Points of inflexion

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

Although only three constants appear explicitly in Eq. (9-91), two further constants are imphed by the choice of zero as the lower bound of y and the point of inflexion at y = c/2. The usual use of Eq. (9-91) is in sales forecasting, in which case y is sales demand andx is time. If such a cui ve alreac exists, the value of c can be read as the upper asymptote and a andZ obtained by the use of an auxiliary variable T where... 

When the estimates are well founded, the skewness may be preserved by using a distribution such as the Gompertz. The median of that curve occurs a.sy = 0.5 c, while the point of inflexion corresponds to the mode at y = c/exp (1) = 0.3679 c. The statistician Karl Pearson suggested as a simple measure of skewness... 

Starting with the Poisson form of the elution equation, the peak width at the points of inflexion of the curve (which corresponds to twice the standard deviation of the normal elution curve) can be found by equating the second differential to zero and solving in the usual manner. Thus, at the points of inflexion, ... 

Thus, the points of inflexion occur after n - Vn and n + Vn plate volumes of mobile phase has passed through the column. It follows, the volume of mobile phase that has passed through the column between the inflexion points will be... 

The peak width at the points of inflexion of the elution curve is twice the standard deviation of the Poisson or Gaussian curve and thus, from equation (8), the variance (the square of the standard deviation) will be equal to (n), the total number of plates in the column. 

Let the distance between the injection point and the peak maximum (the retention distance on the chromatogram) be (y) cm and the peak width at the points of inflexion be (x) cm. If a computer data acquisition and processing system is employed, then the equivalent retention times can be used. 

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. 

The critical temperature is the highest temperature at which a gas may be liquefied by pressure, and, since the pressure increases with the temperature, there will correspond to the critical temperature a critical pressure (pK), which is the greatest pressure which will produce liquefaction. This pressure is given by the ordinate of the critical point K, or point of inflexion, on the critical isotherm. 

The most important case is the critical isotherm on the p, r diagram. This has a point of inflexion at the critical point, there becoming parallel to the volume axis, and everywhere else slopes constantly from right to left upwards (Rule of Sarrau, 1882). 

Le Chatelier (1888) has discussed the general form of the solubility curve in the light of equation (5). If dA/dT is negative (which is usually the case) the curve begins asymptotically to the T axis, and is convex to it. It then passes through a point of inflexion, and is concave up to the maximum where A = 0, df/dT = 0. If A then becomes negative, the solubility... 

In some cases the curves of group (1) showed a point of inflexion (water with methyl alcohol). 

More straightforwardly, the sample may be titrated potentiometrically using hydrochloric acid to two points of inflexion. The first represents sodium hydroxide plus sodium carbonate the second sodium bicarbonate. Clearly there cannot be bicarbonate in the sample if there is sodium hydroxide present. Any second inflexion in this case can be used to determine the carbonate content. Should the titer from the first inflexion to the second be greater than that from start to the first inflexion, then the sample contains only carbonate and bicarbonate. The titer to the first inflexion can be used to estimate carbonate and the difference between twice this titer and the total titer to the second inflexion is a measure of bicarbonate. 

It is now necessary to attend to the second important function of the column. It has already been stated that, in order to achieve the separation of two substances during their passage through a chromatographic column, the two solute bands must be moved apart and, at the same time, must be kept sufficiently narrow so that they are eluted discretely. It follows, that the extent to which a column can constrain the peaks from spreading will give a measure of its quality. It is, therefore, desirable to be able to measure the peak width and obtain from it, some value that can describe the column performance. Because the peak will be close to Gaussian in form, the peak width at the points of inflexion of the curve (which corresponds to twice the standard deviation of the curve) will be determined. At the points of inflexion... 

Then as the retention volume is n(vm + Kvs) and twice the peak standard deviation at the points of inflexion is 2 Vn (vm + Kvs), then, by simple proportion,... 

Allowing a step input disturbance of magnitude Co, as shown in Fig. 2.23, the constants x and to are derived by drawing a tangent to the step response curve at the point of inflexion. 

The maximum flux and the point of inflexion are shown in Figure 5a. The Kynch theory is discussed fully in Section 5.2.3. 

Now we consider the relationship between the effective concentration(reff) and the surface pressure(tt) at the air/water interface. Ideally, the surface pressure is directly proportional to the concentration of surfactants. However, as the actual it-A isotherms show several specific effects, such as limiting area and points of inflexion, we shall assume the following relationships ... 

It is the distance from the mean to the point of inflexion of the normal distribution curve. In comparison to the average deviation the standard deviation is usually considered to be much more useful and meaningful statistically. For a finite number of values it is normally symbolised as S , and may be expressed as follows ... 

The simplest and the most commonly used method is to plot the cell voltage E, millivolts (mV), versus the volume (ml) of titrant added. Ultimately, the end-point is determined from the point of maximum slope of the curve i.e., the point of inflexion, as depicted in Figure 16.1 (a). However, the degree of accuracy and precision with which this point of inflexion can be located from the plotted graph largely depends on the individual number of data points observed in the close proximities of the end-point. 

Figure 16.1 (b) is obtained by plotting AE/AV against V which is termed as the first derivative curve. It gives a maximum at the point of inflexion of the titration curve i.e., at the end-point. 

Figure 16.1 (c) is achieved by plotting the slope of the frst derivative curve against the volume of titrant added i.e., by plotting A2E/AV2 Vs V and is known as the second derivative curve. Thus, the second derivative becomes zero at the point of inflexion and hence, affords a more exact measurement of the equivalence point. 

It is shown in Section 5.3.3 that, for coarse particles, the point of inflexion does not occur at a concentration which would be obtained in practice in a suspension, and therefore the particles will settle throughout at a constant rate until an interface forms between the clear liquid and the sediment when sedimentation will abruptly cease. With a highly flocculated suspension the point of inflexion may occur at a very low volumetric concentration. In these circumstances, there will be a wide range of concentrations for which the constant rate sedimentation is followed by a period of falling rate. 

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