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.

One-component case

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

The chemical potential of the fluid can be written as a sum of an ideal-gas contribution and an excess part: $\mu =\mu _{id}+\mu _{ex}$. 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 $R_{0}$ is equivalent to creating a cavity of radius $R_{0}+R$ in the hard-sphere fluid. The SPT theory gives an approximate expression for this work $W(R_{0})$. In case of inserting a molecule $(R_{0}=R)$ it is

${\frac {\mu _{ex}}{kT}}={\frac {W(R)}{kT}}=-\ln(1-\eta )+{\frac {6\eta }{1-\eta }}+{\frac {9\eta ^{2}}{2(1-\eta )^{2}}}+{\frac {p\eta }{kT\rho }}$,

where $\eta \equiv {\frac {4}{3}}\pi R^{3}\rho$ is the packing fraction, $k$ is the Boltzmann constant.

This leads to the equation of state

${\frac {p}{kT\rho }}={\frac {1+\eta +\eta ^{2}}{(1-\eta )^{3}}}$

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