Inflexion points, detection

The lower part of Fig. 6.5 shows the electric signal obtained with the aid of the described method. An exact differentation of the upper curve is obtained by the selective amplification of the fundamental frequency of the modulation. In fact, at the inflexion point of the upper curve, a sinusoidal variation of the azimuth causes a practically sinusoidal variation of intensity. At any other place, the variation of the intensity can only be described by a Fourier series with the same basic frequency. The higher frequencies are not detected electronically. The amplitude ratio AIJA

absolute value of the curve. This is the reason, why sharp edges are observed in the lower curve at the extinction positions. This forms a welcome increase of the accuracy of the determination. The advantages of this method clearly follow from Fig. 6.514. 

In Figure 4 the equilibrium concentration of dodecyl sulfate ions is plotted against the total surfactant concentration for the PEO-NaDS system in the presence of 0.1 M NaN03. While the PEO-NaDS complex formation starts around 1 mmol.kg, the micelles appear in a detectable amount only if the free surfactant ion concentration exceeds the value of 1.3 mmol.kg" (see the 0 PEO curve). The upper limit of the free DS concentration is the value of Cjyj = 1.44 mmoLkg". In the presence of the polymer, the equilibrium DS concentration shows two inflexion points, the position of which is shifted to higher surfactant concentration with increasing polymer content. From the analysis of Equation 5 at constant counter-ion activity, it follows that... 

The inflection point of the curve /9 is then located at the equivalence. In this case, it is easy to demonstrate that the slope of the curve is maximum at this point. The sharpness index of the titration curve E < is maximum at the equivalence point. We can deduce from these results that the equivalence point may be detected by locating the inflexion point and that by following this strategy, the perpetrated titration error is negligible (see below). Indeed, this strategy is very often followed when potentiometry is used to detect the equivalence point (see electrochemistry). 

The heating rate has only a small effect when a fast reversible reaction is considered. The points of inflexion B and C obtained on the thermogravimetric curve for copper sulphate pentahydrate (Fig. 11.2) may be resolved into a plateau if a slower heating rate is used. Hence the detection of intermediate compounds by thermogravimetry is very dependent upon the heating rate employed. 

Displacement reactions may be treated as neutralizations from the standpoint of the foregoing discussion. If the acid or base displaced is moderately weak, i.e., ka or kh is about 10 the displacement reaction is equivalent to the neutralization of a very weak base or acid, with kh or ka equal to 10 , respectively no indicator is likely to give a satisfactory end-point in aqueous solution, although one may possibly be obtained in an alcoholic medium (cf. p. 396). If the acid or base being displaced is very weak, e.g., carbonic acid from a carbonate or boric acid from a borate, there is a marked pH inflexion at the equivalence-point which can be detected with fair accuracy by means of an indicator. 

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