### Liquid-Phase Fugacity

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

For a liquid, a reference state ($f_i^o$) of an ideal gas is a
poor choice. Instead, we choose what is called an ideal
solution.

##### Definition:

An **ideal solution** is a solution where all of
the intermolecular interactions are essentially the same. The two
ways that this could be accomplished is:

- having any composition of mixture with components that are
molecularly similar
- having a very dilute (or very concentrated) solution

In both of these cases the intermolecular interactions are the
same.

##### Outcome:

Identify conditions when a liquid or solid mixture would form an
ideal solution.

The two choices of reference state for the liquid phase fugacity
are therefore:

- Lewis-Randall State: a state where a-a type interactions are
dominant. This would be the choice for both (all) components in a
molecularly similar mixture or the concentrated component in a
concentrated/dilute mixture
- Henry State: a state where the a-b type interactions are
dominant. This would be the choice for the dilute component in a
concentrated/dilute mixture

##### Outcome:

Explain when Lewis-Randall versus Henry ideal solution reference
states are appropriate.

#### Lewis-Randall

For an a-a dominant ideal solution (Lewis-Randall solution), the
proper choice of reference state is the pure-species fugacity
$\hat f_i^o = x_if_i$ (Note the lack of a hat.) As we will see,
under certain conditions this can reduce to the saturation pressure
of the pure substance.

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

#### Henry

For an a-b dominant ideal solution (Henry solution), the proper
choice of reference state is the so-called Henry's constant for
the species $\hat f_i^o = x_i \mathcal{H}_i$. This quantity can
be found tabulated in a variety of places.

$\mu_i - \mu_i^o = RT \ln \left [\frac{\hat f_i}{x_i
\mathcal{H}_i} \right ]$

For both reference states, it is convenient to define a new
quantity ...

##### Definition:

The **activity coefficient**, $\gamma_i$ is
defined as the ratio of the species fugacity in the liquid mixture
to the ideal solution reference state fugacity:

$\gamma_i = \frac{\hat f_i}{\hat f_i^o}$

L-R: $\gamma_i = \frac{\hat f_i}{x_if_i}$

Henry: $\gamma_i = \frac{\hat f_i}{x_i \mathcal{H}_i}$

##### Outcome:

Use the activity coefficient to calculate the liquid (or solid)
phase fugacity.