Effect of Inert Gas Pressure on Reversible Solid-State Decomposition

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Effect of Inert Gas Pressure on Reversible Solid-State Decomposition Victoria L. Stanford, Tatsiana Liavitskaya, and Sergey Vyazovkin*

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Department of Chemistry, University of Alabama at Birmingham, 901 S. 14th Street, Birmingham, Alabama 35294, United States ABSTRACT: This work explores the effect of an inert gas pressure on the kinetics of reversible thermal decomposition of solids. The nature of this effect is diffusional. Theoretical analysis of the effect suggests that the process rate should decelerate with increasing inert gas pressure and that the deceleration should occur at the expense of a decrease in the pre-exponential factor. The effect is illustrated by applying high-pressure differential scanning calorimetry to the process of the thermal dehydration of lithium sulfate monohydrate (Li2SO4·H2O). An increase in nitrogen pressure from 0.1 to 7 MPa shifts the reaction to higher temperature by more than 10 °C. Kinetic analysis by means of the Kissinger and advanced isoconversional methods indicates that the activation energy remains practically unchanged by changing pressure. The pre-exponential factor has been found to decrease in proportion to an increase in pressure.

1. INTRODUCTION Thermal decomposition of many ionic solids occurs in a reversible manner in accordance with the following equation1−3 A s V Bs + Cg

example, a high-pressure differential scanning calorimetry (HP DSC) study12 on the reversible decomposition of semiclathrate hydrate demonstrates that the decomposition temperature increases with increasing pressure of tetrafluoromethane. On the other hand, dehydration of mascallisterite, inderite, and kurnkovite has shown13 a decrease in the decomposition temperature with increasing nitrogen pressure. The absence of any systematic studies in this area is easy to understand. From a purely thermodynamic standpoint, no such effect should be expected. An inert gas is the gas that is not reactive toward either reactant or products. Therefore, its pressure cannot affect equilibrium, that is, either P or P0 in eq 2 and, thus, should not affect the rate of decomposition. Nevertheless, such effect can be expected from the kinetic standpoint. Imagine that the decomposition occurs in vacuum, that is, in the absence of any gaseous environment. Then, the product gas molecules formed can easily leave the surface without resistance from the environment. However, if the decomposition occurs under pressure of an inert gas, the product gas molecules would experience collisional (diffusional) resistance of the gaseous environment that would slow their escape from the surface of the reacting solid. That is, the decomposition can be expected to slow with increasing inert gas pressure. To some extent, the effect should be similar to that observed for the Stefan flow,14−16 when the rate of vaporization of a liquid decreases with increasing pressure of an inert gas. The effect has a purely diffusional nature. As already stated, there have been only two studies12,13 of the effect of an inert gas pressure on reversible decomposition. If one study12 confirms the expected effect, the second13 contradicts it. Nonetheless, there are some studies that hint indirectly at the existence of such effect. These are the studies of reversible decomposition under inert gases of different molecular weights. Keeping in mind that the effect of an inert

(1)

where As and Bs, respectively, are the solid reactant and product, and Cg is the product gas. In addition to temperature, T, and extent of conversion, α, the rate of such reactions depends strongly on the partial pressure of the product gas, P. In its simplest form, the respective rate equation can be written as4 ij dα P yz i −E zy zzf (α)jjj1 − zzz = A expjjj j dt P0 z{ k RT { k

(2)

where t is the time, A and E are the Arrhenius parameters, the pre-exponential factor and activation energy, respectively, R is the gas constant, f(α) is the reaction model, and P0 is the equilibrium pressure of the product gas. The particular form of the pressure term used in eq 2 has been derived originally by Benton and Drake.5 The effect of the partial pressure of the product gas is straightforward. This gas is a reactant in the reverse reaction. Increasing its amount accelerates the reverse reaction so that the overall rate of decomposition slows. This means that maintaining the same reaction rate at higher partial pressure would require higher temperature. To put it simply, applying higher partial pressure of the product gas shifts reversible decomposition to higher temperature. This effect is well documented.6 The effect of the partial pressure on the Arrhenius parameters has been reported in a number of studies7−11 that demonstrate a significant increase in the effective activation energy and pre-exponential factor of reversible decomposition with increasing the partial pressure of the product gas. However, reports on the effect of an inert gas pressure on reversible decompositions are scarce and contradictory. For © XXXX American Chemical Society

Received: July 1, 2019 Revised: July 30, 2019

A

DOI: 10.1021/acs.jpcc.9b06272 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

was pressurized to a selected pressure. The pressures were selected to be 0.1 and 7 MPa. All runs were performed twice to ensure repeatability.

gas pressure is collisional by its nature, it should be expected that reversible decomposition will slow and shift to higher temperature not only with increasing pressure but also with increasing molecular weight of an inert gas. Indeed, it has been reported17 that CaCO3 decomposes under nitrogen at markedly higher temperature than under helium. It has also been observed18 that CdCO3 decomposes in argon at higher temperature than in nitrogen. Likewise, decomposition of CaCO3 shifts to progressively higher temperatures when separately switching an inert gas from helium to nitrogen, argon, or krypton.19 Putting the aforementioned arguments together, it should generally be expected that an increase in an inert gas pressure will slow reversible decomposition and shift it to higher temperatures. The shift can be expected to be quite significant. The reported values12,17−19 range from 5 to 50 °C. Such effects should manifest themselves as changes in the values of kinetic parameters. For example, a shift to higher temperature can be due to an increase in the effective activation energy as well as a decrease in the effective pre-exponential factor. The only piece of information related to this issue, albeit indirectly, is found in the report 18 that the effective activation energy for decomposition of CdCO3 did not change when switching the inert gas from nitrogen to argon, although the respective decomposition temperature increased by ∼10 °C. Unfortunately, the pre-exponential factors have not been reported, and the computations have been conducted by using single-heating rate methods that are presently known4 to be unreliable. Clearly, there is a significant gap in the current knowledge of the effect of an inert gas pressure on the kinetics of reversible reactions. To close this gap, systematic studies of this effect are needed. The present paper aims to provide a proof of concept that systematic studies of the aforementioned effect would be feasible and worthwhile. More specifically, it offers theoretical insights into this effect and supports them with some preliminary experimental results for the thermal dehydration of lithium sulfate monohydrate as measured using HP DSC under the pressure of nitrogen. The reaction in question has been selected for the following reasons. First and foremost, it is known to be reversible, and previous experience has been gained in studying this reaction by regular DSC20 and thermogravimetric analysis (TGA)21 under ambient pressure of flowing nitrogen. Another advantage of choosing this reaction is that it, just as many other dehydrations, occurs at relatively low temperatures that fit well within the temperature range of the DSC instrumentation. This would not be the case of typical reversible decompositions of carbonates, oxides, or hydrides.

3. COMPUTATIONS Kinetic computations were carried out in accordance with the recommendations of the ICTAC Kinetic Committee.4 For instructive reasons, the computations were conducted by employing two approaches: a simplified and an advanced one. The former made use of the most popular kinetic method by Kissinger,22,23 which is based on the following equation ji β zy AR E lnjjjj 2 zzzz = ln( − f ′(αp)) − j Tp z E RTp k {

(3)

where β is the heating rate, f′(α) is the derivative of the reaction model, and the subscript p denotes the value related to the position of a DSC peak. Equation 3 affords estimating the pre-exponential factor and activation energy from the intercept and slope of a plot of the left hand side against the reciprocal peak temperature. The treatment is simplified because it assumes that any process can be described by a single pair of the Arrhenius parameters. This assumption does not hold for multistep processes because they tend to demonstrate the Arrhenius parameters that vary with conversion and temperature.24 An advanced treatment made it possible to determine variation of the Arrhenius parameters with conversion. The conversion dependence of the effective activation energy or, simply, isoconversional activation energy, Eα, was evaluated by using an advanced isoconversional method.25 The method utilizes flexible integration26,27 that assumes the constancy of Eα over a very narrow range of conversions, Δα, that does not exceed 0.01 or 0.02. Such integration eliminates a systematic error in Eα that is found in the methods that rely on rigid integration, which assumes the constancy of Eα in the whole range of conversions from 0 to α.26,27 For each value of α, the Eα values are estimated by minimizing the function n

Ψ(Eα) =

n

∑∑ i=1 j≠i

J[Eα , Ti(tα)] J[Eα , Tj(tα)]

(4)

ÄÅ É ÅÅ −Eα ÑÑÑ Å ÑÑdt expÅÅÅ J[Eα , Ti(tα)] ≡ Ñ ÅÅÇ RTi(t ) ÑÑÑÖ tα −Δα

where

∫

tα

(5)

The minimization is conducted on the set of data obtained at n temperature programs, T(t). The integral was estimated via the trapezoid rule. A minimum of eq 4 was found by using the COBYLA nongradient method28 from the NLOPT library.29 The Monte Carlo bootstrap method30 was applied to evaluate the uncertainties (standard deviations) in the Eα values. The isoconversional pre-exponential factor was evaluated by substituting the values of Eα into the equation of the compensation effect4

2. EXPERIMENTAL SECTION All HP DSC measurements were obtained with a heat flux HP DSC 1 instrument from Mettler-Toledo. Indium and zinc standards were used to perform temperature, heat flow, and tau-lag calibrations under constant pressure of ultrahigh purity (99.999%) nitrogen gas supplied by Airgas. Lithium sulfate monohydrate (Li2SO4·H2O) (99+% pure) was purchased from ACROS Organics. The samples had the mass of 3.6 ± 0.1 mg. They were placed in open 40 μL aluminum pans and heated from 25 to 200 °C at the heating rates of 2.0, 3.0, 4.4, 6.6, and 10 °C·min−1. Upon loading a sample in HP DSC, the pressurizable chamber was purged for 3 min with nitrogen at a flow rate exceeding 200 mL min−1. After this, the instrument

ln Aα = a + bEα

(6)

The parameters a and b were determined by fitting the pairs of ln Ai and Ei into eq 6. The respective pairs were determined by substituting several reaction models, f i(α), into the linear form of the basic rate equation4 B

DOI: 10.1021/acs.jpcc.9b06272 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C E i dα y lnjjj zzz − ln[fi (α)] = ln Ai − i RT k dt {

(7)

For each given reaction model, ln Ai and Ei values were found, respectively, from the intercept and slope of the linear plot of the left-hand side of eq 7 versus the reciprocal temperature. Based on the previous analysis,20 four f(α) functions that represent the power law (P2, P3, P4) and Avrami-Erofeev (A2) models are sufficient for accurate estimation of the preexponential factor.

4. RESULTS AND DISCUSSION 4.1. Theoretical Insights. By its origin, the effect of surrounding inert gas and its pressure is diffusional. Therefore, it must be analyzed how the rate of diffusion depends on pressure as well as on other relevant factors. The use of the molecular kinetic theory of gases permits the derivation of the following equation for the diffusion coefficient31 D∝

T 3/2 Ptot(a1 + a 2)2

1 1 + m1 m2

Figure 1. Fits of temperature dependence of the diffusion coefficients35 to the power (squares and dash line) and exponential (circles and solid line) equations.

(8)

diffusion of water in nitrogen is displayed in Figure 1. The estimated ED value is 5.6 ± 0.1 kJ mol−1. It is noteworthy that the exponential dependence has given somewhat better goodness of fit (larger correlation coefficient and smaller residual sum of squares) than the power dependence. For a given combination of gases such as water and nitrogen, the a and m values in eq 8 are constant so that the diffusion coefficient is a function of temperature and pressure only. Then, by replacing the power temperature dependence with the exponential one, eq 8 can be converted into the following form

where a1 and a2 are the radii, and m1 and m2 are the masses of molecules. The first thing to notice is that eq 8 indicates that the rate of diffusion of one gas decelerates with increasing the molecular mass of another gas. This can certainly explain the aforementioned effect in which the temperature of reversible decomposition shifts to higher value in the atmosphere of an inert gas that consists of heavier molecules. In other words, the product gas molecules would escape the reacting surface at slower rate in the presence of inert gas molecules that exert stronger collisional resistance. Equation 8 also indicates that the diffusion rate decreases in direct proportion to an increase in the total pressure. Such dependency has been observed experimentally in multiple studies32−34 and in the references therein. This dependence is of direct relevance to the effect sought after in the present work. It suggests that the product gas molecules would escape the reacting surface at slower rate with increasing an inert gas pressure. Now, eq 8 can be used to arrive at a certain form of a kinetic equation for the rate of decomposition. Note that the diffusion coefficient depends on temperature as a power function with the exponent 1.5. This dependence is not very accurate, and the actual values of the exponents are typically found to be between 1.5 and 2.0.32 As an example, Figure 1 shows the temperature dependence of the diffusion coefficient for diffusion of water in nitrogen.35 According to eq 8, the actual value of the exponent can be estimated as the slope of the linear plot ln D versus ln T. Fitting the data set presented in Figure 1 yields the exponent value 2.08 ± 0.05. Alternatively, the temperature dependence of the diffusion coefficient can be presented in the Arrhenius (i.e., exponential) form i −E y D = D0 expjjj D zzz k RT {

D=

D0′ i −E y expjjj D zzz Ptot k RT {

(10)

where D0′ is the pre-exponential factor. This equation describes the rate at which the product gas molecule C diffuses away from the reaction zone. On the other hand, for this molecule to form, the reactant molecule A has to overcome an energy barrier, E. The rate constant for this process can be presented in the usual Arrhenius form

i −E yz zz k = A expjjj (11) k RT { Note that the exponential functions in both equations represent the probability of the respective events, that is, a diffusion jump and a reaction act. The probability, Pr, of the product molecule forming and immediately leaving the reaction zone equals to the product of the probabilities of the respective events that give rise to a two-exponent equation

i −E y i −E yz zz Pr ∝ expjjj D zzzexpjjj (12) k RT { k RT { This type of two-exponent equations is well known in the kinetics of physical and chemical processes. One such example is self-diffusion in liquids that occurs by crossing the energy barriers associated with the formation of a vacancy (hole) and a jump to occupy it.38 Another example is heterogeneous decomposition, whose total energy barrier equals to the sum of the heat of adsorption and activation energy of decomposition on the surface.39 Lastly, the condensed state phase transitions are described by two-exponent equations that combine the energy barriers of nucleation and diffusion.40−42

(9)

where ED is the activation energy of diffusion, and D0 is the pre-exponential factor. Equation 9 is used broadly32,36 for describing the temperature dependence of the diffusion rate. In particular, it has been shown26,37 to work well for diffusion of various gases. The respective ln D versus T−1 plots demonstrate nearly perfect linearity and yield small values of ED, typically around 4−6 kJ mol−1.26,37 The Arrhenius plot for C

DOI: 10.1021/acs.jpcc.9b06272 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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effect of shifting reversible decomposition to higher temperature with increasing inert gas pressure does take place.

The aforementioned arguments allow us to propose a possible form of the effective rate constant for the overall process that combines decomposition with diffusion. To convert the total probability (eq 12) to the rate constant, the former must be complemented by a pre-exponential term. Considering that the pre-exponential term of eq 10 contains the reciprocal pressure, the similar dependence also should be retained in the equation for the effective rate constant. Therefore, the final equation should take the following form k=

i E + E yz Z zz expjjj− D Ptot k RT {

(13)

where Z is the pre-exponential factor. The logarithmic form of eq 13 is ED + E (14) RT where Z′ = Z/Ptot is the effective pre-exponential factor. According to eq 14, the intercept of a ln k against T−1 plot should yield ln Z′, whose value decreases with increasing inert gas pressure. However, the decrease should be relatively small, for example, a tenfold increase in pressure would diminish the ln Z′ value by roughly 2 units. Equation 14 also suggests that the slope of the respective Arrhenius plots should yield the effective activation energy that is independent of pressure and larger than the activation energy determined in vacuum by the ED value. As discussed earlier, the latter should be about 4−6 kJ mol−1.26,37 On the other hand, the activation energy of the reaction should be closer to 100 kJ mol−1 so that the increase in question is likely to be within the magnitude of experimental uncertainty. In other words, the effect of an inert gas pressure on the kinetics of reversible decomposition should most likely reveal itself as a change in the value of the pre-exponential factor. 4.2. Practical Analysis. Figure 2 presents HP DSC data for the thermal dehydration of Li2SO4 H2O at the pressures of ln k = ln Z′ −

Figure 3. HP DSC peak temperatures of Li2SO4·H2O dehydration under different heating rates and pressures (open circles 0.1 MPa, solid circles 7 MPa).

A valid point of concern in studying the reactions of reversible dehydration is the inescapable presence of water vapor from the sources other than the reaction itself. In the experiments conducted here, there are three possible sources of water vapor: atmospheric air humidity, sample-absorbed moisture, and water impurity in nitrogen gas. The pressurizable chamber of HP DSC naturally fills with atmospheric air when the instrument is open to load a sample. The volume of this chamber is 300 mL. Upon loading a sample, the chamber was closed and purged for 3 min with nitrogen gas at a flow rate in excess of 200 mL min−1 before starting a heating run. This procedure should displace virtually all air (and moisture) from the instrument. To test its efficiency, runs with longer purge time, 10 min, have been conducted. As seen in Figure 2 (open circles and squares), the respective HP DSC runs are practically identical, except for some small deviation (∼3 °C) at the high-temperature end of the peaks. It should be noted that the shape of the DSC peaks is a subject to some unavoidable fluctuations due to the uncertainty in selecting a baseline. The effect is always more pronounced in either the high or low temperature ends of the DSC peak.43 Such fluctuations are of no particular concern for further kinetic analysis because they have been demonstrated43 to result in negligible errors in the Arrhenius parameters. Overall, it can be concluded that the use of longer purge time does not show any obvious signs of improving the efficiency of air removal. Solid compounds are frequently found to contain absorbed moisture. Its presence can be detected in TGA runs as a mass loss occurring below 100 °C. However, the previous TGA study21 of Li2SO4·H2O dehydration did not reveal any such loss. In the present study, a TGA run on a sample that was exposed to atmospheric air for 24 h has been performed. This experiment also did not show any detectable mass loss below 100 °C. In addition, an HP DSC run, in which the sample was first heated at 5 °C min−1 to 70 °C under nitrogen flowing in excess of 200 mL min−1, has been conducted. Upon reaching the final temperature, nitrogen flow was shut off, and the instrument was cooled to an ambient temperature. After this, a regular heating run was conducted. The resulting HP DSC curve (stars, Figure 2) shows some small (∼3 °C) deviation from the two other curves at the lower temperature end. As

Figure 2. HP DSC curves of Li2SO4·H2O dehydration measured at 10 °C min−1 under different pressures. The numbers 0.1 and 7 denote, respectively, 0.1 and 7 MPa pressure. Solid and open circles represent regular runs performed after 3 min purging with nitrogen. Run after 10 min purging is represented by squares. Stars designate a run performed after a heating−cooling cycle.

0.1 and 7 MPa. It is seen that the 70-fold increase in the pressure of nitrogen causes the reaction to shift to higher temperature by more than 10 °C. Similar trend is observed at all other heating rates (Figure 3). Therefore, the expected D

DOI: 10.1021/acs.jpcc.9b06272 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C explained earlier, such fluctuations commonly arise from the uncertainty in selecting the baseline. Beyond this, the curve does not show any obvious difference with the curve from a regular run. Therefore, the samples studied are not likely to contain any significant amounts of moisture. Note that the above tests do not necessarily prove that water from atmospheric air humidity or sample absorbed moisture is removed completely. The results of these tests may also be taken to mean that water may still be present in some residual amount that is difficult to reduce by more extensive purging or heating. Obviously, this residual amount could generate some partial pressure of water during the experiments that would affect the rate of dehydration (eq 2). However, even if this is the case, the resulting partial pressure would be independent of the nitrogen pressure and, thus, cannot be a factor contributing to the temperature shift seen in Figures 2 and 3. Lastly, nitrogen gas always contains some moisture as an impurity. Nitrogen used in the present work has been of ultrahigh purity. According to the manufacturer assay, it contains water in the amount of 1 ppm by volume. This impurity would generate the partial pressure of water in the amount of 0.1 and 7 Pa at the nitrogen pressure of 0.1 and 7 MPa. These pressures are much smaller than the partial pressure of water generated by dehydration of Li2SO4·H2O. It can be estimated that a completely decomposed 4 mg sample would generate approximately 230 Pa of water pressure in the 300 mL volume of the HP DSC chamber. It means that the partial pressure of the product gas would exceed 0.1 Pa at 0.04% decomposition and 7 Pa at 3% decomposition. In other words, the water impurity in nitrogen cannot play any important role throughout most of the reaction. This argument is further enhanced by the fact that water molecules generated by the reaction generally play a much more important role in a reversible reaction. This is because they are generated directly in the reaction zone, whereas the water impurity molecules are spread uniformly throughout the whole HP DSC chamber so that only a small fraction of them is found in the reaction zone. Figure 4 illustrates the application of the Kissinger method to the peak temperature data shown in Figure 3. Both plots

values is comparable to the experimental uncertainty. Even if the uncertainty is disregarded, it should be noticed that the activation energy has decreased with increasing pressure. Obviously, a decrease in the activation energy can only cause a decrease in the dehydration temperature. This is opposite to the actually observed effect. Then, the origin of the latter should be sought in the value of the pre-exponential factor. A decrease in this value with increasing pressure would be consistent with the observed increase in the dehydration temperature. In the Kissinger method, the pre-exponential factor is a part of the intercept (eq 3) of the linear plot (Figure 4). It is seen in Figure 4 that the linear plot for 7 MPa is positioned lower along the vertical axis than the one for 0.1 MPa. This is indicative of the smaller intercept value in the former case. The actual difference between the intercept values for 0.1 and 7 MPa data is Δ = 2.1. This value can be used to estimate the difference in the pre-exponential factors, Δln A. From eq 3, this difference is Δln A = Δ + Δln E − Δ[−Rf ′(αp)]

(15)

We can assume that the reaction model does not change with pressure. This assumption is supported by the absence of any obvious changes in the DSC peak shape with pressure (Figure 2) as well as by previous studies44 of Li2SO4·H2O dehydration that have demonstrated that the reaction model does not change with the partial pressure of water. By virtue of this assumption, the last addend in eq 15 is zero. Then, by substituting the values of the activation energy and the difference in the intercepts in the second and first addend, respectively, the Δln A value is estimated to be 2.1. This confirms that the observed increase in the dehydration temperature is linked to a decrease in the pre-exponential factor with increasing pressure. An advanced kinetic treatment by means of isoconversional techniques reveals the complexity of the dehydration process in the form of the effective activation energy and preexponential factor that vary with conversion (Figures 5 and 6).

Figure 5. Isoconversional activation energies for Li2SO4·H2O dehydration under different pressures (open circles 0.1 MPa, solid circles 7 MPa).

Figure 4. Kissinger plots for Li2SO4·H2O dehydration under different pressures (open circles 0.1 MPa, solid circles 7 MPa).

The dependencies are decreasing. A decreasing dependence of Eα on α for dehydration of Li2SO4·H2O has been repeatedly reported in previous studies conducted at ambient pressure under a flow of an inert gas.20,21,45,46 This type of dependence has also been observed for dehydration of calcium oxalate monohydrate and calcium sulfate dihydrate.47 It has been explained20 that such dependence is generally the characteristic

demonstrate statistically significant linearity. The correlation coefficients are 0.985 (0.1 MPa) and 0.994 (7 MPa). The plots are nearly parallel, which indicates that the respective activation energies are practically the same. Indeed, the values are 101 ± 6 and 97 ± 4 kJ mol−1 for dehydration under 0.1 and 7 MPa, respectively. The difference between the two E

DOI: 10.1021/acs.jpcc.9b06272 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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investigation into the practically unexplored effect of an inert gas pressure on the kinetic parameters of reversible decomposition. To better understand this effect, further systematic studies are needed in two directions. First, different types of decompositions (dehydration, decarbonation, dehydrogenation, dehydroxylation, etc) need to be studied under an inert gas pressure. Second, the effect of different kinds (i.e., different molecular weight) of an inert gas on the kinetic parameters of reversible decomposition needs to be explored. The theoretical analysis presented in this paper can be expanded to demonstrate that this effect should reveal itself in the form of decreasing the pre-exponential factor with increasing the molecular weight of an inert gas.

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Figure 6. Isoconversional pre-exponential factors for Li2SO4·H2O dehydration under different pressures (open circles 0.1 MPa, solid circles 7 MPa).

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

of reversible decompositions as it follows directly from eq 2. As seen from Figure 5, the Eα dependencies are decreasing at both pressures. The Eα values for 7 MPa are somewhat smaller than those for 0.1 MPa. The difference is comparable to the uncertainty in Eα values. This result is similar to that obtained via the Kissinger method. Again, it should be emphasized that any decrease in the activation energy could only be consistent with a decrease in the dehydration temperature, which is exactly opposite to the effect observed experimentally (Figures 2 and 3). On the other hand, the effect can be linked to a decrease in the pre-exponential factor with increasing pressure. Figure 6 demonstrates that the ln Aα values for 7 MPa are consistently smaller than those for 0.1 MPa. The average difference in the respective ln Aα values is 4.5 ± 0.6. It is noteworthy that this difference is similar to the value expected from eq 14. It suggests that the effective value of the pre-exponential factor should decrease as many times as an inert gas pressure increases. In the present case, it should decrease about 70 times, which means that the natural logarithm of the preexponential factor should decrease by 4.1 units. Of course, this is limited evidence to confirm the validity of eq 14, especially considering that the Kissinger analysis has yielded a decrease in the pre-exponential factor by only 2.1 units on the natural logarithm scale. Such decrease would be consistent with ∼8 times increase in an inert gas pressure, that is, it would rather be proportional to Ptot than Ptot.

Sergey Vyazovkin: 0000-0002-6335-4215 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge Mettler-Toledo for loaning the HP DSC instrument. REFERENCES

(1) Young, D. A. Decomposition of Solids; Pergamon: Oxford, 1966. (2) Brown, M. E.; Dollimore, D.; Galwey, A. K. Reactions in the Solid State; Elsevier: Amsterdam, 1980. (3) Galwey, A. K.; Brown, M. E. Thermal Decomposition of Ionic Solids; Elsevier: Amsterdam, 1999. (4) 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, 1−19. (5) Benton, A. F.; Drake, L. C. Kinetics of Reaction and Adsorption in the System Silver−Oxygen. J. Am. Chem. Soc. 1934, 56, 255−263. (6) Criado, J.; González, M.; Málek, J.; Ortega, A. The Effect of the CO2 Pressure on the Thermal Decomposition Kinetics of Calcium Carbonate. Thermochim. Acta 1995, 254, 121−127. (7) Dollimore, D.; Jones, T. E.; Spooner, P. Thermal Decomposition of Oxalates. Part XI. Dehydration of Calcium Oxalate Monohydrate. J. Chem. Soc. A 1970, 0, 2809−2812. (8) Maciejewski, M.; Bałdyga, J. The Influence of the Pressure of the Gaseous Product on the Reversible Thermal Decomposition of Solids. Thermochim. Acta 1985, 92, 105−108. (9) Catti, M.; Ghaani, M. R.; Pinus, I. Overpressure Role in Isothermal Kinetics of H2 Desorption- Absorption: the 2LiBH4Mg2FeH6 System. J. Phys. Chem. C 2013, 117, 26460−26465. (10) Perejón, A.; Sánchez-Jiménez, P. E.; Criado, J. M.; PérezMaqueda, L. A. Magnesium Hydride for Energy Storage Applications: The kinetics of Dehydrogenation under Different Working Conditions. J. Alloys Compd. 2016, 681, 571−579. (11) Koga, N.; Favergeon, L.; Kodani, S. Impact of Atmospheric Water Vapor on the Thermal Decomposition of Calcium Hydroxide: A Universal Kinetic Approach to a Physico-Geometrical Consecutive Reaction in Solid−Gas Systems under Different Partial Pressures of Product Gas. Phys. Chem. Chem. Phys. 2019, 21, 11615−11632. (12) Sugahara, T.; Machida, H. Dissociation and Nucleation of Tetra-n-butyl Ammonium Bromide Semi-Clathrate Hydrates at High Pressures. J. Chem. Eng. Data 2017, 62, 2721−2725. (13) Ximing, Y.; Zuowei, S.; Yuanjun, T. DSC Studies on the Kinetics of Decomposition of Some Mg-Containing Borates under High Pressures. Thermochim. Acta 1991, 191, 13−20.

5. CONCLUSIONS The theoretical insights provided in this work suggest that an increase in pressure of an inert gas should decelerate the rate of reversible decompositions. They also suggest that deceleration should occur primarily at the expense of a decrease in the preexponential factor. These inferences have been probed experimentally by applying HP DSC to the thermal dehydration of Li2SO4·H2O. The deceleration of this reaction has been detected in the form of its shift to higher temperature by more than 10 °C as nitrogen pressure increased from 0.1 to 7 MPa. The kinetics of the process has been analyzed by employing multiple heating rate techniques, viz., the Kissinger method and advanced isoconversional method. Both methods have demonstrated that the activation energy of the process remains practically independent of nitrogen pressure. The preexponential factor, however, has been found to decrease with increasing pressure. The present work is just an initiatory F

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Article

The Journal of Physical Chemistry C

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DOI: 10.1021/acs.jpcc.9b06272 J. Phys. Chem. C XXXX, XXX, XXX−XXX