This "law" is just a special case of dimensional analysis in which an equation containing 6 dimensional quantities, p , v , T , a , b , R , and 3 dimensions, [p], [v], [T], must be expressible in terms of 6 − 3 = 3 dimensionless groups. Here v ∗ ∗ = b is a characteristic molar volume, p ∗ ∗ = a / b 2 a characteristic pressure, and T ∗ ∗ = a / ( R b ) a characteristic temperature, and the 3 dimensionless groups are p / p ∗ ∗ , v / v ∗ ∗ , T / T ∗ ∗ . The reduced properties defined previously are p r = 27 ( p / p ∗ ∗ ) , v r = ( 1 / 3 ) ( v / v ∗ ∗ ) , and T r = ( 27 / 8 ) ( T / T ∗ ∗ ) . Recent research has suggested that there is a family of equations of state that depend on an additional dimensionless group, and this provides a more exact correlation of properties. Nevertheless, as Boltzmann observed, the van der Waals equation provides an essentially correct description.