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I've been trying to prove the relation of a and b according to critical points of PR EoS:

prove that a and b are: a = 0.457 (R^2T_c^2)/p_c , b=0.0778(RT_c)/p_c

I tried to prove it by doing the partial derivative of pressure according to volume equals to zero and the second partial derivative of pressure according to volume equals to zero but unfortunately couldn't find any relation anywhere close to what the literature shows.

Also, I couldn't mathematically prove why the compressibility factor equals to 0.307.

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