The Prout-Tompkins Equation

In Chapter 2, the effect of a product of a reaction functioning as a catalyst was examined. It was shown that this type of behavior resulted in a concentration versus time curve that is sigmoidal in shape. It appears that there is autocatalysis in the early stages of some reactions in the solid state, 

This equation is based on a homogeneous reaction where the product can catalyze the reaction of particles of the reactant. The derivation presented here for this equation follows closely that presented by Young (1966). 

If No is the number of nuclei present at the beginning of the reaction, the change in number of nuclei with time, dN/dt, can be expressed as 

For reactive sites that are linear nuclei, the fraction of the sample reacted will vary with the number of nuclei as 

In order to arrive at a final equation that expresses a as a function of time, it is necessary to obtain a relationship between the constants in Eq. (7.55). For a symmetrical sigmoidal curve, there will be an inflection point,at 0.5. At the inflection point, da/dt has its maximum value (the second derivative changes sign) because at that point the second derivative is equal to zero, and, therefore, at that point, fei = 2- Therefore, at the inflection point k2 = kia/ui. Substituting this result for k2 in Eq. (7.55) we find that 

---

This is known as the Prout—Tompkins equation and has found application to many systems, in addition to the thermal decomposition of potassium permanganate [465] with which it is often associated. The kinetic behaviour of silver permanganate was somewhat different and in a variation of... 

The kinetics and mechanism of vacuum decomposition of AgMn04 at 378—393 K [466] are believed to differ from the behaviour of KMn04 in that the effective chain branching coefficient diminishes with time and this leads (Chap 3, Sect. 3.2) to the modified form of the Prout—Tompkins equation... 

Bircumshaw and Edwards [1029] showed that the rate of nickel formate decomposition was sensitive to reactant disposition, being relatively greater for the spread reactant, a—Time curves were sigmoid and obeyed the Prout—Tompkins equation [eqn. (9)] with values of E for spread and aggregated powder samples of 95 and 110 kJ mole-1, respectively. These values are somewhat smaller than those subsequently found [375]. The decreased rate observed for packed reactant was ascribed to an inhibiting effect of water vapour which was most pronounced during the early stages. 

References to a number of other kinetic studies of the decomposition of Ni(HC02)2 have been given [375]. Erofe evet al. [1026] observed that doping altered the rate of reaction of this solid and, from conductivity data, concluded that the initial step involves electron transfer (HCOO- - HCOO +e-). Fox et al. [118], using particles of homogeneous size, showed that both the reaction rate and the shape of a time curves were sensitive to the mean particle diameter. However, since the reported measurements refer to reactions at different temperatures, it is at least possible that some part of the effects described could be temperature effects. Decomposition of nickel formate in oxygen [60] yielded NiO and C02 only the shapes of the a—time curves were comparable in some respects with those for reaction in vacuum and E = 160 15 kJ mole-1. Criado et al. [1031] used the Prout—Tompkins equation [eqn. (9)] in a non-isothermal kinetic analysis of nickel formate decomposition and obtained E = 100 4 kJ mole-1. 

The Prout—Tompkins equation [eqn. (9)] was obeyed (E = 169 kJ mole-1). The first step in the reaction was identified as... 

The rate equation [eqn. (26)], given above for the reaction of magnesium oxalate, is also obeyed [1012] by the decomposition of zinc oxalate (620—646 K), although here the catalytic (second) term is dominant, so that behaviour approximated to the Prout—Tompkins equation [eqn. (9)]. The value of E (201 8 kJ mole 1) was the same as that found... 

The a—time curves for the vacuum decomposition at 593—693 K of lanthanum oxalate [1098] are sigmoid. Following a short induction period (E = 164 kJ mole-1), the inflexion point occurred at a 0.15 and the Prout—Tompkins equation [eqn. (9)] was applied (E = 133 kJ mole-1). Young [29] has suggested, however, that a more appropriate analysis is that exponential behaviour [eqn. (8)] is followed by obedience to the contracting volume equation [eqn. (7), n = 3]. Similar kinetic characteristics were found [1098] for several other lanthanide oxalates and the sequence of relative stabilities established was Gd > Sm > Nd > La > Pr > Ce. The behaviour of europium(III) oxalate [1100] is exceptional in that Eu3+ is readily reduced... 

In an inert atmosphere, the decomposition at 573—623 K of uranyl(VI) oxalate [1101] obeys the Prout—Tompkins equation [eqn. (9)] with E = 261 4 kJ mole-1. The residual product is U02 and, under low pressure accumulatory conditions, the final CO2/CO ratio is 9. In air, the reaction proceeds in two stages. The initial process obeys the Prout—Tompkins equation and is identified as a surface reaction. Thereafter, decomposition fits the Avrami—Erofe ev equation [eqn. (6), n = 2], involving isolated disc-like grains of reactant, to yield amorphous U03 as the final product. Values of E for both stages of reaction are close to that found for reaction in the inert atmosphere ( 260 kJ mole-1) and comparable with theoretical predictions [88],... 

The predominant gaseous products of the decomposition [1108] of copper maleate at 443—613 K and copper fumarate at 443—653 K were C02 and ethylene. The very rapid temperature rise resulting from laser heating [1108] is thought to result in simultaneous decarboxylation to form acetylene via the intermediate —CH=CH—. Preliminary isothermal measurements [487] for both these solid reactants (and including also copper malonate) found the occurrence of an initial acceleratory process, ascribed to a nucleation and growth reaction. Thereafter, there was a discontinuous diminution in rate (a 0.4), ascribed to the deposition of carbon at the active surfaces of growing copper nuclei. Bassi and Kalsi [1282] report that the isothermal decomposition of copper(II) adipate at 483—503 K obeyed the Prout—Tompkins equation [eqn. (9)] with E = 191 kJ mole-1. Studies of the isothermal decompositions of the copper(II) salts of benzoic, salicylic and malonic acids are also cited in this article. 

Silver oxide An early (1905) study by Lewis [34] of the kinetics of decomposition of Ag20 was a notable contribution. The dissociation in oxygen (760 Torr, 593 to 623 K) showed a long induction period followed by a sigmoid nr-time curve which fitted the Prout-Tompkins equation with = 133 kJ mol. Benton and Drake [35] studied the kinetics of the reversible dissociation using a sample of finely-divided active metal. The rate of reaction at 433 K fitted the expression ... 

Rienaecker and Werner [36] suggested that Baj(Mn04)2 is the initial decomposition product, by analogy with results for KMn04- Isothermal ar-time curves for vacuum decomposition at 413 to 463 K were sigmoid [37] following an initial (or= 0.04) burst of gas. The acceleratory process was fitted by the Prout-Tompkins equation ( , = 151 kJ mol ) and the power law with = 2. Fragmentation into thin platelets was observed. The deceleratory period could be described by the first-order equation with , = 124 kJ mol. ... 

The kinetics of decomposition of AgMn04 in vacuum between 378 and 393 K [38] differ fi om the behaviour of KMn04 in that the acceleratory periods of the ar-time curves are described by the modified form of the Prout-Tompkins equation ... 

The first two reactions give sigmoid ur-time curves that are well expressed by the Prout-Tompkins equation. These are believed to involve a branching-chain mechanism. The third process is fitted by the contracting volume equation, possibly due to decomposition through ion-pair evaporation. The thermal stabihty of N02CJ04 is decreased [42] by the incorporation of impurities which increase the mnnber of cation vacancies, whereas the creation of anion vacancies has the opposite effect. 

The disproportionation of Re02(- Re207 and Re) was also deceleratory and the contracting volume equation applied from 0 < n < 0.5 with E, = 423 kJ mol and (somewhat surprisingly) the Prout-Tompkins equation was applied to the rather flat decay period (E, = 318 kJ mol ). 

The kinetics of decomposition of nickel formate [6,7] are sensitive both to the experimental conditions [8] and the reactant structure, ar-time curves for the isothermal decomposition (about 450 K) are usually [8], though not invariably [9], sigmoid and there is microscopic evidence [6] that reaction proceeds through nucleation and growth. The induction period [6] and the shape of the subsequent acceleratory process [8] are influenced by the rapidity with which product water vapour is removed from the vicinity of the reactant. Data fit the Prout-Tompkins equation with , about 100 kJ mol". ... 

The kinetics of the decomposition [17] of thorium tetraformate to ThOj can be described by the Prout-Tompkins equation with = 150 kJ mol" from 498 to 553 K. The autocatalytic process was ascribed to participation of the oxide in breakdown of the carboxyl groups at the reaction interface to yield ThOj, formaldehyde and carbon dioxide as the primary products of reaction. The volatile products could, however, react further on the surface of the active solid to yield a number of secondary products amongst which the following gases were identified Hj, CO, HjO, CHjOH, HCOOCHj, HCOOH and (CHj). Addition of nickel formate to the reactant not only accelerated decomposition but also influenced the composition of the gases evolved, yielding predominantly CO, COj and H2 (which are the main products of nickel formate decomposition). 

Though salt dehydration was not accompanied [27] by particle disintegration, the anhydrous pseudomorph was shown by X-ray diffiaction measurements to be very poorly crystallized (a characteristic feature of many nickel carboxylates). Decomposition in air (554 to 631 K) proceeded at a constant rate (0.1 < nr < 0.8 and = 96 kJ mol" ), ascribed to the operation of an autocatalytic mechanism. The reaction in vacuum (562 to 610 K) gave a sigmoid ar-time curve which was fitted by the Prout-Tompkins equation. Because the activation energy was the same as that for reaction in air, it was concluded that the same mechanism operated. The reaction in air yielded residual nickel oxide, while reaction in vacuum gave the carbide with excess carbon and some oxide. In addition to carbon dioxide, the volatile products of decomposition included water and acetic acid. 

The r-time curves for the decomposition of anhydrous cobalt oxalate (570 to 590 K) were [59] sigmoid, following an initial deceleratory process to a about 0.02. The kinetic behaviour was, however, influenced by the temperature of dehydration. For salt pretreated at 420 K, the exponential acceleratory process extended to flr= 0.5 and was followed by an approximately constant reaction rate to a = 0.92, the slope of which was almost independent of temperature. In contrast, the decomposition of salt previously dehydrated at 470 K was best described by the Prout-Tompkins equation (0.24 < a< 0.97) with 7 = 165 kJ mol . This difference in behaviour was attributed to differences in reactant texture. Decomposition of the highly porous material obtained from low temperature dehydration was believed to proceed outwards from internal pores, and inwards from external surfaces in a region of highly strained lattice. This geometry results in zero-order kinetic behaviour. Dehydration at 470 K, however, yielded non-porous material in which the strain had been relieved and the decomposition behaviour was broadly comparable with that of the nickel salt. Kadlec and Danes [55] also obtained sigmoid ar-time curves which fitted the Avrami-Erofeev equation with n = 2.4 and = 184 kJ mol" . The kinetic behaviour of cobalt oxalate [60] may be influenced by the disposition of the sample in the reaction vessel. 

Danforth and Dix [75] compared the behaviours of both zinc and magnesiimi (see below) oxalates. Kinetic measurements were made by independent determinations of the yields of carbon monoxide and of carbon dioxide at appropriate reaction times. The kinetics of reaction of the zinc salt were fitted by the Prout-Tompkins equation (620 to 645 K) and i , = 196 kJ mol. The activation step was identified as the formation of the radical ion 204, following the transfer of an electron to a cation. The decomposition of this activated species was then accelerated by the product zinc oxide. The term accelerated rather than catalysed was preferred because the magnitude of was not decreased by the presence of the oxide product. 

---