While on an isotherm T > T c c this requirement is satisfied everywhere so all states are gas, those states on an isotherm, T < T c which lie between the local minimum, v m i n , and local maximum, v m a x , for which ∂ ∂ p / ∂ ∂ v | T > 0 (shown dashed gray in Fig. 1), are unstable and thus not observed. This is the genesis of the phase change; there is a range v m i n ≤ ≤ v ≤ ≤ v m a x , for which no observable states exist. The states for v < v m i n are liquid and for v > v m a x > v m i n 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 κ κ T = ∞ ∞ ) as T s p = 2 a ( v − − b ) 2 R v 3 = T c ( 3 v r − − 1 ) 2 4 v r 3 p s p = a ( v − − 2 b ) v 3 = p c ( 3 v r − − 2 ) v r 3 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.