Delving into the Kinetics of Reversible Thermal ... - ACS Publications

Jul 10, 2017 - and cooled at the cooling rates of 0.05, 0.1, 0.2, ... Carlo bootstrap method27 ... the partial pressure, and γ is a fitting parameter...
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Delving into the Kinetics of Reversible Thermal Decomposition of Solids Measured on Heating and Cooling Tatsiana Liavitskaya and Sergey Vyazovkin* Department of Chemistry, University of Alabama at Birmingham, 901 S. 14th Street, Birmingham, Alabama 35294, United States ABSTRACT: This work explores the differences in the kinetics of reversible thermal decomposition measured respectively during heating and cooling. The kinetics of the process is measured by differential scanning calorimetry (DSC) and thermogravimetric analysis (TGA). The thermal dehydrations of calcium oxalate monohydrate (CaC2O4· H2O) and calcium sulfate dihydrate (CaSO4·2H2O) are studied as examples of reversible decomposition. The kinetics is analyzed by an advanced isoconversional method that demonstrates that on cooling the activation energy decreases with decreasing temperature, whereas on heating it decreases with increasing temperature. This qualitative difference can be understood by modifying an earlier proposed kinetic model to account for a dependence of the equilibrium pressure on conversion. The model is applied to the thermal dehydration of CaC2O4·H2O, CaSO4·2H2O, and lithium sulfate monohydrate (Li2SO4·H2O) studied previously. The results indicate that in the reversible decomposition on cooling the equilibrium pressure has much stronger dependence on conversion than for the same process on heating. The dramatic difference in the evolution of the equilibrium pressure explains the qualitative difference in the temperature dependencies of the activation energy evaluated respectively from the cooling and heating data. Li2SO4·H2O,8−10 CaSO4·2H2O,11 Ni3(PO4)2·8H2O,12 NiC2O4· 2H2O,13 La2(C2O4)3·10H2O,14 and La2(CO3)3·3.4H2O.15 As discussed further, there are several models10,16−18 that predict that the effective activation energy should decrease with increasing temperature (i.e., during heating) because the reversible reaction departs from equilibrium. Recently, we have demonstrated10 that the thermal decomposition can be performed not only on heating but also on cooling. An unexpected outcome of that study is that the thermal dehydration of Li2SO4·H2O demonstrated the temperature dependence of the effective activation energy that is qualitatively different from that observed on heating. On cooling, the activation energy decreases with decreasing temperature, whereas on heating it decreases with increasing temperature. Unfortunately, none of the aforementioned models predicts properly the temperature dependence of the effective activation energy observed experimentally for the reversible thermal decomposition taking place on cooling. In this work we demonstrate that the qualitatively different dependencies of the activation energy on temperature that are observed respectively in cooling and heating experiments are not unique to the thermal dehydration of Li2SO4·H2O. The same effect is observed for presently studied dehydrations of CaC2O4·H2O and CaSO4·2H2O and, therefore, is a general phenomenon. To better understand this phenomenon, we introduce some important modifications to one of the models and demonstrate that the modified model is capable of

1. INTRODUCTION Reversible reactions are a subject of continuous research interest not only because they are broadly encountered in nature and widely used in industry but also because they are unavoidably complex. Even the simplest reversible reaction involves two steps that proceed in opposite directions. Since the rate of the steps generally depends on the state variables such as temperature and pressure, their changes can change the direction of the reaction or bring it to a halt, i.e., equilibrium. The complexity of the reaction behavior gives rise to intricate kinetics. In particular, the rate of a process that involves reversible reactions can have a convoluted temperature dependence.1,2 The latter is known to yield experimentally an effective (or apparent) activation energy. This is a composite parameter, whose value commonly is a function of various parameters, including the rate constants and activation energies of the individual reaction steps. The effective nature of the activation energy manifests itself in the fact that its value may depend on temperature and/or reaction progress as readily revealed by isoconversional kinetic analysis.3 For a process involving a reversible reaction, the effective activation energy can even turn negative,1,2 which means the process rate would be decreasing with increasing temperature. The present work focuses on reversible thermal decomposition of solids. There are many solids that decompose in a reversible manner, perhaps the most common case being the thermal dehydration of inorganic hydrates. Isoconversional kinetic analysis has frequently demonstrated that for this process the effective activation energy decreases with the reaction progress and/or temperature. Examples of such behavior include the thermal dehydration of CaC2O4·H2O,4−7 © XXXX American Chemical Society

Received: May 25, 2017 Revised: June 21, 2017

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DOI: 10.1021/acs.jpcc.7b05066 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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cooling data than the effect of overshooting. Therefore, it is essential to use faster heating rate to reach a turning temperature T* without a significant mass loss. For DSC experiments on heating 3.9−4.2 mg of the sample was placed in open 40 μL aluminum pans and heated at 0.75, 1.25, 2.5, 5, 7.5, 10, 12.5, 15, 17.5, and 20 °C min−1. The use of the heating rates slower than 1 °C min−1 was found to result in a poor signal-tonoise ratio, which is not surprising because the DSC signal is proportional to the reaction rate, and the latter decreases with decreasing the heating rate. On the other hand, TGA provides an integral signal whose intensity is independent of the heating rate that allows one to use slower heating rates and yet avoid the problem with the poor signal-to-noise ratio. Thus, slower heating rate data were obtained by using TGA measurements. The runs were performed on 5.0−5.5 mg of the CaC2O4·H2O samples placed in open 40 μL aluminum pans. The heating rates used were 0.3, 0.5, 0.8, 1.25, 2.5, 5, 7.5, 10, 12.5, and 15 °C min−1.

predicting properly the temperature dependence of the effective activation energy on both cooling and heating. Furthermore, we use this model to obtain insights into the origin of the qualitatively different dependencies of the activation energy on temperature that are observed in cooling and heating experiments, respectively.

2. EXPERIMENTAL SECTION All calorimetric measurements were performed with a heat flux differential scanning calorimeter (DSC) (Mettler-Toledo, 822e). Thermogravimetric (TGA) measurements of CaC2O4· H2O were carried out using Mettler Toledo TGA/DSC 3+ instrument. For both instruments, indium and zinc standards were used to perform temperature and tau-lag calibrations. All experiments were performed in the atmosphere of nitrogen flow. The flow rate was maintained at 80 mL min−1 as recommended by the manufacturer. CaSO4·2H2O (white crystalline powder, 98% pure) was purchased from Acros Organics. 3.9−4.1 mg of the sample was placed in open 40 μL aluminum pans and heated to 190 °C at the heating rates of 0.15, 0.3, 0.4, 0.5, and 1.25 °C min−1. The sample mass for all samples was selected to be as small as possible to avoid any significant thermal gradients but large enough to have a good signal-to-noise ratio. In the experiments on cooling, 3.9−4.2 mg of CaSO4·2H2O was heated from 25 to the turning temperature T* = 95 °C at 30 °C min−1 and cooled at 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5 °C min−1 to the temperature where the reaction rate is negligibly small. The turning temperature T* is the temperature at which heating is switched to cooling. The decomposition kinetics of CaSO4·2H2O is known to be of autocatalytic or sigmoid type,19,20 meaning that the reaction rate is minimal at the beginning of the reaction. In the experiments on cooling, it is essential to reach T* as fast as possible to avoid partial decomposition occurring on heating. In the previous experiments performed on DSC,10 relatively fast heating rates 100−300 °C min−1 were used. The use of faster heating rates may lead to overshooting the T* and prolonged stabilization period. Normally, overshooting occurs during the switching of temperature program from fast heating to slow cooling. For a short period of time, DSC continues to increase temperature even though the temperature program is switched to cooling. During this period, it is practically impossible to obtain DSC data suitable for kinetic analysis. Since for the sigmoid types of kinetics the rate is minimal at the initial stages of the reaction, it is possible to use slower heating rates and yet avoid significant decomposition before switching to cooling. The use of slower heating rates to reach T* in the cooling experiments allows one to avoid overshooting associated with fast heating rates. CaC2O4·H2O (white crystalline powder, 99% pure) was purchased from Alfa Aesar. For cooling experiments taken on DSC, 5.3−5.7 mg of the sample was placed in open 40 μL aluminum pans, heated to T* =120 °C at the heating rate of 100 °C min−1, and cooled at the cooling rates of 0.05, 0.1, 0.2, 0.3, 0.4, and 0.8 °C min−1. The decomposition kinetics of CaC2O4·H2O is of decelerating type;21,22 i.e., its reaction rate is maximal at the beginning of decomposition. Because of that, one has to use fast heating rate to reach the turning temperature. The use of slower heating rates leads to decomposition during heating. The decomposition during the heating segment in cooling experiments was already observed for the decelerating process of NH4NO3 decomposition.28 The partial mass loss has a greater effect on kinetic analysis of

3. COMPUTATIONS For the DSC measurements on cooling, a baseline was subtracted according to the previously described procedure.10 Kinetic parameters evaluation was performed in accord with the recommendations of the ICTAC Kinetic Committee.23 The effective activation energy was evaluated by an advanced isoconversional method.7 This method is applicable to data obtained on heating and cooling. Another advantage of the method is that it eliminates a systematic error in Eα when the value varies with the extent of conversion (α). For the DSC measurements, α was evaluated as a ratio of the current heat change to the total heat absorbed during the reaction. This evaluation is based on the common assumption that the conversion rate, dα/dt, is directly proportional to the heat flow rate measured calorimetrically. The validity of this assumption has been justified by Borchardt and Daniels.24 It holds for uncomplicated processes whose thermal effect is independent of the rate of the temperature change. The latter condition has been confirmed for all the processes examined in the present study on heating as well as on cooling. For the TGA measurements, the extent of conversion was estimated as the fractional mass loss. The advanced isoconversional method makes use of the data obtained at n different temperature programs. Evaluation of the activation energy, Eα, is made by assuming the value being constant within a small interval of conversion Δα = m−1, where m typically is selected to be 50 or 100. Within each Δα, Eα is evaluated by minimizing the function7 n

Ψ(Eα) =

n

∑∑ i=1 j≠i

J[Eα , Ti(tα)] J[Eα , Tj(tα)]

(1)

⎡ −E ⎤ α exp⎢ ⎥ dt ⎣ RTi(t ) ⎦ α −Δα

(2)

where J[Eα , Tj(tα)] ≡

∫t

tα

The integral was estimated numerically by the trapezoid method. A minimum in eq 1 was found by using the COBYLA nongradient method25 from the NLOPT library.26 The Monte Carlo bootstrap method27 was applied to evaluate the uncertainties (standard deviations) of the Eα values. In this work we focus on exploring the effects revealed through the values of the isoconversional activation energy. The B

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the effective activation energy is constant and equal to E2 + ΔHads + γΔHr. Vyazovkin and Linert17 have used a simpler model to explain a decrease in Eα for reversible thermal decompositions. They considered this process as a reversible reaction followed by an irreversible one:

discovered effects are easy to extend to the preexponential factor, whose values are readily evaluated23 through the values of the activation energy. Such evaluations have been performed in our previous publications10,28 dealing with comparison of the decomposition kinetics on cooling and heating. Since those studies10,28 have revealed complete similarity in the behavior of the isoconversional activation energies and preexponential factors, we have decided not to pursue evaluation of the latter in the present work.

K

kef =

(3)

where As is the solid reactant, Bs is the solid product, Cg is the gaseous product, and k1 and k2 are the Arrhenius rate constants for the forward and reverse reaction, respectively. This type of decomposition is sensitive to the pressure of the gaseous product, which influences not only the temperature of decomposition but also the effective activation energy of the process. There are several models that have been employed to explain this phenomenon. One of the earliest models has been proposed by Pawlutschenko and Prodan.16 This model is based on the assumption of Cg absorption on the solid surface and an equilibrium between the forward and reverse reactions in eq 3. The effective rate constant is expressed by the equation16 ⎛ ΔHads ⎞ γ ⎛ E ⎞ ⎟(P − P γ ) kef = A 2 exp⎜ − 2 ⎟L exp⎜ − ⎝ RT ⎠ ⎝ RT ⎠ 0

d ln kef dT −1

= E2 + ΔHads + γ ΔHr

(4)

⎛ ⎛ E ⎞ dα P⎞ ⎟f (α )⎜1 − = A exp⎜ − ⎟ ⎝ ⎠ dt RT P0 ⎠ ⎝

(5)

(10)

where f(α) is the reaction model and A is the preexponential factor. This model has been in use since at least the 1930s.29 Reading et al.30 have demonstrated that this relationship can be arrived at from transition state theory where the forward and reverse reactions go through the formation of the activated intermediate. Barret31 has arrived at the same pressure dependence for the reversible decomposition through the formation of an adsorbed intermediate. Searcy and Beruto32 have derived a similar equation by assuming that the diffusion of the gaseous product is a limiting step of a reversible decomposition. By taking the isoconversional derivative of the rate in eq 10, one also arrives at a variable value of the effective activation energy:10

P0γ P0γ − P γ

(9)

where E is the activation energy of the irreversible reaction and ΔHr is the enthalpy of the reversible reaction. With the reaction progress, K becomes ≫1; the effective activation energy decreases and approaches E. This model also suggests that the effective activation energy should decrease with departure from equilibrium, i.e., with increase in temperature. Tan et al.18 have also proposed a model for reversible thermal decomposition of solids. They considered the influence of several factors such as porosity and tortuosity, pressure of gaseous product, diffusion, and nuclei growth of solid product. They demonstrated that the partial pressure and structural parameters of the solid along with conversion have significant influence on the rate of the reaction and, therefore, the effective activation energy of thermal decomposition process. The derived model predicts a decreasing dependence of the effective activation energy on temperature. Moreover, with increasing pressure, the Eα vs T trend becomes steeper, and for the early stages of the reaction the effective activation energy tends to infinity. Although a number of models has been proposed to account for the influence of the product pressure on the reversible decomposition, one of the most widely used3 is based on the equation

Equation 4 leads to the following expression for the effective activation energy of the process: Eef = −R

(8)

Eef = E + ΔHr

where A2 is the pre-exponential factor for the reverse reaction, E2 is the activation energy of the reverse reaction, L is a constant, ΔHads is the adsorption enthalpy of the gaseous product on the solid surface, P0 is the equilibrium pressure, P is the partial pressure, and γ is a fitting parameter (0 < γ < 1). The pressure of the product Cg remains constant during the experiment, and therefore, the term Pγ is constant and independent of temperature. For the reversible reaction 3, the equilibrium pressure P0 is equal to the equilibrium constant K: ⎛ ΔG 0 ⎞ r ⎟ P0 = K = exp⎜ − RT ⎝ ⎠

Kk 1+K

Detailed derivations of eq 8 can be found in ref 3. At the beginning of the reaction when K ≪ 1, the effective activation energy is

k1

k2

(7)

The effective rate constant for this process is expressed as

4. THEORETICAL BACKGROUND The reversible thermal decomposition of many solids can be described by the equation A s ⇔ Bs + Cg

k

A⇔B→C

(6)

The third term on the right-hand side of eq 6 determines the dependence of the effective activation energy on pressure. When the reaction starts, P0 slightly rises above P, and therefore, the reaction system demonstrates the smallest deviation from equilibrium. Since P0 is only slightly greater than P, the denominator P0 − P is a small quantity and the P0/ (P0 − P) is large, which makes the effective activation energy large. While the reaction progresses, P0 − P increases and the reaction departs further from equilibrium. The term P0/(P0 − P) is getting smaller, and Eef decreases. Overall, the Pawlutschenko−Prodan model predicts that the effective activation energy should decrease with reaction progress. One can also notice that in the high-vacuum conditions where P ∼ 0

⎡ ∂ ln(dα /dt ) ⎤ P Eα = −R ⎢ ⎥ = E + ΔHr ⎣ ∂T −1 ⎦α P0 − P C

(11)

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and autocatalytic (sigmoid) kinetics by introducing the exponent m as follows:

Again, eq 11 predicts that the effective activation energy would decrease with temperature. At the initial stages of the reaction, P0 is nearly equal to P so that Eα takes on a large value. On heating, when the temperature increases, P0 raises progressively according eq 5. As the reaction progresses, P0 − P becomes larger and Eα decreases continuously. Despite the fact that all aforementioned models predict a decrease in the effective activation energy with increasing temperature, they cannot explain the variation in the activation energy experimentally observed in cooling experiments. The latter have demonstrated10,28 that the effective activation energy decreases with decreasing temperature. That is, the heating and cooling runs demonstrate qualitatively different trends in the activation energy with respect to temperature. On heating, Eα is a decreasing function of T, whereas on cooling, Eα is an increasing function of T. The failure of the aforementioned model to predict the correct trend for cooling experiment is easy to demonstrate by using eq 11. If the reaction occurs on cooling, P0 decreases with decreasing temperature, and the reaction is moving toward equilibrium. The difference P0 − P becomes smaller, and we should expect increasing dependence of the activation energy. Altogether, eq 11 predicts that the effective activation energy dependencies obtained on heating and cooling should be parts of a single trend: Eα decreasing with increasing T. This obvious contradiction between the model prediction and the actual experimental observations10,28 has prompted us to look for a modification of the model (eq 10) that would be able to predict correctly the Eα trends observed on both heating and cooling. It should be noticed that in eq 10 P0 is only a function of T in accord with eq 5. Equation 5 holds for equilibrium, i.e., the state when the forward and reverse reactions occur at the same rate. That is the amount of the gaseous product does not increase, and, therefore, P0 remains constant. However, reaching equilibrium takes time, during which the gaseous product is produced and its partial pressure rises until reaching its final equilibrium value. In other words, P0 increases with reaction progress. This fact can be accounted for by modifying eq 5 as follows: ⎛ ΔGr ⎞ ⎟ P0′ = α nP0 = α n exp⎜ − ⎝ RT ⎠

m ⎛ ⎛ E ⎞ dα P⎞ ⎟f (α )⎜1 − = A exp⎜ − ⎟ ⎝ RT ⎠ dt P0′ ⎠ ⎝

(13)

As seen in Figure 1, when m = 1 the pressure term, (1 − P/ P0′)m exhibits a decelerating behavior. However, when m > 1,

Figure 1. Equilibrium pressure dependencies of the pressure term in eq 13 for different values of m.

the term reveals an autocatalytic behavior with the initial accelerating phase (“induction period”) being longer for larger m. Applying the isoconversional derivative to eq 13 leads to the following expression for the effective activation energy: Eα = E + mΔHr

P P0′ − P

(14)

As discussed further (section 5.1), eq 14 predicts correctly the trend for the temperature dependence of the effective activation energy obtained by applying an isoconversional method to experimental data obtained on cooling as well as on heating. This provides yet another example of the utility of applying the isoconversional principle as a single-step approximation to multistep kinetics.33,34

(12)

where α is the extent of conversion and n is the parameter that describes the strength of the pressure dependence of the conversion. Another important fact about eq 10 is that the particular form of the pressure term (1 − P/P0) holds true only when both forward and reverse reactions are of first order. The term can be adjusted readily to nth-order kinetics. However, this adjustment would still keep this term suitable for the deceleration type of kinetics only. Another important type of kinetics that the pressure term should be suitable for is that of the autocatalytic (sigmoid) type. A reasoning behind this is quite simple. When the gaseous product Cg is formed (eq 3), it cannot immediately go into the gaseous phase. It remains adsorbed on the solid surface held by various forces such as electrostatic, dipole−dipole, van der Waals interactions, and so on. Since the adsorption capacity is limited, at some point the gaseous product would have to leave the solid surface and go into the gaseous phase. Thus, one should generally expect an induction period in the behavior of the pressure term. The pressure term can be made suitable to cover both decelerating

5. RESULTS AND DISCUSSION 5.1. Simulation. To get insights into the phenomenon of reversible decomposition and to verify the proposed model, kinetic simulations of the reaction occurring on heating and cooling have been performed. The simulation has been designed to imitate the condition of actual experiments. As parameters of the simulation, a hypothetical reaction with ΔHr = 70 kJ mol−1 and ΔSr = 110 J mol−1 K−1 was chosen. The simulation of the heating experiments was performed from 25 °C to the temperatures of complete decomposition at 1, 2, 3, and 4 °C min−1. The simulation of the cooling experiments was conducted from 110 °C at 0.2, 0.4, 0.6, and 0.8 °C min−1 cooling rates. The Arrhenius parameters for eq 13 were taken as E = 64 kJ mol−1 and A = 1.5 × 106 s−1. The contracting sphere reaction model (R3),23 f(α) = 3(1 − α)2/3, was used for the simulations on both heating and cooling. This model was selected as a typical example of decelerating kinetics, i.e., the kinetics characteristic of the thermal decomposition of CaC2O4· H2O. Simulations for autocatalytic (sigmoid) kinetics, which is D

DOI: 10.1021/acs.jpcc.7b05066 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C similar to the thermal decomposition kinetics of CaSO4·2H2O, have been performed in our previous study.10 Parameters n and m from eqs 12 and 13 were chosen as 0.25 and 4, respectively. To examine the influence of the pressure term on the kinetics of decomposition, P was chosen to be close to the initial value of P0′ in each run. Integration of eq 13 was performed numerically. The simulated dα/dt vs T curves for heating and cooling are presented in Figure 2. On cooling, the reaction starts at the

Figure 4. Temperature dependences of the effective activation energy estimated from the simulated data on heating (circles) and cooling from 110 °C (diamonds).

Large values of the activation energy at α < 0.2 show that on cooling as well as on heating the reaction starts at equilibrium. This fact indicates the existence of higher and lower T equilibria (Figure 4) which is the ultimate reason for the qualitative differences in Eα vs T trends obtained for heating and cooling experiments. As shown in Figure 4, for heating and cooling simulations the effective activation energies continuously decrease from respectively lower and higher T equilibria, and these dependences are not parts of one trend. For the cooling experiments, the influence of pressure appears to be larger due to several factors. There are two parameters that determine position of the reaction with respect to equilibrium: α and T (see eq 12). Continuous decrease in T decreases P0′ and moves the reaction toward the low temperature equilibrium. On the other hand, with the reaction progress, α rises which moves the reaction away from the equilibrium. On cooling, these two parameters work in opposite directions which means that the rate of departure from the high temperature equilibrium is slower so that the reaction remains closer to equilibrium conditions for a longer period of time. It can be easily seen from eqs 12 and 13. While decrease in T decreases P0′, increase in α makes P0′ larger, which keeps difference (P0′ − P) smaller than during heating and Eef larger. At the first stage of the reaction, the influence of α parameter is greater even though decreased by the T dependence. At some point, the temperature dependence offsets the conversion dependence and P0′ starts to decrease driving the reaction closer to a lower temperature equilibrium which raises E of the process. It can be seen in Figure 3 that on cooling the activation energy is slowly rising at the latest stages of the reaction (α > 0.8). On heating, both parameters α and T increase, making P0′ and therefore (P0′ − P) in eq 14 continuously larger that moves the reaction away from equilibrium. Although both heating and cooling runs demonstrate a decrease in Eα with increasing α (Figure 3), it can be noticed that for the whole range of α the effective activation energy on heating is smaller than on cooling. This further emphasizes the fact that on cooling the reaction proceeds closer to equilibrium than on heating. 5.2. Calcium Oxalate Monohydrate. The decomposition kinetics of CaC2O4·H2O is of a decelerating type,21,22 meaning that the maximum rate of the reaction is at the beginning of the process. This feature makes it challenging to study the dehydration on cooling. It implies that the heating rate to reach a turning temperature T* should be fast which can potentially lead to a significant overshooting. When a reaction

Figure 2. Simulated data at different cooling and heating rates. The numbers by the line type represent the cooling (0.2, 0.4, 0.6, and 0.8) and heating (1) rates in °C min−1.

highest temperature which continuously decreases during experiments. For slower cooling rates, the process remains longer at higher temperatures and, therefore, occurs faster. This is demonstrated by the very narrow peak with the highest intensity for the experiment at the cooling rate 0.2 °C min−1 (Figure 2). For the faster cooling rates, the process shifts faster from the higher temperatures and decelerates faster. Consequently, the peak becomes shorter and broader. The results of isoconversional analysis of the simulated data are presented in Figures 3 and 4. It is important to notice that the activation energies for heating and cooling exhibit features similar to those observed previously for the experimental data.10,28 First, the activation energy decreases as the reaction progresses from α = 0 to α = 1 which, as discussed earlier, is associated with the effect of pressure on the reaction kinetics.

Figure 3. Isoconversional values of the effective activation energy estimated from the simulated data on heating (circles) and cooling from 110 °C (diamonds). E

DOI: 10.1021/acs.jpcc.7b05066 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C on cooling is studied, the reaction rate continuously decreases due to decreasing T. In the case of reactions with decelerating model types, the process rate on cooling is suppressed even faster than for autocatalytic processes. This feature can lead to incomplete decomposition on cooling and forces one to increase the turning temperature. Even though the use of faster heating rates distorts a DSC signal, a partial decomposition on heating has a stronger detrimental effect on the kinetic data quality than overshooting. As a result, the use of faster heating rates 100−300 °C min−1 to reach a turning temperature T* is a necessary condition to successfully perform kinetic analysis of decelerating processes on cooling. Dehydration of CaC2O4·H2O is expressed by the following reaction: CaC2O4 ·H 2O ⇔ CaC2O4 + H 2O

(15)

Figure 6. Isoconversional values of the effective activation energy estimated for the thermal decomposition of CaC2O4·H2O upon heating (circles) and cooling from 120 °C (diamonds).

The DSC curves for this process obtained on heating and cooling are presented in Figure 5. On heating, decomposition

process and keeping it closer to equilibrium. The obtained trends are in agreement with the proposed model as well as with previously reported data for Li2SO4·H2O.10,28 It is interesting to note that at the later stages of the reaction on cooling (α > 0.7) the activation energy starts to increase. Similar increase has been observed for the simulated data (Figure 3) and the later stages of Li2SO4·H2O decomposition on cooling.28 As discussed in section 5.1, at the later stages of the process on cooling, the reaction is already far away from its high T equilibrium. The value P0′ − P is getting smaller due to the decrease in T so E increases. Consequently, at the larger α, the reaction starts approaching the lower T equilibrium. The effective activation energy data plotted against temperature for the heating and cooling experiments are presented in Figure 7. Figure 5. DSC curves for the thermal decomposition of CaC2O4·H2O upon heating and cooling with the turning temperature of 120 °C. The numbers by the line type represent the cooling (0.05, 0.1, 0.2, 0.3, 0.4, and 0.8) and heating (1.25) rates in °C min−1.

starts at ∼85 °C. Note that the curves in Figure 5 show the features that are characteristic of a decelerating process on cooling and consistent with our simulations for this type of a process (Figure 2). For the runs taken at slower cooling rates the process happens within very narrow temperature range