Smoothed equations

The key step to this first reported case of the highly diastereoselective addition of a fluorinated enolate in an aldol process is the selective formation of the enolate a,a-Difluonnated enolates prepared by a metallation process employing either a zinc-copper couple [S] or reduced titanium species [9] undergo aldol condensation smoothly (equation 9) (Table 5)... 

Numerical data and coeffficients in smoothing equations for 80 compounds... 

Review of theory. Tables of data and parameters in smoothing equations for 57 solutes in water... 

Solutions in hand for the reference pairs, it is useful to write out empirical smoothing expressions for the rectilinear densities, reduced density differences, and reduced vapor pressures as functions of Tr and a, following which prediction of reduced liquid densities and vapor pressures is straightforward for systems where Tex and a (equivalently co) are known. If, in addition, the critical property IE s, ln(Tc /Tc), ln(PcVPc), and ln(pcVPc), are available from experiment, theory, or empirical correlation, one can calculate the molar density and vapor pressure IE s for 0.5 < Tr < 1, provided, for VPIE, that Aa/a is known or can be estimated. Thus to calculate liquid density IE s one uses the observed IE on Tc, ln(Tc /Tc), to find (Tr /Tr) at any temperature of interest, and employs the smoothing relations (or numerically solves Equation 13.1) to obtain (pR /pR). Since (MpIE)R = ln(pR /pR) = ln[(p /pc )/(p/pc)] it follows that ln(p7p)(MpIE)R- -ln(pcVpc). For VPIE s one proceeds similarly, substituting reduced temperatures, critical pressures and Aa/a into the smoothing equations to find ln(P /P)RED and thence ln(P /P), since ln(P /P) = I n( Pr /Pr) + In (Pc /Pc)- The approach outlined for molar density IE cannot be used to rationalize the vapor pressure IE without the introduction of isotope dependent system parameters Aa/a. 

A wide variety of alcohols and alkenes can be utilized in this process. Numerous functional groups in either the alcohol or the alkene can be accommodated during alkoxymercuration. For example, alkenes differing in their nucleophilicity as much as vinylic ethers419 and a,(3-unsaturated aldehydes or ketones420,421 react smoothly (equations 259 and 260). 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

It is preferable, when using the Kamack equation, to smooth out the experimental data. This can be done manually by plotting Q against d on log-probability paper or using a smoothing equation during computer data collection. 

To prepare an a-diazo ester from the ester (129) itself, formylation is effective. Alternatively, it has been reported that trifluoroacetylation of an ester proceeds smoothly (equation 53).The trifluoro-methylacetoacetate (130) would be expected to react directly with methanesulfonyl azide and sodium hydride to give the a-diazo ester. 

With trichloroacetimidates, 1,3-conversion of allylic alcohols to allylic amines proceeds smoothly (equation 40). 

The best known example of this kind is the rearrangement of allyl acetates, a synthetically very useful reaction, especially in the case of tertiary systems (equation 10). These undergo isomerization sometimes even under the conditions of acetylation. OAerwise, Lewis acid catalysis, preferably with PdCl2(MeCN)2, will bring about the isomerization smoothly (equation 11). ... 

Coupling reactions of alkyl Grignard reagents lacking -H substitution are exemplified by the stereoselective synthesis of a trideuteriomethylated alkene (equation 10) and of allylsilanes (equations 11 and 12). ° Cyclic bromoalkenes also undergo coupling smoothly (equation 13). ... 

The values in this table are taken almost exclusively from the International Union of Pure and Applied Chemistry Solubility Data Series . Unless noted, they comprise evaluated data fitted to a smoothing equation. The data at each temperature are then derived from the smoothing equation which expresses the mole fraction solubility of the gas in solution as ... 

Wilson s equation corresponds to the special case of equation (41) with only one ij pair. The extended form has apparently never been used. Wilson s equations are useful as smoothing equations and because the parameters derived for binary mixtures can be applied directly to multicomponent mixtures. Their chief defect is their inability to give C large enough to lead to phase separation. This defect is usually remedied when the equations are used near a liquid-liquid region by including an arbitrary third parameter to give... 

This simplified version of the exponential smoothing equation is often faster to compute on a calculator and is easier to understand intuitively. This equation states that the forecast for the next period (F +i) is equal to the forecast for this period (F ) plus a weight (a) times the error in this period s forecast (d—F). 

By comparing these two time-smoothed equations with Eqs. (3.6-24) and (3.10-16) we see that the time-smoothed values everywhere replace the instantaneous values. However, in Eq. (3.10-20) new terms arise in the set of brackets which are related to turbulent velocity fluctuations. For convenience we use the notation... 

Evaluation Procedure. All the data were examined and evaluated by using the method described by Cohen-Adad (13). Only experimentally obtained data were evaluated. Data obtained from smoothing equations or by extrapolation were excluded from consideration. The data calculated on the basis of sodium determination (1) and the 298 K values reported by others (3, 4) are clearly incorrect and were not included. Some data (4) were reported as mol dm d and could not be used here because no density information was given and the values could not be recalculated in terms of mole fraction. All the other data are consistent with each other and were evaluated together. 

Several papers report solubility data for this system (1-6). Empirical smoothing equations describing the temperature dependence of this solubility have also been reported (6,7). Solubility data for this system have also been reported as part of a study of multicomponent systems having KH.PO, as one component (8-24). Some of these values are clearly incorrect and were rejected immediately (8-10). The same is true for some of the solubility values reported for 323 K (11,12). In another report (13) the solubility of KH2P0 at 298 K is given as 20.3% and also as 21.6%. The latter value is an obvious error. All the other data from the studies of multicomponent systems were evaluated together with the data in refs. (1-6). 

Table III is a list of the recommended solubility values calculated from equation [1] The other smoothing equations that have been suggested (6,7) give values that are 8-10% larger than the recommended values (Table III) in the 273-288 K temperature interval.

The KHqPO/.-KCI-HyO system. Several investigators have published solubility values for this system (4, 10, 13, 19-22). The solubility of KH2PO4 at 298 K may be expressed by the smoothing equation of Kirgintsev (31) where mj is the molality of KH2PO4 and y is its solute mole... 

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