The maximum of this, quadratic, curve occurs, with z 2 = T / ( 2 T ∗ ∗ ) , for D z p / p ∗ ∗ = 4 − − 6 ( T / 2 T ∗ ∗ ) 1 / 2 = 0 which gives ( T / 2 T ∗ ∗ ) m a x 1 / 2 = 2 / 3 , or T m a x = 8 T ∗ ∗ / 9 , and the corresponding p m a x = p ∗ ∗ / 3 . The zeros of the inversion curve 3 z 2 − − 4 z + 1 = 0 , are, making use of the quadratic formula, z = ( 4 ± ± 16 − − 12 ) / 6 , or z = 1 / 3 and 1 ( T / T ∗ ∗ = 2 / 9 = 0. 2 ¯ ¯ and 2 ). In terms of the dimensionless variables, T r , p r the zeros are at T r = 3 / 4 and 27 / 4 , while the maximum is p r m a x = 9 , and occurs at T r m a x = 3 . Note from Fig. 5 that there is an overlap between the saturation curve and the inversion curve plotted there. This region is shown enlarged in the right hand graph of the figure. Therefore a van der Waals gas can be liquified by passing it through a throttle under the proper conditions; real gases are liquified in this way.