Points inflection

B3.5.7.3 BIFURCATION OF THE REACTION PATH AND VALLEY-RIDGE INFLECTION POINTS... 

It has been shown that for most acid-base titrations the inflection point, which corresponds to the greatest slope in the titration curve, very nearly coincides with the equivalence point. The inflection point actually precedes the equivalence point, with the error approaching 0.1% for weak acids or weak bases with dissociation constants smaller than 10 , or for very dilute solutions. Equivalence points determined in this fashion are indicated on the titration curves in figure 9.8. 

The principal limitation to using a titration curve to locate the equivalence point is that an inflection point must be present. Sometimes, however, an inflection point may be missing or difficult to detect, figure 9.9, for example, demonstrates the influence of the acid dissociation constant, iQ, on the titration curve for a weak acid with a strong base titrant. The inflection point is visible, even if barely so, for acid dissociation constants larger than 10 , but is missing when is 10 k... 

Another situation in which an inflection point may be missing or difficult to detect occurs when the analyte is a multiprotic weak acid or base whose successive dissociation constants are similar in magnitude. To see why this is true let s consider the titration of a diprotic weak acid, H2A, with NaOH. During the titration the following two reactions occur. 

Two distinct inflection points are seen if reaction 9.3 is essentially complete before reaction 9.4 begins. 

The most obvious sensor for an acid-base titration is a pH electrode.For example, Table 9.5 lists values for the pH and volume of titrant obtained during the titration of a weak acid with NaOH. The resulting titration curve, which is called a potentiometric titration curve, is shown in Figure 9.13a. The simplest method for finding the end point is to visually locate the inflection point of the titration curve. This is also the least accurate method, particularly if the titration curve s slope at the equivalence point is small. 

Another method for finding the end point is to plot the first or second derivative of the titration curve. The slope of a titration curve reaches its maximum value at the inflection point. The first derivative of a titration curve, therefore, shows a separate peak for each end point. The first derivative is approximated as ApH/AV, where ApH is the change in pH between successive additions of titrant. For example, the initial point in the first derivative titration curve for the data in Table 9.5 is... 

The equivalence point of a complexation titration occurs when stoichiometri-cally equivalent amounts of analyte and titrant have reacted. For titrations involving metal ions and EDTA, the equivalence point occurs when Cm and Cedxa are equal and may be located visually by looking for the titration curve s inflection point. 

Where Is the Equivalence Point In discussing acid-base titrations and com-plexometric titrations, we noted that the equivalence point is almost identical with the inflection point located in the sharply rising part of the titration curve. If you look back at Figures 9.8 and 9.28, you will see that for acid-base and com-plexometric titrations the inflection point is also in the middle of the titration curve s sharp rise (we call this a symmetrical equivalence point). This makes it relatively easy to find the equivalence point when you sketch these titration curves. When the stoichiometry of a redox titration is symmetrical (one mole analyte per mole of titrant), then the equivalence point also is symmetrical. If the stoichiometry is not symmetrical, then the equivalence point will lie closer to the top or bottom of the titration curve s sharp rise. In this case the equivalence point is said to be asymmetrical. Example 9.12 shows how to calculate the equivalence point potential in this situation. 

The second important parameter is the chromatographic peak s width at the baseline, w. As shown in Figure 12.7, baseline width is determined by the intersection with the baseline of tangent lines drawn through the inflection points on either side of the chromatographic peak. Baseline width is measured in units of time or volume, depending on whether the retention time or retention volume is of interest. 

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. 

The procedure outlined in this example needs only one modification to be applicable to the critical point for solution miscibility. In Fig. 8.2b we observe that there are two inflection points in the two-phase region between P and Q. There is only one such inflection point in the two-phase region of the van der Waals equation. The presence of the extra inflection point means that still another criterion must be added to describe the critical point The two inflection points must also merge with each other as well as with the maximum and the minima. 

The mathematical behavior of the critical point is characterized by the two minima, the maximum, and the two inflection points all merging into a common point so that the entire function displays the smooth features seen outside the PQ region. 

In Fig. 8.2b the minima and the maximum are described by (dAGn /dXj) = 0 and the inflection points by (9 AGj /dXj ) =0. If the second derivative describes the inflection point, the third derivative describes the displacement of the inflection point. Hence at the point where the two inflection points merge, (9 AGj /9x )y = 0. 

The inflection point of this function—where the second derivative changes sign-occurs at z = 1 hence the experimental analogs of Fig. 9.11 are examined for the location of their inflection points (subscript infl). The distance through which the material has diffused at this point is therefore given by... 

This shows that Schlieren optics provide a means for directly monitoring concentration gradients. The value of the diffusion coefficient which is consistent with the variation of dn/dx with x and t can be determined from the normal distribution function. Methods that avoid the difficulty associated with locating the inflection point have been developed, and it can be shown that the area under a Schlieren peak divided by its maximum height equals (47rDt). Since there are no unknown proportionality factors in this expression, D can be determined from Schlieren spectra measured at known times. 

Fig. 16. Correlation of the Hquid concentration at which the inflection point of the nonisothermal equihbrium occurs (45).

The titration curve of phosphoric acid in the presence of sodium hydroxide is shown in Figure 1. Three steps, corresponding to consecutive replacement of the three acidic hydrogens, and two inflection points, near pH = 4.5 and 9.0, are evident. Dissociation constants are = 7.1 x 10 = 6.3 x 10 ... 

The alkalinity is determined by titration of the sample with a standard acid (sulfuric or hydrochloric) to a definite pH. If the initial sample pH is >8.3, the titration curve has two inflection points reflecting the conversion of carbonate ion to bicarbonate ion and finally to carbonic acid (H2CO2). A sample with an initial pH <8.3 only exhibits one inflection point corresponding to conversion of bicarbonate to carbonic acid. Since most natural-water alkalinity is governed by the carbonate—bicarbonate ion equiUbria, the alkalinity titration is often used to estimate their concentrations. 

An inflection point is a point at which a function changes the direction of its concavity. 

The condition of minimum reflux for an equihbrium curve with an inflection point P is shown in Fig. 13-102. In this case the minimum internal reflux is... 

The height of the peak and area of the peak ai e traditionally used for calibration techniques in analytical chemistry. Peak maximum can also be evaluated by the height of a triangle formed by the tangents at the inflection points and the asymptotes to the peak branches. We propose to apply the tangent method for the maximum estimation of the overlapped peaks. 

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. 

When estimates are being made by hand calculations, Eq. (19-2) is frequently applied until cr = 0.8L. This will cause an inflection point in a plot of concentrations with distance. 

The measurement of efficiency is important, as it is used to monitor the quality of the column during use and to detect any deterioration that might take place. However, to measure the column efficiency, it is necessary to identify the position of the points of inflection which will be where the width is to be measured. The inflection points are not easily located on a peak, so it is necessary to know at what fraction of the peak height they occur, and the peak width can then be measured at that height. 

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. 

However, the matrix and dispersed material are isotropic, so Vm < 1/2 and Vd<1/2 (the usual limit on Poisson s ratio for an isotropic material as seen in Section 2.4). Thus, upon substitution of these values for v and Vrf, the value of 3 U /3v is seen to be always positive (even when 3U /3v is not zero) becanjselhFtypIcanefnr(l is always positive when b < 1/2. Finally, because 3 U /3v is always positive, the value of U when Equation (3.61) is used, corresponding to a minimum, maximum, or inflection point on the curve for U as a function of v, is proved to be a minimum, and in fact, the absolute minimum. 

The sign of the second derivative of X with respect to m and n must be examined in order to determine whether a minimum, maximum, or inflection point is obtained by the stationary value procedure that many erroneously call minimization. Actually, the determination of such de-... 

Such a stationary value of V can be a relative maximum, a relative minimum, a neutral point, or an inflection point as shown in Figure B-1. There, Equation (B.1) is satisfied at points 1, 2, 3, 4, and 5. By inspection, the function V(x) has a relative minimum at points 1 and 4, a relative maximum at point 3, and an inflection point at point 2. Also shown in Figure B-1 at position 5 is a succession of neutral points for which all derivatives of V(x) vanish. A simple physical example of such stationary values is a bead on a wire shaped as in Figure B-1. That is, a minimum of V(x) (the total potential energy of the bead) corresponds to stable equilibrium, a maximum or inflection point to unstable equilibrium, and a neutral point to neutral equilibrium. 

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