### Vapor-Phase Fugacity

The reference state to choose in calculating the vapor-phase
fugacity is particularly simple: an ideal gas mixture. In this way,
the denominator in our fugacity definition becomes the partial
pressure of an ideal gas at the pressure of interest
($P_{tot}$).

$\mu_i - \mu_i^o = RT \ln \left [\frac{\hat
f_i}{y_iP_{tot}} \right ]$

##### Note:

At low pressures the fugacity approaches that of an ideal gas.
That is, it approaches the partial pressure of the species of
interest, which is why we write our reference fugacity as $\hat
f_i^o = y_iP_{tot}$.

Using this expression, we can define the deviation of the vapor
from its ideal reference.

##### Definition:

The **fugacity coefficient**, $\hat \phi_i$ is
defined as the ratio of the species fugacity in the vapor mixture
to the ideal gas reference state:

$\hat \phi_i = \frac{\hat f_i}{y_iP_{tot}}$

If one were interested in **calculating** this fugacity
coefficient at a given pressure $P_{tot}$ (and a temperature of
interest), we would integrate between a low pressure (where the
vapor will behave ideally) and the pressure of interest.

$\int_{P_{IG}}^{P_{sys}} \bar V_i dP = RT \ln \left [\hat
\phi_i \right ]$

When a vapor is behaving ideally, the fugacity coefficient,
$\hat \phi_i$, becomes equal to 1. This is easy to see by
recalling the above expression:

$\int_{P_{IG}}^{P_{IG}} \bar V_i dP = RT \ln \left [\hat
\phi_i \right ]$

If the gas behaves ideally, the integral becomes zero (because
the pressure to achieve ideal behavior approaches $P$).

##### Outcome:

Use the fugacity coefficient to calculate the vapor phase
fugacity.