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.