Inflection point temperature

Sestak (43) compared the kinetic results calculated by five different methods for a system corresponding to the dehydration of -CaS04 0.5H2O. The five methods evaluated mathematically were (1) Freeman and Carroll (83) (2) Doyle (84) (3) Coats and Redfern (85) (4) Horowitz and Metzger (88) and (5) Van Krevelen et al. (87). From these calculations it was found that the deviations of computed values oF E did not differ by more than 10%. Thus, all the methods appear to be satisfactory for the calculation of E within the limits of accuracy required. The errors of each method due to the inaccuracy of visual deduction of values from the TG curves were also calculated. These errors, % and e (errors in calculation of E or n, respectively), were as follows (1) Freeman and Carroll method, eE = 4% and e = 12% (2) Horowitz and Metzger method, ee = 2% (when the correct value of n is assumed) (3) Doyle method. eE = 4%. However, the magnitude of this error depends primarily on the position of the point on the TG curve on which the calculations are being performed. In the case of differential methods, me most accurate data are calculated from the medium-steep parts of the curve. For the approximation method, the accuracy depends on the determination of the curve inflection point temperature. 

A simple preliminary test of the JMAYK applicability to each studied case is worth of mentioning. The simple multiple of temperature, T, and the maximum reaction rate, da/dt, which should be confined to the value of 0.63 0.02, can be used to check its appropriateness. Another handy test is the value of shape index, i.e., the ratio of intersections, b and bz, of the in inflection slopes of the observed peak with the linearly interpolated peak baseline, which should show a linear relationship of the kind = 0.52 + 0.916 Tu/Ts - , where Ti s are the respective inflection-point temperatures [397,530]. 

For the phase separation problem, the maximum and minima in Fig. 8.2b and the inflection points between them must also merge into a common point at the critical temperature for the two-phase region. This is the mathematical criterion for the smoothing out of wiggles, as the critical point was described above. 

Figure 6.5 The appearence of spinodal decomposition as the temperature is lowered from a range of complete solubility, to the separation of two phases. In the range of composition between the inflection points, the equilibrium spinodal phases should begin to separate...

This result means that the reactor is insensitive if the temperature profile is concave toward the reactor length axis, and the inflection point is avoided. If the AT exceeds that permitted by the previous criterion—the limit set by RT /E— an inflection of the temperature vs., tube length will occur and thermal runaway will set in. Just before runway sets in the temperature at the hot spot can be 1.4 times higher than RT /E. 

For example, a temperature-measuring device, having its sensor placed in a protecting rube, is a system of second order. For such a system no single rime constant exists in the same way as a first-order system. The behavior of such a system is often given by a response time. Another concept is to give the apparent time constant t, which can be constructed by placing a line in the inflection point of the step response curve see Fig. 12.14. 

An inflection point in a pH-rate profile suggests a change in the nature of the reaction caused by a change in the pH of the medium. The usual reason for this behavior is an acid-base equilibrium of a reactant. Here we consider the simplest such system, in which the substrate is a monobasic acid (or monoacidic base). It is pertinent to consider the mathematical nature of the acid-base equilibrium. Let HS represent a weak acid. (The charge type is irrelevant.) The acid dissociation constant, = [H ][S ]/[HS], is taken to be appropriate to the conditions (temperature, ionic strength, solvent) of the kinetic experiments. The fractions of solute in the conjugate acid and base forms are given by... 

Earlier studies [14,15] clearly reveal that there is a reaction between two polymers and that the extent of reaction depends on the blend ratio. As 50 50 ratio has been found to the optimum (from rheological and infrared studies) ratio for interchain crosslinking, the higher heat of reaction for the NBR-rich blend may be attributed to the cyclization of NBR at higher temperatures. There is an inflection point at 50 50 ratio where maximum interchain crosslinking is expected. Higher viscosity, relaxation time, and stored elastic energy are observed in the preheated blends. A maximum 50-60% of Hypalon in NBR is supposed to be an optimum ratio so far as processibility is concerned. 

The temperature dependence of Sr shows two or more maxima, especially at low frequencies. The estimated conductivity values indicate that the temperature dependence of this parameter has an inflection point around 450K. 

At the temperature of the critical isotherm (71 = 304.19 K for C02), the coexistence region has collapsed to a single point and represents a point of inflection in the isotherm. From calculus we know that at an inflection point, the first and second derivatives are equal to zero so that... 

Figure 8.17 Vapor fugacity for component 2 in a liquid mixture. At temperature T, large positive deviations from Raoult s law occur. At a lower temperature, the vapor fugacity curve goes through a point of inflection (point c), which becomes a critical point known as the upper critical end point (UCEP). The temperature Tc at which this happens is known as the upper critical solution temperature (UCST). At temperatures less than Tc, the mixture separates into two phases with compositions given by points a and b. Component 1 would show similar behavior, with a point of inflection in the f against X2 curve at Tc, and a discontinuity at 7V...

Tim, temperature of the highest logarithmic decrement X Tg (dso temperature of the inflection point in the DSC diagram... 

At about 100 C, the desorption of radon from charcoal is rapid and efficient enough to fully regenerate a charcoal bed in a reasonable amount of time. The desorption concentration profile also features an inflection point (see Figure 3). Equation (4) can therefore also be applied to find an optimal temperature for the desorption process. 

Some isotherms corresponding to the Van der Waals equation are shown in figure 5. At a certain critical temperature Tc the isotherm has an intermediate form that goes through an inflection point at critical pressure and volume of Pc and Vc. To satisfy this requirement the first and second derivatives must vanish,... 

Ihe limiting values of k and slopes. At what temperature (in terms of EA) does this curve have an inflection point Based on typical values for EA (say 40,000 to 300,000 J mol-1), would this temperature he within the usual chemical range Hence, indicate what part (shape) of the curve would typify chemical behavior. 

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... 

Various pH sensors have been built with a fluorescent pH indicator (fluorescein, eosin Y, pyranine, 4-methylumbelliferone, SNARF, carboxy-SNAFL) immobilized at the tip of an optical fiber. The response of a pH sensor corresponds to the titration curve of the indicator, which has a sigmoidal shape with an inflection point for pH = pK , but it should be emphasized that the effective pKa value can be strongly influenced by the physical and chemical properties of the matrix in which the indicator is entrapped (or of the surface on which it is immobilized) without forgetting the dependence on temperature and ionic strength. In solution, the dynamic range is restricted to approximately two pH units, whereas it can be significantly extended (up to four units) when the indicator is immobilized in a microhetero-geneous microenvironment (e.g. a sol-gel matrix). 

Sketch a temperature profile that would exist in a one-dimensional laminar flame. Superimpose on this profile a relative plot of what the rate of energy release would be through the flame as well. Below the inflection point in the temperature profile, large amounts of H02 are found. Explain why. If flame was due to a first-order, one-step decomposition reaction, could rate data be obtained directly from the existing temperature profile ... 

Behm et al. have measured LEED diffraction intensities for H monolayers on Pd(100) surfaces as function of temperature at different coverages (Fig. 13b). Taking the temperature Ti,2 where the intensity has dropped to 50% of its low-temperature value as estimated for Td ), they constructed the phase diagram shown by crosses in Fig. 13a. (Alternatively using the inflection points of the I vs T curves (Fig. 13b) yields similar results.)... 

Studying the temperature dependence of structure factor data as shown in Fig. 24, Bartelt et also extract effective exponent estimates for y, and V, working in the limits k < 1 and using Eq. (57). They find that above the exponents are within of order 10% of their expected values, while below reliable estimates are more difficult to obtain. This work gives some justification for the notion of identifying with the inflection point of the... 

Plotting ixbase VS. pH gives a sigmoidal curve, whose inflection point reflects the apparent base-pAi, which may be corrected for ionic strength, I, using Equation 6.11 in order to obtain the thermodynamic pATa value in the respective solvent composition. Parameters A and B are Debye-Hiickel parameters, which are functions of temperature (T) and dielectric constant (e) of the solvent medium. For the buffers used, z = 1 for all ions ao expresses the distance of closest approach of the ions, that is, the sum of their effective radii in solution (solvated radii). Examples of the plots are shown in Figure 6.12. 

Akaganeite particles The first inflection point corresponds to a correlation time larger at 37°C (2.28 ns) than at 5°C (1.76 ns). This temperature dependence is typical of an electronic relaxation time (of Fe " "). The difference between the calculated correlation time and the theoretical relaxation time... 

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... 

There are several possibilities for the determination of the critical micellar concentration. If the micelles are formed from charged surfactants, a plot of the electrophoretic current at constant high voltage against the surfactant concentration shows an inflection point at the ccmc. It should be noted that the critical micellar concentration changes with temperature, the kind and concentration of counterions, and other buffer ingredients. 

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