LIMITS OF VALIDITY

The methods of estimation proposed in this chapter are applicable to gases whose pressure is higher than that of the KNUDSEN regime (where the molecular collisions with the reactor walls become more frequent than the collisions in the gas phase), and lower than a pressure defined by the following relationship  

Define the limits of validity of the formulae for the calculation of D, r, X for the case of iso-octane at 1800 K. Knowing the following values  

The ratio T/T = 3.31 actually obeys constraint (34). The pressure must therefore respect the inequality (33)  

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At pressures above the highest real data point, the extrapolated data were generated by the correlation of Lyckman et al. (1965), modified slightly to eliminate any discontinuity between the real and generated data. This modification is small, only a few percent, well within the uncertainties of the Lyckman method. The Lyckman correlation was always used within its recommended limits of validity--that is, at reduced temperatures no greater than 1.5 to 2.0. 

The right-hand portion of the curve is therefore a straight line which would, if produced, pass through zero pressure on the II. axis (x = 0). The rectilinear relation extends up to the limit of validity of (1). 

In Section II, the basic equations of OCT are developed using the methods of variational calculus. Methods for solving the resulting equations are discussed in Section III. Section IV is devoted to a discussion of the Electric Nuclear Bom-Oppenhermer (ENBO) approximation [41, 42]. This approximation provides a practical way of including polarization effects in coherent control calculations of molecular dynamics. In general, such effects are important as high electric fields often occur in the laser pulses used experimentally or predicted theoretically for such processes. The limits of validity of the ENBO approximation are also discussed in this section. 

Within the limits of validity of the theories developed above on the basis of idealized models and of the simplifying assumptions, xi of the preceding treatment may be related to these parameters by comparing Eq. (42) with (43). Then... 

Becker, P.J. and Coppens, P. (1974) Extinction within the limit of validity of the Darwin transfer equations. I. General formalisms for primary and secondary extinction and their application to spherical crystals, Acta Crystallogr., A, 30, 129-147. 

In addition to its rather narrow limits of validity, this expression is strictly applicable only to spherical charges of Composition B. However, until further work is completed, this equation represents the best method available for predicting secondary fragment velocities. 

Actually we are dealing here with a system where the difference in size of the two components is appreciable (p > 0.1) and this might well go beyond the limit of validity of the APM. It is, however, interesting to note that an evaluation of eJB and r B for the present system has been made by Thomaes et a/.33 from second virial coefficients. This leads roughly to ... 

Fisher polynomials can be used only within the T range for which they were created. Extrapolation beyond the T limits of validity normally implies substantial error progression in high-F entropy and enthalpy calculations. For instance, figure 3.4 compares Maier-Kelley, Haas-Fisher, and Berman-Brown polynomials for low albite. As can be seen, the first two interpolants, if extended to high T, definitely exceed the Dulong and Petit limit. The Berman-Brown interpolant also passes this limit, but the bias is less dramatic. 

Standard state entropy values and Maier-Kelley coefficients of heat capacity at constant P, with respective T limits of validity, are listed for the same components in table 5.13. The adopted polynomial expansion is the Haas-Fisher form ... 

What Would Make a Sweep Rate Too Slow One general criterion that can be taken here is that the time in which (nDT)l/2 = 8, an assumption used in the theory ofpotentiodynamic transients, remains valid. It is an experimental fact that 8 at an electrode in a still solution for a long time (> 10 s, say) is about 0.05 cm and constant because natural convection stirs the solution and wipes out the concentration gradient set up by diffusion alone. Hence, one can assume that the limit of validity of 8,= (jiDt)m is a time at which becomes 8 equal to 0.05 cm. Using a typical value of D (= 3 x 10-5 cm2 s-1), one obtains... 

The present chapter, however, is concerned with the traditional approach. We shall carefully stay within the limits of validity, although these limits will only become clear in chapter X. The reader is advised that any doubts he may have must be postponed. 

At this point it seems of interest to include a graph obtained on a quite different polymer, viz. cellulose tricarbanilate. Results from a series of ten sharp fractions of this polymer will be discussed in Chapter 5 in connection with the limits of validity of the present theory. In Fig. 3.5 a double logarithmic plot of FR vs. is given for a molecular weight of 720000. This figure refers to a 0.1 wt. per cent solution in benzophenone. It appears that the temperature reduction is perfect. Moreover, the JeR-value for fiN smaller than one is very close to the JeR value obtained from Figure 3.1 for anionic polystyrenes in bromo-benzene. As in the case of Fig. 3.1, pN is calculated from zero shear viscosity. The correspondence of Figs. 3.1 and 3.5 shows that also the molecules of cellulose tricarbanilate behave like flexible linear chain molecules. For more details on this subject reference is made to Chapter 5. 

Limits of Validity of the it-Approximation of Heteroaromatic Compounds L. Paoloni, Corsi Semin. Chim., 1969,19, 34-36. 

This chapter recalls the principles of the hyperfine coupling between electrons and nuclei in terms of energy and deals with its consequences on chemical shift. The equations for contact and pseudocontact shifts are derived and illustrated in a pictorial way. Their physical/chemical backgrounds are discussed as well as their limits of validity. The mechanisms of spin delocalization are illustrated. The perspectives when high field magnets are used are highlighted. 

Now we show that the validity of the cascade Langevin approach extends beyond the limits of validity of Boltzmann equation. Consider a frequency-dependent noise in a chaotic cavity, i.e. in a metallic island of irregular shape connected to the electrodes L, R via two quantum point contacts of conductances Gl,r e1 /h and arbitrary transparencies / /. . As the dwell time of... 

The aim of this paper is to make measurements with liquids of various physical properties in order to define the effect of the liquid properties and operating conditions on the parameter /q, and the limits of validity of the literature models for the interpretation of mass transfer coefficients in bubble dispersions. The method, which is used for the measurements, was verified in Part I to minimize misinterpretations. 

The validity interval is the time period during which relevant measurement operations are maintained in control with acceptable repeatabilities by each laboratory involved. This important limitation of validity of a traceability link is provided by design or could be im-... 

The value of traceability for laboratories and customers are in many instances closely related to each other. It has to do with the immediate recognition that an accurate value can only be claimed within the limits of the boundaries indicated in the statement on uncertainty. This helps to avoid over-interpretation of the data and gives a clear view on the limits of validity. Failures in traceability potentially undermine the trust in the professional integrity of analytical chemists. Embarrassing results from these failures could be avoided by paying more attention to the nature and limitations of the traceability of references and of the measurement process itself. 

The simplest treatment of the properties of gases starts with the following assumptions, which will determine the limits of validity of the ideal gas equation P V = nRT ... 

To sum up, although each criterion has limits of validity, nevertheless all the criteria agree in establishing the following order of decreasing ground state aromaticities benzene > thiophene > pyrrole > furan. 

In analogy to indentation experiments, measurements of the lateral contact stiffness were used for determining the contact radius [114]. For achieving this, the finite stiffness of tip and cantilever have to be taken into account, which imposes considerable calibration issues. The lateral stiffness of the tip was determined by means of a finite element simulation [143]. As noted by Dedkov [95], the agreement of the experimental friction-load curves of Carpick et al. [115] with the JKR model is rather unexpected when considering the low value of the transition parameter A(0.2Further work seems to be necessary in order to clarify the limits of validity of the particular contact mechanics models, especially with regard to nanoscale contacts. 

While there are many applicable methods reported in the literature, this paper was restricted to a subjective selection of testing procedures in the hope that it would stimulate the development of better analytical techniques, recognizing that there is always a limitation of validity for each method and evaluation of results. Analysis of carbon materials and especially of the carbonization process will never lose its attraction as a fascinating application of the art of black magic. 

Fig. 4 to Fig. 8 show the severe divergence for activity coefficients such as given here for calcium, chloride, sulfate, sodium and water ions, calculated with different equations. The activity coefficients were calculated applying Eq. 13 to Eq. 17 for the corresponding ion dissociation theories, whereas the values for the PITZER equations were gained using the program PHRQPITZ. The limit of validity of each theory is clearly shown. 

The question of the detailed limits of validity of the reptation model thus remains a pending question. What appears puzzling is the fact that, on one hand, the reptation model and the Doi - Edwards description of the linear viscoelasticity work so well both qualitatively and quantitatively for some experiments, while, on the other hand, they seem unable to account for all the existing data. This may suggest that the reptation model does not contain the whole story of linear polymer dynamics, and that one needs to learn more on other possibly competing processes. 

The plots in Fig. 2 suggest that the limits of validity of transition state theory may be fairly narrow or altogether nonexistent, contrary to the prediction made by Kramers. The nonequilibrium effects duly treated for extremely low friction may start being felt before the TST plateau is approached. This is more likely to occur for the lower barriers and larger ratios... 

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