Point of inflection

These points of inflection ate typical for the K-diagram. They are not found in concentration-time diagrams. 

It is difficult to define general rules. However, one is able to simulate K-diagrams using a desk-top computer for assumed mechanisms and to analyse the curves for points of inflection. The principle approach was discussed in Section 2.3.2. Points of inflection are allowed in K- or X-diagrams only, if more than two linear independent steps of reaction s 2) take place. 

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Inside the point of inflection of equation (31) equation (32) is identical to MM2 with the cubic stretch term turned on. At very long bond distances, it is identical to MM2 with the cubic stretch term turned off. 

As is seen from Fig. 2.L, the BET equation yields an isotherm which (so long as c exceeds 2) has a point of inflection this point is close to, but not necessarily coincident with, the point where the amount adsorbed is equal to the BET monolayer capacity. 

Insertion of this value into Equation (2.12) gives the value, X, ofX at the point of inflection ... 

In Fig. 2.4, the location of the point of inflection thus calculated is plotted for different values of c. Clearly, the value of n/n at the point of inflection may deviate considerably from unity. At the one value of c = 9 the value of n/n is actually equal to unity and the point of inflection then coincides with the point corresponding to the monolayer capacity but for values of c... 

Fig. 2.4 The BET equation. Plot ofX, calculated for different values of c, against Y,. X, is the value of n/ii at the point of inflection in the isotherm F, is the relative pressure at the point of inflection. Each point on the curve is marked with the corresponding value of c in brackets.

Both Type III and Type V isotherms are characterized by convexity towards the relative pressure axis, commencing at the origin. In Ty )e III isotherms the convexity persists throughout their course (Fig. 5.1(a), whereas in Type V isotherms there is a point of inflection at fairly high relative pressure, often 0-5 or even higher, so that the isotherm bends over and reaches a plateau DE in the multilayer region of the isotherm (cf. Fig. 5.1 (b)) sometimes there is a final upward sweep near saturation pressure (see DE in Fig. 5.1(b)) attributable to adsorption in coarse mesopores and macropores. 

As already pointed out, a Type III isotherm results from the BET equation when the value of c is less than 2 (p. 46). For c = 3, the isotherm is no longer strictly of Type III, but the point of inflection, at about 0Olp°, is barely perceptible, and at first glance the isotherm appears to be a genuine Type 111—a fact of some consequence because the value c 3 is relatively common amongst isotherms which are apparently of Type III. 

In Fig. 3-25 the locational dependence of t/, and is shown together. For practical applications and because of possible disturbance by foreign fields (e.g., stray currents) and t/g are less amenable to evaluation than f/g, which can always be determined by a point of inflection between two extreme values [50]. Furthermore, it should be indicated by Fig. 2-7 that there is a possibility of raising the sensitivity by anodic polarization which naturally is only applicable with small objects. In such cases care must be particularly taken that the counter electrode is sufficiently far away so that its voltage cone does not influence the reference electrodes. 

To find and monitor the point of inflection is difficult in the industry while the hot spot (the highest temperature) is somewhat easier to find and monitor. Consequently, an expression was derived for this value. This comes from the fact that at the maximum AT, all the heat is transferred through the wall, therefore the heat generated must equal the heat transferred ... 

If a sample contains groups that can take up or lose a proton, (N//, COO//), then one must expect the pH and the concentration to affect the chemical shift when the experiment is carried out in an acidic or alkaline medium to facilitate dissolution. The pH may affect the chemical shift of more distant, nonpolar groups, as shown by the amino acid alanine (38) in neutral (betaine form 38a) or alkaline solution (anion 38b). The dependence of shift on pH follows the path of titration curves it is possible to read off the pK value of the equilibrium from the point of inflection... 

There is an important practical distinction between electronic and dipole polarisation whereas the former involves only movement of electrons the latter entails movement of part of or even the whole of the molecule. Molecular movements take a finite time and complete orientation as induced by an alternating current may or may not be possible depending on the frequency of the change of direction of the electric field. Thus at zero frequency the dielectric constant will be at a maximum and this will remain approximately constant until the dipole orientation time is of the same order as the reciprocal of the frequency. Dipole movement will now be limited and the dipole polarisation effect and the dielectric constant will be reduced. As the frequency further increases, the dipole polarisation effect will tend to zero and the dielectric constant will tend to be dependent only on the electronic polarisation Figure 6.3). Where there are two dipole species differing in ease of orientation there will be two points of inflection in the dielectric constant-frequency curve. 

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. 

Therefore, if (f) is the fraction of the height (h) at which the points of inflection occur, then... 

Equation (13) shows that the points of inflection occur at 0.6065 of the peak height. It follows that the peak width, at that height, will be equivalent to two standard deviations (2o) of the Gaussian curve. 

Reiterating the conditions for a chromatographic separation once again, for two solutes to be resolved their peaks must be moved apart in the column and maintained sufficiently narrow for them to be eluted as discrete peaks. However, the criterion for two peaks to be resolved (usually defined as the resolution) is somewhat arbitrary and is usually defined as the ratio of the distance between the peak maxima to half the peak width (a) at the points of inflection. To illustrate the various degrees of resolution that can be obtained, the separation of a pair of solutes 2o, 3o, 4o, 5o and 6o apart are shown in Figure 12. Although, for baseline resolution, it is clear that the peak maxima should be separated by at least 6o for most quantitative analyses. 

To demonstrate the effect in more detail a series of experiments was carried out similar to that of volume overload, but in this case, the sample mass was increased in small increments. The retention distance of the front and the back of each peak was measured at the nominal points of inflection (0.6065 of the peak height) and the curves relating the retention data produced to the mass of sample added are shown in Figure 7. In Figure 7 the change in retention time with sample load is more obvious the maximum effect was to reduce the retention time of anthracene and the minimum effect was to the overloaded solute itself, benzene. Despite the reduction in retention time, the band width of anthracene is still little effected by the overloaded benzene. There is, however, a significant increase in the width of the naphthalene peak which... 

The front and back of the peaks were measured at the points of inflection... 

Because a phase change is usually accompanied by a change in volume the two-phase systems of a pure substaiice appear on a P- V (or a T- V) diagram as regions with distinct boundaries. On a P- V plot, the triple point appears as a horizontal line, and the critical point becomes a point of inflection of the critical isotherm, T = T (see Figure 2-78 and Figure 2-80). 

FIGURE 11.17 Symmetrical and asymmetrical dose-response curves, (a) Symmetrical Hill equation with n = 1 and EC5o = 1.0. Filled circle indicates the EC50 (where the abscissa yields a half maximal value for the ordinate). Below this curve is the second derivative of the function (slope). The zero ordinate of this curve indicates the point at which the slope is zero (inflection point of the curve). It can be seen that the true EC50 and the inflection match for a symmetrical curve, (b) Asymmetrical curve (Gompertz function with m = 0.55 and EC50= 1.9). The true EC50 is 1.9, while the point of inflection is 0.36. 

The standard deviation s is the square root of the variance graphically, it is the horizontal distance from the mean to the point of inflection of the distribution curve. The standard deviation is thus an experimental measure of precision the larger s is, the flatter the distribution curve, the greater the range of. replicate analytical results, and the Jess precise the method. In Figure 10-1, Method 1 is less precise but more nearly accurate than Method 2. In general, one hopes that a and. r will coincide, and that 5 will be small, but this happy state of affairs need not exist. 

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...

Although not shown in Figure 8.17, the vapor fugacity of component 1 also goes through a point of inflection at the same Tc and x2.c- This can be shown by noting that since the slope is horizontal at point c, then... 

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