Inflection-point analysis

Another important feature of Fig. 18.9 is that in a narrow pH range from 8 to 9, a rapid increase in complexed enzyme activity was observed, whereas the native enzyme activity gradually decreased. In order to understand such abnormal activity on the part of complexed BT, the potentiometric titration curve with NaOH for the acidic groups (COOH and phenolic OH) remaining in the complex was examined. As shown in Fig. 18.10, the titration curve is characterized by two inflection points. Analysis of the titration data using the same method as for evaluating the number of ionizable groups from the values shows that 12... 

Systematic classification of transitions by inflection-point analysis... 

Inflection-point analysis ot the inverse temperature fi (f) and its derivative y (f) as tunctions otenergy E tor a system exhibiting a first- and a second-order transition, the maxima ot y (E) indicate transitions between the phases A and B at 4 and B and C at E. The associated points (Eff) = pff and P(E ) = define the transition temperatures Tff = and According to the classification scheme otinflection-point... 

Figure 2.6 illustrates the procedure for the identification of the transitions by means of inflection-point analysis. Plotted are the inverse temperature (F") and its energetic derivative y E). There are two regions, where the weak-sensitivity condition applies to E). One is located around the inflection point at E and the other is the backbending regime surrounding the central inflection point at The latter exhibits the already well-described features of a first-order transition y is positive and the intersection points of the inverse transition temperature /S with the E) curve define the coexistence region. The width is interpreted as the latent heat, which is obviously nonzero > 0. The behavior is qual-... 

It has been demonstrated in Section 2.7.3 that microcanonical inflection-point analysis offers the possibility to study all stmctural transitions in a finite system uniquely and systematically. We here employ this method to estimate the transition points for the entire set of elastic Leimard-Jones homopolymers with N= 13,..., 309 monomers [61]. 

The inflection-point analysis for fi E) can also easily be applied to the coil-globule transition, which occurs at much higher temperatures (for this reason, transition points are not inserted in Fig. 6.14). 

Transition temperatures TniN) of conformational transitions for small elastic polymers with chain lengths W = 13,..., 309 in the liquid-solid and solid-solid transition regimes, obtained from inflection-point analysis. First-order transition points are marked by symbols, second-order transition points by symbols x. Also shown is a fit for the liquid-solid transition temperature towards the thermodynamic limit W oo (dashed line). From [61]. 

In inflection-point analysis, we refer to it as a third-order transition. 

Inflection-point analysis of the aggregation process of four GNNQQNY peptides. Shown are the inverse temperature curve p (f) (original data and Bezier smoothed curve, see Section 4.3) and its derivative y (E) (obtained by B ier smoothing). 

The results were obtained by microcanonical inflection-point analysis of data generated in extensive replica-... 

A frequently encountered pH-rate profile exhibits a bell-like shape or hump, with two inflection points. This graphical feature is essentially two sigmoid curves back-to-back. By analogy with the earlier analysis of the sigmoid pH-rate curve, where the shape was ascribed to an acid-base equilibrium of the substrate, we find that the bell-shaped curve can usually be accounted for in terms of two acid-base dissociations of the substrate. The substrate can be regarded, for this analysis, as a dibasic acid H2S, where the charge type is irrelevant we take the neutral molecule as an example. The acid dissociation constants are... 

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... 

In routine analysis, often a one-dimensional so-called end-point titration can be automatically carried out up to a pre-set pH or potential value and with a previously chosen overall titration velocity in order to avoid overshoot, the inflection point should be sufficiently sharp and the titrant delivery must automatically diminish on the approach to that point in order to maintain equilibrium, and stop in time at the pre-set value. For instance, the Metrohm 526 end-point titrator changes both the dosing pulse length and its velocity by means of a pulse regulator in accordance with the course of the titration curve in fact, the instrument follows the titration two-dimensionally, but finally reports only a one-dimensional result. The Radiometer ETS 822 end-point titration system offers similar possibilities. However, automated titrations mostly represent examples of a two-dimensional so-called eqilibrium titration, where the titration velocity is inversely proportional to the steepness of the potentiometric titration curve hence the first derivative of the curve can usually also be recorded as a more accurate means of determining the inflection... 

Provided that the time-temperature curve obtained from the calorimetric experiments is wholly of first-order, or comprises a first-order section, usually after the inflection point of sigmoid reaction curves, a conventional analysis yields a first-order rate constant ku which is related to the concentration of monomer, m, and the initial concentration of initiator, c0, by the equations... 

The results obtained with PhViE in the extra dry PhN02 are listed in Table 3 and are plotted in Figures 4-6. All the reaction curves were sigmoid and kx was evaluated by the usual first-order analysis of the reaction curves beyond the inflection point (line A in Figure 4) and also from the relation... 

Under ideal conditions, the determination of the endpoint of a titration is simple. It can be accomplished by using an appropriate indicator or by straightforward analysis of a pH titration curve, e.g. through the detection of the inflection point of the pH vs. addition curve. Often the requirement of ideal conditions is not met, and so application of the above methods will result in approximations only. Proper numerical analysis of titration curves is possible and will result in significantly improved outcomes. 

The dynamic mechanical thermal analysis also indicated molecular mixing of the two components. The Tg of the thermoplastic was lower than the composite, as determined from the inflection point of the E curve seen in Figures 13 1. The PEOX glass transition temperature appeared at 62°C, while that for PVP showed up at ll5°C. The discrepancy in the Tg of PVP had already been addressed. The Tg extrapolated from the DMTA thermogram was reported since no artifacts were to be introduced. The samples exemplified for all DMTA thermograms were processed in a normalized fashion. Any thermal or hygroscopic Influences were eliminated. Some water may be trapped in the neat thermoplastics but this was minimized since they are hot pressed at 150°C and stored under calcium... 

In [LJ-control maps the substitution of one ligand by another one results in a change of the range of existence of the manifold intermediates. This change can be expressed by the ligand-property imluced shift of the titration curves identified by the relative position of their inflection points Lq s on the log (lL o/[Ni)Q) scale. These characteristic shifts provide information on the thermodynamic selectivity governed by the association processes only. This type of analysis is designated by . 

We require all three right-hand sides to equal zero simultaneously. These three equalities fix the values of the three quantities x, tres, and P0, at which the hysteresis loop just unfolds, uniquely. We find po = g, which is the result found by more traditional analysis in 6.2.4, with the vertical inflection point in the stationary-state locus occurring at x = and tres = 64/27. 

The slope of the plot is a function of the reaction stoichiometry although the functional dependence is complicated. In addition, the thermodynamic point, i.e., the point which corresponds to Keq (which is at 1 X 10"5 for all curves given here) does not correspond to the point of inflection, and even for the simplest case, dimerization, the point of inflection does not correspond to the midpoint. In addition, the curves are not symmetrical about the midpoint. Thus, no simple analysis of curve shape based on midpoint value or inflection point can give the thermodynamic or stoichiometric numbers. 

For materials generally, change in expansion (or density) by dilatometry was traditionally the most often used method for measuring Tg. Thermal properties, for example specific heat, are also widely used, particularly the methods of differential thermal analysis". A method for rubbers using DSC is being developed in ISO TC 45 as ISO 22768, but is not yet published. The inflection point on the heat input - temperature curve is usually obtained automatically by the analyser s software but, if obtained manually, is best found from the derivative of the curve. 

In a series of studies, summarized in Ref 18, Merzhanov and co-workers examined conditions under which thermal explns may be quenched. This theoretical analysis provided two characteristic dimensionless parameters which determine the conditions for quenching. These are Kcr obtained from inflection points of 0M vs k plots (0M is the max value of 6), and a quantity e = [(6m/k) (deM/dK)] =Kcr,whichisa measure of the uncertainty in the determination of the critical conditions. Both CI and are functions of 0 and 7, which (as well as k and 0) are defined as follows ... 

In a DSC analysis of a semi-crystalline polymer, a jump in the specific heat curve, as shown in Fig. 2.22, becomes visible. The glass transition temperature, Tg, is determined at the inflection point of the specific heat curve. The release of residual stresses as a material s temperature is raised above the glass transition temperature is often observed in a DSC analysis. 

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