Back extrapolation

Determination of transformation enthalpies in binary systems. Just as consistent values of for elements can be obtained by back-extrapolation from binary systems, so it is possible to obtain values of by extrapolating the enthalpy of mixing vs composition in an alloy system where the phase has a reasonable range of existence. The archetypal use of this technique was the derivation of the lattice stability of f.c.c. Cr from the measured thermodynamic properties of the Ni-based f c.c. solid solution (7) in the Ni-Cr system (Kaufman 1972). If it is assumed that the f.c.c. phase is a regular solution, the following expression can be obtained ... 

In one particular case, a motorcyclist hit a car and was severely injured. At the hospital, approximately one hour after the accident, the cyclist s blood alcohol content (BAG) was 0.021%. Back extrapolation (see Chapter 7) to the time of the accident could increase that value to 0.036%. His urine tests revealed use of cocaine and marijuana. Marijuana metabolites were confirmed by gas chromatography/mass spectrometry (GC/MS). All these results were used to try and prove that the cyclist was impaired. However, the presence of drug in urine is not a good indicator of drug concentration in blood or of impairment at the time of the accident, and the BAG was probably too low to cause impairment. At best, one could say that the cyclist does use drugs and uses several of them at the same time, but use of the drugs may not have played a role in causing the accident. The case was settled. 

Examples of actual cases involving alcohol include DWl, motor vehicle accidents (involving alcohol in either pedestrian or driver), rape, personal injury (tripping or falling), burglary, theft, assault, and attempted murder. In many cases, it was necessary to calculate the BAG at the time of the accident or other event that had occurred hours before the breath or blood sample was taken (a process known as back extrapolation ). 

Graphically, the determination can be achieved as follows The carbonyl chloride concentration of the test sample can be read from the calibration graph by back extrapolation to the x-axis where the magnitude of the intercept Z is equal to the carbonyl chloride concentration. 

The effect of monomer concentration on molecular weights was investigated at various temperatures. When the temperature lines are back-extrapolated toward lower monomer concentrations (Fig. 4, in... 

In this case, it is the initial part of the isotherm which corresponds to monolayer-multilayer adsorption on the mesopore walls. If the corresponding section of the as-plot is linear and back-extrapolates to the origin, the slope provides a measure of a(S) which is now the total surface area. We may also conclude that there are no... 

Once the micropores have been filled, both plots in Figure 8.3 become linear, provided that capillary condensation is absent (or only detectable at high p/p°). The low slope signifies that multilayer adsorption has occurred on a relatively small external surface. Back-extrapolation of the linear multilayer section gives the specific micropore capacity, np(mic), as the intercept on the n axis. 

The as-plots in Figures 9.18-9.22 have been constructed with the aid of standard adsorption data obtained with Elftex 120 and other non-porous carbon blacks (Carrott et al., 1987, 1988a). As noted earlier, each as-plot has two linear sections (Section 8.2.1). The first linear section (at as < 1.0 pjp° < 0.4) can be attributed to adsorption on the walls of the supermicropores and therefore its back-extrapolation to as = 0 gives the ultramicropore capacity. The second linear section is obtained at as > 1.0 and is associated with multilayer adsorption on the external surface and the intercept gives the total micropore capacity. 

The as-plots for sample DC(1200)6 were found to be linear over the recorded ranges of both isotherms. This correspondence of isotherm shape is to be expected since the adsorbent structure has not been appreciably changed as a result of the 6-hour calcination at 1200°C. The multilayer sections of the other as-plots were for the most part linear, but the monolayer sections all exhibited significant deviation. The fact that the linear multilayer plots can be back-extrapolated to the origin was an indication that the multilayer development had not been affected to any great extent by the change in structure of the adsorbent. 

Each comparison plot in Figure 10.27 has two linear sections. Back-extrapolation of the first linear section gives a zero intercept whereas back-extrapolation of the second (multilayer) section gives a positive intercept. The interpretation of the features is based on the principles introduced in Chapter 8. An analysis of the isotherm data is given in Table 10.15. 

The nitrogen isotherm was replotted by Cases et al. (1992) in the usual BET coordinates and as a t plot. The derived BET area of 43.3 m2 g"1 appeared to be not far removed from the value of 45.9 m2 g 1 obtained from the amount adsorbed at Point B. The r-plot was constructed in the manner originally proposed by de Boer et al. (1966), which involved adopting a standard isotherm with the same value of C, which in this case was 485. It was not easy to interpret the /-plot, although three short linear sections were identified. From the initial slope, the total surface area appeared to be c. 50 m2 g-1. Back-extrapolation of a linear region at higher p/p° gave an apparent micropore volume of c. 0.01 cm3 g 1. 

The nitrogen isotherm data on non-porous hydroxylated silica in Table 10.1 (Bharabhani et al, 1972) have been used to construct the as-plot in Figure 12.6. Since the initial linear section can be back-extrapolated to the origin, we are reasonably sure that monolayer—multilayer adsorption has occurred on the mesopore walls before the onset of pore filling at / //>° = 0.41 and therefore that there was no detectable primary micropore filling at low / // - Similar results have been obtained by Kruk et al (1997b) and Sayari et al. (1997). 

For a moderately fast reaction, it is often difficult to make a direct measurement of p. It is then best to back-extrapolate the early pressure data to zero time to obtain a value of p. Another possibility is to allow the reaction to go to completion (assuming there is no back-reaction) and measure p", which is equal to 2p° or for the particular reactions under consideration. This method is usually not so reliable, since side reactions that have little effect on the early stages of the reaction may influence the final pressure. (The experimental value of is often found to be sUghtly less than the expected value.)... 

As shown in Figure 2.5 the back-extrapolated estimate of Co can be used to calculate the apparent volume (Vdiextrap)) hypothetical single compartment into which digoxin distribution occurs ... 

FIGURE 2.5 Siimilation of plasma (solid line) and tissue (heavy dashed line) digoxin concentrations after intravenous administration of a 0.75-mg loading dose to a 70-kg patient with normal renal function. Cq is estimated by back extrapolation (dotted line) of elimination-phase plasma concentrations. is calculated by dividing the administered drug dose by this estimate of Cq, as shown. Tissue concentrations are referenced to the apparent distribution volume of a peripheral compartment that represents tissue distribution. (Reproduced with permission from Atkinson AJ Jr, Kushner W. Annu Rev Pharmacol Toxicol 1979 19 105-27.)... 

FIGURE 2.10 Plot of drug concentrations vs. time on semilogarithmic coordinates. Back extrapolation (dashed line) of the elimination-phase slope (solid line) provides an estimate of Cq. The elimination half-life (h/2) can be estimated from the time required for concentrations to fall from some point on the elimination-phase line (Cj) to C2 = jCj, as shown by the dotted lines. In the case of digoxin, C would be in units of ng/mL and t in hours. 

FIGURE 3.8 "Curve-peeling" technique used to estimate the coefficients and exponents of Equation 3.5. Data points ( ) are plotted on semilogarithmic coordinates and the points for the cc-curve (o) are obtained by subtracting back-extrapolated -curve values from the experimental data. 

Values for the data equation parameters can be obtained by the technique of curve peeling" that was illustrated in Figure 3.8. After plotting the data, the first step is to identify the terminal exponential phase of the curve, in this case termed the j3--phase, and then back-extrapolate this line to obtain the ordinate intercept (BO- It is easiest to calculate the value of by first calculating the half-life of this phase. The value for j3 then can be estimated from the relationship j3 = ln2/b/2 - The next step is to subtract the corresponding value on the back-extrapolated jS-phase line from each of the data point values obtained during the previous exponential phase. This generates the Q -line from which the a-slope and A intercept can be estimated. 

After calculating the normalized intercept values A and B, the rate constants for the model can be obtained from Equations 3.11, 3.14, and 3.15. The volume of the central compartment is calculated from the ratio of the administered dose to the back-extrapolated value for Co (which equals A + BO as follows ... 

Estimates of Vii extrap) are also based on a singlecompartment model in which drug distribution is assumed to be infinitely fast. However, slowing of intercompartmental clearance reduces estimates of B, the back-extrapolated 8-curve intercept in Figure 3.8, to a greater extent than it prolongs elimination half-life. As a result, Vd(extrap) calculated from the equation... 

A major breakthrough in the study of gas and v or transport in polymer membranes was achieved by Daynes in 1920 He pointed out that steady-state permeability measurements could only lead to the determination of the product EMcd and not their separate values. He showed that, under boundary conditions which were easy to achieve experimentally, D is related to the time retired to achieve steady state permeation throu an initially degassed membrane. The so-called diffusion time lag , 6, is obtained by back-extrapolation to the time axis of the pseudo-steady-state portion of the pressure buildup in a low pressure downstream receiving vdume for a transient permeation experiment. As shown in Eq. (6), the time lag is quantitatively related to the diffusion coefficient and the membrane thickness, , for the simple case where both ko and D are constants. 

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