For the reaction \(2 \mathrm{H}(g) \longrightarrow \mathrm{H}_{2}(g)\), the sign of \(\Delta H\) and \(\Delta S\) respectively are : (a) \(+,-\) (b) \(+,+\) (c) \(-,+\) (d) \(-,-\)

Enthalpy change, denoted as \(\Delta H\), is a measure of the total energy change in a chemical system during a reaction at constant pressure. In the context of our exercise, the enthalpy change involves considering whether energy is absorbed or released when chemical bonds are formed or broken.

Typically, when a reaction forms stronger bonds than those broken, it releases energy to the surroundings, making it an exothermic process with a negative enthalpy change (\(\Delta H < 0\)). Conversely, if the process requires more energy to break bonds than is released upon bond formation, the reaction is endothermic with a positive enthalpy change (\(\Delta H > 0\)).

In the reaction \(2 \mathrm{H}(g) \longrightarrow \mathrm{H}_2(g)\), hydrogen atoms combine to form a hydrogen molecule. Strong covalent bonds are formed in the process, resulting in the release of energy, which is indicative of an exothermic reaction, corresponding to a negative sign of \(\Delta H\).

Entropy change, symbolized as \(\Delta S\), is an indication of the change in disorder or randomness in a system as it undergoes a chemical reaction. A key point in understanding entropy is that, generally, a greater number of particles or a higher volume of gases corresponds to higher entropy or disorder, while fewer particles indicate lower entropy.

In the given reaction \(2 \mathrm{H}(g) \longrightarrow \mathrm{H}_2(g)\), two moles of hydrogen atoms (\(2\times \mathrm{H}\)) consolidate to form a single mole of a diatomic hydrogen molecule (\(\mathrm{H}_2\)). This transformation results in a system that has fewer individual particles and therefore less disorder, indicating that entropy decreases. Hence, the sign of \(\Delta S\) in this reaction is negative. This aligns with the general understanding that a decrease in the number of gas particles results in a decrease in system disorder.

Exothermic reactions, such as the one presented in the exercise \(2 \mathrm{H}(g) \longrightarrow \mathrm{H}_2(g)\), are characterized by the release of energy, usually in the form of heat, to the surroundings. This means that the reactants have more stored chemical energy than the products, and the difference in this energy is given off in the reaction.

Energy Transfer in Exothermic Reactions

During these reactions, the energy required to break existing bonds is less than the energy released when new bonds are formed. This transfer of energy not only influences the temperature of the surroundings but also has important implications in the field of thermochemistry, as exothermic reactions are generally favored for their ability to generate heat.

It is crucial for students, especially those preparing for competitive exams like the Physical Chemistry JEE, to identify exothermic reactions by the negative sign of \(\Delta H\) and understand their role in chemical and real-world applications, like combustion and metabolism.

Physical Chemistry for the Joint Entrance Examination (JEE) often involves applying theoretical concepts to practical problems. Understanding the principles behind enthalpy and entropy changes, as well as exothermic and endothermic reactions, is essential for success in JEE and other competitive exams.

In preparing for such high-stakes tests, students must develop a thorough grasp of how to determine the signs for \(\Delta H\) and \(\Delta S\), as seen in the given exercise. Mastering these principles can greatly affect problem-solving strategies, enable students to make predictions about reaction favorability, and understand energy dynamics in chemical processes.

Examining problems that involve both enthalpy and entropy changes can help students deeply comprehend energy and disorder's role in the spontaneity of reactions and the world of thermodynamics in physical chemistry.