Fractional decomposition

Measurements of overall reaction rates (of product formation or of reactant consumption) do not necessarily provide sufficient information to describe completely and unambiguously the kinetics of the constituent steps of a composite rate process. A nucleation and growth reaction, for example, is composed of the interlinked but distinct and different changes which lead to the initial generation and to the subsequent advance of the reaction interface. Quantitative kinetic analysis of yield—time data does not always lead to a unique reaction model but, in favourable systems, the rate parameters, considered with reference to quantitative microscopic measurements, can be identified with specific nucleation and growth steps. Microscopic examinations provide positive evidence for interpretation of shapes of fractional decomposition (a)—time curves. In reactions of solids, it is often convenient to consider separately the geometry of interface development and the chemical changes which occur within that zone of locally enhanced reactivity. 

Two alternative methods have been used in kinetic investigations of thermal decomposition and, indeed, other reactions of solids in one, yield—time measurements are made while the reactant is maintained at a constant (known) temperature [28] while, in the second, the sample is subjected to a controlled rising temperature [76]. Measurements using both techniques have been widely and variously exploited in the determination of kinetic characteristics and parameters. In the more traditional approach, isothermal studies, the maintenance of a precisely constant temperature throughout the reaction period represents an ideal which cannot be achieved in practice, since a finite time is required to heat the material to reaction temperature. Consequently, the initial segment of the a (fractional decomposition)—time plot cannot refer to isothermal conditions, though the effect of such deviation can be minimized by careful design of equipment. 

Changes in the composition of gaseous products as reaction proceeds may make definition of the fractional decomposition, a, difficult. For example, product CO and residual carbon may be capable of reducing a metallic oxide, particularly at high a and the catalytic properties of an accumulating solid product may result in promotion of secondary gas reactions. 

K did not produce tiny new paramagnetic species, despite FTIR observations confirming appearance of IR features attributable to adsorbed NjO (2234 and 1256 cm in Fig. 4a) upon contact with N2O at 300 K. Stepwise decreases in magnitude of those IR features were, however, observed in each of a sequence of FTIR spectra taken after separate NjO adsorptions at increasing adsorption temperatures (TJ up to 573 K (Fig. 4b-d). From these FTIR observations it could be inferred that increased T, for contact between N,0 and vacuum-outgassed CeOj resulted in increased fractional decomposition of the N O introduced. FTIR spectra did not show bands due to peroxide species after N,0 adsorption. 

For single reactions with uncomplicated kinetics and with availability of a truly representative sample, the DSC can be used with different scan speeds (temperature/time) to determine kinetic constants in the Arrhenius equation. This method, proposed by Ozawa [83] has been accepted by the ASTM Method E698. After determining kinetic constants by this method, it is desirable to check the constants by running an isothermal DSC aging test for a period of time followed by a DSC scan to see if the predicted fraction decomposition... 

Below is an integral that is quite difficult to do by hand. It is not found in standard tables in its given form although it may transform to a recognized case. It is especially difficult to do by hand unless one notices a trick that involves performing a partial fraction decomposition of the integrand with respect to LOG(X). However, KACSYMA handles it readily. 

Thus when n0 = 1, e can be identified with the fractional decomposition of the water vapor. 

Mean degradation temperature is the average of fractional decomposition temperatures over 0.1, weight fraction increments from 0.1-0.9, and heating rates 5°, 10°, 20°, 40°, and 80°C/min. 

D(GG) in dibenzyl may be derived from the activation energy for the pyrolysis of toluene to be 47 kcal. Horrex and Miles 23 2 attempted to measure the dissociation of dibenzyl by equilibrium methods, but were unsuccessful. They also report a study of the pyrolysis of dibenzy] at low partial pressures and using fractional decompositions. They obtained a first order rate constant... 

The plot gives the fractional decomposition of NOj fed versus the ratio of reactor volume V (in cm ) to the NOj feed rate, Ff,Oj Cg mol/h), at different feed concentrations of NOj (in parts per million by weight). 

Deafness and inherited hearing loss Decimal fraction Decomposition Deer... 

The conventional method of measuring the amount of decomposition which has taken place is by the dimensionless fractional decomposition, a, which ranges from 0.00 for the pure reactant, to 1.00 signifying complete decomposition. On the assumption that both solid and gaseous products maintain a constant composition, a at any time during the decomposition may be measured directly from ... 

The fractional decomposition, on the assumption that nuclei grow without interference, a, is obtained from a (f) = F(t)/Ff where Ff is the volume of product formed when decomposition is complete. 

As discussed above, these rate equations make no allowance for the restrictions on growth of the nuclei. It is necessary to relate the unrestricted fractional decomposition, nr, to the true value, nr. A general but complicated solution to the problem has been provided [1,8]. For the simpler case of three-dimensional growth of randomly-distributed nuclei on large crystals, Avrami [21] has shown that nr and... 

Jin J, Zheng X, Yan YJ (2008) Exact dynamics of dissipative electronic systems and quantum transport hierarchical equations of motion approach. J Chem Phys 128 234703 Mathews J, Walker R (1970) Mathematical methods of physics. Benjamin, New York Croy A, Saalmann U (2009) Partial fraction decomposition of the Fermi function. Phys Rev B 80 073102. doi 10.1103/PhysRevB.80.073102. http //link.aps.org/doi/10.1103/ PhysRevB.80.073102... 

Methods depending on the differences in basicity are of several types, such as fractional precipitation, fractional decomposition, etc. As an illustration of the method of fractional precipitation we may consider adding a precipitant like NH4OH to a solution of a mixture of salts. If only a small amount of ammonia is added, the precipitate will contain more of the less basic constituents than the mother liquid. If the precipitate is filtered out and more ammonia added to the filtrate, another precipitate may be obtained. By proceeding in this fashion, the mixture may be separated into any desired number of... 

I With samarium-europium samples, other cerium earths are extracted from the amalgam phase only after the bulk of it has decomposed. Fractional decomposition of the amalgam is thus desirable. 

A statement of the sample weight and weight scale for the ordinate. Weight loss should be plotted as a downward trend, and deviations from this practice should be clearly marked. Additional scales (such as fractional decomposition or molecular composition) may be used for the ordinate where desired. 

RAY/BHA] Ray, H. S., Bhat, B. G., Reddy, G. S., Biswas, A. K., Thermodynamic and kinetic studies in the systems alkali chloride-zirconium (or hafnium) tetrachloride. Part I - Vapour pressure measurements over hexachloro compounds and use of vapour pressure data in fractional decomposition, Trans. Indian Inst. Met., 31, (1978), 177-186. Cited on page 170. 

The X-ray diffraction studies of Krause failed to reveal the presence of crystalhne Tl. Krause suggested that diffraction effects from T1 were not observed because the metal is amorphous. However, even if the colloidal particles are crystalline, sufficiently smaU particle size results in X-ray diffraction lines broadened enough to be undetectable. Before concluding that the absence of Tl diffraction lines means that the Tl is amorphous, it is also necessary to know the fractional decomposition and so the expected intensities of the Tl lines. 

Decomposition and related T1 concentrations of the order of a few precent may be necessary to detect metallic T1 in the presence of TIN3 by X-ray diffraction. While Krause did not measure the fractional decomposition (e.g., by observing nitrogen evolution), the percent changes in lattice parameter are much smaller. It is highly desirable to determine the colloidal metal particle size by optical techniques and to couple these studies with X-ray diffraction and fractional decomposition studies to determine if the colloidal T1 is in amorphous or crystalline form. 

The average fractional decomposition at low (P/Z) for the nascent alkyl replacement products in CH8CHF2 exhibits the value 0.24 0.03. The corresponding result for the substitution products is 0.79 0.02. Similarly, the alkyl replacement species undergo complete collisional stabilization below 10 atm, whereas the average fractional stabilization of the activated substitution products at 190 atm is only 0.45 0.10. The excitation levels accompanying primary substitution Reactions 18-20 apparently are much larger than those for alkyl replacement Reactions 21 and 22. 

For CH3CF3 only 0.27 0.04 average fractions of the nascent alkyl replacement products are capable of decomposing, and these unimolecular processes are subject to complete collisional stabilization above approximately 12 atm. The corresponding average fractional decomposition and stabilization (P/Z) values for CH3CHF2 are 0.24 it 0.03 and approximately 10 atm. 

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