Rough hard-sphere theory

The binary diffusion coefficient of liquid extract in supercritical C02 is calculated with correlations based on the rough-hard-sphere-theory [7], Within the particle structure diffusion is determined by various effects. First, the diffusion can occur only in the void fraction of the particle. Secondly, the diffusion path is given by the contorsion of the pores. 

Rough hard-sphere theory for polyatomic fluids... 

Chandler, D. (1975). Rough hard sphere theory of the self-diffusion coefficient for molecular liquids. J. Chem. Phys., 62,1358-1363. 

Erkey, C., Rodden, J.B., Matthews, M.A. Akgerman, A. (1989). Application of rough hard-sphere theory to diffusion in n-alkanes. Int. J. Thermophys., 10,953-962. 

In the application of this rough hard-sphere theory for the interpretation of transport properties of dense pseudo-spherical molecules, it is assumed that equations (10.21) and (10.22) are exact. Reduced quantities for diffusion and viscosity, similar to those defined by equations (10.11) and (10.12), are given by... 

If a reaction involves a considerable loss of entropy when the activated complex is formed from the reactants (i.e., is negative) the factor will be a small fraction and the frequency factor will be correspondingly small. A positive AS will be associated with a larger frequency factor. If A5 is zero, the frequency factor is simply ekT//i, and it is of interest to note that the magnitude of this is roughly that given by the simple hard-sphere theory of collisions. We shall later consider some of the factors which influence the magnitudes of entropies of activation, and hence of frequency factors. 

The extent of the agreement of the theoretical calculations with the experiments is somewhat unexpected since MSA is an approximate theory and the underlying model is rough. In particular, water is not a system of dipolar hard spheres.281 However, the good agreement is an indication of the utility of recent advances in the application of statistical mechanics to the study of the electric dipole layer at metal electrodes. 

Figure 8.6 Excess chemical potentials of model hard-sphere solutes of sizes roughly comparable to Ne, Ar, methane (Me), and Xe as a function of temperature. The hard-sphere diameters used were 2.8 A, 3.1 A, 3.3 A, and 3.45 A, respectively. The lines indicate the information theory model results and the symbols are the values computed directly with typical error bars (Garde et al, 1996).

Unfortunately, real molecules differ significantly from hard spheres, so Equation (1.12) to (1.14) are not directly useful for real fluids. Additional correction factors can be added to these equations for fairly realistic spherically symmetric interactions these can represent nonpolar fluids that are roughly spherical, such as the noble gases and CH4. However, most molecules of interest are far from spherical, and kinetic theory is still intractable for molecular interactions that are not spherically symmetric. Therefore, the direct applicability of kinetic theory for calculating transport properties of real fluids is limited. However, kinetic theory plays an important role in guiding the functional form of semiempirical correlations such as those discussed below. 

The idea of representing a liquid as a system of hard spheres moving in a uniform, attractive potential well is an old one suffice here to recall the van der Waals equation. Roughly one ean thus regard perturbation methods as attempts to improve the theory of van der Waals in a systematic fashion. 

Dense fluid transport property data are successfully correlated by a scheme which is based on a consideration of smooth hard-sphere transport theory. For monatomic fluids, only one adjustable parameter, the close-packed volume, is required for a simultaneous fit of isothermal self-diffusion, viscosity and thermal conductivity data. This parameter decreases in value smoothly as the temperature is raised, as expected for real fluids. Diffusion and viscosity data for methane, a typical pseudo-spherical molecular fluid, are satisfactorily reproduced with one additional temperamre-independent parameter, the translational-rotational coupling factor, for each property. On the assumption that transport properties for dense nonspherical molecular fluids are also directly proportional to smooth hard-sphere values, self-diffusion, viscosity and thermal conductivity data for unbranched alkanes, aromatic hydrocarbons, alkan-l-ols, certain refrigerants and other simple fluids are very satisfactorily fitted. From the temperature and carbon number dependency of the characteristic volume and the carbon number dependency of the proportionality (roughness) factors, transport properties can be accurately predicted for other members of these homologous series, and for other conditions of temperature and density. Furthermore, by incorporating the modified Tait equation for density into... 

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