Time-dependent expressions

A kinetic equation was developed by inserting a time-dependent expression into the Freundlich equation. The Freundlich equation can be written as,... 

Using the time-dependent expression for the total spectrum, Equation (4.7), we obtain... 

Rearranging terms and using Equations 19, 21 and 22, we obtain a time dependent expression for 7 ... 

The deactivation of catalysts concerns the decrease in concentration of active sites on the catalyst Nj. This should not be confused with the reversible inhibition of the active sites by competitive adsorption, which is treated above. The deactivation can have various causes, such as sintering, irreversible adsorption and fouling (for example coking or metal depositions in petrochemical conversions). It is generally attempted to express the deactivation in a time-dependent expression in order to be able to predict the catalyst s life time. An important reason for deactivation in industry is coking, which may arise from a side path of the main catalytic reaction or from a precursor that adsorbs strongly on the active sites, but which cannot be related to a measurable gas phase concentration. For example for the reaction A B the site balance contains also the concentration of blocked sites C. A deactivation function is now defined by cq 24, which is used in the rate expression. 

The time-dependent expression of photo-orientation is derived by considering the elementary contribution per unit time to the orientation by the fraction of the molecules dC (Ll), whose representative moment of transition is present in the elementary solid angle dQ near the direction Q(0, ) relative to the fixed laboratory axes (see Figure 3.4). This elementary contribution results from orientational hole burning, orientational redistribution, and rotational diffusion. The transitions are assumed to be purely polarized, and the irradiation light polarization is along the Z axis. The elementary contribution to photo-orientation is given by ... 

N. G. Gourmala, M. Buttini, S. Limonta, A. Sauter, H. W. G. M. Boddeke, Differential and Time-Dependent Expression of Monocyte Chemoattractant Protein-1 mRNA by Astrocytes and Macrophages in Rat Brain Effects of Ischemia and Peripheral Lipopolysaccharide Administration, Journal of Neuroimmunology, 74 (1997) 35-44. 

Second, the dynamic nature of protein expression means that it s always cell- and time-dependent. Expression levels in early stage tumor cells will probably vary considerably from those in later stage tumors, and both may have little in common with protein expression levels in, say, endothelial cells that line the tumor vasculature. A tumor cell from an aerobic environment (near blood vessels) will have a different proteomic profile from the one buried deep inside a tumor in an anaerobic environment. These kinds of issues can complicate the interpretation of protein expression in any type of tissue. 

Applying this mathematical procedure with the assumption that reaction occurs immediately at every A-B collision (for instance), the boundary condition is that as molecule B approaches molecule A the local second-order rate constant tends to infinity and so the local concentration of B (cb) tends to zero. Carrying through the mathematics [6], one obtains eventually a time-dependent expression for k equivalent to the one given by Noyes (cf. Equation (2.14), Section 2.2.3) ... 

As for the absorption cross section, it is possible to derive an alternative time-dependent expression. The time-dependent formulation of Raman scattering is computationally simpler and provides a better means of extracting dynamics information from the emission spectrum. Its derivation from the time-independent expression is rather simple and basically exploits the equality ... 

The z axis is the molecular beam axis (as in Figures 2 and 3) is the local terminal gas temperature at the onset of collisionless flow m is molecular mass and k is the Boltzmann constant. In practice, as the experimental data are obtained (initially) in time rather than velocity space, an analogous time-dependent expression P(tJ may also be used to fit the TOA profiles (as in Figure 4). These nonlinear fits provide values of and u. Similar results are obtained from the slope of a linear fit of the dependence of the time corresponding to the maximum intensity TOA values on the species molecular weights (see Figure 5). The general derivation of the relationship between... 

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