Selecting and Evaluating the End Point

Earlier we made an important distinction between an end point and an equivalence point. The difference between these two terms is important and deserves repeating. The equivalence point occurs when stoichiometrically equal amounts of analyte and titrant react. For example, if the analyte is a triprotic weak acid, a titration with NaOH will have three equivalence points corresponding to the addition of one, two, and three moles of OH for each mole of the weak acid. An equivalence point, therefore, is a theoretical not an experimental value. 

An end point for a titration is determined experimentally and represents the analyst s best estimate of the corresponding equivalence point. Any difference between an equivalence point and its end point is a source of determinate error. As we shall see, it is even possible that an equivalence point will not have an associated end point. 

Where Is the Equivalence Point We have already learned how to calculate the equivalence point for the titration of a strong acid with a strong base, and for the titration of a weak acid with a strong base. We also have learned to sketch a titration curve with a minimum of calculations. Can we also locate the equivalence point without performing any calculations The answer, as you may have guessed, is often yes  

It has been shown that for most acid-base titrations the inflection point, which corresponds to the greatest slope in the titration curve, very nearly coincides with the equivalence point. The inflection point actually precedes the equivalence point, with the error approaching 0.1% for weak acids or weak bases with dissociation constants smaller than 10 , or for very dilute solutions. Equivalence points determined in this fashion are indicated on the titration curves in figure 9.8. 

The principal limitation to using a titration curve to locate the equivalence point is that an inflection point must be present. Sometimes, however, an inflection point may be missing or difficult to detect, figure 9.9, for example, demonstrates the influence of the acid dissociation constant, iQ, on the titration curve for a weak acid with a strong base titrant. The inflection point is visible, even if barely so, for acid dissociation constants larger than 10 , but is missing when is 10 k 

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Quasi-Newton methods start out by using two points xP and jfl spanning the interval of jc, points at which the first derivatives of fix) are of opposite sign. The zero of Equation (5.9) is predicted by Equation (5.10), and the derivative of the function is then evaluated at the new point. The two points retained for the next step are jc and either xP or xP. This choice is made so that the pair of derivatives / ( ), and either/ (jc ) or/ ( ), have opposite signs to maintain the bracket on jc. This variation is called regula falsi or the method of false position. In Figure 5.3, for the (k + l)st search, x and xP would be selected as the end points of the secant line. 

Conceptual models link anthropogenic activities with stressors and evaluate the relationships among exposure pathways, ecological effects, and ecological receptors. The models also may describe natural processes that influence these relationships. Conceptual models include a set of risk hypotheses that describe predicted relationships between stressor, exposure, and assessment end point response, along with the rationale for their selection. Risk hypotheses are hypotheses in the broad scientific sense they do not necessarily involve statistical testing of null and alternative hypotheses or any particular analytical approach. Risk hypotheses may predict the effects of a stressor, or they may postulate what stressors may have caused observed ecological effects. 

To date, there are few experimental approaches using the ABTS + radical cation for evaluating or predicting the antioxidative potency of individual biocompounds or complex matrices such as fruits, vegetables, and beverages. For decolorizing-type assays, the reaction between the preformed ABTS + and AOXs might proceed until the disappearance of these radicals. The extent of decolorization of the ABTS radical cation is determined as a function of concentration and time. This actually makes the selection of the end-point time of an assay more arbitrary and hence more difficult. 

The presence of multiple stressors also influences the selection of assessment end points. When it is possible to select one assessment end point that is sensitive to many of the identified stressors, yet responds in different ways to different stressors, it is possible to consider the combined effects of multiple stressors while still discriminating among effects. For example, if recruitment of a fish population is the assessment end point, it is important to recognize that recruitment may be adversely affected at several life stages, in different habitats, through different ways, by different stressors. The measures of effect, exposure, and ecosystem and receptor characteristics chosen to evaluate recruitment provide a basis for discriminating among different stressors, individual effects, and their combined effect. 

The last step in the problem formulation phase is the development of an analysis plan or proposal that identifies measures to evaluate each risk hypothesis and that describes the assessment design, data needs, assumptions, extrapolations, and specific methods for conducting the analysis. There are three categories of measures that can be selected. Measures of effect (also called measurement end points) are measures used to evaluate the response of the assessment end point when exposed to a stressor. Measures of exposure are measures of how exposure may be occurring, including how a stressor moves through the environment and how it may co-occur with the assessment end point. Measures of ecosystem and receptor characteristics include ecosystem characteristics that influence the behavior and location of assessment end points, the distribution of a stressor, and life history characteristics of the assessment end point that may affect exposure or response to the stressor. These diverse measures increase in importance as the complexity of the assessment increases. 

The Consultative Committee for Amount of Substance (CCQM) has set up a definition of primary methods [1, 2] and has selected some methods with the potential of being primary , from the viewpoint of the end user. From the point of view of metrology, methods used for linking the chemical measurements with the SI system at the highest level should not refer to other amount of substance standards. This requirement excludes methods which are relative in their principle. Some other methods identified as having the potential to be primary yield information expressed as amount fraction. This is essential for evaluation of purity, but in order to convert it to a value useful for transfer of the unit, additional information on the identity (molar mass) and content of the impurities is required. This additional information is needed to convert the result into amount content or similar quantities. 

Two selected rich boundary lines are shown in Figure 1. One is based on the NOa emission goal of the present work of 1.0 g N02/kg of total fuel, and the other is based on 3.0 g NOa/kg of total fuel that has been suggested as a standard for cruise operation. The upper and lower end points of the boundary lines were evaluated for kinetically controlled NOa production in one-dimensional flow with a 10.0-msec dwell time, using 100% jet fuel and 100% hydrogen, respectively. The data for intermediate mixed-fuel composition were interpolated for illustrative purposes. Both of these rich operating boundaries fall to the lean side of the flammability limit for jet fuel alone. 

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