Abstract
For dehydration of CaC_{2}O_{4}·H_{2}O and thermal dissociation of CaCO_{3} carried out in Mettler Toledo TGA/SDTA851^{e}/STAR^{e} thermobalance similar experimental conditions was applied: 9–10 heating rates, q = 0.2, 0.5, 1, 2, 3, 6, 12, 24, 30, and 36 K min^{−1}, for sample mass 10 mg, in nitrogen atmosphere (100 ml min^{−1}) and in Al_{2}O_{3} crucibles (70 μl). There were analyzed changes of typical TGA quantities, i.e., T, TG and DTG in the form of the relative rate of reaction/process intended to be analyzed online by formula (10). For comparative purposes, the relationship between experimental and equilibrium conversion degrees was used (for ). It was found that the solid phase decomposition proceeds in quasiequilibrium state and enthalpy of reaction is easily “obscured” by activation energy. For small stoichiometric coefficients on gas phase side (here: ν = 1) discussed decomposition processes have typical features of phenomena analyzable by known thermokinetic methods.
Introduction
 (1) modified van ‘t Hoff's isobar for thermal dissociation of chemical compounds in solid state [1–5],
 (2) van Laar–Planck's isotherm [6, 7].
It is proved that reaction enthalpy (Δ_{r}H)—as an average quantity—is independent of temperature (T) and pressure (P), when P < 3 MPa [8].
In practice, in TGA analysis, the most commonly used is experimental degree of conversion (α) at a given temperature, which must be lower than equilibrium one (α_{eq}) [9–11], but not always so.
Identical results were presented in works [6, 7].
According to Eq. 7 intensive collection of gaseous products producing vacuum () moves equilibrium line α_{eq} over the course deriving from isobaric conditions (4) together with (5).
It is common to carry out studies on thermal processes just in discussed way—the example is presented in Fig. 1 quoted from [12].
Scope of the work
Basing on the analysis of thermal decomposition of two model chemical compounds (CaC_{2}O_{4}·H_{2}O and CaCO_{3}), often used as test substances, in inert atmosphere in dynamic conditions at different heating rates (q), own assessment of thermodynamic conditions of decomposition using the relative rate of reaction/process has been presented.
Basing on threeparametric equation [5], the relative rate of reaction/process was proposed for dynamic conditions [13], which results from considerations on the equilibrium course of chemical reaction of solid phase dissociation, when . It means that the term versus T should be constant, i.e., r = r_{eq} = const. (r in K).
Therefore there was the study carried out, in which it was assumed that relationships r versus T will be analyzed without any data selection procedure in order to analyze graphic images obtained in dynamic conditions, electronically recording the course of thermal dissociation of selected two chemical compounds in the relation: mass versus temperature.
Assumptions
Equation 9 represents straight line parallel to temperature axis (T), limited by perpendicular at the point with coordinates [0, T_{eq}] (e.g., see Fig. 3 in [14]).
Further considerations follow from comparison of temperature of maximum reaction rate T_{m} determined experimentally and calculated by formula (14).
According to Ref. [14], temperature T_{r} is represented by slope of straight lines (11) and (9). To some extent this is the analogue of isokinetic temperature, because (in theory) r = const. Analysis of experimental data consists in determination of linear relationships (11), while the relative rate of reaction/process is determined by formula (10). Next, formulas in forms (14) and (15) were used.
The considerations were carried out in three areas: thermokinetic, thermodynamic, and relative rate. This approach follows from the fact that such considerations prevail in such order in literature.
The basic element of analysed thermoanalytical results consists in evaluation, whether the relative rate of reaction/process r calculated according to Eq. 10 versus T presents relationship (9), of which graphical image is rectangle area bounded by temperature T_{eq}, or linear relationship (11) with a negative slope, expressed by coefficient a_{2}.
Calcium oxalate monohydrate
Thermokinetics
Thermodynamics
Equation 9 is plotted on the basis of thermodynamic data for standard conditions:

T = 298.15 K and P ≈ 0.1 MPa for dehydration according to Ref. [18]:

ΔH = 36.4 kJ mol^{−1}, (according to Ref. [25] enthalpy of vaporization of water in 298.15 K is 44.0 kJ mol^{−1})

ΔG = 2.2 kJ mol^{−1},

ΔC_{p} = −10.2 J (K mol)^{−1}—originates from summing up values: _{298}C_{p}, Table 1 in [18] and entropy in these conditions:

ΔS = 0.1147 kJ (K mol)^{−1} (acc. [26]: ΔS = 0.156 kJ (K mol)^{−1})
From the calculations follows the average enthalpy of reaction in the temperature range of 298–318 K: Δ_{r}H = 36.3 kJ mol^{−1}. Figure 2 shows relationship between degree of conversion and temperature for ten heating rates, and Fig. 3—between the relative rate of reaction/process and temperature according to Eq. 10, giving the coefficient of determination (r^{2}) with equilibrium line (9) limited by straight line T_{eq} = 318 K (45 °C) for selected heating rates (q = 0.2, 3, and 30 K min^{−1}).
Equation 19 was determined on the basis of thermodynamic data [18] in standard conditions, T = 298 K, and thus, referring to Eq. 5, equilibrium conversion degree (for ν = 1) in natural way concerns this reference condition. Under these considerations, Eq. 19 should be regarded as acceptable estimation, assuming that in considered temperature range thermodynamic quantities are close to standard ones.
The difference can be attributed to slightly differing values of entropy. The issue, what may be surprising here, is the fact that this is thermodynamic or maximum value at given temperature and at atmospheric pressure P = 0.1 MPa. This does not mean that at T = 298 K, the decomposition occurs and is detectable, because in the way here may be conditions of kinetic nature, e.g., very slow evaporation or atmospheric equilibrium. For this reaction, Δ_{r}H ≈ 36 kJ/mol is determined by the activation energy E = 80 kJ/mol (or higher) and by the heat of vaporization of water, more than 40 kJ/mol.
In turns, Fig. 4 shows experimental results that show straight lines (11). From Fig. 4 follows that each straight line (11) together with (20) intersect at a wide temperature range and not only at determined temperature T_{r} = 390.2 K. With the exception of q = 0.2 and 1 K min^{−1}, each straight line (for q = 0.5, 2, 3, 6, 12, 18, 24, and 30 K min^{−1}) intersects at higher temperature, i.e., 399 K.
Relative rate
Table 1 presents calculated coefficients of Eq. 11 and temperature of maximum reaction rate T_{m} determined experimentally and calculated by Eq. 14—Fig. 5 compares the two temperatures toward q.
The results of calculations for dehydration of CaC_{2}O_{4}·H_{2}O
Entry  q/K min^{−1}  a _{1}/K  a _{2}  T _{m}/K acc. to Eq. 14  T _{m} ^{a}/K 

1  0.2  255758  631.7  388.8  391.2 
2  0.5  140231  325.7  406.7  401.6 
3  1  104803  233.3  419.9  412.9 
4  2  48539  95.8  455.2  419.9 
5  3  41190  77.4  471.8  430.3 
6  6  31565  53.5  510.3  441.9 
7  12  26579  40.9  549.3  454.8 
8  18  24246  34.8  580.6  462.1 
9  24  22163  29.8  609.7  467.8 
10  30  21049  27.1  631.0  472.8 
^{ a }Experimental value
According to Fig. 2 experimental degrees of conversion α satisfy inequality α_{eq} > α for all heating rates q, and according to Kissinger law (as well as in version (16)) temperatures of maximum reaction rate T_{m} increase with increasing heating rate q. However, from Fig. 3 follows that virtually all the points showing the relative rate of reaction r in terms of temperature are outside the rectangle (20). It demonstrates relative dynamic reaction course for small heating rates (according to Table 1) for q = 0.2, 0.5, 1 K min^{−1}, moderated course (q = 1 and 3 K min^{−1}) and close to equilibrium, when a_{2} < 100 for high heating rates (q ≥ 6 K min^{−1}). Relativity of dynamics here means a reference to temperature, as well to maximum thermal rate (dα/dT) obtained at low heating rates (q), as to temperature ranges, in which conversion degree achieve the end of reaction (α → 1).
However, courses similar to the parallel (a_{2} = small) are greater in value than the expression r_{eq} = Δ_{r}H/νR = 4366.13 K and closer to the ratio E/R ≅ 10000 K.
Characteristic temperatures in these analyses are of very important control meaning, namely: T_{eq} and T_{m}, T_{f} as well as T_{r}, which are determined experimentally. For large heating rates, thermal dissociation approaches to quasiequilibrium with a large temperature shift of the final reaction temperature T_{f} in relation to T_{eq} (T_{f} > T_{eq}). Instead, for small heating rates high values of coefficient a_{2} are observed, what means that even for the lowest heating rate, often regarded as a pseudoisothermal (q = 0.2 K min^{−1}), occurs the largest deformation of the equilibrium distribution according to Eq. 7 by factor (Eq. 13), which gradually reaches value of 1 for a_{2} = 0 [16].
According to Fig. 5 and Table 1 the lower coefficient a_{2}, the differences between fixed temperature T_{m} calculated by Eq. 14 become very large—only for large values of a_{2}, so for very small heating rates formula (14) leads to values compatible with experiment.
Again, one may also note that constant term in Eq. 21 is much higher than the one resulting from enthalpy (r_{eq} = 4366.13 K)—the same observation is included in Ref. [16].
Calcium carbonate (calcite)
Thermokinetics
Calcium carbonate was within the frame of the project described in the report [12]—the results were analyzed and discussed further in subsequent works [27–29], and then in Ref. [16] in terms of significance of Eqs. (12–13). Results of studies on thermal dissociation of CaCO_{3} discussed at the ICTAC Conference [12, 27–29], also summarized in Tables 1 and 2 in Ref. [16], were used applying Kissinger law in version (16).
It was obtained:
 • for decomposition in nitrogen: E = 201.0 kJ mol^{−1}, ln A = 13.46 (A in s^{−1}) (r^{2} = 0.9865, sl = 0.00001), according to Ref. [12] similar values was obtained by H. O. Desseyn: ln A = 13.58 (A in s^{−1}) and E = 199 kJ mol^{−1}, but only in this one case),
 • for decomposition in vacuum: E = 111.9 kJ mol^{−1}, ln A = 6.05 (A in s^{−1}) (r^{2} = 0.9435, sl = 0.00122)—in Ref. [12] several pairs are similar.
Relationship between conversion degree and temperature at variable heating rates and on the background of equilibrium conversion degree is presented in Fig. 6.
Thermodynamics
Compared to the ICTAC data presented in Fig. 1, results of these studies (Fig. 6) point to an even greater shift of experimental curves of conversion degree α to the left in relation to the equilibrium curve (7) for , i.e. α_{eq} < α.
Relative rate
In Ref. [16], it was found that a_{2} coefficients are: for decomposition in nitrogen a_{2} = 208 → 63 and for heating rate q = 1–25 K min^{−1} (Table 1 in Ref. [16]), but they are much lower in case of decomposition in vacuum: a_{2} = 661–317, q = 1.8 to 10 K min^{−1} (Table 2 in Ref. [16]).
Figure 7 presents—similarly as in Fig. 3—relationships of the relative rate of reaction/process according to Eq. 10, giving determination coefficient (r^{2}) together with the equilibrium line (23) limited by T_{eq} = 1162.7 K. Table 2 compiles determined coefficients of linear relationship (11) and temperature T_{m} (experimental) calculated by Eq. 14.
The results of calculations for thermal dissociation of CaCO_{3}
Entry  q/K min^{−1}  a _{1}/K  a _{2}  T _{ m }/K acc. to Eq. (14)  T _{ m } ^{a}/K 

1  0.2  56644  34.9  1064.7  876.7 
2  0.5  50285  26.0  1087.3  902.7 
3  1  53636  29.0  1109.5  925.7 
4  3  82805  59.3  1140.1  964.0 
5  6  113190  88.7  1142.0  996.0 
6  12  96902  71.1  1202.6  1019.7 
7  24  92751  65.1  1248.8  1048.5 
8  30  89552  61.6  1270.2  1063.8 
9  36  89622  61.0  1282.4  1068.2 
^{ a }Experimental value
In Fig. 7 was indicated temperature T_{r} following from Eq. 25, because it relates to small values of a_{2} and thus exposes range closer to equilibrium. Figure 8 presents again data from Fig. 7, where straight lines (11) present experimental results.
From Fig. 8 follows that each straight line (11) together with (24) intersect at a wide temperature range, and not only at determined temperature T_{r}. We note again that constant term in Eq. 25 is higher than the one resulting from enthalpy (r_{eq} = 21039.6 K).
Referring to Table 2, it was again established that for small values of a_{2} experimental temperature T_{m} is much lower than the one calculated by Eq. 14. Observed thermal dissociation of CaCO_{3} can be defined as proceeding under quasiequilibrium conditions in relation to anticipated range of characteristic temperature according to Eq. 7 for . Maybe this is the reason of observing more chaotic arrangement of points of Fig. 8 in comparison to Fig. 4. Approximately, when a_{2} → 0, then Eq. 13 simplifies to relationship known as temperature criterion [24, 30]: ln α = const. − E/RT, then Δ_{r}H → E.
Discussion
The results of investigations on thermal dissociation of two chemical substances frequently used as test compounds (standards) in thermal analysis:
 • dehydration of CaC_{2}O_{4}·H_{2}O,
 • thermal decomposition of CaCO_{3} (calcite).
The investigations were carried out under identical test conditions. Previously, trial and error method was used for selection of experimental conditions—mainly heating rates, nitrogen flow, crucibles—in order to a_{2} ratio was greater than 100. According to Tables 1 and 2 in case of calcite, it was manage to get it completely, but for dehydration of oxalate—only in a large extent.
Observing the experimental conversion degrees α toward temperature profile T, one may conclude that both compounds behave differently in relation to the equilibrium curves:
 • dehydration proceeds under curve resulting from relationship (19), α < α_{eq},
 • thermal dissociation of carbonate proceeds above relationship (22), α > α_{eq}.
In detailed studies, the relative rate of reaction/process was explicitly determined from experimental data according to the formula (10), i.e., using T, TG, DTG, and m_{i}—deliberately it was not used the option of smoothing by Savitzky–Golay method [31], which very useful for studies of complex organic mixtures such as coal tar pitches [15].
In the range of 0 < α < 0.05–0.1 one always must take into account the occurrence of oscillations arising from expression appearing in formula (10) by the ratio −DTG/TG ∝ dα/αdT, i.e., one is close to an indeterminate form 0/0 and using filters of type [31] does not help much here.
Returning to the Eq. 28 is easy to notice that one obtains this formula, if both Eq. 8 and condition dα_{eq}/dT = 0 are used, thus this is the maximum occurring far beyond the scale where α_{eq} ⋙ 1 (compare with Ref. [32]). For these reasons, the differences [Δ_{r}H/2νR − (T_{m})_{exp}] must be very large, and they decreases, when a_{2} = large.
Thus, one may note that the quasiequilibrium conditions prevent accurate determination of enthalpy of reaction Δ_{r}H, because dynamic conditions directly dictate relationships dominated by activation energy E. Thermal analyses of complex organic compounds in the form of salts containing anion halogens (Cl, Br and J), as described by Błażejowski et al. [3, 4, 33–36], behave differently due to the formation of a significant volume of gas (ν ≥ 2). These compounds are characterized by thermal decomposition, combined with the total volatilization of gaseous products. From this comparison, one may even get the feeling that the presence of solid phase (ν = 1) inhibits the process of destruction and it is necessary helping it through physical occurrence (vacuum, intense collection of gaseous products, volatilization).
Conclusions
 1. On the basis of dehydration of calcium oxalate monohydrate and thermal dissociation of calcium carbonate (calcite) in specially selected conditions, it was found that the reactions proceed in a quasiequilibrium conditions in relation to modified van ‘t Hoff's isobars expressed by Eq. 7 when . In the case of CaCO_{3} were observed small values of coefficient a_{2} (maximum a_{2} = 88.65), and the experimental curves of conversion degree toward temperature profile satisfy the relation α > α_{eq}, what means an intense collection of gaseous product from reaction zone and, in consequence, T_{eq} > T_{f}. In the case of dehydration of CaC_{2}O_{4}·H_{2}O were observed very intensive courses for small heating rates q = 0.2 to 1 K min^{−1}, what also translates to deformation equilibrium curve (7) by significant factor (Eq. 13), but at higher heating rates also importance of a_{2} decreases. The reference to equilibrium decomposition is consistent with expectations, because α<α_{eq}.
 2. Quasiequilibrium conditions prevent precise determination of enthalpy of reaction (Δ_{r}H), because dynamic conditions impose relationships dominated by activation energy (E). Despite the very significant correlation between coefficients a_{1} and a_{2} (see Refs. [37–40]), the constant term in Eq. 15 is however greater than the expression (Δ_{r}H/νR). For two reaction analysed in current paper it was not obtained values of a_{2} = 0 and at the same time T_{f} = T_{eq}. Temperature T_{r}, which is an analogue of isokinetic temperature according to Arrhenius law (here: r = r_{eq} = const.), may be determined by correlation—Figs. 4 and 8 indicate that the effect of convergence in one point (for one coordinate) is blurred. It should, however, be borne in mind that similar situation also concerns isokinetic temperature in isokinetic/compensation effect—for example in Ref. [41].
 3. Formula (14)—characteristic for the model of relative rate of reaction/process (12, 13) and (11) becomes less important, when a_{2} = 0 or a_{2} = small for calculation of T_{m}.
 4. In the case of dehydration of CaC_{2}O_{4}·H_{2}O at low heating rates q = 0.2–1 K min^{−1} is observed high dynamics of decomposition determined by coefficient a_{2}. It disappears at higher heating rates, approaching values close to but higher than r_{eq} = 4366.13 K (formula (20)), but significantly shifted above equilibrium temperature above T_{f} ⋙ T_{eq}. This effect is related to phase transformation of the reaction product—water that goes into the gas phase with a variable rate depending on the time and temperature. At lower temperature, but during the very long time (q = 0.2 K min^{−1}), there are more favorable conditions for the evacuation of water from solid surface. In a very short time (high heating rate q ≥ 6 K min^{−1}) high temperatures are necessary.
 5. Formula (10) is very simple and enables online analysis, and in particular its linear relationship with temperature (11) becomes clearer with increasing heating rate q.
Experimental methodology
Thermal decomposition of calcium oxalate monohydrate CaC_{2}O_{4}·H_{2}O (Aldrich, Cat. No. 28,9841) and calcium carbonate CaCO_{3} (Mettler—Test Sample) was carried out using Mettler Toledo TGA/SDTA851^{e}/STAR^{e}, for weight samples of 10 mg, in atmosphere of nitrogen (100 ml min^{−1}), in Al_{2}O_{3} crucible (70 μl), for 10 heating rates: q = 0.2; 0.5; 1; 2; 3; 6; 12; 18; 24 and 30 K min^{−1} (in the case of CaC_{2}O_{4}·H_{2}O) and for 9 heating rates: q = 0.2; 0.5; 1; 3; 6; 12; 24; 30 and 36 K min^{−1} (in case of CaCO_{3}).
Temperatures indicated in the text by T relate to temperature of the reacting sample.
In relation to ICTAC research [12, 27–29] the scope of research was broadened on both lower (q = 0.2 K min^{−1}—it is treated as pseudoisothermal conditions) and higher heating rates (q = 36 K min^{−1}).
List of symbols
Variables
α 
Conversion degree, 0 ≤ α ≤ 1 
α_{eq} 
Equilibrium conversion degree, 0 ≤ α_{eq} ≤ 1, P = const 
α _{ eq } 
Equilibrium conversion degree, 0 ≤ α_{eq} ≤ 1, T, P = var 
a _{0}, a_{1} and a_{2} 
Coefficients of threeparametric equation, acc. to Eq. 12 
A 
Preexponential factor in Arrhenius equation, s^{−1} 
ΔC_{p} 
Heat capacity of reaction, J (K mol)^{−1} 
E 
Activation energy, J mol^{−1} 
f(α) and g(α) 
Kinetic functions toward conversion degree α 
Δ_{r}H 
Average reaction enthalpy, J mol^{−1} 
ΔG 
Free enthalpy, J mol^{−1} 
ΔS 
Entropy, J (K mol)^{−1} 
K 
Thermodynamic equilibrium constant 
m _{ i } 
Initial mass of sample, mg 
P 
Pressure, Pa 
Standard pressure, 

q 
Heating rate, K min^{−1} 
r 
Relative rate of reaction/process, K 
R 
Universal gas constant, R = 8,314 J (K mol)^{−1} 
T 
Temperature, K 
T _{r} 
Temperature–slope in Eq. 15, K 
T _{eq} 
Equilibrium temperature for α_{eq} = 1, , K 
ν 
Stoichiometric coefficient 
Statistical symbols
r ^{2} 
Determination coefficient for linear function, 0 ≤ r^{2} ≤ 1 
sl 
Significance level 
Subscripts
298 
Standard state 
α 
Equilibrium conversion degree 
eq 
Equilibrium state 
exp 
Experimental 
f 
Final state 
fp 
Flex point 
i 
Initial state 
m 
Point of the maximal rate of reaction/process 
r 
Reaction 
We feel grateful to Dr. Wojciech Balcerowiak from ICSO “Blachownia” (KędzierzynKoźle, Poland) for providing the results of the investigations for a series of chemical compounds and thermal decomposition of calcium carbonate and calcium oxalate monohydrate.
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