Bulk concentration

Polymers typically exhibit a high-affinity adsorption isotherm as shown in Fig. XI-5 here the adsorbed amount increases very rapidly with bulk concentration and then becomes practically independent of concentration. 

As evident from Fig. XI-6, the mean field produces concentration profiles that decay exponentially with distance from the surface [66]. A useful approximate solution to Eq. XI-18 captures the exponential character of the loop concentration profile [67], Here a chain of length iV at a bulk concentration of (j>b has a loop profile that can be estimated by... 

Therefore, in tire limiting case—tire surface concentration of tire reacting species is zero as all tire arriving ions immediately react—tire current density becomes voltage independent and depends only on diffusion, specifically, on tire widtli of tire Nerstian diffusion layer S, and of course tire diffusion coefficient and tire bulk concentration of anions (c). The limiting current density (/ ) is tlien given by... 

If we assume that the rates of adsorption and desorption are both large compared with the surface migration rate, the surface and bulk concentrations of each species will be almost in equlibriura, and hence will be... 

A rather simpler situation arises when the bulk concentrations are sufficiently small that the adsorption isotherms approach linearity. Then (7,4), for example, shows that... 

A representation of the various concentrations and driving forces in a.j—x diagram is shown in Eigure 4. The point representing the interfacial concentrations x ) must He on the equiHbrium curve since these concentrations are at equiHbrium. The point representing the bulk concentrations (y, Xj may be anywhere above the equiHbrium line for absorption or below it for desorption. The slope of the tie line connecting the two points is given by equations 4 and 5 ... 

Essences generally are stored separately from the bulk concentrates for stabiHty, and their addition prior to retail packaging is essential to restoring much of the natural fresh flavor of the starting juice otherwise lost during processing. Unlike citms, which affords both an aqueous and an oil-phase essence, only an aqueous-phase essence is obtained for deciduous fmit. Virtually no essential oil is present in the peel or juice in the latter. 

This deposit is composed of suspended particles similar to conventional filter cakes, and more importantly, a slime that forms as retained solutes exceed their solubility. The gel concentration 6 is a function of the feed composition and the membrane-pore size. The gel usually has a much lower hydrauHc permeabihty and smaller apparent pore size than the underlying membrane (27). The gel layer and the concentration gradient between the gel layer and the bulk concentration are called the gel-polarization layer. 

Fig. 5. Concentration polarization = concentration at membrane wall, Cj, = bulk concentration, Cj,. = bulk concentration of species i, J = flux, and...

The chronoamperometric technique illustrates the principle that analytically useful current responses depend critically on the efficiency of analyte mass transport within the solution. The analyte mass transport in turn depends on the efficiency with which an appHed voltage can maintain the surface concentrations of oxidized and reduced species at values specified by the Nemst equation. It is generally the case in chronoamperometry that the bulk concentration of one of the species is zero whereas the surface concentration of the other species is forced to zero by the appHed potential, but this is not always so. 

For systems in which the solute concentrations in the gas and hquid phases are dilute, the rate of transfer may be expressed by equations which predic t that the rate of mass transfer is proportional to the difference between the bulk concentration and the concentration at the gas-liquid interface. Thus... 

If the equilibrium relation y° = F Xi) is sufficiently simple, e.g., if a plot of yfversus Xi is a straight hne, not necessarily through the origin, the rate of transfer is proportional to the difference between the bulk concentration in one phase and the concentration (in that same phase) which would be in equilibrium with the bulk concentration in the second phase. One such difference isy — y°, and another is x° — x. In this case, there is no need to solve for the interfacial compositions, as may be seen from the following derivation. 

Experimentally observed rates of mass transfer often are expressed in terms of overall transfer coefficients even when the eqmlibrium lines are curved. This procedure is empirical, since the theory indicates that in such cases the rates of transfer may not vary in direct proportion to the overall bulk concentration differences y — y°) and (x° — x) at all concentration levels even though the rates may be proportional to the concentration difference in each phase taken separately, i.e., Xi — x) and y — y ). 

Figure 14-10 illustrates the gas-film and liquid-film concentration profiles one might find in an extremely fast (gas-phase mass-transfer limited) second-order irreversible reaction system. The solid curve for reagent B represents the case in which there is a large excess of bulk-liquid reagent B. The dashed curve in Fig. 14-10 represents the case in which the bulk concentration B is not sufficiently large to prevent the depletion of B near the liquid interface and for which the equation ( ) = I -t- B /vCj is applicable. 

Evaluation of the integral in Eq. (14-86) requires a knowledge of the liquid-phase bulk concentration of B as a function of y. This relationship is obtained by means of a material balance around the tower, as shown in Eq. (14-73). Numerical integration by a quadrature method such as Simpson s nrle normally will be required for this calculation. 

For manganese which has a vapour pressure of 4.57 x 10 atmos at 1873 K, this depletion amounts to about one half of the bulk concentration, thus lowering the rate of manganese evaporation by half. These equations may be used to derive tire condition for the preferential removal of a solute. A, from liquid iron... 

The effectiveness of a porous catalyst T] is defined as the actual diffusion-limited reaction rate divided by the reaction rate that could have been achieved if all the internal surface had been at bulk concentration conditions. 

From these equations, the fraction of the bulk concentration inside the pores as a function of the pore depth x can be calculated to be ... 

Modern materials have a complex three-dimensional internal structure with many different phases. Although for these samples quantification is not possible, technologists are often interested in relative differences between several samples, or they already know the bulk concentrations and are only interested in the element distribution. 

Taking into account that E -a = f the bulk concentration of element A is given by ... 

The ratio Db/Da is a so-called relative sensitivity factor D. This ratio is mostly determined by one element, e. g. the element for insulating samples, silicon, which is one of the main components of glasses. By use of the equation that the sum of the concentrations of all elements is equal to unity, the bulk concentrations can be determined directly from the measured intensities and the known D-factors, if all components of the sample are known. The linearity of the detected intensity and the flux of the sputtered neutrals in IBSCA and SNMS has been demonstrated for silicate glasses [4.253]. For SNMS the lower matrix dependence has been shown for a variety of samples [4.263]. Comparison of normalized SNMS and IBSCA signals for Na and Pb as prominent components of optical glasses shows that a fairly good linear dependence exists (Fig. 4.49). 

Whenever die rich and the lean phases are not in equilibrium, an interphase concentration gradient and a mass-transfer driving force develop leading to a net transfer of the solute from the rich phase to the lean phase. A common method of describing the rates of interphase mass transfer involves the use of overall mass-transfer coefficients which are based on the difference between the bulk concentration of the solute in one phase and its equilibrium concentration in the other phase. Suppose that the bulk concentradons of a pollutant in the rich and the lean phases are yi and Xj, respectively. For die case of linear equilibrium, the pollutant concnetration in the lean phase which is in equilibrium with y is given by... 

FIG. 11 Adsorbed amount as a function of bulk concentration for a non-interacting (empty symbols) and adsorbing (full symbols) wall. Diamonds and triangles correspond to a system with semi-rigid chains, circles and squares for flexible chains [28]. 

Now at some pH comparable to pK, two waves are observed, corresponding to the reduction of both HA and A. The currents are proportional to the concentrations of the electroreducible species. Because the pH and pK are known, the concentrations of HA and A in the bulk solution can be calculated. It is then found that the observed polarographic currents cannot be accounted for on tbe basis of the known bulk concentrations. It is concluded that the ratio of the concentrations at the electrode surface is different from the ratio of bulk concentrations, and this is a consequence of the coupling between the chemical and electrode processes. In the pyruvic acid system, HA can be converted to the hydroxy acid by the electrode... 

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