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Problems and faulty uses with the Prout–Tompkins description of autocatalytic reactions and the solutions

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Abstract

The investigation of materials containing higher energy, such as peroxides, highly oxidized salts, nitrated organic compounds, leads very quickly to the need for autocatalytic description of decomposition data. The autocatalytic behaviour shows generally a sigmoid change of the conversion. There are several types of autocatalytic equations—one is the Prout–Tompkins (PT) equation. The PT equation is a reaction kinetic model belonging to a series of models with a single reaction rate constant, which are favourably used in the field of reaction kinetic-based thermal analyses. The PT equation was developed and published in the 1940s by E.G. Prout and F.C. Tompkins, which tried to describe reaction kinetically their observations on the thermal decomposition of potassium permanganate. They recognized the autocatalytical behaviour of this decomposition and developed step by step the full autocatalytic description. But, in spite of using the full description with two reaction rate constants, they simplified the expression and kept only the autocatalytic part of the complete reaction scheme and of the rate equation by arguing that this part is much faster than the primary or intrinsic part and determines nearly totally the conversion of the substance. This can be right, depending on the system considered. But this simplification has resulted in a lot of application problems and also in problems to formulate the model in a correct way. In this paper, the problem is analysed and the viable solution will be given, which is called the improved PT model. Further, a lot of papers and applications can be found in the literature, which are a commingling of several concepts. The corresponding equations have an appearance as the PT equation, but in reaction kinetic terms, they are applied in a faulty way. This point is discussed also to dissolve the commingling and to foster the proper application of an autocatalytic description. For completion of the discussed equations and their usefulness in understanding the equations, the characteristic quantities ‘time to reach a certain degree of degradation’, ‘time to maximum rate’ and ‘conversion at maximum rate’ are presented. Finally, some other equations are discussed, which are able to describe autocatalytic behaviour in that they are transformed to the PT equation by choosing the appropriate reaction orders. However, not every sigmoid change in conversion is caused by autocatalysis, and in spite of this fact autocatalytic descriptions are used for such cases and one example of such a faulty use is discussed.

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Abbreviations

A :

Reactant A, its molar concentration

B :

Autocatalytic reaction product, its molar concentration

F 0 :

Factor considering the presence of B at time t = 0, F 0 = B(0)/A(0), F 0 is an initial kinetic condition

A r :

Normalized reactant concentration of A, A r(t) = A(t)/A(0)

A r0 :

Initial integration condition in integrating the rate equation dA r(t)/dt

α A :

Conversion of A, αA(t) = 1 − A r(t)

α A0 :

Initial integration condition in integrating the conversional rate equation dαA(t)/dt, α A0 = 1 − A r0

cPT:

Classical Prout–Tompkins equation

iPT:

Improved Prout–Tompkins equation

Ea1 :

Activation energy of the intrinsic decomposition of A

Ea2 :

Activation energy of the autocatalytic decomposition of A

NC:

Substance nitrocellulose (cellulose nitrate)

NCR :

Residual NC after its partial decomposition

NC :

Concentration of nitrocellulose; in the calculation of the nitrate ester group concentration ONO 2 is used

ONO 2 :

Concentration of the nitrate ester groups in the sample

S :

Concentration of stabilizer

Sr :

Normalized stabilizer concentration, Sr(t) = S(t)/S(0)

Sro:

Start value of Sr, normally set to 1, values smaller than one are possible

P :

Concentration of autocatalytically active product formed by decomposition of NC

erf(x):

Error function of x

K C :

Catalytic constant in the autocatalytic models C1-X and Cn-X; K C = Z 2/Z 1

k 1(T):

Reaction rate constant of intrinsic decomposition of A, in not normalized form, in 1/time

k 2(T):

Reaction rate constant of autocatalytic decomposition of A in not normalized form, unit in concentration/time

k A1(T):

Reaction rate constant of intrinsic decomposition of A, in normalized form, unit in 1/time

k A2(T):

Reaction rate constant of autocatalytic decomposition of A, in normalized form, unit in 1/time

k PT(T):

Effective reaction rate constant of autocatalytic decomposition of A, in normalized form, in the classical Prout–Tompkins expression, it deviates from k A2(T), unit in 1/time

k Acc(T):

Effective reaction rate constant of autocatalytic decomposition of A, in normalized form, in the crude simplified expression for autocatalysis using commingling concepts, it deviates from k A2(T), unit in 1/time

k AK(T):

Reaction rate constant of intrinsic decomposition of A in the simplified description of full autocatalysis in the models C1-X and Cn-X, unit in 1/time

k SB(T):

Reaction rate constant in Šesták–Berggren equation. This equation is the attempt to generalize the description of data with models using the concept of one-reaction rate constant. The Šesták–Berggren equation contains the classical Prout–Tompkins equation as one form, unit in 1/time

k Ng(T):

Reaction rate constant in Ng equation. This equation is the attempt to generalize the description of data with models using the concept of one-reaction rate constant. The Ng equation contains the classical Prout–Tompkins equation as one form, unit in 1/time

k AE(T):

Reaction rate constant in Avrami–Erofeev equation of crystallization. This is not an autocatalytic equation. k AE(T) deviates from k A2(T), unit in 1/time

k NC(T):

Reaction rate constant of decomposition of NC according to first-order reaction, unit in 1/time

k S(T):

Reaction rate constant of bimolecular stabilizer reaction with product P, unit in concentration/time

k S0(T):

Normalized reaction rate constant in bimolecular stabilizer reaction with product P, unit in 1/time

y A :

Degree of degradation of A, y A = A r(ty A) = 1 − α A(ty A)

ty A :

Time to reach the preset degree of degradation y A

w :

Autocatalytic coupling constant, in correct way it is a function of temperature

z :

Ratio Z 1/Z 2, used in reforming the general autocatalytic description, in correct way it is a function of temperature

Z 1 :

Pre-exponential factor of the intrinsic decomposition of A

Z 2 :

Pre-exponential factor of the autocatalytic decomposition of A

References

  1. Roginsky S. Über die Zersetzung von Sprengstoffen bei niedrigen Temperaturen (On the decomposition of high explosives at low temperatures). Physikalische Zeitschrift der Sowjetunion. 1932;1:640–99 See especially page 660, in German.

    Google Scholar 

  2. Manelis GB, Rubtsov YI, Smirnov LB, Dubovitskii FI. Kinetics of the Thermal Decomposition of Pyroxilin. Kinetika i Kataliz. 1962;3:42–8. Translation in: Kinetics and Catalysis 3, (1962) 32–37.

    CAS  Google Scholar 

  3. Eisenreich N, Pfeil A. Non-linear least-squares fit of non-isothermal thermoanalytical curves. Reinvestigation of the kinetics of the autocatalytic decomposition of nitrated cellulose. Thermochim Acta. 1983;61:13–21.

    Article  CAS  Google Scholar 

  4. Bohn MA. Kinetic description of mass loss data for the assessment of stability, compatibility and aging of energetic components and formulations exemplified with ε-CL20. Prop Explos Pyrotech. 2002;27:125–35.

    Article  CAS  Google Scholar 

  5. Opfermann J. Kinetic analysis using multivariate non-linear regression. I. Basic concepts. J Therm Anal Calorim. 2000;60:641–58.

    Article  CAS  Google Scholar 

  6. Hong-Kun Z, Cao T, Dao-Sen Zh, Wen-Lin X, Ya-Qong W, Qi-Shu Q. Study on the non-isothermal kinetics of decomposition of 4Na2SO4·2H2O2·NaCl. J Therm Anal Calorim. 2007;89:531–6.

    Article  CAS  Google Scholar 

  7. Prout EG, Tompkins FC. The thermal decomposition of potassium permanganate. Trans Faraday Soc. 1944;40:488–98.

    Article  CAS  Google Scholar 

  8. Prout EG, Tompkins FC. The thermal decomposition of silver permanganate. Trans Faraday Soc. 1946;42:468–72.

    Article  CAS  Google Scholar 

  9. Jacobs PWM, Tompkins FC. Classification and theory of solid reactions. Ch. 7. In: Garner WE, editor. Chemistry of the solid state. London: Butterworth Scientific Publication; 1955.

    Google Scholar 

  10. Šesták J, Berggren G. Study of the kinetics of the mechanism of solid state reactions at increasing temperatures. Thermochim Acta. 1971;3:1–12.

    Article  Google Scholar 

  11. Ng W-L. Thermal decomposition in the solid state. Australian J Chem. 1975;28:1169–78.

    Article  CAS  Google Scholar 

  12. Brown ME. The Prout–Tompkins rate equation in solid-state kinetics. Thermochim Acta. 1997;300:93–106.

    Article  CAS  Google Scholar 

  13. Brown ME, Glass BD. Pharmaceutical applications of the Prout–Tompkins rate equation. International J Pharmaceutics. 1999;190:129–37.

    Article  CAS  Google Scholar 

  14. Jacobs PWM. Formation and growth of nuclei and the growth of interfaces in the chemical decomposition of solids—new insights. J Phys Chem B. 1997;101:10086–93.

    Article  CAS  Google Scholar 

  15. Brown ME, Galway AK. Kinetic background to thermal analysis and calorimetry, Chapter 3. In: Brown ME, editor. Handbook of Thermal Analysis and Calorimetry’, Vol. 1, Principles and Practice. Amsterdam: Elsevier; 1998.

  16. Burnham AK, Braun RL, Coburn TT, Sandvik EI, Curry DJ, Schmidt BJ, Noble RA. An appropriate kinetic model for well-preserved algal kerogens. Energy Fuels. 1996;10:49–59.

    Article  CAS  Google Scholar 

  17. Avrami M. Kinetics of phase change. I general theory. J Chem Phys. 1939;7:1103–12.

    Article  CAS  Google Scholar 

  18. Avrami M. Kinetics of phase change. II transformation–time relations for random distribution of nuclei. J Chem Phys. 1940;8:212–24.

    Article  CAS  Google Scholar 

  19. Avrami M. Granulation, phase change, and microstructure kinetics of phase change III. J Chem Phys. 1941;9:177–83.

    Article  CAS  Google Scholar 

  20. Erofeev B.V.; Dokl Akad Nauk, SSSR 52, (1946) 511. (in Russian).

    CAS  Google Scholar 

  21. Bohn M.A.; Description of consumption of stabilizers in gun propellants showing pseudo-sigmoid decrease. Paper 24; pages 24-1 to 24-26 in printed proceedings and pages 24-1 to 24-40 (extended version) in CD proceedings of the 44th International Annual Conference of ICT on ‘Energetic Materials—Characterization and Modelling of Ignition Process, Reaction Behavior and Performance’, June 25 to 28, 2013, Karlsruhe, Germany. ISSN 0722-4087. Fraunhofer-Institut fuer Chemische Technologie (ICT), D-76318 Pfinztal. Germany, 2013.

  22. Bohn MA, Eisenreich N. Kinetic modelling of the stabilizer consumption and of the consecutive products of the stabilizer in a gun propellant. Prop Explos Pyrotech. 1997;22:125–36.

    Article  CAS  Google Scholar 

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Acknowledgements

My colleague at Fraunhofer ICT, Dr. Norbert Eisenreich is thanked for helpful discussions. Dr. Bertrand Roduit from AKTS AG, Switzerland is thanked for critical but clarifying statements.

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  1. Fraunhofer Institut fuer Chemische Technologie (ICT), 76318, Pfinztal, Germany

    Manfred A. Bohn

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Bohn, M.A. Problems and faulty uses with the Prout–Tompkins description of autocatalytic reactions and the solutions. J Therm Anal Calorim 116, 1061–1072 (2014). https://doi.org/10.1007/s10973-013-3509-1

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