Demonstrate that Eqs. `(2. 6a)` and `(2 . 6b)` are valid for a substance, obeying the Van der Qaals equation, in critical state
Make use of the fact that the critical state corresponds to the point of inflection in the isothermal curve `p (V)`.

Explanation

The Vandar Waal's equation can be written as (for one mole)
`p(V) = (RT)/(V - b) - (a)/(V^2)`
At the critical point `((del p)/(del V))_(T)` and `((del^2 p)/(del V^2))` vanish. Thus
`0 = (RT)/((V - b)^2) + (2 a)/(V^3)` or `(RT)/((V - b)^2) = (2a)/(V^3)`
`0 = (2 RT)/((V - b)^3) - (6 a)/(V^4)` or `(RT)/((V - b)^3) = (3a)/(V^4)`
Solving these simultaneously we get on division
`V - b = (2)/(3) V, V = 3 b ~~ V_(M Cr)`
This is the critical molar volume. Putting this back
`(R T_(Cr))/(4 b^2) = (2 a)/(27 b^3)` or`T_(Cr) = (8a)/(27 bR)`
Finally `P_(Cr) = (RT_(Cr))/(V_(MC r) - b) - (a)/(V_(MCr)^2) = (4 a)/(27 b^2) - (a)/(9 b^2) = (a)/(27 b^2)`
From these we see that `(P_(Cr) V_(MCr))/(RT_(Cr)) = (a//9b)/(8a//27 b) = (3)/(8)`.