Isothermal evolution

The arrows show the isotherm evolution for continual addition of dissolved Me. The initial isotherm with the slope of 1 (in the double logaritmic plot) corresponds to a Langmuir isotherm (surface complex formation equilibrium). [Me]S0 = solubility concentration of Me for the stable metal oxide [Me]p = solubility concentration of Me for a metastable precursor (e.g., a hydrated Me oxide phase). 

Figure 13.29. Schematic sorption isotherms of a metal ion (Me) on an oxide (XO ) at constant pH (a) adsorption only (H) adsorption and surface precipitation via ideal solid solution (c) adsorption and heterogeneous nucleation in the absence of a free energy nucleation barrier (AG 0) adsorption and heterogeneous nucleation of a metastable precursor (e) same as in (3) but with transformation of the precursor into the stable phase. The arrows show the isotherm evolution for continual addition of dissolved Me. The initial isotherm with the slope of 1 (in the double logarithmic plot) corresponds to a Langmuir isotherm (surface complex formation equilibrium). [Me]s , = solubility concentration of Me for the stable metal oxide [Me]p = solubility concentration of Me for a metastable precursor (e.g., a hydrated Me oxide phase). (From Van Cappellen, 1991.)...

Fig. 2. Isothermal evolution at -8°C, as a function of time, of small-angle XRD peaks recorded just after quenching of cream from 50°C. Insert evolution of ak relative intensities vs. time (100% corresponds to the strongest peak) (+) 70.4 A, (A) 47 A, (O) 35.8 A, ( ) 23.7 A. 2L, double chainlength 3L, triple chainlength. See Figure 1 for other abbreviation.

The simplest volume recovery experiment performed is the down-jump. In this experiment, a material initially above Tg and at equilibrium is subjected to a temperature down-jump to an aging temperature Ta below Tg. The isothermal evolution of volume at T, as indicated by the downward arrow in Figure 1, is monitored with time via length or volume dilatometry. Figure 2 shows tjq)i-cal data replotted from Kovacs data (9) for a series of aging temperatures for poly(vinyl acetate). These curves, called intrinsic isotherms, are plotted as the relative departure from equilibrium 5 versus the logarithm of time, with S defined as... 

Figure 14 Schematics of the additivity principles used for a model liquid-solid transformation. (a) The thermal path followed hy the sample is cut into small isothermal plateaus of duration 5f, where the isothermal data of the TTT diagram can be applied. ti d is the induction time, estimated for the transformation using an additivity principle. Once find has elapsed, phase transformation begins. EstimaAon of its progress is presented in (b). (b) Calculation of the evolution offs-fs is known at time step i - 1. The fictitious time tf corresponding to this solid fracAon on the curveis calculated, where /,7 (t) is the isothermal evolution of the solid fraction at the temperature T, of the sample at timestep i. The increment of sohd fracAon at time step i is then given by fs.Ti(f + 0 -fs.T,(tn...

Figure 31 Isothermal evolution at T = 30 °C for poly(vinyl acetate) showing memory effect (1) quench from 40 °C to 30 °C (2) quench from 40 °C to 10 °C for 160 h followed by rapid heating to 30 °C (3) quench from 40 °C to 15 °C for 140 h followed by rapid heating to 30 °C and (4) quench from 40 C to 25 °C for 90 h followed by rapid heating to 30 °C (after ref. 13, with...

The theoretical description of a non-isothermal viscoelastic flow presents a conceptual difficulty. To give a brief explanation of this problem we note that in a non-isothennal flow field the evolution of stresses will be affected by the... 

The thermal efficiency of the process (QE) should be compared with a thermodynamically ideal Carnot cycle, which can be done by comparing the respective indicator diagrams. These show the variation of temperamre, volume and pressure in the combustion chamber during the operating cycle. In the Carnot cycle one mole of gas is subjected to alternate isothermal and adiabatic compression or expansion at two temperatures. By die first law of thermodynamics the isothermal work done on (compression) or by the gas (expansion) is accompanied by the absorption or evolution of heat (Figure 2.2). 

The techniques referred to above (Sects. 1—3) may be operated for a sample heated in a constant temperature environment or under conditions of programmed temperature change. Very similar equipment can often be used differences normally reside in the temperature control of the reactant cell. Non-isothermal measurements of mass loss are termed thermogravimetry (TG), absorption or evolution of heat is differential scanning calorimetry (DSC), and measurement of the temperature difference between the sample and an inert reference substance is termed differential thermal analysis (DTA). These techniques can be used singly [33,76,174] or in combination and may include provision for EGA. Applications of non-isothermal measurements have ranged from the rapid qualitative estimation of reaction temperature to the quantitative determination of kinetic parameters [175—177]. The evaluation of kinetic parameters from non-isothermal data is dealt with in detail in Chap. 3.6. 

Although there are experimental and interpretative limitations [189, 526] in the kinetic analysis of non-isothermal data, DTA or DSC observations are particularly useful in determining the temperature range of occurrence of one or perhaps a sequence of reactions and also of phase changes including melting. This experimental approach provides, in addition, a useful route to measurements of a in the study of reactions where there is no gas evolution or mass loss. The reliability of conclusions based on non-isothermal data can be increased by quantitatively determining the... 

The kinetics of the contributory rate processes could be described [995] by the contracting volume equation [eqn. (7), n = 3], sometimes preceded by an approximately linear region and values of E for isothermal reactions in air were 175, 133 and 143 kJ mole-1. It was concluded [995] that the rate-limiting step for decomposition in inert atmospheres is NH3 evolution while in oxidizing atmospheres it is the release of H20. A detailed discussion of the reaction mechanisms has been given [995]. Thermal analyses for the decomposition in air [991,996] revealed only the hexavanadate intermediate and values of E for the two steps detected were 180 and 163 kJ mole-1. 

The corresponding chromium compounds [Cr(en)3]X3 evolve ethylenediamine [1131] and the values of E determined using non-isothermal measurements were 105 and 182 kJ mole 1 for X = Cl" and SCN", respectively. Hughes [1132] reported a value of E = 175 kJ mole"1 for X = Cl" and showed that the decomposition rate is sensitive to sample disposition. Amine evolution from both the (en) and propenediamine (pn) compounds was catalyzed by NH4C1 [1132,1133] or NH CN [1133,1285], addition of small amounts of these substances resulting in a substantial reduction of E. The influence of NH4C1 is ascribed [1132] to the dissociation products, since HC1 promoted the reaction but NH r and NH4I showed no such effect. 

There is an extensive amount of data in the literature on the effect of many factors (e.g. temperature, monomer and surfactant concentration and types, ionic strength, reactor configuration) on the time evolution of quantities such as conversions, particle number and size, molecular weight, composition. In this section, EPM predictions are compared with the following limited but useful cross section of isothermal experimental data ... 

Since the pressure build up is primarily due to the evolution of CO as MDI is being decomposed to carbodiimide, the thermodynamic relationship PV = nRT may be applied to convert the pressure profiles to plots of moles of CO2 generated vs. time. This is shown for the 225 °C isotherm in Figure 3. The theoretical curve obtained through the application of zero-order kinetics is also shown in this plot and the data seem to be well accommodated by this rate law throughout the majority of the run. 

Figure 3. Carbon Dioxide Evolution from Isothermal Decomposition of MDI at 225 °C.

Table III. Summary of Zero-order Rate Constants for the Isothermal Decomposition of MDI in terms of CO2 Evolution...

A full development of the rate law for the bimolecular reaction of MDI to yield carbodiimide and CO indicates that the reaction should truly be 2nd-order in MDI. This would be observed experimentally under conditions in which MDI is at limiting concentrations. This is not the case for these experimements MDI is present in considerable excess (usually 5.5-6 g of MDI (4.7-5.1 ml) are used in an 8.8 ml vessel). So at least at the early stages of reaction, the carbon dioxide evolution would be expected to display pseudo-zero order kinetics. As the amount of MDI is depleted, then 2nd-order kinetics should be observed. In fact, the asymptotic portion of the 225 C Isotherm can be fitted to a 2nd-order rate law. This kinetic analysis is consistent with a more detailed mechanism for the decomposition, in which 2 molecules of MDI form a cyclic intermediate through a thermally allowed [2+2] cycloaddition, which is formed at steady state concentrations and may then decompose to carbodiimide and carbon dioxide. Isocyanates and other related compounds have been reported to participate in [2 + 2] and [4 + 2] cycloaddition reactions (8.91. 

Figure 6. Evolution of the lamella thickness as it transports down the periodic pore for the conjoining/disjoining pressure isotherm of Figure 4. Three capillary pressures are considered in curves 1 through 3. These capillary-pressure values are also labelled in Figure 4. Curve 2 defines the critical or marginally...

Fig. 11 Monomer distributions of 32-mers with Ef/Ec = 0.1 at Ec/k /T =0.174 vs. variable crystalline-stem lengths changing with time during isothermal crystallization at a specific temperature. The evolution time is denoted by the numbers (times 1000 Monte Carlo cycles) near the curves. The curves are shifted vertically with an interval of 300 for clarity. We can see that with time the peak shifts from one third to half of the chain length [56]...

Rate of Isothermal Heat Evolution of Lignocellulosic Sheet Materials in an Air Stream... 

The rate of isothermal heat evolution in lignocellulosic sheet material was studied at temperatures between 150 and 230°C using a labyrinth air flow calorimeter and commercial hardboards, medium density boards and laboratory hardboards of holocellulose, bleached kraft and groundwood, the latter with and without fire retardants. 

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