Derive relationship between critical constants and Vander walls constants

To derive the relationship between critical constants and Vander Waals constants, we can start by considering the Vander Waals equation of state:

(P + a/V^2)(V - b) = RT

Where P is the pressure, V is the volume, T is the temperature, R is the gas constant, and a and b are the Vander Waals constants.

At the critical point, the gas and liquid phases become indistinguishable, and the equation of state simplifies to:

(Pc + a/Vc^2)(Vc - b) = RTc

Where Pc is the critical pressure, Vc is the critical volume, and Tc is the critical temperature.

To find the relationship between the critical constants and Vander Waals constants, we can rearrange the equation of state at the critical point:

PcVc - Pb - a/Vc^2 + ab/Vc^3 = RTc

Since at the critical point, the gas and liquid phases become indistinguishable, the pressure and volume at the critical point can be considered as the critical constants. Therefore, we can rewrite the equation as:

PcVc = RTc + Pb + a/Vc^2 - ab/Vc^3

From this equation, we can see that the product of the critical pressure and critical volume (PcVc) is equal to the sum of the gas constant multiplied by the critical temperature (RTc), the product of the pressure at the boiling point (Pb) and the Vander Waals constant a, and the difference between the Vander Waals constant a divided by the critical volume squared (a/Vc^2) and the Vander Waals constant b divided by the critical volume cubed (ab/Vc^3).

Therefore, the relationship between the critical constants (Pc and Vc) and the Vander Waals constants (a and b) can be expressed as:

PcVc = RTc + Pb + a/Vc^2 - ab/Vc^3