For the gaseous reaction; \(2 \mathrm{NO}_{2} \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{4}\), calculate \(\Delta \mathrm{G}^{\circ}\) and \(K_{\mathrm{p}}\) for the reaction at \(25^{\circ} \mathrm{C}\). Given \(\mathrm{G}_{j}^{\circ} \mathrm{N}_{2} \mathrm{O}_{4}\) and \(\mathrm{G}_{f}^{\circ} \mathrm{NO}_{2}\) are \(97.82\) and \(51.30\) \(\mathrm{kJ}\) respectively. Also calculate \(\Delta \mathrm{G}^{\circ}\) and \(K_{\mathrm{p}}\) for reverse reaction.

In the study of chemical reactions, thermodynamics plays a crucial role that connects the energy changes involved with the feasibility of reactions. Chemical thermodynamics is the area that looks at the relationships between chemical reactions and energy changes involving heat and work. A key concept in this field is the thermodynamic potential known as Gibbs free energy, symbolized by \( G \).

Gibbs free energy is a vital concept because it incorporates both the enthalpy (heat content) and entropy (a measure of disorder) of a system, providing valuable information about the spontaneity of a reaction under constant pressure and temperature. It's represented by the equation \( G = H - TS \), where \( H \) is the enthalpy, \( T \) the absolute temperature, and \( S \) the entropy.

A reaction's \( \Delta G^\circ \) value, which is the change in Gibbs free energy under standard conditions, can be determined from the Gibbs free energies of formation of the reactants and products. If \( \Delta G^\circ \) is negative, it indicates that the reaction can occur spontaneously under standard conditions. The calculations in our exercise exemplify how these values are used to determine the feasibility of a reaction and its balance between products and reactants.

Equilibrium in a chemical system is the state where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time. The position of equilibrium for a gaseous reaction can be numerically defined by the equilibrium constant \(K_p\), which exclusively considers the partial pressures of the gaseous components.

The equilibrium constant is derived from the law of mass action, which asserts that at a constant temperature, a chemical reaction's equilibrium state will have a constant ratio of product concentrations to reactant concentrations, each raised to the power of their stoichiometric coefficients. For the reaction in our exercise, \(K_p\) is particularly useful because all involved substances are gases.

Mathematically, calculating \(K_p\) involves using the standard change in Gibbs free energy, \(\Delta G^\circ\), through the equation \(\Delta G^\circ = -RT \ln K_p\), where \(R\) is the universal gas constant and \(T\) is the absolute temperature. By rearranging this equation and plugging in the correct values, students can find the numerical value of \(K_p\) for both the forward and the reverse reactions, which is a critical step in understanding the chemical dynamics at play.

Spontaneity is a term used in thermodynamics to describe whether a process can occur by itself without needing to be driven by an external force. Reaction spontaneity is not about the speed of the reaction but rather its inherent tendency to occur under given conditions.

The criterion for spontaneity in a chemical reaction is the sign of the Gibbs free energy change, \(\Delta G\). If \(\Delta G\) is negative, the reaction is said to be spontaneous, meaning it has the potential to proceed without external intervention. In contrast, a positive \(\Delta G\) indicates that the reaction is non-spontaneous and requires the input of energy to proceed.

It's also important to remember that spontaneity is temperature-dependent. The relation between Gibbs free energy and the equilibrium constant as shown in our exercise demonstrates how changing conditions such as temperature can shift the balance between reactants and products, thereby affecting the spontaneity of the reaction. When discussing reaction spontaneity, including both forward and reverse reactions provides a complete picture of the system's thermodynamics and helps students understand how equilibria can shift in response to external conditions.