Non-linear calibration curves

The response of the ECD is not linear over a large concentration range, and the curvature varies with the compound. Therefore, calibration curves have to be determined, one for each halocarbon. again each one with at least seven points. The calibration values can be fitted to polynomial equations e.g., Microsoft Excel Solver). The curve fitting should be done stepwise, first assuming a second order equation and then refining to higher orders if needed, until the deviations of standard points from the curve are within acceptable limits. 

Chromatographic integration software usually has facilities to evaluate peaks, even if the calibration curve is quadratic or cubic. If not, it is recommended that the axes are switched and the concentration is plotted against detector signal as in Fig. 23-5 B. 

The shape of the caUbration curve for a speciGc compound changes less with time than the detector response. It is thus possible to use the same curve for calibration over a reasonably long period, as long as it is adjusted for variations of detector response. The method is called stretching. This is especially useful when time does not permit a seven to nine point standard curve to be determined often enough. A few standards are measured to obtain the correction factor, and the concentration is calculated as in Fig. 23-5. 

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Non-linear curves may be treated using Equation 9 directly, using the techniques of non-linear least squares, when appropriate. (Note that a non-linear calibration curve does not necessarily imply non-linear least squares. The latter is necessary only if the problem is non-linear in the estimated parameters (16). For example, y = a+bx+cx and y = a+bx are both non-linear functions, but only the latter is non-linear in the parameters.)... 

Non-linear Hydrodynamic Volume Calibration Curve. The hydrodynamic calibration curve, log. V shown in Figure lb, is generated using the commercially available narrow MWD polystyrene standards listed in Table 1 and published values (28, 29) of the Mark-Houjjink parameters K and a for polystyrene in THF a 25°C, (K=1.6 x 10, o = 0.706 for > 10,000 and K = 9.0 x 10, a = 0.5 for Mw <10,000). The experimental data points composing the non-linear calibration curve were fitted with the phenomenologically based Yau-Malone equation.(30) This equation is derived from diffusion theory and is expressed as follows ... 

The non-linear calibration models are also acceptable. Figure 4.6 shows an example of a non-linear calibration approximated with a quadratic fit. Non-linear calibration curves are not acceptable if used to compensate for the detector saturation at a high concentration level. 

In analytical applications of liquid chromatography the most common causes of peak asymmetry are mixed mechanisms of retention, incompatibility of the sample with the chromatographic mobile phase, or development of excessive void volume at the head of the column. In preparative applications of liquid chromatography and related techniques, column overload can also contribute to peak asymmetry. The causes of severe peak asymmetry in analytical applications should be identified and corrected because they are frequently accompanied by concentration-dependent retention, non-linear calibration curves and poor precision. In addition, peak asymmetry can significantly compromise column efficiency leading, in turn, to reduced resolution and lower peak capacity (see sections 2.5 and 2.6). 

As in thin-layer chromatography the calibration curves are usually only linear in a limited range, various linearisation functions are provided in the software program of this system, but also non-linear calibration curves for working in a wider concentration range 15). 

If an alcohol such as methanol can be separated by ion-exclusion chromatography using water as the eluent, why not do the reverse and separate water using a methanol eluent Stevens et al. [12] did just this. They added a small amount of sulfuric acid to the eluent and detected the chromatographic water peak by a decrease in conductivity. The main drawback with this method was a non-linear calibration curve with very poor detection sensitivity in some concentration regions. 

The NMR experiment makes the direct observation of atoms possible. The integral of an NMR signal is strictly linearly proportional to the amount of atoms in the probe volume. The signals are a measure of molar ratios of molecules, independently of the molecular weight. There are no response factors such as those in UV-detection caused by varying extinctions dependent on molecular structures non-linear calibration curves such as those found with light-scattering detectors are unknown to NMR spectroscopy. 

Ma and Kim (1995) developed an on-line LC/TS/MS method for the analysis of PL molecular species in rat brain. After total lipid extraction, the extract was subjected to analysis with on-line reversed-phase HPLC and filament-on TS/MS. By using non-conventional HPLC conditions, partial separation of individual PL classes (PS, PI, PE and PC) and partial separation of molecular species within each class were achieved. By monitoring the retention time and the characteristic fragment ions (DG ions) formed in the filament-on TS process, individual molecular species in each PL class could be identified. Although non-linear calibration curves were observed for all DG ions monitored, even in the presence of an internal standard, semiquant-itative and quantitative results could still be obtained for a mixture of PLs. 

An ideal transducer would have an exactly linear relationship between pressure and output voltage. In reality, there will always be some deviation from the ideal linear relationship this is referred to as non-linearity. A best straight line (BSL) is fitted to the non-linear curve. The deviation from BSL is quoted in the specifications and expressed as a percent of full scale (FS). The non-linear calibration curve is determined in the ascending direction, i. e., with the pressure going from zero to full rating. 

For optimum sensor performance, the electron transfer process should be rapid and chemically reversible. With conventional metal electrodes it is unlikely that reversible electron transfer processes will change with analyte concentration. However, with CMEs it is likely this will occur. If so, then an extra degree of complexity is added to the interpretation of the CME sensor response in that non-linear calibration curves as well as non-linear standard addition curves will be obtained. Furthermore, the electron transfer process is also dependent on the chemical environment. For example, the nature and concentration of the counter-ion in conductive polymers have a significant influence. In such cases, matrix matching between standards and samples must be carried out. 

Figure 24.2 Non-linear calibration curve for biotin by post-column avidin reaction.

In some special analyses it may be necessary to extend the absorbance measurements beyond the linear range (i.e., beyond the validity of Beer-Lamberts s law). In these cases a non-linear calibration curve has to be generated by measuring concentrations versus spectrophotometer readings. The sample concentrations are then calculated according to the non-linear calibration function or corresponding correction terms applied to the linearly calculated sample concentrations. 

For example, a tandem MS analysis through PIS of mIz 184 was performed for analysis of PC species present in a lipid extract of Chinese hamster ovary cells in which an equimolar mixture of four PC species was used as internal standards [22]. Since this fragment ion represents the protonated phosphocholine under experimental conditions, the PIS detects all the protonated choline-containing GPL species in the solution. However, the spectrum clearly showed the decreases in the ion peak intensities of these internal standards as their molecular weights increased. A calibration curve from the ion peak intensities of the standards was derived and used to correct the experimental ion abundances to quantify individual PC species in this approach [22]. Accordingly, a built-in (non)linear calibration curve from two or more internal standards can be determined from their peak intensities. The concentration of each species of the class can then be derived from its ion peak intensity of the tandem MS mass spectrum by comparison with the calibration curve. 

Fluorescence has an immediate advantage over absorption in that it is not a relative technique. Provided that the sample is optically thin, the ratio of fluorescent signal to laser intensity (in LIF) gives the absolute species concentration. The quantitative use of fluorescence is restricted for the following reasons (i) quenching (by impurities) (ii) temperature-dependence (Hi) non-linear calibration curves and... 

The most sensitive analytical lines for palladium are at 244.791 nm and 247.642 nm with characteristic concentrations of cq = 0.064 mg/L and cq = 0.10 mg/L, respectively, in a very fuel-lean air / acetylene flame, whereby the former exhibits a highly non-linear calibration curve. The characteristic mass at the second line, using a transversely heated graphite tube atomizer, is about mo = 22 pg. 

In many instances time-consuming sample preparations such as filtrations, distillations or extractions can be avoided by employing ion-selective electrodes. From an instrumental standpoint, no expensive, sophisticated instrumentation, i.e. integrators, compensators for non-linear calibration curves, etc. are needed. After the appropriate calibration, the concentration can be read directly from a voltmeter with a high impedance input. 

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