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.  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

1. 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.
• See my new addendum. Jun 10 '19 at 11:59

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.

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.
• That's not my take on this. The first equation I wrote represents the basic relationship for the Residual Gibbs free energy of a mixture, and the second relationship follows directly from the definition of a partial molar quantity of a species in solution, in this case $\phi_i$. Getting the final equation in the form that they have written it must involve writing VdP=d(PV)-PdV, but I still haven't been able to work out the mathematics. Apr 21 '19 at 12:07
• Actually, the partial molar quantity is $\ln{\phi_i}$ Apr 21 '19 at 14:39