Van der Waals supercritical fluid: exact formulas for special lines

Area of Science:

• Thermodynamics
• Statistical Mechanics
• Fluid Dynamics

Background:

• The van der Waals model is a fundamental equation of state for real gases.
• Supercritical fluids exhibit unique properties between liquid and gas phases.
• Understanding critical phenomena is crucial for chemical engineering and materials science.

Purpose of the Study:

• To derive analytical expressions for thermodynamic property extrema in the supercritical region of van der Waals fluids.
• To investigate the behavior of Widom lines, Batschinski lines, and the pseudo-Gruneisen parameter.
• To determine the conditions under which different thermodynamic ridges merge into a single Widom line.

Main Methods:

• Application of the van der Waals model.
• Analytical derivation of expressions for thermodynamic loci.
• Analysis of critical phenomena and fluid behavior.

Main Results:

• Analytical expressions for ridges of heat capacity, thermal expansion coefficient, compressibility, density fluctuation, and sound velocity were obtained.
• Widom lines merge only at T < 1.07T(c) and P < 1.25P(c), becoming smeared at higher reduced temperatures and pressures.
• The pseudo-Gruneisen parameter (γ) for a van der Waals fluid at the critical point is 8/3, consistent with a soft sphere system (n=14).

Conclusions:

• The Widom line is not a universal feature across all supercritical conditions for van der Waals fluids.
• The study provides a theoretical framework for understanding critical behavior in real fluids.
• The findings offer insights into the relationship between macroscopic thermodynamic properties and microscopic particle interactions.