# Why is the fugacity of a pure component not equal to the pressure calculated by an equation of state for that pure component?

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, if we already have the Virial-Coefficients, 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?

The pressure calculated from an equation of state is just the pressure (for a real gas, if applicable). One useful definition is $$P\equiv\left(\frac{\partial G}{\partial V}\right)_{T,N}$$.
Indeed, the chemical potential relative to that at a reference state is $$\mu=RT\ln a$$, where the dimensionless activity $$a$$ for a gas is the fugacity divided by a reference pressure. For ideal mixtures, the activity is the concentration (specifically, the molar fraction). For ideal gases, the fugacity is the pressure.