Parameter Correlation Equations

VII. Multi-parameter Correlation Equations All authors who have seriously considered the scope and limitations of the linear free-energy relationships have recognized the existence of real deviations. Frequently, the limitations of the Hammett eq. (1) for certain substituents in certain situations were considered to be indicative of a duality of u-constants. Hammett noticed that the reactions of anilines and phenols required a special value for aJt NOt, 1.27, in contrast to the value, 0.778, derived from benzoic acids. An example is the increased acidity of p-nitrophenol over that expected on the basis of the a constant based on benzoic acid. Resonance interaction between the substituent and the side-chain is presumed to be responsible ... 

Another problem that has been tackled by multivariate statistical methods is the characterization of the solvation capability of organic solvents based on empirical parameters of solvent polarity (see Chapter 7). Since such empirical parameters of solvent polarity are derived from carefully selected, strongly solvent-dependent reference processes, they are molecular-microscopic parameters. The polarity of solvents thus defined cannot be described by macroscopic, bulk solvent characteristics such as relative permittivities, refractive indices, etc., or functions thereof. For the quantitative correlation of solvent-dependent processes with solvent polarities, a large variety of empirical parameters of solvent polarity have been introduced (see Chapter 7). While some solvent polarity parameters are defined to describe an individual, more specific solute/solvent interaetion, others do not separate specific solute/solvent interactions and are referred to as general solvent polarity scales. Consequently, single- and multi-parameter correlation equations have been developed for the description of all kinds of solvent effects, and the question arises as to how many empirical parameters are really necessary for the correlation analysis of solvent-dependent processes such as chemical equilibria, reaction rates, or absorption spectra. 

The general SPP scale of solvent dipolarity/polarizability and the specific SB and SA scales of solvent HBA basicity and HBD acidity, respectively, are orthogonal to one another and they can be used in the correlation analysis of solvent effects in single- or, in combination with the others, in two- or three-parameter correlation equations, depending on the solvent-influenced process under consideration see also Section 7.7. Examples of the correlation analysis of a variety of other solvent-dependent processes by means of SPP, SB, and SA values, including those used for the introduction of other solvent polarity parameters, can be found in references [335-337, 340-342]. In particular, comparisons with Kamlet and Taft s n scale [340] and Winstein and Grunwald s Y scale [341] have been made. 

The multiparameter equation (7-54) seems to be rather difficult to apply. However, in practice, most of the linear solvation energy relationships that have been reported are simpler than indicated by Eq. (7-54) since one or more terms are inappropriate. For example, if the solute property A does not involve the creation of a cavity or a change in cavity volume between initial and activated or excited states (as is the case for solvent effects on spectral properties), the term is dropped from Eq. (7-54). If the solvent-dependent process under study has been carried out in non-HBD solvents only, the a term drops out. On the other hand, if the solutes are not hydrogen-bond donors or Lewis acids, the P term drops out of Eq. (7-54). Thus, for many solvent-dependent processes, Eq. (7-54) can be reduced to a more manageable one-, two- or three-parameter correlation equation by a judicious choice of solutes and solvents [226],... 

It has been tried to overcome this drawback by the use of multi-parameter correlation equations. One approach involves the Gutmann donor number (DN) [81]. By this the absorption maximum (v ) observed for a dye in a certain liquid can be calculated from the absorption maximum of the dye in a reference medium (Vmax.o) according to [82-85]... 

Fig. 3. Temperature-density diagram for the [Cn-mim] flf2N] family, from n=l (right) to n=14 (left), excepting n=ll and n=13. [C2-rnim] Tf2N] to [C8-mim]fTf2N] compounds were used in the molecular parameters fitting procedure, while the rest of the family is predicted with the molecular parameters correlations (equations (8)-(10)).

Extensive tabulations of Antoine parameters are available for many chemicals of importance to engineers, chemists, and environmental scientists (9,19,20). Caution is in order when using tabulated Antoine constants because several forms of the correlating equation are found in the Hterature. In particular, there are variations in the sign before the second term, the units of temperature, and the use of natural or decimal logarithms of the vapor pressure. 

The data-reduction procedure just desciiDed provides parameters in the correlating equation for g that make the 8g residuals scatter about zero. This is usually accomphshed by finding the parameters that minimize the sum of squares of the residuals. Once these parameters are found, they can be used for the calculation of derived values of both the pressure P and the vapor composition y. Equation (4-282) is solved for yjP and written for species 1 and for species 2. Adding the two equations gives... 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

B = parameter in correlating equation In = natural logarithm log = logarithm to the base 10 N = actual theoretical stages required for a given separation... 

In view of the above arguments, we have continued to use the CTr substituent constants as a delocalized effect parameter. Correlations in this paper have been made therefore with Oj and or constants. The ai values were generally taken from the compilation of Charton (29), and Or values were obtained from the equation... 

The whole concept based on parameters, although used several times (3, 57, 155, 156, 201) and advocated particularly by Good and Stone (200), has a principal defect. The results are dependent not only on experimental rate constants, but also on the values of the parameter and on the form of the correlation equation used. Furthermore, the procedure does not give any idea of the possible error. Hence, it could be acceptable only in an unrealistic case, that in which the isokinetic relationship itself and the correlations with the parameter are very precise. 

The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. It is necessary to decide what equation of state to use for the vapor phase (ideal gas, Redlich-Kwong-Soave, Peng-Robinson, etc.) and what model to use for liquid activity coefficients [ideal solutions, solubility parameters, Wilson equation, nonrandom two liquid (NRTL), UNIFAC, etc.]. See Sec. 4, Thermodynamics. It is necessary to consider mixtures of chemicals, and the interaction parameters must be predictable. The best case is to determine them from data, and the next-best case is to use correlations based on the molecular weight, structure, and normal boiling point. To validate the model, the computer results of vapor-liquid equilibria could be checked against experimental data to ensure their validity before the data are used in more complicated computer calculations. 

This can be empirically modified by introducing additional parameters to give the three-parameter Antoine equation by replacing T with (T + C), where C is a constant, which is the most common vapor pressure correlation used to represent experimental data (Zwolinski and Wilhoit 1971, Boublik et al. 1984, Stephenson and Malanowski 1987, and other handbooks). 

It has already been shown that the Cone calorimeter smoke parameter correlates well with the obscuration in full-scale fires (Equation 1). At least four other correlations have also been found for Cone data (a) peak specific extinction area results parallel those of furniture calorimeter work [12] (b) specific extinction area of simple fuels burnt in the cone calorimeter correlates well with the value at a much larger scale, at similar fuel burning rates [15] (c)maximum rate of heat release values predicted from Cone data tie in well with corresponding full scale room furniture fire results [16] and (d) a function based on total heat release and time to ignition accurately predicts the relative rankings of wall lining materials in terms of times to flashover in a full room [22]. 

Correlation equation An equation with which a data set is correlated by simple (one parameter) or multiple (two or more parameters) linear regression analysis. 

The double-scale four-parameter enthalpic equation proposed in 1965 (54) and successfully developed by Drago et al seems to be the best tool so far available for correlating and predicting the formation enthalpies of Lewis adducts in the gas phase or, if really necessary, in solution. 

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