For a process to be spontaneous at constant \(T\) and \(P\) : (a) \((\Delta G)_{\text {system }}\) must be negative (b) \((\Delta G)_{\text {system }}\) must be positive (c) \((\Delta S)_{\text {system }}\) must be positive (d) \((\Delta S)_{\text {system }}\) must be negative

Gibbs free energy, denoted as \(\Delta G\), is the key to understanding whether a chemical process will occur spontaneously under constant pressure and temperature. In simple terms, it's a measure of the usable energy that can do work in a system. The equation \(\Delta G = \Delta H - T\Delta S\) fuses together the variables of enthalpy (\(\Delta H\)), temperature (\(T\)), and entropy (\(\Delta S\)) to make this prediction.

When \(\Delta G\) is negative, the process is spontaneous, indicating it favors the formation of products without external intervention. A positive \(\Delta G\) means the process is non-spontaneous and requires additional energy to proceed. In essence, a negative \(\Delta G\) reflects a process that naturally moves towards equilibrium, releasing free energy and creating stability within the system.

Thermodynamics is the branch of physical science that deals with the relationships between heat and other forms of energy. In essence, it's all about energy conversions and the laws that govern these transformations.

In the context of chemical reactions, thermodynamics helps us to determine the feasibility of a process. It considers various state functions such as enthalpy (\(\Delta H\)), entropy (\(\Delta S\)), and Gibbs free energy (\(\Delta G\)). The first law of thermodynamics asserts that energy cannot be created or destroyed, only transformed, and it's this concept that underpins the conservation of energy in chemical reactions. The second law introduces the concept of entropy, signaling that the natural direction of energy is towards dispersion or distribution in a system.

Entropy, represented by \(\Delta S\), is a fundamental concept in thermodynamics that describes the level of disorder or randomness within a system. In broader terms, it's a measure of how much the energy of the system is spread out or how many different ways the components of a system can be arranged.

A process that leads to an increase in \(\Delta S\) is favorable as it suggests a natural trend towards disordered states, which is in accordance with the second law of thermodynamics. However, it's important to understand that entropy is not the sole factor in determining spontaneity; rather, it is one component of the Gibbs free energy formula which balances both energy content (\(\Delta H\)) and disorder (\(\Delta S\)) to predict whether a process will occur spontaneously.

Enthalpy, denoted as \(\Delta H\), is a thermodynamic quantity that reflects the total heat content of a system, often related to the bond energies within a substance. It is used to calculate the heat change in a reaction taking place at constant pressure.

Changes in enthalpy can tell us a lot about a reaction. For example, a negative \(\Delta H\) indicates an exothermic process where heat is released to the surroundings, often making it more likely for the process to be spontaneous. Conversely, a positive \(\Delta H\) characterizes an endothermic reaction that absorbs heat, which could imply a barrier to spontaneity unless balanced by a sufficiently large increase in entropy (\(\Delta S\)), as reflected in the \(\Delta G\) equation. Understanding the interplay of enthalpy with entropy and temperature provides crucial insights into the energetic aspects of a reaction.