Concentration discontinuity

Stress concentration. Stress concentration refers to physical discontinuities in a metal surface, which effectively increase the nominal stress at the discontinuity (Fig. 9.7). Stress-concentrating discontinuities can arise from three sources ... 

Figure 8.9 A multi-valued relation between x and C, is unstable and produces a concentration discontinuity (front). The hatched areas on both sides of the vertical line must be equal, which defines the position of the front.

In nonporous membranes, diffusion occurs as it would in any other nonporous solid. However, the molecular species must first dissolve into the membrane material. This step can oftentimes be slower than the diffusion, such that it is the rate-limiting step in the process. As a result, membranes are not characterized solely in terms of diffusion coefficients, but in terms of how effective they are in promoting or limiting both solubilization and diffusion of certain molecular species or solutes. When the solute dissolves in the membrane material, there is usually a concentration discontinuity at the interface between the membrane and the surrounding medium (see Figure 4.55). The equilibrium ratio of the solute concentration in one medium, c, to the solute concentration in the surrounding medium, C2, is called the partition coefficient, K12, and can be expressed in terms of either side of the membrane. For the water-membrane-water example illustrated in Figure 4.55,... 

The space charge in the liquid junction [1]. By liquid junction or the liquid junction potential we mean the diffusion potential developing in an electrically insulated electrolyte solution with differing ionic diffu-sivities and an initial concentration discontinuity. Besides its conceptual importance as probably the simplest nonequilibrium electro-diffusional situation, the dynamics of liquid junction is important to understand for applications, such as salt bridges, etc. 

Fig. 10.24. Gel electrophoresis in discontinuous polyacrylamide gel (3.5 and 10% gels set in tandem, with the concentration discontinuity at the point indicated) of mitochondrial RNA after exposure to the inhibitor cordycepin (cytoplasmic RNA synthesis previously stopped with actinomycin D). From top to bottom, analyses of ( H)-uridine-labeled RNA, 30, 90 and 180 min after addition of cordycepin (Penman...

Equation 6.49 is strictly valid only for the disperse part of the peak (Chapter 2.2.3). Depending on the shape of the isotherm, this is the rear part ( Langmuir ) or the front part ( anti-Langmuir ) of the peak (Fig 2.6). The sharp fronts ( Langmuir ) or tails ( anti-Langmuir ) of the peaks are called concentration discontinuities or shocks. To describe the movement of these shocks, the differential in Eq. 6.49 has to be replaced by discrete differences A, the secant of the isotherm, which describe the amplitudes of the concentration shocks in the mobile and stationary phases ... 

As observed by DeVault [6], there can be only a single value of the concentration in any given point of the (t, z) space. In the framework of the ideal model, in which the column efficiency is infinite, this propagation phenomenon results in a concentration discontinuity or shock appearing at the band front. If the isotherm is convex downward, which occms less frequently, the derivative d q/dC is positive, then the velocity associated with a concentration decreases with increasing concentrations. Therefore, the converse effect occurs a shock appears on the rear part of the band profile, since the low concentrations now move faster than the high ones but cannot pass them either. For this t5q>e of isotherm, the profile obtained is a diffuse front and a rear discontinuity. 

The mathematical origin of these concentration discontinuities or weak solutions of Eq. 7.1 has been explained by Courant and Friedrichs [24] and by Lax [25]. The mathematical backgroimd has been reviewed in cormection with the discussion of the numerical solution of the equilibrium-dispersive model given by Rou-chon et al. [26]. In the traditional theory of partial differential equations, a solution should be continuous. Lax [27] generalized the concept of solution to include weak solutions, which are not continuously differentiable. A solution of Eq. 7.1 that includes a continuous part, or diffuse bormdary, and a concentration shock is a weak solution of this equation [1,26-28]. A serious problem then arises, since there is no unique weak solution of Eq. 7.1. It is necessary to define the weak solution that is acceptable for the physical problem in order to achieve the determination of the band profile. This solution must make physical sense and prevent the crossing of the characteristics. Oleinik has suggested a selection rule that can... 

In the following section (Section 7.3, we will analyze the trajectory of a stable concentration discontinuity injected into the column, i.e., as one of the boxmdaries of a rectangular injection profile. This derivation could be done directly in the case of a Langmuir isotherm [1,2,31] (see Section 7.3.3). It can also be derived as a particular case of a more general solution. First, however, in a last subsection, we determine the time it takes for a concentration shock to form on a simple continuous profile. 

The concentration is 0 at t = te, on a diffuse profile. A concentration discontinuity takes place at t = Ir. The sign in Eq. 7.46 is negative for a convex-upward isotherm e.g., Langmuir), positive for a convex-downward isotherm. Depending on the sign of the isotherm curvature at the origin, te is longer (convex-upward isotherm) or shorter (convex-downward isotherm) than 1r. Between these two times, te and tR, the asymptotic solution is... 

When the second component front appears, the concentration of the second component jumps from 0 to C, while at the same time the concentration of the first component falls from to C°. This is a simultaneous concentration discontinuity. It corresponds to the segment AF in the hodograph plot (Figure 8.2). Afterward, the concentrations of both components in the eluent remain constant and equal to their concentrations in the feed until the end of the injection plateau and the beginning of the diffuse rear profiles. This second front shock moves at the velocity associated with the concentration discontinuity AC2 = in the presence of a concentration of the first component (see Figure 8.1, conditions at the rear of the second shock). From Eq. 8.15 and since Aq2 is equal to 2(Cj, C ), this velocity is... 

Thus, 7 is always larger than unity and At2 is always positive. During the time between f and tj, the concentration of the second component cannot decrease it is "too early" for the elution of a continuous rear profile of the band of the first component. A concentration discontinuity is impossible, as a rear discontinuity would be unstable. The concentration cannot increase either, which, in isocratic. 

As we have shown, the heights of the two concentration discontinuities decrease constantly during their migration, as soon as the plateau at the feed concentration has vanished. The velocity associated with the maximum concentration of the band on the continuous (rear) side is always larger than the shock velocity. This is why the shock erodes continuously. The trajectory of the second shock can be obtained and its retention time calculated. It is longer than the retention time of the same shock in the case of a wide rectangular injection because the shock is less high, hence its velocity is lower (Eq. 8.15). 

We have shown in Chapters 7, 8, and 9 that the ideal model gives a good first approximation of the band profiles observed under conditions of strongly nonlinear behavior of the isotherm. This approximation is better at high column efficiency. Even then, however, the profiles predicted by the ideal model are angular and have no rormd comers like those observed experimentally. Indeed, experimental profiles do not show concentration discontinuities but shock layers of various thicknesses (see Chapter 14), while the other features of the solutions are smoothed out, eroded by the influence of axial dispersion and of the resistances to mass transfer in the column. A more accurate model is needed to account for the actual band profiles and to predict them when needed. 

The ideal model (Chapter 7) assiunes an infinite column efficiency. This makes the band profiles that it predicts unrealistically sharp, especially at low concentrations. This sharpness is explained by the fact that the ideal model propagates concentration discontinuities or shocks. For a hnear isotherm, the elution profile would be identical to the input profile, clearly an unacceptable conclusion. The effects of a nonideal column are significant in three parts of the band profile. The shock is replaced by a steep boimdary, the shock layer, whose thickness is related to the coefficients of the column HETP (axial dispersion and mass transfer resistance see Chapter 14). The top of the band profile is roimd, instead of being... 

It is easily shown that, if we assume that the concentrations of the highest concentration plateaus of the two components at the end of period n, at time nt, and the locations of the new concentration discontinuities formed are given by the following equations at the end of period n. 

Shock Concentration discontinuity arising at the front of a chromatographic band when the isotherm is convex upward, at its rear when it is convex downward, if the column efficiency is infinite (ideal model. Chapter 7). The discontinuity is stable and forms because in this case a velocity is associated with each concentration, and this velocity increases with increasing concentration for a convex upward isotherm. Points on the front profile at high concentrations move faster than points at low concentrations, and pile up at the front of the band. However, the area of the band is proportional to the sample size and is finite. So, a discontinuity must form. 

Under poor mixing conditions (dominance of component A), there are sharp concentration discontinuities instead of a true gradient function along the flowing sample. A myriad of transient liquid mirrors is established within the poorly mixed fluid elements with different concentrations, and hence refractive indices. The incident beam then undergoes attenuation, mainly by reflection, leading to a decrease in the power of transmitted radiation and therefore to an increase in, e.g., absorbance. 

Signal subtraction is more efficient when Schlieren component B prevails and the catalytic determination of iodide in table salts [119], highlighted in 3.1.2.2, illustrates this aspect. After sample passage through a mixing chamber, concentration discontinuities were eliminated and the influence of Schlieren component A became insignificant. Net absorbance values were precisely quantified on top of a very high (0.4 absorbance) yet reproducible blank value. Precise results (r.s.d. <4%) were then obtained. 

As shown in Figure 9.1 for a nonporous membrane, there is a solute concentration discontinuity at both liquid-membrane interfaces. Solute concentration c 0 is that in the feed liquid just adjacent to the upstream membrane surface, whereas c 0 is that in the membrane just adjacent to the upstream surface. The two are related by a thermodynamic equilibrium partition coefficient Kp defined by... 

Donnan potential associated with the concentration discontinuities at the membrane interfaces. 

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