Volume, molar, critical estimation

Chueh s method for calculating partial molar volumes is readily generalized to liquid mixtures containing more than two components. Required parameters are and flb (see Table II), the acentric factor, the critical temperature and critical pressure for each component, and a characteristic binary constant ktj (see Table I) for each possible unlike pair in the mixture. At present, this method is restricted to saturated liquid solutions for very precise work in high-pressure thermodynamics, it is also necessary to know how partial molar volumes vary with pressure at constant temperature and composition. An extension of Chueh s treatment may eventually provide estimates of partial compressibilities, but in view of the many uncertainties in our present knowledge of high-pressure phase equilibria, such an extension is not likely to be of major importance for some time. 

The major differences between behavior profiles of organic chemicals in the environment are attributable to their physical-chemical properties. The key properties are recognized as solubility in water, vapor pressure, the three partition coefficients between air, water and octanol, dissociation constant in water (when relevant) and susceptibility to degradation or transformation reactions. Other essential molecular descriptors are molar mass and molar volume, with properties such as critical temperature and pressure and molecular area being occasionally useful for specific purposes. A useful source of information and estimation methods on these properties is the handbook by Boethling and Mackay (2000). 

To model the solubility of a solute in an SCF using an EOS, it is necessary to have critical properties and acentric factors of all components as well as molar volumes and sublimation pressures in the case of solid components. When some of these values are not available, as is often the case, estimation techniques must be employed. When neither critical properties nor acentric factors are available, it is desirable to have the normal boiling point of the compound, since some estimation techniques only require the boiling point together with the molecular structure. A customary approach to describing high-pressure phenomena like the solubility in SCFs is based on the Peng-Robinson EOS [48,49], but there are also several other EOS s [50]. 

To design a supercritical fluid extraction process for the separation of bioactive substances from natural products, a quantitative knowledge of phase equilibria between target biosolutes and solvent is necessary. The solubility of bioactive coumarin and its various derivatives (i.e., hydroxy-, methyl-, and methoxy-derivatives) in SCCO2 were measured at 308.15-328.15 K and 10-30 MPa. Also, the pure physical properties such as normal boiling point, critical constants, acentric factor, molar volume, and standard vapor pressure for coumarin and its derivatives were estimated. By this estimated information, the measured solubilities were quantitatively correlated by an approximate lattice equation of state (Yoo et al., 1997). 

Partial molar volumes and the isothermal compressibility can be calculated from an equation of state. Unfortunately, these equations require properties of the components, such as critical temperature, critical pressure and the acentric factor. These properties are not known for the benzophenone triplet and the transition state. However, they can be estimated very roughly using standard techniques such as Joback s modification of Lyderson s method for Tc and Pc and the standard method for the acentric factor (Reid et al., 1987). We calculated the values for the benzophenone triplet assuming a structure similar to ground state benzophenone. The transition state was considered to be a benzophenone/isopropanol complex. The values used are shown in Table 1. 

The corresponding diffusion coefficient in the liquid state (L), DLderived from Ds.ji and Dc. Starting from the reference class of n-alkanes, diffusion coefficients can be estimated for any solute and matrix with corresponding specific values for the critical temperature and critical pressure of the matrix and the critical molar volume of the solute. 

Estimate the molar volume of isopropyl alcohol vapor at 10 atm (1013 kPa) and 473 K (392°F) using the Redlich-Kwong equation of state. For isopropyl alcohol, use 508.2 K as the critical temperature Tc and 50 atm as the critical pressure Pc. The Redlich-Kwong equation is... 

Under conditions below the critical temperature, Tc, of the adsorbate, Vm is assumed equal to the molar volume of the saturated liquid at system temperature. Above Tc the adsorbed phase is ill-defined, and this has led to different approximations been proposed for Vm. Likewise, in the supercritical region the concept of vapor pressure does not exist and Ps in Eq. (2) must be replaced a pseudo-vapor pressure. In the present work we followed the suggestions of Agarwal and Schwarz, and estimated and Vm at temperatures above Tc as follows ... 

There have been a number of modeling efforts that employ the concept of clustering in supercritical fluid solutions. Debenedetti (22) has used a fluctuation analysis to estimate what might be described as a cluster size or aggregation number from the solute infinite dilution partial molar volumes. These calculations indicate the possible formation of very large clusters in the region of highest solvent compressibility, which is near the critical point. Recently, Lee and coworkers have calculated pair correlation functions of solutes in supercritical fluid solutions ( ). Their results are also consistent with the cluster theory. 

For biomaterials that are thermally unstable and decompose before reaching the critical temperature, several estimation techniques are available. We have used the Lydersen group contributions method ( ). Other techniques available for predicting critical properties have been reviewed and evaluated by Spencer and Daubert ( ) and Brunner and Hederer Qfi). It is also possible to determine the EOS parameters from readily measurable data such as vapor pressure, and liquid molar volume instead of critical properties (11). We used the Lydersen method to get pure component parameters because the vapor compositions we obtained were in closer agreement with experiment than those we got from pure component parameters derived by Brunner s method. The critical properties we used for the systems we studied are summarized in Table II. 

Some other proposed equations for >12 better fit extended experimental data and point to a stronger temperature dependence of the coefficient, like >1,2 175. A concise informative survey and critical discussion of them was presented by Giddings [5]. Unfortunately, these more sophisticated equations usually require knowledge of some molecular parameters, which are known only for a limited number of gaseous species. No values are available for the molecular compounds of metals. Thus, we are forced to adhere to the Gilliland s prescription and, if necessary, to estimate the unknown molar volumes based on the available data for the compounds of similar stoichiometry. 

For the parallel Diels-Alder addition of methyl acrylate and cyclopentadiene, Kim and Johnston estimated the difference in the partial molar volumes of the endo and exo transition states in SCCO2, and found that the selectivity was related to the t(30) polarity parameter of the solvent [26]. Large negative partial molar volumes were measured near the critical point for several systems by Eckert et al. [41]. Further work is needed to understand the unusual reaction rates and selectivities that can appear near the critical point. 

In this section the experimental liquid phase compositions and liquid molar volumes determined for the P-diketone/C02 systems studied will be presented coupled with the modeling results using the Peng-Robinson EOS with van der Waals-1 mixing rules. Table II lists the estimated critical constants, Tc and Pc, and the acentric factor, co, needed for modeling these systems. 

The %AARD values for all the other P-diketone systems ranged from only 0.86% to 4.59%, further showing the ability of the Peng-Robinson EOS to accurately model these systems using estimated critical properties. Figure 2 shows the experimental saturated liquid volumes for the THD/CO2 system at 30.0°C, 35.0 C, and 45.0 C. Note that the Peng-Robinson EOS tends to overpredict the liquid molar volumes for this system with relative deviations ranging from 9.44% to 10.41%. This is not entirely unexpected as the adjustable parameter in the model was only fit to the liquid phase compositions (see equation 2). 

According to quantum mechanics, isolated molecules do not have a finite boundary, but rather fade away into the regions of low electron density. It has been well established, however, from properties of condensed matter and molecular interactions, that individual molecules occupy a finite and measurable volume. This notion is at the core of the concept of molecular structure. 33 A number of physical methods yield estimations of molecular dimensions. These methods include measurements of molar volumes in condensed phases, critical parameters (lattice spacings and bond distances), and collision diameters in the gas phase. 34 From these results, one derives values of atomic radii from which a number of empirical molecular surfaces can be built. Note that the values of the atomic radii depend on the physical measurement chosen. 35-i37... 

At temperatures well below critical, liquids are practically incompressible and we may assume tc o. In this approximation, an isotherm on the PVgraph is almost perpendicular to the volume axis. Accordingly, the molar volume of a compressed liquid is essentially the same as that of the saturated liquid at the same temperature. The volume of saturated liquid is often tabulated, or it can be calculated from empirical equations such as the Rackett equation. Therefore, the volume of compressed liquid can be estimated quite accurately from the volume of the saturated liquid at the same pressure. This approximation breaks dovra close to the critical point where the liquid phase becomes quite compressible. 

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