Scaled particle theory

The Scaled Particle Theory (SPT) is an equilibrium theory of hard-sphere fluids which gives an approximate expression for the equation of state of hard-sphere mixtures and for their thermodynamic properties such as the surface tension.[1][2]

One-component case

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Consider the one-component homogeneous hard-sphere fluid with molecule radius  . To obtain its equation of state in the form   (where   is the pressure,   is the density of the fluid and   is the temperature) one can find the expression for the chemical potential   and then use the Gibbs–Duhem equation to express   as a function of  .[3]

The chemical potential of the fluid can be written as a sum of an ideal-gas contribution and an excess part:  . The excess chemical potential is equivalent to the reversible work of inserting an additional molecule into the fluid. Note that inserting a spherical particle of radius   is equivalent to creating a cavity of radius   in the hard-sphere fluid. The SPT theory gives an approximate expression for this work  . In case of inserting a molecule   it is

 ,

where   is the packing fraction,   is the Boltzmann constant.

This leads to the equation of state

 

which is equivalent to the compressibility equation of state of the Percus-Yevick theory.

References

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  1. ^ Reiss, H.; Frisch, H. L.; Lebowitz, J. L. (1959-08-01). "Statistical Mechanics of Rigid Spheres". The Journal of Chemical Physics. 31 (2): 369–380. Bibcode:1959JChPh..31..369R. doi:10.1063/1.1730361. ISSN 0021-9606.
  2. ^ Reiss, Howard; Frisch, H. L.; Helfand, E.; Lebowitz, J. L. (January 1960). "Aspects of the Statistical Thermodynamics of Real Fluids". The Journal of Chemical Physics. 32 (1): 119–124. Bibcode:1960JChPh..32..119R. doi:10.1063/1.1700883. ISSN 0021-9606.
  3. ^ Hansen, Jean-Pierre (2006). Theory of simple liquids. Ian R. McDonald (3rd ed.). London: Elsevier Academic Press. ISBN 978-0-12-370535-8. OCLC 162573508.