Isothermal solid-state reactions

Based on the standard equation for the analysis of nucleation and growth processes, the kinetics of isothermal solid-state reactions can be represented by the following equation [15] ... 

Equation (5) is the well-known Jander equation relating the fraction of reaction completed to time, where kj is the rate constant. In order to determine the rate constant for an isothermal solid-state reaction, the fraction of material reacted must be determined as a function of time, and then, according to the Jander model, a plot is made of [1 — (1 — versus time. 

Solid-state reactions have usually been studied either by isothermal or by non-isothermal methods, with few attempts to combine the advantages of these alternative and sometimes complementary approaches. For reasons stated in Chap. 3, the kinetic information obtained from isothermal studies appears to be more accurate and reliable, and these studies are emphasised in this review. Wherever appropriate, however, account is taken of non-isothermal studies as a valuable source of complementary information. 

However, if we set the furnace temperature just slightly greater than T2, we would obtain a reaction limited to that of A - B, and thus could identily the intermediate reaction product, B. This technique is called isothermal thermogravimetry. Thus, we can follow a solid state reaction by first surveying via d3mamic TGA. If there are any intermediate products, we can isolate each in turn, and after cooling (assmning each is stable at room temperature) cam identify it by x-ray analysis. Note that we can obtain an assay easily ... 

An XRPD system equipped with a heatable sample holder has been described, which permitted highly defined heating up to 250°C [55]. The system was used to study the phase transformation of phenan-threne, and the dehydration of caffeine hydrate. An analysis scheme was developed for the data that permitted one to extract activation parameters for these solid-state reactions from a single non-isothermal study run at a constant heating rate. 

Chapters 6 and 7 dealt with solid state reactions in which the product separates the reactants spatially. For binary (or quasi-binary) systems, reactive growth is the only mode possible for an isothermal heterogeneous solid state reaction if local equilibrium prevails and phase transitions are disregarded. In ternary (and higher) systems, another reactive growth mode can occur. This is the internal reaction mode. The reaction product does not form at the contacting surfaces of the two reactants as discussed in Chapters 6 and 7, but instead forms within the interior of one of the reactants or within a solvent crystal. 

The foregoing classification is not without ambiguity. For example, it is common practice to call the reaction A - B +C° (see Fig. 6-1) induced by decreasing the temperature a phase transformation. The similar (peritectoid) reaction C = a+fi (Fig. 12-2) induced by a temperature increase, however, is named a decomposition reaction. In addition, the isothermal reaction AO = A+j02, which occurs if the intensive variable fio2 is decreased so that AO decomposes, is called a metal oxide reduction. It is thus categorized as a genuine heterogeneous solid state reaction (the... 

The kinetics associated with the thermally induced phase transformations of phenanthrene and caffeine monohydrate were studied using hot-state quantitative XRPD.29 Using a single non-isothermal experiment conducted at a constant heating rate, it was possible to obtain the activation parameters for the solid-state reactions. In another study, quantitative XRPD was used to study the tetrahydrate to monohydrate transition of the sodium salt of 5-(4-oxo-phenoxy-4H-quinolizine-3-carboxamide)-tetra-zolate.30... 

The rate equations which have found most widespread application to solid state reactions are summarized in Table 3.3. Other functions can be found in the literature. The expressions are grouped according to the shape of the isothermal a-time curves as acceleratory, sigmoid or deceleratory. The deceleratory group is further subdivided according to the controlling factor assumed in the derivation, as geometrical, diffusion or reaction order. 

Boldyreva [26] has criticized the use of the NIK approach for the determination of kinetic parameters and reaction mechanisms in the absence of more direct studies, and states that in certain technological situations, e.g. processes carried out under non-isothermal conditions, the rapidity with which the information is obtained and the similarities between laboratory and process conditions "may compensate for the absence of a physical meaning". Maciejewski [27] has also provided critical discussions of the usefulness of kinetic data for solid state reactions and has warned of the dangers of regarding measured kinetic parameters as being characteristic of the compound being studied, without reference to the experimental conditions used. 

There is no doubt that isothermal microcalorimetry has the potential to be very useful in studies of chemical degradation. The success will be greatest for solution-state reactions for which the reacting concentration is known, but even these can be limited by physical artifacts by, for example, the slow diffusion of oxygen into the liquid from the head space. The use of this technique for solid-state reactions is certainly not impossible, but is an area where great care is needed. The technique may therefore be used readily in a preformulation environment for solution systems, but is perhaps better applied in a QC (quality control) role for well-characterized solid-state processes. 

No single crystals could be manufactured due to principal difficulties (the compound is formed by a peritectoid solid-state reaction which prevents a growth from the melt). The discrepancy between the bulk spontaneous moment [(0.4-0.45))iB/U] obtained for polycrystalline samples at 4.2 K, and the microscopic moment (0.8/xB) derived from neutron diffraction (Frings and Franse 1985a) point to a high uniaxial magnetic anisotropy. This may account for the lack of saturation of the magnetic isotherms in fields up to 35 T. 

Brown, M.E. Galway. A.K. Arrhenius parameters for solid-state reactions from isothermal rate-time curves. Anal. Chem. 1989. 61, 1136-1139. 

Since the traditional kinetic models of solid-state reactions are often based on a formal description of geometrically well defined bodies treated under strictly isothermal conditions, they are evidently not appropriate to describe the real process, which requires accoimt to be taken of irregularity of shape, polydispersity, shielding and overlapping, unequal mixing anisotropy and so on, for sample particles under reaction. One of the measures which has been taken to solve the problem is to introduce an accommodation function a a) [32]. The discrepancy between the idealized /(a) and the actual kinetic model function h a) can be expressed as... 

Li, C.R. Tang, T.B. (1999). A new method for analyzing non-isothermal thermoanalytical data from solid-state reactions, Thermochimica Acta, Vol. 325, pp. 43-46 ISSN 0040-6031... 

In principle, there are several methods for determining the kinetics of a solid-state reaction. It is possible to conduct either an isothermal experiment by keeping the thermal status of the sample and detecting the charge state or the phase composition... 

The kinetics of thermal degradation have generally been studied using isothermal and nonisothermal methods. In earlier literature, isothermal methods were mostly employed for the study of the kinetics of solid-state reactions. During the past three decades, however, nonisothermal methods, for example, the Doyle method [17, 18], Freeman and Carroll method [19], Coats and Redfem method [24], Ozawa method [20], Flynn and Wall method [21, 22], Friedman method [25], and Kissinger method [26], have received more attention. 

The loss of the guest molecules corresponds to endothermic processes with low enthalpic values ( AH(jec l3-30 KJ rool l). The rate constants for such processes were evaluated for each compound at several temperatures, by fitting isothermal TG curves to different kinetic physical mechanisms of solid state reactions (diffusion, nucle-ation, growth, nucleation-growth and homogeneous)The kinetic parameters (Ko, Ea) were calculated from an Arrenhius plot of the rate constants. The declathration physical mechanisms were assigned on the basis of agreement between these calculated kinetic parameter and those determined from non-isothermal TG curves by mean of Coats--Redfern method. 

The activation energy of a solid-state reaction can be determined, no matter the mechanism of degradation, by isothermal or dynamic methods. After determining the E, the... 

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