In the Mond process for the purification of nickel, CO is passed over metallic nickel to give \(\mathrm{Ni}(\mathrm{CO})_{4}\) : $$\mathrm{Ni}(s)+4 \mathrm{CO}(g) \Longrightarrow \mathrm{Ni}(\mathrm{CO})_{4}(g)$$Given that the standard free energies of formation of \(\mathrm{CO}(g)\) and \(\mathrm{Ni}(\mathrm{CO})_{4}(g)\) are \(-137.3 \mathrm{~kJ} / \mathrm{mol}\) and \(-587.4 \mathrm{~kJ} / \mathrm{mol},\) respectively, calculate the equilibrium constant of the reaction at \(80^{\circ} \mathrm{C}\). (Assume \(\Delta G_{\mathrm{f}}^{\circ}\) to be independent of temperature.

Understanding the standard free energy change is crucial for predicting the direction of a chemical reaction under standard condition. It's symbolized by \( \Delta G^\circ \) and is a measure of the amount of energy available to do work when a chemical reaction occurs at constant temperature and pressure. When \( \Delta G^\circ \) is negative, the process is spontaneous, meaning it can occur without external input. The free energy of formation for pure elements in their most stable form is considered to be zero.

In the context of the Mond process, this concept is applied to determine the spontaneity of converting nickel into its tetracarbonyl complex. By using the formula \( \Delta G^\circ = \Sigma \Delta G_f^\circ[products] - \Sigma \Delta G_f^\circ[reactants] \), we can calculate the standard free energy change for the reaction. This step is essential as it sets the foundation for later determining the equilibrium constant, which further dictates how the reaction proceeds under given conditions.

The equilibrium constant, represented by \( K \), plays a vital role in understanding chemical reactions at equilibrium. It is a dimensionless quantity that provides the ratio of concentrations of products to reactants, raised to the power of their stoichiometric coefficients in a balanced chemical equation. The equilibrium constant is temperature-dependent and can be calculated using the Gibbs free energy equation in connection with the standard free energy change.

For the Mond process example, the equilibrium constant at \(80^{\circ}C\) or \(353.15 K\) is calculated using the relationship \( K = e^{(-\Delta G^\circ / RT )} \) where \( R \) is the universal gas constant and \( T \) is the absolute temperature in Kelvin. A large equilibrium constant indicates the formation of \( Ni(CO)_4 \) is highly favored under these conditions, highlighting the efficiency of the Mond process for nickel purification.

The Gibbs free energy equation is fundamental in thermodynamics and chemical kinetics. It relates the spontaneity of a reaction to its energetic favorability through the equation \( \Delta G = \Delta G^\circ + RT \ln K \), where \( \Delta G \) is the Gibbs free energy change under non-standard conditions, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( K \) is the equilibrium constant.

In the state of equilibrium, \( \Delta G \) becomes zero because the reaction has achieved a state where no further net change occurs. When solving for \( K \) in the equation provided, it becomes apparent that the sign and magnitude of \( \Delta G^\circ \) are instrumental in determining the favorability of the reaction and its equilibrium position. In the example of the Mond process, the negative value of \( \Delta G^\circ \) affirms that the formation of the nickel tetracarbonyl complex is spontaneous and that the reaction is likely to proceed until reaching equilibrium.