Scaled particle theory for hard sphere pairs. I. Mathematical structure

Area of Science:

• Statistical Mechanics
• Thermodynamics
• Physical Chemistry

Background:

• Scaled Particle Theory (SPT) provides a framework for understanding fluid behavior.
• The hard sphere pair correlation function, g(r), is fundamental in describing particle interactions.
• Existing methods for predicting g(r) have limitations, particularly at higher densities.

Purpose of the Study:

• To develop an extended Scaled Particle Theory (SPT) for predicting the hard sphere pair correlation function, g(r).
• To establish smooth connection conditions between small and large cavity regimes for accurate g(r) prediction.
• To derive and analyze a nonlinear integral equation for g(r) based on cavity creation work.

Main Methods:

• Extension of the Reiss-Frisch-Lebowitz scaled particle theory.
• Analysis of reversible cavity creation work for single and double spherical cavities.
• Development of closure conditions leading to a nonlinear integral equation for g(r).
• Derivation of a power series solution for g(r) in terms of density.

Main Results:

• The extended SPT provides a predictive method for the hard sphere pair correlation function g(r).
• The derived power series solution accurately reproduces the exact second and third virial coefficients.
• The predicted fourth virial coefficient shows a deviation of approximately 0.6% from the exact value.

Conclusions:

• The developed extension of SPT offers a promising predictive tool for g(r).
• The method successfully bridges microscopic cavity properties with macroscopic thermodynamic behavior.
• Further numerical analysis of the integral equation is required and will be presented in subsequent work.