The Maxwell crossover and the van der Waals equation of state

TL;DR

This paper introduces the Maxwell crossover (M-line) as a universal, EoS-independent feature that extends the Widom line into the coexistence region, providing new insights into phase transitions and an analytical solution for vapor-liquid equilibrium.

Contribution

It reveals the M-line's role in unifying phase space division, clarifies the relation to the Widom line, and offers an accurate analytical solution for VLE with the van der Waals EoS.

Findings

The M-line is EoS-independent and relates to the coexistence curve diameter.

The M-line extends the Widom line into the coexistence region.

An analytical solution for VLE with vdW EoS is derived.

Abstract

The well-known Maxwell construction[1] (the equal-area rule, EAR) was devised for vapor liquid equilibrium (VLE) calculation with the van der Waals (vdW) equation of state (EoS)[2]. The EAR generates an intermediate volume between the saturated liquid and vapor volumes. The trajectory of the intermediate volume over the coexistence region is defined here as the Maxwell crossover, denoted as the M-line, which is independent of EoS. For the vdW or any cubic[3] EoS, the intermediate volume corresponds to the unphysical root, while other two corresponding to the saturated volumes of vapor and liquid phases, respectively. Due to its unphysical nature, the intermediate volume has always been discarded. Here we show that the M-line, which turns out to be strictly related to the diameter[4] of the coexistence curve, holds the key to solving several major issues. Traditionally the coexistence curve with two branches is considered as the extension of the Widom line[5,6-9]. This assertion causes an inconsistency in three planes of temperature, pressure and volume. It is found that the M-line is the natural extension of the Widom line into the vapor-liquid coexistence region. As a result, the united single line coherently divides the entire phase space, including the coexistence and supercritical fluid regions, into gas-like and liquid-like regimes in all the planes. Moreover, along the M-line the vdW EoS finds a new perspective to access the second-order transition in a way better aligning with observations and modern theory[10]. Lastly, by using the feature of the M-line, we are able to derive a highly accurate and analytical proximate solution to the VLE problem with the vdW EoS.