s
data f o r
solution
of
published
Th i s
is
g l a s s was not
i n T a b l e 23 wer e p r o p o s e d .
T a b l e " 23. Excess e n t h a l p i e s o f m i x i n g o f MgO and Si Og f o r
v a r i o u s model s a t a mol e f r a c t i o n o f 0 . 5 .
Mode I
He ( k J / m o l e )
Li n - P e l t o n
T h i s wor k
Michels
Hoch
-18.7
-26.6
-27.
-0.7
Figure
e n t h a l p y at
shallowest
20 shows f o u r
2123 K i n t h i s
mi ni mum and t h e
Toop- Sami s1 c a l c u l a t i o n
calculated
s y s t e m.
least
with
curves f o r . t h e
The c u r v e w i t h
curvature
is
excess
the
a
the enthal py given
by t h e
X Si O2
O
.2
F i g u r e 20. V a r i o u s c a l c u l a t e d
phase d i a g r a m.
.4
.6
excess e n t h a l p i e s
.8
for
t h e MgO-Si O^
I
90
number o f mol es o f 0 ” m u l t i p l i e d
equilibrium
slightly
of Lin
constant
positive
at
and P e l t o n , ^
ou r f i n a l
selected
calculated
The c u r v e t h a t
concentrations
curve.
curvature
curves
e n t h a l p y curves
It
to
the behavior
of mi xi ng.
shown i n F i g u r e
Our
20 ar e
p a r a me t e r s
i n F i g u r e 20 r e p r e s e n t
seriously
being consi der ed
have more
The e x t r e me b e h a v i o r s
eliminated
a modified
illustrate
for
f r o m what
is
t h e r ange o f
for
this
information
d e p t h o f t h e minumum and on t h e e x t e n t
be e a s i l y
is that
x < 2 / 3 and x > 2 / 3 .
is de sira ble to
region.
goes
t h e de e p e s t
f r o m t wo s e t s o f R e d T i c h - K i s t e f
The f o u r
Si Og
a t x = 1/ 3 i s
selected
enthalpy values
the separate regi ons
this
of
The c u r v e w i t h
more n e g a t i v e e n t h a l p i e s
selected
s y s t e m.
hi gh
calculation
expected f o r
final
of K = 0.103716.
and an
and t h e o t h e r m i d d l e c u r v e r e p r e s e n t s
mi ni mum and l a r g e s t
Lin-Pelton
by 2000 J / mo l e
on t h e
of the c u r v a t u r e
shown i n F i g u r e
known o f
in
20 can
t h e excess f r e e
energy.
The t e m p e r a t u r e o f 2123 K was s e l e c t e d
since t h i s
is
eutectic.
At t h i s
fraction
the temperature o f
near 0 . 7
MgO and Mg2Si O4 .
with
a hi gh
function
in e q u i l i b r i u m with
In o r d e r t o
F i g u r e 21
the M g O f o r s t e r i t e
temperature a l i q u i d
is
for
with
an MgO mol e
t h e t wo s o l i d s
show t h e f r e e
e n e r g y da t a
d e gr e e o f a c c u r a c y .we have chosen t o
plot
the
SiO 2
.2
O
.4
.6
.8
___I________ I________ I________ I________ I
- I 130 —
- I 140 —
Li
♦
♦
♦
8
corr
kJ mol
I ♦
F i g u r e 21. C o r r e c t e d Gi bbs f r e e e n e r g y
Si O2 f o r t h e syst em MgO- Si O2
( G / T - 3 8 2 . 14)
v e r s u s mol e f r a c t i o n
of
G - 3 8 2 . 1 4 x Mgp = - Y T / 1 0 0 0 t
wher e G i s
t h e Gi bbs f r e e
the e n t h a l p y ,
fraction
Y is
o f MgO.
energy
i n J / mo l e o f m e t a l ,
t h e Pl anck, f u n c t i o n
The v a l u e s
and Si Og a r e f r o m p r e v i o u s
Si n c e 2123 K i s
of the curve of the
0.69,
H29g - 3 8 2 . 1 4 x Mg0
for
the s o l i d s
discussions
the e u t e c t i c
liquid
must pass t h r o u g h
b e c a u s e , t h e y ar e a t
in t h i s
at t h i s
Its
enthalpy of
intercept
fusion
must be a t
o f MgO.
This
has a br o a d mi ni mum and i t
reasonable curve f o r
the
meet s t h e s e c o n d i t i o n s
final
curve.
at
for
a point
to
curve;
t h e MgSiOg
d e t e r m i n e d by t h e
indicates
that
sketch
t he curve
a
A smoot h c u r v e t h a t
come c l o s e t o m a t c h i n g our
shown as di amonds a r e c a l c u l a t e d
t wo s e t s o f R e d l i c h - K i s t e r
Enthalpy values
the t angent
temperature.
possible to
liquid.
will
The p o i n t s
from our f i n a l
is
MgO
thesis.
MgO and Mgg4.
T h i s t a n g e n t must pass bel ow t h e p o i n t s
solids
t h e mol e
a t a b o u t X^gO "
the p o i n t s
H is
enstatite,
temperature,
ph as e,
equilibrium
and x i s
( 97)
can be e s t i m a t e d f r o m f r e e
d i f f e r e n t temperatures
coefficients.
energy curves
through the Gi bbs-Helmholtz
equation
( d ( G/T) / d T ) p = -H/T2 .
However * t h i s
AHj-
TUS
for
yields
larger
.= 99000 ± 36000 f o r
protoenstatite.
u s a b l e e n t r o p y mo del .
( 98)
uncertainties.
forsterite
In o r d e r
to
For exampl e
and 93000
do b e t t e r
± 40000
we need a
93
As a f i r s t
considered
mol es o f
approximation
as an i d e a l
liquid
of mixing o f
t h e x ^ q = 0 . 7 can be
solution
MggSi O^.
1. 87 J / K .
o f 0. 1
Thi s y i e l d s
will
uncertainty.
be p o s i t i v e ,
bu t
less
in
u n l e s s t h e he at
of fusion
give
-24
-30
t han t he
state
introduces
a
5 . 0 8 J / mo l e K .
energy i s
fixed.
substantial
But, i t
These v a l u e s
is
in
combi ne t o
t h e excess e n t h a l p y a t
a c c u r a t e enough t o
in
value
o f MgO i s
± 3 k J/mole.
and l o w e s t c u r v e s
Si Og l i q u i d
t h e excess f r e e
± 8 k J / mo l e f o r
x MgO = 0 . 7 0 ,
entropy
N e v e r t h e l e s s , t h e ex c e s s e n t r o p y
The u n c e r t a i n t y
t h e r a nge o f
a positive
Co n v e r t i n g from a r e f e r e n c e
o f pu r e u n d i s s o c i a t e d MggSi O^ t o
substantial
mol e MgO i n 0. 30
Figure
eliminate
both t he h i g h e s t
20.
The E n t h a l p y o f F u s i o n o f Magnesi um Oxi de
-If the f r e e
in Figure
21,
e n e r g y c u r v e has a br oad mi ni mum as shown
then the
s h o u l d be r e a s o n a b l y
I iquidus
symmetri c,
MgO a t t h e 2123 K e u t e c t i c
and 0 . 7 0 .
Fl ood- Knapp
solution
However ,
model w i t h
Thus ,
f or.sterite ■
and t h e mol e f r a c t i o n
of
s h o u l d be bet ween x ^ q = 0. 69
Some d i s p r o p o r t i o n a t i o n
probably present.
adequate.
curve f o r
of the o r t h o s i l i c a t e
at these basic compositions
three
species
we can c o n s i d e r
the
s h o u l d be
liquid
o f MgO, MggSi O^ and Mg^Si gOy. .
enthalpy of fusion
for
disproportionation
of o r t h o s i l i c a t e
as an i d e a l
Wi thout
an
MgO t h e amount o f
and t h e e n t h a l p y ar e
is
a
94
uncertain.
But,
the r e a c t i o n
.ZMg2 Si O4 M ) = Mg3 S i 2O7 + MgO( I )
should
zero,
have a p o s i t i v e
and t h u s ,
z is the
AH,
a AS a p p r o x i m a t e l y e q u a l
an e q u i l i b r i u m
constant
Mg2 Si O4 p r e s e n t
r e a c t i on
Thus,
is
in the
liquid;
the e q u i l i b r i u m
If
then t her e
- x - 2z mol es o f
with
a total
constant
for
of
2x -
I
t h e above
.
K = ( 3x - 2■+ z) ( z ) / ( I - x - 2 z ) 2 .
Therefore,
to
l e s s t h a n one.
number o f mol es o f Mg3S i 2O7 p r e s e n t
ar e 3x - 2 + z mol es o f MgO and I
mo l e s .
( 9 9)
( 100)
when x ^ q = 0 . 7 and assumi ng t h a t
equilibrium
constant.is
greater
the
t h a n z er o and l e s s t han
one t h e n z can r a n g e f r o m 0 t o 0 . 0 8 as shown i n Ta bl e 24.
The a c t i v i t y
fraction
o f MgO l i q u i d
o f MgO p r e s e n t
i s d e t e r m i n e d f r o m t h e mol e
in the
liquid.
x MgO = aMgO = ( 3* ~ 2 + z ^
Thus,
2x - D
UOD
whi ch r e d u c e s t o
aMg 0 ( l i q u i d )
and a^ q ( l i q u i d )
= 0. 18421 + 2 . 6 3 1 6 z a t x ^ g , = 0 . 6 9 .
Therefore,
the a c t i v i t y
the e u t e c t i c
3105 K f o r
In
liquid
= 0.25 + 2.5z
o f MgO i s
a t 2123 K.
at x ^ g
bet ween 0 . 2 5 and 0. 4 5
Usi ng t h e m e l t i n g
MgO and t h e G i b b s - H e l m h o l t z
the a c t i v i t i e s
enthalpy of fusion
agrees w e l l
with
for
the
o f MgO a t
MgO o f
value
60.97
57. 650
point
in
of
equation
( K 2ZK1 ) = ( A H Z R ) MZ T 2 - . T 1 )
along w i t h
= 0.7
( 102)
2123 y i e l d s
an
± 16 k J Z m o l e .
Thi s
± 8 k JZmol e f r o m Howal d
95
and Chang.
*
C e r t a i n l y values
e x c l u d e d and t h e J a n a f v a l u e o f
barely
with
acceptable.
calculations
Thus,
above 90 k J / m o l e ar e
77.4
we f e e l
± 15 k J / m o l e
justified
based upon t h e 5 7 . 65
t
in
3 2
is
proceeding
8 k J / mo l e
value.
T a b l e 24. C a l c u l a t e d e q u i l i b r i u m c o n s t a n t s f o r r e a c t i o n :
ZMggSi O^ ( I ) = MggSi gOy + MgO( I )
a t x MgO = 0 . 7 , wher e z i s t h e number o f mol es o f MggSi gOy.
a (MgO)
. Z5
. Z7 5
. 30
. 3Z5
. 35
. 375
. 40
. 4Z5
. 45
■ . 475
z .
K99
0.0
.01
.OZ
. 03
. 04
. 05
. 06
. 07
. 08
. 09
0.0
. 0140
. 0355
. 0677
. 115
. 1875
. Z963
. 4648
. 7347
1. 1875
The E n t r o p y o f M i x i n g
Wi t h t h e p r o p e r t i e s
previously
eutectic
This
selected
o f MgO l i q u i d
v a l u e s 104 t h e excess G/ T f o r
can be c a l c u l a t e d
can be d i v i d e d
conf i r med at
into
to
be - 1 4 . 2 6 1 3 a t
x^
t he
the
q
= 0.70.
excess e n t r o p y and an excess
e n t h a l p y t e r m i n v a r i o u s ways.
Assumi ng AS = 0 f o r
the
reaction
SMg2Si O4 .+ Si O2 = ZMg3S i 2O7 ,
t h e F l o o d - K n a p p 2 model
ex c e s s
enthalpy to
calculates
( 103)
t he. excess e n t r o p y and
be 3 . 1 8 9 J / mo Te K and - Z3500 J / mo l e
96I
respectively.
The T o o p - S a mi s model
J / mo l e K and - 6015 J / m o l e f o r
enthalpy of mixing.
overestimates
model
3
mixing
for
of negative
Lin-Pelton
mi xi ng o f t he t h r e e types
i n t h e Toop- Sami s model *
the ent r opy of mi xi ng.
corrects
this
by u s i n g
centers
equation
seriously
The L i n - P e l t o n
s e p a r a t e t e r ms f o r t h e
and p o l y m e r i c
anions.
The
is
Se = R[ [ ( x - z ) I n ( ( x - z - ) / ( I
+ (I
11 . 42 8
t h e excess e n t r o p y and
The i d e a l
o f oxygen at om p r e s e n t
yields
- x)ln((l
- x)/(l
- z ))
- z)
.
+ (2 - 2x - z)
I n ( (2 - 2x - z ) ( I - z ) / ( (2 - 2x) ( I
+ [(2
- 2x ) ( I -x.) / ( I - z ) - 2x ) ( I
- x ) / ( I - z)
/((2.-
2x)(l
x)/(l
wher e x i s t h e mol e f r a c t i o n
The f i r s t
for
negative centers
S i Og n " ^ i o n and ( x - z)
present,
yielding
1- z t o t a l
S i - O - S i ( n $ i _s i ) e q ual
a much s i m p l e r
to
a t by u s i n g
we c a l l
a modified
t wo t e r ms
and O
the
account
wher e
The l a s t
(I
- x)
O ^
t wo t er ms ar e
number o f mol es o f
(2 - 2x) ( I - x ) / ( I - z ) .
equation f o r
arrived
1/ 2 t he
i s t h e number o f
mo l e s .
m i x i n g o f O™2 , O0 and 0 ” w i t h
- 2x - z ) )
( 104)
o f MgO, and z i s
present.
the mi xi ng of
- ■( 2
- z))]]
number o f 0™ i o n s
is the
(2 - 2x - z ) ]
In[((2
-
- x) ) )
Ho we v e r ,
t h e ex c es s e n t r o p y can be
N(Si-Si)
Lin-Pelton
= (2 - 2 x ) .
mode I w i t h
Thi s y i e l d s
t h e excess
what
97
e n t r o p y being
Se = - R[ ( x - z ) I n ( ( x - z ) / ( I - z ))
+ (I
- x)ln((l
- x)/(l
- z)
+ (2 - 2x - z ) l n ( (2 - 2x - z ) / ( 2
+ zln(z/(2
2x ) ) ;
wher e 2 - 2x - z = m o l e s . o f
p a r a m e t e r z can be e v a l u a t e d
function
with
- 2x).)
( 105)
O0 and z = ( 1 / 2 ) 0 " .
by m i n i m i z i n g
the f r e e energy
t h e excess e n t h a l p y p r o p o r t i o n a l
c h o i c e o f He = - 3 9 0 0 5 z g i v e s
G/ T = - 1 4 . 2 6 1
at
with
Se = 5 . 4 9 4 4 8 J / m o l e K and He = - 1 8 6 1 1 .
with
this
of fusion
paramet er
uncertainty
liquid
all
bring
forsterite
and
wi t h the
Lin-Pelton
G/T t o
model
is
and T =
3 . 9 8 0 5 J / mo l e
the e n t h a l py
- 1 4 . 2 6 1 3 J / m o l e K.
is
Se = 5 , ± 4
r e a s o n a b l e e s t i m a t e o f t h e ex c e s s e n t r o p y
near x^ q = 0 . 7 .
model
compositions
Redlich-Kister
considered,
enthalpies
protoenstatite
4 . 5 0 0 0 J / mo l e K i f
A s i m p l e model
Lin-Pelton
= 0.70
q
C o mp u t a t i o n s
The excess e n t r o p y a t x = 0 . 7
the o r i g i n a l
J/ mol e K i s a
the
x^
The
due t o t h e u n c e r t a i n t y o f ± 4 J / m o l e K i n t h e
increasing to
adjusted to
± 8000 f o r
± 8000 J / m o l e f o r
ex c e s s e n t r o p y .
2123 K w i t h
t o z.
1830 K and 2163 K y i e l d
o f AH^us =118500
AHf us = 60400
K,
at
The
like
can n o t
t h e one p a r a me t e r m o d i f i e d
d e s c r i b e t h e excess e n t h a l p y a t
and t e m p e r a t u r e s .
coefficients
can f i t
and so t h e y can f i t
However ,
a set of
each o f t h e model s
the d i f f e r e n c e s
bet ween
in
98
various
exact
model s and p r e s u ma b l y c o r r e c t
values.
negative
For exampl e
curvature
phase d i a g r a m ,
immiscibility
so t h a t
G/ I
the
ent r opy of mi xi ng
in t h i s
of
silica
Redlich-Kister
t wo humps f o r
increasing
end o f t he
temperature.
model s pr opo s ed f o r
have a t
a t y p e o f M shaped c u r v e .
22.
coefficients
t he
least
Sever al
Usi ng s e p a r a t e
for
the
nonideal
t h e ex c es s e n t r o p y g i v e s
any c u r v e o f t h i s
a hint
sets
of
of
p a r t s o f the
us t h e f l e x i b i l i t y
type.
Capacity
The d i s p r o p o r t i o n a t i o n
t e m p e r a t u r e dependence f o r
equilibria
give
the e n t h a l p y
in
n e i g h b o r h o o d o f t h e mi ni mum a t x ^ q = 0 . 7 .
o b v i o u s l y be r e f l e c t e d
capacity of mixing.
Lin-Pelton
composition
However ,
introduce
Iiqu i d - l i q u i d
syst em a l l
t h e s e ar e shown i n F i g u r e
The Heat
q u i t e easy t o
t h e amount
various
t h e model s t o
mi ni mum ne ar t h e o r t h o si I i c a t e c o m p o s i t i o n o f
x MgO = 2 / 3 f o r
to f i t
is
t o t he hi gh
decreases w i t h
Similarly,
of a local
for
it
any o f
model
to
i n t e r ms o f
We used t h e
calculate
Cpe f o r
o f MgO and Si O2 l i q u i d s
MgSi O3 ( I i q u i d )
substantially
of
gives
T h i s must
t h e excess heat
single
p a r a me t e r m o d i f i e d
part
of t he heat
x = .5 t o t he heat
capacity.
capacities
a heat c a p a c i t y f o r
Cp = 100 J / mo l e K , whi c h
l ess than t h a t
the
t h e t e m p e r a t u r e and
dependence o f t h i s
adding t h i s
a substantial
reported
is
by Wh i t e ,
115
X Si O2
F i g u r e 22. E n t r o p y o f m i x i n g v e r s u s t h e m o l e f r a c t i o n o f Si Og i n t he s y s t em
MgO- Si Og• C i r c l e s , open s q u a r e s and f i l l e d s q u a r e s a r e f r o m L i n and
F e l t o n , i d e a l m i x i n g and t h i s wor k r e s p e c t i v e l y .
100
Cp6 = 13 0. 7 J / m o l e K .
addi ng
d i s c r e p a n c y was c o r r e c t e d
25 J / m o l e K t o t h e f i r s t
t h e ex c e s s
he at
capacity.
v e r s u s mol e f r a c t i o n
demonst r at e t h a t
is
Thi s
shaped l i k e
calculated
it
at
is
Redlich-Kister
The r e s u l t i n g
2123 K i s
by
term f o r .
curve of
Cp
shown i n F i g u r e 23 t o
a reasonable
choice.
The Cpe c u r v e
t h e excess e n t r o p y c u r v e , s i n c e
f r o m t h e excess e n t r o p y f r o m t h e
it
is
Lin-Pelton
model.
Wi t h t h e h e a t c a p a c i t y
can be combi ned w i t h
heat
capacity of
vitrification
the glass
co nsistent with
the
It
for
is worth
noting
for
corresponds
1830 K .
fix
a heat o f f u s i o n
Thi s
is
just
that
l ower
value f o r
the e n t h a l p y curve
using
for
the
of
barely
a higher
Figure 20.
enthalpy of fusion
A he at
MgSi Og o f 4 8 . 8 ± 3 k J / m o l e f o r
determines
our f i n a l
in
d i s c r e p a n c y worse.
t o He = - 2 3 . 9 1
This
113
it
v a l u e o f 6 0 . 4 ± 15 k J / m o l e g i v e n
MgO o n l y makes t h i s
of fusion
to
MgSi Og.
We have a c c e p t e d t h e
and t o
phase cho s en,
t o c o n v e r t t h e measur ed heat o f
of Navrotsky^ ^
1830 f o r
calculations
liquid
t h e measur ement s o f Wh i t e
48. 8 k J/ mol e at
above.
of the
MgSi Og
± 4 k J / mo l e a t x = 2 / 3 and
our f i n a l
selected
values
of
He = - 2 5 . 1 9 9 ± 4 k J / m o l e and Se = - 2 . 2 9 5 J / m o l e a t x - 2 / 3
and 2123 K , and an e n t h a l p y o f
A Hf us = 9 2 . 8 8 ± 12 k J / m o l e a t
fusion for
2156 K .
forsterite
of
5.
4.
Cg ( J / m o l e )
3.
2.
101
I.
0.
F i g u r e 23 . Excess h e a t c a p a c i t y v e r s u s m o l e f r a c t i on o f Si O2 a t 2123 K f o r
MgO- Si O2 s y s t e m .
t he
102 ■
The Phase D i a g r a mWi t h
heat
choices
made f o r
capacity of mixing,
equilibria
The r e s u l t i n g
a l r e a d y been d i s p l a y e d
at
o f Kambayashi
Ex c e p t
for
solid
22.
coefficients
Also,
and
the c a l c u l a t e d
as shown i n F i g u r e s
the probl em a l r e a d y e v i d e n t
o f MgO by t h e i r
failure
t h e agr ee men t
is
at
fair.
24 and 25.
hi gh
to f i t
measur ement s ar e more r e l i a b l e
t h e known
The phase
than the
me a s u r e me n t s . .
Ther e a r e r e l i a b l e
by J . F . Ri eb l i n g ,
115
d e n s i t y measur ement s o f t h e
so t h a t
higher
d i a g r a ms a t
.1 and 2000 MPa ar e shown i n F i g u r e s
T a b l e 25 g i v e s
both t he a c i d i c
pressures.
The c a l c u l a t e d
the R e d l i c h - K i s t e r
and b a s i c
sides
phase
The t h e r mo d y n a mi c p r o p e r t i e s
coefficients
stoichiometric
phases.
of
for
Binary
o f the a l u m i n a - s i l i c a
b i n a r y have been mo d e l l e d by Howal d and E l i e z e r
equations
26 and
o f t h e phase d i a g r a m.
The A l u m i n a - S i l i c a
We ar e u s i n g t h e i r
liquid
t h e phase d i a g r a m can be
extended t o
27.
liquid
ex c e s s e n t r o p y v a l u e s have
in Figure
and Ka t o * ^
phase e q u i l i b r i a ,
effusion
the a c t i v i t y
of
2000 K can be compar ed w i t h t h e measur ement s
concentrations
equilibria
e n t h a l p y and t he
e v e r y measur ement
gives a value f o r
the e n t r o p y .
activities
t h e excess
state for
The R e d l i c h - K i s t e r
116
in
1978.
the various
coefficients
a (MgO)
103
^ ^
♦
—
♦
4.0 X (SiOZ) 6.0
F i g u r e 24. A c t i v i t i e s o f MgO. F i l l e d di amonds ar e c a l c u l a t e d f r om our
R e d l i c h - K i s t e r c o e f f i c i e n t s , open c i r c l e s ar e f r o m S. Kambayashi and
E. K a t o .
I .2
-T
0.8
-
•
a (SiOZ) 104
0.2
-
0.4 X (SiOZ) 0.6
F i g u r e 25. A c t i v i t i e s o f Si Og. Open c i r c l e s ar e c a l c u l a t e d f r o m our
R e d l i c h - K i s t e r c o e f f i c i e n t s , f i l l e d di amonds ar e f r o m S. Kambayashi and
E. K a t o .
2300
2100
T (K)
1900
105
1700
1500
0.0
0.2
0.4
0.6
0.8
X(SiO2)
F i g u r e 26. The c a l c u l a t e d
MgO- Si O2 s y s t e m.
phase d i a g r a m a t 0. 1 MPa f o r
the
1.0
2500
2300
T (K)
106
2100
1900
0.0
0.2
0.4
0.6
0.0
X ( Si O2)
F i g u r e 27. The C a l c u l a t e d
MgO- Si O2 s y s t em.
phase d i a g r a m a t
1000.
MPa f o r
the
1.0
T a b l e 25.
Redlich-Kister
A
coefficients
for
B
acidic
C
and b a s i c MgO- Si O2 D
E
F
-4.284510
-.328548
-5.0782545
-37.588266
-.7017948
: 87.678091
5.5773431
• -78.884975
6.9039075
22.121726
2.197768
2.197768
a) H
b) H
-106333.59
-16426.745
-106101.23
-818763.25
-21837.49
1850296.
171288.1
-1459206.2
238689.37
238689.37
100752.46
100752.46
a) Cp
b) Cp
11.870530
46.80464
15.174986
427.70785
14.355571
-918.31275
-20.8517
649.06314
-65.6362
-65.6362
-38.9678
-38.9678
c) dCp/dT
.002539050
.002055004
.0132324
-.023815
-.003681462
.0096971585
c ) d 2Cp/dT2
-4.225033E-7
-1.992977E-6
8.4855E-7
3.8824667E-6
-1.025983E-6
-1.55113E-6
c) V
-5.7079297
-9.6020113
-3.206035
22.275335
. 10.892596
.-15.26825
c ) dV/dT
-.0010398796
.0097664152
.0032609281
-.022656
-.011079
.0155296
C) dV/dP
.00010398796
-.9766415E-3
-.3260928E-3
.00226567
.0011079
-.00155296
a) a c id ic c o e ffic ie n ts
b) basic c o e ffic ie n ts
c) c o e ffic ie n ts fo r both the acid and the base
107
a) log Y
b) log Y
108
for
the
liquid
E l i e z e r 116 ar e
capacity
changed.
listed
of the
liquid
Ho we v e r , t h e heat
necessary to
for
1 1 7 '
made t h e s e changes
shown i n T a b l e
the
in
coefficients
change t h e
liquid.
1982.
for
(s)
log Y
enthalpy
(I)
l og Y
Chang , e t
enthalpy
(I)
l og Y
this
enthalpy
(S)
l og Y
enthalpy
aI .
■
for
aI.
wor k
.
the A
B
l ^ - Si Og
C
1. 330516
26778.
-0.212719
. 096297
3. 1 34 095
35974
-9.233145
99035.
10. 646443
1. 330266
26853. 8
-.211929
. 0959393
. 8634264
9000. 0
-.2353337
. 1233324
3. 1 34 095
35974
-9.233145
99035.
10. 646443
S i n c e t h e n we have f o u n d t h a t
large.
These v a l u e s f o r
A /
Re f .
l og y
Howal d , e t
enthalpy
much t o o
Howal d
26.
(I)
Redlich-Kister
Chang,
l og y and e n t h a l p y ar e
Ta b l e 26. R e d l i c h - K i s t e r c o e f f i c i e n t s
s o l i d and l i q u i d s y s t e ms .
Phase Q u a n t i t y
by Howal d and
has been r e f i n e d ; and t h e e q u a t i o n
coefficients
the R e d l i c h - K i s t e r
also
phases c a l c u l a t e d
i n T a b l e 26.
T h i s made i t
Redlich-Kister
and Roy
and s o l i d
c o e f f i cent f o r
the
the e n t h a l p y
liquid
in t h i s
binary
is
We have r e d u c e d t h e e n t h a l p y c o e f f i c i e n t
f r o m 2 6 8 5 3 . 8 1 18 t o
9000 J / m o l e and have r e c a l c u l a t e d t h e
l og Y c o e f f i c i e n t s
for
the
liquid
as shown i n T a b l e 26.
109
The f i r s t
Coefficients
step
for
the e q u i l i b r i u m
in
calculating
the a l u m i n a - s i l i c a
constants
A l O 1 g ( c , c or undum)
Si02(c,
for
liquid.
state
o f the r e a c t a n t s
to calculate
( 1 06)
= Si O2 ( I )
bet ween m u l l i t e
can e a s i l y
is
= Al O 1 ^ ( I )
cristobalite)
Thi s
binary
the r e ac t i on s
a t t h e t e m p e r a t u r e s wher e Dav i s
equilibrium
the' R e d l i c h - K i s t e r
( 107)
and P a s k *
measur ed t h e
and t h e a l u mi n a and s i l i c a
be done f r o m t h e e q u a t i o n s
and p r o d u c t s
in reactions
of
( 106)
and
(107).
The e q u i l i b r i u m
is
given
constant
for
the m e l t i n g
of m u l l i t e
by
('a S i 0 2 ( l ) ) 1/ 4 ( a A1 0 1 <5 ( l ) ) .
K
( 108)
= ------------------- ■
----------------------------mullite
The a c t i v i t y
of m u l l i t e
is
g i v e n by t h e e q u a t i o n
1/ 4,
mullite
so t h a t
( 109)
( aA l O 1 i 5 ( s ) ) ’
( a Si O2 ( S) )
the e q u i l i b r i u m expression
becomes
( a S i 0 2 ( l ) ) 1/ 4 ( a A10K 5 ( l ) )
K
( HO)
= ------------------------------------------------( aS i 0 2 ( s ) ) 1 / 4 ( a A10.l i 5 ( s)
The a c t i v i t i e s
of the s o l i d s
RedI i c h - K i s t e r
coefficients
i n T a b l e 26.
)
can be c a l c u l a t e d
for
The e q u i l i b r i u m
l og y f o r
constant,
the
from the
solid
Kg q , can be
phase
1 10
calculated
from the e q u i l i b r i u m
o f Si O2 and A l O 1 ^ .
constants
The e q u a t i o n
for
the melting
is
(
N o r m a l l y once t h i s
fractions
allowing
of the
for
complicates
for
liquid
can be c a l c u l a t e d .
the c a l c u l a t i o n
substantially.
equilibria
The t wo e q u i l i b r i u m
reactions
( 1 06)
concentrations,
fraction
o f the
solid
fraction
of the
liquid
al umi na,
activities
a7 ( c ),
a7 ( I ),
subscripts
However ,
alumina;
phase
We must a l l o w
bet ween t h e s o l i d
These can be chosen as t h e m e l t i n g
and cor undum as shown f o r
four
and mol e
n o n - s t o i c h i o m e t r y of the m u l l i t e
t wo s e p a r a t e
phase.
i s done t h e a c t i v i t i e s
111)
and l i q u i d
of c r i s t o b a l i t e
and ( 1 0 7 ) .
t h e mol e
X g ( c ) ,
and Xg ( I ) ,
t h e mol e
ar e t h o s e wh i c h g i v e t he
a 6 ( c ) and a 6 ( I ) ; wher e t h e
6 and 7 r e p r e s e n t A l O 1 5 and Si O2 r e s p e c t i v e l y .
These a c t i v i t i e s
must
satisfy
t h e t wo e q u i l i b r i u m
constant
expressions
Xg(l)/Xg(c)
= Kg
X 7( I ) Z X 7(C)
=
(
112)
and
(113.)
Ky
wher e Kg and K7 ar e t h e e q u i l i b r i u m
reactions
(112)
for
and ( 1 1 3 ) .
The c o n c e n t r a t i o n s
a selected
constants
temperature
and a c t i v i t i e s
for
e q u i l i b r i u m at
and p r e s s u r e can be o b t a i n e d by
successive approximations
by any o f a v a r i e t y
of
I
i
Ill
mathemati cal
energy.
procedures
Conv er genc e
for
the m i n i m i z at i o n
in these
no t
current
c o mp u t e r pr ogr am uses a f o r m o f t h e Mur t agh and
efficient
constrained minimization
for
calculations
solutions.
a
probl em f o r
119
non i d e a l
is
trivial
Sargent
highly
calculations
of the f r e e
whi ch
is
o f mol e f r a c t i o n s
from a c t i v i t i e s
c o n v e r g e n c e pr ogr am i n c o r p o r a t i n g
features
to
o f Wh i t e
I ?n
for
t h e t wo
Eventually a
some o f t he
and Mu r t a g h and S a r g e n t
119
will
need
be w r i t t e n .
Once c o n v e r g e n c e
the
n o t as
t h e t wo component syst ems as t h e a l t e r n a t e
phases as used by Howal d and E l i e z e r . * ^
better
Our
liquid
ar e c a l c u l a t e d ,
the a c t i v i t y
silica
has o c c u r r e d and t h e a c t i v i t i e s
coefficient
can be c a l c u l a t e d
t h e change i n t h e
for
l o g a r i t h m of '
b o t h t h e a l u mi n a
and t he
from the equation
41°g r = U newZao l d )
( 114)
wher e a
i s t h e new a c t i v i t y
activity.
These changes
Redlich-Kister
giving
in
coefficients
t h e change i n
These c o e f f i c i e n t s
of
and ao ^
l o g Y can be f i t
by a l e a s t
the R e d l i c h - K i s t e r
can t h e n
to
squares c a l c u l a t i o n
coefficients.
be added on t o t h e o l d
coefficients
for
l og Y t o g i v e
coefficients
for
the
liquid
is the old
t h e new R e d l i c h - K i s t e r
shown i n T a b l e 26.
112
THE TERNARY SYSTEM:
MAGNESI A- SI LI CA-ALUMI NA
Wi t h t h e R e d l i c h - K i s t e r
component
coefficients
s y st ems and t h e e q u a t i o n s o f
compounds p r e s e n t
in t h i s
s y s t e m,
for
states
t h e two
for
we can now pr oc ee d t o
d e s c r i b e t h e t h r e e component syst em Mg O- Si Og- Al ^
first
step
Kister
in t h i s
coefficients
o f met hods
for
interpolation
it.
but
121
and T o o p .
calculating
ternary
the Redl i c h-
"power
because as x ^ a p p r o a c h e s one Xg +
Toop
122
x ^/ ( Xg + Xg)
a l s o d e v e l o p e d a met hod o f
h i s met hod has one s e t o f
t e r ms
interpolation
Redlich-Kister
and i s e a s i l y
extended R e d l i c h - K i s t e r
in the
i s t h e Toop i n t e r p o l a t i o n
i n the K o hl e r form d r o p p e d .
Thi s
in c a l c u l a t i n g
di agr ams,
122
o f t h e f o r m x ^/ ( Xg + Xg ) i n
we have a m o d i f i e d Toop or a T o o p - Mu g g i anu
useful
the Redl i ch-
t h e most commonl y used
Thus, t he term
o f t e r ms
The
Ther e a r e a v a r i e t y
h o we v e r ,
What we ar e u s i n g
i n t e r p o l a t i o n . 123
g.
c a n n o t be e x p r e s s e d as a f i n i t e
zero.
interpolation,
the set
the t e r n a r y .
has t e r ms
infinity.
Kohler form.
to calculate
procedure f o r
i n mol e f r a c t i o n ,
Xg a p p r o a c h e s
with
17 I
coefficients
approaches
is
met hods ar e by K o h l e r
These t e r ms
series
for
doing t h i s ;
The K o h l e r
Kister
process
t he
notation.
■
met hod i s
very
coefficients
expressed
Thus,
for
i n t e r ms o f
The To op- Muggi anu
is
113
exactly
a loop
interpolation
on-e b i n a r y edge
is
The m o d i f i e d
ternary
binary
ideal
loop
or
if
the s o l u t i o n
parallel
interpolation
RedTich-Kister c o e f f ic ie n t s
RedTich-Kister
these c o e f f i c i e n t s
Redlich-Kister
coefficients
so t h a t
pseudobinary is
a line
v e r t e x , holding
the r a t i o
straight
for
through
calculates
solution.
the
from t he a p p r o p r i a t e
and a l s o c a l c u l a t e s
through
the p s e u d o b i n a r i e s .
A,
t h e t e r n a r y t h r o u g h one
o f t wo o f t h e component s
v a r y i n g the t h i r d .
line
an i d e a l
they give well-behaved
coefficients
constant while
to
f or med a l o n g
Thus, i t
t h e apex o f one o f
component s and t h r o u g h t h e b a s e l i n e
f or ms
the
a
pur e
connecting the ot her
t wo c o m p o n e n t s .
For e x a m p l e , assume we want t o
Redlich-Kister
|w,y,z>.
If
can e a s i l y
coefficients' for
the b i n a r y
calculate
the pseudobi nar y
corresponds to
| w, y>
the
is
the R e d l i c h - K i s t e r
|w2 y , z > .
a straight
■
l",Z>
=
for
ideal
through
the b i n a r i e s
edge.
The R e d l i c h - K i s t e r
■
coefficients
can be c a l c u l a t e d
|w2 y , z>
pu r e z and t he
| y , z > = - | A y z , By z , Cy z , 0y z > ■
| w2 y , z >
al ong
ar e
I a W Z - b W Z - c W Z - 1V
The R e d l i c h - K i s t e r
t h e n we
coefficients
The p s e u d o b i n a r y
line
the
imaginary t e r n a r y
close to
p o i n t W2 y a l o n g t h e w , y b i n a r y
coefficients
calculate
lll5 )
( 1 16)
along the pseudobi nary
from the eq uat i o n
114
j w2 y , z >
= | 2/ 3 w, z> + 1 / 3 | y , z >
( 117)
wh i c h . b e c o me s
I * , y,z>
= I( 2 / 3 ) +
(l/3)Ayz,(2/3)Bwz.+ (1/3)8^;,
( 2 / 3 ) Cw; + ( l / 3 ) C y z ,
The m o d i f i e d
loop
component t e r ms
give t h i s
interpolation
as l i s t e d
behavior
pseudobinaries.
Red I i c h - K i s t e r
Thi s
calculates
i n Tabl es
i n t h e t e r ms
. . .>
( 118)
the t hr ee
I and 3 so t h a t
calculated
for
i s done by m u l t i p l y i n g
coefficients
for
m a t r i x MTOOP shown in. Ta b l e 27.
the
the
t h e t wo b i n a r i e s
This
they
equation
* '
by t h e
is
By w ’ Cy w ’
( 119)
Thi s m a t r i x
seventh
is
pr ogr ammed i n t o
power t e r ms
i n mol e f r a c t i o n ,
n i n e component
s y s t e ms .
the
for
subscripts
pseudobinaries
■e q u a t i o n
| N^
Redlich-Kister
and i t
up t o
h a n d l e s up t o
The c o mp u t e r can e a s i l y
t h e t h r e e component t e r ms t o
along
=
t h e c o mp u t e r f o r
another ax i s .
Thi s
is
and CMM i s
the
gi ve the
done by t h e
|CMM | I N g S l ^ ' wher e N i s t h e
coefficients
rotate
vector of
15 by 15 m a t r i x
shown i n T a b l e 28.
In the
s y s t em Mg O- S i C ^ - A l O 1 5 t h e component s
Al O 1 g b o t h a c t
as a c i d s .
Thi s
us t o
Red I i c h - K i s t e r
coefficients
pseudobinaries
I A l O 1 g x Si Og y , Mg 0 > w i t h
interpolation.
for
l eads
Wi t h t h e m a t r i x
Si Og and
calculate
the
t he t e r n a r y al ong the
a m o d i f i e d Toop
CMM we can r o t a t e
the
115
T a b l e 27. The m a t r i x MTOOP f o r
Red I i c h - K i s t e r c o e f f i c i e n t s .
calculating
ternary
I
-2
3
-4
5
I.
-2
3
-4
5
0
7/2
- 21/2
21
-35
0
7/2
- 21/2
21
-35
0
1/2
- 3/ 2
3
-5
0
- 1/2
3/2
-3
5
0
0
37/4
-37
185/2
0
0
37/4
-37
185/2
0
0
10/4
-10
25
0
0
-10/4
10
-25
0
0
1/4
-I
5/2
0
0
1/4
-I
5/2
0
0
0
175/8
-875/8
0
0
0
175/8
-875/8
0
0
0
67/8
-335/8
0
0
0
-67/8
335/8
0
0
0
13/8
-65/8
0
0
0
13/8
-65/8
0
0
0
1/8
-5/ 8
0
0
0
- 1/8
5/8
0
0
0
0
781/16
0
0
0
0
781/16
0
0'
0
0
376/16
0
0
0
0
-376/16
0
0
0
0
106/16
0
0
0
0
106/16
0
0
0
0
16/16
0
0
0
0
-16/16
0
0
0
0
1/16
0
0
0
0
1/16
the
subscripts
that
subscripts
want .
t o get
For t h i s
coefficients
used t o
any o r d e r
ternary
that
calculate
it
for
is the
:
we
I 6 , 7 , 4> t e r n a r y
ar e used by t h e c o mp u t e r .
They can be
t h e ex c e s s e n t h a l p y ( He ) a l o n g t h e
14 - 6 , 7> p s e u d o b i n a r y ,
wher e 4,
6 and 7 r e p r e s e n t
MgO, A l O l i 5
T a b l e 28.
vector
The t r a n s f o r m a t i o n m a t r i x ,
N^23 t o t h e v e c t o r
I
I
-I
I
0
-d /2 )
(3/2)
-I
0
-(1/2)
-(1/2)
-I
-I
CNN, f r o m t h e R e d l i c h - K i s t e r
Ng1 2 -
I
I
-I
I
-I
I
2
-3
-(3/2)
(5/2)
-(7/2)
(9/2)
-2
3
0
I
-(3/2)
(1/2)
(1/2) -(3/2)
-2
I
O
-I
2
(9/4)
(3/4)
-(7/4)
(15/4) -(27/4)
-3
(11/2)
-9
(27/2)
-3
I
3
-9
-0 /2 )
O
(3/2)
O
0 -
(1/4)
-(3/4)
0
0
0
(1/2)
(1/2) -(3/2)
(3/2)
-(3/2)
-0 /2 )
(9/2)
O
0
0
(1/4)
(1/4)
(1/4)
(3/4)
(1/4)
-0 /4 )
-(3/4)
0
0
0
0
0
0
-(1/8)
(3/8)
-(9/8)
O
0
0
O
0
0
-(3/8)
(5/8)
O
0
O
O
0
0
-(3/8)
(1/8)
(5/8)
(9/8)
O
O
P
O
0
0
-0 /8 )
-0 /8 )
-0 /8 )
-0 /8 )
0
0
0
0
0
0
O
O
O
O
0
0
0
0
0
O
O
O
O
0
0
0
0
0
0
O
O
O
0
0
0
0
O
O
0
0
0
0
0
0
O
(3/2)
3
-4
-I
I
-6
5
(3/2)
O
(27/8)
-0 /2 )
(5/4)
-3
-(27/2)
-(27/2)
-(3/8) -(27/8)
-(3/2)
(9/4)
-2
-(9/4)
(27/4)
-(3/2)
(3/4)
I
-(3/4)
-(9/2)
O
0 /4 )
0 /2 )
(1/16) -(3/16)
(9/16)
(27/16)
(81/16)
O
0 /4 ) -0 /2 )
(3/4)
O
-(27/4)
O
O .
(3/8) -(3/8)
-0 /8 )
(9/8)
(27/8)
O
O
O
(1/4)
-0 /4 )
-0 /2 )
-(3/4)
O
O
O
(1/16) (1/16)
- 0 / 2 ) -(1/4)
O
(1/16)
0/16)
(1/16)
116
0
I
-I
117 ■
and Si Og r e s p e c t i v e l y .
coefficients
for
interpolation
selected
ar e
The c a l c u l a t e d
Redlich-Kister
excess e n t h a l p y f r o m t h e l o o p
shown
i n T a b l e 29 a l o n g w i t h
the f i n a l
coefficients.
Ta b l e 29. C a l c u l a t e d R e d l i c h - K i s t e r c o e f f i c i e n t s f o r
ex c es s e n t h a l p y f r o m t h e To op- Muggi anu i n t e r p o l a t i o n
w i t h our f i n a l s e l e c t e d val ues f o r t he
Mg O- Si Og - Al O1 5 t e r n a r y a t 1800 K .
C oefficient
■
• H6
Toop- Muggi anu
-He
Final
Val ues
Ba
61 840. 73
58 082 . 73
Ca
36 477. 38
-2173532.6
Cb
-77664.28
79 358 3. 7 2
D3
-27662.38
1590207. 6
Db
50 574. 75
-710619.25
DC
-58146.47
5 3 5 63 6. 5 3
Ea
104838. 4
10 4838. 4
Eb
-346939.8
-346939.8
EC
33 9367. 6
33 9 3 6 7 . 6
Ed
15143. 46
15143. 46
Fa
-72416.5
-72416.5 .
Fb
36 2082. 9
36 2082. 9
Fc
-724165.9
-724165.9
Fd
72 4165. 9
72 416 5. 9
Fe
-362082.9
-362082.9
al ong
118
We c o r r e c t e d
in the f o l l o w i n g
the t e r n a r y
manner .
Redlich-Kister
First,
we needed t o d e t e r m i n e
t h e ma g n i t u d e o f t h e change needed t o
Redlich-Kister
coefficients.
pseudobinary through
XjyigQ = 0 . 5 .
is
straight
correct
We l o o k e d a t
pur e Al O^
along the a c i d i c
Thus,
if
side to
a first
t h e ex c e s s e n t h a l p y a t
X4 = 0 . 5 and X7 = 0 . 5 a l o n g t h e
then at
X4 = 1/ 3 t h e excess e n t h a l p y
16 , 4> b i n a r y .
We can dr aw a s t r a i g h t
X g 9X45X7 = 0 , 1 / 3 , 2 / 3 ;
c e n t e r o f the graph)
at
Xg
the
^ and t h e MgO-Si Og edge a t
Xg = 0,
mullite
t he
The excess e n t h a l p y a l o n g t h e MgO-Si Og b i n a r y
approximation.
points
coefficients
composition
is
the p o i n t
| 6, 4> b i n a r y i s
2/3 z al ong the
line
through
and X g 5X49X7 = 2 / 3 , 1 / 3 , 0 .
9X 4 9 X 7 = 1 / 3 , 1 / 3 , 1 / 3
t he
X g 9X4 sX7 = i / 3 , 1 / 3 , 1 / 3
t h e excess e n t h a l p y
is
If
(the
at the
Y, He = Y, t h e n
t h e excess e n t h a l p y s h o u l d be
He = ( 1 / 2 ) Y '+ ( I / 3 ) Z + A.
The A c o n t r i b u t i o n
compar ed t o
Si Og.
( 120)
t o t h e ex c e s s e n t h a l p y
Y and Z because
is
s ma l l
it
is
t h e m i x i n g o f A l O^ g and
Thus t h e A c o n t r i b u t i o n
is
negligible.
Al s o a l o n g
passes t h r o u g h
the c e n t e r ,
the pseudobinary
|6,7-4>,
wh i c h
t h e ex c e s s e n t h a l p y a t t h e p o i n t
s h o u l d be a p p r o x i m a t e l y
We r e p e a t e d t h i s
points
z,
|6,7,4> =
11/ 3 , 1/ 3 , 1/ 3>
(2/3)Y.
calculation
i n t he t e r n a r y , di agr am.
at
seven d i f f e r e n t
These p o i n t s
ar e
listed
in
119
T a b l e 30,
the l oop
along w i t h
t h e excess e n t h a l p i e s
interpolation's
the f i n a l
values
coefficients
from the f i n a l
chosen f o r
Redlich-Kister
Redlich-Kister
this
coefficients
and t h e n
the R e d l i c h - K i s t e r
interpolation.
symmetri cal
ternary
ternary.
we f i t
a r e used t o
is
H o we v e r , t h e y do n o t
serious
order
p r o b l e ms w i t h
ternary
t e r ms w i t h
within
that
the
for
ternary
Figures
six
Ba , Ca , C*3, Da , D*3
MgO-Si Og w i t h
E and F
ternary.
One a v o i d s
by no t t r y i n g
to
fit
hi gh
data a v a i l a b l e .
coefficients
for
by a m o d i f i e d
l og
Y i n t he
Toop i n t e r p o l a t i o n .
f r o m t h e known e q u i l i b r i a
phase d i a g r a m u s i n g a met hod s i m i l a r
coefficients
di agr am.
t h e phase f i e l d s
in
the f i r s t
t h e excess e n t h a l p y c o r r e c t i o n s .
Redlich-Kister
this
limited
l og Y v a l u e s
the t e r n a r y
f r o m t h e Toop
i n c l u d e t hem i n t h e
ternaries
wer e c a l c u l a t e d
We c o r r e c t e d
like
on t o
t h e seven s e l e c t e d
have t o be a d j u s t e d .
The R e d l i c h - K i s t e r
ternary
calculated
coefficients:
necessary to
i n t he
Redlich-Kister
adjust
When one has b i n a r i e s
it
t h e changes
The changes d e s i r e d a t
points
and
To change t he
added t h e s e c o e f f i c i e n t s
coefficients
Redlich-Ki star
and Dc .
terms,
just
coefficients
f r om
set of R e d l i c h - K i s t e r
excess e n t h a l p y t h a t we want ed t o
coefficients
calculated
for
28 and 29,
ar e
listed
The c a l c u l a t e d
The l og
in Tabl e
contour
Y
31 f o r
lines
and
t h e t e r n a r y phase d i a g r a m a r e shown
respectively.
to
T a b l e 30. C a l c u l a t e d excess e n t h a l p i e s a t 1800 K u s i n g t he
R e d l i c h - K i s t e r c o e f f i c i e n t s f r o m t h e Toop- Muggi anu
i n t e r p o l a t i o n and ou r f i n a l s e l e c t e d v a l u e s f o r t he
MgO- Si Og- Al O^ g t e r n a r y
He
Mo I e f r a c t i o n
Toop- Muggi anu
MgO
Si O2
.5
. 25
. 25
-13451.336
-22124.990
■ .4
.4
.2
-7857.474
-17292.748
.4
, .2
.4
-13019.940
-20210.219
■ A10l .5
Final
val ues
1/ 3
1/ 3
1/ 3
-8193.079
-17319.420
. 25
.5
. 25
-4158.164
-10178.852
. 25
. 25
.5
-7260.797
-11241.445
.2
.4.
.4
-4042.353
-52789.555
121
T a b l e 31. R e d l i c h - K i s t e r t e r ms t h r o u g h F f r o m o u r comput er
f i l e TI 75 f o r t h e t e r n a r y s y s t em A l O 1 5 - Si Og-MgO.
■6 7 4
Ba
Ca
.
l og
Y
H
Cp '
16. 274239
56848 1. 3 3
-6.637291
-56.611459
-2158662.5
-15.02179
Cb
18. 364036
764703.
23. 56041
Da
39 . 58 694 2
1558140.
57 . 7 5 1 0 4
■ -23.388194
-645556.5
-121.9810
16. 585357
50 452 9. 5
Ea
1.81443'
164321. 0
-90.23578
Eb
-5.075334
-556531.3
321. 9720
EC
2. 160231
■ 58 2914. 3
-385.5388
Ed
5. 8 3 1 2 0 7
-52767.02
127. 1332
Fa
-2.1809064
-100752.4
38 . 96 796
50 376 2. 4
-194.8391
-100752.5
38 9 . 6 7 8 0
Db
. Dc
Fb
.
'
10. 9065889
. Fc
-21.814092
Fd
2 2 . 32 109 3
1007525
Fe
-10.9068664
-503762.4
58. 41545.
-389.6780
194. 8390
122
F i g u r e 28. The c a l c u l a t e d c o n t o u r
MgO- Si Og- Al O^ g t e r n a r y s y s t em.
lines
for
the
123
SiO2
F i g u r e 29. The c a l c u l a t e d phase f i e l d s
s y s t em
MgO-Si Op-Al Oi , 5 .
a i s the c r i s t o o a l i t e f i e l d
b is the c o r d e r i t e f i e l d
c i s the e n s t a t i t e f i e l d
d is the m u l l i t e f i e l d
e i s the f o r s t e r i t e f i e l d
f is the s pi nel f i e l d
g i s t h e cor undum f i e l d .
for
the t e r n a r y
124
THE FeO- FeO1 g - S i O ^ - A l j
Although
it
not
i s wort h w h i l e
central
5-CaO SYSTEM
t o t h e mai n body o f t h i s
decribing
t h e wor k t h a t
we d i d on t he
syst em FeO- FeO1 ^ - S i O g - A l O 1 ^-CaO sy s t em.
It
n e c e s s a r y t o d e s c r i b e t h e mai n component s o f
and t h e
stoichiometric
compounds o f
A l O 1 g and many o t h e r
involved.
must a l s o
However ,
oxides
t h e syst em
We have
compounds
and c a l c i u m o x i d e s
be t h e r m o d y n a m i c a l l y d e s c r i b e d a l o n g ,
stoichiometric
compounds:
FegSi O^ and a n o r t h i t e ,
compounds e x i s t
outside
iron
in t h i s
t h e ar ea
hercynite,
CaAl gSi gOg.
FeAl gO^ ;
Several
phase d i a g r a m,
i n wh i c h we ar e
agai n
f r o m Si Og and
of the s t o i c h i o m e t r i c
the
is
interest.
a l r e a d y o b t a i n e d t h e r mo d y n a mi c p r o p e r t i e s
work,
but
wi t h the
fayalite,
other
t h e y ar e
interested.
The FeO- FeO1 g Syst em
The f i r s t
component
for
sy s t em was o b t a i n i n g
wustite,
Fe0(c).
stoichiometric
Although,
necessary,
this
p r o b l e m we had t o
crystalline
one can g e t
it
Th i s
is
is
solve
in the
t h e r mo d y n a mi c p r o p e r t i e s
a probl em i n
that
FeO does not e x i s t
close to
never q u i t e
five . .
pur e
i n nat ur e. .
the c o n c e n t r a t i o n s
reached..
p r o b l e m we had t o c a l c u l a t e
In o r d e r
Redlich-Kister
to solve
125
coefficients
extrapolate
for
t h e FeO- FeO1 5 s yst em and t h e n
back t o
The e x i s t i n g
pu r e F e O( c ) .
phases p r e s e n t
ar e m a g n e t i t e , Fe3O4 ( C ) ;
i n t h e FeO- FeO1 5 syst em
hematite,
FeO1
FeO1 g ( c , Y ) ; FeO1 g ( I ) and FeO( I ) .
properties
of
Fe3O4 l i s t e d
G e o l o g i c Su r v e y B u l l e t i n
wustite,
FeO(c);
The t h e r mo d y n a mi c
i n Ta b l e 32 wer e t a k e n
I 45240 and t h e
heat
from t he
capacity
I 9 E
equation
and H39g ar e f r o m NBS.
The h e a t c a p a c i t y
!
'
equation f o r
melting
Fe3O4 i s
point
t h e r mo d y n a mi c
valid
The h e a t
c a p a c i t y and
properties
o f FeO1
hematite,
f r o m 950 t o
The h e a t
be t h e
capacity
1452,
4 0
and t h e he at
1805 K , t h e m e l t i n g
equation f o r
same as t h a t
point
FeO1 5 ( c * Y)
o f FeO1 g ( c , h e m a t i t e ) .
and e n t r o p y o f FeO1 g ( c , Y) wer e c a l c u l a t e d
model
for
Fe3O4 , assumi ng t h a t
FeO and FeO1 g ( c , Y)
The h e a t
is very
et
10.% l a r g e r
than t h a t
AHf us d a t a , and t h e
of the
is
o f FeO1 g .
is
estimated
to
The e n t h a l p y
I
from a l a t t i c e
of
68 . 2 J / m o I e - K a t
and i s
1650
held constant
up
o f FeO1 g ( I ) was assumed t o
solid
both
Plank's
is
aI .,* ^
The h e a t c a p a c i t y
298. 15 f o r
capacity
|
small.
t o 3000 K .
at
i n T a b l e 32
t he e n t h a l p y o f mi xi ng
c a p a c i t y o f F e O( I )
as measur ed by C o u g h l i n
enthalpies
1870 K , t h e
o f Fe3O4 .
ar e a l s o f r o m B u l l e t i n
valid
f r o m 850 K t o
FeO1 g ( c ) .
liquids
be
'I
The
were c a l c u l a t e d
f u n c t i o n was a d j u s t e d
from
to f i t
■s
,1
126
T a b l e 32. The h e a t c a p a c i t y e q u a t i o n s and t h e r mo d y n a mi c
p r o p e r t i e s o f v a r i o u s s t o i c h i o m e t r i c compounds i n t he
FeO- FeO1 ^ - S i O g - A l O 1 g-CaO s y s t e m.
HEAT CAPACITY EQUATIONS
B
A
C
D
E
FeO1 5 (C, GAMMA)3
74.245
.136201
Fe (C, GAMMA)
FeO1 5 (C, HEMATITE)3
32.3431
74.245
.837930E-2
.136201
-.174110 E-7
FeO (C, WUSTITE)
Fe3 O4 (C, MAGNETITE)3
56.709
205.97
.0251173
.052733
.377231E-4
.117640E-6 -.92509E-10
5.6413E+7
Fe2SiO4 (C1FAYALITE)
188.14
.0414165
2.241 IE-5
-3.6299E+6
FeO1 5 (LIQUID)3
74.245
FeO (LIQUID)
FeAI2O4 (C, HERCYNITE)
68.2
90.016
CaAI2Si2O8 (ANORTHITE) 21.89929
THERMODYNAMIC
-51042186.
16953.56
-51042186.
16953.56
-51042186.
.146200
16953.56
.037055
-.574449E-4
.352436E-7
.0527382
.141137E-4
-.457359E-7 .564705E-10
QUANTITIES
Y1OOO
J MOL'1
H1000*H298
JM OL'1
H298
J MOL'1
FeO1 5 (C, GAMMA)
82.12558
50275.5
-401175.94
Fe (C, GAMMA)
FeO1 5 (C, HEMATITE)
41.9488
76.08
28405.
50275.5
0.0
-412320.
9.41399
44.24774
7.2118
10.0
FeO (C, WUSTITE)
Fe3O4 (C1MAGNETITE)
87.292906I
241.16403I
35875.90
147986.
-264522.47
-1118400.
45.75706
146.14
12.4874
45.83
Fe2 SiO4 (C1FAYALITE)
228.4400
118426.
-1477896.
148.32
47.369
FeO1 5 (LIQUID)
96.78606
50275.5
-371240.
61.77637
10.0
47866.17
115478.
-245660.
-1983946.0
58.83758
106.3
41.486
-4231800.
199.30
101.76
93.50423
FeO (LIQUID)
FeAI2O4 (C1 HERCYNITE) 181.71161
CaAI2Si2O8 (ANORTHITE) 332.26016 201018.5
S298
. V1000
J MOL'1 K'1 CM3
. 50.29333
10.0
THE HEAT CAPACITY EQUATIONS ARE GIVEN BY Cp=A+B(T-1000)+C(T-1000)2+D(T-1000)3 +
E(T-IOOO)4. FOR THOSE COMPOUNDS MARK WITH (a), THE D AND E TERMS ARE
D(1/T'2 - 10*6) AND E(T5 - 1 0OO-5). ANORTHITE ALSO HAS THE ADDITIONAL EINSTEIN TERM
38 E(T/700)
127
t he known s o l i d
liquid
equilibrium within
the
FeO-FeOj
^
s y s t e m.
Ther e
FeOj
is
heat
c a p a c i t y data a v a i l a b l e
g , Fe^OzJ and FeO 5 4 7 *
able to
calculate
3 2 40
’
for
Usi ng t h i s
Redlich-Kister
the s o l i d s
d a t a we were
coefficients
for
c a p a c i t y and t h e t e m p e r a t u r e dependence o f t h e
capacity
for
values f o r
with
t he heat
calculating
enthalpy.
for
stoichiometric
the
R. W. G u r r y
solution
at
is
oxide
12 7
c a p a c i t y we wer e t he n
the R e d l i c h - K i s t e r
Ther e
iron
a substantial
s y s t e m.
heat
T a b l e 33.
Wi t h
able to
pr oceed
coefficients
for
amount o f e n t h a l p y da t a
12 6, 1 27 ^ ^ 5 e Dar ken and
have measur ed t h e e n t h a l p y o f t h e s o l i d
1523 K w i t h
f r o m x = . 111932 t o
amount o f
FeO shown i n
t h e heat
x = 0.32.
good d a t a f o r
so we wer e a b l e t o
mol e f r a c t i o n s
o f FeOj
g ranging
is a sub s ta n ti al
32 40
Fe^O^ and FeOj g e n t h a l p i e s
’
,
calculate
Ther e
Redlich-Kister
coefficients
.
at
1523 K f o r
t h e e n t h a l p y and t h e n c o n v e r t
coefficients
to
coefficients
for
with
the
heat
capacity.
l o g Y and Cp ar e
of
12 7
listed
These c o e f f i c i e n t s
i n Tabl e
have measur ed v a l u e s
C0? and CO o v e r
pressures
ranging
f r o m XpeQ = . 8988 t o XpeQ = . 6406 a t
t he. e q u i l i b r i u m
F e ( c , y)
constants
+ CO2 = FeO( w u s ) + CO
iron
for
al ong
33.
partial
for
j
1000 K u s i n g t h e R e d l i c h - K i s t e r
Dar ken and G u r r y
values
these
I
o f the
oxide
solutions
1573 K . Wi t h
j
the r e a c t i o n s
( 121)
I
128
Ta bl e 33. R e d l i c h - K i s t e r
Solid Binary.
A
coefficients
for
the
B
C
D
1.541358
-4.835708
7.388763
FeO-FeO
E
log
-.535413
He
-10430.564 58374.976 -175289.548 256811.822 -129030.652 ■
-3.670011
1.156545
Cp
dCp/dT
-.0226686
d2Cp/ dT2
-.746594E-5
and
FeO( w u s ) + 1/ 2 CO2 = FeO1 g + 1/ 2 CO
we can c a l c u l a t e
fractions
for
activity
and t h i s
reactions
=1.33667.
( 121)
coefficients
temperature.
and ( 122)
( 122)
a t t h e s e mol e
The e q u i l i b r i u m c o n s t a n t s
ar e K121 = 3 . 2 0 1 4 3 and K122
Therefore .
K122 = . 133667 = aFe0/ a F e 0 ^
( C
O
/ C
O
g ) ( 123)
and
K121 = 3. 2 01 43 = aFe0/ a F e ( C i 7 , ( C 0 / C 0 2 )
The a c t i v i t y
o f F e ( c , Y ) is- I a t t h e end o f t h e r ange wher e
Fe(c,Y)
is
present.
o f the
iron
oxides
sufficient
for
( 124)
to
l og Y f o r
Knowl edge o f dependence o f a c t i v i t i e s
as a f u n c t i o n
calculate
solid
o f mol e f r a c t i o n ,
the R e d l i c h - K i s t e r
FeO^ shown i n Ta b l e 33.
is
coefficients
129
Ther e
liquid
is
very
little
i n t h e FeO-FeOi1 5 s y s t e m .
enthalpy of fusion
of
We were a b l e t o
coefficients
•an i n i t i a l
slope
concentrations
for
1697,
hi gh
concentrations
liquid
-4.0017
wer e c a l c u l a t e d
the mel t in g of
calculated
We assumed
l ow
with
and - 27
t h e excess
i n T a b l e 34 f o r
;|
t he
from t he e q u i l i b r i u m
the v ar i o us
from the R e d l i c h - K i s t e r
solids
at T -
o f the
coefficients
in
34.
T a b l e 34. R e d l i c h - K i s t e r
liq u id binary.
coefficients
Y
He
the
FeO- FeO. r
C
-.8557493
-.0237277
. 1814582
-20633.607
-7500.000
1133. 606
The R e d l i c h - K i s t e r
binaries
for
B
A
l og
.67.*^
k J / mo l e o f m e t a l .
coefficients
present
in t h i s
coefficients
i
Redlich-Kister-
1644 and 1870 K and f r o m t h e a c t i v i t i e s
solids
Tabl e
for
calculate
FeO o f - 12 k J / m o l e o f me t a l
at
on t he
= 3300 0 ' c aI / mo Te 128 and •
t h e excess e n t h a l p y a t
The R e d l i c h - K i s t e r
constants
data
f r o m - Xf e 0 = . 92 t o
data t o
Xf e Q = 2 / 3 o f
y of the
is
the enthalpy of t h e . l i q u i d .
for
k J / m o l e o f me t a l
e n t h a l p y at
liquid
use t h i s
for
Ther e
Fe3O4 , AHf u s
on t h e e n t h a l p y o f t h e
l og
d a t a on t h e e n t h a l p y o f t h e
for
the ot h er
syst em wer e c a l c u l a t e d
from
I
I
130
enthalpy of
within
fusion
the various
wer e c a l c u l a t e d
adjusting
the
d a t a and t h e known e q u i l i b r i a
b i n a r i e s . ' The t e r n a r y
with
Toop Muggi anu
activities
somet i mes r e q u i r e d
to f i t
adjusting
phase d i a g r a ms
interpolations,
the e q u i l i b r i a . .
the b i n a r i e s
l og Y o f t h e t e r n a r y phases d i d
unrealistic
when a Toop i n t e r p o l a t i o n
Figures
slices
phase d i a g r a ms
30 t h r o u g h
at
reported
34.
for
this
0% CaO up t h r o u g h
the
n o t become
was done.
syst em ar e
These f i g u r e s
Thi s
so t h a t
e n t h a l p y or
calculated
occurring
The
shown as
ar e p s e u d o t e r na r y
20% CaO.
The i r o n
oxide
is
i n t h e s e d i a g r a ms as w e i g h t % FeO^ g , however t h e
computations
correspond to
fact
t h e aver age, o x i d a t i o n
that
in e q u i l i b r i u m with
( H2O)ZH2 ) = 1 . 3 .
substantial
state
water
of
vapor
FeO p r e s e n t .
In
i r o n was p i c k e d t o be
and h y d r o g e n w i t h
131
We i g h t f r a c t i o n
o f Si O0
Cristobal I t e
t r Idvmite
iull I t e
£ 7 fayal I t e
rcyn/te
We i g h t
fraction
FeOx c a l c u l a t e d
as FeO1 g
F i g u r e 30. C o n t o u r l i n e s i n t h e A l O i ^ g - S i O g - F e O * syst em
v e r s u s w e i g h t f r a c t i o n c a l c u l a t e d as F e Oi . g f o r HgO/Hg =
1.3.
T e mp e r a t u r e s ar e g i v e n i n 200 degr ee F a h r e n h e i t
intervals.
132
We i g h t f r a c t i o n
o f Si O0
Wei gh t
fraction
FeOx c a l c u l a t e d
as FeO1 5
F i g u r e 31. Contour l i n e s in the A l O i . g - S i O g - F e O x
at 5% CaO by wei ght and H2OZH2 = 1 . 3 .
system
133
We i g h t f r a c t i o n
o f Si O0
<WuJ I Ice
EaVallce
We i g h t
fraction
:
FeOx c a l c u l a t e d
as FeOj
^
F i g u r e 32. Contour l i n e s in the A l O i i 5 " s i 0 2 "FeOx system
at 10% CaO by we i ght and HgO/Hg = 1 . 3 .
134
We i gh t f r a c t i o n
o f Si O0
We i g h t
fraction
FeOx c a l c u l a t e d
as FeO1 5
F i g u r e 33. Contour l i n e s in the A l O i . g - S i O g - F e O x system
at 15% CaO by we i ght and H2OZH2 = 1 . 3 .
135
We i g h t f r a c t i o n
o f Si O0
SOO0F
We i g h t
fraction
FeOx c a l c u l a t e d
as FeO^ g
F i g u r e 34. Contour l i n e s in t he A l O i . g - S i O g - F e O x
at 15% CaO by wei ght and HgO/Hg = 1 . 3 .
system
136
SUMMARY
Equations of
A l Oj
5 and i t s
developed.
various
inherent
for
relationship
in t h i s
the t e r n a r y
stoichiometric
In c a l c u l a t i n g
solids
of thermal
state
equations
syst em MgO-Si Ogcompounds have been
of st at e
for
t he
syst em we d e v e l o p e d a u s e f u l
bet ween t h e b u l k modul us and t h e c o e f f i c i e n t
expansion.
' The e q u a t i o n
is
Ko T = Ko T 0 ( V T 0 / V T ) *
We wer e a b l e t o
the
solids
show t h e r e l i a b i l i t y
MgO and A l Oj
g.
many o f t h e s t o i c h i o m e t r i c
with
the
good r e s u l t s .
Thi s equat i on
compounds
However ,
l ambda t r a n s i t i o n
of t h i s
this
al ong w i t h
also
used f o r
in the t e r n a r y
f orm i s
syst em
n o t a c c u r a t e near
in quartz.
The P i p p a r d t h e o r y o f second o r d e r
was used,
is
equation f o r
phase t r a n s i t i o n s
X - r a y measur ement s o f t h e vol ume and
measur ement s o f t h e s l o p e o f t h e
l ambda t r a n s i t i o n
ver s us
p r e s s u r e t o r e s o l v e t h e 380 J / mol e d i s c r e p a n c y bet ween t h e
standard compi l at i on s
aI . ^
H
j o o o
o f R o b i e,
et aI
and S t u l l ,
et
The JANAF v a l u e o f 45354 ± 150 J./mol e f o r
-
H2gg
for
quartz
be 45452 ± 70 J / m o l e .
by R i c h e t , e t
has been c o n f i r m e d and r e f i n e d t o
Also,
independent a n a l y s i s
a l . based on dr op c a l o r i m e t r y y i e l d s
- H298 = 45579 ± 150 J / mo l e f o r
quartz.
H j 000
137
From a n a l y s i s
t h e r mo d y n a mi c
fo rsterite
o f t h e phase e q u i l i b r i a
properties
and e n s t a t i t e
vitrification
for
o f magnesi um o x i d e ,
along wi t h
enstatite,
respectively.
coefficients
binary
the a c i d i c
and b a s i c
Procedur es f o r
five
component
here,
tackle
of t h r e e ,
lattice
four,
and
in t h i s
most s l a g
thesis
pr obl ems.
equilibria
and p r e s s u r e s .
now makes
it
the s o l i d s
over
a
The d e v e l o p me n t
fairly
We wer e a b l e t o
t h e MgO- Si Og- Al O^
silicate
model s f o r
met hods o f d e a l i n g w i t h
phase d i a g r a ms f o r
five
and t h e phase
met hods o f d e a l i n g w i t h t h e
wi d e r a n g e o f t e m p e r a t u r e s
summar i zed
bot h
1000 MPa.
the c a l c u l a t i o n
The d e v e l o p me n t o f
also provides
for
s y st ems have been d e v e l o p e d and r e f i n e d
along w i t h
liquid's.
have been c a l c u l a t e d
up t o
for
Al s o R e d l i c h - K i s t e r
s i des o f t he di agr am,
d i a g r a m has been c a l c u l a t e d
of fusion
t o be 4 8 . 8 ± 3 k J / mo l e
and 9 2 . 9 ± 12 k J / m o l e
this
silica,
a measur ed e n t h a l p y o f
the e n t h a l p i e s
MgSi Og and MggSi O^ were c a l c u l a t e d
for
and f r o m t he
easy t o
calculate
g t e r n a r y and t he
component s yst em FeO-FeO ^ J - C a O - S i O g - A l O j . ^ f r om
analysis
o f t h e phase e q u i l i b r i a
and f r o m t h e
t h e r mo d y n a mi c measur ement s o f t h e component s o f t h e phase
di agr ams.
138
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