This calculation can be done algebraically by noting that the vdW equation can be written as a cubic in v , which at the critical point is, p c c v 3 − − ( p c b + R T c ) v 2 + a v − − a b = 0 . Moreover, at the critical point all three roots coalesce so it can also be written as ( v − − v c ) 3 = v 3 − − 3 v c v 2 + 3 v c 2 v − − v c 3 = 0 Then dividing the first by p c , and noting that these two cubic equations are the same when all their coefficients are equal gives three equations b + R T c / p c = 3 v c a / p c = 3 v c 2 a b / p c = v c 3 whose solution produces the previous results for p c , v c , T c .