Although the gap in v delimited by the two spinodal points on an isotherm (e.g. T r = 7 / 8 shown in Fig. 1) is the origin of the phase change, the spinodal points do not represent its full extent, because both states, saturated liquid and saturated vapor coexist in equlilbrium; they both must have the same pressure as well as the same temperature. Thus the phase change is characterized, at temperature T s , by a pressure p m i n < p s < p m a x that lies between that of the minimum and maximum spinodal points, and with molar volumes of liquid, v f < v m i n and vapor v g > v m a x . Then from the vdW equation applied to these saturated liquid and vapor states p s = R T s v f − − b − − a v f 2 and p s = R T s v g − − b − − a v g 2 These two vdW equations contain 4 variables, p s , T s , v f , v g , so another equation is required in order to specify the values of 3 of these variables uniquely in terms of a fourth. Such an equation is provided here by the equality of the Gibbs free energy in the saturated liquid and vapor states, g f = g g . This condition of material equilibrium can be obtained from a simple physical argument as follows: the energy required to vaporize a mole is from the second law at constant temperature q v a p = T ( s g − − s f ) , and from the first law at constant pressure q v a p = h g − − h f . Equating these two, rearranging, and recalling that g = h − − T s produces the result.