The Van der waal factor

Chemistry Level pending

If I change the definition of compressibility factor of one mole of a real gas to $$\frac{pv}{RT}$$ where

p designates the corrected pressure according to Van der waal's equation

v designates the corrected volume according to Van der waal's equation

So the question is what is the value of the new compressibility factor at critical conditions?

Details and Assumptions

The compressibility factor for one mole of a real gas at critical conditions is $$\frac{PV}{RT}$$ is $$\frac{3}{8}$$ .

If Pressure at critical conditions is $$\frac{a}{27b^{2}}$$

If Volume at critical conditions if 3b

If Temperature at critical conditions is $$\frac{8a}{27bR}$$

Here R is the gas constant and a and b are Van der waal's constants.

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