AUTHOR=Yu J. S. , Zhou X. , Chen J. F. , Du W. K. , Wang X. , Liu Q. H. 

TITLE=Local Shape of the Vapor–Liquid Critical Point on the Thermodynamic Surface and the van der Waals Equation of State

JOURNAL=Frontiers in Physics

VOLUME=9

YEAR=2021

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2021.679083

DOI=10.3389/fphy.2021.679083

ISSN=2296-424X

ABSTRACT=<p>Differential geometry is a powerful tool to analyze the vapor–liquid critical point on the surface of the thermodynamic equation of state. The existence of usual condition of the critical point <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>p</mml:mi><mml:mo>/</mml:mo><mml:mo>∂</mml:mo><mml:mi>V</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>T</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> requires the isothermal process, but the universality of the critical point is its independence of whatever process is taken, and so we can assume <inline-formula id="inf2"><mml:math id="m2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>p</mml:mi><mml:mo>/</mml:mo><mml:mo>∂</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>V</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>. The distinction between the critical point and other points on the surface leads us to further assume that the critical point is geometrically represented by zero Gaussian curvature. A slight extension of the van der Waals equation of state is to letting the two parameters <italic>a</italic> and <italic>b</italic> in it vary with temperature, which then satisfies both assumptions and reproduces its usual form when the temperature is approximately the critical one.</p>