Do you expect \(\Delta S\) for the following reaction to be positive or negative? Explain your reasoning. $$ \mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g}) \rightleftarrows 2 \mathrm{NO}_{2}(\mathrm{~g}) $$

Gibbs free energy, symbolized as \( G \), is a central concept in chemical thermodynamics, bridging the gap between the energetics and spontaneity of reactions. It represents the maximum amount of work that a system can perform during a process at constant temperature and pressure, and is defined by the equation \( G = H - TS \), where \( H \) is the enthalpy, \( T \) is the temperature in Kelvin, and \( S \) is the entropy.

For a chemical reaction to be spontaneous, the change in Gibbs free energy, denoted as \( \Delta G \), must be negative. When \( \Delta G < 0 \), the reaction occurs without needing external energy. Conversely, if \( \Delta G > 0 \), the reaction is non-spontaneous and requires additional energy to proceed. For the equilibrium state, \( \Delta G \) is exactly zero, indicating that there is no net change occurring in the system.

Chemical thermodynamics is a subset of thermodynamics focused on chemical processes and the energy and entropy changes associated with them. It provides a quantitative understanding of energy transfers and the directions in which chemical reactions proceed.

In the context of the given exercise, the principles of thermodynamics assist in predicting whether the dissassociation of \( N_2O_4(g) \) into \( NO_2(g) \) is favorable or not. The reaction will spontaneously move towards the side that lowers the Gibbs free energy of the system, ultimately aiming for a point of equilibrium.

Reaction spontaneity refers to whether a chemical reaction will occur naturally without any external intervention. Several factors govern this spontaneity, primarily the change in Gibbs free energy (\( \Delta G \) ), as well as enthalpy (\( \Delta H \) ) and entropy (\( \Delta S \) ).

To improve understanding, it's necessary to note that spontaneity is not directly related to the speed of a reaction—some spontaneous reactions may be quite slow. For the exercise provided, an increase in entropy suggests the reaction is more likely to be spontaneous, as it aligns with the second law of thermodynamics, which states that the entropy of the universe tends to increase over time.

Entropy, symbolized as \( S \), is a measure of the disorder or randomness within a system. An increase in entropy typically corresponds to an increase in molecular disorder and is a natural trend in the universe, as per the second law of thermodynamics.

Applying this to the provided exercise increases in the number of gaseous molecules, from one \( N_2O_4(g) \) to two \( NO_2(g) \) molecules, results in higher randomness and thus, greater entropy. In thermodynamic terms, \( \Delta S > 0 \) for the reaction, which is favorable for spontaneous processes. Understanding how molecules moving from a more ordered state (fewer molecules) to a less ordered state (more molecules) impacts entropy helps grasp why some reactions proceed forward spontaneously.