Schwartz plot

Vb / Va) (A / Cb), and next to it compute the quantity [H+] Vb / Va, then make the Schwartz plot. It will save time and effort to scale both through division by the constant Va> because you already have a column for VbIVa. 

You will notice that the Schwartz plot is somewhat more affected by the noise than the Gran plots, but that all three are still quite serviceable as long as the noise amplitude is not too large. 

Schwartz has published some hypothetical data for the titration of a 1.02 X ICr" M solution of a monoprotic weak acid (pXa = 8.16) with 1.004 X ICr M NaOH. " A 50-mL pipet is used to transfer a portion of the weak acid solution to the titration vessel. Calibration of the pipet, however, shows that it delivers a volume of only 49.94 ml. Prepare normal, first-derivative, second-derivative, and Gran plot titration curves for these data, and determine the equivalence point for each. How do these equivalence points compare with the expected equivalence point Comment on the utility of each titration curve for the analysis of very dilute solutions of very weak acids. 

Schwartz, L. M. Advances in Acid-Base Gran Plot Methodology, /. Chem. Educ. 1987, 64, 947-950. 

Useful reviews of the kinetics of autocatalytic reactions have recently been published by Mata-Perez and Perez-Benito (22) and by Schwartz Q2) For an autocatalytic reaction a plot of r/c, where r is the rate of reaction and c the concentration of reactant, as a function of c should be linear with a negative slope (22). When this analysis is applied to the possibility of autocatalysis of liquids production by LOG, no dependence of r/c on c was found. In fact, least uares "correlation coefficients were in the range 0.01 - 0.03. Although the initid hypothesis of autocatalysis by thiols is shown to be untenable, an alternative is the possibility of autocatalysis by H2S. 

Since the development of the equation, it has been tried to derive further information from it. Rees and Rue [129] determined the area under the Heckel plot. Duberg and Nystrom [137] used the nonlinear part for characterization of particle fracture. Paronen [138] deduced elastic deformation from the appearance of the Heckel plot during decompression. Morris and Schwartz [139] analyzed different phases of the Heckel plot. Imbert et al. [134] used, in analogy to Leuenberger and Ineichen [14], percolation theory for the compression process as described by the Heckel equation. Based on the Heckel equation, Kuentz and Leuenberger [135,140] developed a new derived equation for the pressure sensitivity of tablets. 

A phase change scheme similar to those described above was proposed by Schwartz and Schmidt 141,142) on the basis of LEED experiments, and by Schiith and Wicke (91,101) on the basis of IR measurements for the oscillatory CO/NO reaction on Pt(lOO). The experiments of Schwartz and Schmidt demonstrated that the transition from the high- to the low-reaction-rate state was accompanied by a change from the 1 x 1 to the hex phase in LEED patterns. The position of the L-CO band in the IR spectra recorded during oscillations varied between the high- and low-reaction-rate states with a relatively high absorption band below 2050 cm present in the low-reaction-rate state, which is characteristic of CO on the Pt(lOO) hex surface. Further proof was provided by Clausius-Clapeyron plot of the conditions for the occurrence of oscillations, which yielded points near the isostere associated with the hex 1 x 1 phase transition (Fig. 14). 

Some examples of simple and complex Freundlich and Langmuir isotherm plots are shown in Fig. 10.12 from Domenico and Schwartz (1990). 

Vary the amplitude reaofthe noise, and observe the results. While the value of nawill also affect the progress and titration curves, and the Schwartz and Gran plots, you will notice that it has a much more dramatic effect on the first derivative. In fact, the theoretical shape of the first derivative is visible only for na < 0.001, whereas the linear (Schwartz and Gran) plots barely show the effects of noise. At na = 0.01 the linear plots are noisy but can still be used, especially when combined with a least-squares line, but the derivative fails miserably to indicate the position of the equivalence point. At na = 0.03, only the Gran2 plot is still serviceable. 

Fig. 4.5-1 The progress and titration curves, the Schwartz and Gran plots, and the first derivative of the titration curve, for the titration of 0.1 M acetic acid (p Ka = 4.7) with 0.1 M NaOH, without Gaussian noise added, na = 0. The first derivative is computed with a 13-point moving parabola.

The approaches we have discussed so far to determine the precise location of the equivalence point use more than just one point, and are therefore in principle less prone to experimental error. The Schwartz and Gran plots rely on a linearization of the titration curve unfortunately, for samples that contain more than one monoprotic acid or base, linearization is no longer possible, nor is it (in general) for polyprotic acids and bases. And as for the alternative, we have seen that taking the derivative is easily overwhelmed by experimental noise. Is there no more robust yet general way to determine the equivalence volume with better precision ... 

This criterion for gas-phase diffusion limitation is illustrated in Figure 12.7 for a series of droplet diameters and an arbitrarily chosen sg = 0.1. In Figure 12.7 the inequality (12.85) corresponds to the area below and to the left of the lines. For a given situation if the point (k, H K) is to the left of the corresponding line in Figure 12.7, then gas-phase mass transport limitation does not exceed 10%. Similar plots, introduced by Schwartz (1984), provide an easy way to ascertain whether there is a mass transport limitation for a given condition of interest. 

D. Eisenberg, E. Schwartz, M. Komaromy. and R. Wall, Analysis of membrane and surface protein sequences with the hydrophobic moment plot, J. Mol. Biol. 179, 125-142 (1984). 

A large value of m indicates that the all samples of the material fail over a narrow range of apphed stresses. Therefore the stress required to cause failure is more predictable than it would be if the failures occurred over a wide range of applied stresses, as would be indicated by a low value of m. Saito has published several plots for specimens prepared differently, e.g., ground with different mesh abrasives, which demonstrate the sensitivity of this parameter to surface preparation. Schwartz compiled data over a period of years showing how Weibull modulus and mean tensile strength depend on sample preparation. 

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