Determination of Shape Factors

The theoretical basis for the molecular shape factors was derived in section 6.2. That analysis, which led to temperature-dependent shape factors, represents an idealized case where the non-spherical potential parameters may be incorporated with the spherical parameters through angle averaging. Although that approach is correct in certain circumstances, it is of limited practical use since the intermolecular potential function for real fluids is not known precisely. Hence, one is forced to use macroscopic thermodynamic measurements to determine the shape factors and then try to develop a generalized correlation for them which depend on known molecular parameters. We shall refer to the shape factors determined from experimental data as the apparent or exact shape factors and their generalized correlation as the correlated shape factors. 

The shape factors are weak functions of temperature and, in principle, density and can be visualized as distorting scales that force the two fluids to conformality. Although there is no direct theoretical evidence for the density dependence of the shape factors, mathematical solutions for exact shape factors found by equating the dimensionless residual compressibility factor and Helmholtz energy of two pure-fluids exhibit weak density dependence. 

The first attempt to find exact shape factors is due to Leach,who equated the residual compressibility factor and fugacity coefficient of two fluids, with 

As emphasized by Leland and Chappelear, the shape factors determined from the solutions to eq 6.22 depend on both density and temperature and, as such, cannot be related to any sort of intermolecular pair potential. An explanation for the apparent density dependence can be partially attributed to the role of three-body intermolecular forces which are not considered in the basic corresponding-states model. In particular, it has been shown that if one wish to simultaneously represent gas phase and condensed phase properties three body forces must be included in the calculations. One method of achieving a simultaneous representation of properties is, however, the use of an effective 

Using methane as reference and a large number of pure normal hydrocarbons from CH4 to C15H32, Leach obtained solutions to these systems of equations and empirically fitted the results in terms of the acentric factor and the critical parameters. The set of correlated shape factors that were obtained is given by  

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Qualitative terms [10] may be used to give some indication of particle shape but these are of limited use as a measure of particle properties ( Fable 2.4). Such general terms are inadequate for the determination of shape factors that can be incorporated as parameters into equations concerning particle properties where shape is involved as a factor. In order to do this, it is necessary to be able to measure and define shape quantitatively. 

The dimensionless geometric parameter VA/A is proposed as an alternate parameter for determination of shape factors of complex convex bodies. 

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