Over the parameter range 0 ≤ ≤ y < ∞ ∞ , f ( y ) decreases monotonically from f ( 0 ) = 1 and approaches 0 as f ( y ) ∼ ∼ 2 ( y − − 1 ) e − − y in the limit y → → ∞ ∞ . Therefore ϱ ϱ f ( 0 ) = ϱ ϱ g ( 0 ) = 1 / 2 and in the limit y → → ∞ ∞ , ϱ ϱ f ( y ) ∼ ∼ 2 ( y − − 1 ) and ϱ ϱ g ( y ) ∼ ∼ 2 ( y − − 1 ) e − − 2 y . The behavior of p s ( y ) and T s ( y ) follow from the equations. Both these properties also decrease monotonically from p s ( 0 ) = p ∗ ∗ / 27 and T s ( 0 ) = 8 T ∗ ∗ / 27 , and approach 0 as p s ∼ ∼ p ∗ ∗ ϱ ϱ g / ϱ ϱ f and T s ∼ ∼ T ∗ ∗ / ϱ ϱ f in the limit y → → ∞ ∞ . Note from these that p s ∼ ∼ ( p ∗ ∗ / T ∗ ∗ ) T s ϱ ϱ g ∼ ∼ ( p ∗ ∗ v ∗ ∗ / T ∗ ∗ ) T s ρ ρ g = ρ ρ g R T s ; the van der Waals saturated vapor is an ideal gas in this limit. To paraphrase Sommerfeld, it is remarkable that the theory due to van der Waals is able to predict that when s g − − s f ≫ ≫ 2 R the saturated vapor behaves like an ideal gas; the saturated vapor of real gases behave exactly this way.