Quantitative curves

The VBSSA rule can be used to select relevant VB structures and construct the semiquantitative and quantitative curve-crossing VB diagram. Combining the MO method and VB calculations, we investigated atom exchange reactions H + HLi —> H2 + Li and H + LiH—>HLi + H, and discussed the effect of energy differences between the VB structures on the activation energy and properties of the transition state [70],... 

While CH5 is a high-energy transition state in the nucleophilic attack of CH4 by H" (vide supra), SiHs is a stable intermediate which has been characterized in the gas phase [70]. These stability differences do not seem to originate from the absence or availability of d orbitals. Indeed a recent study by Reed and Schleyer [71] demonstrates that SiH remains stable even if calculated in a basis set devoid of d orbitals. Quantitative curve crossing diagrams provide a vivid account of the difference betwen the SiH and CH5 species in terms of an intermediate configuration, not based on d-AO"s participation, which is stable for Si and very unstable in the carbon analog [12 b, c]. 

Fig. 10 Quantitative curve resolving of free water in lipids [230].

Now let us take a closer look at the two most commonly used heat engines (steam and gas turbines) to see whether they achieve this efficiency in practice. To make a quantitative assessment of any combined heat and power scheme, the grand composite curve should be used and the heat engine exhaust treated like any other utility. 

The preceding appropriate placement arguments assume that the process has the capacity to accept or give up the reactor heat duties at the given reactor temperature. A quantitative tool is needed to assess the capacity of the background process. For this purpose, the grand composite curve can be used and the reactor profile treated as if it was a utility, as explained in Chap. 6. 

Appropriate placement can be assessed quantitatively using the grand composite curve. The streams associated with the reactor can be represented as a grand composite curve for the reactor and then matched against the grand composite curve for the rest of the process. 

The appropriate placement of distillation columns when heat integrated is not across the pinch. The grand composite curve can be used as a quantitative tool to assess integration opportunities. 

The predicted cumulative cash-flow curve for a project throughout its life forms the basis for more detailed evaluation. Many quantitative measures or indices have been proposed. In each case, important features of the cumulative cash-flow curve are identified and transformed into a single numerical measure as an index. 

Many groups are now trying to fit frequency shift curves in order to understand the imaging mechanism, calculate the minimum tip-sample separation and obtain some chemical sensitivity (quantitative infonuation on the tip-sample interaction). The most conunon methods appear to be perturbation theory for considering the lever dynamics [103], and quantum mechanical simulations to characterize the tip-surface interactions [104]. Results indicate that the... 

When structure-property relationships are mentioned in the current literature, it usually implies a quantitative mathematical relationship. Such relationships are most often derived by using curve-fitting software to find the linear combination of molecular properties that best predicts the property for a set of known compounds. This prediction equation can be used for either the interpolation or extrapolation of test set results. Interpolation is usually more accurate than extrapolation. 

Quantitative studies of solid-state organic reactions were performed by Glazman (267. 268). Equal amounts of acetic anhydride and 2-aminothiazole (grain diameter 0.15 mm) were mixed for 20 rain, and the mixture was heated in a glycerol bath at 0.5°C per minute. Heating curves showed that the reaction starts in the solid phase the use of an eutectic composition of organic reactants increases the yields. 

A second spectrophotometric method for the quantitative determination of Pb + levels in blood gives a linear normal calibration curve for which... 

An external standardization allows a related series of samples to be analyzed using a single calibration curve. This is an important advantage in laboratories where many samples are to be analyzed or when the need for a rapid throughput of samples is critical. Not surprisingly, many of the most commonly encountered quantitative analytical methods are based on an external standardization. 

A fifth spectrophotometric method for the quantitative determination of the concentration of Pb + in blood uses a multiple-point standard addition based on equation 5.6. The original blood sample has a volume of 1.00 mb, and the standard used for spiking the sample has a concentration of 1560 ppb Pb +. All samples were diluted to 5.00 mb before measuring the signal. A calibration curve of Sjpike versus Vj is described by... 

In a titrimetric method of analysis the volume of titrant reacting stoichiometrically with the analyte provides quantitative information about the amount of analyte in a sample. The volume of titrant required to achieve this stoichiometric reaction is called the equivalence point. Experimentally we determine the titration s end point using a visual indicator that changes color near the equivalence point. Alternatively, we can locate the end point by recording a titration curve showing the titration reaction s progress as a function of the titrant s volume. In either case, the end point must closely match the equivalence point if a titration is to be accurate. Knowing the shape of a titration... 

A quantitative analysis for NH3 in several household cleaning products is carried out by titrating with a standard solution of HGl. The titration s progress is followed thermometrically by monitoring the temperature of the titration mixture as a function of the volume of added titrant. Household cleaning products may contain other basic components, such as sodium citrate or sodium carbonate, that will also be titrated by HGl. By comparing titration curves for prepared samples of NH3 to titration curves for the samples, it is possible to determine that portion of the thermometric titration curve due to the neutralization of NH3. 

Equations 10.4 and 10.5, which establish the linear relationship between absorbance and concentration, are known as the Beer-Lambert law, or more commonly, as Beer s law. Calibration curves based on Beer s law are used routinely in quantitative analysis. 

Quantitative Analysis for a Single Analyte The concentration of a single analyte is determined by measuring the absorbance of the sample and applying Beer s law (equation 10.5) using any of the standardization methods described in Chapter 5. The most common methods are the normal calibration curve and the method of standard additions. Single-point standardizations also can be used, provided that the validity of Beer s law has been demonstrated. 

Standardizing the Method Equations 10.32 and 10.33 show that the intensity of fluorescent or phosphorescent emission is proportional to the concentration of the photoluminescent species, provided that the absorbance of radiation from the excitation source (A = ebC) is less than approximately 0.01. Quantitative methods are usually standardized using a set of external standards. Calibration curves are linear over as much as four to six orders of magnitude for fluorescence and two to four orders of magnitude for phosphorescence. Calibration curves become nonlinear for high concentrations of the photoluminescent species at which the intensity of emission is given by equation 10.31. Nonlinearity also may be observed at low concentrations due to the presence of fluorescent or phosphorescent contaminants. As discussed earlier, the quantum efficiency for emission is sensitive to temperature and sample matrix, both of which must be controlled if external standards are to be used. In addition, emission intensity depends on the molar absorptivity of the photoluminescent species, which is sensitive to the sample matrix. 

In most quantitative analyses we are interested in determining the concentration, not the activity, of the analyte. As noted earlier, however, the electrode s response is a function of the analyte s activity. In the absence of interferents, a calibration curve of potential versus activity is a straight line. A plot of potential versus concentration, however, may be curved at higher concentrations of analyte due to changes in the analyte s activity coefficient. A curved calibration curve may still be used to determine the analyte s concentration if the standard s matrix matches that of the sample. When the exact composition of the sample matrix is unknown, which often is the case, matrix matching becomes impossible. 

The slope of the calibration curve is found to be -57.98 mV per tenfold change in the concentration of F , compared with the expected slope of-59.16 mV per tenfold change in concentration. What effect does this have on the quantitative analysis for %w/w Snp2 in the toothpaste samples ... 

Quantitative Calculations Quantitative analyses are often easier to conduct with HPLC than GC because injections are made with a fixed-volume injection loop instead of a syringe. As a result, variations in the amount of injected sample are minimized, and quantitative measurements can be made using external standards and a normal calibration curve. 

If the rate is measured at a fixed time, then both k and are constant, and a calibration curve of (rate)f versus [A]o can be used for the quantitative analysis of the analyte. 

In a quantitative flow injection analysis a calibration curve is determined by injecting standard samples containing known concentrations of analyte. The format of the caK-bration curve, such as absorbance versus concentration, is determined by the method of detection. CaKbration curves for standard spectroscopic and electrochemical methods were discussed in Chapters 10 and 11 and are not considered further in this chapter. 

For a two-factor system, such as the quantitative analysis for vanadium described earlier, the response surface is a flat or curved plane plotted in three dimensions. For example. Figure 14.2a shows the response surface for a system obeying the equation... 

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