# FEC: Ideal Gases vs Ideal Solutions

### 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.