Equilibrium theory of hard-sphere fluids
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.
One-component case.
Consider the one-component homogeneous hard-sphere fluid with molecule radius formula_1. To obtain its equation of state in the form formula_2 (where formula_3 is the pressure, formula_4 is the density of the fluid and formula_5 is the temperature) one can find the expression for the chemical potential formula_6 and then use the Gibbs–Duhem equation to express formula_3 as a function of formula_4.
The chemical potential of the fluid can be written as a sum of an ideal-gas contribution and an excess part: formula_9. 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 formula_10 is equivalent to creating a cavity of radius formula_11 in the hard-sphere fluid. The SPT theory gives an approximate expression for this work formula_12. In case of inserting a molecule formula_13 it is
formula_14,
where formula_15 is the packing fraction, formula_16 is the Boltzmann constant.
This leads to the equation of state
formula_17
which is equivalent to the compressibility equation of state of the Percus-Yevick theory.
References.
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