  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.
