First-order precision

Because the diffusion profiles at the next time step are calculated directly from initial and boundary conditions, this method is called the explicit method. The method is stable only when a <0.5, i.e., Ax/(A ) < 0.5, and has only first-order precision because the expression for (dCldx) has only first-order precision. Hence, given A, it is necessary to choose a small Ax. 

Doppler) tuning capability, making it possible resonantly to interact on two optical transitions using only one laser field, retroreflected along the fast accelerated particle beam. It will be shown, that high optical resolution, Doppler free to first order, precise velocity control and high time resolution can be obtained in three-level fast beam laser spectroscopy. 

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

The conditions chosen make the reaction appear to be first-order overall, although the reaction is really not first-order overall, unlessjy and happen to be 2ero. If a simple exponential is actually observed over a reasonable extent (at least 90—95%) of decay the assumptions are considered vaUdated and is obtained with good precision. The pseudo-first-order rate constant is related to the k in the originally postulated rate law by... 

Example 6 Calculation of Probability of Meeting a Sales Demand A store that is open 5 days a week is to promote a new product. The manager heheves that not more than 5 units will he sold in any one day, hut he cannot he more precise about the probable sales pattern. Stocks are dehvered once per week. What size should the first order he to give a 95 percent certainty of meeting demand ... 

Or, more precisely, a firsl-order saddle paint, where the order indicates the number of dimensions in which the saddle point is a maximum. A second-order saddle point would be a maximum in two dimensions and a minimum in all others. Transition structures are first-order saddle points. 

The aim of the present study is precisely to investigate the thermodynamical properties of an interface when the bulk transition is of first order. We will consider the case of a binary alloy on the fee lattice which orders according to the LI2 (CuaAu type) structure. 

Some conseivation measures do not easily fit into the form of an initial investment followed by a stream of energy savings because there will be other costs and benefits occurring during the measure s operating life. Furthermore, it is difficult to incorporate peak power benefits into the CCE approach. In these situations, a more precise analysis will be necessary. However, the CCE and supply cuive approaches provide first-order identifications of cost-effective conseivation, those... 

It seems likely that other polymerizations will be found to depart from Bemoullian statistics as the precision of tacticity measurements improves. One study12 indicated that vinyl chloride polymerizations are also more appropriately described by first order Markov statistics. However, there has been some reassignment of signals since that time. 4 25... 

The values used in plotting Figs. 2-1 and 2-2 can be used to illustrate the method for first-order and second-order data. Plots of t/E versus time are shown in Fig. 2-9. The second-order data define a precise straight line, and those for n = 1 are linear to E < 0.4. The latter graph has a slope of 0.6, giving n = 1.2. 

Wilkinson s method for the estimation of the reaction order is illustrated for first-order (left) and second-order (right) kinetic data. The first-order reaction is the decomposition of diacetone alcohol (Table 2-3 and Fig. 2-4) data to about 50 percent reaction are displayed. The slope gives an approximate order of 1.2. The second-order data (Fig. 2-2) give a precise fit to Eq. (2-59) and an order of two exactly. 

Although many reaction-rate studies do give linear plots, which can therefore be easily interpreted, the results in many other studies are not so simple. In some cases a reaction may be first order at low concentrations but second order at higher concentrations. In other cases, fractional orders as well as negative orders are obtained. The interpretation of complex kinetics often requires much skill and effort. Even where the kinetics are relatively simple, there is often a problem in interpreting the data because of the difficulty of obtaining precise enough measurements. ... 

This may be understood more fully by reference to Fig. 11.2. Curve A shows the type of response which would be obtained if the lethal process followed precisely the pattern of a first-order reaction. Some experimental curves do, in fact, follow this pattern quite closely, hence the genesis of the original theory. 

Consider the case when the equilibrium concentration of substance Red, and hence its limiting CD due to diffusion from the bulk solution, is low. In this case the reactant species Red can be supplied to the reaction zone only as a result of the chemical step. When the electrochemical step is sufficiently fast and activation polarization is low, the overall behavior of the reaction will be determined precisely by the special features of the chemical step concentration polarization will be observed for the reaction at the electrode, not because of slow diffusion of the substance but because of a slow chemical step. We shall assume that the concentrations of substance A and of the reaction components are high enough so that they will remain practically unchanged when the chemical reaction proceeds. We shall assume, moreover, that reaction (13.37) follows first-order kinetics with respect to Red and A. We shall write Cg for the equilibrium (bulk) concentration of substance Red, and we shall write Cg and c for the surface concentration and the instantaneous concentration (to simplify the equations, we shall not use the subscript red ). 

In (9.2), AEy is the bandwidth of the incoming radiation and Cei is the electronic absorption cross section. The exponential decay is modulated by the square of a Bessel function of the first order (/j), giving rise to the aforementioned dynamical beats. The positions of their minima and maxima (i.e., the slope of the envelope of the time-dependent intensity) can be determined with high accuracy and thus give precise information about the effective thickness of the sample. 

The first analysis is one with AS-level precision, the second with TIMS-level precision. The first order 2a error for the resulting 331 ka age is 96 ka, but examination of the distribution of a Monte Carlo simulation (Fig. 2) shows that the actual age distribution is strongly asymmetric, with 95% confidence limits of 158/-79 ka. For either younger ages or more-precise analyses, however, the first-order age errors are more than adequate, as shown by the Monte Carlo results for the same data, but with TIMS-level precision (Fig. 2B). 

Figure 2. Histograms of Monte Carlo simulations for two synthetic analyses (Table 1) of a 330 ka sample. The lower precision analysis (A) has a distinctly asymmetric, non-Gaussian distribution of age errors and a misleading first-order error calculation. The higher precision analysis (B) yields a nearly symmetric, Gaussian age distribution with confidence limits almost identical those of the first-order error expansion.

It is necessary to calibrate the 14C time scale for greater dating accuracy. However, the second-order variations are at least as important as the first-order constancy of atmospheric 14C. For example, they provide a record of prehistoric solar variations, changes in the Earth s dipole moment and an insight into the fate of C02 from fossil fuel combustion. Improved techniques are needed that will enable the precise measurement of small cellulose samples from single tree rings. The tandem accelerator mass spectrometer (TAMS) may fill this need. 

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