Proving a relation for fugacity of component i in term of independent variables (T,V) In Molecular Thermodynamics of fluid phase equilibria by Prausnitz et al. [1] the authors recommend to use the following equation which gives the fugacity of component $i$ in terms of independent variables $V$ (volume) and $T$ (temperature):

$$RT\ln\phi_i = \int_v^\infty\left(\left(\frac{\partial p}{\partial n_i}\right)_{T, V, n_i} - \frac{RT}{V}\right)\mathrm{d}V - RT\ln z$$

I know that I should use chemical potential definition to prove this equation, but there are many different ways to calculate it like using Helmholtz free energy or Gibbs free energy. 

I would appreciate someone give me an idea or any references to help me proving this equation. 

References


Prausnitz, J. M.; Lichtenthaler, R. N.; Azevedo, E. G. de. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd edition; Prentice Hall: Upper Saddle River, N.J, 1998. ISBN 978-0-13-977745-5.
I haven't been able to work out all the details of the mathematics yet, but the derivation of this equation must start out from the following equation:  

$$nRT\mathrm{d}\ln{\phi} = (V - V^{ig})\mathrm{d}P,$$

where $\phi$ is the fugacity coefficient of the mixture, $n$ is the total moles of the mixture, $V$ is the volume of the mixture $znRT/P$ and $V^{ig}$ is the volume of the mixture under ideal gas conditions: $nRT/P$.  Once the equation is integrated to get $RTn\ln{\phi}$, we get the fugacity of the $i$th component of the mixture by evaluating the following partial derivative:

$$RT\ln{\phi_i} = RT\frac{\partial (n\ln{\phi})}{\partial n_i}$$ 

Hope that this helps a little.

ADDENEUM

If we write, $$d(nV)=\left(\frac{\partial (nV)}{\partial n_i}\right)_{p,n_k}dn_i+\left(\frac{\partial (nV)}{\partial p}\right)_{ni,n_k}dp$$

then it follows that:  $$\left(\frac{\partial (nV)}{\partial n_i}\right)_{p,n_k}=-\left(\frac{\partial (nV)}{\partial p}\right)_{ni,n_k,V}\left(\frac{\partial (p)}{\partial n_i}\right)_{n_k,V}$$

The rest is easy.  The subscript $n_i$ in the OP equation is a typo.  It should be $n_k$, representing all other species being held constant.