Half-slopes

The logarithmic form of Eq. (25) is a hyperbolic curve, just like Eq. (24), with an apparent pK associated with the pH at half-slope positions (cf Fig. 3.4, solid-line curves). 

A relatively simple and effective way to determine if a selected reexpression procedure linearizes the data can be completed with EDA pencil-paper techniques (Figure 2.28). It is known as the method of half-slopes in EDA parlance. In practice, it is suggested, when reexpressing a data set to approximate a straight line, that this EDA procedure be used rather than the Fc- test for lack of fit. 

A connnon approach has been to measure the equilibrium constant, K, for these reactions as a fiinction of temperature with the use of a variable temperature high pressure ion source (see section (Bl.7.2)1. The ion concentrations are approximated by their abundance in the mass spectrum, while the neutral concentrations are known from the sample mlet pressure. A van t Hoff plot of In K versus /T should yield a straight Ime with slope equal to the reaction enthalpy (figure B1.7.11). Combining the PA with a value for basicityG at one temperature yields a value for A.S for the half-reaction involving addition of a proton to a species. While quadnipoles have been tire instruments of choice for many of these studies, other mass spectrometers can act as suitable detectors [19, 20]. 

Figure 6.3 shows some data which constitute a test of Eq. (6.26). In Fig. 6.3a, Rp and [M] are plotted on a log-log scale for a constant level of redox initiator. The slope of this line, which indicates the order of the polymerization with respect to monomer, is unity, showing that the polymerization of methyl methacrylate is first order in monomer. Figure 6.3b is a similar plot of the initial rate of polymerization—which essentially maintains the monomer at constant con-centration—versus initiator concentration for several different monomer-initiator combinations. Each of the lines has a slope of indicating a half-order dependence on [I] as predicted by Eq. (6.26).

Typically the plate is half-moon in shape and is sloped slightly in the direction of liquid flow. Gas contacts the liquid as it showers from the plate, and a serrated lip or weir at the edge of the plate can be used to improve the distribution of liquid in the shower. 

When Ki, square root of Ki, or loguj , is proportional to one-half log A". Thus, this relationship is shown by a straight line with a slope of 0.5 in Figure 5-15. When Ki, > Kr, the total effective natural frequency is equal to the natural rigid-body frequency. The actual curve lies below these two straight lines as shown in Figure 5-15. 

In addition to the elimination rate constant, the half-life (T/i) another important parameter that characterizes the time-course of chemical compounds in the body. The elimination half-life (t-1/2) is the time to reduce the concentration of a chemical in plasma to half of its original level. The relationship of half-life to the elimination rate constant is ti/2 = 0.693/ki,i and, therefore, the half-life of a chemical compound can be determined after the determination of k j from the slope of the line. The half-life can also be determined through visual inspection from the log C versus time plot (Fig. 5.40). For compounds that are eliminated through first-order kinetics, the time required for the plasma concentration to be decreased by one half is constant. It is impottant to understand that the half-life of chemicals that are eliminated by first-order kinetics is independent of dose. ... 

Evidently the reaction shown in Fig. 2-8 is first-order (or pseudo first-order) over the period of time (about 2.5 half-lives) plotted in the figure. From the slope of the line the rate constant is calculated to be 4.90 x 10 s . ... 

The E/GDP for the United States has sloped steadily downward from 18,000 to 11,000 Btu/. Europe and Japan are typically only half as energy-intensive as the United States. An explanation is that, during their development. Western Europe and Japan were petroleum-poor compared to the United States, so energy use was perceived to imply imports (and risk of supply disruption) and trade deficits. Thus, they adopted tax policies to conseive energy. The United States took the opposite path to stimulate economic growth, domestic oil and gas production was subsidized. 

Slope the trays downward from liquid inlet to outlet, with the total drop from inlet to outlet weir not exceeding one-half the calculated gradient. 

Drug elimination may not be first order at high doses due to saturation of the capacity of the elimination processes. When this occurs, a reduction in the slope of the elimination curve is observed since elimination is governed by the relationship Vmax/(Km- -[conc]), where Vmax is the maximal rate of elimination, Km is the concentration at which the process runs at half maximal speed, and [cone] is the concentration of the drug. However, once the concentration falls below saturating levels first-order kinetics prevail. Once the saturating levels of drugs fall to ones eliminated via first-order kinetics, the half time can be measured from the linear portion of the In pt versus time relationship. Most elimination processes can be estimated by a one compartment model. This compartment can... 

FIGURE 11.17 Symmetrical and asymmetrical dose-response curves, (a) Symmetrical Hill equation with n = 1 and EC5o = 1.0. Filled circle indicates the EC50 (where the abscissa yields a half maximal value for the ordinate). Below this curve is the second derivative of the function (slope). The zero ordinate of this curve indicates the point at which the slope is zero (inflection point of the curve). It can be seen that the true EC50 and the inflection match for a symmetrical curve, (b) Asymmetrical curve (Gompertz function with m = 0.55 and EC50= 1.9). The true EC50 is 1.9, while the point of inflection is 0.36. 

Figure 9.5 Extrapolation of the cell emf data of G. A. Linhart [/. Am. Chem. Soc., 41, 1175-1180 (1919)] to obtain E° for the Ag/AgCl half-cell. The dashed line gives the limiting slope as predicted by Debye-Hiickel theory. LHS = Left-hand side of equation see text.

As an example, consider the polymerization of methyl methacrylate, initiated by a,a -azobisisobutyronitrile.4 The dependence of the initial rate on the concentration of the initiator is displayed in Fig. 1-1, which shows them on a double logarithmic scale. The points define a straight line with a least-squares slope of0.496. Clearly the reaction is half-order with respect to the concentration of the initiator. 

A plot of the initial reaction rate versus concentration, on logarithmic scales. The reaction is the polymerization of methyl methacrylate, and the concentration is that of the initiator, azobisisobutyronitrile. The slope is 0.496, showing that the reaction is half-order with respect to the initiator concentration. 

All quantities on the right are time-independent save the last. Thus, a plot of In F, - Yr+T versus time is linear with slope —k. Each point appearing in the plot depends upon two readings, as before, but now upon two different readings. For best accuracy, r is chosen as two to three half-times. 

The first of these reactions was carried out in 1,4-cyclohexadiene over a temperature range of 39 to 100 °C. It is fairly slow the half-times were 20 h and 3.4 min at the extremes. Reaction (7-11) is quite fast the second-order rate constant, kn, was evaluated over the range 6.4 to 47.5 °C. Values of feio and fen are presented in Table 7-1. The temperature profiles are depicted in Fig. 7-1 from their intercepts and slopes the activation parameters can be obtained. A nonlinear least-squares fit to Eq. (7-1) or... 

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