### Ideal Gases vs Ideal Solutions

Since we will be using ideal gases (which we are already
familiar with) as a reference for gas phase fugacities and ideal
solutions for liquid phase fugacities, it is useful to examine the
similarities and differences between the two.

##### Note

An **ideal gas** is one whose molecules occupy no
volume and have no intermolecular interactions. *It is generally
assumed that gases at low pressure approximate this
behavior*.

##### Note

An **ideal solution** is one whose molecules
exhibit essentially the same intermolecular interactions between
all constituents. *It is generally assumed that liquid mixtures
that are highly concentrated (dilute) or mixtures of
molecularly-similar materials approximate this behavior*.

These ideal mixtures exhibit no change in intermolecular
interactions upon mixing, so we can conclude the following:

ideal gas |
ideal solution |

$\Delta v_{mix}^{ig} = 0$ |
$\Delta v_{mix}^{is} = 0$ |

$\Delta h_{mix}^{ig} = 0$ |
$\Delta h_{mix}^{is} = 0$ |

$\Delta s_{mix}^{ig} = -R\sum_i y_i\ln[y_i]$ |
$\Delta s_{mix}^{is} = -R\sum_i x_i\ln[x_i]$ |

$\Delta g_{mix}^{ig} = RT\sum_i y_i\ln[y_i]$ |
$\Delta g_{mix}^{is} = RT\sum_i x_i\ln[x_i]$ |

We can expand the $\Delta g_{mix} as:$

$\Delta g_{mix}^{ig} = \sum_i (\bar g_i - g_i)^{ig} =
RT\sum_i y_i\ln[y_i]$

$\Delta g_{mix}^{is} = \sum_i (\bar g_i - g_i)^{is} =
RT\sum_i x_i\ln[x_i]$

If we consider the $\Delta g_{mix}$ on a component-by-component
basis, we can write:

$(\bar g_i - g_i)^{ig} = RT \ln[y_i]$

$(\bar g_i - g_i)^{is} = RT ln[x_i]$

Rearranging and recalling the definition of $\mu$:

$\mu_i^{ig} = \bar g_i^{ig} = g_i^{ig} + RT \ln[y_i]$

$\mu_i^{is} = \bar g_i^{is} = g_i^{is} + RT \ln[x_i]$

##### Note

Despite the fact that the two versions of ideal solution differ
in what limit they are applicable, they *both* yield the
same (above) analysis for property changes upon mixing.