# FEC: Vapor-phase Fugacity

### 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.