Fluctuation in concentration

As noted at the end of the last section, it is fluctuations in concentration 5c2 rather than density which act as the scattering centers of interest for solutions of small molecules. There is nothing in the forgoing theory that prevents us from placing 6p by 6c2, the solute concentration in mass volume" units. Therefore we write for a solution of small molecules... 

In the next section, we consider the application of Eq. (10.49) to scattering from fluctuations in concentration. 

To reinforce the connection with time—domain kinetics, linewidth studies can be understood as monitoring the spontaneous fluctuations in concentrations that occur when a chemical system is at equiUbrium ... 

As shown in Fig. 22, since the dissolved metal ions are locally enriched near the surface, the fluctuation in concentration takes a positive value. In this fluctuation process, the passive film does not provide the absolute condition for protecting the substrate dissolution because, as shown in the preceding section, a breakdown in local passivity prior to... 

A hold tank is installed in an aqueous effluent-treatment process to smooth out fluctuations in concentration in the effluent stream. The effluent feed to the tank normally contains no more than 100 ppm of acetone. The maximum allowable concentration of acetone in the effluent discharge is set at 200 ppm. The surge tank working capacity is 500 m3 and it can be considered to be perfectly mixed. The effluent flow is 45,000 kg/h. If the acetone concentration in the feed suddenly rises to 1000 ppm, due to a spill in the process plant, and stays at that level for half an hour, will the limit of 200 ppm in the effluent discharge be exceeded ... 

They recorded such a polarization curve for zinc, for copper in the presence of gelatin and for silver in nitrate solution. Under this mechanism, a negative fluctuation in concentration drives the current density up, resulting in further reduction in interfacial concentration. For this instability to be expressed, the surface concentration must be free to respond to variations in current. As a result, the instability is seen only far from the limiting current, where the interfacial concentration is pinned at zero. At high Peclet numbers, the concentration disturbance is propagated downstream by convection, and the striations follow the streamlines. 

During a seven-month period (from January to July 2000) of this study a seasonal fluctuation in concentrations of all target compounds... 

Fluctuations in refractive index occur within a volume element and arise from variations in density and concentration. The former are responsible for scattering by the solvent and may be ignored in the present context since solution and solvent scatterings are subtracted. Regarding the dissolved polymer, it is only necessary, therefore, to consider scattering caused by local fluctuations in concentration. 

Or the mean value of these fluctuations in concentration will be... 

There have been many theories of metal-ammonia solutions that differ from that presented here. Cohen and Thomson (1968) and more recently Cohen and Jortner (1973) supposed that large fluctuations in concentration occur over large distances, sufficient to invalidate the pseudogap model and substitute a semiclassical percolation theory. For the reasons given above, we think that this is only likely to be so within a few degrees of the critical point. 

It is certainly more constant than that of sediments being introduced into the basin. This fact is due to the greater mobility of material in solution which tends to even out local fluctuations in concentration through the action of waves and currents. The sediment is much less subjected to such a mechanical homogenization process and tends, therefore, to attain equilibrium by localized mineral reaction. The type of thermodynamic system operative is most likely to be "open", where each point of sediment has some chemical variables fixed by their concentration in the sediment (inert components due to their low solubility in the solution) and other chemical components, which are soluble, have their concentration in the sediment a function of their activity in the aqueous solution. The bulk composition of the resulting sediment will be largely determined by the composition of the waters in which it is sedimented and the length of time it has reacted with this environment. The composition of the aqueous solution is, of course, determined to a minor extent by these reactions. 

Because of the length and complexity of Pecoras calculation, it will not be reproduced here. Instead, we will perform a somewhat less rigorous, but plausible, evaluation of Cse(r) to reach the same result. In doing this, it must be assumed that the time dependence of Cse(r) is the same as that observed if a large fluctuation in concentration were artificially imposed on the system at time t = 0 and allowed to decay according to Ficks second law of diffusion (58) where... 

For a unit volume, the intensity of scattering will be equal to the total scattering from all < > volume elements which it contains. Since 0) = 1 /SV and since the fluctuations in electron density are the consequence of fluctuations in concentration of component 2, A C2e, 2, where the concentration Ce>i is given as the number of electrons per unit volume (A3), we have... 

Danckwerts defined a degree of segregation J based on the fluctuations in concentration which occur. He studied a chemical reaction in a binary mixture of two components A and B, in which a is the volume fraction of A at a certain point and (1 — a) the volume fraction of B at this same... 

In the pheromone orientation system it has been shown how important time aspects are. Several species will not be attracted to a pheromone source unless the stimulus arrives in a pulsed fashion, mimicking the filamentous structure of a natural odor plume. Correlates to this requisite have been found among AL neurons, where both fast neurons, able to code fast fluctuations in concentration, and slow neurons, seemingly only coding qualitative aspects of the plume, are present. 

While actual chemical events involved in nucleation and crystal growth are not known a phenomenological treatment (gives some insight. Willard Gibbs (9J considered processes of phase separation of two extreme kinds. In the first, fluctuations in concentration occur which are minute in volume but large in extent of departure from the mean (the case of binodal phase separation). In the second the volume of the fluctuation is large but the deviation from the mean for the solution is minute (responsible for spinodal phase separation). In nucleation of zeolites one is conerned only with fluctuations of the first kind. 

For non-ideal polymer solutions where there are interactions between the polymer molecules, Einstein and Debye showed independently that if the solute is uniformly distributed throughout the solution, no light is scattered by the solution because light scattered by one particle will interfere destructively with light scattered by the neighbouring particle. Random Brownian motion causes fluctuations in concentration, the extent of fluctuations is inversely proportional to the osmotic pressure developed by the concentration difference. It is found that... 

For individual constituents of drinking-water, health-based targets are established. These targets represent a health risk from long-term exposure, in a situation where fluctuations in concentration are small or occur over long periods, It is important that such targets are defined by the relevant local authority, are realistic under local operating conditions and are set to protect and improve public health,... 

At first sight, this simple model appears to have the capability of accounting only for axial mixing effects. It will be shown, however, that this approach can compensate not only for problems caused by axial mixing, but also for those caused by radial mixing- and other nonflat velocity profiles These fluctuations in concentration can result from different flow velocities and pathways and from moleeular and turbulent diffusion. 

Boundary Conditions There are two cases that we need to consider boundary conditions for closed vessels and open vessels. In the case of closed-closed vessels we assume that there is no dispersion or radial variation in concentration either upstream (closed) or downstream (closed) of the reaction section, hence tliis is a closed-closed vessel. In an open vessel, dispersion occurs both upstream (open) and downstream (open) of the reaction section hence this is an open-open vessel. These two cases are shown in Figure 14-8, where fluctuations in concentration due to dispersion are superimposed on the plug-flow velocity profile. A closed-open vessel boundary condition is one in which there is no dispersion in the entrance section but there is dispersion in the reaction and exit sections. 

As noted previously, fluctuations in concentration estimates about the true value arise from multiple, independent, random errors. Each of the independent errors (erf) is statistically additive, such that the total assay error cr = Ecrf. Similarly, the CV% of the assay will be the square root of the sum of the squared CV%... 

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