Calculate the change of entropy, \(\Delta_{r} S^{\circ}\) at \(298 \mathrm{~K}\) for the reaction in which urea is formed from \(\mathrm{NH}_{3}\) and \(\mathrm{CO}_{2}\). \(2 \mathrm{NH}_{3}(g)+\mathrm{CO}_{2}(g) \longrightarrow \mathrm{NH}_{2} \mathrm{CONH}_{2}(a q)+\mathrm{H}_{2} \mathrm{O}(l) .\) The standard entropy of \(\mathrm{NH}_{2} \mathrm{CONH}_{2}(a q), \mathrm{CO}_{2}(g), \mathrm{NH}_{3}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(l)\) are \(174.0\), \(213.7,192.3\) and \(69.9 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) respectively.

Standard entropy values, denoted as \(S^{\textcircled{\tiny R}}\), are pivotal in assessing the spontaneity and equilibrium state of chemical reactions from a thermodynamic perspective. They represent the absolute entropy of a substance at a reference state, which is usually 1 bar of pressure and a specified temperature, often taken as 298 K (25°C).

When you encounter a problem that asks you to calculate the entropy change of a reaction, as in the case of the urea formation from \(NH_3\) and \(CO_2\), you must discern the standardized entropy values for each reactant and product involved. These values are intrinsic properties and can be found in thermodynamic tables or databases. In this problem, the standard entropy for \(NH_3(g)\), \(CO_2(g)\), \(NH_2CONH_2(aq)\), and \(H_2O(l)\) has been provided.

Understanding that entropy is a measure of disorder, where higher values correspond to greater disorder, allows students to appreciate the shift from reactant to product entropy. The calculation of total standard entropies for the reactants and products involves multiplying the individual entropies by their respective mole coefficients in the balanced chemical equation and then summing them up.

Thermodynamics plays an essential role in explaining the behavior of systems in chemical reactions, and it lies at the core of understanding Physical Chemistry. The change in entropy, or \(\Delta S\), is just one aspect of this broader field. It is one of the state functions used to determine the spontaneity of a process, along with enthalpy (\(H\)) and temperature (\(T\)).

In predictive thermodynamics, we operate under the assumption that processes tend to occur in a direction that increases the overall entropy of the universe, which is a statement derived from the Second Law of Thermodynamics. In the context of the textbook problem at hand, calculating the entropy change for the formation of urea is crucial to anticipate if the reaction occurs naturally under standard conditions.

Understanding the relationship between entropy, enthalpy, and the Gibbs free energy equation, \(G = H - T\Delta S\), can further the students' comprehension of reaction feasibility. For instance, negative \(\Delta G\) values suggest that a reaction is spontaneous, while positive values indicate non-spontaneous reactions unless external energy is added.

The Indian Institutes of Technology Joint Entrance Examination (IIT-JEE) is a highly competitive test, with Physical Chemistry representing a significant portion of the chemistry syllabus. The problems routinely combine theoretical concepts with practical calculations, aimed at assessing the students' ability to integrate their knowledge and apply it to solve complex questions.

In the context of our example problem, students are expected to not only grasp the calculation of entropy changes but to also understand the underlying theories of thermodynamics that govern such calculations. Mastering these concepts is essential for success in IIT-JEE Chemistry. Such problems demand careful attention to detail in every step—from balancing the chemical reaction to the precise calculation of thermodynamic quantities.

For thorough preparation, students should practice a multitude of problems involving entropy calculations, enthalpy changes, free energy, and equilibrium constants. Conceptual clarity combined with problem-solving skills can help students excel in tackling IIT-JEE Physical Chemistry problems, contributing significantly to their overall performance in the examination.