The Working Curve

In atomic absorption the attenuation of a signal from the source is a measure of the concentration of the analyte element in the sample solution. The signal attenuation obeys Beer s law, given in equation (10-1). The form of Beer s law that is more useful for preparation of analytical working curves is that presented in equation (10-3). Since a and / of equation (10-3) are constants for any particular instrument and element, the equation reduces to 

FIGURE 10-37. Calibration curves of atomic absorption signals of zinc using absorbance and percentage absorption versus zinc concentration in /ig/ml. [From W. G. Schrenk, Evaluation of Data, in Flame Emission and Atomic Absorption Spectroscopy, Vol. 2, Edited by J. A. Dean and T. C. Rains, Marcel Dekker, New York (1971), Chap. 12. Used by permission of Marcel Dekker, Inc.] 

Use of a recorder with an amplifier allows for scale expansion, a technique for amplifying weak signals for ease of reading, but which places zero absorbance off the chart. Such a technique may make the determination of precise absorbance units indefinite. However, data also may be plotted in terms of signal intensity or in some arbitrary units, such as millimeters on the chart vs. concentration. If these data are plotted on semilogarithmic paper, a nearly linear and very useful calibration results. When scale expansion is used, noise also is amplified, so measurements of signal intensities will be less accurate. 

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In a molded polymer blend, the surface morphology results from variations in composition between the surface and the bulk. Static SIMS was used to semiquan-titatively provide information on the surface chemistry on a polycarbonate (PC)/polybutylene terephthalate (PBT) blend. Samples of pure PC, pure PBT, and PC/PBT blends of known composition were prepared and analyzed using static SIMS. Fn ment peaks characteristic of the PC and PBT materials were identified. By measuring the SIMS intensities of these characteristic peaks from the PC/PBT blends, a typical working curve between secondary ion intensity and polymer blend composition was determined. A static SIMS analysis of the extruded surface of a blended polymer was performed. The peak intensities could then be compared with the known samples in the working curve to provide information about the relative amounts of PC and PBT on the actual surface. 

The working curves in Figure 7-8 were obtained on known samples,... 

The parameter k° is generally calculated from the working curve shown in Figure 9.8... 

Should it be impossible to measure td, the problem can be overcome by using the working curve reported in Figure 13.9... 

The conditions in which ipT/ipr assumes values between 0.45 and 0.95 are the best for calculating the rate constant of the following reaction. In fact, from the working curve reported in Figure 16, based on the current ratio ipT/ipf, one can determine the value of /Cf x. 9 Since x is easily calculated, being the time necessary to move from E° (calculated under the conditions where ipr/ipt =1) to the inversion potential E(, k is determined. 

The second method is based on the measurement of the current ratio Zpr/Jpf at a given scan rate. Through the use of the working curve reported in Figure 18, one is able to determine k210... 

When the value of kf is such that one does not obtain an S-like curve, kf can be determined from the working curve reported in Figure 22a,9... 

However, if the redox couples Ox/Red and Ox /Red have sufficiently different standard potentials, can be also calculated using the working curve reported in Figure 16. In fact, considering the process simply as a reversible electron transfer followed by an irreversible first-order chemical reaction (see Section 1.4.2.2), one measures only the current ratio /pr//pf of the first couple Ox/Red. Obviously, the return peak must be recorded before the second process begins to appear this means that the direction of the potential scan must be reversed immediately after having traversed the first forward peak. 

The second way is to evaluate the ratio zP(H)/zP(i) at a given scan rate and subsequently determine kf from the working curve reported in Figure 16 (valid for an irreversible chemical reaction following a reversible electron transfer Section 1.4.2.2). 

The data partly give a quantitative presentation of the preceding discussion and can also be converted into the working curve shown in Figure 33.15 The use of this curve enables an immediate evaluation of AEo1 through the measurement of A p. 

Relative error values for the elements ranged from zero to a high of 18% with most values being 5% or less. The 18% relative error was obtained for indium and is attributable to the low concentration of this element in the solution analyzed. Moreover, the indium values were obtained on the lower end of the working curve where the sensitivity is greatly reduced. Standard deviations and coefficients of variation for the elements of interest are at acceptable levels (less than 1% standard deviation and around 5% coefficient of variation) for this technique. Again it should be pointed out that the original purpose of the subject method was to develop a rapid routine analysis for the major and minor constituents in coal ash and related materials without the necessity of several preconcentration steps, solvent extraction techniques, or pH adjustments. 

In practice, experimental data are usually plotted in conjunction with the theoretical working curve for the particular mechanism that the process in question is believed to follow. Whether or not the experimental data are consistent with the mechanism is then concluded on the basis of how well the data fit the working curve. Very often this aspect of the use of the theoretical working curve is highly subjective and it can be difficult for the reader of a paper to assess just how good the mechanism assignment really is. 

D.. .. to here, where the oxidation of I occurs at a diffusion-controlled rate. After a prescribed electrolysis time t, the potential is stepped back to the initial potential, C. Only the reduction of the unreacted carbonium ion II occurs at this potential. The value of the pseudo-first-order rate constant, k[, is then determined from a dimensionless working curve that relates the ratio of the cathodic and anodic currents to k,t. Details for the construction of the working curves (each ratio of tr/t requires a different working curve) and their subsequent use may be found in the literature [8]. 

Fig. 6.7 The double potential step chronoamperometry working curve for the eCei, mechanism (full line) and experimental data for the protonation of the anthracene radical anion by phenol (points). The scale at the top corresponds to the working curve and the scale at the bottom to the experimental data. (The parameter 6 in the figure corresponds to ff in the text.) Note that the data for the variation of ff and [PhOH] have been plotted on the same working curve. Reprinted with permission [35].

An example of the working curve approach, resulting from a study of the stereochemistry of the electrohydrodimerisation of cinnamic acid esters [38], is shown in Fig. 6.16. 

Here the experimental data for the dimerisation of the (-)-bornyl cinnamate radical anion (Scheme 6.9) are fitted to the working curve for the eC(dim) mechanism. 

For the ErQ mechanism K = kcd2/D. Figure 10a represents the working curve, K versus I S/I T, along with the simulated data from Ref. [21]. The numerical results fit the analytical approximation... 

For many mechanisms, the steady-state Eia or N tt value is a function of just one or two dimensionless parameters. If simulations are used to generate the working curve (or surface) to a sufficiently high resolution, the experimental response may be interpolated for intermediate values without the need for further simulation. A free data analysis service has been set up (Alden and Compton, 1998) via the World-Wide-Web (htttp //physchem.ox.ac.uk 8000/wwwda/) based on this method. As new simulations are developed (e.g. for wall jet electrodes), the appropriate working surfaces are simulated and added to the system. It currently supports spherical, microdisc, rotating disc, channel and channel microband electrodes at which E, EC, EC2, ECE, EC2E, DISP 1, DISP 2 and EC processes may be analysed. 

Commercially available microdisc electrodes of radii 0.6-70/xm may be used for steady-state measurements without problems associated with natural convection. Dimensionless rate constants for spherical and microdisc electrode were interpolated from the working curves of Alden and Compton (1997a). 

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