Sigmoid shape

The titration curve in Figure 9.1 is not unique to an acid-base titration. Any titration curve that follows the change in concentration of a species in the titration reaction (plotted logarithmically) as a function of the volume of titrant has the same general sigmoidal shape. Several additional examples are shown in Figure 9.2. 

The heat removal line is unehanged irrespeetive of the kineties, and the fraetional eonversion Xa has a qualitatively similar sigmoidal shape for seeond-order kineties. Numerieal values as in Example 6-11 have been put into both the mass balanee and heat balanee equations. Fraetional eonversion Xa from both the mass balanee and heat balanee equations at effluent temperatures of 300, 325, 350, 375, 400, 425, 450, and 475 K, respeetively, were determined using the Mierosoft Exeel Spreadsheet (Example6-12.xls). Table 6-8 gives the results of... 

An increase in co from 400 to 1600rpm tints results in a twofold increase of the signal. A deviation from linearity of a plot of z) vs. col/1 suggests some kinetic limitations. In addition, at veiy low rotation speeds (0-100 rpm), a slight upward bend is observed due to contribution by natural convection. The voltammetric wave has a sigmoidal shape for reversible systems it is identical to that common in DC polarography (described in Section 3-2), and independent of to. 

Typical decay curves for various ionic masses are shown in Figure 3b, where the effect of differing thermal velocity is clearly evident. Neglect of such decay characteristics can lead to serious discrimination effects. Note that all the experimental curves have the sigmoid shape... 

The horizontal portion of the plot in Figure 23.6 represents the equilibrium mass uptake—the amount absorbed at equilibrium whose magnitude is influenced by the solubility parameter 8 (Section 23.4.3.1). In reality, coefficient D quite often varies with concentration, so that the overall plot takes on a sigmoid shape (see Section 23.4.4.3), although a horizontal portion is still usually achieved eventually if it is not, the elastomer is possibly a two-phase material (a blend), with one phase much slower at absorbing the incoming liquid. 

As a result of fliese temperature - programmed runs, additiond experiments were conducted isothermally at temperatures between 723 - 873 K. Based on tbe second phase peak (peak 2), the conversion - time profiles at different temperatures exhibited a characteristic Sigmoid shape and may therefore be used to interpret the solid carburization kinetics. As detailed in Brown [5], the transient convrasion data may be d cribed by... 

The concentration of the unknown is then read off the standard curve opposite its B/Bq value. This sigmoid shaped standard curve, because of its linear portion, simplifies data handling. A mathematical transform of the B/Bq vs log dose is shown in Figure 2. This logit of B/Bq vs log dose is a widely used method of standard curve presentation (5,6,7). Logit B/B is defined as follows ... 

OS 59] [R 34] [P 42] The furan and dimethoxylated product concentrations were monitored as a function of the cell voltage [71]. The product concentration follows the sigmoidal shape of the bromide oxidation current. The furan concentration shows the inverse behavior. For a cell voltage of 3 V and a current of 30 mA, a concentration of 50% is approached. The product formation is limited by mass transfer of bromine generation. 

From comparable systems with a particle-size distribution width of 7%, experimental and computational results for temperature-dependent conductivity measurements were reported on by Remade et al. [57]. At low temperatures, the I(U) characteristics have a sigmoid shape and are non-linear (Figure 25). 

For the amino-borane dehydrocoupling using [Rh(l,5-cod)(p-Cl)]2 as starting catalyst, an induction period and a sigmoid-shaped kinetic curve (plot of substrate conversion versus time) were also observed, consistent with metal-particle formation. But, for Ph2PH BH3... 

Permeability-pH profiles, log Pe - pH curves in arhficial membrane models (log Pjpp - pH in cehular models), generally have sigmoidal shape, similar to that of log Dod - pH cf. Fig. 3.1). However, one feature is unique to permeabihty profiles the upper horizontal part of the sigmoidal curves may be verhcally depressed, due to the drug transport resistance arising from the aqueous boundary layer (ABL) adjacent to the two sides of the membrane barrier. Hence, the true membrane contribution to transport may be obscured when water is the rate-limiting resistance to transport. This is especially true if sparingly soluble molecules are considered and if the solutions on either or both sides of the membrane barrier are poorly stirred (often a problem with 96-well microhter plate formats). 

Cohen and Coon observed that the response of most uncontrolled (controller disconnected) processes to a step change in the manipulated variable is a sigmoidally shaped curve. This can be modelled approximately by a first-order system with time lag Tl, as given by the intersection of the tangent through the inflection point with the time axis (Fig. 2.34). The theoretical values of the controller settings obtained by the analysis of this system are summarised in Table 2.2. The model parameters for a step change A to be used with this table are calculated as follows... 

The kinetics of soybean lipoxygenase-1 in a biphasic medium is different from the kinetics in an aqueous system [Fig. 5(a),(b)]. The kinetic curve in the two-phase system has a sigmoid shape, which is due to surface active properties of LA and HP [25]. When initial LA concentration is small in the organic phase (0-5 mM) its transfer is poor and bioconversion in the aqueous phase is slow. 

Furthermore, in the system with coupled lipase and lipoxygenase, the production rate of HP is governed by the first enzymatic reaction and mass transfer. When TL,- is small (0 to 1 mM equiv. 3 mM LA), the kinetic curve has a sigmoid shape due to surface active properties of LA and HP [25]. Hydrolysis of TL and the increase of LA favor the transfer of LA. Such a transfer allows the lipoxygenase reaction to progress. Since lipox-ygenation consumes LA and produces HP, catalysis and transfer demonstrates a reciprocal influence. 

A transformation of the peak voltammogram to the sigmoidal shape shown in the preceding section, Fig. 5.13, is achieved by the convolution analysis method proposed by K. Oldham. The experimental function j(t) = j[T(E — Ej)/v] is transformed by convolution integration... 

Because of the sigmoid shape of the thermal energy generation curve, intersections at high or very low degrees of conversion are more probable than intersections at intermediate levels. 

The kinetic curve has a sigmoidal shape and can be intuitively divided into four stages (a) induction period, (b) stage of accelerated hydroperoxide formation, (c) stage of retarded ROOH formation, and (d) stage with prevalence of ROOH decomposition after the point... 

The probit relationship of Equation 2-4 transforms the sigmoid shape of the normal response versus dose curve into a straight line when plotted using a linear probit scale, as shown in Figure 2-10. Standard curve-fitting techniques are used to determine the best-fitting straight line. 

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