Linear segment curve

Curve 6 is a linear segment curve 11 is a step function. 

In this chapter and several that follow, we deal exclusively with sigmoidal titration curves. We explore linear-segment curves in Section 26A-5. 

Amperometric titration A method based on applying a constant potential to a working electrode and recording the resulting current a linear segment curve is obtained. 

Linear segment curve A titration curve in which the end point is obtained by extrapolating linear regions well before and after the equivalence point useful for reactions that do not strongly favor the formation of products. 

Concentration is not the only property that may be used to construct a titration curve. Other parameters, such as temperature or the absorbance of light, may be used if they show a significant change in value at the equivalence point. Many titration reactions, for example, are exothermic. As the titrant and analyte react, the temperature of the system steadily increases. Once the titration is complete, further additions of titrant do not produce as exothermic a response, and the change in temperature levels off. A typical titration curve of temperature versus volume of titrant is shown in Figure 9.3. The titration curve contains two linear segments, the intersection of which marks the equivalence point. 

Carter, K. N., Scott, D. M., Salomon, J. K., and Zarcone, G. S., Confidence Limits for the Abscissa of Intersection of Two Least-Squares Lines Such as Linear Segmented Titration Curves, Anal. Chem. 63, 1991, 1270-1278. 

Suppose that the actual behavior of temperature versus enthalpy is known and is highly nonlinear, as shown in Figure 19.4. How can the nonlinear data be linearized so that the construction of composite curves and the problem table algorithm can be performed Figure 19.4 shows the nonlinear streams being represented by a series of linear segments. The linearization of the hot streams should... 

Figure 5.3 Schematic representation of the penetration profile for bulk, grain boundary, and dislocation diffusion in a polycrystalline solid. The initial part of the curve is bell shaped, and the part due to short-circuit diffusion is made up of linear segments. The insets show the distribution of the tracer in the sample.

There are a number of ways to model calibration data by regression. Host researchers have attempted to describe data with a linear function. Others ( 4,5 ) have chosen a higher order or a polynomial method. One report ( 6 ) compared the error in the interpolation using linear segments over a curved region verses using a curvilinear regression. Still others ( 7,8 ) chose empirical or spline functions. Mixed model descriptions have also been used ( 4,7 ). 

The linear segment of the universal calibration curve is customarily expressed as... 

Replicate tests were conducted at 3, 25 and 75% RH and good repeatability was observed. The elastic modulus as function of RH calculated from these stress-strain curves is also shown. The shape of the stress-strain curve can be approximated by two linear segments. It is clear that RH affects the elastic modulus and the yield stress of these MEAs with Nation-type membrane. Note that the elastic modulus more than doubled when the MEA was dried from 75 to 3% RH. However, the yield strain and the slope of the second linear segment are affected to a lesser degree it is notable that the 3% RH condition exhibited the lowest strain-to-failure. Despite some variations, the MEAs tested at all four RH levels were found to be fairly ductile, with strain-to-failure exceeding 100%. The yield stress varies from approximately 12 MPa to 17.5 MPa and the strain-to-failure varies from 86.4 to 152.7%. This is indicative of the initial non-uniformity of the MEA and the presence of initial random defects in the as-fabricated membrane or MEAs. 

Table 2.1 lists and defines the terminology of mechanical stress-strain testing. Table 2.2 shows the values for procine skin. The typical stress-strain relationship for human skin is shown in Fig. 2.6 and the E for modulus value is shown as the slope on the linear segment of the curve. 

Fig. 22 a Measured force curves of linear segmented poly(N-isopropylacrylamide-seg-styrene) (PNIPAM-seg-St) copolymer chains adsorbed on a hydrophobic PS substrate in water, b Statistics of the distance between two adjacent peaks in the measured force curves [97]... 

Linear titration curve — A type of -> titration curve in which a variable that is directly proportional to the concentration of the titrand and/or -> titrant, and/or a product of their chemical reaction is plotted as a function of the volume of titrant added. Thus, a linear titration curve generally consists of two linear segments that have to be extrapolated to intersect at a point that is associated with the equivalence point. The measurements are performed below and above the zone of the equivalence point and preferably away from this last point where nonlinear behavior is commonly found [i]. Linear titration curves are typical for - amperometric titrations, and - conductometric titrations, whereas - poten-tiometrc titrations yield nonlinear curves (- logarithmic titration curve). 

Large Debond Energy. For LDE (Fig. 1.23), when cr< crs, the unloading modulus depends on both r and T, (Fig. 1.26). There are also linear segments to the unloading and reloading curves. These segments can be used to establish... 

In a NaDoS foam solution formed in the absence of an electrolyte, there is no linear segment that crosses the abscissa axis and the VF /Vg = f(C) curve is S-shaped without a clearly expressed emin> (Fig. 10.9, curve 3). 

The lowest residual concentration of lysozyme found by extrapolation of the linear segment of the VFIVg = f(C) dependence gives the value of 2.1-10 6 mol dm 3 which is close to the concentration of the onset of black spot formation (Fig. 10.10, curve 2). At pH values less than the isoelectric point that corresponds to the formation of thick films from solutions of such surfactants [96], the VF/Vg = f(C) dependence has an S-shaped course. 

Ll Cl" " (Figure 12) was the most abundant species, and hence the easiest to measure. A reasonably linear segment is apparent over ca. one decade, although this curve is less steep than in the case of Li. This may reflect the more complex Boltzmann population of states for this tetratomic molecule, where a pre-exponential factor is necessary. The departure from linearity is also less abrupt, and occurs at ca. 10.20 eV. We take this to be the adiabatic I.P. of Li2Cl2. Our earlier photoelectron spectroscopic studies (18) yielded 10.22 and 10.17 eV for this quantity, employing two alternative methods of extrapolation. Hence, the inference seems quite plausible that departure from linearity on the semi-logarithmic plot yields a value very close to the adiabatic ionization potential. 

Attempts are also known to relate the type of time dependence of viscosity in the curing process to the kinetics of the reaction. Thus, upon curing of diglycidyl ester of Bisphenol A by triethanolamine, the viscosity curve Tj (t) was approximated by two linear segments [40]. The appearance of an inflection point is explained by the authors on the basis of the formation of a meshing network an the linearity of the tj (t) dependence in the first party by the fact that a curing reaction is of a zeroth order. 

The change in absorbance of a solution may be used to follow the change in concentration of a radiationabsorbing constituent during a titration. The absorbance is directly proportional to the concentration of the absorbing constituents. A plot of absorbance versus titrant consists, if the reaction is complete, of two straight lines that intersect at the endpoint. If the reaction is appreciably incomplete, extrapolation of the two linear segments of the titration curve establishes the intersection and endpoint volume. 

When effect can be measured for a wide concentration range, the relationship between effect and concentration is often observed to be curvilinear. A semi-logarithmic plot of effect versus log concentration commonly linearizes these data within the approximate range 20-80% of maximal effect. This log transformation of the concentration axis facilitates a graphical estimation of the slope of the apparently linear segment of the curve ... 

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