Tangent method

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. 

Our calculations show that the systematic errors for the evaluation of the triangle height are lower then for the peak height and peak ar ea. It is to be noted that tangent method allows estimating of the latent peak in the overlapped signals when peak area and peak maximum determination is impossible. 

To sum up, in some instances the proposed tangent method and procedure of systematic error correction allows excluding the necessity of mathematical or chemical resolution of overlapped peak-shaped analytical signals. 

A differential variant of the tangent method was used for the processing of the kinetic data, because a linear correlation exists between the absorbance at 700 nm and time during the first 6 min after mixing. 

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) ... 

Fig. 7.85 Common tangent method applied to AC" ref The value of the free...

The partial molar volumes for water in the ethanol solutions can be calculated by analogous procedures. An interesting alternative method is the tangent method [2]. 

The efficiency of a column, N, is a number that describes peak broadening as a function of retention and is dependent on the entire chromatographic system. The most common method for calculating N is the tangent method. 

TANGENT METHOD AREA METHOD HALF-WIDTH METHOD... 

Before dismissing the tangent method, we should mention a slight variant which becomes possible when small concentrations of product are experimentally detectable (as, for example, in a reaction yielding a coloured product with a very high extinction coefficient). In these circumstances, the appearance of the product can be followed under conditions where the concentrations of the reactants remain effectively constant at their initial values and so we can write... 

Calculate the IBW (4 a) using the tangent method as shown in Figure 4.19b. An IBW of -60 pL was found for this HPLC. Figure 4.20 shows a chart plotting column efficiencies of 150-mm long 3-pm columns of various inner diameters versus retention factors (k) on an HPLC system with a 60-pL IBW. Note that efficiency loss from instrumental dispersion can be severe for small inner diameter columns where peak volumes are smaller. Note that peaks of low k are affected more by system dispersion since peak volumes are proportional to k. 

Also to be taken into account is the background absorption which is often visible for a substantial part of the spectrum. In particular it depends upon the manner in which the sample has been prepared. As absorbances are additive, that not due to the compound under study must be subtracted from the total absorbance. This is evaluated by the tangent method (Figure 10.23). 

In the tangent method, two lines are drawn between two consecutive maxima or minima, and the distance between this tangent and the intermediate maximum or minimum is calculated this method allows correcting any variation of the base line due to a matrix effect. 

When the medium has a wide and nonlinear spectrum, it is often necessary to use a higher derivative degree in order to remove its contribution. However, in order to avoid an important loss of sensitivity, the second derivative is used, together with the tangent method, to remove the background noise. 

The peak width is measured in the same time units as retention time. The factor / depends on how the peak width is measured. It is, for example, 16 for the tangent method, S.S4S for the half-heigfrt method, and 25 for the 5[Pg.13]

Tangent method Intersection of tangents with basdine 16.00... 

Fig. L(a) Three common working definitions of the yield point for metals. (1) Load maximum (2) tangent method, (3) firoofistress" or "strain-offset method. (The proof-strain is commonly taken to be 0 /%, but is quite arbitrary.) (b) Load elongation curves for polymers. (I) Brittle, (2) strain softening, (3) cold-drawing, (4) strain-hardening, (5) rubbery. Typical definitions of the yield point are marked by arrows on curves (2), (3) and (4). Any one polymer can show behaviour raiding from (1) to (5) depending on test conditions, e.g. temperature, strain-rate, tension...

The pioneering analytical solution by Eshelby [59], for an ellipsoidal inclusion embedded in an infinite elastic medium, has been extended to nonlinear cases in the literature. For example, the secant approach by Berveiller and Zaoui [63] and the self-consistent tangent method by HiU [64] and Hutchinson [65] are generalizations of this method for elastoplastic problems. The limitation of these analytical methods persists in their inability to simulate complex material stractures, which result in inelastic responses that are too stiff [62,66]. Also, accurate stress redistribution in an inelastic analysis cannot be captured by these models [67]. Several models have been developed to resolve these issues in the literature, such as the above-mentioned tangent [64,66,68,69], secant [63,70], and affine [67,71] methods. 

Figure 4.8 Permeation curves of dichloromethane and methanol vapour in Hyflon AD80X membranes at 25 °C. Time lag calculation by the tangent method (top) and by a direct least squares fit of the entire permeation curve according to Eq. (4.6) (bottom). The thick dark line represents the experimental data the thin brighter line gives the tangent (top) or the least squares fit. The experimental and fitted lines superimpose completely in the case of DCM, and only one curve can be distinguished, whereas MeOH gives a poor fit. See text for further explanation...

With this equation the diffusion coefficient can be obtained from the time lag if the membrane thickness is known. This way to calculate the diffusion coefficient will be further referred to as the Tangent method and the procedure is schematically displayed in Figure 4.8, top section. The permeability follows from the steady state pressure increase rate and can be defined as ... 

In this work preliminary vapour permeation measurements were carried out with two different species, the rather bulky dichloromethane (DCM) molecules and the much smaller methanol molecules. Two typical permeation curves are displayed in Figure 4.8. The transport parameters, determined on the basis of the tangent method and Equations (4.9)-(4.11), are listed in Table 4.3. It contains the parameters dehned above as well as solubility C in the membrane in equilibrium with the feed pressure of penetrants. 

More careful analysis of the permeation curves show that the two vapours actually display completely different behaviour. The tangent method can be applied without problems to the DCM permeation curve, resulting in a time lag of ca. 400 s. hi contrast, this method shows that methanol has an unusually wide transient period. While the extrapo-... 

The different behaviour between the two vapours is even more evident if we fit the entire experimental curve directly with Equation (4.6) after expansion into 10 terms (n = 1, 2, 3,. .. 10). The DCM curve yields a nearly perfect fit (Figure 4.8, bottom), indicating that the DCM transport can be described well by the simple Fickian diffusion with a single diffusion constant, independent of time or concentration. In this case the values of D and S are directly obtained from the curve fit and they agree within an error of a few percent with the values in Table 4.3, obtained by the tangent method. 

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