\(.\) The standard entropies for the following substances are 26.9 \(\mathrm{J} / \mathrm{K} \bullet\) mol for \(\mathrm{MgO}(s)\) 213.8 \(\mathrm{J} / \mathrm{K} \bullet \mathrm{mol}\) for \(\mathrm{CO}_{2}(g),\) and 65.7 \(\mathrm{J} / \mathrm{K} \bullet \mathrm{mol}\) for \(\mathrm{MgCO}_{3}(s) .\) Determine the entropy for the reaction below. $$ \mathrm{MgCO}_{3}(s) \rightarrow \mathrm{MgO}(s)+\mathrm{CO}_{2}(g) $$

When it comes to thermal dynamics, every substance has an intrinsic energy-related property known as standard entropy (denoted as S°), which refers to the entropy content of a substance at a standard state, typically 1 bar of pressure. It signifies the level of disorder or randomness within the substance at the atomic or molecular level. The measurement unit for entropy is joules per kelvin per mole \(\mathrm{J/K\cdot mol}\).

For instance, the standard entropies for substances like \(\mathrm{MgO}(s)\), \(\mathrm{CO}_{2}(g)\), and \(\mathrm{MgCO}_{3}(s)\) are provided in the exercise, signaling how much disorder these substances have at a baseline level. Understanding these values is crucial for predicting the behavior of a substance in a reaction, as it influences the reaction's spontaneity and direction.

The Second Law of Thermodynamics plays a fundamental role in our understanding of energy conversion and entropy. It postulates that the total entropy of an isolated system can never decrease over time. For any spontaneous process, the entropy of the universe (system plus surroundings) always increases.

This law is especially relevant in chemical reactions, as it helps predict whether a reaction is favorable. When entropy increases during a chemical reaction (\(\Delta S > 0\)), it means the disorder within the system has increased, and the reaction is spontaneous under constant temperature and pressure. On a practical level, this indicates that it's more likely for \(\mathrm{MgCO}_{3}(s)\) to decompose into \(\mathrm{MgO}(s)\) and \(\mathrm{CO}_{2}(g)\) than the reverse process to occur, based solely on entropy considerations.

The exercise revolves around the entropy change calculation, which is critical in determining the spontaneity of a reaction. The change in entropy \(\Delta S\) during a reaction is the quantitative measurement of the difference in disorder between the products and the reactants.

To calculate \(\Delta S\) for a particular reaction, one must look at the standard entropies of reactants and products as outlined in the exercise. The formula given by the Second Law of Thermodynamics makes it straightforward: \(\Delta S = \sum S_{\text{products}} - \sum S_{\text{reactants}}\), where \(\sum S_{\text{products}}\) and \(\sum S_{\text{reactants}}\) are the sums of the standard entropies of products and reactants, respectively. By inputting the known values for standard entropy, as demonstrated in the step-by-step solution, you obtain the overall entropy change for the reaction.

Thermochemistry is the branch of chemistry that deals with the energy changes involving heat, particularly the study of thermal energy transfer during chemical reactions. It includes understanding concepts like enthalpy, entropy, and free energy. These principles allow us to predict whether a reaction will release or absorb heat (\(\Delta H\)), decrease or increase disorder (\(\Delta S\)), or be spontaneous (\(\Delta G\)).

Applying thermochemistry to the provided exercise helps us comprehend the energy flow of the decomposition of \(\mathrm{MgCO}_{3}(s)\). Analysis of entropy changes is a segment of thermochemistry that provides insights into the disorder level of the system, indicating if the energy spread out (increased entropy) or became more concentrated (decreased entropy). These indications are vital for chemists to assess reactions' feasibility and optimize them for practical applications, like making the process more energy-efficient.