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Experimental methods useful in the kinetic modelling of heterogeneous reactions
__ _IB
Js
SOLID STATE
ELSEVIER
IONKS
Solid State Ionics 95 (1997) 33-40
Experimental
methods useful in the kinetic modelling of heterogeneous reactions M. Soustelle*,
CRESA-SHIV,
EC& des Mines,
158 cows
M. Pijolat
Faurief, 42023 Saint-Etienne
Cedex 2, France
Abstract A theorem on the separation of models is reported which shows how it may be possible to describe the kinetics of a reaction involving a solid: a physico-chemical model in which ‘reactivity’ has been defined, and a geometrical model to which a ‘space function’ corresponds. Then a series of rigorous experimental methods based on this theorem are presented. Their use improves and validates the kinetic modelling of a heterogeneous reaction. Keywords:
Modelling;
Heterogeneous
reaction;
Reaction
kinetics;
Reactance
1. Introduction
+L.
Let us consider a heterogeneous reaction of a phase A which is transformed into a phase B according to:
%A+(gw,J =B +(+G)
(1)
where G, and G, are the eventually present gases. It is possible to define an absolute rate of the reaction by means of the derivative of the extent of transformation, 5, with respect to time. It depends on all the extensive and intensive variables fixed by the experimental conditions. It is also possible to use the fractional conversion, A, defined from the extent of conversion, and its derivative with respect to time, that we will call the ‘reactance’, %, in the following. When the solid A is chosen for the expression of the extent of conversion, we have: *Corresponding
author.
0167-2738/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SOl67-2738(96)00562-O
n,(O) ’
S = n,(O) $ dt
;
(2) where n,(O) is the initial number of moles of A. To obtain the kinetic modelling of this reaction we consider that it results from the combination of various elementary steps which occur in different regions [l]. All these steps provide us with the mechanism of the reaction. There can be either reactions located at interfaces or diffusions through bulk regions from an interface to another. Representing this mechanism theoretically leads to the expression of the absolute rate (or of the reactance) as a function of the intensive stresses and of the time. Generally it is not possible to find a solution to the mathematical problem, so useful approximations are used like the assumption of the rate-limiting step for which one of the steps is slow while the other ones are at equilibrium.
M. Soustelle,
34
2. Theorem
of the separation
M. Pijolat I Solid State Ionics 95 (1997) 33-40
of models
2.1. The theorem
The following theorem, whose demonstration can be found elsewhere [2], may be used to succeed in the kinetic modelling of a heterogeneous reaction: When the experiments are conducted in isothermal conditions and at fixed partial pressures of all the gases, and on the assumption of a rate-limiting step, the reactance can be expressed with the product of two functions: 4(Y), which is a function of the intensive stresses only (Y: temperature, partial pressures) and thus is not dependent on time, and E(x) which is a function related to the shape and size of the region in which the rate-limiting step occurs. It is thus possible to write: % = 4(Y)E(x) = ~(kJz(t,
Y.(O,t)).
(3)
We will call c5 the ‘reactivity’ and E the space function. E depends on the size of the limiting region, and thus on time and on the initial shape and size of the solid A, as well as on the intensive stresses fixed from t =0 to t. In heterogeneous reactions two reactivities may be found, one for the growth and one for the nucleation. When not mentioned, the reactivity will be that of the growth process. Thus the relation Eq. (3) may be used in a general manner and Table 1 reports the expression taken by E and R in various cases; the following notations have been used: y is the number of nuclei formed by unit of time and of nucleation area S,; v is the number of moles of B due to the growth by unit of time and of interfacial area S,; G, is a function of the diffusion region and the flux takes the following form: J=-
DAC I0
Table 1 Expressions Rate-limiting Nucleation Growth
(4)
GD
of the reactivities
interface
interface
step
step
Growth diffusion through B
2.2. Experimental models
validation
of the separation
of
In order to verify the validity of application of the theorem of the separation of models for a given reaction, it is sufficient to show that the reactivity is not dependent on the time of reaction. This is possible by doing a first experiment in fixed conditions, for example a partial pressure Y1, and at a time t, changing suddenly its value from Y, to Y2.In a second experiment the same change is done at a time t,, different from t,, as illustrated in Fig. 1.
and of the space functions Specific fate
step
As a consequence of this theorem it will be possible to establish separate models for the kinetics of a reaction: those corresponding to the space function will be called ‘geometrical models’ and those corresponding to the processes of nucleation and growth will be called ‘physico-chemical models’. For a reaction in which one of these two processes is infinitely rapid, it is obvious that only one of the two physico-chemical models will be considered. It is useful to note that this theorem can be extended to the mixed (steady-state) kinetics involving two elementary steps, provided that these two steps occur in the same region. It will thus be valid for two interfacial reactions occurring at the same interface, but not for mixed diffusion-reaction kinetics. In the most general cases of kinetics (unsteady state, Bodenstein or others), the theorem of the separation of models cannot be used. It is then necessary to build a single model by taking into account the physico-chemical and the space aspects, according to possible pseudo-steady state approximations [3], which nevertheless leads to a complex problem.
Reactivity
4
Space function E
Reactance
%
M. Soustelle, M. Pijolat
I Solid State tonics 95 (1997)
33-40
35
3. The various forms of the space function Since the reactivity is not dependent on time, the space function obtained in isothermal and isobaric conditions varies directly as the absolute rate does with time. Thus, two forms of the space function can be found, depending on the presence or absence of a maximum: continuously decreasing (or increasing) curves, or curves with a maximum which corresponds to a sigmoidal curve for the extent of conversion. The form of the space function will depend on various factors: Fig. 1. Experimental
validation
of the separation
of models.
Let us consider the ratios of the absolute rates before and after the sudden change in the stress Y. It is easy to show that
(5) only if the theorem can be applied. We have achieved such a verification in the case of the reaction of oxidation of the cerium (III) hydroxycarbonate at 220°C [4] according to: 4CeOHC0,
+ 0, = 4Ce0,
+ 2H,O + 4C0,.
The partial pressures in water vapour and carbon were fixed to 667 Pa, and the partial pressure in oxygen was suddenly changed from 667 to 6667 Pa. Table 2 reports the values of the reactance measured, and it can be observed that their ratios remain constant. To be absolutely rigorous, this test should be accompanied with another test which demonstrates that the kinetic behaviour is in agreement with the assumption of pseudo-steady state [2].
Table 2 Experimental
validation
of the separation
The nature of the processes (nucleation or growth or both). The nature of the region in which the rate-limiting step of the growth process occurs (interface step or diffusion step). The direction of the development of the phase B (internal or external development). The initial geometry and size of the solid A.
3.1. Examples of space functions In the most simple cases, a well-known example is the sphere contracting model [5] in which the limiting step occurs at the internal interface and for an internal development, in the case of an infinite rate of germination. The space function is then given by:
E=;
[1-~t]2+(l-~)2’3
(6)
where V,,,, is the molar volume of A. For the same sphere with a diffusion through the layer of B as the rate-limiting step [6], the space function is given by:
of models
Time of sudden change (s)
Fractional conversion
Reactance % (right side)
Reactance %’ (left side)
!H/!H’
2700 3600
0.17 0.40
3.11x1o-4 3.88X 1O-4
1.96x 10m4 2.48 X 1O-.4
1.59 1.56
M. Soustelle,
36
M. Pijolat I Solid State Ionics 9.5 (1997) 33-40
3NOvmA(l - A)“3 [l - (1 -z)A]“3 E=
r,{l-(1-z)h]“3
- (1 - A)1’3} .
(7)
It can be seen that when the nucleation (or the growth) is supposed to occur with an infinite rate, the space function is fully determined by means of A, the fractional conversion, independent of the way chosen to reach this value with respect to Y; this shows that A is a state variable of the function E. For a given value of A, the space function is not dependent on the physico-chemical stresses, which is not true if the comparison is done for a given time. Moreover, if the initial particles constituting the solid A are not porous, the rate always varies monotonously with time of reaction. In a more complex case, when the nucleation and growth are competitive processes, it is necessary to use dimensionless numbers: the fractional conversion and the reduced time, 0, defined by: d-t
6=
vm,F
0
(8)
in which +g is the growth reactivity (interfacial reaction, v, or diffusion, A). The space function involves then a parameter of modelling which depends on the initial shape and size of the grains of A, given by:
(9)
in which so is the initial surface of the grains, x0 is characteristic of the size of the grains (for example it is their radius if they are spherical or cylindrical, or the half-width if they are plates). Then Eq. (3) of the rate becomes:
g
= AE(A, 0).
E(A, 0) = 3
K
i+$+-$
>
(1 -exp(-f?A))
(11) It must be noticed that the space function does not depend only on the fractional conversion as in the previous examples (Eqs. (6) and (7)), because of the parameter A which involves the two physico-chemical rates. Therefore, the fractional conversion does not fix the value of the space function nor that of the absolute rate under the related physico-chemical conditions; thus, the space function depends on the physico-chemical stresses that were fixed before the time t. It can be shown that this result may be generally found for all the reactions in which nucleation and growth rate are not instantaneous, and we will call these the complex cases. 3.2. Experimental approximation
validation
of the limiting ease
It is possible to verify experimentally the validity of the approximation of the limiting case when the curve A(t) does not exhibit a sigmoidal shape, otherwise it is ascertained that the case is complex. The method consists in doing two successive experiments, as illustrated in Fig. 2: the first one is achieved with the physico-chemical conditions Y, , the second one is started with the conditions Y, until the time t = t, then the conditions are suddenly changed from Y2 to Y,. To the time t, corresponds an extent of conversion 5, and a reactance R, measured
(10)
For example, in the case of spherical particles, on the assumptions of anisotropic growth (infinite tangential and finite radial growth rate), internal interfacial rate-limiting step, and internal development, it can be shown that the space function takes the following form [7]:
t Fig. 2. Experimental
validation
w
of the limiting case approximation.
M. Sousrelle, M. P&tat
I Solid State Ionics 95 (1997)
on the right side of the sudden change point, on the curve of the second experiment. This value is compared to !hi, measured at the time t = t, to which corresponds the extent of conversion 6, on the curve of the first experiment. It can be shown that if the values of !h, and X2 are found to be the same, the approximation of the limiting case is valid; otherwise the case is complex, and nucleation and growth rates have to be considered. 3.3. Experimental determination of the space function
of the variations
We have previously pointed out that the isothermal and isobaric curve of the reactance versus time (or versus A) varies directly as the curve of the space function versus time (or versus A). To obtain the variations of the space function with the intensive variables (temperature, partial pressures), a series of experiments started in various conditions Y,, Y,,..., Y, of the chosen variable, until the time t= t, are then all continued with the same condition, for example Y, (Fig. 3). The reactance, M,, measured in each experiment on the right side of the curve at t = t, is given by:
33-40
37
versus the chosen physico-chemical stress. These functions display usually very intricate forms. 3.4. Determination
of a geometrical
model
The determination of a geometrical model in accordance with the experimental results is aimed at the study of the values of the nucleation and growth reactivities, respectively y and 4, and their variations with the intensive stresses (partial pressures, temperatures). This can be achieved using a method of tests and errors [8] and computer simulations, and leads finally to the values of the physico-chemical rates of nucleation and growth, y and C$(or v for an interfacial step). For a series of experiments conducted with various values of a physico-chemical stress Yj (while the other stresses are maintained at the same value), it will be possible to obtain the variations of the rates of nucleation and growth with Y, respectively y(Y) and v(Y). In the next section, it will be shown how to verify experimentally that the variations of v
4. Study of the reactivity function
$71, =
($),=,<, =4V, )E(t,t T).
(12)
The study of the variations of SO versus q allows us to determine, to the factor +(Y,) (which is not dependent on Y,), the variations of the space function
The modelling of the reactivity can be done from the knowledge of the specific rates by considering that the solid exhibits plane interfaces of unit area at any time of the reaction. A mechanism involving elementary steps in their respective regions may then be proposed and solved. This gives the expression of the reactance, in a system chosen so that the reactance and the specific rate are identical. 4.1. Mathematical
form of the reactivity
This section concerns the reactivities as well the nucleation and the growth. Let us consider the following transformation (a nucleation or a growth process is represented by the same chemical equation even if the mechanisms are different): O=Y,A,+Y~A~+...+YA,+...Y~A~_ Fig. 3. Experimental function.
determination
of the variations
of the space
(13)
It can be shown that for a rate-limiting step i in a mechanism, the reactivity c#+~,is expressed by:
38
M. Soustelle,
M. Pijolat I Solid State lonics
n
P,)
=ki(z-)J,‘(z-,Pj)
L 1 1- +
[1
-exp(%) 1 (14)
in which Zj 5~~ represents the affinity and K represents the equilibrium constant. In the product ki(t)$(T,Pj), k,(t) is the kinetic constant of the limiting step i (or the ratio DO/X if the limiting step is a diffusion) and J(T,P,) is an expression which involves the equilibrium constants of the steps preceding the rate-limiting step in the mechanism, and also some partial pressures of the reactants or of the products, or of gaseous catalysts that appear in the steps preceding the rate-limiting step and this step itself. The term in the brackets of Eq. (14) is the expression of the deviation from equilibrium of the chosen experimental conditions. We can see that the product k;(t)J,‘(T,P,) is the reactivity that could be calculated in conditions far from equilibrium, by neglecting the rate of the inverse reaction in the rate-limiting step, which means neglecting all the steps occurring after the limiting one. For a given value of the space function, the ratio R can be defined as:
R=
R”
1 - l-J,P,YK
=k;(T)J(T,
Pj)E(tOt
33-40
appear as a reactant in one of the steps that precede the rate-limiting one, or in the rate-limiting step itself, without being produced by one of the preceding steps.
P,?
&] = k,(t)jp,
95 (1997)
‘0)’
(15)
This ratio varies as the product k#)J(T,P,) versus the physico-chemical stresses. If the steps preceding the rate-limiting step in the mechanism consist of a linear sequence, some properties of the ratio R can be deduced, such as: The temperature and partial pressure variables are separated. There is an order with respect to the partial pressures involved in the expression of A. The Arrhenius law is followed with respect to temperature. If a catalytic effect of a gas is revealed, it must
Therefore by determining experimentally the variations of this product ki(t)J(T,Pj) versus the partial pressures, it will be possible to find out the reactants or the products or the catalysts which are involved in the elementary steps, such as the rate-limiting step and the steps occurring just before it.
4.2. Direct determination of the variations growth reactivity with the stresses
of the
To obtain directly from the experiments the variations of the growth reactivity with a stress q, we may from Eq. (3) compare the absolute rates measured for various values of Y, and the same value of the space function. In the limiting cases, since the space function is perfectly determined by the fractional conversion, we compare the absolute rates measured for various values of Y, and for the same fractional conversion. In the complex cases of nucleation and growth, the previous method based on the comparison at the same value of the fractional conversion is not any more valid, and the isolation method may be used [9] as depicted below. A series of experiments is done with the same initial conditions Y, until the time t =t,. At this moment in each experiment a new value Y is fixed by a sudden change. The rate measured on the right of time t, is given by:
(5d5>
Ro=
I=,,,
=
dgy,Mt,,
Y,).
(16)
Thus, the variations of R, with 5 allow us to determine those of +g with 5. Then the corresponding values of the ratio R, Eq. (15), are plotted versus q, which gives the shape of the variations of the product k,(t)~(T,P,) with 5. Fig. 4 illustrates this method in the case of the oxidation of cerium hydroxycarbonate when the partial pressure in oxygen was varied [4].
M. Soustelle,
h
M. Pijolat I Solid State Iorzics 95 (1997) 33-40
39
pH_O=pCO,= 667 Pa
1
dh/dt,,,, 0.8
(s-l)
3.5 10-4:
3 m4 __
0.4
r .a-
po2=1333 Pa
l’
upo2=2666
Pa poz= 5333 Pa
--A--
2.5 1O-4 -
.
0.2
3Km
Fig. 4. Determination
of the variations
4.3. Experimental model
7200
t 6)
validation
10800
of the growth
reactivity
pH,O=pCO,=667 Pa, 220°C
of a geometrical
5. Consequence
6-l)
= (sI/no>* V
?’
,’ ,
’
method (left).
of models
. I.r ’ c
.
2,5 10‘4 -
A5
(S-l)
.
3 10-4 -
Geometrical model
0.’ / /
of the separation
“Suddenchange”method
I ., I
(right) from the isolation
When the separation of models can be used, we can see that the reactivities of growth and nucleation are appropriate to describe the chemical reaction (in the range of the stresses under study), and that the space function takes a form which depends only on the reactional topography. Thus, for the same reaction, changing the reactional topography must not change the reactivities. For example in the study of the dehydration of the lithium sulfate monohydrate, we have obtained the results reported in Fig. 6 in the case of a powder and in the case of a single grain with the same geometrical shape as the grains of the powder. The curves which represent the reactance versus time in the same physico-chemical conditions, are different since the sigmoidal shape is only obtained with the powder. The growth reactivities
3,5 10-4 -
I ,* , .
I 2oooo
15000
(d~ldt>(,=,,) = (sjno> * ”
“Sudden change” method
3 lo4
5&
vs. the oxygen partial pressure
We have previously shown that the geometrical model provides the variations of the reactivity of growth versus the physico-chemical stresses. The comparison between the results obtained from the model and those obtained directly from experiments the (isolation method) allows us to validate geometrical model. As a consequence the results on the nucleation reactivity (its variations with the stresses) are also validated. The validation may be achieved by plotting the values obtained by the experimental method versus those obtained from the geometrical model. When a straight line can be drawn as shown in Fig. 5 in the case of the oxidation of cerium hydroxycarbonate [4], the geometrical model is validated.
(dtidt>(,,,)
2 ‘o-‘o’*
/’
/ ’
/
01 0
3.5 1o-4
/ ,*
.’
0.6
/’
I /’
,’
,
, * .
Geometrical model
d’ . , ‘a.
’
c
I 10-6
’
’
1.4 10-6
v (mole.mT2. s-l)
’
2 10-4 0’ 2 10-6
4 10-6 (mo*e.m-2.6s”~ v
Fig. 5. Experimental validation of the geometrical model: growth reactivity vs. growth specific hydroxycarbonate at 220°C. Influence of oxygen (left) and carbon dioxide (right) partial pressures.
rate (v) for the oxidation
8 1o-6
of cerium
M. Soustelle, M. Pijolat I Solid State Ionics 95 (1997) 33-40
40
-B-
0
l&h-IO
2oom
20QPa
30000
time (s)
0.8
0.6
modelling of heterogeneous reactions. This methodology points out the need for well-defined experiments in order to succeed in the modelling. The theorem of the separation of models allows a simplification of the comprehensive work. However, it does not imply that the models are unrelated, in particular as far as the growth process, the physicochemical model and the geometrical one are concerned, since the assumptions on the interface where the rate-limiting step occurs, and on the direction of development of the new phase must be considered in both models. The theorem of the separation of models is also useful when the transformation involves a modification of the texture of the initial solid. Finally, it is the only way to obtain information for the prediction of the behaviour of solids in non-isobaric and nonisothermal conditions.
0 0
50x104
l.OxlOS
1.5x105
2.0x105
2.5x105
3.0x105
3.5x105
References
the(s)
Fig. 6. Fractional conversion vs. time for the dehydration of lithium sulfate monohydrate in the case of: (a) a powder, and (b) a single grain.
deduced from a geometrical model in the two experiments were found to be identical [7]. Such a consequence provides justification of the term ‘reactivity’ that we have previously given to the function f$ since it is really an intrinsic property of the chemical reaction.
6. Conclusion We have reported a series of rigorous experimental methods that appear to be very useful for the kinetic
[l] M. Soustelle, C.R. Acad. Sci. Paris 270 C (1970) 2032. [2] M. Soustelle, Modelisation Macroscopique des Transformations Physico-chimiques (Masson, Paris, 1990) pp. 287-288. [3] M. Soustelle, Modelisation Macroscopique des Transformations Physico-chimiques (Masson, Paris, 1990) p. 256. [4] J.P. Viricelle, M. Pijolat and M. Soustelle, J. Chem. Sot. Farad. Trans. 91 (1995) 4437. [5] P.W.M. Jacobs and F.C. Tomkins, in: W.E. Gardner, ed. Chemistry of the Solid State (Butterworths, London, 1955) p, 201. [6] G. Valensi, C.R. Acad. Sci. 272 C (1970) 1917. [7] F. Valdivieso, V Bouineau, M. Pijolat and M. Soustelle, presented to XIIIth ISRS, Hamburg, Sept. 1996; Solid State Ionics, in press. [8] B. Delmon, Introduction a la Cinetique Htterogtne (Technip, Paris, 1969). [9] J.C. Colson, D. Delafosse and P. Barret, Bull. Sot. Chim. (1964) 687.
Js
SOLID STATE
ELSEVIER
IONKS
Solid State Ionics 95 (1997) 33-40
Experimental
methods useful in the kinetic modelling of heterogeneous reactions M. Soustelle*,
CRESA-SHIV,
EC& des Mines,
158 cows
M. Pijolat
Faurief, 42023 Saint-Etienne
Cedex 2, France
Abstract A theorem on the separation of models is reported which shows how it may be possible to describe the kinetics of a reaction involving a solid: a physico-chemical model in which ‘reactivity’ has been defined, and a geometrical model to which a ‘space function’ corresponds. Then a series of rigorous experimental methods based on this theorem are presented. Their use improves and validates the kinetic modelling of a heterogeneous reaction. Keywords:
Modelling;
Heterogeneous
reaction;
Reaction
kinetics;
Reactance
1. Introduction
+L.
Let us consider a heterogeneous reaction of a phase A which is transformed into a phase B according to:
%A+(gw,J =B +(+G)
(1)
where G, and G, are the eventually present gases. It is possible to define an absolute rate of the reaction by means of the derivative of the extent of transformation, 5, with respect to time. It depends on all the extensive and intensive variables fixed by the experimental conditions. It is also possible to use the fractional conversion, A, defined from the extent of conversion, and its derivative with respect to time, that we will call the ‘reactance’, %, in the following. When the solid A is chosen for the expression of the extent of conversion, we have: *Corresponding
author.
0167-2738/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SOl67-2738(96)00562-O
n,(O) ’
S = n,(O) $ dt
;
(2) where n,(O) is the initial number of moles of A. To obtain the kinetic modelling of this reaction we consider that it results from the combination of various elementary steps which occur in different regions [l]. All these steps provide us with the mechanism of the reaction. There can be either reactions located at interfaces or diffusions through bulk regions from an interface to another. Representing this mechanism theoretically leads to the expression of the absolute rate (or of the reactance) as a function of the intensive stresses and of the time. Generally it is not possible to find a solution to the mathematical problem, so useful approximations are used like the assumption of the rate-limiting step for which one of the steps is slow while the other ones are at equilibrium.
M. Soustelle,
34
2. Theorem
of the separation
M. Pijolat I Solid State Ionics 95 (1997) 33-40
of models
2.1. The theorem
The following theorem, whose demonstration can be found elsewhere [2], may be used to succeed in the kinetic modelling of a heterogeneous reaction: When the experiments are conducted in isothermal conditions and at fixed partial pressures of all the gases, and on the assumption of a rate-limiting step, the reactance can be expressed with the product of two functions: 4(Y), which is a function of the intensive stresses only (Y: temperature, partial pressures) and thus is not dependent on time, and E(x) which is a function related to the shape and size of the region in which the rate-limiting step occurs. It is thus possible to write: % = 4(Y)E(x) = ~(kJz(t,
Y.(O,t)).
(3)
We will call c5 the ‘reactivity’ and E the space function. E depends on the size of the limiting region, and thus on time and on the initial shape and size of the solid A, as well as on the intensive stresses fixed from t =0 to t. In heterogeneous reactions two reactivities may be found, one for the growth and one for the nucleation. When not mentioned, the reactivity will be that of the growth process. Thus the relation Eq. (3) may be used in a general manner and Table 1 reports the expression taken by E and R in various cases; the following notations have been used: y is the number of nuclei formed by unit of time and of nucleation area S,; v is the number of moles of B due to the growth by unit of time and of interfacial area S,; G, is a function of the diffusion region and the flux takes the following form: J=-
DAC I0
Table 1 Expressions Rate-limiting Nucleation Growth
(4)
GD
of the reactivities
interface
interface
step
step
Growth diffusion through B
2.2. Experimental models
validation
of the separation
of
In order to verify the validity of application of the theorem of the separation of models for a given reaction, it is sufficient to show that the reactivity is not dependent on the time of reaction. This is possible by doing a first experiment in fixed conditions, for example a partial pressure Y1, and at a time t, changing suddenly its value from Y, to Y2.In a second experiment the same change is done at a time t,, different from t,, as illustrated in Fig. 1.
and of the space functions Specific fate
step
As a consequence of this theorem it will be possible to establish separate models for the kinetics of a reaction: those corresponding to the space function will be called ‘geometrical models’ and those corresponding to the processes of nucleation and growth will be called ‘physico-chemical models’. For a reaction in which one of these two processes is infinitely rapid, it is obvious that only one of the two physico-chemical models will be considered. It is useful to note that this theorem can be extended to the mixed (steady-state) kinetics involving two elementary steps, provided that these two steps occur in the same region. It will thus be valid for two interfacial reactions occurring at the same interface, but not for mixed diffusion-reaction kinetics. In the most general cases of kinetics (unsteady state, Bodenstein or others), the theorem of the separation of models cannot be used. It is then necessary to build a single model by taking into account the physico-chemical and the space aspects, according to possible pseudo-steady state approximations [3], which nevertheless leads to a complex problem.
Reactivity
4
Space function E
Reactance
%
M. Soustelle, M. Pijolat
I Solid State tonics 95 (1997)
33-40
35
3. The various forms of the space function Since the reactivity is not dependent on time, the space function obtained in isothermal and isobaric conditions varies directly as the absolute rate does with time. Thus, two forms of the space function can be found, depending on the presence or absence of a maximum: continuously decreasing (or increasing) curves, or curves with a maximum which corresponds to a sigmoidal curve for the extent of conversion. The form of the space function will depend on various factors: Fig. 1. Experimental
validation
of the separation
of models.
Let us consider the ratios of the absolute rates before and after the sudden change in the stress Y. It is easy to show that
(5) only if the theorem can be applied. We have achieved such a verification in the case of the reaction of oxidation of the cerium (III) hydroxycarbonate at 220°C [4] according to: 4CeOHC0,
+ 0, = 4Ce0,
+ 2H,O + 4C0,.
The partial pressures in water vapour and carbon were fixed to 667 Pa, and the partial pressure in oxygen was suddenly changed from 667 to 6667 Pa. Table 2 reports the values of the reactance measured, and it can be observed that their ratios remain constant. To be absolutely rigorous, this test should be accompanied with another test which demonstrates that the kinetic behaviour is in agreement with the assumption of pseudo-steady state [2].
Table 2 Experimental
validation
of the separation
The nature of the processes (nucleation or growth or both). The nature of the region in which the rate-limiting step of the growth process occurs (interface step or diffusion step). The direction of the development of the phase B (internal or external development). The initial geometry and size of the solid A.
3.1. Examples of space functions In the most simple cases, a well-known example is the sphere contracting model [5] in which the limiting step occurs at the internal interface and for an internal development, in the case of an infinite rate of germination. The space function is then given by:
E=;
[1-~t]2+(l-~)2’3
(6)
where V,,,, is the molar volume of A. For the same sphere with a diffusion through the layer of B as the rate-limiting step [6], the space function is given by:
of models
Time of sudden change (s)
Fractional conversion
Reactance % (right side)
Reactance %’ (left side)
!H/!H’
2700 3600
0.17 0.40
3.11x1o-4 3.88X 1O-4
1.96x 10m4 2.48 X 1O-.4
1.59 1.56
M. Soustelle,
36
M. Pijolat I Solid State Ionics 9.5 (1997) 33-40
3NOvmA(l - A)“3 [l - (1 -z)A]“3 E=
r,{l-(1-z)h]“3
- (1 - A)1’3} .
(7)
It can be seen that when the nucleation (or the growth) is supposed to occur with an infinite rate, the space function is fully determined by means of A, the fractional conversion, independent of the way chosen to reach this value with respect to Y; this shows that A is a state variable of the function E. For a given value of A, the space function is not dependent on the physico-chemical stresses, which is not true if the comparison is done for a given time. Moreover, if the initial particles constituting the solid A are not porous, the rate always varies monotonously with time of reaction. In a more complex case, when the nucleation and growth are competitive processes, it is necessary to use dimensionless numbers: the fractional conversion and the reduced time, 0, defined by: d-t
6=
vm,F
0
(8)
in which +g is the growth reactivity (interfacial reaction, v, or diffusion, A). The space function involves then a parameter of modelling which depends on the initial shape and size of the grains of A, given by:
(9)
in which so is the initial surface of the grains, x0 is characteristic of the size of the grains (for example it is their radius if they are spherical or cylindrical, or the half-width if they are plates). Then Eq. (3) of the rate becomes:
g
= AE(A, 0).
E(A, 0) = 3
K
i+$+-$
>
(1 -exp(-f?A))
(11) It must be noticed that the space function does not depend only on the fractional conversion as in the previous examples (Eqs. (6) and (7)), because of the parameter A which involves the two physico-chemical rates. Therefore, the fractional conversion does not fix the value of the space function nor that of the absolute rate under the related physico-chemical conditions; thus, the space function depends on the physico-chemical stresses that were fixed before the time t. It can be shown that this result may be generally found for all the reactions in which nucleation and growth rate are not instantaneous, and we will call these the complex cases. 3.2. Experimental approximation
validation
of the limiting ease
It is possible to verify experimentally the validity of the approximation of the limiting case when the curve A(t) does not exhibit a sigmoidal shape, otherwise it is ascertained that the case is complex. The method consists in doing two successive experiments, as illustrated in Fig. 2: the first one is achieved with the physico-chemical conditions Y, , the second one is started with the conditions Y, until the time t = t, then the conditions are suddenly changed from Y2 to Y,. To the time t, corresponds an extent of conversion 5, and a reactance R, measured
(10)
For example, in the case of spherical particles, on the assumptions of anisotropic growth (infinite tangential and finite radial growth rate), internal interfacial rate-limiting step, and internal development, it can be shown that the space function takes the following form [7]:
t Fig. 2. Experimental
validation
w
of the limiting case approximation.
M. Sousrelle, M. P&tat
I Solid State Ionics 95 (1997)
on the right side of the sudden change point, on the curve of the second experiment. This value is compared to !hi, measured at the time t = t, to which corresponds the extent of conversion 6, on the curve of the first experiment. It can be shown that if the values of !h, and X2 are found to be the same, the approximation of the limiting case is valid; otherwise the case is complex, and nucleation and growth rates have to be considered. 3.3. Experimental determination of the space function
of the variations
We have previously pointed out that the isothermal and isobaric curve of the reactance versus time (or versus A) varies directly as the curve of the space function versus time (or versus A). To obtain the variations of the space function with the intensive variables (temperature, partial pressures), a series of experiments started in various conditions Y,, Y,,..., Y, of the chosen variable, until the time t= t, are then all continued with the same condition, for example Y, (Fig. 3). The reactance, M,, measured in each experiment on the right side of the curve at t = t, is given by:
33-40
37
versus the chosen physico-chemical stress. These functions display usually very intricate forms. 3.4. Determination
of a geometrical
model
The determination of a geometrical model in accordance with the experimental results is aimed at the study of the values of the nucleation and growth reactivities, respectively y and 4, and their variations with the intensive stresses (partial pressures, temperatures). This can be achieved using a method of tests and errors [8] and computer simulations, and leads finally to the values of the physico-chemical rates of nucleation and growth, y and C$(or v for an interfacial step). For a series of experiments conducted with various values of a physico-chemical stress Yj (while the other stresses are maintained at the same value), it will be possible to obtain the variations of the rates of nucleation and growth with Y, respectively y(Y) and v(Y). In the next section, it will be shown how to verify experimentally that the variations of v
4. Study of the reactivity function
$71, =
($),=,<, =4V, )E(t,t T).
(12)
The study of the variations of SO versus q allows us to determine, to the factor +(Y,) (which is not dependent on Y,), the variations of the space function
The modelling of the reactivity can be done from the knowledge of the specific rates by considering that the solid exhibits plane interfaces of unit area at any time of the reaction. A mechanism involving elementary steps in their respective regions may then be proposed and solved. This gives the expression of the reactance, in a system chosen so that the reactance and the specific rate are identical. 4.1. Mathematical
form of the reactivity
This section concerns the reactivities as well the nucleation and the growth. Let us consider the following transformation (a nucleation or a growth process is represented by the same chemical equation even if the mechanisms are different): O=Y,A,+Y~A~+...+YA,+...Y~A~_ Fig. 3. Experimental function.
determination
of the variations
of the space
(13)
It can be shown that for a rate-limiting step i in a mechanism, the reactivity c#+~,is expressed by:
38
M. Soustelle,
M. Pijolat I Solid State lonics
n
P,)
=ki(z-)J,‘(z-,Pj)
L 1 1- +
[1
-exp(%) 1 (14)
in which Zj 5~~ represents the affinity and K represents the equilibrium constant. In the product ki(t)$(T,Pj), k,(t) is the kinetic constant of the limiting step i (or the ratio DO/X if the limiting step is a diffusion) and J(T,P,) is an expression which involves the equilibrium constants of the steps preceding the rate-limiting step in the mechanism, and also some partial pressures of the reactants or of the products, or of gaseous catalysts that appear in the steps preceding the rate-limiting step and this step itself. The term in the brackets of Eq. (14) is the expression of the deviation from equilibrium of the chosen experimental conditions. We can see that the product k;(t)J,‘(T,P,) is the reactivity that could be calculated in conditions far from equilibrium, by neglecting the rate of the inverse reaction in the rate-limiting step, which means neglecting all the steps occurring after the limiting one. For a given value of the space function, the ratio R can be defined as:
R=
R”
1 - l-J,P,YK
=k;(T)J(T,
Pj)E(tOt
33-40
appear as a reactant in one of the steps that precede the rate-limiting one, or in the rate-limiting step itself, without being produced by one of the preceding steps.
P,?
&] = k,(t)jp,
95 (1997)
‘0)’
(15)
This ratio varies as the product k#)J(T,P,) versus the physico-chemical stresses. If the steps preceding the rate-limiting step in the mechanism consist of a linear sequence, some properties of the ratio R can be deduced, such as: The temperature and partial pressure variables are separated. There is an order with respect to the partial pressures involved in the expression of A. The Arrhenius law is followed with respect to temperature. If a catalytic effect of a gas is revealed, it must
Therefore by determining experimentally the variations of this product ki(t)J(T,Pj) versus the partial pressures, it will be possible to find out the reactants or the products or the catalysts which are involved in the elementary steps, such as the rate-limiting step and the steps occurring just before it.
4.2. Direct determination of the variations growth reactivity with the stresses
of the
To obtain directly from the experiments the variations of the growth reactivity with a stress q, we may from Eq. (3) compare the absolute rates measured for various values of Y, and the same value of the space function. In the limiting cases, since the space function is perfectly determined by the fractional conversion, we compare the absolute rates measured for various values of Y, and for the same fractional conversion. In the complex cases of nucleation and growth, the previous method based on the comparison at the same value of the fractional conversion is not any more valid, and the isolation method may be used [9] as depicted below. A series of experiments is done with the same initial conditions Y, until the time t =t,. At this moment in each experiment a new value Y is fixed by a sudden change. The rate measured on the right of time t, is given by:
(5d5>
Ro=
I=,,,
=
dgy,Mt,,
Y,).
(16)
Thus, the variations of R, with 5 allow us to determine those of +g with 5. Then the corresponding values of the ratio R, Eq. (15), are plotted versus q, which gives the shape of the variations of the product k,(t)~(T,P,) with 5. Fig. 4 illustrates this method in the case of the oxidation of cerium hydroxycarbonate when the partial pressure in oxygen was varied [4].
M. Soustelle,
h
M. Pijolat I Solid State Iorzics 95 (1997) 33-40
39
pH_O=pCO,= 667 Pa
1
dh/dt,,,, 0.8
(s-l)
3.5 10-4:
3 m4 __
0.4
r .a-
po2=1333 Pa
l’
upo2=2666
Pa poz= 5333 Pa
--A--
2.5 1O-4 -
.
0.2
3Km
Fig. 4. Determination
of the variations
4.3. Experimental model
7200
t 6)
validation
10800
of the growth
reactivity
pH,O=pCO,=667 Pa, 220°C
of a geometrical
5. Consequence
6-l)
= (sI/no>* V
?’
,’ ,
’
method (left).
of models
. I.r ’ c
.
2,5 10‘4 -
A5
(S-l)
.
3 10-4 -
Geometrical model
0.’ / /
of the separation
“Suddenchange”method
I ., I
(right) from the isolation
When the separation of models can be used, we can see that the reactivities of growth and nucleation are appropriate to describe the chemical reaction (in the range of the stresses under study), and that the space function takes a form which depends only on the reactional topography. Thus, for the same reaction, changing the reactional topography must not change the reactivities. For example in the study of the dehydration of the lithium sulfate monohydrate, we have obtained the results reported in Fig. 6 in the case of a powder and in the case of a single grain with the same geometrical shape as the grains of the powder. The curves which represent the reactance versus time in the same physico-chemical conditions, are different since the sigmoidal shape is only obtained with the powder. The growth reactivities
3,5 10-4 -
I ,* , .
I 2oooo
15000
(d~ldt>(,=,,) = (sjno> * ”
“Sudden change” method
3 lo4
5&
vs. the oxygen partial pressure
We have previously shown that the geometrical model provides the variations of the reactivity of growth versus the physico-chemical stresses. The comparison between the results obtained from the model and those obtained directly from experiments the (isolation method) allows us to validate geometrical model. As a consequence the results on the nucleation reactivity (its variations with the stresses) are also validated. The validation may be achieved by plotting the values obtained by the experimental method versus those obtained from the geometrical model. When a straight line can be drawn as shown in Fig. 5 in the case of the oxidation of cerium hydroxycarbonate [4], the geometrical model is validated.
(dtidt>(,,,)
2 ‘o-‘o’*
/’
/ ’
/
01 0
3.5 1o-4
/ ,*
.’
0.6
/’
I /’
,’
,
, * .
Geometrical model
d’ . , ‘a.
’
c
I 10-6
’
’
1.4 10-6
v (mole.mT2. s-l)
’
2 10-4 0’ 2 10-6
4 10-6 (mo*e.m-2.6s”~ v
Fig. 5. Experimental validation of the geometrical model: growth reactivity vs. growth specific hydroxycarbonate at 220°C. Influence of oxygen (left) and carbon dioxide (right) partial pressures.
rate (v) for the oxidation
8 1o-6
of cerium
M. Soustelle, M. Pijolat I Solid State Ionics 95 (1997) 33-40
40
-B-
0
l&h-IO
2oom
20QPa
30000
time (s)
0.8
0.6
modelling of heterogeneous reactions. This methodology points out the need for well-defined experiments in order to succeed in the modelling. The theorem of the separation of models allows a simplification of the comprehensive work. However, it does not imply that the models are unrelated, in particular as far as the growth process, the physicochemical model and the geometrical one are concerned, since the assumptions on the interface where the rate-limiting step occurs, and on the direction of development of the new phase must be considered in both models. The theorem of the separation of models is also useful when the transformation involves a modification of the texture of the initial solid. Finally, it is the only way to obtain information for the prediction of the behaviour of solids in non-isobaric and nonisothermal conditions.
0 0
50x104
l.OxlOS
1.5x105
2.0x105
2.5x105
3.0x105
3.5x105
References
the(s)
Fig. 6. Fractional conversion vs. time for the dehydration of lithium sulfate monohydrate in the case of: (a) a powder, and (b) a single grain.
deduced from a geometrical model in the two experiments were found to be identical [7]. Such a consequence provides justification of the term ‘reactivity’ that we have previously given to the function f$ since it is really an intrinsic property of the chemical reaction.
6. Conclusion We have reported a series of rigorous experimental methods that appear to be very useful for the kinetic
[l] M. Soustelle, C.R. Acad. Sci. Paris 270 C (1970) 2032. [2] M. Soustelle, Modelisation Macroscopique des Transformations Physico-chimiques (Masson, Paris, 1990) pp. 287-288. [3] M. Soustelle, Modelisation Macroscopique des Transformations Physico-chimiques (Masson, Paris, 1990) p. 256. [4] J.P. Viricelle, M. Pijolat and M. Soustelle, J. Chem. Sot. Farad. Trans. 91 (1995) 4437. [5] P.W.M. Jacobs and F.C. Tomkins, in: W.E. Gardner, ed. Chemistry of the Solid State (Butterworths, London, 1955) p, 201. [6] G. Valensi, C.R. Acad. Sci. 272 C (1970) 1917. [7] F. Valdivieso, V Bouineau, M. Pijolat and M. Soustelle, presented to XIIIth ISRS, Hamburg, Sept. 1996; Solid State Ionics, in press. [8] B. Delmon, Introduction a la Cinetique Htterogtne (Technip, Paris, 1969). [9] J.C. Colson, D. Delafosse and P. Barret, Bull. Sot. Chim. (1964) 687.