Inflection point method

The inflection point method At the inflection point, d afidfi = 0 and... 

When applicable, this method is the least demanding in terms of experimental accuracy. It is merely necessary to estimate the slope of what should be a straight line when In a/ is plotted versus t. By comparison, the inflection point method requires estimating the slope at an earlier time before it is constant. 

It has to be noted that the half-height and inflection-point methods do not give reliable results if the isotherm is concave upward and ascending concentration steps are performed. The same is true for a convex upward isotherm and descending concentration steps. The reason for this is that, in these cases, a diffuse breakthrough profile is obtained and, consequently, errors are made in the accurate determination of the retention volumes when they are derived from the half-height or the inflection point. The diffuse profile can, however, be used for the determination of isotherms by the frontal analysis by characteristic points method (FACT). 

The same authors used the inflection point of titration curves to find the PZC from a charging curve for one ionic strength. The dtro/dpH is plotted versus pH and the maximum indicates the PZC. The point of zero charge corresponds to the inflection point of titration curves (second derivative of uq versus pH = 0). [66]. Sometimes the cTo-pH dependence is linear [67], and in such case the infection point method to find the PZC cannot be applied. An example of application of the inflection point method to authentic experimental data (IniOj and In(OH)3), Hamada et al., cf Table 3.1) is given in Figs. 3,7 and 3.8 (first and second derivative). The match between originally claimed PZC and that obtained from the first and second derivatives of [Pg.83]

FIG. 3.7 Inflection point method (first derivative). The arrows indicate the PZC from titration. Calculated from uncorrected titration curves published by Hamada et al. (1990). 

A potentiometric titration curve often has an inflection point at the PZC (Section 2.6.3). This property has been proposed as a method to determine the PZC [673]. The inflection point method gained some popularity after a publication by Zalac and Kallay [670]. Also, the differential potentiometric titration described in [674] is equivalent to the inflection point method. This method is not recommended by the present author as a standalone method to determine the PZC, but a few results obtained by the inflection point method, usually in combination with other methods, are reported in the tables in Chapter 3 (as Inflection in the Methods columns). In [675], the potentiometric titration curve of one sample had two inflections, and the inflection at the lower pH was assumed to be the PZC. The potentiometric titration curves of other samples had one inflection each. Reference [676] reports an inflection point in the titration curve of niobia at pH 8, which is far from the pHg reported in the literature. A few examples of charging curves without an inflection point or with multiple inflection points are discussed in Section 2.6.3. 

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... 

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. 

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. 

Tangents Method. The tangents on either side of the peak are drawn through the inflection points until the baseline. For an ideal Gaussian peak the resulting base line interval Wi, is equal to 4or. Equation (1) becomes with the width (Wb) ... 

It is still possible to enhance the resolution also when the point-spread function is unknown. For instance, the resolution is improved by subtracting the second-derivative g x) from the measured signal g x). Thus the signal is restored by ag x) - (7 - a)g Xx) with 0 < a < 1. This llgorithm is called pseudo-deconvolution. Because the second-derivative of any bell-shaped peak is negative between the two inflection points (second-derivative is zero) and positive elsewhere, the subtraction makes the top higher and narrows the wings, which results in a better resolution (see Fig. 40.30). Pseudo-deconvolution methods can correct for sym-... 

For a completely symmetrical curve, the end-point can be easily established as the inflection point through which a tangent can be drawn here for convenience the "rings method (Fig. 2.23) can be used, where the inflection point is obtained by intersection of the titration curve with the line joining centres of fitting circles (marked on a thin sheet of transparant plastic see ref. 61). 

Figure 6.2. Illustration of fitting Eq. (6-2, solid curve) to open-loop step test data representative of self-regulating and multi-capacity processes (dotted curve). The time constant estimation shown here is based on the initial slope and a visual estimation of dead time. The Ziegler-Nichols tuning relation (Table 6.1) also uses the slope through the inflection point of the data (not shown). Alternative estimation methods are provided on our Web Support.

Experimental results obtained at a rotating-disk electrode by Selman and Tobias (S10) indicate that this order-of-magnitude difference in the time of approach to the limiting current, between linear current increases, on the one hand, and the concentration-step method, on the other, is a general feature of forced-convection mass transfer. In these experiments the limiting current of ferricyanide reduction was generated by current ramps, as well as by potential scans. The apparent limiting current was taken to be the current value at the inflection point in the current-potential curve. 

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. 

An additional point worth mentioning is that the potentiometric method can monitor several partially soluble salts at once. For example, if a solution contains chloride, bromide and iodide ions, then a plot of emf against the volume of cation (e.g. Ag ) will contain three inflection points (see Figure 4.8), one for each of the three silver halides. for Agl is smaller than that for AgCl, while (AgBr) has an intermediate value, so the first inflection point represents the precipitation of Agl, the second represents formation of AgBr and the third represents the formation of insoluble AgCl. ... 

Mg in fertilizers is based on such proceedings thereof has been applied on multiple occasions. In milk fermentation, where the samples were dried, calcined in a furnace at 600 °C, the ash was dissolved in 0.03 M HCl, the solution was centrifuged and the supernatant was thus analyzed . The complexometric method for determination of Ca(II) and Mg(II) can be carried out in a single titration with EDTA in alkaline solution, using a Ca-ISE for potentiometric determination of two endpoints. This is accomplished on digitally plotting pCa values measured by the ISE as a function of the volume V of titrant added to the aliquot of analyte the first and second inflection points of the curve mark the Ca(II) and Mg(n) equivalences, respectively. ... 

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