text: According to the extremum principle of thermodynamics  and , namely that at equilibrium the entropy is a maximum. This leads to a requirement that . This mathematical criterion expresses a physical condition which Epstein described as follows: 
   "It is obvious that this middle part, dotted in our curves [the place where the requirement is violated, dashed gray in Fig. 1 and repeated here], can have no physical reality. In fact, let us imagine the fluid in a state corresponding to this part of the curve contained in a heat conducting vertical cylinder whose top is formed by a piston. The piston can slide up and down in the cylinder, and we put on it a load exactly balancing the pressure of the gas. If we take a little weight off the piston, there will no longer be equilibrium and it will begin to move upward. However, as it moves the volume of the gas increases and with it its pressure. The resultant force on the piston gets larger, retaining its upward direction. The piston will, therefore, continue to move and the gas to expand until it reaches the state represented by the maximum of the isotherm. Vice versa, if we add ever so little to the load of the balanced piston, the gas will collapse to the state corresponding to the minimum of the isotherm"
  While on an isotherm  this requirement is satisfied everywhere so all states are gas, those states on an isotherm,  which lie between the local minimum, , and local maximum, , for which  (shown dashed gray in Fig. 1), are unstable and thus not observed. This is the genesis of the phase change; there is a range , for which no observable states exist. The states for  are liquid and for  are vapor; due to gravity the denser liquid lies below the vapor. The transition points, states with zero slope, are called spinodal points. Their locus is the spinodal curve that separates the regions of the plane for which liquid, vapor, and gas exist from a region where no observable states exist. This spinodal curve is obtained here from the vdW equation by differentiation (or equivalently from ) as
  A projection of this space curve is plotted in Fig. 1 as the black dash dot curve. It passes through the critical point which is also a spinodal point.