Order transformation

In another study Milehev and Landau [27] investigated in detail the transition from a disordered state of a polydisperse polymer melt to an ordered (liquid erystalline) state, whieh oeeurs in systems of GM when the ehains are eonsidered as semiflexible. It turns out that in two dimensions this order-disorder transition is a eontinuous seeond-order transformation whereas in 3d the simulational results show a diseontinuous first-order transformation. Comprehensive finite-size analysis [27] has established... 

Simple models are used to Identify the dominant fate or transport path of a material near the terrestrial-atmospheric Interface. The models are based on partitioning and fugacity concepts as well as first-order transformation kinetics and second-order transport kinetics. Along with a consideration of the chemical and biological transformations, this approach determines if the material is likely to volatilize rapidly, leach downward, or move up and down in the soil profile in response to precipitation and evapotranspiration. This determination can be useful for preliminary risk assessments or for choosing the appropriate more complete terrestrial and atmospheric models for a study of environmental fate. The models are illustrated using a set of pesticides with widely different behavior patterns. 

Concurrent first-order changes invariably arise from the decay of an excited state A which can undergo a number of first-order changes. In the presence of an excess of added B, pseudo first-order transformations can also occur ... 

We now distinguish solid state transformations as first-order transitions or lambda transitions. The latter class groups all high-order solid state transformations (second-, third-, and fourth-order transformations see Denbigh, 1971 for exhaustive treatment). We define first-order transitions as all solid state transformations that involve discontinuities in enthalpy, entropy, volume, heat capacity, compressibility, and thermal expansion at the transition point. These transitions require substantial modifications in atomic bonding. An example of first-order transition is the solid state transformation (see also figure 2.6)... 

Alpha-beta (a-jS) transitions of the condensed forms of Si02 quartz, try-dimite, and cristobalite may all be regarded as lambda transformations. Their kinetics are higher than those of quartz-trydimite, quartz-cristobahte, and quartz-coesite, which are first-order transformations. Figure 2.7 plots in detail the evolution of enthalpy, entropy, heat capacity, and volume at the transition zone... 

If kepi, k-epi are much larger than the rates k, ki of substitution, the enantiomeric ratio Hepi-1 is similar to kxjk (path C, dynamic kinetic resolution . Both mechanisms are performing when the rates of the two steps are similar. Since rates and equilibrium are temperature-dependent, enhancement of stereoselectivities can be achieved by sophisticated protocols (see Section m.E). The equilibrium 6/epi-6 is determined by the difference of free energy A AG. This effective energy difference is enlarged if it can be coupled with a second order transformation such as the selective crystallization of one diastere-omer dynamic thermodynamic resolution ). In fact, this applies to the first successful (—)-sparteine-mediated deprotonation (Section FV.C.l). 

In alloys, structural ordering transformations can proceed with either a continuous or discontinuous change in entropy at the transformation temperature, while in... 

The possibility of simulating the actual BWG ordering energy, rather than Cp, using a polynomial approximation was also examined by Inden (1976) using the disordered solid solution as a reference state. The following expression was suggested for a continuous second-order transformation such as A2/B2 ... 

Given that 5 can be described using Eq. (8.4), it may be asked whether a corresponding value for can be obtained by combining with Tg. If magnetic ordering was a first-order transformation with a critical temperature, T, it would then follow that (Fig. 8.3). 

However, since magnetic ordering is a second-order transformation, this has to be... 

Figure 8.3. Relation of an effective first-order transformation temperature (T ) to the teal second-order Curie temperature (T ) (from Miodownik 1977).

Continuous transition between first- and second-order transformations. When examined more closely, the Nishizawa Horn represents a situation where there is a continuous transition between a first- and second-order transformation. This remaikable situation is not restricted to systems which exhibit a miscibility gap (Inden 1981a) (Fig. 8.10), and it therefore remains to be seen whether it is possible to maintain a hard and fast distinction between these two types of transformation... 

Informations on the vibrational and electron mean free path properties. Such analysis is possible only if the interface phase is very well defined, and if temperature dependent measurements are done and compared. Debye Waller effects can be tangled with ordering transformation of the interface phase as a function of temperature and so on. If a single phase interface with order at least to the second nearest neighbour is recognised, then a temperature dependent Debye Waller, and mean free path analysis can be attempted. 

There are still other causes of nonlinearities than (apparent or real) higher-order transformation kinetics. In Section 12.3 we discussed catalyzed reactions, especially the enzyme kinetics of the Michaelis-Menten type (see Box 12.2). We may also be interested in the modeling of chemicals which are produced by a nonlinear autocatalytic reaction, that is, by a production rate function, p(Q, which depends on the product concentration, C,. Such a production rate can be combined with an elimination rate function, r(C,), which may be linear or nonlinear and include different processes such as flushing and chemical transformations. Then the model equation has the general form ... 

The system is shown in Fig. 21.7. It is described by two concentrations (state variables), CA and CB, by two zero-order input functions, JA and JB (input per volume and time), by two first-order output functions, kACA and kBCB (output per volume and time), and by the first-order transformations from A to B and vice versa. The inputs and outputs can be the sum of two or more processes, for instance, the sum of the input through different inlets and from the atmosphere (as in Eq. 21-7a), or the sum of the output at the outlet and by exchange to the atmosphere (as in Eq. 21-7b.). 

All internal transformations (chemical, photochemical, biological) in which the chemical is consumed, are described by first-order processes. The in situ production of the compound is described as a zero-order process, j. This includes the case in which the compound may be produced by (e.g., first-order) transformation of another compound, provided that the concentration of the latter is approximately constant. 

Caex81 116 Analcite. The most persuasive example of a structurally related discontinuity in dehydration behavior was found in the investigation of Caex 116 analcite. Figure 7 shows the TGA data in which the dehydration and rehydration curves describe a nearly closed hysteresis loop. The break at 425°C is correlated with a first-order transformation detected by x-ray at 453°C. The transformation is accompanied by a 10-20% reduction in cell volume as the last water is rejected. 

According to transition state theory, if the transmission coefficient k = 1, T and ET will be transformed to products at the same rate. Thus, if the mechanisms of the nonenzymatic and enzymatic reactions are assumed the same, the ratio of maximum velocities for first-order transformation of ES and S will be given by Eq. 9-85. For some enzymes the ratio... 

TWo limiting cases for the rate laws of second-order transformations can easily be distinguished. If q) is small and the higher order terms are neglected,... 

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