Separable kinetics

Another important characteristic of the late stages of phase separation kinetics, for asynnnetric mixtures, is the cluster size distribution fimction of the minority phase clusters n(R,z)dR is the number of clusters of minority phase per unit volume with radii between R and + cW. Its zeroth moment gives the mean number of clusters at time r and the first moment is proportional to die mean cluster size. 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

The fiindamental problem of understanding phase separation kinetics is then posed as finding the nature of late-time solutions of detemiinistic equations such as (A3.3.57) subject to random initial conditions. 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

Fraai]e J G E M 1993 Dynamic density functional theory for micro-phase separation kinetics of block copolymer melts J. Chem. Phys. 99 9202... 

Consecutive Reactions. The prototypical reaction is A B C, although reactions like Equation (6.2) can be treated in the same fashion. It may be that the first reaction is independent of the second. This is the normal case when the first reaction is irreversible and homogeneous (so that component B does not occupy an active site). A kinetic study can then measure the starting and final concentrations of component A (or of A and A2 as per Equation (6.2)), and these data can be used to fit the rate expression. The kinetics of the second reaction can be measured independently by reacting pure B. Thus, it may be possible to perform completely separate kinetic studies of the reactions in a consecutive sequence. The data are fit using two separate versions of Equation (7.8), one for each reaction. The data will be the experimental values of for one sum-of-squares and b ut for another. 

Partial rate factors may be obtained by separate kinetic measurements of the overall rate constants /cC(H Y and fcC(,H6 under analogous... 

In fact, the ammination reaction forming [Cu(NH3)4]2+ occurs stepwise, with first one ammonia ligand bonding to the copper ion, then a second, and so forth until the tetra-amminated complex is formed. And if there are four separate reaction steps, then there are four separate kinetic steps - one for each ammination step, with each reaction having its own rate constant k - we call them k(i), k(2), k(3 and k(4. This observation helps explain why the increase in reaction rate is not 16-fold when we double the concentration of ammonia. 

The mutual termination of growing chains which prevails in radical polymerizations must be ruled out for all ionic systems in which the opposite ions form separate kinetic units because of the electrostatic repulsion between like ions. However, in solvents of low DC in which the growing end of the polymer chain consists of an ion pair, a mutual termination by interaction of two such ion pairs is at least conceivable. 

The rate constants for the KIEs were measured using UV spectroscopy in separate kinetic runs using the undeuterated and deuterated substrates. Although this normal secondary /3-deuterium KIE could be due to hyperconjugation, the authors, like Streitwieser and Van Sickle (1962), preferred to attribute it to an inductive effect. 

The secondary /3-deuterium KIEs observed for the reaction of the same substrate with hydroxide ion and with tris(hydroxymethyl)methylamine in aqueous solution at 25°C were small, i.e. kH/kD = 1.09 0.01 and 1.10 0.01, respectively. While Kresge argued that the EIE was primarily due to hyperconjugation, the secondary /3-deuterium KIEs were attributed partly to hyperconjugation and partly to a polar (inductive) effect. The rate constants for the evaluation of both the EIE and the KIEs were determined in separate kinetic runs by following the increase in the absorbance due to the nitronate ion by UV spectroscopy. 

Conceptually, the simplest way to measure a kinetic isotope effect (KIE) is to use a non-competitive method, in which two separate kinetic runs are carried out, each starting with a different isotopomer of the reactant. The rate constants for both species are determined and the kinetic isotope effect (KIE) is the ratio of the two rate constants. This procedure is frequently referred to as the direct method . 

Unlike for an atom, a shell correction expansion up to high orders is possible for an electron bound in a harmonic-oscillator potential [16]. However, this system is characterized by only one parameter and, hence, does not readily allow to separate kinetic from other contributions. 

Unlike desferrioxamine analogs designed for specific therapeutic purposes described above, chiral DFO analogs that form conformationally unique complexes with iron(lll) were designed to serve as chemical probes of microbial iron(lll) uptake processes. As mentioned above, ferrioxamine B can form a total of five isomers when binding trivalent metal ions, each as a racemic mixture. Muller and Raymond studied three separate, kinetically inert chromium complexes of desferrioxamine B (N-cis,cis, C-cis,cis and trans isomers), which showed the same inhibition of Fe-ferrioxamine B uptake by Streptomyces pilosus. This result may indicate either that (i) ferrioxamine B receptor in this microorganism does not discriminate between geometrical isomers, or that (ii) ferrioxamine B complexes are conformationally poorly defined and are not optimal to serve as probes. 

The rate of phase separation after extraction in AOT-RMs is slow [167]. Keeping this in view, there is a need to study in detail the phase separation kinetics of this reverse micellar system in order to evolve means to enhance the phase separation rate. This is a very important aspect as far as industrial adaptability of RME is concerned, since the slower separation rate may become a bottleneck as in the case of ATPE. One possible approach to enhance phase separation is the application of external fields such as electric, acoustic, and microwave to reverse micellar systems. These are shown to enhance the phase separation rate in the case of ATPE [346-348]. Employing reverse micellar systems which phase separate quickly without the need for any external effort could also be a plausible solution. Some examples of such systems are DTDPA-RMs [237], sugar esters DK-F-110 RMs [239], and NaDEHP-RMs [167,243]. 

We study ensembles of systems with a given graph and independent and well-separated kinetic constants fcy. This means that we study asymptotic behavior of ensembles with independent identically distributed constants, log-uniform distributed in sufficiently big interval log ke[a., ft], for a — 00, ft- CO, or just a log-uniform distribution on infinite axis, logfc e M. 

For deriving of the auxiliary discrete dynamical system we do not need the values of rate constants. Only the ordering is important. Below we consider multiscale ensembles of kinetic systems with given ordering and with well-separated kinetic constants ( (i) k(,(2) > > for some permutation cr). 

In analyzing reactions over deactivating catalysts, we can follow separable or non-separa-ble kinetics. According to separable kinetics, Szepe and Levenspiel used two different terms one for the reaction kinetics (independent of time) and another for activity (time dependent). Consequently,... 

Determination of the kinetic constant for a bi-substrate reaction is carried out in a similar manner to that for single substrate reactions. This is achieved by investigating only one substrate at a time, while the other is kept at a set concentration which is usually its saturation concentration. Thus, to determine the Km and Kmax of substrate A, B is kept constant at a saturating level while the reaction of A is investigated at different concentrations. The experimental conditions are then reversed to determine the kinetic constants of B. Thus, the kinetic constants for a bi-substrate reaction are determined using two separate kinetic plots, as discussed previously for the conditions where concentrations of A or B limit the rate of the reaction. Clearly, the conditions under which the rates are determined must be quoted for any determination. 

Active nitrogen has been a favorite subject of investigation by physicists for many years and, more recently, interest has also been aroused in its chemical behavior. However, serious difficulties have been experienced in this connection as a result of the presence of other active species besides nitrogen atoms. These have been difficult to identify with certainty and to separate kinetically. Substantial progress seems now to have been made in this respect and also it appears that complications are reduced when a microwave discharge is used to activate... 

Vasanthavada, M., W. Tong, Y. Joshi, and M. S. Kislalioglu. 2005. Phase behavior of amorphous molecular dispersions II Role of hydrogen bonding in solid solubility and phase separation kinetics. Pharm Re 2 440-448. 

Finally, we would like to emphasize that, besides obtaining values of ccfrom the Marcus analysis, another great advantage is that one can separate kinetic and thermodynamic contributions to the parameters in linear free-energy relations. In this article we have done this for the Swain-Scott nucleophilicity parameter n, for m describing the change of reaction rate with solvent, and for the Hammett p-values. 

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