Physics:Scaled particle theory

Short description: 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 [math]\displaystyle{ R }[/math]. To obtain its equation of state in the form [math]\displaystyle{ p=p(\rho,T) }[/math] (where [math]\displaystyle{ p }[/math] is the pressure, [math]\displaystyle{ \rho }[/math] is the density of the fluid and [math]\displaystyle{ T }[/math] is the temperature) one can find the expression for the chemical potential [math]\displaystyle{ \mu }[/math] and then use the Gibbs–Duhem equation to express [math]\displaystyle{ p }[/math] as a function of [math]\displaystyle{ \rho }[/math].

The chemical potential of the fluid can be written as a sum of an ideal-gas contribution and an excess part: [math]\displaystyle{ \mu=\mu_{id}+\mu_{ex} }[/math]. 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 [math]\displaystyle{ R_0 }[/math] is equivalent to creating a cavity of radius [math]\displaystyle{ R_0+R }[/math] in the hard-sphere fluid. The SPT theory gives an approximate expression for this work [math]\displaystyle{ W(R_0) }[/math]. In case of inserting a molecule [math]\displaystyle{ (R_0=R) }[/math] it is

[math]\displaystyle{ \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} }[/math],

where [math]\displaystyle{ \eta\equiv\frac{4}{3}\pi R^3\rho }[/math] is the packing fraction, [math]\displaystyle{ k }[/math] is the Boltzmann constant.

This leads to the equation of state

[math]\displaystyle{ \frac{p}{kT\rho}=\frac{1+\eta+\eta^2}{(1-\eta)^3} }[/math]

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