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I don't get why fugacity coefficients, $\phi = f/p$, of pure components are usually calculated via integrating an eos over a pressure or volume range. For example, when using a pressure explicit eos (such as the Virial-Eos for example), one can write: $$ RT \ln \phi = \int_0^p(v-\frac{RT}{p})dp $$ I was wondering, when we already know the real gas parameters such as the Virial-Coefficients, for example, why not calculate the actual pressure straight away? Isn't the fugacity some kind of "real pressure", with $\phi$ serving as a conversion factor, $f = \phi p$. And isn't the pressure calculated from eos (PR, VdW, Virial etc.) also some kind of "real pressure" aswell. But why are they not equal?

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No. Fugacity is related to the Gibbs free energy variation with pressure.$$dG=VdP=\frac{zRT}{P}dP=RTd\ln{f}$$So, $$RTd\ln{(f/P)}=\frac{(z-1)RT}{P}dP$$

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