Sigmoid yield-time curves

The relative ease of nucleation exerts an important influence on the kinetics of the overall reaction. When the nucleation step occurs infrequently, sometimes ascribed to a large activation energy, a sigmoid yield-time curve is observed. In contrast, if nucleation is rapid, all surfaces of the reactant are rapidly converted to product early in the reaction and the interface so formed advances inwards. Yield-time curves are then predominantly or entirely deceleratory. 

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. 

The behaviour [821] of AgN02 is closer to that expected of a nitro compound than a nitrite. Decomposition (308—363 K) yields Ag metal and N02. a—Time curves are sigmoid with a prominent linear region (0.15 < a < 0.45) but the Arrhenius plot was curved at >333 K. This was attributed to inadequate gaseous product removal. In contrast to the behaviour observed for most other solids, pre-irradiation with 7-rays inhibits subsequent thermal decomposition [829]. 

Kadlec and Rosmusova [1153] believe that both Ni and Co oxalates initially yield product oxide and that the proportion of metal increases with a. Since nickel oxalate decomposes at temperatures 60 K lower than those for CoC204, even a small proportion of Ni2+ markedly increases the rate of decomposition of cobalt oxalate. The effect was attributed to the catalytic properties of the preferentially formed Ni metal. The a—time curves were generally sigmoid and showed only slight deviations in shape with changes in the Ni Co ratio. In the decomposition of a mechanical... 

Figure 4. The typical sigmoid form of curves of yield of zeolite plotted against time. The illustration is for ZSM-5 (Silicalite 1). 150°C 160°C 170°C 180°C (Reproduced with permission from Ref. 11. Copyright 1984 Butterworths.)...

Several hydrated transition metal nitrates (Co, Cu, Cr, Zn and Ni) undergo aqueous fusion [62] prior to hydrolysis which yields basic salts. Vratny and Gugliotta [63] reported sigmoid -time curves for the decomposition of Pb(N03)2 between 523 and 708 K, and concluded that the cation was not further oxidized. Margulis et al. [64] identified both dissociation and autoxidation steps during the decomposition of lead nitrate. The first step involved melting of the eutectic formed from the reactant and the initial product, 2Pb0.Pb(N03)2. 

Iron(n) oxalate dihydrate loses water above 420 K to yield the anhydrous salt [53] in an inert atmosphere, but in the presence of oxygen FejOj may be formed [54]. At about 590 K the anhydrous oxalate decomposed slowly to yield magnetite and iron but no wustite was detected [53]. Kadlec and Danes [55] showed that the ar-time curves are sigmoidal and Zr, = 167 kJ mol. During decomposition of the mixed (Fe-Mg) oxalates, magnesium may stabilize the wustite product against either oxidation or reduction [56]. 

Broadbent et al. [69] showed that ar-time curves for the decomposition of copper(II) oxalate (503 to 533 K) were sigmoidal and that data for the vacuum reaction fitted the Avrami-Erofeev equation with values of = 2.9 initially and later n = 3.5 ( , = 136 kJ mol ). Electron transfer was identified as the step controlling the reaction. There was no evidence from X-ray diffraction studies for the intervention of the Cu salt the orthorhombic structure was present until disappearance of the reactant and product copper metal was detected. However, many metal carboxylates, chilled after dehydration, yield anhydrous salts that are amorphous to X-rays or poorly crystalline, see, for example [70]. 

Isothermal (593 to 693 K) ar-time curves for the decomposition of lanthanum oxalate [81] were sigmoid but asymmetrical with the decay period being the more pronounced. The Prout-Tompkins equation applied in two linear regions with , = 132 kJ mol. A branching chain mechanism was proposed during which channels fi om the surface penetrate the crystals as reaction proceeds. The predominant initial product was carbon monoxide, which disproportionated to yield carbon dioxide, and the residual solid contained carbonate and finely divided carbon. 

Decomposition of lithium oxalate [84] (742 to 765 K) yielded the carbonate and was accompanied by an initial large increase in surface area, followed by extensive sintering. The nr-time curve was sigmoid with no induction period and = 223 kJ... 

The decomposition of mercury(II) oxalate [96] yields the metal, which is volatile under reaction conditions, mercury(II) oxide and both CO and COj. nr-time curves were sigmoid (453 to 478 K). The reaction initially spread rapidly over the surfaces of crystalhtes (ii, = 107 kJ mol ) from a number of reaction centres. Subsequently the rate was controlled by the inward advance of a reaction interface (f, = 155 kJ mol ). The reaction rate was not markedly influenced by the presence of gaseous or sohd products. The initial process is accelerated by radiation, which is believed to yield the mercury(I) salt following electron transfer. 

Time dependence of the reaction products can be seen more clearly in the time-yield curves of oligomerization in methylene chloride at —40° (Fig. 4). The yield of mixture of the cyclic tetramer and hexamer (mostly the latter), passed through a maximum value of about 40% and then decreased to nearly zero after 48 hours. On the other hand, the yield of the cyclic dimer increased rather sigmoidly with reaction time. 

Provided that the time-temperature curve obtained from the calorimetric experiments is wholly of first-order, or comprises a first-order section, usually after the inflection point of sigmoid reaction curves, a conventional analysis yields a first-order rate constant ku which is related to the concentration of monomer, m, and the initial concentration of initiator, c0, by the equations... 

Also, as the total area of growing crystals increases so does the rate at which chemical nutrients are consumed. The slope of the curve of yield against time therefore increases. Later, as the supply of nutrients becomes more and more exhausted, the slope of the curve decreases to zero. As a result curves of yield vs. time are sigmoid in contour, as illustrated in Figure 4 (11). 

We have previously pointed out that, under the appropriate conditions, the sigmoidally shaped fluorescence induction curves should also be observed when the PS II reaction centers are partially closed by short, pulsed light flashes and when the fluorescence yield is measured with a weak probe light flash delivered at some time 6t (30 - 100 ps) after the variable - intensity pump flash (3). This follows from the assumption that under either steady-state or flash-excitation conditions, the fraction of closed reaction centers q should depend simply on the number of photons absorbed by PS II In both cases. However, using pump flashes of less than 1 /is in duration, the fluorescence induction curves measured by the pump-probe technique have been shown to be exponential in shape [3.4]. Similar obsenrations have been made by Mauzerall and his co-workers [5.6] who concluded that the probability of escape of an exciton from a PS II unit with a closed reaction center to a unit with an open one. is less than 0.25 and that the apparent optical cross-section of PS II with open and closed traps is constant to within + 10 % [7]. The exponentiaiity of the pump-probe fluorescence Induction curves implies that the variable fluorescence Fy = (F[l ] - Fo)/(Fmax " Fq) is proportional to q under these conditions, where 1 represents the fiuence of the pump flash expressed in units of incident photons/cm. ... 

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