Abstract
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 (∂p/∂V)T=0 requires the isothermal process, but the universality of the critical point is its independence of whatever process is taken, and so we can assume (∂p/∂T)V=0. 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 a and b in it vary with temperature, which then satisfies both assumptions and reproduces its usual form when the temperature is approximately the critical one.
Highlights
In thermal physics, a critical point is the end point of a phase equilibrium curve, the pressure–temperature curve that designates conditions under which a liquid phase and a vapor phase can coexist
We propose that the critical point is geometrically represented by zero Gaussian curvature on the thermodynamic equation of state (EoS) surface, together with some physical assumptions
One is why we assumeV 0 that is complementary to the first formula of Eq (1) and why we propose that the critical point is geometrically represented by zero Gaussian curvature on the thermodynamic EoS surface
Summary
A critical point is the end point of a phase equilibrium curve, the pressure–temperature curve that designates conditions under which a liquid phase and a vapor phase can coexist. The strong resemblance of these two facts suggests that a geometrical description of the critical point is advantageous Based on this observation, we propose that the critical point is geometrically represented by zero Gaussian curvature on the thermodynamic EoS surface, together with some physical assumptions. One is why we assume (zp/zT)V 0 that is complementary to the first formula of Eq (1) and why we propose that the critical point is geometrically represented by zero Gaussian curvature on the thermodynamic EoS surface. Another is to use the above assumptions to discuss the long-standing problem within thermodynamics. It is worth mentioning that, in contrast to the realistic experiments which seem hard to measure these two response functions near the critical point, the computer simulations are more feasible [8,9,10,11], which shows that the critical slowing down is really an overall phenomenon no matter what path is chosen to approach the critical point
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