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The occurrence of a maximum value of the rate constant of oxidation of a metal or a metal oxide as a function of temperature
Scripta METALLURGICA
Vol. 7, pp. 573-578, 1973 Printed in the United States
Pergamon Press, Inc.
THE OCCURRENCEOF A MAXIMUM~ALUE OF THE RATE CONSTANTOF OXIDATION OF A METAL OR A METAL OXIDE AS A FUNCTION OF TEMPERATURE G. Simkovich and S. Kertoatmodjo Metallurgy Section, Department of Material Sciences The Pennsylvania State U n i v e r s i t y , U n i v e r s i t y Park, Pennsylvania
16802
(Received March 30, 1973) In 1933 ( I ) Wagner derived the general r e l a t i o n s h i p e x i s t i n g between the e l e c t r i c a l properties of binary i o n i c type crystals and t h e i r rate of growth under an o x i d i z i n g potential At a l a t e r time, 1951 (2) Wagner obtained the same growth r e l a t i o n s h i p s in terms of the s e l f d i f f u s i o n c o e f f i c i e n t s of the i o n i c species in the growing compound when the e l e c t r i c a l transport number of electrons in the compound is almost one; that i s , in an e l e c t r o n i c conductor. One form of the equations derived by Wagner (2) f o r a d i f f u s i c n c o n t r o l l e d process i s :
E Ja r = z2 C2
(T2 D1 + D2 ) dlna 2
]
[I] t3=l
where k r is the r a t i o n a l rate constant, z i is the absolute charge, a i is the thermodynamic activity,
Di
is the s e l f - d i f f u s i o n c o e f f i c i e n t , the superscript single prime indicates an
e q u i l i b r i u m value at the metal-compound i n t e r f a c e while the superscript double prime indicates a value at the compound-gas i n t e r f a c e , and c i is the concentration in terms of g-atoms cm-3. Cations, anions, and electrons are indicated by subscripts I , 2, and 3 respectively. In most c r y s t a l s , one of the ions usually has a much larger d i f f u s i o n c o e f f i c i e n t than the other ion.
Let us assume that D1 >> D2
Hence, eq. [ I ] may be reduced to: k r = z I c2
ja;
D1
[2]
dlna 2
Let us consider that the compound formed is an oxide. the i n t e g r a t i o n l i m i t s to oxygen pressures.
kr
Hence, i t is convenient to change
Performing t h i s , one obtains
P'02
-
[3]
ZlC2 f 2 JP'
O2 573
* D1 dlnP 0
2
[4]
574
RATE CONSTANT OF OXIDATION AS A FUNCTION OF TEMPERATURE
Vol.
7, No. 6
Let i t be assumed further that diffusion of the cation in MeO is primarily via a vacancy and that the s e l f - d i f f u s i o n c o e f f i c i e n t , D1 , of the cations at unit oxygen pressure has been measured as a function of temperature.
The resulting diffusion c o e f f i c i e n t is usually expressed
as :
1 : D1 exp ( ~ )
1 atm
[5]
where the symbols have t h e i r usual meaning. The s e l f - d i f f u s i o n
coefficient,
a function of oxygen pressure.
in addition to being a function of temperature, is also
This variation is given as: o
D1 = P02 I/n D1 exp ( ~-&E )
[6]
where n is a positive number related to the point defect structure of MeO. Insertion of D1
into eq. [4] gives upon integration
kr =
[nzl~]
•
[e
-&E ] IFP"02 / I/n n ]- P ~02 ' xp (R--T) • l
[7]
I t may be noted from consideration of eq. [7] that a plot of In k r vs. I/T is linear i f AE is constant, but also only i f PG2I/n << P"021/n" In this case, the slope and intercept at infinite
temperature on the In k r axis are found to be: Slope = - (~-~)
[8]
and ° Intercept = In
_
I/n 2
[9]
Inspection of eq. [7] with respect to temperature variations reveals the following. Ist term in brackets on the right-hand side of eq. [7] is a constant, while the 2nd term
The
increases exponentially with an increase in temperature. The t h i r d term in brackets is the difference between the oxidizing potential and the equilibrium potential, with each potential raised to the I/n power, and i f oxidation is carried out at a constant oxygen potential then, since the equilibrium potential generally increases exponentially with temperature, this term w i l l decrease with an increase in temperature. Thus, an increase in temperature of oxidation at a constant oxidizing potential w i l l i n i t i a l l y result in an increase in the rate of oxidation, attain a maximum rate at a p a r t i c u l a r temperature, Tmax, a f t e r which a further increase in temperature w i l l
result in lower rates.
To obtain Tmax, P'02 must be expressed as a function of temperature. consideration is given as: Me + I/2 02 = MeO
The reaction under
[10]
Vol.
7, No.
6
RATE CONSTANT OF O X I D A T I O N AS A FUNCTION OF T E M P E R A T U R E
575
and the Gibbs Standard Free Energy of Formation of MeO may be w r i t t e n as: AG° = -RT In ( aMeO/aMe.P02 I/2 ) o
= RT In P02 I/2 = AH° - TAS°
[II]
o
where AH and AS are the standard enthalpy and entropy of formation. in eq. [ I I ] gives:
p, I/n = exp (2AG°I '-n-RT-' = exp (AN° ~ 02
Solving f o r P' I/n 02
AS°.) 2-) R (n
[12]
Inserting this value into eq. [7] gives: kr =
=
o}[ ,it. nzlc2Dl 2
"
-AE exp ( - ~ -
"
-AE exp (--RT
.
,2~G°,l
p,, I/n j 02 - e x p , n---~,r,I .J
p,, I/n _ exp ( ~ L 02
R -) (n2-)
[13]
In order to determine the temperature at which k is maximized, eq. [13] is d i f f e r e n t i a t e d r o o with respect to temperature and set equal to zero in the usual manner assuming that P" , AE , AH o AS are constants. Upon dropping the constant m u l t i p l i e r , one obtains: 02
I
A RT-2-) E
P"02I/n
fro
o
7
exp (--RT -AE + L(RT ~ - ) (2) - (AE RT2] . xp ( AH - R T - TAS ) ( 2 n - ) - (-RT-) AE [14] =0
Solving for Tmax in eq. [14] gives: o
Tmax
AH
:
n.AE-P
o _
AS + (9_~)in
I/n
[15]
_2 [11_AE_ _- ;_2AH
which permits calculation of the temperature at which a maximum rate of oxidation occurs under a given o x i d i z i n g p o t e n t i a l . o
I f desired Tmax may be put in terms of AGmax, the Gibbs Standard Free Energy of Formation of MeO at Tmax, which r e s u l t s in the r e l a t i o n s h i p : o
Tmax
=
AG
max n.AE.P" I/n
[16]
• AE - 2AH Use of eq. [ I I ] in eq. [16] permits one to obtain the r a t i o of the o x i d i z i n g p o t e n t i a l to that of the e q u i l i b r i u m p o t e n t i a l e x i s t i n g between Me-MeO at Tmax. This r e l a t i o n s h i p is
576
RATE CONSTANT OF OXIDATION AS A FUNCTION OF TEMPERATURE
P" 02
:
P6
Vol. 7, No. 6
(nAEn~E2AHo in
[17]
2(eq-max) For an n-type conductor where cation diffusion occurs via i n t e r s t i t i a l s ,
k r may be
expressed as o
kr :
exp ( T )
L 02
02
]
where AE is now the activation energy for diffusion of Me ions in Me0 when Me0 is in equilibrium with the metal phase Me and n is, as previously, a positive integer which again reflects the point defect structure of the n-type semiconductor being formed. U t i l i z i n g eq. [ I I ] formation
'
one may obtain P' - I / n in terms of the Gibbs Standard Free Energy of 02 o
p, - I / n : e x p ( ~ ) ( ~ 1 O2
[19]
Inserting B' -I/n from eq. [19] into eq. [18], one obtains O2 o k
: [.nzl~ 2DI ] "
~xp(Z-~#-)] • ~ x p ( ~ ) ( ~ ) -
P" " I / n l 02 ]
r Further u t i l i z i n g
[20]
the standard enthalpy and entropy of formation of Me0, one obtains: o
o
kr =
•
exp(~)
• xp
o
- ~)
-
02
]
[21]
Consideration of eqs. [ 1 4 and [21] reveals that a In k r vs. I/T w i l l e x h i b i t a linear behavior only when AE is a constant and when P" >> P' similar to the conditions for a p-type 02 02 conductor. However, the values of the slope and intercept in the case of an n-type conductor are found to be: 2 (AE + AN°) slope = - (~-~) o
Intercept = I n ( ~ ) + o
[22] o
2AS nR
[23]
o
Assuming again that P# , AE, AS , and AH are constants as a function of temperature u2 of eq. [21] with respect to temperature, equating the d i f f e r e n t i a l to zero
differentiation
and solving for Tmax, one obtains o
Tmax
:
AH
° nAE l AS - (9_~) I n [ P" i/n(nA E + 2AHo) ] O2
[24]
Vol.
7, No.
6
RATE CONSTANT OF O X I D A T I O N AS A FUNCTION OF T E M P E R A T U R E
577
In terms of AGmax, eq. [24] becomes: o
-AG = max [ nAE o] Tmax (~-~) In P" I/n(nAE + 2AH O2
[25]
Uti|izing eq. [ I I ] in eq. [25] and solving for the ratio of oxidizing potential to the equilibrium potential existing between Me-Me0, one obtains: p. 02
p~ 2(eq-max)
=~
nAE
o nAE + 2AH
)n
[26]
Discussion The equations derived permit one to calculate the temperature at which a maximum parabolic oxidation rate occurs whenever oxidation is carried out at a fixed oxidation potential.
In
addition, the equations i d e n t i ~ the slopes and intercepts obtained from a In k versus I/T r plot and l i s t s the conditicns which j u s t i ~ a linear relationship in this plot. Depending upon the type of compound being formed, i . e . whether cation vacancy diffusion (p-type MeO) or cation i n t e r s t i t i a l motion (n-type MeO) predominates, one obtains somewhat different relations for Tmax, PG2/P~2(eq.max), and the slopes and intercepts of In kr versus I/T plots. From a practical point of view, the equations relating the parabolic oxidation kinetics to a temperature at which the rate is maximized are of consiaerable interest.
For example, a
metal which might be considered unsuitable for a high temperature application because i t s rate of oxidation is too high even though other properties of the metal might be appropriate for the application, may become usable i f the oxidizing potential in the gas phase is lowered s u f f i ciently to become comparable to P02' or at least to lower the oxygen potential to such an extent that Tmax lies below the temperature of application. The l i t e r a t u r e provides at least one example of a temperature at which parabolic oxidation is a maximum (3,4).
In particular, the reaction in a i r of 3 CoO + I/2 02 = Co304
[27]
attains a maximum rate at approximately 800°C. At both higher and lower temperatures the rates were less than that at ~O°C.
Unfortunately, the activation energy of diffusion and the point
defect structure of Co304 are unknown at the present time so that the derived relationships presently cannot be u t i l i z e d or confirmed for this reaction. Studies (5,6) concerned with the reaction Cu20 + I/2 02 = 2CuD
[28]
578
RATE CONSTANT OF OXIDATION AS A FUNCTION OF TEMPERATURE
Vol.
also imply that Tmax may have been approached at two d i f f e r e n t oxygen potentials.
7, No. 6
In the study
by Hauffe and Kofstad (5) under one atmosphere of oxygen a plot of In k3 vs. I / T , where k3 was taken as the cubic oxidation rate constant, the authors obtained a'so-called " a c t i v a t i o n energy" of 2 kcal/mole at temperatures above 750°C indicating an approach to Tmax, while Meijering's studies (6) which were conducted in a i r gave a Tmax at about 900°C and, hence, a negative "activation energy" above this 900°C. This Tmax was at a lower temperature than Hauffe's studies (5) in q u a l i t a t i v e accord with the derived equations. Although these studies indicated cubic behaviour for the growth of CuO which Hauffe attempted to explain in terms of a space charge e f f e c t and Meijering attempted to explain in terms of an ageing e f f e c t , the growth was s u f f i c i e n t l y close to parabolic behaviour that the analysis given here may be applicable to future studies on this system especially i f the Cu20 samples u t i l i z e d are more c a r e f u l l y prepared so that pores in the Cu20 are eliminated.
In such
a case, where parabolic kinetics are obtained, the slopes of a In kr vs. I/T plot in the vicinity^~ of Tmax are more complex functions than those presented here and do not represent (9_~) only.
Such slopes may be obtained by manipulation of e i t h e r eq. [13] or eq.
[20]
depending upon the ionic point defect structure of CuO which is unknown at the present time. Acknowledgements Sincere thanks are extended to Professors W. R. B i t l e r and Go W. Healy for helpful discussions and especially to Professor L. S. Darken whose comment, during a luncheon discussion concerning the Co304 l i t e r a t u r e , of the "C" curve nature of oxidation kinetics was the key to our analysis. References I.
C. Wagner, Z. Physik. Chemie, BE], 25 (1933).
2. 3.
c. Wagner, in "Atom Movements", ASM, Cleveland, Ohio, U.S.A., 153 (1951). C. R. Johns and W. M. Baldwin, J r . , Metal Trans., 185, 720 (1949).
4.
M. G. Vallee and M. J. Paidassi, Corr et a n t i - c o r r . , I0, 132 (1962).
5.
K. Hauffe and P. Kofstad, Z. f u r Elektrochem., 59, 399 (1955).
6.
J. L. Meijering and N. L. Verheijk~, Acta Metal, 2, 331 (1959).
Vol. 7, pp. 573-578, 1973 Printed in the United States
Pergamon Press, Inc.
THE OCCURRENCEOF A MAXIMUM~ALUE OF THE RATE CONSTANTOF OXIDATION OF A METAL OR A METAL OXIDE AS A FUNCTION OF TEMPERATURE G. Simkovich and S. Kertoatmodjo Metallurgy Section, Department of Material Sciences The Pennsylvania State U n i v e r s i t y , U n i v e r s i t y Park, Pennsylvania
16802
(Received March 30, 1973) In 1933 ( I ) Wagner derived the general r e l a t i o n s h i p e x i s t i n g between the e l e c t r i c a l properties of binary i o n i c type crystals and t h e i r rate of growth under an o x i d i z i n g potential At a l a t e r time, 1951 (2) Wagner obtained the same growth r e l a t i o n s h i p s in terms of the s e l f d i f f u s i o n c o e f f i c i e n t s of the i o n i c species in the growing compound when the e l e c t r i c a l transport number of electrons in the compound is almost one; that i s , in an e l e c t r o n i c conductor. One form of the equations derived by Wagner (2) f o r a d i f f u s i c n c o n t r o l l e d process i s :
E Ja r = z2 C2
(T2 D1 + D2 ) dlna 2
]
[I] t3=l
where k r is the r a t i o n a l rate constant, z i is the absolute charge, a i is the thermodynamic activity,
Di
is the s e l f - d i f f u s i o n c o e f f i c i e n t , the superscript single prime indicates an
e q u i l i b r i u m value at the metal-compound i n t e r f a c e while the superscript double prime indicates a value at the compound-gas i n t e r f a c e , and c i is the concentration in terms of g-atoms cm-3. Cations, anions, and electrons are indicated by subscripts I , 2, and 3 respectively. In most c r y s t a l s , one of the ions usually has a much larger d i f f u s i o n c o e f f i c i e n t than the other ion.
Let us assume that D1 >> D2
Hence, eq. [ I ] may be reduced to: k r = z I c2
ja;
D1
[2]
dlna 2
Let us consider that the compound formed is an oxide. the i n t e g r a t i o n l i m i t s to oxygen pressures.
kr
Hence, i t is convenient to change
Performing t h i s , one obtains
P'02
-
[3]
ZlC2 f 2 JP'
O2 573
* D1 dlnP 0
2
[4]
574
RATE CONSTANT OF OXIDATION AS A FUNCTION OF TEMPERATURE
Vol.
7, No. 6
Let i t be assumed further that diffusion of the cation in MeO is primarily via a vacancy and that the s e l f - d i f f u s i o n c o e f f i c i e n t , D1 , of the cations at unit oxygen pressure has been measured as a function of temperature.
The resulting diffusion c o e f f i c i e n t is usually expressed
as :
1 : D1 exp ( ~ )
1 atm
[5]
where the symbols have t h e i r usual meaning. The s e l f - d i f f u s i o n
coefficient,
a function of oxygen pressure.
in addition to being a function of temperature, is also
This variation is given as: o
D1 = P02 I/n D1 exp ( ~-&E )
[6]
where n is a positive number related to the point defect structure of MeO. Insertion of D1
into eq. [4] gives upon integration
kr =
[nzl~]
•
[e
-&E ] IFP"02 / I/n n ]- P ~02 ' xp (R--T) • l
[7]
I t may be noted from consideration of eq. [7] that a plot of In k r vs. I/T is linear i f AE is constant, but also only i f PG2I/n << P"021/n" In this case, the slope and intercept at infinite
temperature on the In k r axis are found to be: Slope = - (~-~)
[8]
and ° Intercept = In
_
I/n 2
[9]
Inspection of eq. [7] with respect to temperature variations reveals the following. Ist term in brackets on the right-hand side of eq. [7] is a constant, while the 2nd term
The
increases exponentially with an increase in temperature. The t h i r d term in brackets is the difference between the oxidizing potential and the equilibrium potential, with each potential raised to the I/n power, and i f oxidation is carried out at a constant oxygen potential then, since the equilibrium potential generally increases exponentially with temperature, this term w i l l decrease with an increase in temperature. Thus, an increase in temperature of oxidation at a constant oxidizing potential w i l l i n i t i a l l y result in an increase in the rate of oxidation, attain a maximum rate at a p a r t i c u l a r temperature, Tmax, a f t e r which a further increase in temperature w i l l
result in lower rates.
To obtain Tmax, P'02 must be expressed as a function of temperature. consideration is given as: Me + I/2 02 = MeO
The reaction under
[10]
Vol.
7, No.
6
RATE CONSTANT OF O X I D A T I O N AS A FUNCTION OF T E M P E R A T U R E
575
and the Gibbs Standard Free Energy of Formation of MeO may be w r i t t e n as: AG° = -RT In ( aMeO/aMe.P02 I/2 ) o
= RT In P02 I/2 = AH° - TAS°
[II]
o
where AH and AS are the standard enthalpy and entropy of formation. in eq. [ I I ] gives:
p, I/n = exp (2AG°I '-n-RT-' = exp (AN° ~ 02
Solving f o r P' I/n 02
AS°.) 2-) R (n
[12]
Inserting this value into eq. [7] gives: kr =
=
o}[ ,it. nzlc2Dl 2
"
-AE exp ( - ~ -
"
-AE exp (--RT
.
,2~G°,l
p,, I/n j 02 - e x p , n---~,r,I .J
p,, I/n _ exp ( ~ L 02
R -) (n2-)
[13]
In order to determine the temperature at which k is maximized, eq. [13] is d i f f e r e n t i a t e d r o o with respect to temperature and set equal to zero in the usual manner assuming that P" , AE , AH o AS are constants. Upon dropping the constant m u l t i p l i e r , one obtains: 02
I
A RT-2-) E
P"02I/n
fro
o
7
exp (--RT -AE + L(RT ~ - ) (2) - (AE RT2] . xp ( AH - R T - TAS ) ( 2 n - ) - (-RT-) AE [14] =0
Solving for Tmax in eq. [14] gives: o
Tmax
AH
:
n.AE-P
o _
AS + (9_~)in
I/n
[15]
_2 [11_AE_ _- ;_2AH
which permits calculation of the temperature at which a maximum rate of oxidation occurs under a given o x i d i z i n g p o t e n t i a l . o
I f desired Tmax may be put in terms of AGmax, the Gibbs Standard Free Energy of Formation of MeO at Tmax, which r e s u l t s in the r e l a t i o n s h i p : o
Tmax
=
AG
max n.AE.P" I/n
[16]
• AE - 2AH Use of eq. [ I I ] in eq. [16] permits one to obtain the r a t i o of the o x i d i z i n g p o t e n t i a l to that of the e q u i l i b r i u m p o t e n t i a l e x i s t i n g between Me-MeO at Tmax. This r e l a t i o n s h i p is
576
RATE CONSTANT OF OXIDATION AS A FUNCTION OF TEMPERATURE
P" 02
:
P6
Vol. 7, No. 6
(nAEn~E2AHo in
[17]
2(eq-max) For an n-type conductor where cation diffusion occurs via i n t e r s t i t i a l s ,
k r may be
expressed as o
kr :
exp ( T )
L 02
02
]
where AE is now the activation energy for diffusion of Me ions in Me0 when Me0 is in equilibrium with the metal phase Me and n is, as previously, a positive integer which again reflects the point defect structure of the n-type semiconductor being formed. U t i l i z i n g eq. [ I I ] formation
'
one may obtain P' - I / n in terms of the Gibbs Standard Free Energy of 02 o
p, - I / n : e x p ( ~ ) ( ~ 1 O2
[19]
Inserting B' -I/n from eq. [19] into eq. [18], one obtains O2 o k
: [.nzl~ 2DI ] "
~xp(Z-~#-)] • ~ x p ( ~ ) ( ~ ) -
P" " I / n l 02 ]
r Further u t i l i z i n g
[20]
the standard enthalpy and entropy of formation of Me0, one obtains: o
o
kr =
•
exp(~)
• xp
o
- ~)
-
02
]
[21]
Consideration of eqs. [ 1 4 and [21] reveals that a In k r vs. I/T w i l l e x h i b i t a linear behavior only when AE is a constant and when P" >> P' similar to the conditions for a p-type 02 02 conductor. However, the values of the slope and intercept in the case of an n-type conductor are found to be: 2 (AE + AN°) slope = - (~-~) o
Intercept = I n ( ~ ) + o
[22] o
2AS nR
[23]
o
Assuming again that P# , AE, AS , and AH are constants as a function of temperature u2 of eq. [21] with respect to temperature, equating the d i f f e r e n t i a l to zero
differentiation
and solving for Tmax, one obtains o
Tmax
:
AH
° nAE l AS - (9_~) I n [ P" i/n(nA E + 2AHo) ] O2
[24]
Vol.
7, No.
6
RATE CONSTANT OF O X I D A T I O N AS A FUNCTION OF T E M P E R A T U R E
577
In terms of AGmax, eq. [24] becomes: o
-AG = max [ nAE o] Tmax (~-~) In P" I/n(nAE + 2AH O2
[25]
Uti|izing eq. [ I I ] in eq. [25] and solving for the ratio of oxidizing potential to the equilibrium potential existing between Me-Me0, one obtains: p. 02
p~ 2(eq-max)
=~
nAE
o nAE + 2AH
)n
[26]
Discussion The equations derived permit one to calculate the temperature at which a maximum parabolic oxidation rate occurs whenever oxidation is carried out at a fixed oxidation potential.
In
addition, the equations i d e n t i ~ the slopes and intercepts obtained from a In k versus I/T r plot and l i s t s the conditicns which j u s t i ~ a linear relationship in this plot. Depending upon the type of compound being formed, i . e . whether cation vacancy diffusion (p-type MeO) or cation i n t e r s t i t i a l motion (n-type MeO) predominates, one obtains somewhat different relations for Tmax, PG2/P~2(eq.max), and the slopes and intercepts of In kr versus I/T plots. From a practical point of view, the equations relating the parabolic oxidation kinetics to a temperature at which the rate is maximized are of consiaerable interest.
For example, a
metal which might be considered unsuitable for a high temperature application because i t s rate of oxidation is too high even though other properties of the metal might be appropriate for the application, may become usable i f the oxidizing potential in the gas phase is lowered s u f f i ciently to become comparable to P02' or at least to lower the oxygen potential to such an extent that Tmax lies below the temperature of application. The l i t e r a t u r e provides at least one example of a temperature at which parabolic oxidation is a maximum (3,4).
In particular, the reaction in a i r of 3 CoO + I/2 02 = Co304
[27]
attains a maximum rate at approximately 800°C. At both higher and lower temperatures the rates were less than that at ~O°C.
Unfortunately, the activation energy of diffusion and the point
defect structure of Co304 are unknown at the present time so that the derived relationships presently cannot be u t i l i z e d or confirmed for this reaction. Studies (5,6) concerned with the reaction Cu20 + I/2 02 = 2CuD
[28]
578
RATE CONSTANT OF OXIDATION AS A FUNCTION OF TEMPERATURE
Vol.
also imply that Tmax may have been approached at two d i f f e r e n t oxygen potentials.
7, No. 6
In the study
by Hauffe and Kofstad (5) under one atmosphere of oxygen a plot of In k3 vs. I / T , where k3 was taken as the cubic oxidation rate constant, the authors obtained a'so-called " a c t i v a t i o n energy" of 2 kcal/mole at temperatures above 750°C indicating an approach to Tmax, while Meijering's studies (6) which were conducted in a i r gave a Tmax at about 900°C and, hence, a negative "activation energy" above this 900°C. This Tmax was at a lower temperature than Hauffe's studies (5) in q u a l i t a t i v e accord with the derived equations. Although these studies indicated cubic behaviour for the growth of CuO which Hauffe attempted to explain in terms of a space charge e f f e c t and Meijering attempted to explain in terms of an ageing e f f e c t , the growth was s u f f i c i e n t l y close to parabolic behaviour that the analysis given here may be applicable to future studies on this system especially i f the Cu20 samples u t i l i z e d are more c a r e f u l l y prepared so that pores in the Cu20 are eliminated.
In such
a case, where parabolic kinetics are obtained, the slopes of a In kr vs. I/T plot in the vicinity^~ of Tmax are more complex functions than those presented here and do not represent (9_~) only.
Such slopes may be obtained by manipulation of e i t h e r eq. [13] or eq.
[20]
depending upon the ionic point defect structure of CuO which is unknown at the present time. Acknowledgements Sincere thanks are extended to Professors W. R. B i t l e r and Go W. Healy for helpful discussions and especially to Professor L. S. Darken whose comment, during a luncheon discussion concerning the Co304 l i t e r a t u r e , of the "C" curve nature of oxidation kinetics was the key to our analysis. References I.
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