Substitutes for

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

The maximum-likelihood method, like any statistical tool, is useful for correlating and critically examining experimental information. However, it can never be a substitute for that information. While a statistical tool is useful for minimizing the required experimental effort, reliable calculated phase equilibria can only be obtained if at least some pertinent and reliable experimental data are at hand. 

Consider now thermal coupling of the prefractionator arrangement from Fig. 5.116. Figure 5.16a shows a prefi-actionator arrangement with partial condenser and reboiler on the prefractionator. Figure 5.166 shows the equivalent thermally coupled prefractionator arrangement sometimes known as a Petlyuk column. To make the two arrangements in Fig. 5.16 equivalent, the thermally coupled prefractionator requires extra plates to substitute for the prefractionator condenser and reboiler. 

Now recalculate the network area target substituting for in Fig. 7.5. Table 7.2 is revised to the values shown in Table 7.3 ... 

A mixture of polymethylene glycols of the type (CH20) ,xH20 where n is 6- 50. It is a white, amorphous powder having the odour of meth-anal, m.p. I20-130°C. The commercial product contains 95% methanal and is obtained in while flocculent masses when solutions of methanal are evaporated or allowed to stand. When heated it is converted to methanal. Used as a convenient solid substitute for methanal. 

In this instance a second aromatic ring can be substituted for two adjacent hydrogen atoms giving condensed polynuclear aromatics, for... 

Or by a naphthenic ring which can also be substituted for two adjacent hydrogen atoms forming a naphthene aromatic such as tetralin or tetra.hydronaphthaiLene. ... 

Other methods for analyzing combustion products can be substituted for chromatography. Gravimetry can be used, for example, after a series of absorption on different beds, as in the case of water absorption in magnesium perchlorate or CO2 in soda lime infra-red spectrometry can be used for the detection of CO2 and water. 

Hydrorefining can substitute for extraction processes such as furfural where it integrates perfectly into the conventional process scheme. 

The second tenu in the Omstein-Zemike equation is a convolution integral. Substituting for h r) in the integrand, followed by repeated iteration, shows that h(r) is the sum of convolutions of c-fiinctions or bonds containing one or more c-fiinctions in series. Representing this graphically with c(r) = o-o, we see that... 

Plenary 21B. A C Albrecht et al, e-mail address aca7 comell.edu (I CRS) (substituting for plenary 21 A). Discusses four new applications using a third approach to the Class II spectroscopies (see above). Raman... 

Eleetron energy-loss speetroseopy is used for obtaining speetroseopie data as a eonvenient substitute for optieal speetroseopy, and, taking advantage of diflferenees in seleetion rules, as an adjimet to optieal speetroseopy. In addition, eleetron speetroseopy has many applieations to ehemieal and stnietural analysis of samples in the gas phase, in the solid phase, and at the solid-gas interfaee. 

If the atom has nonzero nuclear spin /tlien F = S + I substitutes for S in equation (Cl.4.49), tire g-factor generalizes to... 

Yet another form for the equations, this time expressed in terms of the can be obtained by substituting for N and the from equations (5.5) into (5.7) and (5,8). After a little rearrangement the result can be written... 

Substituting for the from equation (5,44) and simplifying then gives... 

We may substitute for p from this into equations (12.24) and (12.25) then, on passing to the limit - only terras of zeroth order in need be retained." Consequently the flux relations reduce to... 

We now introduce the atomic orbital expansion for the orbitals i/), and substitute for the corresponding spin orbital Xi into the Hartree-Fock equation,/,(l)x,(l) = X (1) ... 

When the Coulomb and exchange operators are expressed in terms of the basis functions and the orbital expansion is substituted for xu then their contributions to the Fock matrix element take the following form ... 

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