# 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

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

## 1 Answer

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.

• It very much looks like one of the thermodynamic equations after being subjected to some maxwell relation partial derivation. Or perhaps starting from the definition of fugacity and then just iterating towards T and V as variables. Oh and BTW, fugacity and ideal gas are mutually exclusive. Fugacity is a property only real gases have. Apr 21 '19 at 11:59
• 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