Gibbs Free Energy for Mixing of Ideal Gases

In a system where gases mix without any chemical reactions, the thermodynamic quantities, particularly the Gibbs free energy, are essential for understanding the driving forces behind the mixing process. For ideal gases, mixing is associated with a change in the Gibbs free energy, which determines the spontaneity of the process.

When discussing ideal gas mixtures, it’s crucial to remember that each gas behaves independently of others due to the lack of intermolecular forces (by definition of an ideal gas). As a result, the overall behavior of the mixture is the sum of the behaviors of the individual components. The Gibbs free energy change for mixing provides insight into how this distribution affects the system's energy and entropy.

The Gibbs free energy of mixing ($\Delta G_{\text{mix}}$) of two ideal gasses A and B can be expressed as a difference of the chemical potentials of the gases in the mixtures and their chemical potentials as pure gases.

We start with the general expression for the chemical of a gas:

\[\mu_i(T, p) = \mu_i^{\circ}(T) + RT \ln \frac{p}{p^{\circ}}\]

The Gibbs free energies before (i) and after (f) mixing are:

\[G_i = G_A + G_B = n_A \mu_A^{\circ} + n_B \mu_B^{\circ} + n_A RT \ln \frac{p_A}{p^{\circ}} + n_B RT \ln \frac{p_B}{p^{\circ}}\]

\[G_f = n_A (\mu_A^{\circ} + RT \ln \frac{p_A}{p^{\circ}}) + n_B (\mu_B^{\circ} + RT \ln \frac{p_B}{p^{\circ}})\]

Therefore, the Gibbs free energy change upon mixing, $\Delta G_{\text{mix}}$ is given by:

\[\Delta G_{\text{mix}} = G_f - G_i = n_A RT \ln \frac{p_A}{p} + n_B RT \ln \frac{p_B}{p}\]

Substituting the partial pressures in terms of mole fractions:

\[\Delta G_{\text{mix}} = n RT (x_A \ln x_A + x_B \ln x_B) = nRT \sum x_i \ln x_i\]

Since for all mole fractions \(x_i\), \(0 < x_i \leq 1\), the term \(\ln x_i \leq 0\), leading to:

\[\Delta_{\text{mix}} G \leq 0\]

This equation shows that \Delta G_{\text{mix}} is always negative, indicating that mixing ideal gases is a spontaneous process under constant temperature and pressure.

 

Entropy of Mixing

The entropy of mixing, $\Delta S_{\text{mix}}$, is a measure of the increase in disorder as gases mix. For ideal gases, this entropy change is positive, reflecting the tendency of gases to distribute evenly within a container.

\[\Delta_{mix} S = - \left( \frac{\partial \Delta_{mix} G}{\partial T} \right)_P = - \left( \frac{nRT \sum x_i \ln x_i}{\partial T} \right)_P = -nR \sum x_i \ln x_i\]

This expression shows that $\Delta S_{\text{mix}}$ is positive as long as $x_i$ values are between 0 and 1, meaning mixing increases the entropy of the system.

The entropy of mixing drives the spontaneity of mixing for ideal gases. However, for liquids, mixing may not be spontaneous due to intermolecular interactions, especially if the components are immiscible.

Application to Binary Mixtures

In binary mixtures, such as a mixture of two gases, the entropy and Gibbs free energy of mixing can be visualized by plotting them against the mole fraction $x_A$.

For instance, the entropy of mixing reaches a maximum when $x_A = x_B = 0.5$, reflecting the greatest disorder. This is represented by the formula:

\[\Delta S_{\text{mix}} = -nR \left( x_A \ln x_A + (1 - x_A) \ln (1 - x_A) \right)\]

Similarly, the Gibbs free energy of mixing for ideal gases follows:

\[\Delta G_{\text{mix}} = nRT \left( x_A \ln x_A + (1 - x_A) \ln (1 - x_A) \right)\]

This equation helps predict whether the mixing will occur spontaneously, depending on the sign of $\Delta G_{\text{mix}}$. For ideal gases, $\Delta G_{\text{mix}} \leq 0$, making mixing spontaneous.

 

Gibbs Free Energy vs. Entropy of Mixing

The interplay between Gibbs free energy and entropy of mixing can be illustrated by comparing these quantities in a binary mixture. A graph of $\Delta G_{\text{mix}}$ and $\Delta S_{\text{mix}}$ as functions of $x_A$ often shows that the entropy term dominates, especially at higher temperatures.

Since $\Delta H_{\text{mix}} = 0$ for ideal gases, $\Delta G_{\text{mix}}$ depends solely on $T \Delta S_{\text{mix}}$, which is why temperature significantly affects the Gibbs free energy of mixing. This explains why mixing is more favorable at higher temperatures.