Concentration profiles definition

The numerical value of the Rp coefficient for a given chromatographic band can be determined for the maximum value of the concentration profile of the band (which actually is the point at which local concentration of the analyte is the highest). The Rp eoeffident determined according to this definition can be denoted as Rp mmy... 

Finally, an officially updated definition of the retardation factor, R, issued by lUPAC is important to the whole field of planar chromatography (the linear and the nonlinear TLC mode included). The importance of such a definition has two reasons. First, it is promoted by the growing access of planar chromatography users for densitometric evaluation of their chromatograms and second, by the vagueness of the present definition in the case of skewed concentration profiles with the samples developed under mass overload conditions. 

The definition of surface excess, on the other hand, starts with the concentration profile, but involves an integration between limits Kq. (6.65)]. Once the integration is done and limits are inserted, one obtains a number (so many moles per square centimeter) and loses all knowledge of the function c((x). In other words, after the integration is carried out and the surface excess evaluated, the concentration profile... 

By multi-phase transient experiments , we mean those such as cyclic voltammetry (see Chap. 3) or current reversal chronopotentiometry [33] in which some sudden charge is made in the imposed conditions at some definite time (t = r, for example) after the experiment commences. To address such multiphase experiments, one usually treats each phase separately, the solution of the first phase providing details of the concentration profiles at t — t, which then serve as the initial conditions for the second phase. Likewise, the final conditions of the second phase provide the initial conditions for the third phase, if any, and so on. 

There are two interpretations of the statistical quantity A, both being closely related to the geometrical interpretation of the term phase boundary . From a purely macroscopic point of view, the gel of course represents a phase to which thermodynamical functions of state are related. Nevertheless, such a macroscopic image with a sharp boundary can hardly be correct in a PDC-column considering the range of end-to-end distances of the transported coils in the concentration profile, because the transported P-mer and the stationary gel are chemically equal in PDC. The two possible definitions of the quantity A(P) are ... 

The TCE distribution in the control column after 672 shows a decline in TCE concentration from the contaminated zone proximal to the anode (0 cm), vertically downwards in the soil column, and most likely represents the diffusion of TCE (Figure 4). The TCE concentration profiles in ECs define a broad distribution and indicate that the bulk of the TCE has moved towards the cathode in response to electroosmotic fluid flow. After one week of processing, the bulk of contaminant has moved more than 5 cm toward the cathode. After two weeks ofEO, the center of the plume has moved more than 10 cm toward the cathode. After three and four weeks of EO, there are no definite peaks within the column, however, the concentrations of TCE remained highest near the cathode (Figure 4). 

To demonstrate equivalence in plasma concentration profiles, rate and extent of availability must be assessed and compared. The parameters Cmax and AUC are typically used here, and they are regarded as surrogate markers for clinical safety and efficacy. If they are too much higher in the new drug formulation N, they could lead to unwanted side effects. On the other hand, if they are too much lower, the new formulation may be less effective in treating the condition. The definition of too much in this context is not simple, and we will not discuss the details here (see Patterson and Jones, 2006, for more details). 

Figure 9.9 shows a comparison of the simulated and measured gas-phase concentrations of NO and NO2 throughout the whole absorption plant. The zigzag form of the simulated concentration profiles results from switching different sections of each single column (see Ref. [35]). Good agreement between experimental and simulation results can be definitely observed here. 

If the values of ncCE SAE fixed, the parameter a of Eq. 91 has a definite value. The oxygen concentration profiles are then changed by altering the value of the parameter b. By assuming b as 2.5, 4.5 and 6.5, the computed oxygen distribution in the bed is obtained as shown in Figure 5. The effect of these profiles is examined for two values of k3(To),... 

Since the concentration profile is determined by Equation (6.4.32), evaluation of the overall effectiveness factor, rj , is straightforward. By definition, is the observed rate divided by the rate that would be observed at conditions found in the bulk fluid. Recall... 

Note that it may be surprising that a chemical term is considered in Eq. (3), whereas we have supposed that no chemical reaction term was involved within the stagnant layer. As explained in Chapter 1, this stems from the fact that the effect of any chemical reaction within the diffusion layer depends on the relative magnitude of k5 /D versus unity. For usual laboratory conditions, 8 10 cm and D 5 x IC cm s and then 5 /D is of the order of 0.2 s. Thus, provided k is less than approximately 1 s it has no tangible effect on the concentration profiles of the species, whereas it has a definite effect in the bulk solution due to the long reaction times (usually longer or comparable to half an hour). In practice it is important to decide when the simplification in Eq. (2) or (3) applies to a given experimental situation. The discussion just presented affords a simple answer to the problem. Indeed, consider the electron transfer reaction in Eq. (5), possibly followed by a chemical step in Eq. (6). 

The experimental dependence M = /(V), i.e. the classical SEC calibration curve usually obtained by using narrow standards, in such a case can be obtained directly without calibration from the on-line LS detector. By combining the experimental function M = /(V) and the concentration profile (from DRI), one can construct the complete MMD of the HA sample. The differential and cumulative MMD of a high molar mass HA sample (Mw = 652 kDa, D = 2.1) are shown in Fig. (10). Starting from the initial MMD, the molecular weight averages and dispersity index (Mn, Mw, Mz, and D) could be easily calculated using the appropriate definitions. 

When the residence time becomes shorter, this approach becomes questionable for several reasons. For example, the asymptotic state may not have been reached yet, or the peaks may be unsymmetrical. These "short-time" situations may be encountered when trying to apply chromatographic concepts to the study of dispersion in connecting tubes, or in some apphcations, such as hollow-fiber liquid chromatography. Shankar and Lenhoff [77] have derived a solution in the time domain, using series expansion. This solution can be implemented by numerical computation for the determination of concentration profiles inside a tube coated with a retentive layer, when the fluid flow is laminar. This solution is valid for systems that are either short or long after the Taylor-Aris definition. 

The agreement between the concentration profiles predicted using the more exact numerical schemes and that obtained by assuming a normal distribution indicates that this definition of resolution is consistent with the other assumptions made in this model. 

For such large intraparticle diffusion has a large effect on the rate. Practically these conditions mean that diffusion into the pellet is relatively slow, so that reaction occurs before the reactant has diffused far into the pellet. In fact, an alternate definition for is the fraction of the whole surface that is as active as the external surface. If 1, Eq. (11-43) shows that the rate for the whole pellet is the same as the rate if all the surface were available to reactant at concentration Q i.e., the rate at the center is the same as the rate at the outer surface—all the surface is fully effective. In this special case the concentration profile shown in Fig. 11-6 would be horizontal, with C = Cj. In contrast, if 77 1, only the surface near the outer periphery of the pellet is effective the concentration drops from Q to nearly zero in a narrow region near r, e, or high Ar. The latter factor shows that low effectiveness factors are more likely with a very active catalyst. Thus the more effective the active catalyst, the more likely it is that intrapellet diffusion resistance will reduce the rate per pellet. 

In order to estimate the surface excess of any ion in the diffuse layer its concentration profile must be integrated in the diffuse layer. Thus, a general definition of the surface excess is... 

The solubilization process seems to be well understood on a qualitative basis. Quantitatively, however, there appears to be less agreement. First, in reporting the extent of solubilization different authors may use different definitions and concentration units, as we discuss later. Second, for a three-component system both the concentrations of the surfactant and the solute can be varied. This means that we rarely find data that are directly comparable due to variation in concentrations. Often solubilization is reported as single points along the concentration profiles of surfactant and solute. In some cases the method of measurement sets the limits. 

Some parts of the solution in the concentration profile are imaginary, so first we need to consider the definition of the exponential integral function ... 

However, the best way to obtain the concentration profile would be the use of the inverse function of Ei(lnC,(x, y)). According to Pecina [16], by delimiting some intervals of definitions for the variable y, it is possible to obtain the inverse function of the exponential integral function, Ei 1(y). Besides, a rough approximation for Ei(y) is possible [17] and then it will be for the inverse function by means of an empirical formula valid for x > 1.6, such as... 

In this example only the analytes 2-hydroxybenzaldehyde (2-HBA) and 3-hydro-xybenzaldehyde (3-HBA) are discussed. The pK values of these analytes are 8.37 and 8.98, respectively. Hence, both the acidic and basic form of the analytes will be present along the pH gradient. Therefore, the ultraviolet spectrum of the acidic and basic form of each solute produces the signal together with the concentration profiles of the acidic and basic forms. If the spectra of the acidic forms are called sa>2 and sa>3 for 2-HBA and 3-HBA, respectively, and caj2, caj3 are the concentration profiles of the acidic forms of 2-HBA and 3-HBA, and if analogous definitions are used for the basic forms (sb,2, Sb,3, Cb,2, Cb,3) then the responses of the pure solutes 2-HBA and 3-HBA can be written as... 

Further contaminants detected in the sediment core could not be definitely attributed to individual emission sources, because the knowledge about their environmental occurrence and behaviour is very limited. Examples include the high-viscosity lubricant bis(-4-octylphenyl)amine (Vanlube 81 ), 3,6-dichlorocarbazole and limonene. The quantification of limonene revealed a very uniform concentration profile, which cannot be attributed to any other contamination history discussed so far. In contrast, the occurrence of 3,6-dichlorocarbazole, formerly not described as an environmental contaminant, exhibits high similarities to more modem sewage marker compounds (see Tab. 2). 

One can also see from Figure 5.2.1 that the dijfusion layer, that is the zone near the electrode where concentrations differ from those of the bulk, has no definite thickness. The concentration profiles asymptotically approach their bulk values. Still, it is useful to think about the thickness in terms of which has units of length and... 

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