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A kinetic study of the LaNi4.9Al0.1-H2 system
Journal of the Lee-Cordon
199
Metals, 81 (1981) 199 - 206
A KINETIC STUDY OF THE LaNi&I,,r-H,
SYSTEM
L. BELKBIR, E. JOLY and N. GERARD
Laboratoire de Recherches sur la Rdactivit6 des Solides, LA 23, Faculte’ Sciences Mirande, B.P. 138,21004 LItjon Ckdex (France) (Received February 23, 1981)
Summary Formation and decomposition kinetics between the (Y(solid solution) and the fi hydride have been carried out on the LaNi4.9A10+1-H2system. We have investigated the influence of temperature with regard to kinetics near equilibrium conditions, i.e. contributions of activation energy on one hand and the change with temperature of the thermodynamic equilibrium pressure on the other hand. We discuss the meaning of experimental activation energy in terms of fo~ation and decomposition mech~isms and the possible causes of the anomaly obtained in the decomposition curve of reaction rate uersus temperature.
1. Introduction In the family of low substituted LaNi, compounds {in which 0.1 atoms of nickel are replaced by 0.1 atoms of copper, titanium or aluminium), LaNi4,sAl,.i has the lowest equilibrium plateau in the pressure-composition isotherms [l] and the slowest rates of hydride formation and decomposition kinetics [Z] . According to their behaviours, LaNi4.9Al,.1 has been chosen for carrying out detailed formation decomposition kinetics between the ar(solid solution) and the p (defined hydride) phases with the high accuracy volumetric device that we have developed [ 31. The aim of this work is firstly to investigate the role of pressure and especially of temperature on the kinetics of a system near equilibrium, i.e. conditions where temperature changes involve rate changes but also significantly modify the equilibrium pressure P, (and as a result modify the gap between the imposed pressure Pi and P,); therefore the two kinetic parameters temperature and Pi - P, are linked and vary in opposite directions. Secondly the aim is to determine an activation energy for the system and to investigate its meaning in formation and decomposition mechanisms. 0022-5088/81/0000-0000/$02.50
0 Elsevier Sequoia/Printed in The Netherlands
Fig. 1. A family of LaNidgAlo.lHG formations (transformation function of Pg, at PO = 2000
fraction 5 us. time) as a
Pa and T = 291 K: 0, 0.156 MPa; 9%0.171 MPa;r,
0.199
MPa; Q,O.22 MPa; A, 0.24 MPa; 0, 0.30 MPa.
2. Samples LaNi4.BAlo,I was prepared and characterized by Achard and Percheron at the CNRS Rare Earth Laboratory, Meudon, by melting pure component under an argon atmosphere in an induction furnace containing a watercooled copper crucible. The samples were annealed several times. Stoichiometry and homogeneity were tested by microprobe analysis, X-ray diffraction and metallographic observations. The samples were single phase with X-ray parameters a = 5.023 II and c = 3.989 A [1]. Kinetic studies were carried out in a differential volumetric device [ 31 with samples 5 mg in mass. The hydrogen used was of “U” quality supplied by Air Liquide; it contained less than 5 ppm of oxygen and water vapour.
3. Results 3.1. Kinetics of the Q + p transformatbn as a function of the hydrogen PWSSUR? F&we 1 shows a family of sigmoidal fl formations (5 conversion fraction versus time) carried out on an activated sample under hydrogen pressures varying from 0.156 to 0.30 MPa (T = 291 K). Each decomposition was obtained under Pn, = 0.003 MPa (20 Torr) with the same frequency between formations and decompositions. The equilibrium pressure P, is 0.122 MPa when T = 291 K, taken in the middle of the plateau, in the P-n formation (or decomposition) isotherm.
201
The instantaneous reaction rate at the inflection point (t = 0.5; halfreaction time, t = 0.5), given by the slope of the tangents, varies linearly with the hydrogen pressure, according to
(1) in which k, a constant, is a function of temperature T and of the formationdecomposition frequency p. The experimental curves of Fig. 1 become linear with -2.3
log( 1 - 5 ) = Kp, T, r* t2
(2)
The rate constant K* is a function of pressure P, temperature T and frequency p . Equation (2) agrees with the model of Johnson and Mehl [ 41, as given in Delmon [5],
-ln(l- 0!) = Ap(q)&voKiPKf,tP+Q+l
(3)
in which v. is the nuclei density, Kf is the germination rate constant, Ki is the interfacial rate constant, I#J~ is the shape factor, p is a factor which depends on nuclei growth (unidimensional plane, volume) and q is the order of the nucleation reaction. A comparison of eqns. (2) and (3) leads to Ap(q)&voK:Kg,
= K+
and q+p+1=2 which gives p = 1 and q = 0. From these results the p formation can be described as the precipitation of a compound defined in volume from a saturated solid solution (Johnson and Mehl’s model). 3.2. Activation energy of the 01+ 0 reaction The thermodynamic equilibrium pressure P, is dependent on temperature according to AH,+, --RT
AS,+, R
(4)
Thus when the imposed pressure Pi is maintained constant, the effect of a pressure gap variation Pi -P, is combined with the temperature action on the kinetics (according to Arrhenius’ law); as a consequence of the changes in these parameters, which are in opposite directions, the experimental curve d,$/dt = f(T) has a maximum, as shown in Fig. 2. The activation energy E, of the reaction can be calculated in two ways. (1) As the curve in Fig. 2 has a maximum, = Al(r)(Pi
-P,)
exp
(5)
202
Fig. 2. Reaction rate dt/dt US.temperature for Pi = 0.298MPa; the maximum is a consequence of the changes in the activation energy and the equilibrium pressure P, with T.
1:::q-y
;
1.
2.
F
\
0 3.3
3.4
3.5
.
lo%K>
3.6
Fig. 3. Two values of the activation energy can be obtained from the Arrhenius diagram as a consequence of the chosen expression for the PH, contribution: upper curve, E,, = 26.7 kJ mole1 ; lower curve, EA, = 62 kJ mol-‘.
its derivative (which is equal to zero for the T value corresponding to the maximum) gives
=0
E,=-AH-
(6) PC2 Pi -Pe
(2) E, can be obtained from the slope of the curve in an Arrhenius diagram (Fig. 3), which corresponds to
203
As shown in Fig. 3, the experimental values situated after the maximum in the curve of Fig. 2 are not correctly transformed; one reason is that the P, values are calculated from the thermodynamic equation, eqn. (4), and differ from the kinetic P, values, i.e. a pressure P, which gives a reaction rate equal to zero. This difference is important when P, is similar to Pi. It is to be noted that, even though eqn. (1) showed that dt/dt has a linear relationship with Pi -P,, this does not mean that it is not proportional to (Pi - P,)/P,, according to Barret [ 61. We have v=k
Pi -Pe P,
(9)
when the gaseous molecule is directly fixed on the solid and v=k
Pi -P, 1 + k,P,
(10)
when an adsorbed transitory state exists, with a slow adsorption rate; k, is the equilibrium constant for adsorption and k,Pe is small compared with unity. These two expressions for the rate-pressure dependence lead to two different values for the activation energy: = 26.75 kJ mol-’ E‘% with Pi -P, and EAz = 62 kJ mole1 with (Pi - P,)/P,. We note that there is no criterion of choice between these two values and that EA, = EA, + AH,,,. 3.3. p-a! decompositions as functions of an imposed hydrogen pressure As a function of time, decomposition of the fl hydride into the Q solid solution gives continuously ratedecreasing curves which become linear when expressed as
(11) in which the transformed proportion E is defined as the mass of hydrogen lost at time t related to the mass of hydrogen fixed during the formation. Ki is the rate constant and ci is the grain radius (which is constant after the “activating” period). Such an equation shows that the regulating step depends on a geometrical parameter, i.e. the linear rate of the interface progress in spherical coordinates, and the “second-order” desorption kinetics mentioned by Boser [ 71 are not observed when decompositions are carried out under constant imposed pressures.
204
0.106 0.079 -
0.053 .
cLo26 /j!jjjI 0
nk-i a;
0.4
Fig. 4. The change in the rate constant us. temperature plot in the 0 + Q decomposition, showing an anomaly near 300 K. Fig. 5. The first parts of the P-n isotherms, corresponding to CY+ metal decomposition, as a function of temperature: $, 290 K; 0, 295 K;A, 297 K; a. 298 K; 0, 299 K;A, 301 K; 0,303 K; 8, 305 K; +, 308 K.
The change in the rate constant Ki/Ui versus the hydrogen pressure P, Pi or (I’, - Pi )/Pe is linear for a range of pressures Pi which corresponds to practically no change in the hydrogen composition of the OLphase. We observe that the reaction rate in eqn. (11) (according to Barret [ 81) depends on the number of moles which react as written in eqn. (12) and is still linear for pressures where the cwphase composition n, varies mainly with P.
3.4. Temperature dependence of p + (Ydecomposition The rate constant values obtained from the slope of the transformed curves in eqn. (11) as a function of temperature give the curve of Fig. 4 which exhibits an anomaly near 300 K (this anomaly will be explained later). The calculation of the activation energy depends on the chosen expression for the pressure influence, i.e. P, -Pi or (P, - Pi)/Pe (as earlier), and requires an additional correction to the rate constant values plotted against T, which arises from the contribution of the variable n, with T. This result is shown in Fig. 5 which represents the first parts of the P-n, decomposition isotherms. When T varies eqn. (12) is replaced by = ‘(T, r
‘)r (Pe -Pi)pi 3n,lt3
(13)
205 TABLE
1
Activation energies (kJ mol-l)
290K<
T< 309K
from the Arrhenius diagram
301 K<
EA,
41.8
33.4
EA,
78.7
58.8
T< 313K
Pressuredependence p, -Pi
P,-Pi P,
nn /
LaNi
I 290
4.9 * ‘0.1
300
310
TK
’
Fig. 6. The composition in hydrogen atoms per mole of alloy obtained in decomposing the p phase under constant pressure Pi (0.0081 MPa) as a function of temperature.
in which B(T, r), = exp(--E/RT) The Arrhenius diagram ln { 3n, 1’3(Ki/Ui)/(P, -Pi)pi} = f(l/T) gives tW0 straight lines from which two activation energy values are obtained for each side of the anomaly (Table 1). The anomaly (also obtained in the n,-2’ curve of Fig. 6), which is located between two different values of the activation energy for the decomposition reaction, can be interpreted as a transition in the OLphase, as observed by Saba et al. [9] in the Ta-Ha system near 300 K.
4. Discussion From these kinetic results it seems that the formation and decomposition of the p hydride from or into the OLsolid solution involve two different mechanisms. (1) Formation takes place in a relationship with the precipitation from a saturated solid solution (Johnson and Mehl’s mechanism). (2) Decomposition is determined by the progress of the interface in spherical coordinates.
In the two cases the reaction rate is in proportion to the hydrogen pressure P, -Pi or (Pi - Pe)/Pe. When formations, as well as decompositions, are carried out under pressure-temperature conditions which give rather slow kinetics (to be sure that reactions are not via transitory states), the reaction itself seems to be the regulating step. Even though the activation energy values obtained are of the same order of magnitude as those given by Karlicek [lo] for hydrogen diffusion in LaNi,, it is not evident that hydrogen diffusion is the regulating step. Throughout this paper we have insisted on the complex aspect in the determination of E, from experimental results and on the fact that this activation energy is an apparent value and probably not the activation energy of the regulating step. More significant is the variation in E, with the reaction conditions which confirms a change in mechanism or a compound transition as we have shown. Steps proposed by Schlapbach et al. [ll],which occur on metal hydride surfaces, are coherent and surely essential steps in hydride formations and decompositions, but in the pressuretemperature domains investigated here they are not regulating steps. The results that we have given do not exclude the fact that under differing conditions (i.e. faster reactions) the rate-determining steps proposed by Wallace et al. [12] may be probable, but we did not find a second-order reaction rate in the p f (Yreaction even though we obtained this second order in the OL+ metal decomposition [2].
References 1 2 3 4 5 6 7 8 9
Diaz, The’se d’Zng&ieur-Docteur, University of Paris, September 20, 1978. L. Belkbir, Ph.D. Thesis, University of Dijon, 1979 (in French). N. Gerard, L. Belkbir and E. Joly, J. Phys. E, 12 (1979) 476 - 477. W. A. Johnson and R. F. Mehl, Trans. AZME, 135 (1939) 416 - 458. B. Delmon, Introduction to Heterogeneous Kinetics, Technip, Paris, 1969 (in French). P. Barret, C. R. Acad. Sci., Ser. C, 226 (1968) 856. 0. Boser, J. Less-Common Met., 46 (1976) 91. P. Barret, Cine’tique H&e’rog&e, Gauthier-Villars, Paris, 1973. W. G. Saba, W. E. Wallace, M. Sandmo and R. S. Craig, J. Chem. Phys., 35 (2) (1961) 148. 10 R. F. Karlicek, Jr., and I. J. Lowe, J. Less-Common Met., 73 (1980) 219 - 225. 11 L. Schlapbach, A. Seiler, F. Stucki and H. C. Siegmann, J. Less-Common Met., 73 (1980) 145 - 160. 12 W. E. Wallace, R. F. Kariicek, Jr., and H. Imamma, J. Phys. Chem., 83 (13) (1979) 1708 - 1712.
199
Metals, 81 (1981) 199 - 206
A KINETIC STUDY OF THE LaNi&I,,r-H,
SYSTEM
L. BELKBIR, E. JOLY and N. GERARD
Laboratoire de Recherches sur la Rdactivit6 des Solides, LA 23, Faculte’ Sciences Mirande, B.P. 138,21004 LItjon Ckdex (France) (Received February 23, 1981)
Summary Formation and decomposition kinetics between the (Y(solid solution) and the fi hydride have been carried out on the LaNi4.9A10+1-H2system. We have investigated the influence of temperature with regard to kinetics near equilibrium conditions, i.e. contributions of activation energy on one hand and the change with temperature of the thermodynamic equilibrium pressure on the other hand. We discuss the meaning of experimental activation energy in terms of fo~ation and decomposition mech~isms and the possible causes of the anomaly obtained in the decomposition curve of reaction rate uersus temperature.
1. Introduction In the family of low substituted LaNi, compounds {in which 0.1 atoms of nickel are replaced by 0.1 atoms of copper, titanium or aluminium), LaNi4,sAl,.i has the lowest equilibrium plateau in the pressure-composition isotherms [l] and the slowest rates of hydride formation and decomposition kinetics [Z] . According to their behaviours, LaNi4.9Al,.1 has been chosen for carrying out detailed formation decomposition kinetics between the ar(solid solution) and the p (defined hydride) phases with the high accuracy volumetric device that we have developed [ 31. The aim of this work is firstly to investigate the role of pressure and especially of temperature on the kinetics of a system near equilibrium, i.e. conditions where temperature changes involve rate changes but also significantly modify the equilibrium pressure P, (and as a result modify the gap between the imposed pressure Pi and P,); therefore the two kinetic parameters temperature and Pi - P, are linked and vary in opposite directions. Secondly the aim is to determine an activation energy for the system and to investigate its meaning in formation and decomposition mechanisms. 0022-5088/81/0000-0000/$02.50
0 Elsevier Sequoia/Printed in The Netherlands
Fig. 1. A family of LaNidgAlo.lHG formations (transformation function of Pg, at PO = 2000
fraction 5 us. time) as a
Pa and T = 291 K: 0, 0.156 MPa; 9%0.171 MPa;r,
0.199
MPa; Q,O.22 MPa; A, 0.24 MPa; 0, 0.30 MPa.
2. Samples LaNi4.BAlo,I was prepared and characterized by Achard and Percheron at the CNRS Rare Earth Laboratory, Meudon, by melting pure component under an argon atmosphere in an induction furnace containing a watercooled copper crucible. The samples were annealed several times. Stoichiometry and homogeneity were tested by microprobe analysis, X-ray diffraction and metallographic observations. The samples were single phase with X-ray parameters a = 5.023 II and c = 3.989 A [1]. Kinetic studies were carried out in a differential volumetric device [ 31 with samples 5 mg in mass. The hydrogen used was of “U” quality supplied by Air Liquide; it contained less than 5 ppm of oxygen and water vapour.
3. Results 3.1. Kinetics of the Q + p transformatbn as a function of the hydrogen PWSSUR? F&we 1 shows a family of sigmoidal fl formations (5 conversion fraction versus time) carried out on an activated sample under hydrogen pressures varying from 0.156 to 0.30 MPa (T = 291 K). Each decomposition was obtained under Pn, = 0.003 MPa (20 Torr) with the same frequency between formations and decompositions. The equilibrium pressure P, is 0.122 MPa when T = 291 K, taken in the middle of the plateau, in the P-n formation (or decomposition) isotherm.
201
The instantaneous reaction rate at the inflection point (t = 0.5; halfreaction time, t = 0.5), given by the slope of the tangents, varies linearly with the hydrogen pressure, according to
(1) in which k, a constant, is a function of temperature T and of the formationdecomposition frequency p. The experimental curves of Fig. 1 become linear with -2.3
log( 1 - 5 ) = Kp, T, r* t2
(2)
The rate constant K* is a function of pressure P, temperature T and frequency p . Equation (2) agrees with the model of Johnson and Mehl [ 41, as given in Delmon [5],
-ln(l- 0!) = Ap(q)&voKiPKf,tP+Q+l
(3)
in which v. is the nuclei density, Kf is the germination rate constant, Ki is the interfacial rate constant, I#J~ is the shape factor, p is a factor which depends on nuclei growth (unidimensional plane, volume) and q is the order of the nucleation reaction. A comparison of eqns. (2) and (3) leads to Ap(q)&voK:Kg,
= K+
and q+p+1=2 which gives p = 1 and q = 0. From these results the p formation can be described as the precipitation of a compound defined in volume from a saturated solid solution (Johnson and Mehl’s model). 3.2. Activation energy of the 01+ 0 reaction The thermodynamic equilibrium pressure P, is dependent on temperature according to AH,+, --RT
AS,+, R
(4)
Thus when the imposed pressure Pi is maintained constant, the effect of a pressure gap variation Pi -P, is combined with the temperature action on the kinetics (according to Arrhenius’ law); as a consequence of the changes in these parameters, which are in opposite directions, the experimental curve d,$/dt = f(T) has a maximum, as shown in Fig. 2. The activation energy E, of the reaction can be calculated in two ways. (1) As the curve in Fig. 2 has a maximum, = Al(r)(Pi
-P,)
exp
(5)
202
Fig. 2. Reaction rate dt/dt US.temperature for Pi = 0.298MPa; the maximum is a consequence of the changes in the activation energy and the equilibrium pressure P, with T.
1:::q-y
;
1.
2.
F
\
0 3.3
3.4
3.5
.
lo%K>
3.6
Fig. 3. Two values of the activation energy can be obtained from the Arrhenius diagram as a consequence of the chosen expression for the PH, contribution: upper curve, E,, = 26.7 kJ mole1 ; lower curve, EA, = 62 kJ mol-‘.
its derivative (which is equal to zero for the T value corresponding to the maximum) gives
=0
E,=-AH-
(6) PC2 Pi -Pe
(2) E, can be obtained from the slope of the curve in an Arrhenius diagram (Fig. 3), which corresponds to
203
As shown in Fig. 3, the experimental values situated after the maximum in the curve of Fig. 2 are not correctly transformed; one reason is that the P, values are calculated from the thermodynamic equation, eqn. (4), and differ from the kinetic P, values, i.e. a pressure P, which gives a reaction rate equal to zero. This difference is important when P, is similar to Pi. It is to be noted that, even though eqn. (1) showed that dt/dt has a linear relationship with Pi -P,, this does not mean that it is not proportional to (Pi - P,)/P,, according to Barret [ 61. We have v=k
Pi -Pe P,
(9)
when the gaseous molecule is directly fixed on the solid and v=k
Pi -P, 1 + k,P,
(10)
when an adsorbed transitory state exists, with a slow adsorption rate; k, is the equilibrium constant for adsorption and k,Pe is small compared with unity. These two expressions for the rate-pressure dependence lead to two different values for the activation energy: = 26.75 kJ mol-’ E‘% with Pi -P, and EAz = 62 kJ mole1 with (Pi - P,)/P,. We note that there is no criterion of choice between these two values and that EA, = EA, + AH,,,. 3.3. p-a! decompositions as functions of an imposed hydrogen pressure As a function of time, decomposition of the fl hydride into the Q solid solution gives continuously ratedecreasing curves which become linear when expressed as
(11) in which the transformed proportion E is defined as the mass of hydrogen lost at time t related to the mass of hydrogen fixed during the formation. Ki is the rate constant and ci is the grain radius (which is constant after the “activating” period). Such an equation shows that the regulating step depends on a geometrical parameter, i.e. the linear rate of the interface progress in spherical coordinates, and the “second-order” desorption kinetics mentioned by Boser [ 71 are not observed when decompositions are carried out under constant imposed pressures.
204
0.106 0.079 -
0.053 .
cLo26 /j!jjjI 0
nk-i a;
0.4
Fig. 4. The change in the rate constant us. temperature plot in the 0 + Q decomposition, showing an anomaly near 300 K. Fig. 5. The first parts of the P-n isotherms, corresponding to CY+ metal decomposition, as a function of temperature: $, 290 K; 0, 295 K;A, 297 K; a. 298 K; 0, 299 K;A, 301 K; 0,303 K; 8, 305 K; +, 308 K.
The change in the rate constant Ki/Ui versus the hydrogen pressure P, Pi or (I’, - Pi )/Pe is linear for a range of pressures Pi which corresponds to practically no change in the hydrogen composition of the OLphase. We observe that the reaction rate in eqn. (11) (according to Barret [ 81) depends on the number of moles which react as written in eqn. (12) and is still linear for pressures where the cwphase composition n, varies mainly with P.
3.4. Temperature dependence of p + (Ydecomposition The rate constant values obtained from the slope of the transformed curves in eqn. (11) as a function of temperature give the curve of Fig. 4 which exhibits an anomaly near 300 K (this anomaly will be explained later). The calculation of the activation energy depends on the chosen expression for the pressure influence, i.e. P, -Pi or (P, - Pi)/Pe (as earlier), and requires an additional correction to the rate constant values plotted against T, which arises from the contribution of the variable n, with T. This result is shown in Fig. 5 which represents the first parts of the P-n, decomposition isotherms. When T varies eqn. (12) is replaced by = ‘(T, r
‘)r (Pe -Pi)pi 3n,lt3
(13)
205 TABLE
1
Activation energies (kJ mol-l)
290K<
T< 309K
from the Arrhenius diagram
301 K<
EA,
41.8
33.4
EA,
78.7
58.8
T< 313K
Pressuredependence p, -Pi
P,-Pi P,
nn /
LaNi
I 290
4.9 * ‘0.1
300
310
TK
’
Fig. 6. The composition in hydrogen atoms per mole of alloy obtained in decomposing the p phase under constant pressure Pi (0.0081 MPa) as a function of temperature.
in which B(T, r), = exp(--E/RT) The Arrhenius diagram ln { 3n, 1’3(Ki/Ui)/(P, -Pi)pi} = f(l/T) gives tW0 straight lines from which two activation energy values are obtained for each side of the anomaly (Table 1). The anomaly (also obtained in the n,-2’ curve of Fig. 6), which is located between two different values of the activation energy for the decomposition reaction, can be interpreted as a transition in the OLphase, as observed by Saba et al. [9] in the Ta-Ha system near 300 K.
4. Discussion From these kinetic results it seems that the formation and decomposition of the p hydride from or into the OLsolid solution involve two different mechanisms. (1) Formation takes place in a relationship with the precipitation from a saturated solid solution (Johnson and Mehl’s mechanism). (2) Decomposition is determined by the progress of the interface in spherical coordinates.
In the two cases the reaction rate is in proportion to the hydrogen pressure P, -Pi or (Pi - Pe)/Pe. When formations, as well as decompositions, are carried out under pressure-temperature conditions which give rather slow kinetics (to be sure that reactions are not via transitory states), the reaction itself seems to be the regulating step. Even though the activation energy values obtained are of the same order of magnitude as those given by Karlicek [lo] for hydrogen diffusion in LaNi,, it is not evident that hydrogen diffusion is the regulating step. Throughout this paper we have insisted on the complex aspect in the determination of E, from experimental results and on the fact that this activation energy is an apparent value and probably not the activation energy of the regulating step. More significant is the variation in E, with the reaction conditions which confirms a change in mechanism or a compound transition as we have shown. Steps proposed by Schlapbach et al. [ll],which occur on metal hydride surfaces, are coherent and surely essential steps in hydride formations and decompositions, but in the pressuretemperature domains investigated here they are not regulating steps. The results that we have given do not exclude the fact that under differing conditions (i.e. faster reactions) the rate-determining steps proposed by Wallace et al. [12] may be probable, but we did not find a second-order reaction rate in the p f (Yreaction even though we obtained this second order in the OL+ metal decomposition [2].
References 1 2 3 4 5 6 7 8 9
Diaz, The’se d’Zng&ieur-Docteur, University of Paris, September 20, 1978. L. Belkbir, Ph.D. Thesis, University of Dijon, 1979 (in French). N. Gerard, L. Belkbir and E. Joly, J. Phys. E, 12 (1979) 476 - 477. W. A. Johnson and R. F. Mehl, Trans. AZME, 135 (1939) 416 - 458. B. Delmon, Introduction to Heterogeneous Kinetics, Technip, Paris, 1969 (in French). P. Barret, C. R. Acad. Sci., Ser. C, 226 (1968) 856. 0. Boser, J. Less-Common Met., 46 (1976) 91. P. Barret, Cine’tique H&e’rog&e, Gauthier-Villars, Paris, 1973. W. G. Saba, W. E. Wallace, M. Sandmo and R. S. Craig, J. Chem. Phys., 35 (2) (1961) 148. 10 R. F. Karlicek, Jr., and I. J. Lowe, J. Less-Common Met., 73 (1980) 219 - 225. 11 L. Schlapbach, A. Seiler, F. Stucki and H. C. Siegmann, J. Less-Common Met., 73 (1980) 145 - 160. 12 W. E. Wallace, R. F. Kariicek, Jr., and H. Imamma, J. Phys. Chem., 83 (13) (1979) 1708 - 1712.