Equilibrium Position in a Gas Phase Chemical Reaction

Equilibrium Position and Gibbs Free Energy

In a chemical reaction, we are often interested in determining the specific concentrations (or partial pressures) of reactants and products at equilibrium. At equilibrium, the system’s Gibbs free energy is minimized, meaning it’s at its most stable state under given conditions of temperature and pressure. This state is where the reaction "wants" to be, as any shift away from it would result in a higher (less favorable) Gibbs free energy, and the reaction would naturally return to equilibrium.

Mathematically, at equilibrium:

\[ dG = \sum \mu_i \, d n_i \]

where: $ \mu_i $ is the chemical potential of component $i$. The chemical potential is essentially the "driving force" for each component in the reaction, representing how much Gibbs free energy would change if a small amount of component $i$ were added to the system (i.e. for a small change in the mol number $d n_i$).

 

Using the Extent of Reaction

To describe the progress of a reaction quantitatively, we use the extent of reaction, represented as$ \xi $. This parameter tells us how far a reaction has gone in a particular direction. We link the mole change in any component $i$ to this extent by using the stoichiometric coefficient $ \nu_i $, so:

\[ d n_i = \nu_i d \xi \]

Substituting this into the expression for $ dG $, we get:

\[ dG = \sum \mu_i \nu_i d \xi \]

This leads us to the expression:

\[\left( \frac{\partial \Delta_r G}{\partial \xi} \right)_{p,T} = \sum_i \nu_i \mu_i = \Delta \mu\]

The formula represents the relationship between the change in Gibbs free energy for a reaction and the reaction's equilibrium position. At equilibrium, the differential change in Gibbs free energy with respect to the extent of reaction must be zero:

\[ \left( \frac{dG}{d\xi} \right)_{P,T} = 0 \]

This equation tells us that the sum of the chemical potentials, weighted by their stoichiometric coefficients, is zero at equilibrium.

 

Extent of Reaction and Minimization of Gibbs Free Energy

The equilibrium position corresponds to the point where the Gibbs free energy of the system is at its minimum for the given conditions. This minimum is achieved when the chemical potentials of the reactants and products balance each other out, so no net reaction occurs in either direction. Imagine a valley: the reaction progresses downhill until it reaches this valley’s lowest point, which represents equilibrium.

Consider a general reaction:

\[ aA + bB \leftrightarrow cC + dD \]

The following equations describe the Gibbs free energy for unmixed and mixed states, specifically for a binary mixture of components A and B:

\[G_{\text{unmixed}} = x_B \mu_B^\circ(T) + (1 - x_B) \mu_A^\circ(T)\]

This represents the Gibbs free energy when the components are in their pure, unmixed states. Here $x_B$ is the mole fraction of component B, and $x_A = (1 - x_B)$ is the mole fraction of component A, $\mu_B^\circ(T)$ and $\mu_A^\circ(T)$ are the standard chemical potentials of components B and A at temperature T.

When the components are mixed, the Gibbs free energy becomes:

\[G_{\text{mixture}} = x_B \mu_B^\circ(T) + x_B RT \ln x_B + x_A \mu_A^\circ(T) + x_A RT \ln x_A\]

Expanding this, we get:

\[G_{\text{mixture}} = x_B \mu_B^\circ(T) + x_B RT \ln x_B + (1 - x_B) \mu_A^\circ(T) + (1 - x_B) RT \ln (1 - x_B)\]

The additional terms

\[x_B RT \ln x_B\]
and
\[(1 - x_B) RT \ln (1 - x_B)\]
represent the Free energy of mixing for the components, which is negative for a spontaneous process.

Thus we find that:

\[ G = x_B \mu_B + (1 - x_B) \mu_A \]

where # x_B$ is the mole fraction of $B$. At the minimum point, we find:

\[ \frac{dG}{d\xi} = 0 \]

Graphically, this is represented as the lowest point on a Gibbs free energy vs. extent of reaction plot, showing that equilibrium corresponds to the minimum Gibbs free energy.