S-shaped curve

For T Waals equation yields at temperatures below the critical (figure A2.5.6). As in the van der Waals case, the dashed and dotted portions represent metastable and unstable regions. For zero external field, there are two solutions, corresponding to two spontaneous magnetizations. In effect, these represent two phases and the horizontal line is a tie-line . Note, however, that unlike the fluid case, even as shown in q., form (figure A2.5.8). the symmetry causes all the tie-lines to lie on top of one another at 6 = 0 B = 0). 

Remember the units involved here For f they are length time for N, length and for t, time. Therefore the exponent is dimensionless, as required. The form of Eq. (4.24) is such that at small times the exponential equals unity and 6 = 0 at long times the exponential approaches zero and 0 = 1. In between, an S-shaped curve is predicted for the development of crystallinity with time. Experimentally, curves of this shape are indeed observed. We shall see presently, however, that this shape is also consistent with other mechanisms besides the one considered until now. 

The decrease in amorphous content follows an S-shaped curve. The corresponding curve for the growth of crystallinity would show a complementary but increasing plot. This aspect of the Avrami equation was noted in connection with the discussion of Eq. (4.24). 

If Fj = F2, with both values greater than unity, an S-shaped curve passing through the point (0.5, 0.5) would also result, but in this case reflected across the 45° line compared to item (3). 

Fig. 3. Typical nonionic amphiphile—oil—water—temperature phase diagram, illustrating (a) the S-shaped curve of T, M, and B compositions, (b) the lines of plait points, (c) the lower and upper critical end points (at and respectively), and (d) the lower and upper critical tielines.

Fig. 3. Passage of the adsorption wave through a stationary bed during the course of an adsorption cycle. The progressing S-shaped curves indicate the nonadsorbed vapor concentration by position in the bed at different time periods. represents the maximum permissible oudet concentration for release...

Volumetric heat generation increases with temperature as a single or multiple S-shaped curves, whereas surface heat removal increases linearly. The shapes of these heat-generation curves and the slopes of the heat-removal lines depend on reaction kinetics, activation energies, reactant concentrations, flow rates, and the initial temperatures of reactants and coolants (70). The intersections of the heat-generation curves and heat-removal lines represent possible steady-state operations called stationary states (Fig. 15). Multiple stationary states are possible. Control is introduced to estabHsh the desired steady-state operation, produce products at targeted rates, and provide safe start-up and shutdown. Control methods can affect overall performance by their way of adjusting temperature and concentration variations and upsets, and by the closeness to which critical variables are operated near their limits. 

S-Shaped Curves K. D. Tocher (The Ait of Simulation, rev. ed., English Universities Press, London, 1967) presented a comprehen-... 

In all such S-shaped curves the range of x is from —oo to so that... 

Powder Insulation A method of reahzing some of the benefits of multiple floating shields without incurring the difficulties of awkward structural complexities is to use evacuated powder insulation. The penalty incurred in the use of this type of insulation, however, is a tenfold reduction in the overall thermal effectiveness of the insulation system over that obtained for multilayer insulation. In applications where this is not a serious factor, such as LNG storage facihties, and investment cost is of major concern, even unevacuated powder-insulation systems have found useful apphcations. The variation in apparent mean thermal conductivity of several powders as a function of interstitial gas pressure is shown in the familiar S-shaped curves of Fig. 11-121. ... 

No industrially significant reaction has 3 > 0,3 (or with y = 20, and D/a = 1, 5 > 5) and only above this value are the interesting S-shaped curves possible. Of the three values of T, the effectiveness at one value of the Thiele modulus < >, the middle one is an artificial, non-existent solution. The two other values for T show the possibility of discontinuity inside the pellet. While this is possible, it is very unlikely to occur. 

Sigmoid, the characteristic S-shaped curves defined by functions such as the Langmuir isotherm and logistic function (when plotted on a logarithmic abscissal scale). 

Two types of redox systems (Fig. 7) are used for batteries [14]. The standard potential (E°) of MnO 2 should be a good representation of the total energy of the oxide. For two-phase systems such as Pb02, Ag20, HgO, etc., the initial potential (E ) and middle potential (Em ) are equal to E°, from which we can calculate AF (-nFE°). For MnOz, a one-phase system, as shown in Fig. 7(A), the E (initial potential) cannot be used as E°. Ko-zawa proposed the middle potential (Em) of the S-shaped curve to be used as the E°... 

Figure 2.4 illustrates the course of the reaction for various values of bolao. Inflection points and S-shaped curves are characteristic of autocatalytic behavior. The reaction rate is initially low because the concentration of the catalyst, B, is low. Indeed, no reaction ever occurs if bo = 0. As B is formed, the rate accelerates and continues to increase so long as the term ab in Equation (2.28) is growing. Eventually, however, this term must decrease as component A is depleted, even though the concentration of B continues to increase. The inflection point is caused by depletion of component A. 

The relation between E and t is S-shaped (curve 2 in Fig. 12.10). In the initial part we see the nonfaradaic charging current. The faradaic process starts when certain values of potential are attained, and a typical potential arrest arises in the curve. When zero reactant concentration is approached, the potential again moves strongly in the negative direction (toward potentials where a new electrode reaction will start, e.g., cathodic hydrogen evolution). It thus becomes possible to determine the transition time fiinj precisely. Knowing this time, we can use Eq. (11.9) to find the reactant s bulk concentration or, when the concentration is known, its diffusion coefficient. 

In order to prove the S-shaped character of the polarization curve, the system was studied galvanostatically. The model predicts that the sandwiched branch of the polarization curve should be stable, and therefore measurable under galvanostatic conditions. Figure 6.10 shows the results of the experiment depending on the scan rate, an S-shaped curve can be observed in the back scan, i.e., from high to low current. At low... 

Mixed-solvent solutions of various cosolvent-water proportions are titrated and psKa (the apparent pKa) is measured in each mixture. The aqueous pKa is deduced by extrapolation of the psKa values to zero cosolvent. This technique was first used by Mizutani in 1925 [181-183]. Many examples may be cited of pKa estimated by extrapolation in mixtures of methanol [119,161,162,191,192,196,200], ethanol [184,188-190,193], propanol [209], DMSO [212,215], dimethylformamide [222], acetone [221], and dioxane [216]. Plots of psKa versus weight percent organic solvent, Rw = 0 — 60 wt%, at times show either a hockey-stick or a bow shape [119]. For Rw > 60 wt%, S-shaped curves are sometimes observed. (Generally, psKa values from titrations with Rw > 60 wt% are not suitable for extrapolation to zero cosolvent because KC1 and other ion pairing interferes significantly in the reduced dielectric medium [223].)... 

When the concentration (c + x) is plotted against time t, a S-shaped curve is obtained. (Fig. 1.5). This curve is characteristic of autocatalytic reactions and many growth processes. It can be seen from equation (1.49) that rate will be maximum when... 

When K is small, because of the fast continuous maintaining of the small equilibrium amount of Ox, the complete depletion of Ox at the electrode surface will never be reached, so that the forward profile no longer maintains the peak-shaped form, rather it assumes a sigmoidal S-shaped curve (similar to a polarogram), the height of which remains constant at all the scan rates, Figure 12. 

Diagnostic criteria to identify a chemical reaction preceding a reversible electron transfer. Granted that when the chemical reaction has a large K value, or its rate is slow, the response looks like an unperturbed reversible process, the simplest and most wished-for opportunity to detect the presence of a preceding equilibrium reaction lies on the possible appearance of an S-shaped curve in the forward scan. Otherwise, other criteria are the following. 

Lemke et al. [21] reported that adsorption of zearalenone by organophilic mont-morillonite clay gave an S-shaped curve with at least two plateaus, suggesting additional mechanisms of adsorption. On the other hand, Grant and Philip [135] and Valverde-Garcia et al. [16] reported that binding of aflatoxins on phyllosili-cate clay and pesticides (such as thiram and dimethoate) on soils explained an L-shape isotherm. 

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