The final slope

76) requires a summation from 1 to s. The final slope in eq. (2.76) becomes indefinite at the limit Assuming r, is the smallest eigen- 

In this case the sums have to be taken from 2 to s. Then all the exponents are negative. Consequently at the limit r — o all the exponential functions become zero therefore the final slope is given by 

The slope depends on the value. The following cases can be considered  

Examples for the values p,-, and p, respectively are given in Table 2.6 and were used to calculate the concentration s dependence on time in Appendix 6.4.1. 

The determination of these final slopes is demonstrated using the mechanism of a multi-step consecutive reaction 

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When carbon (S) spins are locked following a CP 90° pulse, the HB term becomes nonsecular and, after these oscillations vanish, the S spin-lock magnetization is fractionally reduced. Thus, the observed decay is generally a sum of two Weibull functions, usually exponentials (Equation 20 p = 1) the initial slope reflects a Tle dominated by a spin-lattice process and the final slope yields a Tt dominated... 

The rate of reduction of cupric ion is illustrated in Fig. 3 (1). Curve 21 represents the hydrogen pressure drop observed in a typical experiment. Curve 23 is a repetition of this reduction, starting, however, with the reduced solution of experiment 21, which was oxidized by molecular oxygen from a red reduced solution to a green oxidized one. The final slopes (reduction rate) are the same for 21 and 23, the only difference being that 23 has a much longer induction period. If the reaction is stopped just before the flat portion of the curve (21 and 23) is reached and the solution filtered through a fine filter paper, no precipitate is found. 

Schaefer et al. performed kinetic studies with radioiodinated apoA-I and apoA-II in two patients with homozygous Tangier disease (S5). ApoA-I was removed from the circulation faster than apoA-II, but both were catabolized much faster than in normal subjects. After the plasma HDL concentration was increased by HDL infusion in one of the Tangier patients, a repeat kinetic study showed that increasing HDL concentration may have influenced the initial removal of HDL tracer from the circulation, but not the final slope of the plasma radioactivity removal curve (S5). 

The derivation of this relation will be treated in a future publication dealing more extensively with heating corrections to diffusion data, finally, it was found that values of D obtained from the final rate of sorption using (8) with n replacing Xj, were very close to the half-time or JA values, which varied in a more systematic manner with concentration. Therefore, the final slope was computed from these values of D. 

When analysing the data in the range of short adsorption times, a y(Vt)-plot is useful (Van Himsel Joos 1987a, Fainerman et al. 1994a). The results from Fig. 5.31 are shown in this form in Fig. 5.32. Only at low concentrations can a reasonable diffusion coefficient be calculated. For higher concentrations, the final slope of y(Vt) is located outside the experimental range, so that D cannot be determined from the plot. 

The final slopes are given in Table 2.7 for a variety of cases. In the first column the concentrations are listed which are graphed versus each other to form the K-diagram The next three columns repre-... 

In these equations, the symbols represents the degree of advancement reduced with respect to the initial concentration. In consequence in the -diagrams (reduced X-diagrams) the initial o and the final slopes are given according to eq. (2.87) as... 

In any case the final slope for this type of reaction in an X-diagram amounts to... 

The constant x was defined by eq. (3.40). Since this x does not contain any initial concentrations, the final slope in an ED-diagram will... 

The residence time for retinol in the test subject was predicted by the compartmental model to be 474 days. The residence time of 474 days is in excellent agreement with the 460 day MST that can be calculated from the data of Song et al. (1995) using the enrichment ratio method (Cobelli and Saccomani, 1992). Also, an MST of 105 to 337 days can be calculated from the half-life values (75 to 241 days) of body vitamin A reported by Sauber-lich et al. (1974) who depleted human subjects with vitamin A-deficient diets. At the same time the empirical description predicted the MST for retinol to be 26 days. While the reason for such a large discrepancy in MST (474 versus 26 days) between the compartmental model and the empirical description prediction is unclear, it is not likely to be accounted for by slight errors in estimating the final slope of the plasma retinol-d4 decay curve. Because the compartmental model embodies several features of retinol metabolism de novo production and release of retinol can occur in unobservable compartments, etc.) in addition to plasma retinol concentrations, its predicted MST is more likely to better reflect the dynamics of retinol metabolism. 

It is essential in measurements of this nature to condition the samples in a vacuum before starting the experiment in order to remove the air gases from the sample. Failure to do this will not affect the final slope of the curve, but the intercept will be meaningless. As an alternative, the sample can be conditioned in the test gas at, say, one atmosphere. After equilibrium has been attained, the pressure in the high pressure side can then be increased to, say. two atmospheres. 

This reaction was found to be slow compared to NO reactions with other organic radicals Upper limits between 0.3 and 1 x 10" cmVs were obtained by careful consideration of interfering reactions NO2 impurities in the NO are undesirable for two reasons, (i) The fast reaction adduct + NO2 mimics a NO reactivity - probably the cause for a higher rate constant in the literature [12], / (benzene-OH + NO) = 1 x 10" cmVs. We successfully purified the NO with FeS04. (ii) Although the concentration of NO2 was measured (in the absence of aromatic) to be at most 10 relative to NO, Le., well below [OHJo, the effect of the OH source term (H + NO2) may still be important in the presence of aromatic, because of the lower level of OH during the late part of the biexponential decays. Model calculations with [NO2] = 10 [NO] showed that the final slope of the decays may even decrease with NO. Finally, the unavoidable reaction OH + NO + M HONO + M 10" cmVs at our pressure and temperature) is formally independent (see above as and ds) t>ut increases the uncertainty. 

The system sulphuric acid-water follows the RAOULT-equation at great dilution of the acid, which is demonstrated by the fact that the final slope of the isotherm is the one representing that for ideal solutions (compare p. 514). 

The surface excess isotherms for methanol adsorption from benzene on untreated montmorillonites and kaolinite were both of type II in the Schay classification. " Pretreatment of montmorillonite with water+isopropanol or with methanol (50 h contact with boiling liquid) lowered the initial part of the isotherm indicating that pretreatment had covered some of the more active sites, but the final slope as x (alcohol) 1 was unchanged showing that the... 

A further model-independent parameter can be found in the final slope of the SAXS pattern. In this region, which depends mainly on the fine structure of the particle and not on the mutual arrangement of particles, the intensity I q) can... 

After extraction of the workable mineral, berms or benches remain on the final slopes, and the correct choice of width for these horizontal ledges is important in connection with the subsequent growth of vegetation on them. As a rule, they should be 3 to 6 m wide, depending on the height above the quarry floor and unless statutory regulations require other dimensions. Against the need for suitably wide berms must be set the requirement that the least possible quantity of workable mineral should be left behind in the quarry. A compromise will therefore have to be effected. 

In cases of lighter particles and/or higher energies, where curves are still shouldered, the final slope of the response curve is usually taken as measure for the cross section.)... 

Bunds may be constructed in one or more layers to their final level. In case the bund is constructed in several layers, the bund is generally raised just above the level of the next fill phase. This method may provide a better control of the final slope, see Figure 6.4. Usually these bunds are compacted by bulldozers. 

Unless a value specific to the polymer of interest is known, a is set equal to 3.4. The limiting power law slope, V, has some effect on the shape of the calculated MWD, as shown by Shaw Tuminello [6]. Eor their studies of a series of linear polyethylenes, Wood-Adams and Dealy [7] found that V = 1 gave the best agreement with SEC results. The selection of a value for the final slope and its effect on the predicted MWD has been the subject of some debate [8,9]. 

Noting the problems arising in the differentiation of data in Eq. 8.24, Liu etal. [13] proposed a method based on the integral Eq. 8.20 together with an assumption of the form of the MWD. They felt that this method was less affected by incomplete data than the differential approach the issue of how best to deal with incomplete data, particularly uncertainty as to the value of the final slope has been the subject of some discussion [8,9]. 

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