Hard-sphere theory

To a first approximation, the activation energy can be obtained by subtracting the energies of the reactants and transition structure. The hard-sphere theory gives an intuitive description of reaction mechanisms however, the predicted rate constants are quite poor for many reactions. 

Two predictions of the LADM for the effective viscosity are shown In Table II. The first was made by using the Enskog hard-sphere theory for the calculation of the local viscosities. It agrees qualitatively with the simulation result In that It predicts a large decrease of the effective viscosity as a result of the density structure. For the second prediction the local... 

Mass Diffusivity in Liquid Metais and Ailoys. The hard-sphere model of gases works relatively well for self-diffusion in monatomic liquid metals. Several models based on hard-sphere theory exist for predicting the self-diffusivity in liquid metals. One such model utilizes the hard-sphere packing fraction, PF, to determine D (in cm /s) ... 

The more obvious and consistent deviations from the hard sphere theory occur, at the low density values, due to the effects of attractive forces in the real system. We can attempt to correct for these effects using a method described previously (27-30) for the analysis of angular momentum correlation times in supercritical CFjj and CFjj mixtures with argon and neon. We replace the hard sphere radial distribution function at contact hs with a function gp (0) which uses the more realistic... 

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. 

Liu and Ruckenstein [Ind. Eng. Chem. Res. 36, 3937 (1997)] studied self-diffusion for both liquids and gases. They proposed a semiem-pirical equation, based on hard-sphere theory, to estimate self-diffusivities. They extended it to Lennard-Jones fluids. The necessary energy parameter is estimated from viscosity data, but the molecular collision diameter is estimated from diffusion data. They compared their estimates to 26 pairs, with a total of 1822 data points, and achieved a relative deviation of 7.3 percent. 

He and Yu [13] presented a semiempirical equation to estimate diffusivities under supercritical conditions that is based on hard-sphere theory. It is limited to pr > 0.21, where the reduced density is pr = pA(T, P)/pcA. They compared their estimates to 107 pairs, with a total of 1167 data points, and achieved lower deviations (7.8 percent) than the Catchpole-King equation (9.7 percent), which was restricted to PrS I-... 

This mean-free-path treatment involves several simplifying and somewhat crude approximations a more sophisticated and rigorous hard-sphere theory gives results identical with Eqs. (10) and (11) except for the numerical factors ... 

Crystallization occurs for many common fluids, such as carbon tetrachloride, benzene, and cyclohexane, at pressures less than 200 MPa, thus their entire fluid range is rather limited. The supercritical noble gases significantly extend this range, achieving a maximum crystallization pressure for helium of P = 11.6 GPa. The viscosity increase prior to all such transitions is modest. The viscosity for a typical dense fluid is 1-100 mPa and this will increase by, at most, about three orders of magnitude. Experimentally, this viscosity and pressure regime is covered by many of the viscometers discussed below, and hard-sphere theory can explain most of the viscosity increase. 

Complications of these kinds are very difficult to treat by a simple modification of the hard-sphere theory of collisions, and an alternative treatment is necessary. 

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. 

Hard sphere theories can be compared in a special sense with "experiment . This is not because any real substance possesses a fluid phase which over its whole existence region could be adequately described by this simple model. The "experimental data, rather, are the result of extensive machine calculations of the equation of state of large collections of hard spheres undertaken by Alder and Wainwright and Wood, Parker, and Jacobson, among others. 

Comparison of the Hard Sphere Theory and Experiment (a) Pure Liquids... 

Calculated from hard-sphere theory ---------Experimental. 

In Fig. 18 we also show the density dependence of the viscosity tj for argon as computed by this modified Enskog theory (MET) at 348 K, where x and b in Eq. (147a) are replaced by x and b as determined from Eqs. (151) and (154), respectively. We see that the MET agrees with experiment to within 10% over a much wider range of densities than does the simple Enskog theory discussed earlier. This improvement appears to be due to the fact that the MET takes into account the actual density dependence of the collision frequency, while the simple Enskog theory approximates the collision frequency by the hard-sphere theory, extrapolated from low densities. 

Matthews, M. A., and Akgerman, A., 1987, Hard - sphere theory for correlation of tracer diffusion of gases and liquids in alkanes , J. Chem. Phys. 87, 2285. 

Donahue and Prausnitz [23] developed the perturbed-hard chain theory (PHCT) based on perturbed hard-sphere theory for small molecules and Prigogine s theory for chain molecules. In order to account for attractive and repulsive forces among molecules, empirical parameters such as the c parameter were introduced by subsequent investigators. More accurate expressions for the repulsive and attractive forces were... 

In all the equations that include the double power-series expansion (or variations thereof), such as the family of perturbed-hard-chain equations, the parameters are related to molecular rather than critical properties, and the mixing and combining rules are quadratic in composition for the attractive term and based on hard-sphere theory for the repulsive term. 

Because of the unphysical nature of a pseudopotential that depends on thermodynamic state, one would like an alternative description valid over the whole liquid range. One approach is to generalize the van der Waals mean field model by incorporating a pressure term of the form F" rather than the van der Waals V term. The expression for the cohesive energy in Eq. (3.28) is then U — —constant/F " with m > 1. An additional improvement of the simple van der Waals model is made by using hard sphere theory. The corresponding equation of state is then... 

In this case a less rigorous approach has to be applied which involves a modification of the Enskog hard-sphere theory for real gases (Enskog 1922 Hanley etal. 1972 Hanley Cohen 1976 Ross et al. 1986 Nieto de Castro et al. 1990). This can be used to find an approximate value for the internal contribution for polyatomic gases, whereas the... 

Smooth hard-sphere theory for monatomic fluids... 

For calculation of transport properties on the basis of hard-sphere theory, the hard-sphere expressions for the dilute-gas transport properties are required. These are given... 

The smooth hard-sphere theory discussed above has been remarkably successful for monatomic fluids, as exemplified by xenon (see Chapter 10). For application to polyatomic fluids, it is necessary to take into account additional considerations ... 

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

In view of the success of the methods based on hard-sphere theories for the accurate correlation and prediction of transport properties of single-component dense fluids, it is worthwhile to consider the application of the hard-sphere model to dense fluid mixtures. The methods of Enskog were extended to mixtures by Thome (see Chapman Cowling 1952). The binary diffusion coefficient >12 for a smooth hard-sphere system is given by... 

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. 

Developments in liquid state theory and empirical studies of transport phenomena in liquids can be exploited by restricting the correlation to the region of saturated or compressed liquids away from the critical point (Brush 1962 Schwen Puhl 1988). The basis of one such class of correlations is the recognition that liquid transport properties, and in particular viscosity, can be correlated well in terms of the difference between the molar volume and a compact packing volume which is fluid-specific and a weak function of temperature. The simplest form of this type of correlation is the fluidity ( ) versus molar volume V = p ) approach described by Batschinski (1913) and Hildebrand (1977) for pure fluids based on experimental observations these variables are linearly related to a very good approximation. Hard-sphere theories such as that proposed by Enskog and several modifications of this approach have also been used for liquids. Current work in this area has evolved significantly and is described in detail in Chapters 5 and 10. 

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... 

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