Just as standard enthalpies of formation (\(\Delta H_f^\circ\)) allow reaction enthalpies to be calculated, standard Gibbs functions of formation (\(\Delta G_f^\circ\)) allow standard reaction Gibbs functions to be determined.
The standard Gibbs function of formation, \(\Delta G_f^\circ\), is defined as the Gibbs free-energy change associated with forming one mole of a compound from its constituent elements in their standard states.
For example, the standard Gibbs function of formation of methane is defined by the reaction
\[ C(\mathrm{gr}) + 2H_2(g) \rightarrow CH_4(g) \]
By convention,
\[ \Delta G_f^\circ = 0 \]
for elements in their standard states. Thus,
\[ \Delta G_f^\circ \left( H_2(g) \right) = 0 \]
and
\[ \Delta G_f^\circ \left( C(\mathrm{gr}) \right) = 0 \]
while compounds generally have nonzero values of \(\Delta G_f^\circ\).
Standard Gibbs functions of formation can be combined in exactly the same manner as standard enthalpies of formation.
For a general reaction
\[ aA+bB \rightarrow cC+dD \]
the standard reaction Gibbs function is
\[ \boxed{ \Delta G_{\mathrm{rxn}}^\circ = \sum \nu_i \Delta G_{f,i}^\circ (\mathrm{products}) - \sum \nu_i \Delta G_{f,i}^\circ (\mathrm{reactants}) } \]
where the \(\nu_i\) are the stoichiometric coefficients appearing in the balanced chemical equation.
As with reaction enthalpies, products contribute positively and reactants contribute negatively.
Once \(\Delta G_{\mathrm{rxn}}^\circ\) has been calculated, its sign can be interpreted in the same way as any other Gibbs free-energy change:
The superscript "\(^{\circ}\)" is easy to overlook, but it carries a great deal of meaning.
The quantity
\[ \Delta G_{\mathrm{rxn}}^\circ \]
is defined only for a very specific set of conditions:
In other words, every reactant and every product must exist in its standard state.
Such conditions are actually quite rare. During most chemical reactions, reactant and product concentrations are continually changing, meaning that the system almost never remains in a standard state.
Consequently,
\[ \Delta G_{\mathrm{rxn}}^\circ \]
is often a poor indicator of the direction in which a reaction will actually proceed under laboratory conditions.
Fortunately, there is a more useful quantity:
\[ \Delta G_{\mathrm{rxn}} \]
which describes the Gibbs free-energy change under the actual conditions experienced by the system.
Looking ahead: In a later module, we will develop methods for calculating \(\Delta G_{\mathrm{rxn}}\) for systems that are not in their standard states. Unlike \(\Delta G_{\mathrm{rxn}}^\circ\), \(\Delta G_{\mathrm{rxn}}\) provides a reliable indicator of the direction of spontaneous change.
Calculate \(\Delta G_{\mathrm{rxn}}^\circ\) at \(25^\circ\mathrm{C}\) for the reaction
\[ CH_4(g)+2O_2(g)\rightarrow CO_2(g)+2H_2O(l) \]
using the following standard Gibbs functions of formation:
| Substance | \(\Delta G_f^\circ\) \(\mathrm{(kJ\,mol^{-1})}\) |
|---|---|
| \(CH_4(g)\) | \(-50.8\) |
| \(O_2(g)\) | \(0\) |
| \(CO_2(g)\) | \(-394.4\) |
| \(H_2O(l)\) | \(-237.1\) |
The standard reaction Gibbs function is calculated from
\[ \Delta G_{\mathrm{rxn}}^\circ = \sum \nu_i\Delta G_{f,i}^\circ(\mathrm{products}) - \sum \nu_i\Delta G_{f,i}^\circ(\mathrm{reactants}) \]
Substitute the product terms:
\[ \sum \nu_i\Delta G_{f,i}^\circ(\mathrm{products}) = (1)(-394.4) + (2)(-237.1) \]
\[ \sum \nu_i\Delta G_{f,i}^\circ(\mathrm{products}) = -868.6\ \mathrm{kJ} \]
Substitute the reactant terms:
\[ \sum \nu_i\Delta G_{f,i}^\circ(\mathrm{reactants}) = (1)(-50.8) + (2)(0) \]
\[ \sum \nu_i\Delta G_{f,i}^\circ(\mathrm{reactants}) = -50.8\ \mathrm{kJ} \]
Therefore,
\[ \Delta G_{\mathrm{rxn}}^\circ = -868.6 - (-50.8) \]
\[ \Delta G_{\mathrm{rxn}}^\circ = -817.8\ \mathrm{kJ} \]
\[ \boxed{ \Delta G_{\mathrm{rxn}}^\circ = -818\ \mathrm{kJ} } \]
Interpretation: The large negative value of \(\Delta G_{\mathrm{rxn}}^\circ\) indicates that methane combustion is strongly product-favored under standard-state conditions.
A reaction will be chosen at random. Use the tabulated \(\Delta G_f^\circ\) values to calculate \(\Delta G_{\mathrm{rxn}}^\circ\).
Big picture: Gibbs free energy provides a practical criterion for spontaneity that depends only on properties of the system. Standard Gibbs functions of formation allow standard reaction Gibbs functions to be calculated, but standard-state conditions are rarely encountered in practice. The next step is to determine how Gibbs free energy changes when reactants and products are not in their standard states, allowing the true direction of spontaneous change to be predicted.