The entropies of \(\mathrm{H}_{2}(\mathrm{~g})\) and \(\mathrm{H}(\mathrm{g})\) are \(130.6\) and \(114.6 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) respectively at \(298 \mathrm{~K}\). Using the data given below calculate the bond energy of \(\mathrm{H}_{2}\) (in \(\mathrm{kJ} / \mathrm{mol}\) ): \(\mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{H}(g) ; \Delta G^{\circ}=406.6 \mathrm{~kJ}\) (a) \(377.2\) (b) \(436.0\) (c) \(425.5\) (d) \(430.5\)

Gibbs free energy, denoted by the symbol \(\Delta G^\circ\), is a thermodynamic quantity that can indicate the spontaneity of a chemical reaction under constant temperature and pressure. It's a measure combining the system's enthalpy (\(\Delta H\)) and the entropy (\(\Delta S\)) changes. The Gibbs free energy equation is expressed as \(\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ\), where \(T\) is the temperature in Kelvin.

In the context of bonding, the \(\Delta G^\circ\) can provide information on the energy change associated with breaking or forming bonds. For example, a positive value for \(\Delta G^\circ\) suggests that energy is absorbed to break the bonds, like the dissociation of \(\mathrm{H}_2(g)\) into atomic hydrogen. This concept directly ties into calculating bond energies, which are crucial for understanding chemical reactivity and stability of molecules.

Entropy, symbolized as \(S\), is a fundamental concept in thermodynamics that represents the level of disorder or randomness within a system. An increase in entropy indicates a more disordered system, while a decrease suggests a more ordered one.

To calculate the entropy change (\(\Delta S^\circ\)) in a reaction, one must subtract the sum of the entropies of the reactants from the sum of the entropies of the products, which is formulated as \(\Delta S^\circ = \Sigma S^\circ(\text{products}) - \Sigma S^\circ(\text{reactants})\). Entropy is particularly important when determining the spontaneity of a reaction alongside enthalpy change, because these two variables compose the Gibb's free energy.

Enthalpy, represented by the symbol \(\Delta H^\circ\), measures the total heat content of a system at constant pressure. It's an indication of the energy that is either released or absorbed by a system during a reaction. In chemical reactions, the change in enthalpy covers energy changes due to bond breaking and formation.

Enthalpy is a crucial part of the Gibbs free energy equation, and when we are calculating bond energies, we are often looking for the enthalpy change associated with the breaking of bonds. The bond energy of a diatomic molecule like \(\mathrm{H}_2\) can be directly related to the enthalpy change needed to break the bond between the two hydrogen atoms.

Thermochemistry is the branch of chemistry that deals with the relationships between chemical reactions and energy changes involving heat. This field of study uses principles of thermodynamics, like enthalpy, entropy, and Gibbs free energy, to predict the energy changes during reactions.

Understanding these interactions allows scientists to estimate bond energies and determine whether reactions will occur spontaneously. Thermochemistry is heavily employed in the analysis of energy changes during bond dissociation, which is an essential concept in predicting the reactivity and stability of different substances.