Critical constants, empirical parameters

As the parameters of each of the two-parameter semi-empirical equations can be evaluated in terms of the critical constants of the gases of interest, we provide in Table 5.3 a set of values of the critical constants for some gases of interest. 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

Table I. Relation Between Empirical Parameters and Critical Point Constants"...

Elevens Constants. The empirical parameters in an equation advanced by Elevens for predicting the critical micelle concentrations (cmc) of surfactants in terms of the number of carbon atoms in the hydrocarbon chain. Elevens constants for numerous surfactants are tabulated in references [17, IS]. 

The critical input parameters are then (1) the grain size, which should be known for each case, (2) the Aci temperature which is calculated from thermodynamics, (3) the effective diffusion activation energy, Qea, and (4) the empirical constants aj for each element. Qea and aj were determined by empirically fitting curves derived using Eq. (11.12) to experimentally observed TTT curves, and the final formula for calculating r was given as... 

Note that the parameter (Z2/A)critical is the ratio of two empirical constants related to the strength of the Coulomb and surface (nuclear) forces. If we take the view that the limit to the size of the periodic table is given by the point at which the heaviest nuclei spontaneously fission... 

Of over one million pounds of Special Purpose lead azide produced, approximately half was not consumed but stockpiled. Later this stockpile material was tested for compliance with the RD 1333 specification and was found to be suitable. After other tests, not covered in the specification and described elsewhere in this volume, the use of Special Purpose lead azide was authorized in place of RD 1333 lead azide in the United States. Since there is such a large stockpile of Special Purpose lead azide, all U.S. CMC-type lead azide needs for several decades could be supplied by this material if it could be preserved for this length of time. The most interesting point brought forward here is that certain process parameters can be varied seemingly without affecting product quality. Others, such as CMC type, have to be held constant to maintain quality. Unfortunately, without empirical studies as conducted by Taylor et al. and Hopper, the knowledge to predict which parameters are critical is not available. 

For high molecular-weight polymer-solvent systems, the polymer critical concentration is close to zero and the interaction parameter has a value close to 0.5. Thus, a good solvent (polymer soluble in the solvent at all proportions) is obtained if < 0.5, whereas values greater than 0.5 indicate poor solvency. Since we mentioned that the model is only an approximate representation of the physical picture and that the FH parameter is often not a constant at all, this empirical rule is certainly subject to some uncertainty. Nevertheless, it has found widespread use and its conclusions are often in good agreement with experiment. 

Solubility parameters for polymers are usually obtained from viscosity or swelling experiments but it has been known for a long time that they also can be calculated from physical constants, such as thermal coefficients, critical pressure and more particularly surface tension, using semi-empirical equations. They generally give values of poor accuracy. 

In a significant departure from conventional practice, Chueh and Prausnitz (11,12) proposed that the critical constraints on the RK equation be relaxed, and that parameters b and c be treated as empirical constants, determined separately for the liquid phase and for the vapor phase of a given substance. The conventional RK expression for (T) was retained the application was to vapor-liquid equilibrium calculations, in which the vapor-phase version of the equation was used for computation of vapor-phase fugacity coefficients, but in which the liquid-phase version was used only for Poynting corrections. Thus, they proposed that... 

Because In y,x varies almost inversely with absolute temperature, % and 6/ are frequently taken as constants at some convenient reference temperature, such as 25°C. Thus, calculation of by regular solution theory involves only the pure species constants Vl and 5. The latter parameter is often treated as an empirical constant determined by back calculation from experimental data. Chao and Seader suggest that the solubility parameters of isomers be set equal. For species with a critical temperature below 25°C, and 6 at 25°C are hypothetical. However, they can be evaluated by back calculation from phase equilibria data. Recommended values of the solubility parameter are included in Appendix I. 

DE) is useful as a theoretical index to the empirical resonance energy (RE), would be erroneous under the circumstances that the symmetrical hexagonal structure of benzene is driven by the o framework alone and the n electrons favor a distorted and localized structure. They then derived a formula for RE with a distance-dependent /I parameter. They could show that the proportionality between RE and DE exists only under the assumption of a constant /I. They criticized the Hess—Schaad justification of the HSRE at the Hiickel level, because their results would show erroneously that the a-compression energy favors a distorted and localized structure. Later they worked out these ideas in more detail ° with the inclusion of o energies. They concluded that... 

A review paper examines the nucleophilic properties of solvents. It is based on accumulated data derived from calorimetric measurements, equilibrium constants, Gibbs free energy, nuclear magnetic resonance, and vibrational and electronic spectra. Parameters characterizing Lewis-donor properties are critically evaluated and tabulated for a large number of solvents. The explanation of the physical meaning of polarity and discussion of solvatochromic dyes as the empirical indicators of solvent polarity are discussed (see more on this subject in Chapter 10). ... 

---