Thermal Decomposition Kinetics of Malonic Acid in the Condensed

Jul 10, 2017 - (3) He linked the effect to a decrease in the activation energy by arguing that the size of the free energy barrier for liquid reactant...
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Thermal Decomposition Kinetics of Malonic Acid in the Condensed Phase Victoria L. Stanford and Sergey Vyazovkin* Department of Chemistry, University of Alabama at Birmingham, 901 South 14th Street, Birmingham, Alabama 35294, United States ABSTRACT: This study reexamines an early literature report of an unusual phenomenon of a dramatic difference in the activation energy for the thermal decomposition of malonic (propanedioic) acid in the liquid and solid states. The study has been carried out via thermogravimetric analysis (TGA) to probe possible differences between the kinetics in three condensed phases: solid, liquid, and supercooled liquid. An advanced isoconversional method has been applied to determine the activation energy, preexponential factor, and reaction model. The activation energy as well as preexponential factor for the decomposition processes in all three condensed phases have been determined to be practically the same: E = 110 ± 10 kJ mol−1 and log(A/min−1) = 13 ± 1. In all phases, the reaction kinetics has been found to follow the reaction order model. The reaction order values have been similar for the liquid and supercooled liquid phases but markedly smaller for the solid-phase decomposition. The results obtained have not confirmed the phenomenon reported in the early literature. A practically important conclusion of the present study is that the liquid-phase kinetic data can be used to obtain a reasonable estimate for the thermal stability of this compound in the solid phase.

1. INTRODUCTION Solids can exist in crystalline, glassy, or intermediate (semicrystalline) state. To initiate decomposition, solids need to be heated. An increase in temperature may cause a crystalline solid to melt or a glassy solid to undergo the glass transition so that decomposition occurs in the liquid phase. For some compounds the phase transition temperature lies within the temperature region of the thermal decomposition that makes it possible to study them in the solid and liquid phases. This type of study is of great practical relevance and fundamental significance. From the practical standpoint, the most important question is whether accelerated (i.e., higher temperature) tests can be used to predict lower temperature stability when the respective temperature regions are separated by a phase transition. This question cannot be answered without fundamental studies probing the effect of the aggregate state on the mechanism and kinetics of thermal decomposition. Although some reactions are known1,2 to proceed faster in the solid than liquid phase, it is commonly accepted, and found experimentally, that the thermal decomposition accelerates in liquid phase. Some insights into the nature of such acceleration have been proposed by Bawn.3 He linked the effect to a decrease in the activation energy by arguing that the size of the free energy barrier for liquid reactant is smaller by the value of the melting enthalpy of the respective solid. Using the Arrhenius law for the temperature dependence of the reaction rate one can write ln

E − El kl A = ln l + s ks As RT

Then the difference between Es and El equals the enthalpy of melting, ΔHm. Assuming that Al ≈ As and taking into account that for many organic solids ΔHm is about 20 kJ mol−1, the ratio of kl/ks at 100 °C should be expected to be ∼600, which suggests significant acceleration on transition from the solid to liquid phase. In reality, the aforementioned ratio rarely exceeds a few decades. This is not unexpected considering the oversimplified nature of the above argument. For one, the assumption of Al ≈ As does not seem justifiable. More importantly, the proposed decrease in the energy barrier would require only the energy of the reactant to increase, whereas the energy of the transition state would have to remain the same for solid and liquid phases. The latter condition is not very likely to be fulfilled, so that the Es and El values are frequently estimated to be insignificantly different. For example, recent extensive work by Manelis et al.4 on the thermal decomposition of solid organics suggests that the Es and El values are usually equal within the typical experimental error. A similar conclusion has been arrived at by Vyazovkin et al.5 in their study of the thermal decomposition of ammonium nitrate in the solid and liquid phases. Marginally smaller values of El have been reported by Rieckmann et al.6 for the thermal decomposition of hexanitrostilbene. Among a number of studies comparing the solid- and liquid-phase kinetics, one stands out as an example of a truly remarkable difference in the values of Es and El. This example originates from Hinshelwood’s paper7 on the thermal decomposition of malonic acid studied in the solid and supercooled liquid phases. This example has been highlighted by Bawn,3 who used the rate

(1) Received: Revised: Accepted: Published:

where k, A, and E are respectively the rate constant, preexponential factor, and activation energy, and the subscripts l and s represent the liquid and solid phases correspondingly. © 2017 American Chemical Society

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May 19, 2017 June 25, 2017 June 26, 2017 July 10, 2017 DOI: 10.1021/acs.iecr.7b02076 Ind. Eng. Chem. Res. 2017, 56, 7964−7970

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Industrial & Engineering Chemistry Research

decomposition at the heating rates 2, 4, and 8 °C min−1. The lower temperature limit (120 °C) for stability of the supercooled liquid was determined from differential scanning calorimetry (DSC) measurements that demonstrated that on cooling at 5 °C min−1 malonic acid starts to crystallize below 115 °C. DSC measurements were taken with a heat flux DSC (Mettler-Toledo, 823e). To compensate for the mass loss associated with decomposition of malonic acid during preparation of supercooled liquid, we used a larger sample (∼10 mg) to ensure that enough sample was left to measure the decomposition kinetics. Both DSC and TGA runs were conducted in open, 40 μL Al pans. All runs were performed in duplicate and demonstrated more than satisfactory repeatability. Indium and zinc standards were used to perform temperature, heat flow, and tau-lag calibrations for both TGA and DSC. Phenyl salicylate was also used for temperature calibrations for TGA. The experiments were performed in the atmosphere of nitrogen flow (80 mL min−1).

constants reported by Hinshelwood7 to estimate the respective activation energies as 253 (Es) and 136 (El) kJ mol−1. Our own analysis of Hinshelwood’s data yields somewhat different results: 252 ± 20 (Es) and 122 ± 5 (El) kJ mol−1. Nevertheless the difference is truly remarkable in either case. This unusual example has inspired us to undertake a new kinetic study of the thermal decomposition of malonic acid. Hinshelwood’s study has been conducted by manometric method, i.e., in a closed volume by monitoring the pressure of released gas. The measured kinetics have been assumed to be of first order. In our work, we measure the kinetics by thermogravimetric analysis (TGA), i.e., by monitoring the mass loss in an open system. There have been a few recent TGA studies8−10 of the kinetics of malonic acid decomposition. However, they all have dealt with the liquid-phase decomposition kinetics. For the most part, those studies have derived kinetic parameters from nonisothermal runs at a single heating rate. Presently, such an approach is strongly disfavored as being incapable of producing reliable kinetic parameters. As one of viable alternatives, the ICTAC kinetics committee recommends11 isoconversional methods that afford reliable kinetic evaluations by simultaneously using data obtained at multiple temperature programs, e.g., at multiple heating rates or temperatures. We collect TGA data on the thermal decomposition of malonic acid in the solid, liquid, and supercooled liquid phases at multiple heating programs. The data are then subjected to comprehensive isoconversional kinetic analysis12 in order to determine the activation energies, preexponential factors, and reaction models. The purpose of this work is to test whether the phenomenon of the dramatic differences in Es and El would be reproduced under the conditions of TGA runs and, if so, to obtain insights into its origin.

3. KINETIC COMPUTATIONS The kinetic triplets (activation energy, preexponential factor, and reaction model) for the decomposition of malonic acid in the solid, liquid, and supercooled liquid phases were determined according to the recommendations of the ICTAC Kinetics Committee.11 The values of the activation energy, Eα, as a function of conversion, α, were determined by an advanced isoconversional method.15 The values of α were evaluated from TGA as the ratio of the difference between the initial and current mass to the difference between the initial and final mass. By its meaning the resulting value of α represents the relative conversion, i.e., it varies from 0 to 1 regardless of the absolute value of the mass loss observed during decomposition. At any given α, the effective activation energy is evaluated by determining the Eα that minimizes the function:15

2. EXPERIMENTAL SECTION Malonic (propanedioic) acid (Sigma-Aldrich) purchased at 99% purity was used without any further purification. The samples were ground up in an agate mortar. Approximately 1 mg of each sample was placed in an open, 40 μL Al pan, for the solidand liquid-phase analysis. A Mettler-Toledo TGA/DSC3+ module was used to measure the mass loss kinetics under both isothermal and nonisothermal conditions. The liquid-phase runs were carried out under nonisothermal conditions, where the sample was heated from 25 to 200 °C at the heating rates 12, 15, 20, and 24 °C min−1. The use of slower heating rates resulted in data for which the initial stages of decomposition occurred in the solid state. On the other hand, under even very slow heating rates such as 0.5 °C min−1 only the first 40% of mass loss occurred before melting. Therefore, the solid-state decomposition was studied under isothermal conditions at the following temperatures: 132, 128, 124, and 120 °C, which fall below the melting point, Tm. The isothermal temperatures were reached at the heating rate of 30 °C min−1. The literature values for Tm are found in the range 134.0−135.1 °C.13 DSC measurements on our sample yielded the temperature and enthalpy of melting to be 135.2 ± 0.2 °C and 21 ± 1 kJ mol−1, respectively. The enthalpy value is also in agreement with the literature value,14 23 ± 1 kJ mol−1. Supercooled liquid samples were prepared directly in the TGA instrument by heating malonic acid to 145 °C at a heating rate of 25 °C min−1 to achieve melt. Heating was followed by cooling to 120 °C at a cooling rate of 5 °C min−1. The resulting supercooled liquid samples were then heated to measure

n

Ψ(Eα) =

n

J[Eα , Ti(tα)] J[Eα , Tj(tα)]

(2)

⎡ −E ⎤ α exp⎢ ⎥ dt RT ( ⎣ i t) ⎦ α −Δα

(3)

∑∑ i=1 j≠i

where J[Eα , Ti(tα)] ≡

∫t

tα

In eq 3, Eα is determined from numerical integration of the Ti(t) data over small time segments that correspond to change in α from α − Δα to α, where Δα = m−1 and m is the number of intervals chosen for computation. The integral, J, in eq 3, is estimated by using the trapezoid rule. Minimization is repeated for each value of α to establish the dependence of Eα on the extent of conversion. Unlike simpler or, more precisely, rigid12 integral methods this method can process data under arbitrary temperature variation, and eliminate systematic error found when Eα varies significantly with α. The obtained activation energy is sometimes termed empirical,16 or effective, apparent, global, overall, total, etc.11,12,17 Considering that the condensed-phase reactions tend to involve multiple steps, the activation energy, derived from an overall property such as mass loss, usually has a meaning of a composite parameter, whose value is a function of the activation energies of the individual reaction steps. As a result, the effective activation energy may demonstrate a dependence on conversion and temperature, which is a sign of multistep kinetics. On the other hand, the effective activation 7965

DOI: 10.1021/acs.iecr.7b02076 Ind. Eng. Chem. Res. 2017, 56, 7964−7970

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plots, α vs t/tα, where tα is the time to reach a given conversion (e.g., α = 0.9) at a given temperature.26 Different reaction models yield respectively different reduced time plots. As a result, a comparison of an experimental plot against the model plots allows one to identify the reaction model. Once the model is established, the preexponential factor is readily estimated as

energy is frequently found to be practically independent of these parameters, which is an indication of the kinetics limited or dominated by a single step. A detailed discussion of the pertinent issues is provided elsewhere12,17 and, thus, lies beyond the scope of this paper. The preexponential factors for nonisothermal data were determined via a compensation effect,11 ln Ai = aEi + b

Aα =

(4)

where a and b are the parameters of the compensation effect, and the Arrhenius parameters Ai and Ei are estimated via substituting different reaction models (Table 1) into an

1 2 3 4 5 6 7 8 9 10 11 12 13

power law power law power law power law one-dimensional diffusion Mampel (first order) Avrami−Erofeev Avrami−Erofeev Avrami−Erofeev three-dimensional diffusion contracting sphere contracting cylinder two-dimensional diffusion

code

g(α)

P4 P3 P2 P2/3 D1 F1 A4 A3 A2 D3 R3 R2 D2

α α1/3 α1/2 α3/2 α2 −ln(1 − α) [−ln(1 − α)]1/4 [−ln(1 − α)]1/3 [−ln(1 − α)]1/2 [1 − (1 − α)1/3]2 1 − (1 − α)1/3 1 − (1 − α)1/2 (1 − α)ln(1 − α) + α

( −RTE )

tα exp

α

(7)

4. RESULTS AND DISCUSSION Although heating of some organic compounds results in vaporization without decomposition, malonic acid is known7−9,27−29 to decompose in accord with the following mechanism:

Table 1. Some of the Kinetic Models Used in the Solid-State Kinetics reaction model

g (α )

CH 2(COOH)2 → CH3COOH + CO2

1/4

(8)

This mechanism does not include possible side reactions. For example gas chromatographic data30 indicate the presence of CO that is likely a product of the thermal decomposition of acetic acid formed in the main step of decomposition. Both the liquid and supercooled liquid state decomposition of malonic acid were carried out using nonisothermal temperature programs. Figure 1 depicts the decomposition

equation of some single heating rate method. Mathematical origins of the compensation effect are well discussed in the literature.18−20 First, by correlating the resulting ln Ai and Ei, the parameters a and b are found, and then the isoconversional estimate of the preexponential factor can be determined by substituting the isoconversional activation energy, Eα, into eq 5 as follows: ln Aα = aEα + b

(5) 21

The approach to determine Aα was originally proposed for a single-step process, but was later demonstrated22 to also work for multistep processes. The technique has been improved by Sbirrazzuoli,23 who demonstrated that the parameters of the compensation effect of only four pairs of Ai and Ei are needed for highly accurate values of the preexponential factor, when estimated by the single-heating-rate method of Tang et al.24 using the reaction models (Table 1) of Mampel (F1) and Avrami−Erofeev (A2, A3, A4). The obtained values of Aα were further used to determine the reaction models for the nonisothermal decomposition of malonic acid. The reaction models were determined in the numerical form of eq 6:11,12 g (α ) =

Aα β

∫0

Tα

⎛ −E ⎞ exp⎜ α ⎟ dT ⎝ RT ⎠

Figure 1. Nonisothermal kinetic curves of the liquid phase of malonic acid decomposition.

kinetics in the liquid phase at four different heating rates. At any of these heating rates, decomposition commences around the melting point and proceeds to 100% mass loss. The decomposition process is observed by the increase in the extent of conversion from α = 0 to α = 1 on the TGA curve starting at ∼140 °C, and reaching total mass loss at ∼200 °C. Figure 2 shows the decomposition kinetics of the supercooled liquid sample of malonic acid. From the figure it can be seen that multiple heating rates are used and decomposition becomes detectable at ∼120 °C, which is above the onset of crystallization of malonic acid observed on cooling. It is noticed that the decomposition of the sample is detectable at a lower temperature than that in the liquid state (Figure 1). This is expected because the supercooled liquid samples have been studied at markedly slower heating rates, and the kinetic curves are known to shift to lower temperatures with lowering the heating rates (cf., Figure 1 or 2). All of the samples decompose completely, and the process proceeds from the supercooled liquid state to the regular liquid state without crystallization. It should be recalled that, by the time the heating segment starts,

(6)

For each value of α the values of Eα, Aα, and Tα are inserted and numerical values of g(α) are yielded, which can be matched to the theoretical g(α) models (Table 1). The temperature integral in eq 6 is evaluated by using the third order Senum− Yang approximation.25 For isothermal decomposition of malonic acid, the reaction model was determined by using the so-called reduced time 7966

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Figure 4. Dependencies of the preexponential factor on the extent of conversion for the thermal decomposition of malonic acid in the solid (diamonds), liquid (circles), and supercooled liquid (pentagons) phases.

Figure 2. Nonisothermal kinetic curves of the supercooled liquid phase of malonic acid decomposition.

the sample is heated to 145 °C at 25 °C min−1 and cooled down to 120 °C at 5 °C min−1. During this period the sample has unavoidably undergone significant decomposition that corresponded to 30% of the absolute mass loss. Since α by its meaning represents the relative mass loss, α = 0 in Figure 2 has been conventionally assigned to the sample mass detected in the beginning of the heating segment at 120 °C. That is, in the case of the supercooled samples α = 0 represents partially decomposed sample. Note that this does not affect the value of Eα as long as α represents the same absolute extent of decomposition at all heating rates regardless of what this absolute value is. The dependencies of the activation energy, Eα, on the extent of conversion for the liquid and supercooled liquid phases of malonic acid decomposition are illustrated in Figure 3. It is seen

The resulting Arrhenius parameters are quite similar to the ones that we derived by using the rate constants reported by Hinshelwood7 for decomposition of supercooled malonic acid: E = 122 ± 5 kJ mol−1 and log(A/min−1) = 13 ± 2. Hinshelwood’s rate constants were determined by assuming first order kinetics of decomposition. The g(α) values that we determined for decomposition in the liquid and supercooled liquid states are shown in Figure 5. The experimental values are

Figure 5. Dependencies of g(α) on the extent of conversion for the models shown in Table 1 (solid lines) and experimental g(α) data for the thermal decomposition of the liquid (circles) and the supercooled liquid (pentagons) phases. Dashed line represents reaction order model with n = 0.5.

plotted against the theoretical g(α) models (Table 1). Although none of these models fit the data well, it is apparent that both the liquid and supercooled liquid phases should follow very similar models. As seen from Figure 5, the data points clustered around the reaction order model with n = 0.5. More accurate values of n have been found by fitting the experimental g(α) values to a reaction order model with an adjustable value of n:

Figure 3. Dependencies of the activation energy on the extent of conversion for the thermal decomposition of malonic acid in the solid (diamonds), liquid (circles), and supercooled liquid (pentagons) phases.

that decomposition in both phases is characterized by an activation energy that is practically independent of the extent of conversion. This constancy of the activation energy on the extent of conversion indicates that the overall kinetics of the thermal decomposition of malonic acid is likely to be governed by a single reaction step. The average activation energy for these two processes is around 110 ± 10 kJ mol−1. The preexponential factors for these two processes are also nearly the same within the experimental error, and practically independent of conversion (Figure 4). The respective average value for these two processes is around log(A/min−1) = 13 ± 1.

g (α ) =

1 − (1 − α)1 − n 1−n

(9)

The reaction order was determined to be n = 0.63 ± 0.02 for the liquid-phase decomposition, and n = 0.48 ± 0.01 for the supercooled liquid decomposition. The values are quite similar and fit with the average value, n = 0.56 ± 0.08. Solid-state decomposition of malonic acid has been studied under isothermal conditions. The respective TGA curves are 7967

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Industrial & Engineering Chemistry Research presented in Figure 6. As is well-known,11 one of the major challenges of isothermal runs is to minimize decomposition

the reaction-order values for the liquid and supercooled liquid phase decomposition. This is not surprising considering that the reaction model is linked to the mechanism of the process. In either phase, the thermal decomposition is initiated via an energy fluctuation, i.e., the reaction occurs when the molecules gain enough energy to jump over the energy barrier. In the liquid phase, the molecules move and collide stochastically throughout the reaction volume and, thus, can acquire sufficient excess energy in any part of the whole volume. It causes the liquid-phase decomposition to occur homogeneously throughout the volume. In the solid crystalline phase, the molecules are locked in the crystalline lattice that severely restricts the molecular mobility. Because of uncompensated chemical bonds and defects, the surface is the most mobile part of a solid.31 For this reason, the surface molecules are the ones that acquire sufficient excess energy to initiate the thermal decomposition. This ultimately causes the thermal decomposition to localize on the solid surface.26 As a result, the solid-phase mechanism is largely determined by such factors as crystalline lattice defects and strains that are of no relevance in the liquid phase. The isoconversional activation energy for the solid-state decomposition is practically constant with the average value around 105 kJ mol−1 (Figure 3). The value is consistent with the activation energies determined for the liquid and supercooled liquid state decomposition. That is, all three values fit well within the 110 ± 10 kJ mol−1 range. Similarly we find that the preexponential factor for the solid-state decomposition (Figure 4) is consistent with the value estimated for the liquid and supercooled liquid state decomposition. Clearly, our results demonstrate that the Arrhenius parameters for the thermal decomposition of malonic acid do not depend to any significant extent on the aggregate state of the compound. A similar conclusion has been arrived at in the study of the thermal decomposition of ammonium nitrate in the liquid and solid states.5 Most importantly, our study does not support the Bawn report3 on the dramatic difference in the activation energy of thermal decomposition of malonic acid in liquid and solid states. As stressed earlier, this type of difference is very unusual because most of the literature,4−6 including other examples discussed by Bawn,3 reports that the difference is usually comparable to experimental error. Although the large difference in the activation energies that one obtains when using Hinshelwood’s data7 does not cause any doubts, its origin remains unexplained. It may have been specific to the closed volume conditions used in that study. Under such conditions the measured gas pressure is affected not only by the amount of gaseous products formed in the primary decomposition of the condensed-phase malonic acid. It is also affected by possible decomposition of malonic acid in the gas phase as well as by secondary reactions of the gaseous decomposition products. It is not unreasonable to assume that some of the secondary reactions may have a large activation energy and become the rate-determining step. However, under our open system conditions the secondary reactions of gaseous products are practically eliminated by continuously blowing an inert gas through the reaction zone. Needless to say, elimination of the secondary reactions simplifies the overall kinetics of decomposition. Remarkably, the kinetics of decomposition of malonic acid appears to be determined by a single step in any aggregate state. This conclusion is supported by the fact that the activation energy is practically independent of conversion in all experiments performed in the present study.

Figure 6. Isothermal kinetic curves for the solid-state decomposition of malonic acid.

during the initial period when sample heats up toward the set temperature. In our runs it has been accomplished by using fast heating at the rate 30 °C min−1. Under such conditions the absolute mass loss during the heat-up has never exceeded 0.5%. It is evident that decomposition began as soon as the temperature stabilized at the isothermal temperature program. It can also be seen that, at the temperatures closer to the melting point of malonic acid, decomposition was accomplished in shorter time, but all isothermal experiments ended in total (100%) decomposition. In order to determine the appropriate reaction model for the solid state, a plot of α as a function of a reduced time variable t/ t0.9 has been used. Figure 7 shows the theoretical reduced time

Figure 7. Reduced time plots for the reaction models (solid curves, for numbering see Table 1) and isothermal experimental data for the solid-state malonic acid decomposition (diamonds). Diamonds correspond to data averaged over four temperatures: 132, 128, 124, and 120 °C. The dashed line represents the reaction-order model with n = 0.13.

plots (solid lines) for the models from Table 1 and the experimental plot (points), which is the average of the thermal decomposition runs performed at four temperatures. It is seen that none of the models match the data well. However, it has been found that the data can be fitted quite well by the reaction-order model with adjustable n (eq 9). The estimated value is n = 0.131 ± 0.001. The value is markedly smaller than 7968

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(5) Vyazovkin, S.; Clawson, J. S.; Wight, C. A. Thermal Dissociation Kinetics of Solid and Liquid Ammonium Nitrate. Chem. Mater. 2001, 13, 960. (6) Rieckmann, Th; Volker, S.; Lichtblau, L.; Schirra, R. Investigation on the Thermal Stability of Hexanitrostilbene by Thermal Analysis and Multivariate Regression. Chem. Eng. Sci. 2001, 56, 1327. (7) Hinshelwood, C. N. The Rate of Decomposition of Malonic Acid. J. Chem. Soc., Trans. 1920, 117, 156. (8) Gyore, J.; Ecet, M. Thermal Transformation of Solid Organic Compounds. J. Therm. Anal. 1970, 2, 397. (9) El-Awad, A. M.; Mahfouz, R. M. Kinetic Analysis of Isothermal and Nonisothermal and Catalyzed Thermal Decomposition of Malonic Acid. J. Therm. Anal. 1989, 35, 1413. (10) Muraishi, K.; Suzuki, Y.; Kikuchi, A. Kinetics of the Thermal Decomposition of Dicarboxylic Acids. Thermochim. Acta 1994, 239, 51. (11) Vyazovkin, S.; Burnham, A. K.; Criado, J. M.; Pérez-Maqueda, L. A.; Popescu, C.; Sbirrazzuoli, N. ICTAC Kinetics Committee Recommendations for Performing Kinetic Computations on Thermal Analysis Data. Thermochim. Acta 2011, 520, 1. (12) Vyazovkin, S. Isoconversional kinetics of thermally stimulated processes; Springer: Heidelberg, 2015. (13) Wilhoit, R. C.; Shiao, D. Thermochemistry of Biologically lmportant Compounds. Heats of Combustion of Solid Organic Acids. J. Chem. Eng. Data 1964, 9, 595. (14) Hansen, A. R.; Beyer, K. D. Experimentally Determined Thermochemical Properties of the Malonic Acid/Water System: Implications for Atmospheric Aerosols. J. Phys. Chem. A 2004, 108, 3457. (15) Vyazovkin, S. Modification of the Integral Isoconversional Method to Account for Variation in the Activation Energy. J. Comput. Chem. 2001, 22, 178. (16) IUPAC Gold Book (https://goldbook.iupac.org/html/A/ A00102.html, accessed June 23, 2017). (17) Vyazovkin, S. A Time to Search: Finding the Meaning of Variable Activation energy. Phys. Chem. Chem. Phys. 2016, 18, 18643. (18) Koga, N. A Review of the Mutual Dependence of Arrhenius Parameters Evaluated by the Thermoanalytical Study of Solid-State Reactions: the Kinetic Compensation Effect. Thermochim. Acta 1994, 244, l. (19) Vyazovkin, S.; Wight, C. A. Model-Free and Model-Fitting Approaches to Kinetic Analysis of Isothermal and Nonisothermal Data. Thermochim. Acta 1999, 340/341, 53. (20) Barrie, P. J. The Mathematical Origins of the Kinetic Compensation Effect: 1. The Effect of Random Experimental Errors. Phys. Chem. Chem. Phys. 2012, 14, 318. (21) Vyazovkin, S. V.; Lesnikovich, A. I. Estimation of the Preexponential Factor in the Isoconversional Calculation of the Effective Kinetic Parameters. Thermochim. Acta 1988, 128, 297. (22) Vyazovkin, S.; Linert, W. False Isokinetic Relationships Found in the Nonisothermal Decomposition of Solids. Chem. Phys. 1995, 193, 109. (23) Sbirrazzuoli, N. Determination of Pre-exponential Factors and of the Mathematical Functions f(α) or G(α) that Describe the Reaction Mechanism in a Model-free Way. Thermochim. Acta 2013, 564, 59. (24) Tang, W.; Liu, H.; Zhang, H.; Wang, C. New Approximate Formula for Arrhenius Temperature Integral. Thermochim. Acta 2003, 408, 39. (25) Senum, G. I.; Yang, T. I. Rational Approximations of the Integral of the Arrhenius Function. J. Therm. Anal. 1977, 11, 445. (26) Brown, M. E.; Dollimore, D.; Galwey, A. K. Reactions in the Solid State, Comprehensive Chemical Kinetics; Elsevier: Amsterdam, 1980; Vol. 22. (27) Wendlandt, W. W.; Hoiberg, J. H. A differential thermal analysis study of some organic acids. Anal. Chim. Acta 1963, 28, 506. (28) Gal, S.; Meisel, T.; Erdey, L. On the thermal analysis of aliphatic carboxylic acids and their salts. J. Therm. Anal. 1969, 1, 159.

To conclude this discussion, it appears appropriate to propose that changes in the aggregate state of a thermally decomposing compound generally should not have a dramatic effect on the value of the activation energy of the process. If possible, such changes may occur when a change of the aggregate state causes a change in the reaction mechanism. As a simplistic example, one could hypothesize a change from monomolecular to bimolecular mechanism. For instance, when molecules are locked firmly in certain positions within a crystalline solid, they may only be able to decompose via monomolecular bond breaking, i.e., through bond vibration. However, in the liquid state when molecules are free to move around, they experience continuous collisions that can initiate the mechanism of bimolecular bond breaking. Since bimolecular reactions generally have lower activation energies than monomolecular ones, the liquid-state decomposition may then demonstrate a decrease in the activation energy relative to the process in the solid state.

5. CONCLUSIONS The thermal decomposition of malonic acid has been studied via TGA. Advanced isoconversional kinetic analysis has been employed to determine the kinetic triplets in the solid, liquid, and supercooled liquid states. Our study has not found any dramatic difference in the activation energies of the liquid- and solid-phase decomposition of this compound. This result contradicts the early report by Bawn. A likely reason between the two different sets of the results is the difference in the conditions of experiments that used open and closed reaction systems. On the other hand, the results of the present study are in line with a number of other reports that suggest that in general the aggregate state has no significant effect on the activation energy of thermal decomposition. Considering that the preexponential factor has also been unaffected by the aggregate state, it is possible to conclude that both liquid- and solid-state decomposition of malonic acid fits a single Arrhenius plot. It means that in such a situation one can use the liquidphase kinetic data to obtain a reasonable estimate for the thermal stability of a compound in the solid phase. Of course, the occurrence of such a situation for malonic acid and, maybe, a few other compounds does not prove this to be a general rule. Rather, it emphasizes the need for systematic comparative studies of the liquid- and solid-phase kinetics.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Sergey Vyazovkin: 0000-0002-6335-4215 Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Pincock, R. E. Reactions in Frozen Systems. Acc. Chem. Res. 1969, 2, 97. (2) Tanaka, K.; Toda, F. Solvent-Free Organic Synthesis. Chem. Rev. 2000, 100, 1025. (3) Bawn, C. E. H. Decomposition of Organic Solids. In Chemistry of the Solid State; Garner, W. E., Ed.; Butterworth: London, 1955; p 254. (4) Manelis, G. B.; Nazin, G. M.; Rubtsov, Y.; Strunin, V. A. Thermal Decomposition and Combustion of Explosives and Propellants; Taylor & Francis: London, 2003. 7969

DOI: 10.1021/acs.iecr.7b02076 Ind. Eng. Chem. Res. 2017, 56, 7964−7970

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Industrial & Engineering Chemistry Research (29) https://www.britannica.com/science/carboxylic-acid (accessed June 24, 2017). (30) Muraishi, K.; Suzuki, Y. The thermal behaviour of dicarboxylic acids in various atmospheres. Thermochim. Acta 1994, 232, 195. (31) Somorjai, G. A. Principles of Surface Chemistry; Prentice Hall: Englewood Cliffs, 1972.

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DOI: 10.1021/acs.iecr.7b02076 Ind. Eng. Chem. Res. 2017, 56, 7964−7970