The initial rate of solid-phase reaction and composition of the reaction product

Article abstract of DOI:10.1134/S1070363209100016

Dependence of initial reaction rate on the initial composition of reaction mixture in the systems of Na₂CO₃-V₂O ₅, Na₂CO₃-Fe₂O₃, and Na₂CO₃-AgCl

Full text of DOI:10.1134/S1070363209100016

ISSN 1070-3632, Russian Journal of General Chemistry, 2009, Vol. 79, No. 10, pp. 2081–2086. © Pleiades Publishing, Ltd., 2009.
Original Russian Text © V.G. Povarov, O.B. Sokolova, 2009, published in Zhurnal Obshchei Khimii, 2009, Vol. 79, No. 10, pp. 1585–1590.
The Initial Rate of Solid-Phase Reaction
and Composition of the Reaction Product
V. G. Povarov and O. B. Sokolova
Hertzen Russian State Pedagogical University, nab. r. Moiki 48, St. Petersburg, 191186 Russia
e-mail: obsokolova@rambler.ru
Received March 26, 2009
Abstract―Dependence of initial reaction rate on the initial composition of reaction mixture in the systems of
Na2CO3–V2O5, Na2CO3–Fe2O3, and Na2CO3–AgCl at different temperature was studied. In all the cases is
found a minimum in the degree of transformation at the stoichiometric ratio of the reagent. A model is
proposed for the description of the obtained dependences based on the representation of the reaction mixture as
randomly moving spherical grains of reagents.
DOI: 10.1134/S1070363209100016
The rate of solid-phase reaction is affected by a
large series of the factors of different nature, like
degree of dispersion of reagents, temperature, pressure,
method of the reagents preparation, the way of the
preparation of reaction mixture and, finally, the
mixture composition [1]. At the study of the influence
of composition on the reaction rate a number of
difficulties appears. First, the ratio of reagents remains
unchanged in the course of reaction only for the
stoichiometric mixture of reagents. In the case of a
mixture of any other composition it grows in favor of
the reagent taken originally in excess. Therefore
kinetic data for the high degree of transformation
correspond to the reagents ratio strongly differing from
the initial ratio [2]. Second, the accumulation of the
solid reaction products also distorts the kinetic data.
Therefore in order to consider correctly the effect of
composition of a reaction mixture on the kinetics of
solid-phase reaction it is expedient to study the
dependence of the initial reaction rate on the initial
composition of system.
of their initial rates. First, in these systems side
reactions are absent. Second, the rate of each reaction
is within the limit which allow to measure its initial
value at several temperatures in a reasonable time.
Third, the degree of the transformation of all three
processes can be determined by measuring the sample
mass loss which is strictly connected with the reaction
progress. Since the content of the reaction product at
the moment of measuring its initial rate is small in
comparison with the masses of reagents, the kinetic
measurements of the initial rate by the determination
of the change in the quantity of product ensure
considerably higher accuracy than by determination of
the content of a reagent.
The rate in all three systems was studied by
measuring the sample mass loss (due to the liberation
of CO2 or CO2 + O2) at intermittent time intervals. The
initial mass of weighted sample was 0.12–0.17 g. In all
systems the rate was measured for five mixtures with
the initial molar ratio of reagents 3:1, 2:1, 1:1, 1:2, 1:3.
Iron oxide and silver chloride are insoluble in water.
Therefore for the systems of Na2CO3–Fe2O3 and
Na2CO3–AgCl was used a “wet” mixing: the reagents
were thoroughly ground in a mortar, then 3–4 drops of
water were added, and the mixture was ground again.
Then the mixture was dried to the constant weight at
200°C. At this temperature the reactions Na2CO3 +
Fe2O3 and Na2CO3 + AgCl do not occur. The oxide of
vanadium is noticeably water-soluble. Therefore for
the system of Na2CO3–V2O5 was used “dry” mixing:
This work consists in the study of the influence of
the initial composition of mixture on the initial rate of
three solid-phase heterogeneous reactions:
Na2CO3 + V2O5 →2NaVO3 + CO2,
Na2CO3 + Fe2O3 → 2NaFeO2 + CO2,
Na2CO3 + 2AgCl → 2NaCl + 2Ag + CO2 + 1/2O2.
All these reactions satisfy a number of the
requirements necessary for the correct measurements
2081

2082
POVAROV, SOKOLOVA
Table 1. Degree of transformation in the systems of
Na2CO3–V2O5, Na2CO3–Fe2O3, and Na2CO3–AgCl at
different initial reagent ratios
Each measurement was repeated no less than 3
times. The relative error of the degree of trans-
formation for all experiences was not higher than 10%.
Besides the kinetic data, for each reaction mixture was
measured its bulk density. The obtained results are
represented in Table 1. For all studied systems the X-
ray phase analysis of samples was done, which
confirmed the formation of the reaction products and
no sode products.
Degree of transformation, %
Molar composition Bulk density,
of the mixture
g ml–1
400°С
450°С
500°С
Na2CO3–V2O5 (experiment duration 2 min)
1Na2CO3:3V2O5
1Na2CO3:2V2O5
1Na2CO3:1V2O5
2Na2CO3:1V2O5
3Na2CO3:1V2O5
1.0
10.7
8.55
6.83
14.1
18.3
43.7
25.9
15.2
20.1
25.0
72.9
54.6
34.7
58.1
61.7
0.93
0.67
0.59
0.57
It is evident from the data of Table 1 that in all
cases the smallest degree of transformation is reached
at the stoichiometric mixture of reagents: In the case of
reaction of sodium carbonate with vanadium oxide and
iron oxide the minimum is observed at the molar ratio
of reagents 1:1, and in the case of the system of
Na2CO3 + 2AgCl at the ratio Na2CO3:AgCl = 1:2.
Analogous data were obtained in [3] for the reactions
of CaCO3 + MoO3 → CaMoO4 + CO2 and 2Li2CO3 +
LiFe5O8 → 5LiFeO2 + 2CO2. For the reaction of
sodium carbonate with iron oxide, at the increase in the
mole fraction of the iron oxide over stoichiometric the
degree of transformation grows more sharply than at
the increase in the content of sodium carbonate. For
the reaction Na2CO3 + AgCl an opposite picture is
observed: Here the degree of transformation increases
more sharply at the increase in the content of silver
chloride in the system. In the case of the system
Na2CO3–V2O5 the influence of the excess of one or
another reagent on the degree of transformation
depends on temperature.
Degree of transformation, %
Molar composition Bulk density,
of the mixture
g ml–1
700°С
750°С
800°С
Na2CO3–Fe2O3 (experiment duration 3 min)
1Na2CO3:3Fe2O3
1Na2CO3:2Fe2O3
1Na2CO3:1Fe2O3
2Na2CO3:1Fe2O3
3Na2CO3:1Fe2O3
0.80
0.69
0.58
0.56
0.55
19.3
12.4
25.6
15.7
41.9
35.7
15.3
19.5
23.6
4.76
9.55
13.0
6.91
12.4
18.5
Degree of transformation, %
Molar composition Bulk density,
of the mixture
g ml–1
350°С
400°С
440°С
Na2CO3–AgCl (experiment duration 5 min)
1Na2CO3:3AgCl
1Na2CO3:2AgCl
1Na2CO3:1AgCl
2Na2CO3:1AgCl
3Na2CO3:1AgCl
1.2
3.95
1.51
5.29
7.70
7.59
5.11
6.77
16.3
9.23
18.2
29.3
48.0
1.1
0.73
0.69
0.66
11.9
34.2
24.7
Thus, already for 5 different reactions the similar
dependences of the degree of trans-formation on the
initial composition of mixture were obtained. Let us
note that the rate of four reactions was studied by the
gravimetric method, and for the reaction 2Li2CO3 +
LiFe5O8 → 5LiFeO2 + 2CO2 the determination of the
degree of transformation was done only by measuring
the losses of the ferromagnetic phase LiFe5O8. That is,
the data obtained were got with the use of different
methods of monitoring the reaction course. All this
indicates existence of general regularity in the kinetics
of solid-phase reactions. In the already mentioned
work [3] a number of equations was suggesten, which
gave the qualitatively correct dependences of the rate
on the composition. They all are based on the analogy
with the law of mass action for the rate of bimolecular
reaction in solutions, V = k[A][B]. If mass fractions of
reagents or any other proportional to them values are
considered instead of the reagents concentrations, then
the derivative of the degree of transformation on time
the reagents weighed in the necessary quantity were
thoroughly ground in a mortar without adding water.
The experiments were carried out at three
temperatures in quartz test tubes. Before performing an
experiment the test tube with the mixture was warmed
thoroughly at a temperature of 200°C for 10 min for
the removal of the possible residual water. Then the
test tube by pincers was placed into the resistance tube
furnace, heated to the temperature of test, and it was
left for a required time. Then the test tube was taken
out from the furnace, cooled and weighed. The degree
of transformation (α) was calculated using the formula:
Δm
,
α =
Δmmax
where Δm is the mass loss of the reaction mixture after
its heating, Δmmax is the mass loss of reaction mixture
at the end of reaction.
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THE INITIAL RATE OF SOLID-PHASE REACTION
2083
will have a minimum at the stoichiometric ratio of
reagents and approach finite values when the mass
fraction of any of two reagents tends to zero. However,
it should be noted that these equations contradict one
of the principal characteristics of solid-phase reactions:
the reaction occurs only in the places of the contact of
the reagent particles. This circumstance is possible to
overcome by the introduction of the molar, volumetric
or specific surface areas of reagents. So it is possible to
write:
value of the derived degree of transformation in time at
the initial time moment it is possible to use the
expression (dα)/(dt) ≈ α/t, finally we obtain:
dα
α
1
ΔM∞
gA gB
.
(1)
≈
= p'
·
·
dt
t
U
U
Here p' is new proportionality coefficient, ΔM∞ is a
number of moles of carbon dioxide liberated from one
mole of the mixture of the given initial composition.
Numerically this value is equal to the mole fraction of
the reagent present in shortage the only exception is
the system Na2CO3 + 2AgCl with a deficiency in silver
chloride. In this case ΔM∞ = XAgCl/2. This equation is
possible to bring to the form which allows the
calculation of the unknown parameters by the least
squares method. Let us write Eq. (1) in the form:
dα
dt
= kSASB,
where SA and SB are surfaces of the reagents per unit of
the volume of mixture. This expression also gives the
qualitatively accurate picture of the phenomenon,
however, it does not consider such a property of
dispersed solid bodies as the presence of the large
difference between the external and full surface area of
the particle. Both the values should be measured in
separate experiments, which strongly increases the
labor consumption. Ideally, a simple model is needed,
which correctly accounts for both circumstances, finite
dimension of the reacting particles and their interaction
only at the points of their contact. In search for the
solution we focused our attention on the model of solid
spheres used often in the theory of nonideal solutions.
Actually, assuming that an average particle (grain) of
the reagent A has spherical form and contains mA
moles of substance, and average particle of the reagent
B cntains mB moles, then knowing the mole fractions
of reagents XA and XB it is possible to find a number of
grains of each reagent in one mole of the mixture:
t
(XA/mA + XB/mB)2U 2
(XA/mA)(XB/mB)
.
= p'
(2)
αΔM ∞
After simple transformations, we obtain the
dependence:
tXAXB
αΔM∞ = √p'm m U(X /m + X /m ).
√
(3)
A
B
A
A
B
B
This expression can be made more convenient, if
we have the data on the dependence of the bulk molar
volume of mixture on the composition. Since we
measured the bulk densities (ρ) of our mixtures, the
bulk molar volume can be found with the formula:
U = (XAMA + XBMB)/ρ.
Here MA and MB are molecular weights of reagents A
and B. If we now devide the left and the right sides of
Eq. (8) by molar volume, then in the right side a linear
function of the mixture composition appears:
XA
XB
mB
.
and
NA =
NB =
mA
tXA(1 – XA)
αΔM∞
√
Knowing the numbers NA and NB, it is possible to
find the fractions, analogous to the mole fraction (gi):
Z(XA) =
= p'[XA(1/mA – 1/mB) + 1/mB]. (4)
U
NA
NA + NB
NB
NA + NB
In Table 2 the initial data are given and inter-
mediate calculations with Eq. (9) for the system of
Na2CO3–V2O5, at 400°C. Calculations were performed
on the PC IBM with a FORTRAN routine, with an
accuracy to 16 significant numbers (double precision).
In Figs. 1 and 2 the results of the calculations are
presented. It can be seen that for the bulk molar
volume linear dependence was onserved. The
intercepts at the ordinate axes from the corresponding
straight lines have a simple physical sense. For the
dependence U(XA) these are the bulk molar volumes of
each of the components, while for the dependence
.
gA =
, gB =
We assumed that the total number of contacts (K)
of such spheres with each other is proportional to the
product of the ratios of their portions to the volume of
the reaction mixture:
gA gB
.
K = p
·
U
U
Here U is the molar volume of reaction mixture, p is
the proportionality coefficient. The number of contacts
must be proportional to the reaction rate. Since for the
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POVAROV, SOKOLOVA
are presented in Figs. 3a–3c and in Table 3. In Table 3
the parameters of linear plots according to Eq. (4) for
all the systems are given. It is possible to note the
fairly good correspondence between calculated curve
and the experimental points. The parameters A0 and A1
in linear Eq. (4) vary monotonically with an increase in
the temperature in each system and in accordance with
the physical sense. According to the data in Table 3 it
is possible to calculate the ratio of the numbers of
moles contained in one grain of reagent at all
temperatures. According to Eq. (4), m(Na2CO3)/
m(X) = A0/( A0 + A1). Here X is the second reagent.
The results are presented in Table 4. It is possible to
see that the computed values of the ratios indicated
noticeably change with an increase in the temperature
for all the systems. This contradicts our model. The
smallest variations are found for the system with
Fe2O3, and greatest, for AgCl. It is possible to assume
that this is connected with the melting points of the
reagents. Among all the studied systems the iron oxide
(mp 1565°C) is the most refractory and the reaction
was conducted at temperatures much lower than its
melting temperature. For silver chloride, mp is 455°C,
while for the oxide of vanadium, mp 685°C, for
sodium carbonate, mp 851°C. It is possible to see that
with an increase in the temperature the particles of the
second reagent are enlarged (ratio the decreases) and
this is especially noticeable for the system with silver
chloride. Thus, this nonconformity between the model
and the experiment can be explained by sintering of
the particles of one of the reagents. It is interesting to
note that the system Na2CO3–Fe2O3 has the lowest
correlation coefficients. This is naturally explained by
Table 2. Results of intermediate calculations of kinetic
parameters according to Eq. (4) for te system Na2CO3–V2O5
at 400°С
X(Na2CO3)
0.250
α
Z(X)
U(X)(ml)
163
0.107
0.0855
0.0683
0.141
0.183
0.0230
0.0234
0.0178
0.0138
0.0131
0.334
168.5
214.9
222.6
219.3
0.500
0.666
0.750
Z(XA) they are the values of p'/mA and p'/mB. Thus,
from the kinetic measurements we can reveal the ratio
of rate constant to the number of moles of the reagent
contained in its single particle (grain). Hence it is
possible to find the ratio mA/mB. Since p'/mA = 0.0069,
and p'/mB = 0.0295, mA/mB = 295/69 = 4.28. This
means that the particles of sodium carbonate contain
four over-hold more formula units, than the particles of
vanadium oxide. To find the value of rate constant we
can only when the real sizes are known of the particles
of at least one of the reagents. This problem can be
solved, but in this investigation we were focused on
the kinetic description of the process. It is important to
reveal from this point of view the correspondence
between the theoretical curve and the experimentally
measured degrees of transformation. Respective data
0.032
0.028
0.024
0.020
α
0.6
Z(X)
0.016
0.012
U(X)
0.5
0.4
0.3
0.2
0.1
0
0.008
0.004
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x(Na2CO3)
Fig. 1. Experimental calculated straight lines for the
dependences
tXA(1 – XA)
0
0.2
0.4
0.6
0.8
1.0
= F(XA) and U(XA).
Z(XA) = (1/U)
αΔM∞
√
x(Na2CO3)
The system of Na2CO3–V2O5, 400°С, reaction duration
2 min, U = 134 + 127.9XA, Z(XA) = 0.0295 – 0.0226XA
with the coefficients found by mean-square method,
correlation coefficients are 0.9265 and 0.9803 respectively.
Fig. 2. Theoretical curve and experimental data for the
degree of transformation. The system of Na2CO3–V2O5,
400°С, reaction duration 2 min.
RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 79 No. 10 2009

THE INITIAL RATE OF SOLID-PHASE REACTION
(a)
2085
(b)
calculation
α
0.8
α
0.95
– 400°C
– 450°C
– 500°C
0.85
0.75
0.65
0.55
0.45
0.35
0.25
0.15
0.05
calculation
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
– 700°C
– 750°C
– 800°C
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
x(Na2CO3)
x(Na2CO3)
(c)
α
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0
– 300°C
– 350°C
– 400°C
0.2
0.3 0.4
0.5 0.6
0.7 0.8
x(Na2CO3)
Fig. 3. Theoretical curve and experimental data for all the studied systems. (а) Na2CO3–V2O5, (b) Na2CO3–Fe2O3, and (c) Na2CO3–
AgCl.
proximity to the unity of the ratio m(Na2CO3)/m
(Fe2O3) and it rather tells in favor of the considered
model. One additional indirect method of validation of
the model consists in the use of the Arrhenius
equation. Actually, if rate constant p' in Eqs. (3) and
(4) does not depend on the composition, we must
observe the identical values of activation energy
independent from the composition of the reaction
product. The activation energies were calculated for
this purpose for all the studied systems. In these
calculation it was considered that the initial rate of
reaction with the fixed initial composition is
proportional to the value dα/dt ≈ α/t. The slope tangent
of the dependence is found by the method of least
squares from the dependence ln α/t = ln (A – Ea/RT)
(T is absolute temperature).
The obtained values of the activation energy of the
studied reactions, the correlation coefficients (R) and
value of ln A are presented in Table 5. An average
relative error in the activation energy is 20%. As can
be seen from Table 5, the data in all cases are
excellently described by Arrhenius equation. For the
systems Na2CO3–Fe2O3 and Na2CO3–AhCl the maximum
RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 79 No. 10 2009

2086
POVAROV, SOKOLOVA
Table 4. Ratio of number of moles in one grain of a reagent
from the data of kinetic measurements (X is second reagent)
Table 3. Coefficients of equations
tXA(1 – XA)
Z(XA) = (1/U)
= A0 + A1XA
αΔM∞
√
System
t, °C
m(Na2CO3)/m(X)
and correlation coefficients for all the studied systems and
temperatures
Na2CO3–V2O5
400
450
4.3
1.2
Correlation
System
t, °C
A0
A1
R
500
700
2.0
Na2CO3+V2O5
Na2CO3+V2O5
Na2CO3+V2O5
Na2CO3+Fe2O3
Na2CO3+Fe2O3
Na2CO3+Fe2O3
Na2CO3+AgCl
Na2CO3+AgCl
Na2CO3+AgCl
400
450
500
700
750
800
300
350
400
0.030
0.013
0.010
0.020
0.017
0.011
0.142
0.086
0.059
–0.023
–0.002
–0.005
0.0017
0.0005
0.0049
–0.14
0.9803
0.4671
0.9255
0.1576
0.0662
0.8513
0.8391
0.9428
0.9091
Na2CO3–Fe2O3
Na2CO3–AgCl
0.92
750
1.0
800
300
0.69
71
350
400
6.1
5.9
–0.072
–0.049
Table 5. Parameters of the Arrhenius equation for the reactions Na2CO3 + V2O5 →, Na2CO3 + Fe2O3 →, Na2CO3 + AgCl
X = V2O5
X = Fe2O3
X = AgCl
Molar composition
of Na2CO3–X
mixture
Ea,
kJ mol–1
84
Ea,
kJ mol–1
67
Ea,
kJ mol–1
58
Rcorr
ln A
12.2
Rcorr
ln A
Rcorr
ln A
1Na2CO3:3X
1Na2CO3:2X
1Na2CO3:1X
2Na2CO3:1X
3Na2CO3:1X
0.98
0.99
0.99
0.95
0.95
0.98
0.94
0.97
0.98
0.99
5.5
7.9
8.2
4.1
3.3
0.98
0.99
0.91
0.93
0.93
6.2
8.8
4.7
7.1
5.9
80
70
60
52
11.3
9.2
91
100
61
75
49
55
48
8.0
6.8
52
value of the activation energy is achieved at the
stoichiometric ratio of reagents (Na2CO3: Fe2O3 = 1:1
and Na2CO3:AgCl = 2:1). At the preparation of these
systems we used “wet” mixing. For the system of
Na2CO3–V2O5 is observed completely different
dependence: here the activation energy practically
linearly decreases with an increase in the mole fraction
of sodium carbonate. During the preparation of this
mixture we used “dry” mixing. The dependence of
activa-tion energy on composition indicates income-
pleteness of the proposed model. We plan to verify
Eqs. (3) and (4) in our further investigations by two
methods. First, we will consider a new class of reac-
tions with other methods of their monitoring. Second,
we will study influence on the rate of the effect of
dilution of the reaction mixture with an inert solvent.
REFERENCES
1. Tret’yakov, Yu.D., Tverdofaznye reaktsii (Solid Phase
Reaction), Moscow: Khimiya, 1978.
2. Del’mon, B., Kinetika geterogennykh reaktsii (Kinetics
of Heterogenous Reactions), Boldyrev, V.V., Ed.,
Moscow: Mir, 1972.
3. Povarov, V.G. and Blyandur, E.P., Kinetika i Kataliz,
1999, vol. 40, no. 4, p. 520.
4. Povarov, V.G., Vest. Sankt-Peterburg. Gos. Univ. Ser.
4, 2001, no. 3(20), p. 54.
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