How to get equation of state from fugacity Given that the fugacity of a substance is $f=Pe^{bT}$, how do I obtain the equation of state of the substance?
This is my approach, but I have not made good headway:
$$RT d (\ln f) = VdP \\
d(\ln f) = \frac{V}{RT} dP \\
bdT = \left( \frac{V}{RT} - \frac{1}{P}\right)dP$$
But what next? I see that
$$\left( \frac{V}{RT} - \frac{1}{P}\right)dP = \frac{(Z-1)}{P}dP$$
But I don't see how I can reduce this further.
My end goal is to find derivatives of pressure wrt temperature keeping volume constant. How do I go about this problem?
That first equation is supposed to be at constant T.  You can rewrite it as $$RTd\ln{(f/P)}=RT(z-1)\frac{dP}{P}$$or $$\ln{\frac{f}{P}}=bT=\int_0^P{(z-1)\frac{dP'}{P'}}$$So if z behaves like z = 1+aPT, this becomes:
$$bT=aPT$$ or b = aP