Estimation of the order

The methods by which the order with respect to A, a, may be estimated are merely modifications of those described in the previous section. Instead of choosing [BJo such that [B]o/[A]q = r as required for the estimation of total order, we now arrange that [BJo [A]q so that the disappearance of A is followed in the presence of an effectively constant concentration of B. More precisely, we arrange that 

Since both k and I are constants and a is known from the superimposition, an estimate of b can be obtained from the slope of the linear plot of scale displacement against logio [AJo,j. 

The fractional life methods can also be used to determine a. As already indicated, the fractional life method of treating the data of a single experiment has few advantages over the superimposition method and consequently need not be considered further. On the other hand, the measurement of a particular fractional life of the reactant A in a series of experiments in which its initial concentration is varied provides a means by which a can be determined which sometimes offers experimental advantages. The only point to remember is that throughout the series of experiments [BJo must be kept constant at such a value that [BJo/[A]o,i Vb/Va. From the condition that 

Finally, it should be obvious that the method of tangents can be equally well applied to the determination of a utilizing eqn. (29) in precisely the same way as eqn. (15). 

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Let us make an estimate of the order of magnitude of these effects. Referring to equation (9,23), the coefficient of the term relating the micropore contribution to and grad p is larger by a factor... 

By a different route the same equation can be obtained for n = 1. A plot of tip vs. t leads to an estimate of the order n from the slope. 

We have employed the Bragg-Williams approximation (BWA) to obtain rough estimates of the ordering/segregation critical temperatures. It is well known that the BWA usually overestimates critical temperatures (approximately by 20 %) in comparison with the exact value obtained from Monte Carlo simulations, or by other highly accurate methods of statistical mechanics. This order of accuracy Is nevertheless sufficient for our present purposes. 

In order to get an estimate of the order of magnitude of the correlation energy, Froman makes it plausible that the correlation energy should be roughly proportional to the total Coulomb repulsion energy of all the electrons within the system, and he suggests the formula... 

The co-equivalence property of homogeneous schemes lies in the main idea behind a new approach to the further estimation of the order of accuracy of a scheme on account of (9) or (10) its coefficients a, d, ip should be compared with coefficients d, d, (p of a simple specimen scheme, the accuracy order of which is well-known (see Section 7). 

We proceed to the estimation of the order of approximation for scheme (II) under the agreement that u = u(x,t) possesses a number of derivatives in X and t necessary in this connection for performing current and subsequent manipulations. Within the notations... 

For the three heat exchangers from Exercise 1, make a first estimate of the order of magnitude of the overall heat coefficients from tabulated values of film transfer coefficients and fouling coefficients. Neglect the resistance from the tube walls. 

The quantification should start with a rough estimation of the order of magnitude of each uncertainty contribution pi and pij. Insignificant one can be neglected because the uncertainty components are added according to a squared model. The significant values should be refined in subsequent stages and converted to parameters u(pf) which correspond to standard deviations. 

We may make a very rough estimate of the order of magnitude of the electric field experienced by the hydrogen atom in a hydrogen bond from the simple electrostatic of an O—H e 0 bond illustrated in... 

Equation (8.3.14) is not an asymptotically exact result for the black sphere model due to the superposition approximation used. When deriving (8.3.14), we neglected in (8.3.11) small terms containing functionals I[Z], i.e., those terms which came due to Kirkwood s approximation. However, the study of the immobile particle accumulation under permanent source (Chapter 7) has demonstrated that direct use of the superposition approximation does not reproduce the exact expression for the volume fraction covered by the reaction spheres around B s. The error arises due to the incorrect estimate of the order of three-point density p2,i for a large parameter op at some relative distances ( f — f[ < tq, [r 2 - r[ > ro) the superposition approximation is correct, p2,i oc ct 1, however, it gives a wrong order of magnitude fn, oc Oq2 instead of the exact p2,i oc <7q 1 (if n — r[ < ro, fi — f[ < ro). It was... 

For an estimation of the order of magnitude of the preexponential factor in the Arrhenius equation for a surface reaction between simple molecules, the terms f and f7A may be taken as equal to 1 since they include only vibrational modes that are degenerate if hv > A B7 v being the vibration frequency) (21). The values of fK for molecules in the gas phase are calculated in the usual manner. 

The specification of ARIMA models is very expensive for the operator who analyzes time series. The first phase is the estimation of the order of three inherent processes, autoregression, integration, and moving average. 

Estimation of the Order of Magnitude of the FED Generated during a Single Pass of a Compacted Particulate Solid Bed over the Rotor Wing-Tip Clearance of a... 

The closest theoretical result, the Unified theory [68], differs by more than 300 times the experimental uncertainty. This discrepancy should be partially removed by analysis including an estimate of the order (Za)4a2mec2 relativistic term and a complete calculation of the two-electron Bethe-logarithm [92]. The 14,i5N5+ 21S o — 23Pi isotope shift was measured to be —1.6623(10) cm-1, in fair agreement with an estimate based on [68]. The hyperfine corrected 3Pq —3Pi... 

The half-life method (in general, the fractional-life method) is very useful in preliminary estimates of the order of the reaction. A series of experimental runs is carried out with different initial concentrations. If an irreversible reaction is considered ... 

In general, it is not possible to obtain exact expressions for s as it requires knowledge of the exact solution of (c)e as well as the solution of the truncated model. However, an estimate of the order of magnitude of the error may be obtained by simply expanding it in a Taylor series around p — 0. For example,... 

Clearly, equation (49) reduces to equation (24) if tr is replaced by the approximation (18) formally it now takes full account of the (rapid) variation of electron density in the atom, in contrast to the semiclassical TF Euler equation. Unfortunately, tr is only presently known in two special cases (i) to low order in gradient expansion corrections to equation (18) as in equation (76) below and (ii) in a perturbative development about the uniform electron assembly.13 Form (i) will be referred to again below. However, as Scott14 was first to argue for the neutral atom, the origin of the Z2 term in equation (48) resides in the inhomogeneity correction to the TF theory, which is formally contained in equation (49). Fortunately, an approximation based on the Coulomb field treatment of Section 4 suffices to gain a useful estimate of the order of the Z2 term in the neutral atom. 

An estimate of the order of magnitude of the iceberg size can be made. For 12% MjO, the radius of an (assumed spherical) iceberg is about 1.9 nm, and at 33% MjO, the iceberg of the iceberg model becomes identical in size with the disaete polyanion of the discrete-polyanion model, which has a radius of about 0.6 nm. 

Since the change in e through the detonation wave is Ae 1, equations (15) and (5-33) imply that a representative dimensionless distance scale Ac is large compared with unity for detonations (it is of the order of unity for deflagrations). An estimate of the order of magnitude of the left-hand side of equation (5-19) then shows that dx/d 1 (since the change in x through the wave is At = 1), and therefore equation (5-19) implies that, approximately. 

Equations (5)-(8) may easily be used to obtain the estimates of the orders of magnitude of the terms in equations (2)-(4) indicated beneath these equations. Since S/l 1, the indicated orders of magnitude imply that the last term in equation (2) is large compared with the other terms on the right-hand side, the last term in equation (3) is small compared with the other terms on the right-hand side, and the third term in L(a) [equation (4)] is small compared with the fourth (last) term. Equation (2) then shows that the dominant (last) viscous term will be of the same order as the convective terms appearing on the left-hand side only if the Reynolds number. 

To enable one to use the results of computer simulation for quantitative evaluation of the interdififiision rate of ions in real systems, it is essential to determine as effectively as possible the accuracy with which one has to specify the magnitude of dissociation constants K,, Kk and diffusion coefficients Dq, D, Dy. To accomplish this, computer simulations were performed to compare results when and Kkq differed by 10-fold and when and were taken to be equal [48,68]. The calculated kinetic curves coincide if the relations K Cq < 10 and Kgg/Cg < are applicable. In this case the concentration of the free ions Q is so low that its variation probably does not affect the electric field in the ion exchanger so that the distribution of the ions in the resin depends only on the ratios Kk /Kkb iind D /Db- this basis, an approximate estimate of the order of exchange isotherm. Also it is evident from Fig. 3 that in the case where the ratio D /Db taken for calculation purposes does not correspond to the real system, an error in determining the process rate will be larger for the unfavorable (relation Kp /KpB < 1) than it is for the favorable (relation > 1) isotherms. 

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