Following the development of the Coats-Redfern method1, in practice

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A: Kinetics, Dynamics, Photochemistry, and Excited States

Isoconversional Methods in Thermodynamic Principles Andrzej Mianowski, and Wojciech Urba#czyk J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b04432 • Publication Date (Web): 30 Jul 2018 Downloaded from http://pubs.acs.org on August 2, 2018

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The Journal of Physical Chemistry

Following the development of the Coats-Redfern method1, in practice, there appeared a few types of integral isoconversional methods referred to by the acronyms KAS, OFW or FWO, and NLN2-4 as well as their modifications (C-KAS, C-FWO).5 The Vyazovkin method is more complicated.2, 3, 6-9 Since 2000, there has been an intense development and at the same time an almost arbitrary attitude towards isoconversional methods of analysis of thermokinetic data due to the kinetic triplet: − − ) )

(1)

the need to look for kinetic functions has been eliminated. The most extensive and convincing method is that of Vyazovkin,3, described in detail elsewhere,

10

6-9

which is

and assumes that the activation energy E can be

determined from changes in the increase of the rate of conversion α. Returning to the impressive achievements of Vyazovkin, in particular to the monograph,10 the following elements of the original methodology can be observed. The basic expression is Eq. (2.15):10

) = , , ) =

(2)

where the temperature integral , , ) is determined by Eq. (2.8).10 The complete method includes the relations (2.18) and (2.19)10 in which, according to Arrhenius, the pre-exponential coefficient A disappears. The explanation for this result uses the existence of KCE (Fig. 2.11, chapter 2, page 43).10 At variable heating rates (q), at the point of intersection of the simple, one value of Eo corresponds to only one value of lnAo. This is important because the coefficient A (according to Arrhenius) is assigned to the reaction with the entropy of activation, and the energy of activation has a close relationship with the enthalpy of activation.12 However, it is worth remembering that this leads to a considerable inaccuracy in the integral isoconversional methods in the case of a variable activation energy.11 2

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2. The aim of this paper The excellent achievements of Vyazovkin and co-workers,10 discussed in chapter 2, unambiguously concludes the long-term discussion of the implications of the Arrhenius law in thermogravimetric analysis, especially for dynamic methods, i.e., a linear temperature increase. However, the thermodynamic aspects of the isoconversional methods of thermal decomposition processes, which are used to discuss other processes but primarily phase transformations, was omitted.10 Regarding chemical reactions of the type A(s) ↔ B(s) + ∑ !" C(g) or A(s) ↔ ∑ !" C(g), and the elements of phenomenological and applied thermodynamics in a wider assessment, the use of the elements of chemical thermodynamics is observed in three dimensions: I.

as relationships between the rates of conversion: the equilibrium rate of conversion for isothermal conditions with dynamic conditions brought to it13 in the form of an equation, later called the Holba-Šesták equation,14

II.

as a result of using semi-empirical methods of quantum chemistry to determine selected values of thermodynamic functions, presented in many papers by Błażejowski and co-workers,15-19

III.

by demonstrating the relationship between the thermodynamic functions of activation (according to Eyring's theory)19-22 and observing enthalpy-entropy compensation (EEC).23-25

Therefore, if we consistently acknowledge the achievement of ending the long-term scientific battle, then there is reflection on which process isoconversional methods should be classified as from the thermodynamic point of view. Next - is the isoconversional method still considered a classical kinetic method? Thus, the book10 is only a starting point for further considerations, because it treats the problem in a definitive way. This paper is intended for providing comments concerning isoconversional methods in thermodynamic principles. It was assumed that the Gibbs free energy of a reaction system is an appropriate criterion for analyzing the course of thermal dissociation reactions of condensed phase compounds, treated as elemental transformations. Particularly noteworthy are the isoconversional variants for dynamic conditions. 3


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3. Thermodynamic model The Gibbs free energy of the thermal dissociation reaction of a condensed phase compound can be expressed by the van't Hoff isotherm:26

∆$ = % &

'

(3)

'

In formula (3), the equilibrium constant of the topochemical reaction K is represented by two formulas:27, 28 &( = ! ln +

for a single gas product

(4)

&( = &φ + . !/ )& +

for multiple gas products

(5)

where 1

&φ = Σ !/ & 312 )

(6)

2

The variable quantity Kα adopts the same mathematical structure as the formulas for the chemical equilibrium constant, which formally means that we simply omit the index (eq). For Eq. (3), Błażejowski determined the expression for Kα as the product of the activities of all components.26 By substituting Eq. (4) or (5) into Eq. (3) and using: ∆$ ∅ = −%&(

(7)

we obtain: ∆$ = !%& + ∆$ ∅

for a single gas product

(8)

and

4



∆$ = %[&φ + .!/ )& ] + ∆$ ∅

for multiple gas products

Page 4 of 35

(9)

For further considerations it is sufficient to use only Eq. (8), because all quantities on the right-hand sides of Eqs. (8) and (9) are known, and the standard Gibbs free energy for thermal dissociation processes can be adopted as a linear relationship, expressed as the average values of thermodynamic functions:28

8888 − ∆9 8888 ∆$ ∅ = ∆7

(10)

In this way, we combine (10) with (8) to obtain the final form, which is valid when α > 0:

8888 − ∆9 8888 + !%& ∆$ = ∆7

(11)

For α = 1, Eq. (11) is identical to (10). Using the rate of conversion α is more beneficial for analysis conducted with thermogravimetric methods, which is particularly troublesome for full identification or quantitative analysis of the gas phase. However, the fundamental difficulty stems from the fact that the phenomenological free energy of the system for isothermal-isobaric conditions, i.e., constant T and P, refers only to the progress of the gas phase reaction and not to the kinetics of the rate of conversion. The vast majority of papers on this topic consider changes in the rate of conversion with reaction conditions, usually observing a linear increase with temperature = :/: 0.99 depending on the heating rate q. The data used is presented in Table 1, and Table 2 presents the results of the analysis of Eq. (28).

9


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Table 1 a) Dynamic and b) isothermal decomposition of calcite under nitrogen - data from Prof. M. Brown (Chem. Dep. Rhodes Univ. Grahamstown, South Africa), converted to unify the value of α. More details about the data are provided elsewhere35a-e

a) α

q=1 K min-1

q=3 K min-1

q=5 K min-1

q=7.5 K min-1

q=10 K min-1

q=15 K min-1

q=25 K min-1

temperature, °C 0.05

621.80

651.99

668.99

683.49

695.98

710.48

731.47

0.1

638.00

672.99

690.99

706.99

718.99

735.99

755.99

0.2

660..00

696.00

716.49

732.49

745.99

763.99

784.99

0.3

674.69

712.50

732.99

749.69

762.99

782.48

803.99

0.4

685.50

723.80

744.80

762.49

775.69

795.76

817.00

0.5

694.00

733.00

755.00

773.00

785.98

805.99

827.98

0.6

700.99

740.49

763.50

780.98

794.29

815.00

836.97

0.7

707.49

747.00

771.20

788.20

801.98

822.70

844.98

0.8

713.00

753.49

777.50

794.39

808.99

829.69

852.00

0.9

718.30

759.52

783.50

800.81

815.70

836.00

859.01

t=700°C

t=710°C

t=719°C

t=732°C

t=740°C

t=750°C

t=770°C

b) α time, min 0.05

2.55

2.12

1.73

1.45

1.25

1.00

0.66

0.1

5.11

4.18

3.42

2.70

2.42

1.97

1.27

0.2

9.92

8.07

6.75

5.38

4.52

3.59

2.45

0.3

14.95

11.90

10.02

7.80

6.68

5.41

3.56

0.4

19.87

15.82

13.35

10.35

8.83

7.18

4.68

0.5

24.75

20.03

16.77

12.73

10.92

8.93

5.76

0.6

29.90

24.26

20.22

15.18

13.10

10.76

6.83

0.7

35.32

28.47

23.85

17.75

15.32

12.68

7.92

0.8

41.03

33.09

27.65

20.48

17.71

14.66

9.07

0.9

46.82

37.83

31.41

23.32

20.19

16.78

10.29

It is surprising that the temperature, present in two terms in Eq. (11), remains apparently unaffected, which basically means that

∆@

= 0. 10



The derivative of Eq. (28) should be properly treated as ∆@

=a

∆@

Page 10 of 35

∆@

= e f ∗ e f:

O

(29)

Table 2 Correlations of Eq. (28) ∆$ = ah + a& , i molZX , 0.05 ≤ ≤ 0.9

Heating rate, q, K min-1 1.0 3.0 5.0 7.5 10.0 15.0 25.0

Bo, J mol-1 25873.4 19839.3 16301.4 13713.6 11649.6 8517.3 5255.0

B, J mol-1 2484.6 2257.4 2023.5 2012.4 2009.1 1829.7 1889.3

Table 3 shows expressions for the derivatives of

∆@

r2 0.9889 0.9941 0.9906 0.9947 0.9919 0.9970 0.9927

for the considered cases.

11


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8888 − ∆9 8888 + !%& Table 3 List of equations for derivatives :Δ$ ⁄: 11) ∆$ = ∆7 No.

Conditions

1a

1b

nop⁄nq =

Eq

8888 + !%& + !% = −∆9

(11)

General

again (11) finally (25)

isothermal T = const

3a

3b

dynamic q = const

3c 4

:& = :

8888 Δ$ − Δ7 :& = + !% :

!%& =

888 ∆$ − 8∆7 8888 + ∆9,

∆$ = 0, 21)

1c

2

remarks

(11)

≡0

(25) with (23), (24) as (26)

/YF = !%TW − 1) or /YF = !%TW & or :& =a ≈ 0, a > 0 :

(A.4) (29)

isoconversion

from (18)

8888 + !%N, = −Δ9 N = & =

-

if

from Eq. (23) U = 0, s s t = S

O

UV

see Appendix

-

12



6. Graphical analysis and variability of thermodynamic functions The study results presented in Table 1 were used to present linear relationships (isothermal conditions (11)) and linearized forms (dynamic (28) and isoconversional conditions (18)). The basic relationship (10) for calcite decomposition was adopted according to:28 ∆$ ∅ = 170 − 0,145) ∗ 10x , J molZX

(30)

888 = 170.0 ∗ 10x J molZX and ∆9 8888 = 145 J molZX K ZX, are The adopted mean values, 8∆7 within the range of acceptable changes (Figs. 2a-c).

35 30

T = 973K (28)

25

q=1

20

(11)

T = 1043K

15

ΔG, kJ mol-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 35

(18)

(18)

10

(11)

5

(28) Teq = Tin = = 1172.4 K

q = 25

0 -5

0.1

1.0

-10

irreversible reaction

-15 -20

lnα

Fig. 1 Graphic illustration of free energy changes in relation to the isothermal (11), dynamic (28) and isoconversion (18) conditions. Only the range of 0.2 < α < 0.8 was deliberately marked (q in K min-1)

In addition, a straight line fit to the equation is presented for this system, when

888 = 170⁄0.145 = 1172.4 (. Above this temperature 8888⁄8∆9 ∆$ ∅ = 0, i.e., + = / = ∆7 the thermal dissociation reaction is irreversible. Fig. 1 shows that the functional scale system (11) is suitable for predicting and analyzing the results of all considered cases. 13 ACS Paragon Plus Environment

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Isothermal conditions indicate a higher angular coefficient than that in the case of dynamic methods, and the isoconversional conditions resulting from dynamic methods indicate a completely different orientation of the thermodynamic path of the reaction, which is consistent with the increase in temperature. Therefore, it is also worth checking the variation of the free energy of the system in relation to the temperature and other conditions. The results are presented in the Table 3. They show that isothermal and dynamic methods are comparable in terms of temperature changes, whereas isoconversional methods represent completely different thermodynamic principles that are impossible to study experimentally. We observe the start of the reaction and the formation of a gas product in the stoichiometric amount, whereby these amounts do not change with increasing temperature, and the reaction does not end. The increase in dissociative pressure is the result of the action of gas laws, with a constant degree of conversion, which is thermodynamically impossible. It seems that the isoconversional methods broaden the knowledge in terms of thermokinetic considerations, which manifests itself as the KCE levels discussed elsewhere.12

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a)

180

160

178

158 156 154

Δ7

174

152

172

150 Enthalpy

148

170 Average enthalpy 168

146

Entropy

166

144

Δ9

Average entropy

142

164 0

200

400

600

ΔS∅, J mol-1 K-1

ΔH∅, kJ mol-1

176

800

1000

140 1400

1200

Temperature, K b)

c)

200 180

ΔH∅, kJ mol-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 35

160

∆$ ∅ > 0

140

Teq = 1172.4 K

120 100

∆$ ∅ = 0

80 60

∆$ ∅ < 0

40 20 0 0

50 `

100

CaCO3 CaO + CO2

150

TΔS∅,

200

kJ

mol-1


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Fig. 2 The dependence of standard thermodynamic functions on the temperature. The mean values adopted are marked and the thermodynamic functions were determined on the basis of Barin tables:32 a) equivalent

relations

of

Eq.

8888 = 171.2 ∗ 10 J mol , ∆7 x

b) control

(10),

ZX 11, pp 210, Eq. (4.77)

diagram

for

for

data

(30)

-

other

values:

determined value experimentally,

EEC,

∆7∅ = 696∆9 ∅ + 68.5; /YF = 696 (,

W = 0.9697) – the correlation indicates enthalpy character with a constant term that is too large in relation to ∆7 ∅, c) control

diagram

for

∆$ ∅ = 0,

enthalpy

domination

∆7 ∅ > 0, ∆9 ∅ > 0

to

+ = 1172.4 K, and entropy domination for + > 1172.4 K

After differentiation with respect to the temperature (18) (also Table 3, no. 4) we can assume that, according to the definition of entropy for isobaric conditions: Z ∆@

= ∆9

(31)

where X

8888 + !%& ) ∆9 = ∆9

(32)

For the total conversion, α = 1, we obtain the lowest value of the entropy. Eq. (31) indicates that, for a constant degree of conversion, a constant decrease in Gibbs free energy with temperature increase is equal to the entropy change, which is slight with respect to the degree of conversion. According to (32), for α > 0, its average value

8888 is acceptable. J9 ≈ J9 In this sense the isoconversional methods should be included with isoentropic methods. On the other hand, for isothermal and dynamic conditions, its growth (Fig. 1 and Table 3, no. 2 and 3) is seemingly independent of temperature and depends on the degree of conversion.



Page 16 of 35

7. Proposal for quantification By dividing the considered methods into isothermal and dynamic ones in relation to isoconversional methods, the presented considerations include derivatives with a completely different meaning in thermodynamic terms. The Gibbs free energy in terms of (11) informs the course of the reaction in relation to the initial state, but this state also needs to be determined for energetic reasons. Because &0 = −∞ is an indefinite value, the initial state is taken as a constant term from the tangent equation to the F coordinate, e.g., F = 0.05. The purpose of the quantitative comparison is to calculate the off-distance, understood as the "end - minus - start" in Eq. (11). The tangent equation for the coordinate [∆$ F ), F ] is as follows (Fig. 3): ∆$F = e

1H \

f ∗ + ∆$ F ) − !%

(33)

That is, the intercept is: [∆$ F ) − !%].

Fig. 3 Geometric interpretation of the off-distance D and the initial condition


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For dynamic conditions, we obtain = ∆$ ∅ − [∆$ h ) − %/ ]: 8888 + !%/ 1 − & F ) D = / − t ∆9

(34)

or 8888 + 4!%/ D ≅ / − t ∆9

F = 0.05

(35)

where for isothermal conditions the initial temperature (/ ) and final temperature (t ) are equal and Eqs. (34) and (35) are converted into the following form: = !% 1 − & F ) ≅ 4!%

F = 0.05

(36)

For isoconversional conditions, we cannot determine the end, and even precisely the beginning, of the process. Therefore, we obtain from Eq. (11) two extreme measurements, for dynamic conditions – the heating rates, and for isothermal conditions – the lowest and highest temperature: 8888 − !%N) D = / − t ∆9

ℎ N = & /

(37)

and for N = & / = &1 = 0, Eq. (37) is reduced to the form: 8888 D = / − t ∆9

(38)

In Eqs. (34) to (38) a formal notation D =∆∆G is used.48, 49 Comparing Eqs. (35) to (38), it can be concluded that the off-distance (35) of the classical dynamic process, in the sense of Coats-Redfern,1 is the sum of two components, the off-distance (38) of the dynamic isoconversional process for a finite time (α ≡ 1) - Eq. (38), and the off-distance of the isothermal process (36). In general, the process described by (37) is therefore impoverished by the compensating, positive isothermal component (36). 18 ACS Paragon Plus Environment


Page 18 of 35

From the scope of the tests performed, it follows that / pertains to the lowest heating rate (dynamics) or temperature rate (isotherm), and t similarly pertains to the highest heating or temperature rate. However, this is a matter of convention, because if we change the order of the adopted temperatures, the signs of the numerical values will change. Table 4 shows the off-distance values for data from Table 1. From the thermodynamic point of view, isothermal and isobaric conditions are clearly defined and thus the isoconversional methods are not meaningful (hence the sign ± in Table 4), in contrast to isoconversional methods under dynamic conditions. However, here the direction of change in the temperature scale is unambiguously determined, assuming a linear increase in temperature (heating rate q > 0). In contrast to the equations for isothermal and dynamic conditions, (34) and (35), respectively, the isoconversional methods retain their separateness, and from the thermodynamic point of view these methods have a virtual character. In turn, Eq. (38) shows that the off-distance is always negative, D < 0. Table 4. List of the off-distance values D for data from Table 1 and for the variants considered


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Table 4 Value of off-distance, D Conditions

Eqs.

Ti*

Tf* or T in K

Isothermal for theoretical case ∆$ = 0 ⋁ ∆$ ∅ = isothermal

Dynamic

(16)

off-distance D, kJ mol-1

remarks

always 0

973.15

973.15

32.36

700°C

1043.15

1043.15

34.69

770°C

874.84

998.55

11.16

q = 1 K min-1, C = 0, α = 1

970.32

1134.68

8.44

q = 25 K min-1, C = 0, α = 1

(37)

967.15

1101.13

-20.20

C = -0.693, α = 0.5

(38)

998.55

1134.68

-19.74

C = 0, α = 1

±10.55

700°C, C = -0.693, α = 0.5 770°C, C = -0.693, α = 0.5

(36)

(35)

dynamic isoconversion

isothermal isoconversion

(37)

973.15 1043.15

Ti*, Tf* - extrapolation

20



Page 20 of 35

In the meaning of Eq. (11), isothermal methods and, surprisingly, dynamic methods occur in one system group although with different energy activities, whereas isoconversional variants of dynamic methods represent opposite effects in this scope. The thermodynamic character of isothermal and dynamic methods is of enthalpy (Fig. 2c) and that of the isoconversional method is of isoentropy, Eqs. (31, 32). The Gibbs free energy of the system is the maximum useful work with the exception of the volume work associated with changes in the volume of the system when the process occurs reversibly at a constant temperature and pressure. ∆G is the maximum amount of energy that can be “freed” from the system to perform useful work. From this point of view, only isothermal conditions satisfy this condition and dynamic conditions do so only under the approximation (28). In turn, the isoconversional in the dynamic conditions indicate the complete inhibition of the chemical reaction, such that the amount of moles formed is also constant at a constant dissociation degree. In other words, from a thermodynamic point of view they are a waste, yet they are interesting for kinetic reasons.

8. Discussion I.

Thermodynamics for the solid phase

The essence of the proposed methodology is to accept the basic Eq. (8), which, when written out, obtains (11). In phenomenological thermodynamics, in the progression of the reaction, the argument of the logarithm refers to the dissociative pressure, also the equilibrium pressure, in vapor pressure. If Eq. (3) is accepted, then for Eqs. (4) and (21)28, 31 the mathematical analogue must be expressed in terms of the degree of conversion α. Analyzing Eq. (13), the linkage in the stoichiometric equation also follows de Donder's numbers (12) and further acc (14). In addition, the introduction of a derivative for the three-parameter equation according to (23) can be reduced to the relation ∆G = 0 (or ≅ 0), which confirms the validity of the accepted theses.


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For a single thermal dissociation reaction, the entire constitution of both phases is determined, regardless of λ, and the condition ∆$ = 0 is valid only for the equilibrium dissociative pressure.33 Consequently,

for

the

it is consistently simplex

A

A∅

emerging

monogas,

when

C+ /C∅ = + ,31

= when C < C∅ .

The assumption of Eq. (11) for isothermal conditions results from the fact that the measurements in thermobalance conditions are quasi-isothermal. It is sufficient to compare the obtained scale variability ∈ [0.05 − 0.9], ] from Table 1b, and the equilibrium

conversion

rate

for

the

temperature

range

700-770°C

is:

+ = 0.028-0.115. These values were calculated from Eq. (30) in the form:18 & + = 17.44 −

Wh.

.

The calculations indicate a very common inequality: > + .

II.

Equilibrium conversion degree

Analyzing the Holba-Šesták13 equation and further aspects of this problem,14 the problem of exceeding the thermodynamic efficiency (η > 1), or off-distance from equilibrium, appears in the experimental conditions of thermogravimetric analysis. Experimentally, this means that the determined conversion rate exceeds the equilibrium rate for the given thermodynamic conditions. This problem also appears in this paper. Apart from the errors resulting from the method for determining the reaction temperature,34 the main reason is the removal of gas products of the reaction, which disturbs the correctness of the assessment of the phenomenon in strict thermodynamic principles. For these reasons, the considerations in terms of recognizing thermodynamic potentials are more favorable as a reasonable point of view.

III.

Argumentation in thermodynamics

One can obtain the impression that the considerations were narrowed down to Eq. (11) and the example concerns the most frequently discussed calcium carbonate.35ae-42



Page 22 of 35

Overall, in thermodynamics, the evidence is accepted even for Bayesian statistics, and it is considered in other branches of the processes of this category. Note that the EEC (exemplified in Fig. 2b) is demonstrated on a very intricate path of mathematical conversions for only one coordinate.43 The deciding role of the EEC effect is played by the Gibbs free energy,44 which can be calculated using Eq. (11). Upon extending these problems to kinetic aspects, it is worth citing the theory of congruent dissociative vaporization acc. L'vov.45,

46

It is proven in thermodynamic

terms, and the characteristic feature is that thermodynamic function values are much higher than classical values.

IV.

Connection of thermodynamic considerations with the dynamic method for the isoconversional variant

According to the graphics mode presented in Fig. 1, in relation to Eq. (38), the isoconversional methods in dynamic conditions are from the thermodynamic point of view impossible for practical implementation. However, their significance results from the fact that for thermal dissociation processes, in elementary cases, important information about the relation between the activation energy and the rate of conversion is obtained, which is achieved by assuming its consistency with increasing temperature. In elementary cases, the energy of activation decreases indirectly with increasing temperature, a classic example being the dehydration of calcium oxalate monohydrate.10, pp 219 For

an

increase

in

the

activation

energy

with

temperature,

there

is

a premise that changes in the solid phase are occurring and the process of pyrolysis may exemplify this. This is particularly evident in the case of plastics pyrolysis.10, pp 98 However, such a division should be approached carefully, because in the cases that are considered elementary, there may be intense physical processes related to diffusion, agglomeration, melting, sublimation, etc.


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V.

The consequences resulting from considerations of coherence in the interpretation

of

thermogravimetric

data

from

the

kinetic

and

thermodynamic point of view In thermogravimetric analysis, isoconversional methods for dynamic conditions, in Vyazovkin's approach, unambiguously solve the problem of searching for a kinetic triplet (1). However, it is already simple to present the results of the same research from a classical approach (q = const, α = var) and isoconversion (q = var, α = const) to order on to the separateness of phenomena as ended processes (Fig. 4) with possible common (or co-ordinates) values.

Fig. 4 KCE, for the same data in Table 1, of thermal decomposition of calcite (T. Radko, Institute of Chemical Processing of Coal, private information, 2018, unpublished)

As a consequence, the variation of the activation energy with increasing temperature47 forced the extended knowledge from a range of chemical kinetics, determining the thermal durability of condensed phases, to be accepted. All isoconversional methods, for the assumed direction of temperature increase, are thermodynamically completely different processes from the classical method in a kinetic sense, and impossible to implement in practice, as indicated by the difference in signs for linear values of the off-distance.



Page 24 of 35

9. Conclusions I.

Eq. (11) was suggested for the thermodynamic principles, assuming the Gibbs free energy of the system for the conditions most commonly used in thermogravimetric analysis, i.e., isothermal and dynamic analysis and their isoconversional variants. They are readily used to determine the activation energy for dynamic conditions. The functional scale system (17) is a linear relation for isothermal conditions and an acceptable approximation for dynamic conditions according to Eq. (28).

II.

In the meaning of Eq. (11), isothermal methods and, surprisingly, dynamic methods, occur in one system group although with different energetic activities, whereas isoconversional in dynamic conditions represent opposite effects in this respect.

III.

The isoconversional variant for dynamic conditions indicates a stable number of moles of gaseous products formed at a constant degree of dissociation. These are the features of stopping the process of the chemical reaction. In other words, they are waste from the thermodynamic point of view but interesting for kinetic reasons because they extend the scope of kinetic interpretations by determining the variation of the activation energy of a reaction with the temperature.

IV.

According to the presented considerations, the proposed value off-distance, Eqs. (35 - 38), individualizes each of the analyzed cases: dynamic isoconversion, classical dynamic and isothermal isoconversion in terms of thermal dissociation. This view shows common elements on a scale, considering the direction of changes: positive effects are obtained for classical methods, and negative effects are obtained for the isoconversional method.

V.

According to the presented considerations, the isothermal and dynamic methods are dominated by enthalpy, and the isoconversional method is dominated by entropy.


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Nomenclature A

pre-exponential factor, s-1,

TF , TX , TW

coefficients of the three-parametric equation, Eq. (22),

B

slope, Eq. (28),

B0

intercept, Eq. (28),

C

constant in the Eq. (21),

D

off-distance, J mol-1,

E

activation energy, J mol-1.

EEC and KCE

Enthalpy-Entropy Compensation and Kinetics Compensation Effect,

) )

kinetics functions,

Δ$, Δ7, Δ9

thermodynamic functions, J mol-1 or J mol-1 K-1,

∆$h

the tangent line the Eq. (33) for the assumed coordinate on the function (17), J mol-1,

, , )

temperature integral in form,10

K

equilibrium constant,

n

number of moles, mole,

P

pressure or dissociative pressure, Pa,

q

heating rate, K min-1,

R

universal gas constant, R = 8.314 J mol-1 K-1,

r2

coefficient of linear determination, 0 ≤ r2 ≤ 1,

T

temperature, K,

t

temperature, °C,

α

conversion degree, 0 ≤ α ≤ 1,

λ

extent of reaction, mole,

!

stoichiometric ratio,

Following the development of the Coats-Redfern method1, in practice

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