The two first partial derivatives of the vdW equation are ∂ ∂ p ∂ ∂ T = R v − − b and ∂ ∂ p ∂ ∂ v = − − R T ( v − − b ) 2 + 2 a v 3 The first equation is α α / κ κ T , while the second is − − ( v κ κ T ) − − 1 , where κ κ T = − − v − − 1 ∂ ∂ p v , the isothermal compressibility, is a measure of the relative increase of volume from an increase of pressure, at constant temperature, while α α = v − − 1 ∂ ∂ T v p , the coefficient of thermal expansion, is a measure of the relative increase of volume from an increase of temperature, at constant pressure. Therefore κ κ T = v 2 ( v − − b ) 2 R T v 3 − − 2 a ( v − − b ) 2 and α α = R v 2 ( v − − b ) R T v 3 − − 2 a ( v − − b ) 2 In the limit v → → ∞ ∞ α α = 1 / T while κ κ T = v / ( R T ) . Since the vdW equation in this limit becomes p = R T / v , finally κ κ T = 1 / p . Both of these are the ideal gas values, which is consistent because, as noted earlier, the vdW fluid behaves like an ideal gas in this limit.