Initial Slope Experiments

Model Long Time Small k Short Time Large k 

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To lessen experimental time, the null-point method may be employed by locating the pulse spacing, tnun, for which no magnetization is observed after the 180°-1-90° pulse-sequence. The relaxation rate is then obtained directly by using the relationship / , = 0.69/t n. In this way, a considerable diminution of measuring time is achieved, which is especially desirable in measurements of very low relaxation-rates, or for samples that are not very stable. In addition, estimates of relaxation rates for overlapping resonances can often be achieved. However, as the recovery curves for coupled spin-systems are, more often than not, nonexponential, observation of the null point may violate the initial-slope approximation. Hence, this method is best reserved for preliminary experiments that serve to establish the time scale for spin-lattice relaxation, and for qualitative conclusions. 

Apart from the separation into two fractions, the propagator can also be interpreted in terms of an average quantity, the second moment of displacements, which is proportional to the dispersion coefficient D (A). Rather than computing D ( A) from the shape of the propagator directly, it is also possible to obtain it from the initial slope of the signal function E(q,A) in a ID NMR experiment [43] ... 

Edwards et al. (1980) used a Cm-242 source to irradiate externally a thin film of blood. The energy of 4 9 MeV of the alpha particles were almost entirely absorbed by the blood. The dicentrics yield was linear from 0.11 to 4 2 Gy. From this resulted a RBE of 17.9 with respect to Co-60 gamma rays. It was, however, only 6.0 at the initial slope. To explain this RBE, which was low compared to that expected from neutron experiments, a model is presented taking into account cell killing and mitotic delay. 

Determination of the segmental adsorption energy. The determina-tion of x ° is also possible. Since x d can be found from Equation 5 if Xsc an the solvency terms are known, we can add xf° and find x ° by Equation 1. The determination of xf° calls for a separate experiment, e.g., an adsorption isotherm of the displacer from solvent, in the absence of polymer. Following such a scheme we used the values of cr obtained from the displacement isotherms of Figure 3 and 4 to determine segmental adsorption energy parameters Xg° for PVP on silica. The required additional information on xdo was obtained from the initial slopes of dis-... 

Figure 6. A plot of the logarithm of the square root of the initial slope of the current time transients determined in the double step experiments against the final potential for the three single crystal surfaces investigated.

These initial slopes are readily evaluated since we can measure the signal corresponding to Igq (or Seq if S is a nuclear spin) and, since the relevant instrumental factor is identical to the one which prevails in the measurement of leq (or Seq). Moreover, initial conditions can be devised for determining separately one of the three relaxation parameters 7 (, or a. The simplest experiment consists of selectively inverting one of the two magnetizations. Consider first the selective inversion of I magnetization for which the following initial conditions hold 72(0) = —leg and S2(0) = Seq. This yields for the initial slopes... 

Except for very simple systems, initial rate experiments of enzyme-catalyzed reactions are typically run in which the initial velocity is measured at a number of substrate concentrations while keeping all of the other components of the reaction mixture constant. The set of experiments is run again a number of times (typically, at least five) in which the concentration of one of those other components of the reaction mixture has been changed. When the initial rate data is plotted in a linear format (for example, in a double-reciprocal plot, 1/v vx. 1/[S]), a series of lines are obtained, each associated with a different concentration of the other component (for example, another substrate in a multisubstrate reaction, one of the products, an inhibitor or other effector, etc.). The slopes of each of these lines are replotted as a function of the concentration of the other component (e.g., slope vx. [other substrate] in a multisubstrate reaction slope vx. 1/[inhibitor] in an inhibition study etc.). Similar replots may be made with the vertical intercepts of the primary plots. The new slopes, vertical intercepts, and horizontal intercepts of these replots can provide estimates of the kinetic parameters for the system under study. In addition, linearity (or lack of) is a good check on whether the experimental protocols have valid steady-state conditions. Nonlinearity in replot data can often indicate cooperative events, slow binding steps, multiple binding, etc. 

Figure 8.9 shows that the concentration of intermediate in reversible series reactions need not pass through a maximum, while Fig. 8.10 shows that a product may pass through a maximum concentration typical of an intermediate in the irreversible series reaction however, the reactions may be of a different kind. A comparison of these figures shows that many of the curves are similar in shape, making it difficult to select a mechanism of reaction by experiment, especially if the kinetic data are somewhat scattered. Probably the best clue to distinguishing between parallel and series reactions is to examine initial rate data—data obtained for very small conversion of reactant. For series reactions the time-concentration curve for S has a zero initial slope, whereas for parallel reactions this is not so. 

Figure 5. An experiment illustrating the change in stoichiometry in the presence of thiocyanate ion and O represent measurements at 380 and 278 mu, respectively the dotted line represents our evaluation of the initial slope.

Derivation of Reaction Schemes Based on Experimental Results. Although numerous methods for evaluating reactions schemes have been developed ( 0-44), most of them (40-42) start with a hypothetical mechanism which is, by means of experiments, either confirmed or rejected. A newly developed method for the systematic elucidation of reaction schemes of complex systems requires no chemical considerations, but concentration-time measurements and system-analytical considerations (45). The method is based on the initial slope of the concentration-time profiles and when necessary the higher derivatives of these curves at t = 0. Reaction steps in which products are formed directly from reactants can be identified in a concentration-time plot by a positive gradient c. at t = 0 (zero order delay). dtJ... 

The initial slope of the reduced absorption curve is denoted by Ia and that of the reduced desorption curve by Id. These are generally functions of the initial concentration cf and the final concentration cf° of a particular experiment. In most work undertaken to determine D, measurements are done in such a way that C]° = 0 for absorption and cf° = 0 for desorption. In these cases, Ia — Io(ci°°) and Id = IJipf0). The methods... 

Values for m,, y/,and F are determined from experimental conditions. The product kg C" can be determined at r 0. For short deactivation times when e - 1) 0, the initial slope gives a method for calculating kg°. Values of k were determined to give the best statistical fit to the experimental data. A discussion of the experiments is given next. 

The correct calculation of the isosteric heat requires the derivation of Qex/ meq). Very often, qsi is calculated by dividing Qexp by the mass uptake Ameq between two experiments. This procedure provides the average isosteric heat of adsorption qsu- qsu, is equal to q, if the mass uptake between two adsorption experiments remains low. This assumption is difficult to check for systems exhibiting type 1 isotherms with a high initial slope. That is the reason why we present the calorimetric results on the basis of Qexp as a fimction of meq. 

A batch reactor by its nature is a transient closed system. While a laboratory batch reactor can be a simple well-stirred flask in a constant temperature bath or a commercial laboratory-scale batch reactor, the direct measurement of reaction rates is not possible from these reactors. The observables are the concentrations of species from which the rate can be inferred. For example, in a typical batch experiment, the concentrations of reactants and products are measured as a function of time. From these data, initial reaction rates (rates at the zero conversion limit) can be obtained by calculating the initial slope (Figure 3.5.1b). Also, the complete data set can be numerically fit to a curve and the tangent to the curve calculated for any time (Figure 3.5. la). The set of tangents can then be plotted versus the concentration at which the tangent was obtained (Figure 3.5.1c). 

The adsorption equilibria were measured using a gravimetric method and were expressed as isotherms. A chromatographic method was used to get the initial slope of the isotherms. In the simulation, GCMC method was used to calculate amounts adsorbed for various conditions. When the experiment result and simulation result of chloroform are compared, the simulation for the acid site model was most agreement with chromatographic data and baratron data. The simulation result of tetrachloroethylene with three models corresponded mostly for the non-polar molecule, and above all the acid site model was the closest to the experiment result. Therefore, to get better coincidence between experimental data and simulation, it was found to be necessary to account for aluminum rather than silanol nest. 

The evaluation of the separation factor enables characterization of the initial slopes of the adsorption isotherm for the product and neighboring impurities under various conditions. The term linear conditions means, under analytical conditions or under conditions where the injection size is small and the injection concentration is in the linear region of the ad.sorption i.sotherm. Retention experiments enable evaluation of the thermodynamics under infinite dilution. 

Regardless of which method is used, the initial slope of the isotherm must be determined with great care. Either the lower bound for integration must be approximated with the lowest recorded concentration or the slope for zero concentration must be determined separately. As the latter task is identical to the determination of the Henry constant, this value can be obtained from pulse experiments with very low injection concentrations using momentum analysis (Section 6.5.7.2). 

Again the initial slope of the isotherm has to be determined by a pulse experiment of B. 

The sorption/desorption experiments were carried out as a function of a both above and below the Tg of the matrix. The data obtained from both experiments closely resembled Fickian diffusion. Below 0.55a (between 0 < Mt and Mf < 0.5) and above 0.55a (between 0.08 < Mf and Mf < 0.75) initial slope of Mf/Mf vs. curves were linear with respect to abscissa (R > 0.98). Diffusion coefficients were obtained using the linear portion of the normalized moisture sorption (Mt/Moo vs. curves from Equation 46.14. 

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