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UBC Theses and Dissertations

Ionic equilibrium in silicate slags Toop, Gerald Wesley 1960

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IONIC EQUILIBRIUM IN SILICATE SLAGS  by  GERALD WESLEY TOOP  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE DEPARTMENT OF MINING AND METALLURGY  We a c c e p t t h i s t h e s i s as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, I960  In the  presenting  requirements  this thesis f o r an  in partial  advanced  fulfilment  degree at  the  of  University may  of  British  it  freely available  agree that for  Columbia,  Library  afesxbcl make  for reference  and  study.  I  copying  gain  shall  or  not  of  for  his  be  publication be  granted, by  of  Metallurgy  2nd, i960.  Columbia,  of  the  It  this thesis  a l l o w e d w i t h o u t my  M i n i n g and  September  copying  representatives.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a . Date  extensive  p u r p o s e s may  o r by  that  Department  the  permission  scholarly  Department  I agree that  further this  Head o f  thesis my  i s understood for financial  written  permission.  - i ABSTRACT  The problem o f d e t e r m i n i n g t h e t o t a l number o f d i s c r e t e anions i n both a c i d and b a s i c s i l i c a t e m e l t s has been i n v e s t i g a t e d i n o r d e r t o a p p l y i o n i c t h e o r y and 'Temkin's r u l e ' t o b i n a r y and t e r n a r y s i l i c a t e With an examination a method was developed  slags.  o f s t r u c t u r e and e q u i l i b r i a i n s i l i c a t e  melts,  f o r d e t e r m i n i n g t h e most probable number o f anions i n  any b a s i c o r a c i d s l a g i n terms o f an e q u i l i b r i u m constant i n v o l v i n g  singly  bonded oxygen, d o u b l y bonded oxygen, and f r e e oxygen ions i n t h e m e l t . was  It  found t h a t f o r an a p p r o p r i a t e v a l u e o f t h e e q u i l i b r i u m c o n s t a n t , v a l u e s  o f Temkin's i o n i c f r a c t i o n o f oxygen i o n s may be o b t a i n e d t h a t a r e c l o s e t o , or equal t o the a c t i v i t i e s of various metal oxides i n binary s i l i c a t e melts.  T h i s o b s e r v a t i o n was c o n s i s t e n t w i t h the thermodynamics o f t h e  systems s t u d i e d and was v e r i f i e d m a t h e m a t i c a l l y by t h e a p p l i c a t i o n o f t h e Gibbs-Duhem r e l a t i o n s h i p t o i o n i c s i l i c a t e  melts.  Some i o n i c forms o f t h e Gibbs-Duhem e q u a t i o n were d e r i v e d and a p p l i e d s u c c e s s f u l l y t o t h e CaO-FeO-Si0 the C u 0 - P b 0 - S i 0 2  2  system c o n t a i n i n g d i v a l e n t  2  c a t i o n s , and t o  system c o n t a i n i n g both d i v a l e n t and monovalent c a t i o n s .  Some t e r n a r y a c t i v i t y v a l u e s o f C u 0 and PbO were determined ex2  perimentally i n t h e Cu 0-PbO-Si0 2  2  system by e q u i l i b r a t i n g s i l i c a s a t u r a t e d  melts o f C u 0 and PbO w i t h c o p p e r - l e a d a l l o y s at 1100°C. 2  ACKNOWLEDGMENT  The author w i s h e s t o express  his gratitude f o r the t e c h n i c a l  a d v i c e and guidance g i v e n by the members o f t h e Department o f M i n i n g and M e t a l l u r g y o f t h e U n i v e r s i t y o f B r i t i s h Columbia.  Sincere  apprec-  i a t i o n i s extended t o D r . C.S. Samis f o r s u p e r v i s i o n and i n s t r u c t i o n r e c e i v e d d u r i n g the p r e p a r a t i o n o f t h i s The  author  thesis.  i s i n d e b t e d t o t h e N a t i o n a l Research C o u n c i l f o r  f i n a n c i a l a s s i s t a n c e g i v e n d u r i n g t h e p a s t academic  year.  — I l l —  TABLE OF CONTENTS Page INTRODUCTION  a  *  o  e  o  »  o  9  4  0  o  o  o  o  o  o  o  o  o  D e f i n i t i o n s o f Terms  o  o  o  o  »  o  o  »  I  o  . . . . . . . . . . .  1  Structure of L i q u i d S i l i c a t e s  1  E q u i l i b r i u m i n S i l i c a t e Melts . . . . . . . . . . . . .  2  . M a t e r i a l Balance C o n s i d e r a t i o n s  . . . . . . . . . . . . . . . .  EVALUATION OF IONIC FRACTIONS OF IONS IN SILICATE MELTS S o l u t i o n o f the E q u i l i b r i u m E q u a t i o n  5  . . . . . .  5  . . . . . . . . . . . . .  5  D e t e r m i n a t i o n o f the Number o f S i l i c a t e Anions i n MS I t S o o o o o o o o a o o e o o o o o o o o o o  SlllCcL"b©  D e t e r m i n a t i o n o f t h e I o n i c F r a c t i o n of Oxygen Ions  &  . . . . . .  APPLICATION OF THE GIBBS-DUHEM EQUATION TO IONIC SILICATE MELTS  13  . . 18  APPLICATION OF IONIC GIBBS-DUHEM EQUATIONS TO THERMODYNAMIC DATA OF THE C a 0 » F e 0 - S i 0 Calculation of a 0  =  SYSTEM . . . . . . . . . . . . . . . .  2  i n t h e CaO~FeO~Si0  C a l c u l a t i o n of yCa and  2  System . . . . . . . . .  Ternary A c t i v i t y  + +  Values f o r CaO  APPLICATION OF IONIC THEORY TO THE C u 0 ~ P b O - S i 0 2  EXPERIMENTAL  a o o o o o o e e a o e o o o  21  2  SYSTEM  22  . . . .  28  . . . . . .  31  o o e o o e o o o v e o  o  3-^-  E q u i l i b r i u m Conditions  . . . . . . . . . . . . . . . . . . . . . .  31  Materials  . . . . . . . . . . . . . . . . . . . .  32  and Equipment  Chemical Analysis  „. . . . . . . . . . . . . . .  . 32  C a l c u l a t i o n o f t h e E q u i l i b r i u m Constant K . . . . . . . . . .  . 33  RESULTS  o < •o o • B O O  of Samples  o*o  o o o a o o o o o o o o o a o o e o o  3^4-  COMPILATION OF THE AVAILABLE THERMODYNAMIC DATA FOR THE BINARY SYSTEMS P b 0 - S i 0 , C u 0 - S i 0 2  2  2  and Cu 0-Pb0 . . . . . . . . 2  C a l c u l a t i o n o f A c t i v i t y ' D a t a from Phase Diagrams The C u 0 - S i 0 2  2  System . . . . . . . .  . . . . . .  . . . . . . . .  . . . . . .  39 39 39  The Cu 0-Pb0 System . . . . . . . . . . . . . . . . . . . . . .  41  The P b 0 - S i 0  43  2  2  System . . . . . . . . . . . . . . . . . . . . . . .  -ivPage CALCULATION OF TERNARY ACTIVITY VALUES OF C u 0 , PbO, and S i 0 2  IN THE C u 0 - F b O - S i 0 2  2  SYSTEM WITH THE AID OF AN IONIC GIBBS-  2  DUHEM EQUATION , . . . . . . . . . . . . . . . . . . . . . . . .  43  INTEGRABILITY OF THE ACTIVITIES OF NEGATIVE IONS IN BINARY SILICATE MELTS  9  «  9  a  oo  01SOUSSION  0  0  0  0  9  CONCLUSIONS  0  0  9  9  0  REFERENCES  o « <> « o o « •  APPENDICES  9  A.  B.  C.  0  «  »  0  0  4  0  o  o  0 0  0  a  0  o  0  0  0  0  «  o  o  o  0  o  0  0  «  «  0  0  o  o  «  «  «  9  0  0  • « • • « • « *  o  0  •  «  e o o  0  a  5-3-  0 5^  •  o o o o o o w o » o o o o » * o o o 5 9  « o o « o o o - » o o <  '  0  0  0  a  o  •  o  o  «  «  o  •  «  o  •  «  «  «  «  «  *  »  «  »  o«6l«  0  ^3  D e r i v a t i o n o f a G e n e r a l I o n i c Gibbs-Duhem E q u a t i o n f o r Two D i v a l e n t M e t a l Oxides CaO and FeO i n a T e r n a r y Melt w i t h S i l i c a , . . , . . . . . . . . . . . . . .  63  C a l c u l a t i o n o f A c t i v i t y Data i n the CopperLead System  66  I n i t i a l and F i n a l Compositions o f E q u i l i b r i u m M e l t s at 1100°C i n t h e C u 0 - P b O - S i 0 System . . .  67  D e r i v a t i o n o f a G e n e r a l I o n i c Gibbs-Duhem E q u a t i o n f o r One D i v a l e n t . M e t a l Oxide PbO, One Monovalent M e t a l Oxide C u 0 , and S i l i c a . . . . . . . . . . . . . . . .  68  2  D.  9  2  2  - V -  LIST OF TABLES Page 1.  Q u a d r a t i c E q u a t i o n s i n Terms o f (0~) and k f o r V a r i o u s Values of N S i 0 . . . . . . .  .  2  2.  7  E q u i l i b r i u m Number of Moles of S i n g l y Bonded Oxygen, Doubly Bonded Oxygen, and F r e e Oxygen Ions p e r Mole of S l a g , C a l c u l a t e d f o r V a r i o u s V a l u e s of N S i 0 and k  •  2  3.  •  9  N u m e r i c a l V a l u e s , E x p r e s s e d per Mole of S l a g , Used i n t h e C a l c u l a t i o n of t h e T o t a l Number o f Negative Ions and Temkin's I o n i c F r a c t i o n of Oxygen Ions f o r V a r i o u s V a l u e s of N S i 0 and k 2  4.  A c t i v i t y Data at 1600°C o f t h e B i n a r y Systems F e O - S i 0 , CaO-SiOg and FeO-CaO, w i t h Respect t o the Pure L i q u i d  5.  C a l c u l a t e d Values o f a 0  6.  C a l c u l a t e d A c t i v i t y Values f o r CaO System  7.  2  Compositions  9.  i n the C a 0 - F e 0 - S i 0  =  i n the  Ca0-Fe0-Si0  Cu  2  and 33  1110° C  I n i t i a l and F i n a l Compositions o f M e l t s H e l d at 1100°C f o r 45, 60, 100 and 130 Minutes  35  E x p e r i m e n t a l T e r n a r y A c t i v i t y Values o f PbO and C u 0 a l o n g t h e S i l i c a S a t u r a t i o n Line o f t h e C u 0 - P b O - S i 0 System 2  1100  11.  2  29  2  10.  26  System  2  o f E q u i l i b r i u m M e l t s o f C u 0 , PbO,  Pb a t 1087°C and 8.  G o  e«  o o o o o o o *  2  o o o o  o « o a « o * * « * « «  A c t i v i t i e s of C u 0 and S i 0 i n the C u 0 - S i 0 System 3."b 1100 C e o a « e c e o e e » o o * ° o o e * e » a * o « o o 2  2  2  A c t i v i t y of S i l i c a i n the C u 0 - S i 0 f o r Low V a l u e s o f N S i 0 2  37  2  Zj.0  System at 1100°C  2  41  2  12.  16  A c t i v i t i e s of PbO and C u 0 i n the PbO-Cu 0 System 3/b 1100 C o o o o o o o o o o o o o o * o o o o o » . o o o * » o 2  2  2$-2  13.  A c t i v i t i e s o f PbO and S i 0 i n the P b 0 - S i 0 System cit/ 1100 C « « » • « o o o o o o e o o o o o o o o o o o o o * * . /t-3  14.  C a l c u l a t e d Values o f a 0 System a t 1100°C  15.  2  17.  and a S i 0  2  i n the  Cu 0-Pb0-Si0 2  2  4*2>  C a l c u l a t e d T e r n a r y A c t i v i t y Values f o r C u 0 a t 1100°C, 2  Obtained 16.  =  2  from Values o f a 0  =  and y C u  A c t i v i t i e s of t h e Negative Ions i n P b 0 - S i 0 Proposed  50  +  2  M e l t s a t 1100°C . .  54  Values o f t h e E q u i l i b r i u m Constant k f o r V a r i o u s  B i n a r y S i l i c a t e Systems  . . . . . . . . . . . .  58  -vi-  LtST OF FIGURES  Page  1.  Proposed D i s c r e t e S i l i c a t e Anions In S i l i c a t e M e l t s .  2.  T e r n a r y P l o t o f t h e P r o p o r t i o n s o f S i n g l y Bonded Oxygen, S i l i c o n , and Doubly Bonded Oxygen i n any D i s c r e t e S i l i c a t e Anion . . . . „ . „ . . . . . . . . . .  10  P l o t of t h e P r o p o r t i o n o f S i n g l y Bonded Oxygen i n any S i l i c a t e A n i o n o r i n any S i l i c a t e M e l t , V e r s u s t h e Number o f S i l i c o n Atoms p e r Ion •  11  Temkin's I o n i c F r a c t i o n o f Oxygen Ions Versus N S i 0 f o r V a r i o u s V a l u e s of k . . . . . .  14  3.  4.  5.  3  2  A c t i v i t i e s o f Various M e t a l Oxides i n B i n a r y M e l t s w i t h S i l i c a w i t h Respect t o the Pure L i q u i d Oxides „  15  6.  I s o - a c t i v i t y L i n e s f o r FeO i n the C a 0 - F e 0 - S i 0  7.  I s o - a c t i v i t y Lines f o r S i 0  8.  I s o - a c t i v i t y L i n e s f o r Oxygen Ions i n the C a 0 - F e 0 - S i 0  9.  at 1600°C . . . . . . . . . . . . . . . . . I s o - A c t i v i t y L i n e s f o r CaO c a l c u l a t e d from V a l u e s o f a 0 and y C a i n . the C a 0 - F e 0 - S i 0 System  2  System  2  i n t h e CaO-FeO-Si0  2  System  System  2  27 =  30  + +  2  10.  I n i t i a l and F i n a l S l a g Compositions at 1100°C, i n t h e C u 0 - P b 0 - S i 0 System . . . . . . . . . . . . .  36  E x p e r i m e n t a l l y Determined R a t i o of a C u 0 t o aPbO versus NCu 0 along t h e S i l i c a S a t u r a t i o n L i n e of the Cu 0-PbOS i 0 System a t 1100°C . . . . . . . . . . . . . . . . . . . . .  38  2  11.  2  2  2  2  2  12.  I s o - a c t i v i t y L i n e s f o r S i l i c a i n the C u 0 - P b 0 - S i 0 1100 C a « 9 o o o o o » a o o o o a c t t 2  cL*t>  13.  0  9  0  a  a  System  2  t  o  o  D  O  I s o - a c t i v i t y L i n e s f o r Oxygen Ions i n t h e C u 0 - P b 0 - S i 0 System at 1100°C . . . . . . . . . . . . . . . . . . . . . . . 2  9  l±5  2  46  14.  I s o - a c t i v i t y L i n e s f o r PbO i n the C u 0 - P b 0 - S i 0  15.  I s o - a c t i v i t y L i n e s f o r C u 0 a t 1100°C c a l c u l a t e d from V a l u e s o f a0= and y C u i n t h e C u 0 - P b 0 - S i 0 System  49  A c t i v i t y o f PbO a t 1100°C i n the P b 0 - S i 0 B i n a r y System Compared w i t h Values o f NO"" c a l c u l a t e d w i t h k = .01 . . . . .  53  2  2  System  2  +  2  16.  2  2  -vii—  Page 17.  18.  Activities at 1100°C  o f the N e g a t i v e Ions i n P b O - S i 0  2  Melts  P o s i t i o n s o f t h e I m m i s c i b i l i t y Gaps o f V a r i o u s B i n a r y S i l i c a t e Phase Diagrams  55  57  IONIC EQUILIBRIUM IN SILICATE SLAGS  INTRODUCTION  D e f i n i t i o n s o f Terms Most o f the symbols used i n t h i s i n v e s t i g a t i o n are d e f i n e d following  list  of d e f i n i t i o n s .  Other terms used a r e d e f i n e d  number o f moles o f oxygen i o n s  i n the  i n the text.  (0~)  = equilibrium  p e r mole o f s l a g .  (0~)  = e q u i l i b r i u m number o f moles o f s i n g l y bonded oxygen p e r mole o f s l a g .  0"  = number o f s i n g l y bonded oxygen.atoms i n any d i s c r e t e anion.  (0°)  = equilibrium of s l a g .  silicate  number o f moles o f d o u b l y bonded oxygen per mole  = number o f d o u b l y bonded oxygen atoms i n any d i s c r e t e anion.  silicate  - (o°)(Q ) =  NSi0 NPbO NFeO  (0~)2 2  = mole f r a c t i o n s o f m e t a l o x i d e s .  = activity. nM  .++  NM'++  = activity coefficient. = number o f moles o f -a g e n e r a l c a t i o n p e r mole o f s l a g . sum of p o s i t i v e nM  ions  sum o f n e g a t i v e  ions  ++  NO"  Structure  of Liquid S i l i c a t e s  The i o n i c c h a r a c t e r o f f u s e d o x i d e s and s i l i c a t e s has become w i d e l y a c c e p t e d w i t h f i r m e x p e r i m e n t a l measurements  of e l e c t r i c a l  conductivity , 1  -2-  e l e c t r o n t r a n s p o r t ^ , v i s c o s i t y - * , d e n s i t y and e x p a n s i v i t y ^ .  The common s l a g  f o r m i n g o x i d e s , FeO,MnO,CaO,MgO,PbO a r e c o n s i d e r e d t o i o n i z e and t o cont r i b u t e m e t a l c a t i o n s and oxygen anions hand,is  t o t h e melt„  S i l i c o n , o n the other  a complexing agent which e x h i b i t s s t r o n g t e t r a h e d r a l c o - o r d i n a t i o n  w i t h oxygen t o form d i s c r e t e c h a i n - t y p e  i o n s , r i n g - t y p e i o n s and complex  r i n g - t y p e i o n s , depending on t h e c o n c e n t r a t i o n and type o f m e t a l oxides  3 present.  Some o f t h e proposed d i s c r e t e s i l i c a t e  anions  a r e shown i n  F i g u r e 1. In t h e fundamental s i l i c a t e t e t r a h e d r o n , S i O ^ " , 4  silicon  i s tetra-  h e d r a l l y c o - o r d i n a t e d w i t h f o u r s i n g l y bonded oxygen atoms and t h e i o n i s expected silica  t o be predominant i n low s i l i c a  content  slags or basic s l a g s .  o f t h e s l a g i s i n c r e a s e d t h e s i l i c a t e anions  As t h e  can p o l y -  merize by l i n k i n g w i t h o t h e r s i l i c a t e a n i o n s , and s i l i c o n becomes -coo r d i n a t e d w i t h d e c r e a s i n g numbers o f s i n g l y bonded oxygen atoms and w i t h i n c r e a s i n g numbers o f doubly bonded oxygen atoms u n t i l , i n p u r e the s i l i c o n i s t e t r a h e d r a l l y c o - o r d i n a t e d w i t h f o u r doubly  silica,  bonded oxygen  atoms and t h e m a t e r i a l i s e l e c t r i c a l l y n e u t r a l . The q u e s t i o n o f t h e degree of p o l y m e r i z a t i o n o f complex i o n s , p a r t i c u l a r l y i n a c i d s l a g s , can c o m p l e t e l y a n i o n i c concentrations i n the melt,  silicate  obscure e v a l u a t i o n o f  A satisfactory solution to this  problem must precede any s u c c e s s f u l a p p l i c a t i o n o f i o n i c t h e o r y t o s l a g s . Equilibrium i n Silicate Any  Melts  attempt t o d e s c r i b e e q u i l i b r i a  i n s i l i c a t e melts  by t h e f a c t t h a t the s t r u c t u r e o f t h e m e l t changes w i t h S i n c e t h e mechanism o f e q u i l i b r i u m i s expected cate compositions,  i s complicated  composition.  t o be t h e same f o r a l l s i l i -  i t f o l l o w s t h a t any proposed e q u i l i b r i u m r e a c t i o n s h o u l d  be o f g e n e r a l form and s h o u l d n o t i n v o l v e s p e c i f i c s i l i c a t e Fincham and Richardson,-'  anions,  r e c o g n i z i n g t h a t oxygen i n s i l i c a t e  o c c u r s i n o n l y t h r e e f o r m s , s i n g l y bonded, doubly  melts  bonded, and f r e e oxygen  -3-  R i n g - t y p e Ion  Complex Ring-type Ion  F i g u r e 1„ Proposed D i s c r e t e Anions i n S i l i c a t e M e l t s . 3  Silicate  i o n s , have suggested  a v e r y simple and p r o b a b l e  2(=Si-0") or  (HSi-O-SiH) +  0  equilibrium  reaction,  =  briefly, 2 0°  '  0° + 0  1  =  E q u a t i o n 1 i s a fundamental r e s u l t tetrahedral co-ordination  of the charge  balance r e q u i r e d  of oxygen w i t h s i l i c o n .  bonded oxygen atoms i n a s i l i c a t e  by the  The number of s i n g l y  a n i o n i s e q u a l t o the v a l e n c e o f t h e  a n i o n and the number o f d o u b l y bonded oxygen atoms i s e q u a l t o the oxygen of the a n i o n minus t h e v a l e n c e . (d) i n c l u s i v e , reduces  to (a)  show t h a t when any s i l i c a t e anions a s s o c i a t e the  s—  2 SiCV*"  Si 0 2  *=- 0° +  2 0"  —  SiCV"" + S i 0 2  0° + 6 7  2  6  3^  "  2 Si 0 3  6 9  0°+0 0  8 1 0  +  —  6  3  Si 0 6  * - 0° +  0  =  =  =  9  0  =  =  0  0° +  ^  s  6 1 5  - 3 +  0  0=  =  conditions, a quasi-chemical equilibrium 1,  C  -  2  ) 2  I t i s proposed t h a t k, as d e f i n e d  i n e q u a t i o n 2, i s constant at a g i v e n  and c h a r a c t e r i s t i c o f the c a t i o n s p r e s e n t i n any b i n a r y  silicate  constant  (0°)(0 ) ( G  ternary  =  2 Si 0 "+3  be w r i t t e n a c c o r d i n g t o e q u a t i o n  temperature  0  _  Under e q u i l i b r i u m  -  =  =s- 12 0 +6 0°+3 0  2 0*  k  G  3  2 0"  may  reaction  =  8 0"+2  18 0~+3 0°  (d)  0  0° +  7  ~+  »_ S i 0  2 0" 3 Si 0  6 7  6 0%  ~  10 0"+0°  (c)  (a) t o  equationl,  8 0"  (b)  The f o l l o w i n g r e a c t i o n s  total  melt.  or  M a t e r i a l Balance  Considerations  F o r one mole o f any b i n a r y o r t e r n a r y s i l i c a t e melt i n v o l v i n g m e t a l o x i d e s which c o n t r i b u t e one oxygen i o n p e r m o l e c u l e , i t may be shown that, NSi0 and  moles o f s i l i c a  2  ( l - N S i 0 ) moles o f 0  =  2  combine t o form t h e s i l i c a t e  anions  ions,  o f the melt p l u s f r e e oxygen i o n s ,  assuming complete d i s s o c i a t i o n of the m e t a l Throughout t h i s  oxides.  i n v e s t i g a t i o n m e t a l o x i d e s which c o n t r i b u t e one  oxygen i o n t o the melt have been c o n s i d e r e d  and t h e above m a t e r i a l  w i l l be r e f e r r e d t o as the b i n a r y o r t e r n a r y m a t e r i a l balance slag.  balance  p e r mole o f  T h i s i s c o n s i s t e n t w i t h b i n a r y and t e r n a r y phase diagrams p l o t t e d on  a mole f r a c t i o n b a s i s ,  EVALUATION OF IONIC FRACTIONS OF IONS IN SILICATE MELTS  S o l u t i o n o f the E q u i l i b r i u m Equation  Equation  2 may be s o l v e d f o r ( 0 ° ) , (0") and ( 0 ) a t any com=  p o s i t i o n of a binary o r ternary s i l i c a t e melt.  The m a t e r i a l balance  states  that, NSi0 and  2  moles of  silica  (1-N) moles o f 0~ i o n s  combine t o form the a n i o n i c p o r t i o n o f t h e m e l t .  A charge balance  gives  the f o l l o w i n g e x p r e s s i o n , 2(0°)  + (0")  =  which may-be r e a r r a n g e d  4N = number o f s i l i c o n  bonds  to g i v e t h e c o n c e n t r a t i o n o f d o u b l y bonded oxygen  atoms i n one mole o f s l a g , (0°)  -  4N - (0-)  2  3  -6M a t e r i a l balance (0 )  «  =  c o n s i d e r a t i o n s g i v e t h e c o n c e n t r a t i o n o f f r e e oxygen i o n s ,  (1-N) - (OjO - 2 - 2N - (0") 2  4  2  = number o f moles o f oxygen i o n s c o n t r i b u t e d by t h e m e t a l minus t h e number o f moles o f oxygen i o n s complexed w i t h Equations in  3 and 4'may be i n s e r t e d i n t o e q u a t i o n  oxides silicon.  2 t o give a q u a d r a t i c  equation  (0"*) i n terms o f k and N S i 0 . 2  4k  =  (4N - ( 0 ~ ) ) ( 2 - 2N - ( 0 " ) )  5  (0-)2 A s e r i e s of quadratic equations  a(0~)2 + b(0") + G was d e r i v e d from e q u a t i o n NSi0 .. 2  =  i n the s t a n d a r d  form  0  5 and i s g i v e n i n Table  1 f o r various values o f  TABLE  1  Q u a d r a t i c E q u a t i o n s i n Terms o f (0~) and k f o r V a r i o u s V a l u e s o f N S i 0 . 2  NSi0  .  2  a(0")  2  2  +  b(0")  +  c  =  0  -  2.1(0")  +  .38  =  0  -  2.2(0")  +  .72  =  '0  -  2.3(0°)  + 1.02  =  '0  -  2.4(0")  + 1.28  =  0  -  2.5(0")  + 1.50  =  0  -  2,6(0")  + 1,68  =  0  -  2.7(0")  + 1.82  =  0 -  -  2.8(0")  + 1.92  =  0  -  2.9(0")  + 1.98  =  0  -  3.0(0")  + 2.00  =  0  .05  (l-4k)(0 :  .10  .2 (l-4k)(0~;  .15  (l-4k)(0~;  .20  ,2 (l-4k)(0~;  .25  (l-4k)(0~;  .30  ,2 (l-4k)(0~;  .35  (l-4k)(0~  .40  (l-4k)(0~;  .45  ,2 (l-4k)(0~,  .50  (l-4k)(0~;  .55  (l-4k)(0~; 2  -  3.1(0")  + 1.98  =  0  .60  (l-4k)(0"  -  3.2(0")  + 1.92  =  0  .65  .2 (l-4k)(0~;  -  3.3(0")  + 1.82  =  0  .70  ,2 (l-4k)(0~:  -  3.4(0")  + 1.68  =  0  .75  (l-4k)(0 ;  -  3.5(0")  + 1.50  =  0  .80  .2 d^kKO"'  -  3.6(0")  + 1.28  =  0  .85  (l-4k)(0 ;  -  3.7(0")  + 1.02  =  0 .  .90  .2 (l-4k)(0~;  -  3.8(0")  +  .72  =  0  .95  (l-4k)(0~;  -  3.9(0")  +  .38  =  0  _  2  ,2  )  2  2  Z  2  _  _  2  )2  The  e q u a t i o n s of Table 1 may  be s o l v e d by the mathematical  solution,  2a and of  v a l u e s of NSi0  (0°), (0°) and  ( 0 ) have been c a l c u l a t e d f o r v a r i o u s v a l u e s =  and k (see T a b l e 2 ) .  2  The v a l u e s of ( 0 " ) , (0°), and m a t e r i a l and  charge  ( 0 ) g i v e n i n Table 2 s a t i s f y the =  b a l a n c e of the s l a g , and t h e problem o f  the t o t a l number of n e g a t i v e i o n s i s reduced silicon,  (0~) and  discrete silicate  t o one  determining  of d e t e r m i n i n g  (0°) a r e assembled a t any s l a g c o m p o s i t i o n t o  how  form  anions.  D e t e r m i n a t i o n of t h e Number of S i l i c a t e  Anions  in Silicate  Melts  With the use of t h e t r i a n g u l a r t e r n a r y phase diagram shown i n F i g u r e 2,  i t was  found t h a t s i l i c a t e  anions can be arranged  polymerization pattern i f s i l i c o n  into a very regular  i s always c o n s i d e r e d t o be  c o - o r d i n a t e d w i t h f o u r oxygen atoms.  The p l o t was  c o n s t i t u e n t s of t h e a n i o n s , s i l i c o n S i ^ - ^ ,  tetrahedrally  made by p l a c i n g the  s i n g l y bonded oxygen 0~,  d o u b l y bonded oxygen 0 ° , a t the c o r n e r s of the t e r n a r y diagram and c e i v a b l e s i l i c a t e anions f a l l on a s t r a i g h t l i n e t h e most h i g h l y i o n i z e d form, SiCr,.  .  4  and  a l l con-  j o i n i n g pure s i l i c a  with  I t should be noted t h a t F i g u r e 2 i s  not a phase d i a g r a m i n the o r d i n a r y sense but i s a g r a p h i c a l i l l u s t r a t i o n o f t h e p r o p o r t i o n s of S i ^ " ^ , 0",  and 0° i n any d i s c r e t e s i l i c a t e  With the use o f F i g u r e 2 a v e r y e f f e c t i v e method may for  d e t e r m i n i n g t h e most probable number o f s i l i c a t e  s i l i c a t e melt  f o r any v a l u e of N S i 0  2  and  k.  anion.  be d e r i v e d  anions p r e s e n t i n a  T h i s i s done by t a k i n g the  l i n e a r i o n i c p l o t of F i g u r e 2 and p l o t t i n g i t as the a b s c i s s a a g a i n s t the number o f s i l i c o n atoms p e r i o n (see F i g u r e 3 ) . bonded oxygen atoms i n any s i l i c a t e given  The p r o p o r t i o n of s i n g l y  a n i o n , v a r y i n g from  zero t o .8,  and  by,  0" + 0° +  S iT T v T  p r o p o r t i o n o f s i n g l y bonded oxygen atoms i n any s i l i c a t e a n i o n .  TABLE 2 E q u i H b r i u m Number o f Moles of S i n g l y Bonded Oxygen, Doubly Bonded Oxygen and Free Oxygen Ions Per Mole o f S l a g , C a l c u l a t e d f o r V a r i o u s Values o f N S i 0 k NSiO;» ( o - )  ,05 .10 .15 .20 .25 :3o .35 .40 .45 .50 .55 .60 .65 .70 .75  (o ) a  (o ) =  .199 ,0002 .850 .398 . 0 0 1 .700  .597 ,002 .786 .007  .552 .407  ,018  .268  ,040 ,120 .230 ,370  .140 .065 .030  .965  1.12  1.17  1,14  1.07 .980 .510 .888 .650 .791 ,800  .694 .950 .597 1,10 .500 1.25  .015  .010 .006 .005 ,003 ,002 .001  (0°)  k = .06  k = .02  k = .01  .005  (0 )  (o-) (0°) ( o )  .199 .0005 .851 .702 .395 ,002 .588 .006 .556  .190 i o o i ;855 .391 .005 :?05  (o-)  .773 . .937 1.66 1.12 1.11 1.04' .963  .013 .031 .067 .140 .244 .379 .518  .880 .660 .784 .808  .692 .954 ,594 1.10 .494 1.25  =  .413 ,281 .168  ,090 ,045 .029 ,019 .010 ,008  .004 ,003 ,002  =  .576 .012 .562 .750 .025 :425 .891 .055 .305 .995 .103 .203 1.05 .175 .125 1.04 .280 ,080 .995 .403 .053 .935 ,533 .033 .853 .673 .023 .772 .814 .014  .679 . 9 6 1 , 0 1 1 .587 1.10 ,007 .494 1.25 ,003  (o-) ( 0 ) e  2  and k.  k = .12  (0")  .197 .003 .852 .375 .013 .713 .539 . 0 3 1 .581 .677 .062 .462 .790 .105 .355 -.869 .165 ,265  .908 .246 .196 .915 .343 ,143 .895 .453 .103  .850 .575 ,075 .796 .702 .052 .723 ,839 .039 .645 .978 ,028 .565 1,12 .018 .480 1,26 .010  (0-)  ( 0 ° ) (0=)  ,192 .005 .854 .356 .022 .722  .500 .615 .702 .760 .798 .807 .798 .769  .050 ,600 .093 .493 .149 .399 .220 .320 .301 .251 ,397 .197  .501 .151 .616 .116 .730 .735 .085 .672 .864 .064 .615 .993 .043  .538 1.13 . 0 3 1 . 4 6 1 1.27 ,020  k = .25  (0-)  (0°)  (o ) =  ,181 .010 .860 .327 .037 .737  ,443 .533 .600 .646 .673 .685 .683 .666  .079 .134 .200 .277 ,364 .458 .559 .666  ,629 .534  .450  .377 .314 .258 .209  .166 .639 .781 .131  .600 .900 .551 1.02 .494 1.15 0 428 1.29  .100 .075 .053 ,036  I  vO  I  Figure 2. Ternary Plot of the pro-  Versus t h e number o f S i l i c o n Atoms p e r I o n .  -12was  used n u m e r i c a l l y as t h e a b s c i s s a i n F i g u r e 3«  value o f N S i 0 any  2  p r o v i d e s the p r o p o r t i o n o f s i n g l y bonded oxygen atoms i n  melt,  (o-) «T) and  The v a l u e o f k and t h e  =  + (0°) + N S i 0  NO  =  p r o p o r t i o n o f s i n g l y tonded oxygen atoms i  n  a  2  f i x e s the c o m p o s i t i o n  n  ^  m  e  l  t  -  a l o n g the a b s c i s s a o f F i g u r e 3»  s i l i c o n atoms p e r i o n i s then d i v i d e d i n t o N S i 0 number o f s i l i c a t e  anions  present  The number o f  t o give the most  2  at that composition.  probable  The p r i n c i p l e used i n  t h i s method i s t h a t the p r o p o r t i o n of s i n g l y bonded oxygen atoms i n t h e e n t i r e melt i s a good measure o f the p r o p o r t i o n o f s i n g l y bonded oxygen atoms i n t h e predominant s i l i c a t e anions  present.  E x a m i n a t i o n o f F i g u r e 2 shows t h a t simple SijrO ^*^ 3X+I  c h a i n i o n s o f the form  occupy v a r i o u s p o s i t i o n s on the t e r n a r y diagram, but t h a t  2  2  simple  x-  r i n g 10ns o f the form S i 0  a l l o c c u r a t one p o i n t , and some o f t h e com-  x  3  X  p l e x r i n g i o n s such as S i ^ 0 ^ ~ and SigO^~Q o c c u r at one p o i n t .  This  intro-  duces some choice i n t h e p o s i t i o n o f t h e curve o f F i g u r e 3. Beginning the f i r s t let  a t t h e Si0£  p a r t o f t h e curve  p o s i t i o n , t h e f o l l o w i n g a n a l y s i s shows t h a t  i s mathematically  the b i n a r y or t e r n a r y m a t e r i a l b a l a n c e ,  fixed.  NSi0  F o r any v a l u e  moles o f s i l i c a  2  o f k,  and ( l - N )  moles o f 0~ i o n s , combine t o form, ( 0 ) moles o f f r e e 0 =  =  ions. 2  y moles o f a simple  c h a i n s t r u c t u r e o f t h e form S i 0 wx  3 x +  Cx+l)]_.  2  u moles o f a simple A m a t e r i a l balance =  =  (0 )+4(xy+uw)+y =  =  structure S i 0 w  3 W  on the t o t a l number o f atoms shows t h a t ,  (0 )+xy+3xy+y+uw+3uw (0 )+4xy+y+4uw  ring  =  3N+1-N  1+2N  =  1+2N  but xy+uw=N= number o f s i l i c o n atoms so 4(xy+uw) and  (0 )+y =  = =  4N 1-2N  6  -13E q u a t i o n 6 shows, i f any number o f moles o f simple i n the presence  chain ions i s considered  o f any number o f moles o f simple r i n g i o n s , t h a t t h e sum o f  t h e 0" i o n s p l u s the c h a i n i o n s i s always e q u a l t o l - 2 N S i 0 .  Equation 6  2  g i v e s t h e e x a c t number o f n e g a t i v e i o n s f o r low s i l i c a melts and up t o the c o m p o s i t i o n where simple r i n g - t y p e i o n s j u s t  s t a r t t o form.  The e q u a t i o n  would g i v e a good e v a l u a t i o n o f the t o t a l number o f n e g a t i v e i o n s i n v e r y basic  slags. I t i s c o n c e i v a b l e t h a t b o t h c h a i n and  ring-type s i l i c a t e  anions  can become v e r y l o n g o r ' i n f i n i t e ' i n l e n g t h , g i v i n g p o i n t s i n F i g u r e 3 t h a t can have v e r y l a r g e numbers o f s i l i c o n atoms per i o n . •infinite*  s t r u c t u r e s are c o n s i d e r e d t o be v e r y improbable  view of the u n i f o r m i t y o f the melts  However, t h e s e from the p o i n t o f  and t h e curve o f F i g u r e 3 was drawn  g i v i n g c o n s i d e r a b l e weight t o the s m a l l e r - s i z e d a n i o n s .  Another u s e f u l  con-  s i d e r a t i o n i n drawing t h e curve o f F i g u r e 3, i s the v a l e n c y o f the  anions,  which may be c a l c u l a t e d by d i v i d i n g  Figure 3  may  (0~) by the number o f a n i o n s .  be c o n s t r u c t e d w i t h n u m e r i c a l v a l u e s g i v e n i n Table 3.  D e t e r m i n a t i o n o f the I o n i c F r a c t i o n o f Oxygen  Ions  I n t h e d e t e r m i n a t i o n of the i o n i c f r a c t i o n s o f p o s i t i v e and n e g a t i v e ions the equations o f Temkin^ have been used i e , NM  ++  = sum  N0~  7  nM++ of p o s i t i v e ions  =  [CQ sum  8  of negative ions  W i t h the v a l u e s o f ( 0 " ) , (0°) and N S i 0  2  g i v e n i n T a b l e 2, t h e number o f  s i l i c a t e anions was c a l c u l a t e d u s i n g F i g u r e 3 and e q u a t i o n 6. f r a c t i o n o f oxygen i o n s was determined k and N S i 0 . 2  Temkin's i o n i c  by e q u a t i o n 8 , f o r v a r i o u s v a l u e s o f  The r e s u l t s are shown i n F i g u r e U, and t h e n u m e r i c a l v a l u e s  are g i v e n i n Table 3. The most s t r i k i n g  and s i g n i f i c a n t f e a t u r e o f F i g u r e 4 i s t h a t the  -14-  F i g u r e 4. Temkin's I o n i c F r a c t i o n o f Oxygen Ions Versus NSi0 f o r V a r i o u s V a l u e s o f k. 2  -15-  F i g u r e 5. A c t i v i t i e s o f V a r i o u s M e t a l Oxides i n B i n a r y M e l t s w i t h S i l i c a w i t h Respect t o the Pure L i q u i d Oxides.  TABLE 3 N u m e r i c a l V a l u e s , Expressed Per Mole o f S l a g , Used i n the C a l c u l a t i o n o f t h e T o t a l Number o f Negative Ions and Temkin's I o n i c F r a c t i o n o f Oxygen Ions f o r V a r i o u s V a l u e s o f NSi0  o  x>  CO  s  •H  NSiO, .05 ,10 ,15  .20 .25  .30 .35 .40 .45 .50  .55 .60 .65 .70 .75  i o  .796 .797 .797 .791 .732 .767 .713 .643 .566 .492 .425 .361 .303 .250 .200  •H  CO  U  U  1.00 1.00 1.01 1.03 1.08 1.15 1.48 2.10 3.05 4.15  5.35 7.20 10.3 15.0 24.0  II  .050 .100 .148 .193 .232 .260 .236 .190 .147 .120 .103  .850 .700 .552 .407 .268 .140 .065 .030 .015 .010 .006 .083 .005 .063 ,003 .047 .002. .031 .001  <  II  .  O  .900 .945 .800 .875 .700 .790 .600 .678 .500 .536 .400 .350 .301 .216 .220 .136 .162 .093 .130 .077 .109 .055 .088 .056 .066 .045 .049 .041 .032 .046  CO  .798 .794 .790 .784 .769 .743 .695 .632 .557 .486 .421 .358 .301 .248 .198  c o  CO  •H  .O  _  U P< a>  1.02 1.02 1.04 1.07 1.14 1.29 1.60 2.23 3.16 4.20 5.45 7.30 10.4 15.2 24.5  U CO  .  CJ  CO X>  H •rl  (0 C cd O -P -rt O C EH  •8  •H  r-i  O  a,  0)  -p cd o  •rl r-i  X>  k=.02  0)  CO  -p cd  c  and k.  k= .01  k=.005  o : ,0) •H .  2  in C  o •ri c  s *  r-i  o co mc  cn C  cd o -p • H  oc  EH  c  •r-i  II  o is  ..  CO  U  .  <0 ft  1.11 .851 .900 .945 .775 .702 .800 .878 .789 1.05 .556 .700 .794 .781 1.09 .413 .600 .688 .769 1.14 .281 .500 .561.745 1.28 .168 .400 .420 .711 1.50 .090 .308 .292 .668 1.87 .045 .224 .201 .604 2.54 .029 .176 .165 .538 3.44 .019 .138 .138 .475 4.35 .010 .111 .090 .411 5.70 .008 .090 .089 .370 6.90 .062 .004 .066 .061 .296 10.7 .046 .003 .049 .061.245 15.6 .031 .002 .034 .082 .198 24.5  .049 .098 .144 .187 .219 .232 .218 .179 .147 .119 .101 .082  43  •H iH •H  CO  .  S  o  •t-i  c  CD  l»  II  o  cd c -P O •rt C EH cd  o  .855 .900 .950 .095 .705 .800 .882 .138 .562 .700 .802 .045  .175 .425 .195 .305 .200 .203 .187 .125 .157 .080  .131 .053 .115 .033 .096 .023 .087 .014 .061 .011 .045 .007 .031 .003  .600 .500 .403 .312 .237 .184 .148 .119 .101 .072 .052 .034  .708  .610 .503  .401 .337 .288 .223 .193  .137 .153 .134 .088  TABLE 3 - CONTINUED CD  k=,06  a> -p cd o  -p cd o  •ri  H c  >  NSi0 05 10 15 20 25 30 35 40 45 c50 .55 .60 .65 .70 .75  O  •H  U 2  0) co  a  to c cd o -P •H o C  CD  1s  H  CD  Ei  42  •rl  cd  791 1.04 .048 852 .900 768 1.15 .087 713 .800 748 1.26 .119 581 .700 721 1.45 .138 .462 .600 690 I.64 .152 .355 .507 651 2.01 .149 .265 .414 604 2.56 .137 .196 .333 552 3.25 .123 .143 .266 498 4.10 .110 .103 .213 .441 5.00 ,100 .075 .175 .388 6.35 .087 .052 .139 .334 8.40 .071 .039 .110 .284 11.7 o055 .028 .083 .237 16.7 .042 .018 .060 .192 2.6.3 .029 .010 .039  II  o  .945 .892 .830 .770 .700 .640 .589 .537 .483 .428 .374 .354 .338 .300 .256  > o  .780 .745 .713 .677 .638 .593 .550 .504 .456 .407 .362 .314 .272 .227 .186  c O •rl  CO  ^  cn H C cd o •H b C cd  43  •H <D to  a.  1.09 1.28 1.50 1.75 2.16 2.70 3.24 3.92 4.70 5.80 7.20 9.40 12.7 18.2 23.0  .046 .078 .100 .114 .116 .111 .108 .102 .096 .086 .076 .O64 .051 .038 .027  .854 .722 .600 .493 .399 .320 .251 .197 .151 .116 .085 .064 .043 .031 .020  .900 .800 .700 .607 .515 .431 .359 .299 .247 .202 .161 .128 ,094 .069 .047  Si  o •H  a>  r-f  CO  k=.25  •s  •H  43  •H  k=.012  > —I O  .950 .902 .857 .813 .775 .743 .700 .660 .611 .574 .527 .500 .457 .450 .425  o  .750 .705 .659 .615 .571 .528 .485 .446 .404 .363 .324 .286 .260 .211 .174  —'  c o -H  u  43  H •rl  CO  to  C cd o -p •H O C cd  43  •H CU CO ft  1.25 1.59 1.92 2.44 2.94 3.53 4.21 4.88 5.77 7.25 9.01 11.5 13.8 21.2 32.6  .040 .063 .078 .082 .085 .085 .083 .082 .078 .069 .061 .052 .047 .033 .023  .860 .737 .629 .534 .450 .377 .314 .258 .209 .166 .131 .100 .075 .053 .036  .900 .800 .707 .616 .535 .462 .397 .340 .287 .235 .192 .152 .122 .086 .059  II O  .955 .921 .890 .867 .841 .815 .791 .759 .728 .706 .682 .657 .615 .615 .610  -18shapes o f the NO"  curves are almost  i d e n t i c a l t o t h e shapes o f t h e  curves o f v a r i o u s metal o x i d e s i n b i n a r y s i l i c a t e melts Comparison of F i g u r e s 4 and of N0  =  may  activity-  (see F i g u r e 5 ) .  5 shows t h a t f o r an a p p r o p r i a t e v a l u e o f k, v a l u e s  be o b t a i n e d which are n e a r l y e q u a l t o t h e v a l u e s o f a c t i v i t y f o r  the v a r i o u s m e t a l o x i d e s .  S i n c e T e m k i ^ s i o n i c f r a c t i o n o f the p o s i t i v e i o n  i n b i n a r y s i l i c a t e melts i s equal t o u n i t y , t h i s  i n d i c a t e s t h a t i n the  ionic  6 definition aMO  o f a c t i v i t y o f a m e t a l oxide = NM yM N0 v0 + +  + +  =  MO, 9  =  t h e product of t h e a c t i v i t y c o e f f i c i e n t s o f the p o s i t i v e i o n and the oxygen i o n , i s approximately equal t o u n i t y .  In a subsequent s e c t i o n , i t w i l l  shown t h a t the a c t i v i t y o f oxygen i o n s , a0~, s h o u l d be e q u a l t o the of t h e m e t a l oxide i n a b i n a r y s i l i c a t e m e l t , and  be  activity  i t f o l l o w s t h a t the  activity  c o e f f i c i e n t o f t h e p o s i t i v e i o n s h o u l d be i d e n t i c a l l y e q u a l t o u n i t y . To used  summarize b r i e f l y , the f o r e g o i n g s t r u c t u r a l a n a l y s i s may  be  to c a l c u l a t e t h e t o t a l number o f n e g a t i v e i o n s i n b i n a r y and t e r n a r y  s i l i c a t e m e l t s , and hence Temkin's i o n i c f r a c t i o n of oxygen i o n s may c a l c u l a t e d i n terms o f N S i 0  2  and k.  be  I n t h i s development, the v a l u e o f k has  not been i n t e r p r e t e d i n terms o f the thermodynamics of t h e d i f f e r e n t  binary  o r t e r n a r y m e l t s , a l t h o u g h i t has been shown t h a t f o r an a p p r o p r i a t e v a l u e o f k, values of N0° may  be  c a l c u l a t e d t h a t are n e a r l y e q u a l t o the  v a l u e s o f v a r i o u s m e t a l oxides i n b i n a r y s i l i c a t e m e l t s .  activity  This r e s u l t i s  s t r o n g l y supported m a t h e m a t i c a l l y by the a p p l i c a t i o n of the Gibbs-Duhem equation to i o n i c s i l i c a t e  APPLICATION OF THE  Temkin's^ concept  melts.  GIBBS-DUHEM EQUATION TO IONIC SILICATE MELTS  of an i o n i c melt i s a random a r r a y o f c a t i o n s  i n t e r l o c k e d w i t h a random a r r a y of anions and  this  concept  suggests t h a t the  a c t i v i t i e s and a c t i v i t y c o e f f i c i e n t s of the p o s i t i v e and n e g a t i v e  components  -19o f an i o n i c melt might be i n t e g r a t e d i n d e p e n d e n t l y by the Gibbs-Duhem equation.  T h i s i d e a o f t h e s e p a r a b i l i t y o f the p o s i t i v e and n e g a t i v e  com-  ponents o f i o n i c melts has been used by Chipman and Chang' i n t h e i r a p p l i c a t i o n o f Temkin's r u l e t o l i q u i d Gibbs-Duhem e q u a t i o n was  i r o n oxide s l a g s .  a p p l i e d to the ions of l i q u i d s i l i c a t e  A g e n e r a l i o n i c Gibbs-Duhem e q u a t i o n was m e t a l o x i d e s , FeO The  and  CaO,  + +  + NCa dlog y C a  + +  + +  melts.  d e r i v e d f o r two  i n a t e r n a r y melt w i t h s i l i c a  general expression i s given  NFe dlog yFe  On t h i s b a s i s , t h e  divalent  (see Appendix A ) .  by, + dlog a0  + +  =  + NSi0  2  dlog aSi0  l-NSi0  =  2  io  0  2  The terms i n e q u a t i o n 10 d e a l i n g w i t h t h e p o s i t i v e i o n i c  components  7 o f the melt are i d e n t i c a l i n form w i t h those used by Chipman and Chang' i n t h e i r treatment NFe  + + +  dlog yFe  Examination  of t h e c a t i o n i c components o f l i q u i d i r o n o x i d e s l a g s i e , + NFe dlog yFe  + + +  + +  =  + +  0  of e q u a t i o n 10 shows t h a t the c h a r a c t e r i s t i c o f the  independent  i n t e g r a b i l i t y o f t h e a c t i v i t i e s and a c t i v i t y c o e f f i c i e n t s o f the p o s i t i v e n e g a t i v e components of s i l i c a t e m e l t s , may r e w r i t i n g the e q u a t i o n i n two NFe dlog yFe + +  dlog a0  =  +  + +  NSi0  dlog a S i 0  2  l-NSi0  2  by  expressions,  + NCa dlog y C a + +  be v e r y e f f e c t i v e l y r e a l i z e d  and  =  + +  0  11  = 0  12  2  I f equations  11 and  12 may  be j u s t i f i e d by Temkin's concept  o f the  s e p a r a b i l i t y o f t h e p o s i t i v e and n e g a t i v e components o f i o n i c m e l t s , i t may be shown t h a t i n a b i n a r y s i l i c a t e melt the a c t i v i t y o f t h e m e t a l p r e s e n t i s e q u a l to the a c t i v i t y o f oxygen i o n s a 0 . =  oxide  I t f o l l o w s t h a t the  a c t i v i t y c o e f f i c i e n t f o r t h e p o s i t i v e i o n s must be i d e n t i c a l l y e q u a l t o u n i t y . C o n s i d e r i n g the F e 0 - S i 0 and  2  b i n a r y system o f t h e C a 0 - F e 0 - S i 0  a p p l y i n g e q u a t i o n 11 w i t h N G a  + +  equal to zero, gives,  2  t e r n a r y system  -20NFe dlog y F e + +  Since  NFe  =  + +  yFe  »  + +  =  + +  0  1  dlog y F e and  =  + +  0  constant  Examination o f equation  12 under t h e same c o n d i t i o n s shows t h a t i n t h e  a n i o n i c p o r t i o n of the melt, (l-NSi0 ) dlog a 0  =  2  but  i n the Fe0-Si0  =  - NSi0  dlog  2  aSi0  2  b i n a r y system,  2  • (l-NSi0 )  -  2  NFeO  and NFeO d l o g aFeO Therefore  i n the F e 0 - S i 0  a0 and  =  =  =  yFe  + +  - NSi0  2  dlog  aSi0  2  binary,  2  aFeO =  constant  =  1.  F u r t h e r s t u d y o f t h e above e q u a t i o n s some r e s t r i c t i o n o n t h e d i v i s i o n  of equation  T h i s becomes e v i d e n t by l e t t i n g a S i 0 dlog a0 and  =  =  2  i n d i c a t e s t h a t t h e r e must be 10 t o form equations  be constant  i n equation  11 and 12.  12, whereupon,  0  t h e a c t i v i t y o f oxygen would be constant  l i n e o f the C a 0 - F e 0 - S i 0  2  t e r n a r y system.  a l o n g an i s o - s i l i c a - a c t i v i t y  T h i s means t h a t a l l i s o - s i l i c a -  a c t i v i t y l i n e s would j o i n p o i n t s on the s i l i c a t e b i n a r i e s where aFeO aCaO, and t h i s although  i s h i g h l y improbable.  equation  system, equations  10 i s v a l i d  equals  Consequently, i t i s proposed t h a t ,  between any two compositions  11 and 12 a r e i n d e p e n d e n t l y  valid  of the ternary  only along  composition  paths where t h e r a t i o , NFe** NCa that  i s , along s t r a i g h t  lines  =  constant  + +  joining  gram w i t h t h e FeO-CaO b i n a r y system.  the s i l i c a apex o f t h e t e r n a r y  dia-  Darken^ Wagner^ and Schuhmann ^, make 1  -21use o f the same composition paths  i n a p p l y i n g t h e Gibbs-Duhem e q u a t i o n t o  t e r n a r y systems, and i t i s proposed  t h a t the same r e s t r i c t i o n s a p p l y t o  the i o n i c forms o f the e q u a t i o n d e r i v e d i n t h i s investigation„ j u s t i f i c a t i o n f o r t h i s assumption equations  A strong  i s g i v e n i n the f o l l o w i n g s e c t i o n , where  11 and 12 are used t o c a l c u l a t e t e r n a r y a c t i v i t y v a l u e s o f CaO  i n the Ca0-Fe0-Si0  2  system, which agree v e r y w e l l w i t h v a l u e s o b t a i n e d b y  the c o n v e n t i o n a l and f a m i l i a r forms o f t h e Gibbs-Duhem r e l a t i o n s h i p . An important of  r e s u l t o f the above a n a l y s i s i s t h a t t h e a c t i v i t y  oxygen i o n s may be d e r i v e d from t h e a c t i v i t y o f s i l i c a ,  along  straight  l i n e s j o i n i n g the s i l i c a apex o f the t e r n a r y diagram t o p o i n t s on the m e t a l oxide b i n a r y system where N 0 The  =  i s u n i t y and where a 0  =  i s d e f i n e d as u n i t y .  r e s u l t t h a t t h e a c t i v i t y o f a m e t a l oxide i n a b i n a r y melt w i t h  i s e q u a l t o the a c t i v i t y o f oxygen i o n s a 0 a n a l y s i s of t h i s  =  silica  agrees w e l l w i t h t h e s t r u c t u r a l  investigation.  APPLICATION OF IONIC GIBBS-DUHEM EQUATIONS TO THERMODYNAMIC DATA OF THE C a 0 - F e 0 - S i 0  On t h e b a s i s o f Temkin's r u l e , equations  2  SYSTEM  I I and 12 d e a l i n g w i t h t h e  p o s i t i v e and n e g a t i v e components o f s i l i c a t e m e l t s , were d e r i v e d from t h e g e n e r a l i o n i c Gibbs-Duhem e q u a t i o n 10. ionic  equations d e r i v e d may be o b t a i n e d by u s i n g measured a c t i v i t y data o f  the CaO-FeO=Si0  2  system.  The, CaO-FeO-Si0 and  An e v a l u a t i o n o f t h e v a l i d i t y o f t h e  system i s w e l l known and important  2  has been t r e a t e d i n some d e t a i l by E l l i o t t ^ .  d i s c r e p a n c i e s between t h e data o f E l l i o t t  i n metallurgy  However, t h e r e a r e some  on t h e C a 0 - S i 0  2  b i n a r y system and  12 t h e more r e c e n t d a t a o f Langenberg and Chipman of  s i l i c a i n the Ca0-Si0  2  „  Consequently  the a c t i v i t y  b i n a r y system was taken from a combination of  d a t a by Langenberg and Chipman (for  ( f o r low v a l u e s o f a S i O ) and by Richardson' a  h i g h v a l u e s o f a S i 0 ) and t h e a c t i v i t y o f CaO  c u l a t e d by a Gibbs-Duhem i n t e g r a t i o n the  i n the system was  2  FeO-Si0  2  a l s o t a k e n from E l l i o t t  All liquid  The a c t i v i t y d a t a o f  b i n a r y system and o f t h e FeO-CaO b i n a r y system was  E l l i o t t " ^ " and i s g i v e n i n T a b l e 4. system was  (see T a b l e 4)«  cal-  t a k e n from  The a c t i v i t y o f FeO i n t h e t e r n a r y and i s shown i n F i g u r e  6,  a c t i v i t y v a l u e s are g i v e n at 1600° C, w i t h r e s p e c t t o the pure  oxides. I t was  proposed t h a t t h e independent i h t e g r a b i l i t y o f the  a c t i v i t i e s and a c t i v i t y c o e f f i c i e n t s of the p o s i t i v e and n e g a t i v e components of  the Ca0-Fe0-Si0  a c t i v i t y of CaO aCaO where y C a , + +  =  t e r n a r y system might be checked by c a l c u l a t i n g t h e  2  i n t h e t e r n a r y system by, NCa yCa aO + +  + +  =  o f t h e c a t i o n i c p o r t i o n o f the melt may  e q u a t i o n 11, and a 0  =  be c a l c u l a t e d from  i n the a n i o n i c p o r t i o n o f t h e melt may be  • i n d e p e n d e n t l y by e q u a t i o n 12.  calculated  The r e s u l t i n g v a l u e s of aCaO i n t h e 11  t e r n a r y system can be checked w i t h t h e d a t a o f E l l i o t t  and by the t e r n a r y  t a n g e n t i n t e r c e p t method of Schuhmann^. C a l c u l a t i o n o f a0~ i n the CaO-FeO-Si0 Although t h e C a 0 - S i 0  of  2  3  System  b i n a r y system was  a l t e r e d , the ternary 11  the i s o - a c t i v i t y l i n e s f o r s i l i c a , g i v e n by E l l i o t t  guide t o c o n s t r u c t a t e r n a r y p l o t o f s i l i c a system  (see F i g u r e 7 ) .  , was  pattern  used as a  a c t i v i t y i n the Ca0-Fe0-Si0  The r e s u l t s were checked by t h e i n t e r c e p t method o f  Schuhmann^, u s i n g the i s o - a c t i v i t y l i n e s f o r FeO given i n F i g u r e  6.  I n t e g r a t i o n o f e q u a t i o n 12 g i v e s an- e x p r e s s i o n f o r t h e a c t i v i t y of  oxygen ions i n terms o f the a c t i v i t y o f s i l i c a  j o i n i n g the s i l i c a system,  2  along s t r a i g h t  lines  apex of the t e r n a r y diagram w i t h the CaO-FeO b i n a r y  -23-  TABLE 4 A c t i v i t y Data a t 1600°C o f t h e B i n a r y Systems, FeO-Si0  FeO-»Si0  C a 0 ~ S i 0 , and FeO-CaO w i t h r e s p e c t t o the Pure L i q u i d O x i d e s .  2 s  2  2  B i n a r y System  aFeO  NSi0  .96  .05 .10 .15 .20 .25 .30 .35  .93  .89 .83 .75  .64  .54  .46 .41 .38 .37 .36  .40 .45 .50 .55 .58  2  aSi0  2  Ca0-Si0 aCaO  .06  .92  .10 .14 .20 .30 .41 .56 .78 .91 .97  .69  .98 .98  .48 .28 a 14 .04 .013  .0068  .0031 .0021 .0012 .0005 .00005  2  B i n a r y System  FeO- CaO B i n a r y System  NSi0  NFeO  aFeO  .000015 .000105 .00038 .0013 .005  .05 .10 .15 .20 .25  .01  .10 .20 '  .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .'85  .14  2  .07 . 18  .26 .33 .39 • 45 .48  .49  aSi0  2  .03  .52  .40  .54 .58 .62 .67  .60 .80  .90 .95  .30  .02 .04 .06 .08 .11 .18 .23 .29  .36 .43  .50 .58  .67  .90  .75 .81 .88  .95  .94  aCaO  .94  .88 .8.1  .74 .66 .58 .50  .43 .36 .29 .25 .20 .15 .12 .09  .06 .04 .02 .01  1  9  W  .1*  .o3  .00^3  CO^P  0  003& .0'  ptv  Ya^ ?a  ooo^  1  £5 .3 .2  -26-  13  where a°0  i s t h e a c t i v i t y o f oxygen i o n s at t h e lower i n t e g r a t i o n  where a S i 0 = a ° S i 0 . 2  2  The v a l u e o f a°0  was chosen  limit  so t h a t t h e i n t e g r a t e d  v a l u e s o f aCT e x t r a p o l a t e d t o a0~ = 1, i n the FeO-CaO b i n a r y system. Three  composition paths  (see F i g u r e 7) were chosen and a r e d e t a i l e d  below, NFe  NCa  + +  a°Si0  + +  a°(f  2  1  .2  .8  ,000105  ,84  Composition  path 2  .5  .5  .000105  .94  Composition  path 3  .8  .2  ,0013  .89  Composition p a t h  The v a l u e s o f a0~ o b t a i n e d a r e g i v e n i n T a b l e 5 and a r e p l o t t e d as i s o a c t i v i t y l i n e s f o r oxygen i o n s i n F i g u r e 8.  TABLE 5 C a l c u l a t e d Values o f aO  aSi0  2  0 .000105 .00038 .0013 .005 .03 .2  .4 .6 .8  .9 .95  Composition -NSi0 2  0 .12 .21 .29  .36 .43 .48 .50 .53  .57 .61 .66  Path a0  1  =  1.0 .84  .65 .43 .23  .071 .015  .0076 .0049 .0034 .0029 .0026  i n t h e CaO-FeO-Si0 System. 2  Composition NSi0 2  0  Path 2 a0 =  1.0  .06 .13  .94 .82  .37  .17 .045 .025  .20 .29  .44  .47 .51 .55 .59  .64  .64 .41  .017 .012 .011 .010 •  Composition Path 3 NSi0 a0~ 2  0 ~-  1.0 —  __  .09 .17 .27  .89  .50  .062  .37 .42 .46 .54 .59  .73 .44 .17 .11 .080 .054 .051  -28C a l c u l a t i o n of yCa  and C a l c u l a t i o n o f T e r n a r y A c t i v i t y V a l u e s f o r CaO  ++  I n t e g r a t i o n o f e q u a t i o n 11 was performed a l o n g t h e same paths  1, 2 and 3, as i n t h e c a l c u l a t i o n o f a 0 . =  composition  The i n t e g r a t e d form o f  e q u a t i o n 11 i s g i v e n by, / log y C a  + +  =  —  yFe  / NFe /  + +  NCa  \J Fe -y°Fe ++  =yFe  dlog y F e  +  + +  log y°Ca  14  + +  + +  ++  K  The v a l u e s o f y F e  were o b t a i n e d w i t h t h e use o f F i g u r e s 6 and 8,  + +  yFe  =  + +  aFeO NFe aO + +  and  integrated values of y C a  =  were c a l c u l a t e d by e q u a t i o n 14.  + +  The t e r n a r y  v a l u e s of aCaO were t h e n c a l c u l a t e d by, aCaO  =  NCa yCa aO + +  + +  =  and t h e v a l u e s are g i v e n i n T a b l e 6 and p l o t t e d  as i s o - a c t i v i t y l i n e s f o r  CaO i n F i g u r e 9. The  c a l c u l a t e d i s o - a c t i v i t y l i n e s f o r CaO g i v e n i n F i g u r e 9 conform  v e r y c l o s e l y to the p a t t e r n t h a t may be d e r i v e d from E l l i o t t ' s t e r n a r y p l o t of l o g yCaO.  1 1  As a check on the d i r e c t i o n s o f the a c t i v i t y l i n e s f o r CaO,  t h e t e r n a r y tangent  i n t e r c e p t method o f Schuhmann"'"^ was a p p l i e d u s i n g t h e i s o -  a c t i v i t y l i n e s of FeO and S i 0 F i g u r e 9 was o b t a i n e d  2  given i n F i g u r e s 6 and 7.  Good agreement w i t h  (see F i g u r e 9 ) .  These r e s u l t s s u b s t a n t i a l l y c o n f i r m t h e v a l i d i t y o f e q u a t i o n s 12.  11 and  The above c a l c u l a t i o n s show t h a t t h e a c t i v i t y o f a component such as  l i m e , i n a t e r n a r y s i l i c a t e m e l t , may be c a l c u l a t e d w i t h the use of two i n dependent e q u a t i o n s , one o f which d e a l s w i t h the a n i o n i c components of t h e melt  and p r o v i d e s a 0 , t h e o t h e r w h i c h d e a l s w i t h t h e c a t i o n i c components o f t h e =  melt and p r o v i d e s values -of y C a CaO a r e c o n s i s t e n t w i t h those o f the Gibbs-Duhem e q u a t i o n .  + +  .  The r e s u l t i n g c a l c u l a t e d a c t i v i t y v a l u e s o f  o b t a i n e d by the c o n v e n t i o n a l and - f a m i l i a r forms  TABLE 6 C a l c u l a t e d T e r n a r y A c t i v i t y Values f o r CaO. Composition  Path 1  NFe =.2  NCa =.8  ++  NSi0 0 .06  .15  .20 .23 .27 .32 .38 .43  .49  .53 .58  2  .060 .10 .20 .30  .40 .50 .50  .40 .30 .25  .20 .15  NFe =.5  ++  aFeO yFe  a0  =  .30 1.0 .54 .93 1.32 .76 2.34 .64 3.57 5.56 7.81 11.8 25.0 62.5  143. 150.  .56  .45 .32 .17 .06 .02 .007 .005  Composition  Composition Path 2  yCa  + +  .93 .81  aCaO .74 .60  .64  .39  .55  .28 .22 .16 .10 .048 .015 .0040 .0011 .00079  .49 .44  .39 .35 .31  .25  .20 .20  NSi0 0 .05 .09 .12 .15 .20  .30 .35 .40 .45 .56  2  + +  .58 .85 1.12 1.45 1.85 2.50 3.79 5.00  8.33  17.8 54.5  aO  =  1.0  „ ++  yCa  .58  aCaO  .29  ++  N S i 0 aFeO y F e  + +  2  0  .02 .19 .05 .14 .83 .24 .099 .15 .76 .19 .071 .23 .29 .64 .14 .044 .37 .088 .016 .34 .0080 .24 .067 .39 .12 .040 .0024 .47 .56 .045.019 .00043 .OH .0062 .000034 .94 .40 .89 .31  NCa =.2  ++  ++  aFeO y F e .29 .40 .50 .60 .70 .80 .70 .60 .50 .40 .30  NFe =.8  NCa =.5  ++  .75 .80  .90 .90 .80 .70 .60  .50 .40  .36  Path 3  aO  .94 1.02 1.19 1.48 1.79 2.13 2.78  1.0 .98 .94 .76 .56 .41 .27  4.16  .15  6.67 8.82  .075 .051  yCa  + +  .30 .21 .11 .047  .022 .011 .0039 .0007  aCaO  .060 .041 .021 .0072 .0025 .00090 .00021 .000021  -31-  APPLICATION OF  One  IONIC THEORY TO THE  of the important  2  account  2  2  SYSTEM  i s taken of the number  p a r t i c l e s c o n t r i b u t e d t o the melt by a g i v e n o x i d e .  view the C u 0 - P b O - S i 0  2  r e s u l t s of applying i o n i c theory to slags i s  t h a t , i n the e x p r e s s i o n o f i o n i c f r a c t i o n s , of  Cu 0-PbO-Si0  system w i t h one  From t h i s p o i n t o f  d i v a l e n t m e t a l oxide PbO,  and  one  monovalent m e t a l o x i d e C u 0 , i s of i n t e r e s t , and a thermodynamic study o f 2  t h i s system was silicate  undertaken i n o r d e r t o i l l u s t r a t e the c h a r a c t e r i s t i c s o f a  system which c o n t a i n s b o t h d i v a l e n t and monovalent c a t i o n s . Although  thermodynamic d a t a i s a v a i l a b l e f o r the b i n a r y systems  PbO-Si0 , C u 0 - S i 0 2  2  and Cu 0-Pb0, no  2  2  v a l u e s i n the t e r n a r y system.  l i t e r a t u r e was  However, a l o n g composition  t e r n a r y system where the a c t i v i t y o f s i l i c a C u 0 and 2  PbO may  equilibrium. and PbO  be  determined  T h i s was  found  giving paths  activity i n the  i s constant, the a c t i v i t y of  e x p e r i m e n t a l l y by measurements o f  slag-metal  done by e q u i l i b r a t i n g s i l i c a s a t u r a t e d melts o f C u 0 2  w i t h c o p p e r - l e a d a l l o y s at 1100°C.  EXPERIMENTAL  E q u i l i b r i u m Conditions I f t e r n a r y melts liquid  of C u 0 , PbO 2  and  Si0  2  are h e l d i n c o n t a c t w i t h a  c o p p e r - l e a d a l l o y , e q u i l i b r i u m i s e s t a b l i s h e d by the r e a c t i o n , C u 0 + Pb  2Cu  2  + PbO  and a c h e m i c a l e q u i l i b r i u m c o n s t a n t , K, a t a g i v e n temperature may  15 be  written, K = a C u aPbO 2  aPb  l  6  aCu 0 2  The  a c t i v i t i e s o f copper and  l e a d i n the metal phase (see Appendix  B)  may  be o b t a i n e d by c h e m i c a l a n a l y s i s of the e q u i l i b r i u m a l l o y , and the r a t i o  of  -32the  a c t i v i t y o f C u 0 t o t h e a c t i v i t y o f PbO i s g i v e n by e q u a t i o n 16, 2  aCu 0 = a Cu  17  2  2  aPbO  aPb K  E q u a t i o n 17 may then be used w i t h the Gibbs-Duhem r e l a t i o n s h i p t o determine the  values o f  a C u 0 and aPbO over a range o f s l a g c o m p o s i t i o n s . 2  However,  it  f o l l o w s t h a t i f t h e Gibbs-Duhem e q u a t i o n i s a p p l i e d t o two components  of  a t e r n a r y system, the a c t i v i t y o f t h e t h i r d , s i l i c a , must be c o n s t a n t . The  r e q u i r e d e x p e r i m e n t a l c o n d i t i o n s were a c h i e v e d b y m e l t i n g  charges o f C u 0 , PbO, S i 0 2  2  and copper m e t a l a t 1100°C,  T h i s p e r m i t t e d the s l a g t o become s a t u r a t e d w i t h s i l i c a by r e a c t i o n w i t h t h e c r u c i b l e w a l l s .  insilica  crucibles.  (constant a S i 0 ) 2  The s l a g - m e t a l e q u i l i b r i u m was ex-  p e c t e d t o be e s t a b l i s h e d by the t r a n s f e r o f l e a d from t h e s l a g t o t h e copper m e t a l , a c c o r d i n g to r e a c t i o n 15. M a t e r i a l s and Equipment A l l m a t e r i a l s used were o f reagent grade.  The s i l i c a  c r u c i b l e s were  made o f two-inch l e n g t h s o f a p p r o x i m a t e l y o n e - i n c h i n s i d e diameter v i t r e o u s s i l i c a t u b i n g , c l o s e d at the bottom r e f r a c t o r y cement.  end w i t h s i l i c a  Examination o f t h e c r u c i b l e s a f t e r use showed no r e -  a c t i o n o f the m e l t s w i t h t h e composite The of  sand and ' S a i r s e t '  bottoms o f t h e c r u c i b l e s .  charges were melted i n a 'Globar' f u r n a c e a t a temperature  1100°C - 10°G.  C h e m i c a l A n a l y s i s o f Samples Both t h e s l a g phase and t h e m e t a l phase were a n a l y z e d f o r copper and l e a d by an X-ray f l u o r e s c e n c e method  developed i n t h e Department o f M i n i n g  and M e t a l l u r g y o f t h e U n i v e r s i t y o f B r i t i s h  Columbia,  The X-ray measurements were made on aqueous s o l u t i o n s o f t h e s l a g and m e t a l phases  and were c a l i b r a t e d w i t h a s e r i e s o f s o l u t i o n s  containing  known q u a n t i t i e s o f copper and lead i n the expected c o n c e n t r a t i o n ranges. The e n t i r e m e t a l phase of each experiment was d i s s o l v e d  and sampled  i n order  -33t o e l i m i n a t e the problem o f s e g r e g a t i o n o f l e a d i n t h e quenched a l l o y s . A n a l y s i s o f t h e c o p p e r - l e a d a l l o y s was c o n s i d e r e d t o be a c c u r a t e t o —5/^ o f t h e weight o f t h e major c o n s t i t u e n t p r e s e n t  (Cu) and t o *2% o f t h e  weight o f t h e minor c o n s t i t u e n t p r e s e n t ( P b ) , A n a l y s i s o f t h e s l a g s f o r copper  and l e a d p r o v i d e d v a l u e s o f  NCu 0 and NPbO i n a g i v e n weight of sample, and N S i 0 2  weight d i f f e r e n c e .  2  was c a l c u l a t e d by  The r e s u l t i n g v a l u e s o f N S i 0 , NPbO, and NCu 0 were 2  2  c o n s i d e r e d t o be a c c u r a t e t o about t2 mole p e r c e n t . C a l c u l a t i o n o f t h e E q u i l i b r i u m Constant K Although  a t h e o r e t i c a l v a l u e o f the e q u i l i b r i u m constant o f r e a c t i o n  15 may be c a l c u l a t e d w i t h t h e r m a l d a t a , t h e a c c u r a c y l i m i t s o f t h e d a t a a v a i l a b l e g i v e s r i s e t o c o n s i d e r a b l e u n c e r t a i n t y i n the v a l u e o f K„  How-  e v e r , a good e v a l u a t i o n ' o f Knuay be d e r i v e d from t h e e x p e r i m e n t a l work o f 15 Pelzel used  on e q u i l i b r i a i n t h e copper-lead-oxygen  system.  Although  Pelzel  c o n c e n t r a t i o n terms i n t h e e v a l u a t i o n o f t h e e q u i l i b r i u m c o n s t a n t , t h e  c h e m i c a l a n a l y s i s o f t h e e q u i l i b r i u m melts g i v e n may be used w i t h t h e a c t i v i t y d a t a o f t h e p r e s e n t work t o o b t a i n a v a l u e o f K i n terms o f a c t i v i t i e s . The  d a t a o f P e l z e l , i n t e r p o l a t e d between 1087°C and 1110°C, was  used t o c a l c u l a t e t h e c o m p o s i t i o n o f Cu 0-PbO s l a g i n e q u i l i b r i u m w i t h 2  Cu-Pb a l l o y a t 1100°C (see T a b l e 7 ) . TABLE 7 Compositions  o f E q u i l i b r i u m Melts o f  C u 0 , PbO, Cu and Pb a t 1087°C and 1110°C from 2  Weight  °c  Pb  %  1 Pelzel.  Mole F r a c t i o n s  PbO  NPb  NCu  NPbO  NCu 0 2  1087  29  86  .11  .89  .80  .20  1110  30.5  85.5  .12  .88  .79  .21  -34With the use o f t h e c o m p o s i t i o n data o f T a b l e 7 and  subsequent  a c t i v i t y d a t a c a l c u l a t e d from the Cu 0-Pb0 and Cu-Pb phase diagrams,  the  2  v a l u e of t h e e q u i l i b r i u m c o n s t a n t a t 1100°C may K « a2Cu(aPbO) = (.90)2(.65) = aPb(aCu 0) 2  (.73)  be c a l c u l a t e d t o  be,  6.0  (.12)  Although t h e a c c u r a c y of t h i s v a l u e o f K depends on the a c c u r a c y of the c a l c u l a t e d a c t i v i t y d a t a , i t was v a l u e a v a i l a b l e and was  used  c o n s i d e r e d t o be the most  i n the e x p e r i m e n t a l  satisfactory  results,  RESULTS  In o r d e r t o determine the r a t e of approach t o e q u i l i b r i u m i n the melts s t u d i e d , a s e r i e s of melts o f constant i n i t i a l  composition were h e l d  at  The r e s u l t s g i v e n i n  1100°C f o r times r a n g i n g from U5 t o 130 minutes.  Table 8 show t h a t t h e r e i s no s i g n i f i c a n t  change i n e i t h e r the s l a g com-  p o s i t i o n or t h e a l l o y composition a f t e r a h o l d i n g time o f more than hour.  A h o l d i n g time of 60 minutes a t 1100°C was  sequent  t h e r e f o r e used f o r the sub-  experimental melts. A s e r i e s o f charges were prepared w i t h pure  various i n i t i a l  s l a g compositions.  c o n s i d e r e d to be c l o s e t o the s i l i c a Pb0-Si0 for  one  2  t e r n a r y system  The  copper m e t a l shot  s l a g compositions  and  chosen were  s a t u r a t i o n l i n e across the Cu 02  (see F i g u r e 10).  The melts were h e l d at 1100°C  one hour and t h e n removed from the f u r n a c e and quenched i n water.  i n i t i a l and f i n a l compositions of t h e melts are shown i n Appendix C. f i n a l s l a g compositions are a l s o p l o t t e d i n F i g u r e 10. t h e e q u i l i b r i u m melts are numbered from 1 t o 14. are the e q u i l i b r i u m time runs of T a b l e  The The  In Appendix C,  Numbers 5, 6, 7 and  8  8.  Examination of t h e i n i t i a l and f i n a l s l a g compositions g i v e n i n Appendix C shows t h a t o x i d a t i o n o f t h e c o p p e r - l e a d a l l o y s has taken p l a c e i n the m e l t s .  This f i r s t  became apparent  e x p e r i m e n t a l l y w i t h a l o s s o f weight  -35-  TABLE 8 I n i t i a l and F i n a l Compositions o f M e l t s Held at  1100°C f o r 45, 60, 100 and 130 M i n u t e s .  SLAG PHASE  Initial Melt Time Number Minutes 5  45  6 7  60 100 130  8  NCu 0 2  .15 .15 .15 .15  Final  Composition NPbO  NSi0  .29 .29 .29 .29  Composition  NCugO  NPbO  NSi0  .56 .56  .15  .57  .56 .56  .16 .15  .28 .27 .27  2  .15  .26  METAL PHASE Initial Melt Number 5  6 7 8  Time Minutes 45 60 100 130  Composition  Final  Composition  NCu  NPb  NCu  NPb  1 1 1 1-  0 0 0 0  .99 .99  .0019  .99 .99  .0027 .0021 .0024  .58  .57 .59  2  .1  .2  .3  .4  .5  .6  .7  .8  .9  -37of  t h e m e t a l phase.  f i n a l slags.  The r e s u l t  i s an i n c r e a s e o f t h e C u 0 content o f t h e 2  However, i t may be shown t h a t e q u i l i b r i u m was e s s e n t i a l l y  m a i n t a i n e d i n the melts b y p l o t t i n g t h e r a t i o o f the a c t i v i t y o f C u 0 t o t h e 2  a c t i v i t y o f PbO v e r s u s NCu 0, a l o n g t h e s i l i c a s a t u r a t i o n l i n e , and t h e r e 2  s u l t s show a smooth c o n t i n u o u s curve as would be expected  (see F i g u r e 11 and  Appendix C ) . In  the Cu 0-Pb0-Si0 2  2  t e r n a r y system,  along the s i l i c a  saturation  l i n e a t 1100°C, t h e a c t i v i t i e s o f C u 0 and PbO a r e r e l a t e d by t h e G i b b s 2  Duhem e q u a t i o n , NCu 0 d l o g a C u 0 2  =  2  - NPbO d l o g aPbO  F i g u r e 11 gives t h e r a t i o o f a C u 0 t o aPbO as a f u n c t i o n o f NCu 0, 2  aCu 0 2  aPbO  =  2  a Cu 2  aPb K  Combination o f these two e q u a t i o n s g i v e s , d l o g aPbO = - NCu 0 2  d l o g asCu  NCu 0 + NPbO 2  1  aPb K  and t h e a c t i v i t y o f PbO may be c a l c u l a t e d by i n t e g r a t i o n o f e q u a t i o n 18 u s i n g t h e v a l u e o f aPbO on the P b 0 - S i 0 gration. aPbO,  2  b i n a r y as t h e lower l i m i t o f i n t e -  V a l u e s f o r aCu 0 may t h e n be o b t a i n e d by t h e r a t i o o f a C u 0 t o 2  2  The r e s u l t s a r e shown i n T a b l e 9. TABLE 9 E x p e r i m e n t a l T e r n a r y A c t i v i t y V a l u e s o f PbO and C u 0 2  along the S i l i c a S a t u r a t i o n Line o f the Cu 0-Pb0-Si0 2  NCu 0 2  NPbO  aHCu aPb K  aPbO  2  System a t 1100°C.  aCu 0 2  •  .01 .05 .10 .15 .20 .25  .30 .35 .40 .45 .50  .35 .33 .30 .28  .26 .24 .21 .19 .17 .14 .12  .83 2.7 5.0 8.3 12.0 16,2  21.2 26.7 32.5 40.0 50.3  .058 .053 .047 .040 .035 .030 .026 .023 .020 .017 .0l4  •  .048 .14 .23 .33 .42 .49 .55 .61  .64 .68 .71  8  NCu 0 Along S i l i c a S a t u r a t i o n L i n e at 1100°C 2  F i g u r e 11„ E x p e r i m e n t a l l y Determined R a t i o o f a C u 0 t o aPbO v e r s u s NCu 0 Along t h e S i l i c a S a t u r a t i o n Line o f t h e Cu 0=-PbO-Si0 System a t 1100°G. 2  2  2  2  - 3 9 -  COMPILATION OF THE AVAILABLE THERMODYNAMIC DATA FOR THE BINARY SYSTEMS Pb0-Si0 , Cu 0-Si0 2  2  C a l c u l a t i o n o f A c t i v i t y Data from Phase  16 Chipman  2  and Cu 0-Pb0 2  Diagrams  18 and Rey  have shown t h a t i n c e r t a i n types o f b i n a r y  phase diagrams, t h e l i q u i d u s curve o f a component may be used'to c a l c u l a t e i t s a c t i v i t y , p a r t i c u l a r l y i f the s o l i d zero or v e r y l i m i t e d i n e x t e n t . l o g a ( a t T°K)  s o l u b i l i t y o f t h e component i s  The e x p r e s s i o n used i s ,  = A S f (T-Tm)  1  9  4.575 (T) where A S f i s t h e e n t r o p y of f u s i o n o f t h e c o n s t i t u e n t and Tm i s t h e m e l t i n g p o i n t i n degrees K„ The method was a p p l i e d t o t h e C u 0 - S i 0 2  2  and Cu 0-Pb0 systems and 2  r e s u l t s o f a t l e a s t approximate v a l i d i t y were e x p e c t e d . The C u 0 - S i 0 a  2  System 19 20  The C u 0 - S i 0 2  2  phase diagram  terminal solid s o l u b i l i t i e s ,  ''  showing a simple e u t e c t i c , no  5  and a wide i m m i s c i b i l i t y gap on t h e S i 0 2  r i c h s i d e , was used t o c a l c u l a t e t h e a c t i v i t i e s o f S i 0  2  and C u 0 i n t h e 2  b i n a r y system. The a c t i v i t y o f s i l i c a was c a l c u l a t e d a l o n g t h e S i 0 by e q u a t i o n 19>  2  liquidus  line  and account was t a k e n o f t h e t r a n s i t i o n o f c r i s t o b a l i t e t o  t r i d y m i t e a t 1743°K u s i n g t h e d a t a g i v e n  below, Tm  Cristobalite  13 Tridymit e  A  s  f  1986°K  1.56 eu.  1953°K  1.81 eu.  I n c a l c u l a t i n g t h e a c t i v i t y o f C u 0 , t h e i o n i c Gibbs-Duhem e q u a t i o n 2  d e r i v e d i n Appendix D f o r t h e C u 0 - P b 0 - S i 0 2  Cu 0-Si0 2  2  2  system, was a p p l i e d t o t h e  b i n a r y and i t was found t h a t t h e i o n i c e q u a t i o n w i t h aG"~ « a C u 0  reduces t o t h e f a m i l i a r  2  form.  -40NCu 0 d l o g a C u 0 2  just  =  2  - NS10  2  dlog aSi0  2  as i n t h e case o f a d i v a l e n t m e t a l o x i d e i n a b i n a r y melt w i t h  silica.  r; Consequently, the  t h e above e q u a t i o n was used, and a c c o r d i n g t o Darken and Gurry  function, logySiQ  2  (l-NSi0 )2 a  was i n t r o d u c e d i n t o t h e e q u a t i o n p e r m i t t i n g e v a l u a t i o n o f the a c t i v i t y of C u 0 i n t h e b i n a r y system from NCu 0 = 1 t o t h e i m m i s c i b i l i t y gap (see 2  2  T a b l e 10). The c o n s t a n c y o f the f u n c t i o n (.837) from NCu 0 » 1 t o about 2  NCu 0 = .16, a l s o p e r m i t t e d an a c c u r a t e e v a l u a t i o n o f t h e a c t i v i t y o f s i l i c a 2  i n t h i s r e g i o n where t h e a c t i v i t y o f s i l i c a  i n c r e a s e s v e r y r a p i d l y (see  T a b l e 11). TABLE 10 A c t i v i t i e s o f C u 0 and S i 0 2  NCu 0  aCu 0  .95  .95 .92 .88 .86 .83 .81 .79 .76  2  2  i n the Cu 0-Si0 2  NSi0  2  .90 .85 .80  .75 .70 .65 .60  .55 .50  2  2  2  .28 .48 .60 .69 .75 .81 .86 .90 .94 .97 .99  .'05  .15 .20 .25  .30 .35 .40 .1*5  .74  S i n c e the C u 0 - S i 0  aSi0  2  .10  .50 .56  .72 .70  .44  System a t 1100°C.  2  phase diagram shows no s o l i d  s o l u b i l i t y on  the C u 0 - r i c h s i d e , a melt i s i n c o n t a c t w i t h pure C u 0 where t h e C u 0 2  2  l i q u i d u s temperature ing of  i s 1100°C.  T h i s p o i n t i s 130 degrees  p o i n t o f Cu 0 and t h e a c t i v i t y o f C u 0 i s .89. 2  2  2  below t h e melt-  The e n t r o p y o f f u s i o n  cuprous o x i d e can t h e r e f o r e be e s t i m a t e d t o be 2.5 eu. by e q u a t i o n 19.  T h i s v a l u e f o r t h e e n t r o p y o f f u s i o n o f C u 0 i s c o n s i d e r a b l y lower 2  t h e v a l u e 7.5 g i v e n b y R i c h a r d s o n  and B i l l i n g t o n . ^ " * "  than  However, i t i s found  -41TABLE 11 A c t i v i t y of S i l i c a i n t h e C u 0 - S i 0 2  System a t 1100°C  2  f o r Low V a l u e s o f N S i 0 . 2  NSi0  logySi0  2  aSi0  2  (l-NSi0 )  2  a  2  .01 .02 .03 .04 .05 .06 .07 .08 .09 ,10 .11 .12 .13 .14 .15 .16  .066 .13 .18 .24 .28 .33 .37 .41 .44 .48 .51 .53 .56 .58 .60 .62  .837 .837 .837 .837 .837 .837 .837 .837 .837 .837 .837 .837 .837 .837 .837 .837  i n t h e c a l c u l a t i o n o f t h e a c t i v i t i e s o f C u 0 and PbO from the Cu 0-Pb0 2  2  phase diagram, t h a t the e n t r o p y of f u s i o n o f C u 0 must n e c e s s a r i l y be low 2  i n o r d e r t o g i v e i n t e g r a t e d v a l u e s o f aPbO compatible w i t h an e n t r o p y o f f u s i o n of PbO o f 6 - .5 c a l . p e r degree f r o m t h e measured l e a d o x i d e a c t i v i t y data o f  ftichardson  and Webb."^  The Cu 0-Pb0 System 2  The Cu 0-Pb0 phase diagram b y Hofmann and Kohlmeyer  was used t o  2  c a l c u l a t e t h e a c t i v i t i e s o f Cu 0 and PbO i n t h e b i n a r y system-. 2  For t h e i n t e g r a t i o n o f t h e a c t i v i t i e s o f Cu 0 and PbO, t h e i o n i c 2  Gibbs-Duhem e q u a t i o n d e r i v e d  i n Appendix D, f o r t h e Cu 0-Pb0-Si0 2  was a p p l i e d t o t h e Cu 0-Pb0 b i n a r y . 2  equation  P u t t i n g a S i 0 O and a0~ 2  s  2  system,  1 i n the  gives, NCu dlog y C u +  +  -  - NPb dlog yPb + +  + +  20  I n t h e Cu 0-PbO b i n a r y system, Temkin's i o n i c f r a c t i o n o f oxygen ions i s 2  u n i t y and t h e a c t i v i t y ' o f oxygen i o n s , a 0  =  i s d e f i n e d as u n i t y .  The a c t i v i t y  c o e f f i c i e n t s f o r t h e c a t i o n s may t h e r e f o r e be o b t a i n e d b y , yCu  +  =  aCu 0 2  NCu  and  yPb  ++  +  aPbO NPb The  ++  a c t i v i t i e s o f C u 0 and PbO a t T°K were c a l c u l a t e d 2  a l o n g t h e C u 0 and PbO l i q u i d u s l i n e s o f t h e phase diagram. 2  o f f u s i o n f o r C u 0 o f 2.5 eu, from t h e Cu 0-Si.0 2  e q u a t i o n 19.  2  system, was used i n  2  V a r i o u s v a l u e s f o r t h e entropy o f f u s i o n o f PbO were t r i e d  the v a l u e s o f y C u 20,  The e n t r o p y  +  and y P b  + +  until  a t 1100°G were m u t u a l l y i n t e g r a b l e by e q u a t i o n  T h i s c o n d i t i o n was s a t i s f i e d w i t h an e n t r o p y o f f u s i o n f o r PbO o f 5.6  eu. i n good agreement w i t h the v a l u e g i v e n by R i c h a r d s o n and Webb^. i n t e g r a t i o n o f e q u a t i o n 20 was extended  The  f o r both c a t i o n s by i n t r o d u c i n g t h e  functions, log yCu  +  (l-NCu )2 +  and  l o g yPb  ++  (l-NPb ) + +  2  from Darken and Gurry17  The r e s u l t s a r e shown i n T a b l e 12, TABLE 12  C a l c u l a t e d A c t i v i t i e s o f PbO and C u 0 i n t h e Pb0-Cu 0 System a t 1100°C. 2  NPbO  NPb  .18  .10 .20 .30 .40 .50 .60 .65 .70  .34 .46 .57 .67 .75 .79 .82 .89 .947  .80  .90  + +  yPb  1.70 1.55 1.37 1.22 1.10 1.00 .98 .97 .97 .99  + +  2  aPbO  NCu 0 2  NCu  .17 .31  ,82 .66  .90 .80  .41 .49 .55 .60 .64  .54 .43 .33  .70  .68 .78  .89  .25 .21 .18  oH .053  +  yCu  +  1.07  1.10 1.15  .60 .50  1.17  .40 .35 .30  1.12 1.03 1.00 1.00 1.00  .20  .10  1.18  aCu 0 2  .92 .78 .65 .49 .35 .20 .13  .09 .04 .01  -43The P b 0 - S i 0  System  2  In t h e P b O - S i 0  system, t h e measured a c t i v i t i e s of PbO  2  1100°C were taken from Richardson and Webb^ ". 1  for  and S i 0  at  2  However, the a c t i v i t y v a l u e s  s i l i c a were r e - e x p r e s s e d i n terms o f a l i q u i d s t a n d a r d s t a t e u s i n g 13  e q u a t i o n 19 and the t h e r m a l data f o r t r i d y m i t e g i v e n p r e v i o u s l y c o r r e c t e d d a t a i s shown i n Table  .  The  13. TABLE 13  Activities  o f PbO  and S i 0  2  i n the P b 0 - S i 0  from R i c h a r d s o n and Webb NPbO  aPbO  .95 .90 .85 .80 .75 .70 .65 .60 .55 .50 .45  .93 .85  .24 .17 .12 .091 .069 .058  2  2  The  2  SYSTEM WITH THE  2  .0012 .0035 .0082  .016 .031 .058 .10 .16 .26 .38 .50  .62  .65  CALCULATION OF TERNARY ACTIVITY VALUES OF Gu 0, PbO, Cu 0-Pb0-Si0  aSi0  2  .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60  .33  .40  System at 1100°C  e  NSi0  .76 .66 .54 .43  .35  2  .68  AND  SiQ  2  IN  THE  AID OF AN IONIC GIBBS-DUHEM EQUATION  g e n e r a l i o n i c Gibbs-Duhem e q u a t i o n d e r i v e d i n Appendix D f o r the  Cu 0-PbO-Si0 2  system was  2  v a l u e s f o r Cu 0, PbO  used as an a i d i n e s t a b l i s h i n g t h e t e r n a r y a c t i v i t y  and S i 0 .  2  The two  2  and n e g a t i v e components o f the melt may and are g i v e n by e q u a t i o n s 20 and NCu dlog yCu +  d l o g a0  =  »  »  +  —  e x p r e s s i o n s d e a l i n g w i t h the  be obtained from the g e n e r a l e q u a t i o n  12,  - NPb dlog yPb + +  NSi0  dlog aSi0  2  l-NSi0  2  positive  2  + +  20 12  -44The  a c t i v i t y d a t a o f the b i n a r y systems P b O - S i 0 , C u 0 - S i 0 , and 2  2  2  Cu 0-PbO, and t h e e x p e r i m e n t a l a c t i v i t y d a t a a c r o s s the s i l i c a s a t u r a t i o n 2  l i n e at 1100°C were p l o t t e d on a t r i a n g u l a r t e r n a r y phase diagram. activity lines  f o r s i l i c a were then drawn t o about the mid p o i n t o f the  diagram u s i n g t h e shape o f the s i l i c a Equation aO  s a t u r a t i o n l i n e , a t 1100°C, as a g u i d e .  12 was i n t e g r a t e d a l o n g t h r e e composition  i n this  Iso-  r e g i o n and t h e i s o - a c t i v i t y  the b i n a r y s i d e s .  lines  paths t o g i v e v a l u e s o f  f o r a0~ were e x t r a p o l a t e d t o  The i n t e g r a t i o n c o n s t a n t s f o r aO" were chosen so t h a t  the i n t e g r a t e d v a l u e s o f aO  e x t r a p o l a t e d t o u n i t y i n t h e Cu 0-Pb0 b i n a r y 2  system. The  v a l u e s o f aO  i n t h e lower p a r t o f t h e diagram were t h e n used i n  e q u a t i o n 12 t o e s t a b l i s h the i s o - a c t i v i t y values of N S i 0 . 2  The c o m p o s i t i o n paths  l i n e s o f s i l i c a f o r the lower  1, 2 and 3, and the i n t e g r a t i o n  c o n s t a n t s used a r e g i v e n below, NCu  NPb  +  + +  a°Si0  a°0~  2  Composition  Path  1  .4  .6  .20  .46  Composition  Path 2  .7  .3  .20  .66  Composition  Path 3  .9  .1  .20  .86  The n u m e r i c a l v a l u e s o f a0~ and activity lines for  for silica  aSi0  o b t a i n e d are g i v e n i n T a b l e  2  The i s o  14.  are shown i n F i g u r e 12 and t h e i s o - a c t i v i t y  lines  oxygen i o n s a r e g i v e n i n F i g u r e 13. The t e r n a r y i s o - a c t i v i t y  lines  f o r PbO were e s t a b l i s h e d by t h e  10 i n t e r c e p t i n t e g r a t i o n method o f Shuhmann silica  g i v e n i n F i g u r e 12.  p o s i t i o n paths  , u s i n g the i s o - a c t i v i t y  lines f o r  I n t e g r a t i o n was performed a l o n g the same com-  1, 2 and 3 as p r e v i o u s l y and the i n t e g r a t i o n c o n s t a n t s f o r  aPbO were t a k e n from the e x p e r i m e n t a l v a l u e s o f aPbO a l o n g t h e s i l i c a a t i o n l i n e at 1100°C. for  The r e s u l t i n g  PbO are shown i n F i g u r e 14.  calculated ternary i s o - a c t i v i t y  saturlines  •p-  I  T A B L E  C a l c u l a t e d V a l u e s o f aCf and a S i 0 aSi0  NSi0  2  .010 .050 .10 .20 .30  a0  2  .12  .23 .29 .44 .49  .50 .60 .65 .68  .99 .93  .08 .15 .20  .33  .26  .46  .36 .40  .40  NSi0  _  .53  .26 .21  .31 .35  .17  .39  .14 .13 .12  .57 .60  2  14  i n the Cu 0-Pb0-Si0 2  2  NSi0  .89 .77 .66 .53 .45  .03 .07  .39 .34 .30  .44 .47 .51  Path 1  a0~  .28 .27  Path 2  2  System a t 1100°C. a0  2  =  .96 .90 .86 .78  .10 .14 .18 .21  .72  .67 .63  .25 .30 .33 .38  .59  .57 .55 Path  3  The t e r n a r y a c t i v i t y v a l u e s f o r C u 0 were drawn w i t h the a i d o f t h e 2  i o n i c e q u a t i o n 20. w i t h F i g u r e s 13 and yPb  + +  =  The a c t i v i t y c o e f f i c i e n t s f o r t h e Pb  i o n were o b t a i n e d  14, aPbO  NPb aO ++  s:  ++  The v a l u e s o f yPb  were i n s e r t e d i n t o e q u a t i o n 20 and t h e e q u a t i o n was  g r a t e d a l o n g c o m p o s i t i o n paths  1,' 2 and 3 t o g i v e v a l u e s o f y C u . +  inte-  Values o f  a0~ were o b t a i n e d from F i g u r e 13 and t h e a c t i v i t y v a l u e s o f C u 0 were d e t e r 2  mined a l o n g t h e t h r e e c o m p o s i t i o n aCu 0 2  The  =  N Cu y Cu a0 2  +  2  +  paths,  =  Cuo0-Pb0 b i n a r y system was used  of values of y C u . +  as t h e lower l i m i t f o r the i n t e g r a t i o n  The r e s u l t i n g c a l c u l a t e d i s o - a c t i v i t y l i n e s f o r C u 0 i n the. 2  t e r n a r y system a r e shown i n F i g u r e 15, and the n u m e r i c a l v a l u e s are g i v e n i n Table 15.  The d i r e c t i o n s o f the i s o - a c t i v i t y l i n e s f o r C u 0 were checked 10  t e r n a r y tangent for Si0  2  by t h e  2  i n t e r c e p t method o f Schuhmann  and PbO g i v e n i n F i g u r e s 12 and 14.  was o b t a i n e d The  using the i s o - a c t i v i t y  lines  Good agreement w i t h F i g u r e 15  (see F i g u r e 15). a p p l i c a t i o n o f t h e i o n i c Gibbs-Duhem e q u a t i o n t o a s i l i c a t e  c o n t a i n i n g b o t h monovalent and d i v a l e n t and i t i s apparent  cations gives s a t i s f a c t o r y  system  results  t h a t t h e i o n i c forms o f t h e e q u a t i o n should be a p p l i c a b l e  TABLE 15 C a l c u l a t e d Ternary A c t i v i t y Values f o r Cu 0 a t 1100°C, 2  Obtained f r o m Values of a0~ and yCut Composition Path 1  Composition Path 2  Composition Path 3  NCu =.4  NCu =.7  NCu =.9  NPb =.6  +  NSi0  "0 .06  .12 .13 .24 .30 .40 .45 .50  2  ++  aPbO  yPb  .60 .53 .45 .36 .27 .19" .094 .065 .050  1.02 .97 .92 .37  + +  .80 .72 .61 .56 .52  a0~ 1 .91 .81 .70 .57 .44 ,26 .20 .16  yCu 1.12 1.20 1.30 1.43 1.62 1,88 2.45 2.79 3.13  +  +  aCu 0  NSi0  .20 .21 .22 .23 .24 .25 .25 .25 .25  0 .19 .26 .32 .38 .44  2  2  NPb =.3  aPbO  yPb  .41 .17 .13 .080 .054 .039  1.37 .81 .77 .62 .51 .44  + +  a0  =  1 .69 .55 .44 .36 .30  NPb =.l  +  ++  yCu  +  1.15 1.33 1.47 1.62 1.76 1.88  aCu C  NSi0  .65 .60 .58 .56 .54 .52  0 .07 .18  2  2  + +  aPbO  yPb  .17 .10 .058  1.70 1.11 .81  + +  aO  yCu  1 .90 .72  1.07 1.11 1.17  +  aCu 0 2  .92 .90 .80  t o any  combination  o f monovalent and  INTEGRABILITY OF THE  d i v a l e n t m e t a l oxides i n s i l i c a t e  ACTIVITIES OF NEGATIVE IONS IN BINARY SILICATE MELTS  A good c o r r e l a t i o n between the s t r u c t u r a l and mathematical ships derived i n t h i s the independent  melts,  i n v e s t i g a t i o n may  relation-  be o b t a i n e d by f u r t h e r a p p l i c a t i o n of  i n t e g r a b i l i t y o f the a c t i v i t i e s o f p o s i t i v e and n e g a t i v e i o n s ,  i m p l i e d i n Temkin's r u l e . The o x i d e s and  i o n i c Gibbs-Duhem e q u a t i o n 10 d e r i v e d f o r two  s i l i c a , may  NM-]_ dlog y M ++  + NM  + + 1  + + 2  metal  be r e w r i t t e n i n the g e n e r a l form, dlog yM  + + 2  + dlog a0  =  +NSi0  of e q u a t i o n 21 shows t h a t i t does not  dlog a S i 0  2  l-NSi0 Examination  divalent  2  = 0  21  2  c o n t a i n an e x p l i c i t  term  f o r t h e d i s c r e t e s i l i c a t e anions p r e s e n t i n the m e l t , but t h a t r e p r e s e n t a t i o n of the s i l i c a t e anions NSi0  dlog aS10  2  l-NSi0  i s i m p l i e d i n the term,  2  2  However, f o r an i o n i c melt  as d e f i n e d by Temkin, i t should be p o s s i b l e t o  w r i t e t h e Gibbs-Duhem e q u a t i o n i n the f o r m , NM-j^dlog a M  + + 1  + N M ^ d l o g aM^*  where a S i 0 y ~ r e p r e s e n t s the U  x  silicate  =  + (1-N0 )dlog =  aSi 0 x  combined a c t i v i t i e s of the predominant  anions present a t the c o m p o s i t i o n  e q u a t i o n s 21 and  considered,,  Examination  U y  ~ = 0  discrete of  22 shows t h a t t h e e x p r e s s i o n s d i f f e r o n l y i n form; whereas  the p r o p e r t i e s o f the s i l i c a t e are e x p l i c i t i n e q u a t i o n 22. 21 and  + N 0 d l o g aO"  anions are i m p l i c i t  i n e q u a t i o n 21,  On the o t h e r hand, the terms o f  they  equations  22 t h a t d e a l w i t h the p o s i t i v e i o n s are i d e n t i c a l s i n c e t h e forms  of the Gibbs-Duhem e q u a t i o n i n v o l v i n g a c t i v i t i e s o r a c t i v i t y  coefficients,  are synonymous. F o r the same reasons t h a t e q u a t i o n 21 was  d i v i d e d i n t o two  ex-  -52p r e s s i o n s a l o n g c o m p o s i t i o n paths where, NM]_ NM  ~ constant  ++  + + 2  e q u a t i o n 22 may  be w r i t t e n as,  NM^+dlog aM-J*  =  - NM^dlog aM  23  + + 2  and, NO dlog a 0 =  E q u a t i o n 24, activity of  =  =  - (1-N0 ) d l o g a S i O =  x  "  24  above, should t h e r e f o r e p r o v i d e a means o f c a l c u l a t i n g the  o f the predominant s i l i c a t e a n i o n s present at any  a b i n a r y s i l i c a t e melt where the a c t i v i t y The work of  fiichardson.  a wide range of a c t i v i t y grability  u y  o f oxygen i o n s a 0  and Webb ^ on the P b 0 - S i 0 1  data t h a t may  A p p l y i n g equation 12 t o the P b 0 - S i 0 =  2  2  i s known.  =  system p r o v i d e s  be used t o i l l u s t r a t e  o f the negative ions i n a b i n a r y s i l i c a t e  aCr  composition  the  inte-  melt.  b i n a r y system g i v e s ,  aPbO  From the above e q u a l i t y , i t i s evident t h a t i n o r d e r f o r the P b 0 - S i 0 to  approximate i d e a l i t y  as c l o s e l y as p o s s i b l e , v a l u e s o f NO"  should  system  2  be  c a l c u l a t e d from a value o f k such t h a t , N0 or  =  NCT  -  aCT  ~  aPbO  F i g u r e 16 shows t h a t when k = (0 )(CT) o  -  .01  (0")2 t h e c a l c u l a t e d v a l u e s o f N0 activity  =  are v e r y c l o s e t o t h e measured l e a d oxide  v a l u e s , at 1100°C, g i v e n by R i c h a r d s o n and Webb  14  . u-  I n t e g r a t i o n o f e q u a t i o n 24 of  a0 , =  g i v e s an e x p r e s s i o n f o r a S i 0 y x  i n terms  -53-  F i g u r e 16. A c t i v i t y of PbO at 1100°C i n the PbO~SiO B i n a r y System. Comparison i s made w i t h Values o f N 0 C a l c u l a t e d w i t h k= ( 0 ) ( 0 ) - .01. a  =  o  (0")2  =  -54-  where the v a l u e of a ° S i O x  c l o s e l y obeys R a o u l t ' s Law  v  ~ was  chosen by c o n s i d e r i n g t h a t s i n c e aO  i n compositions  near the lower  l i m i t of  inte-  ugration, that aSi Oy  a l s o obeys R a o u l t ' s Law  x  The aSi 0y. x  U  i n the same r e g i o n .  n u m e r i c a l v a l u e s of v a r i a b l e s p e r t i n e n t t o t h e c a l c u l a t i o n of  are g i v e n i n T a b l e 16.  The v a l u e s o f a 0  and a S i C y ~ are  =  x  u  p l o t t e d as a f u n c t i o n o f Temkin's i o n i c f r a c t i o n s i n F i g u r e 17 showing t h a t t h e a c t i v i t i e s o f the n e g a t i v e i o n s i n t h e P b O - S i 0  2  b i n a r y system, at  H O C C , are m u t u a l l y i n t e g r a b l e by the i o n i c e q u a t i o n  24.  TABLE 16 A c t i v i t i e s o f the Negative NPbO  aPbO=aO  .95 .90 .85 .80 .75 .70 .65 .60 .55 .50 .45 .40 .35  •fc  Ions i n P b 0 - S i 0  =  .93 .85 .76 .66 .54 .43 ' .33 .24 .17 .12 .091 .069 .058 *  a 0 c  2  M e l t s at 1100°G.  NO"  aS^O/.04 .12 .22 .34 .48 .61 .70 .78 .84 .90 .93 .95 .96  .95 .88 .79 .69 .56 .42 .29 .20 .17 .14 .09 .07 .04  =  Aft a ° S i 0 v  u T r  ' A good c o r r e l a t i o n between the s t r u c t u r a l a n a l y s i s  and the mathematical  a n a l y s i s of t h i s work may  be shown.  t h a t f o r an a p p r o p r i a t e v a l u e o f k, v a l u e s o f NO"  may  F i g u r e 16 shows  be o b t a i n e d t h a t  -55-  F i g u r e 17. A c t i v i t i e s o f t h e Negative Ions i n P b 0 - S i 0 M e l t s a t 1100°C.  2  are n e a r l y e q u a l t o the a c t i v i t y v a l u e s o f PbO i n b i n a r y s i l i c a t e  melts,  and i t was shown m a t h e m a t i c a l l y t h a t i n t h e b i n a r y system, a0  =  =  aPbO  T h i s means t h a t t h e oxygen i o n s a r e behaving n e a r l y i d e a l l y i n t h e a n i o n i c p o r t i o n o f the melt. the s i l i c a t e  I t i s i m p l i e d i n Temkin's r u l e t h a t t h e behaviour o f  anions i n t h e melt should a l s o be n e a r l y i d e a l , and t h i s i s  shown v e r y e f f e c t i v e l y i n F i g u r e 17.  DISCUSSION  In  the a n a l y s i s o f s t r u c t u r e and e q u i l i b r i a i n s i l i c a t e  melts,  the v a l u e o f t h e e q u i l i b r i u m constant k may be i n t e r p r e t e d i n terms o f t h e thermodynamics o f t h e d i f f e r e n t b i n a r y and t e r n a r y systems c o n s i d e r e d . M a t h e m a t i c a l l y i t was shown t h a t i n b i n a r y s i l i c a t e melts t h e a c t i v i t y o f oxygen i o n s a0~, s h o u l d be i d e n t i c a l t o t h e a c t i v i t y o f t h e m e t a l present.  I t f o l l o w s t h a t i n o r d e r f o r t h e b i n a r y system t o approach  a v a l u e o f k should be used to  oxide  such t h a t c a l c u l a t e d v a l u e s o f N 0  t h e a c t i v i t y o f t h e m e t a l oxide as p o s s i b l e .  shown t o be c o n s i s t e n t w i t h t h e a c t i v i t y ,  =  ideality,  a r e as c l o s e  T h i s c h o i c e o f k may be  i m m i s c i b i l i t y , and compound  f o r m a t i o n c h a r a c t e r i s t i c s o f t h e systems s t u d i e d . In  F i g u r e 18, t h e p o s i t i o n s o f the i m m i s c i b i l i t y gaps o f t h e b i n a r y  s i l i c a t e phase diagrams a r e shown.  Comparison o f F i g u r e 18 w i t h F i g u r e 5 shows  t h a t , w i t h the exception o f the Pb0-Si0 oxide, f o r a given S i 0 b i l i t y gap.  2  2  system, t h e a c t i v i t y o f t h e m e t a l  c o n t e n t , i n c r e a s e s w i t h t h e width o f t h e i m m i s c i -  T h i s may be e x p l a i n e d i n terms o f t h e e q u i l i b r i u m c o n s t a n t k.  F o r example, i f a s i l i c a t e melt s i l i c a content, t h e s i l i c a t e gap, s u c h as t h e C u 0 - S i 0 2  2  i s approaching  immiscibility with  a n i o n s i n a system w i t h a wide  increasing  immiscibility  system, would be p o l y m e r i z e d t o a h i g h e r degree  58 f o r a g i v e n s i l i c a content t h a n i n a system w i t h a narrower gap, such as C a O - S i 0 .  immiscibility  A h i g h degree o f p o l y m e r i z a t i o n r e q u i r e s a h i g h v a l u e  2  o f k and s h i f t s e q u a t i o n 1,  20" toward h i g h  +  0°  (0°) and h i g h  0  =  (0~) g i v i n g l a r g e polymers and a tendency toward  i m m i s c i b i l i t y , t o g e t h e r w i t h h i g h oxide  activity.  On t h e o t h e r hand, as t h e i m m i s c i b i l i t y gap narrows w i t h FeO,ZnO, and  CaO t h e systems show an i n c r e a s i n g tendency toward compound f o r m a t i o n ,  which lowers t h e v a l u e o f k by s h i f t i n g e q u a t i o n 1 toward h i g h (0°) and a maximum number o f d i s c r e t e s i l i c a t e a n i o n s . lowers  NO", g i v i n g lower oxide a c t i v i t y .  T h i s consumes oxygen i o n s and  Although t h e P b 0 - S i 0  2  system i s an  e x c e p t i o n t o t h i s g e n e r a l r u l e , t h e d e v i a t i o n i s moderate. A c c o r d i n g t o t h e above r e a s o n i n g , v a l u e s o f k a r e suggested various binary s i l i c a t e  f o r the  systems c o n s i d e r e d and a r e g i v e n i n Table 17.  TABLE 17 Proposed Values o f the E q u i l i b r i u m Constant  kfor  V a r i o u s B i n a r y S i l i c a t e Systems, System  Number o f Compounds  Cu 0-Si0 Fe0-Si0 ZnO-Si0 PbO-Si0 CaO-Si0 2  2  2  l e s s than  2  Although temperature, of the S i 0  2  .005  i t i s suggested  t h a t a r e l a t i o n s h i p may e x i s t between t h e s l o p e  l i q u i d u s curves o f b i n a r y s i l i c a t e  melts behave almost  phase diagrams and t h e  temperature.  I t i s apparent  apparent.  1100 1600 1300 1100 1600  no attempt was made t o e v a l u a t e the v a r i a t i o n o f k w i t h  dependence o f k on  not  0 1 1 2 2  ,25 .06 .01 .01  ?  2  Temperature o f Melt °C  t h a t f r o m an i o n i c p o i n t o f view, b i n a r y  ideally.  silicate  On a molar b a s i s , i d e a l i t y i n t h e s e melts i s  -59In t h e t e r n a r y s i l i c a t e systems CaO=FeO-Si0  2  and C u 0 - P b 0 - S i 0 , t h e 2  2  oxygen i o n tends t o behave as an i d e a l p a r t i c l e , i n t h a t a l l v a l u e s o f are w i t h i n the range o f v a l u e s f i x e d by the b i n a r y s i l i c a t e systems.  a£T On  the  o t h e r hand, examination o f the behaviour o f the c a t i o n s i n the t e r n a r y melts i n d i c a t e s t h a t a l t h o u g h the major c a t i o n p r e s e n t tends t o behave the behaviour o f the minor c a t i o n p r e s e n t may  ideally,  be q u i t e n o n - i d e a l .  A p p l i c a t i o n o f i o n i c Gibbs-Duhem e q u a t i o n s t o t e r n a r y s i l i c a t e  systems  i s q u i t e h e l p f u l i n d e t e r m i n i n g t h e t e r n a r y i s o - a c t i v i t y l i n e s f o r the stituents.  T e r n a r y v a l u e s o f a0~ and a S i 0  2  con-  are m u t u a l l y i n t e g r a b l e by the  i o n i c Gibbs-Duhem e q u a t i o n which d e a l s w i t h t h e a n i o n i c p o r t i o n o f the m e l t . The a c t i v i t i e s o f the m e t a l oxides p r e s e n t may  then be c a l c u l a t e d w i t h the  use o f t e r n a r y i n t e g r a t i o n t e c h n i q u e s and the i o n i c e q u a t i o n d e a l i n g w i t h the p o s i t i v e i o n s i n t h e  melt.  CONCLUSIONS  A method has  been developed f o r d e t e r m i n i n g the most p r o b a b l e number  of a n i o n s i n b a s i c and a c i d s i l i c a t e  s l a g s i n terms o f an e q u i l i b r i u m  s t a n t k i n v o l v i n g s i n g l y bonded oxygen, doubly bonded oxygen, and oxygen i o n s i n the m e l t .  The v a l u e o f k depends on the type and  t r a t i o n o f c a t i o n s p r e s e n t , and i s such t h a t , i n b i n a r y s i l i c a t e  free concenmelts,  Temkin's i o n i c f r a c t i o n of oxygen i o n s i s c l o s e t o , o r e q u a l t o , the of the metal oxide present.  T h i s c o n c l u s i o n was  supported  con-  activity  mathematically  by a p p l i c a t i o n o f t h e Gibbs-Duhem r e l a t i o n s h i p t o i o n i c s i l i c a t e  melts.  Some i o n i c forms o f the Gibbs-Duhem e q u a t i o n were d e r i v e d and were a p p l i e d s u c c e s s f u l l y t o the Ca0-Fe0-Si0  2  system  and t o t h e C u 0 - P b 0 - S i 0 2  system. T e r n a r y a c t i v i t y v a l u e s o f C u 0 and PbO were determined 2  experi-  2  mentally along the s i l i c a  s a t u r a t i o n l i n e of' the C u 0 - P b O ~ S i O  1100°C.  The shape o f the s i l i c a  was  established.  also  2  saturation  l i n e across  g  system at  the t e r n a r y  diagram  -61-  REFERENCES  1.  J.O'M. B o c k r i s , J.A. K i t c h e n e r , and S. I g n a t o w i c z , T r a n s . Faraday S o c , 41, 75 (1952).  2.  J.O'M. B o c k r i s , J.A. K i t c h e n e r , and A.E. D a v i e s , J . Chem, Phys., 12, 255 (1950).  3.  J.O'M. B o c k r i s , J.D. Mackenzie, S o c , 51, 1734 (1955).  4.  J.O'M. B o c k r i s , J.W. Tomlinson, and J . L . White, T r a n s . Faraday S o c , 52, 299 (1956).  5.  C.J.B. Fincham, and F.D. R i c h a r d s o n , P r o c  6.  M. Temkin, A c t a P h y s i c o c h i m . , U.R.S.S., 20, 411 (1945).  7.  J . Chipman, and L. Chang, J . M e t a l s , 1, 191 (1949).  8.  L.S. Darken,  9.  C. Wagner, Thermodynamics o f A l l o y s pp. 19-22.  and J.A. K i t c h e n e r , T r a n s . Faraday  Roy. S o c , A223, 40, (1954).  J . Amer. Chem. S o c , 72, 2909 (1950). (Addison-Wesley P r e s s , 1952),  10.  R. Schuhmann, J r . , A c t a Met., 3_, 219 (1955).  11.  J . F . E l l i o t t , J . M e t a l s , 6, 1&5 (1955).  12.  F.C. Langenberg,  13.  F.D. R i c h a r d s o n , The P h y s i c a l C h e m i s t r y o f M e l t s , I n s t . M i n . Met.,  and J . Chipman, T r a n s . A.I.M.E., 215. 958 (1959).  London (1953) p. 83. 14.  F.D. R i c h a r d s o n , and L.E. Webb, T r a n s . I n s t . Min. Met., 64_, 529  (1954-55).  Metall.,  15.  E . P e l z e l , Z. E r z .  11, 247 (1958).  16.  J . Chipman, D i s c u s s i o n s Faraday S o c ,  17.  L.S. Darken and R.W. G u r r y , P h y s i c a l C h e m i s t r y o f M e t a l s , (McGrawH i l l Book Co., New York, 1953), p . 264.  18.  M. Rey, D i s c u s s i o n s Faraday S o c ,  19.  A.S. B e r e z h n o i , L . I . K a r y a k i n , and I . F . D u d a v s k i i , Doklady Nauk, S.S.S.R., 8_3_, 401, (1952).  20.  E.M. L e v i n , H.F. McMurdie, and F.P. H a l l , Phase Diagrams f o r C e r a m i s t s , Amer. Ceram. S o c , Columbus (1956), p. 36.  4, 23 (1948).  257 (1948). Akad.  -6221.  F „ D . R i c h a r d s o n , and J.C. B i l l i n g t o n , 65_, 273 (1955-56).  22.  W. Hofmann, and J . Kohlmeyer, Z. M e t a l l k u n d e , 4JL, 339 (1954).  23.  P.A. Beck, Metals Handbook, American  24.  O.J. Kleppa, and J.A. W e i l , J . Amer. Chem. S o c , 72,  25.  W.G.  26.  R.G. B u t t e r s , and M. Z o g o v i c , u n p u b l i s h e d r e p o r t , U n i v e r s i t y o f B r i t i s h Columbia (1959).  27.  J . F . E l l i o t t , and M. G l e i s e r , Thermochemistry (Addison-Wesley P r e s s , I960), p. 48.  Davenport,  private  T r a n s . I n s t . Min. Met.,  S o c i e t y f o r M e t a l s , (1948), p. 1200. 4848 (1951).  communication.  f o r Steelmaking,  APPENDIX A  D e r i v a t i o n o f a G e n e r a l I o n i c Gibbs-Duhem E q u a t i o n f o r Two D i v a l e n t M e t a l Oxides CaO and FeO i n a Ternary Melt w i t h S i l i c a . The i o n i c e x p r e s s i o n s aFeO  -  NFe  aO  =  aCaO  =  NCa yCa aO  =  + +  yFe  f o r aFeO and aCaO a r e given by,  + +  + +  + +  S i n c e Temkin's i o n i c f r a c t i o n s o f t h e p o s i t i v e i o n s N F e constant  + +  and N C a  + +  are  a l o n g s t r a i g h t l i n e s j o i n i n g t h e s i l i c a apex o f t h e CaO-FeO-Si0  phase diagram w i t h t h e CaO-FeO b i n a r y system, i t may be shown (see F i g u r e A l ) t h a t f o r any composition NFeO NFe  «  + +  NCaO NCa  =  l-NSi0  i n t h e t e r n a r y diagram,  2  + +  A p p l y i n g the f a m i l i a r Gibbs-Buhem NFeO d l o g aFeO + NCaO d l o g aCaO + N S i 0 and  2  equation,  dlog aSi0  = 0  2  i n s e r t i n g t h e i o n i c forms o f aFeO and aCaO g i v e s ,  NFeO d(NFe++yFe^*aQ'') NFe yFe aO~ + +  +  NCaO d(NCa+ vCa+ aO' ) + NSiO, d l o g a S i 0 NCa yCa aO +  + +  + +  + +  +  = 0  e  2  n i  Introducing, NFeO NFe  =  + +  NCaO NCa  =  l-NSi0  2  + +  gives, d(NFe+ yFe* aO ) + d ( N C a y C a a O ) +  +  =  + +  yFe a0~  + +  yCa aO  + +  ++  =  +  NSiOg d l o g a S i 0 l-NSi0  ffi  The i n d i c a t e d d i f f e r e n t i a l s may be o b t a i n e d , d(NFe yFe aO ) + +  and  + +  =  + +  + +  using, dNCa  ++  +  dNFe  + +  «  0  and (NCa  + +  + NFe )dlog a0 + +  =  =  d l o g aO"  =  0  2  f o r example,  «= N F e y F e d a O " + N F e a O " a y F e + +  2  + +  + yFe aO dNFe + +  =  + +  2  F i g u r e A l . Geometry o f the CaO-FeO-Si0 T e r n a r y Diagram. The f i g u r e s i n brackets i l l u s t r a t e the r e l a t i o n s h i p , NFeO = NCaO = l - N S i 0 a t various compositions. NFe NCa 2  2  + +  CaO  Z  1 .2  i  .4  + +  I  .6  V -  8  - 6  a g e n e r a l i o n i c Gibbs-Duhem e q u a t i o n may be o b t a i n e d f o r two d i v a l e n t o x i d e s i n a t e r n a r y melt w i t h NFe dlog yFe + +  + +  silica,  + NCa dlog y C a + +  + +  + d l o g aCT + N S i 0 d l o g 2  l-NSi0  2  aSi0  2  -  0  metal  5  -  APPENDIX B  C a l c u l a t i o n o f A c t i v i t y Data i n the Copper-Lead The a c t i v i t i e s o f copper and l e a d c u l a t e d from t h e Cu-Pb phase diagram  23  .  s i d e was t a k e n from Kleppa and W e i l ^ .  System.  i n c o p p e r - l e a d a l l o y s were c a l The l i q u i d u s curve o f t h e P b - r i c h The e n t r o p y o f f u s i o n used f o r copper  27 was 2.29 eu.  The r e s u l t s  are shown i n T a b l e B l .  • E x a m i n a t i o n o f T a b l e B l shows t h a t i n v e r y d i l u t e s o l u t i o n s o f l e a d i n copper the a c t i v i t y o f copper i s e q u a l t o i t s mole f r a c t i o n R a o u l t ' s Law).  (obeys  I t f o l l o w s t h a t t h e a c t i v i t y o f l e a d must obey Henry's  ( c o n s t a n t ' a c t i v i t y c o e f f i c i e n t ) i n t h e same c o m p o s i t i o n range.  Law,  The v a l u e o f  t h e c o n s t a n t a c t i v i t y c o e f f i c i e n t f o r l e a d was e v a l u a t e d f o r a p p l i c a t i o n t o the c o p p e r - r i c h a l l o y s t h a t were expected e x p e r i m e n t a l l y . TABLE B l C a l c u l a t e d A c t i v i t i e s of Cu and Pb i n the Cu-Pb System at 1100°C. NCu  aCu  NPb  aPb  .967 .929 .88 .82 .31  .97 .93 .90 .88 .88 .82 .59 .39 .24 .12 .052 .018  .033 .071 .12 .18 .69 .75 .88 .934 .966 .984 .994 .998  .52 .73 .84 .84 .85 .91 .95 .97 .98 .99 .99  .25 .12. .066 .034 .016  .006 .002  .26  A p l o t o f the f u n c t i o n , logyPb (l-NPb)2 v e r s u s NCu was e x t r a p o l a t e d t o i n f i n i t e  d i l u t i o n and i t was found t h a t as  (1-NPb) approaches u n i t y , t h e a c t i v i t y c o e f f i c i e n t f o r l e a d approaches a cons t a n t v a l u e of 8.3.  APPENDIX C I n i t i a l and F i n a l Compositions o f E q u i l i b r i u m M e l t s a t 1100°C. Melt Number  I n i t i a l Compositions S l a g Phas e M e t a l Phase NPb NCu 0 NPbO N S i 0 NCu 2  2  1 2  ,05 .05  .33  3  .10 .10 .15 .15 .15 .15 .20  .30  4 5  6 7  8  9 10 11 12  13  14  .20 .25 .30 .  .33 .30 .29 .29 .29 .29  .26  .26 .24  .56 .56  .56 .54 .54  .51 .48  .45  .35  .40  .62 .62 .60 .60 .56  .18  1 1 1 1 1 1 1  T  _L  1 1 1  1 1  1  0 0 0 0 0 0 0 0 0 0 0 0 0 0  F i n a l Compositions M e t a l Phase S l a g Phas e NPb aCu aPb NCu NCu 0 NPbO N S i 0 2  2  .062 .057 .24 .14  .15 .15 .16 .15 .33 .29  .34 .32 .25 .29 .28 .27 .27 ,26 .23  .49  .12  .43 .50  .15 .14 .13  .46  .60  .99  .62  .99 .99  .51 .57 .57 .58  .57 .59 .44 .48  .39 .42  .36  .41  .99 .99  .99 .99 .99 .99  .0058 .0060 .0011 .0027 .0019 .0027 .0021  .0024  .99  .00070 .0011  .99 .99 .99 .99  .00063 .00039 .00048  .00041  .99 .99  .99 .99 .99  .99 .99 .99 .99  .99 .99 .99 .99  .99  .048 .050 .0091 .022 .016 .022 .017 .020 .0058 .0091 .0034 .0052 .0032 .0040  a Cu aPb 2  20.3 19.7 108. 43.7 62.0 43.7 56.3 49.2 169. 108. 288. 187. 302.  246.  a2Cu aPb K  314 3.3  18.0 7.3 10.3  7.3 9.4 8.2 28.2 18.0 48.0 31.2 50.3 41.0  -68APPENDIXJD  D e r i v a t i o n o f a G e n e r a l I o n i c Gibbs-Duhem E q u a t i o n f o r One D i v a l e n t M e t a l Oxide PbO, One Monovalent M e t a l Oxide C u 0 , and S i l i c a . 2  The  i o n i c e x p r e s s i o n s f o r aPbO and a C u 0 a r e g i v e n by, 2  aPbO and'  -  aCu 0  NPb yPb aO t +  •=  2  + +  B  NSCuVCu+aO"  V a l u e s o f NPbO a r e a f u n c t i o n o f t h e geometry o f t h e t e r n a r y phase d i a NFF gram and a r e shown i n F i g u r e D l . The r a t i o NCugO may a l s o be r e p r e s e n t e d +  P e l ?  g e o m e t r i c a l l y and v a l u e s a r e g i v e n i n F i g u r e D2.  I t may be shown, a l o n g  l i n e s where, NjPb** NCu*"  constant  =  t h a t i s , a l o n g s t r a i g h t l i n e s j o i n i n g t h e s i l i c a apex o f t h e t e r n a r y diagram w i t h the Cu 0-PbO b i n a r y system t h a t , 2  NPbO NPb  »  2NCu  NCu 0 - 2NCu 0  +  2  N«Cu  + +  a  2  NCu  +  +  A p p l y i n g t h e f a m i l i a r Gibbs-Duhem e q u a t i o n , NPbO d l o g aPbO + NCu 0 d l o g a C u 0 + N S i 0 2  and  2  2  dlog aSi0  2  « 0  i n t r o d u c i n g t h e i o n i c e x p r e s s i o n s f o r aPbO and a C u 0 , g i v e s 2  di^CuVCu+acT) + NSi0 dlog a S i 0  NPbO d(NPb**yPb" *aQ ') + N C u 0 H  5  2  NPb yPb aO + +  + +  2  N Cu y2Cu aO  =  2  +  +  M u l t i p l y i n g through by NPb** and NPbO i n t r o d u c i n g e q u a t i o n a, t h e f o l l o w i n g e x p r e s s i o n d(NPb" " yPb" "'aO ) + d(N Cu"V Cu'aO ) + NPb** N S i 0 l  +  l  yPb aO + +  =:  2  2  +  The  2  +  2  d l o g aS10  +  a  2  - 0  NPbO  =  i n d i c a t e d d i f f e r e n t i a l s may be o b t a i n e d n o t i n g f o r example,  d(N Gu y Cu aO ) - N Cu y Cu daO" + N Cu aO 2yCu dyCu 2  » 0  results,  c  2NCu y Cu aO  =  2  =  +  =  2  +  2  +  2  +  S i m p l i f y i n g and u s i n g the r e l a t i o n s h i p , dNPb  ++  + dNCu  +  -  0  s  +  +  + y2Cu aO 2NCu dNCu +  =  +  +  -71a g e n e r a l I o n i c Gibbs-Duhem e q u a t i o n f o r a t e r n a r y melt  o f one d i v a l e n t  m e t a l o x i d e , one monovalent m e t a l o x i d e , and s i l i c a may be o b t a i n e d , NPb dlog y P b + +  + +  + NCu dlog yCu +  + NPb** N S i 0  d l o g aSiO  2  + (NPb +„5NCu )dlog a 0  +  ++  +  =  = 0  ?-  b  NPbO U s i n g Temkin's concept  o f t h e s e p a r a b i l i t y o f t h e p o s i t i v e and  n e g a t i v e components of an i o n i c m e l t , e q u a t i o n b may be w r i t t e n i n two e x p r e s s i o n s proposed NPb""'" 1  NCu for  when,  constant  +  the p o s i t i v e i o n s ,  NPb dlog yPb + +  and  =  1  t o be v a l i d  «  + +  -NCu dlog y C u +  c  +  f o r the negative ions,  (NPb +„5NCu )dlog a 0 ++  +  Rearranging  =  = - NPb * N S i 0 NPbO +  2  dlog aSi0  d  2  e q u a t i o n d,  d l o g aO" = - M P b NSiQ dlog aSi0 NFb0(NPb +.5NCu ) H  2  ++  2  +  and n o t i n g t h a t , NPb* = 1 NPb0(NPb r.5NCu' ) NPbO+NCu 0  -  +  ++  r  2  1 l-NSi0  2  gives, d l o g a0~  «  - NSiQ  2  dlog aSi0  l-NSi0  e  2  2  Comparison o f e q u a t i o n c and e above w i t h e q u a t i o n s r i v e d f r o m the C a 0 - F e 0 - S i 0  2  system, shows .that combinations  11 and 12 deof divalent  m e t a l o x i d e s and monovalent m e t a l o x i d e s i n t e r n a r y melts w i t h  silica,  g i v e i d e n t i c a l i o n i c e q u a t i o n s which d e a l w i t h the p o s i t i v e and n e g a t i v e components o f the m e l t s .  

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