Solution:
$\text{Hint: In stretching of rubber band, entropy decreases because molecules are arranged in ordered form}$
$\text{To determine which reaction involves a decrease in entropy, we need to analyze how molecular order/disorder changes in each process:}$
$\text{1. Boiling of egg:}$
$\text{When an egg is boiled, the proteins in the egg undergo denaturation. This process actually increases entropy as the ordered helical structures of proteins unfold into more random coiled structures. During denaturation, the hydrogen bonds that maintain the ordered secondary structure of proteins break, leading to more disordered arrangements. Therefore, boiling an egg results in an increase in entropy, not a decrease.}$
$\text{2. Combustion of benzene at 127}^{\circ}\text{C:}$
$\text{The combustion reaction of benzene can be represented as:}$
$\text{C}_6\text{H}_6 (l) + \frac{15}{2}\text{O}_2 (g) \rightarrow 6\text{CO}_2 (g) + 3\text{H}_2\text{O} (g)$
$\text{In this reaction, one liquid molecule (benzene) and several gas molecules (oxygen) are converted into a larger number of gas molecules (carbon dioxide and water vapor). Since gases have higher entropy than liquids, and the total number of gas molecules increases in this reaction, the overall entropy increases. Combustion reactions generally lead to an increase in entropy.}$
$\text{3. Stretching of rubber band:}$
$\text{A rubber band consists of polymer chains that are randomly coiled in the relaxed state. When stretched, these polymer chains become more aligned and ordered in the direction of stretching. This alignment reduces the number of possible conformations (}\Omega\text{) that the polymer chains can adopt, leading to a decrease in entropy according to the Boltzmann equation:}$
$\text{S} = k_B \ln \Omega$
$\text{Where}$ $k_B$ $\text{is the Boltzmann constant and}$ $\Omega$ $\text{is the number of possible microstates.}$
$\text{When the rubber band is released, it spontaneously returns to its original shape because this increases entropy (the polymer chains return to more random arrangements). The fact that the relaxation is spontaneous indicates that}$ $\Delta\text{G} < 0$ $\text{for this process, which requires that}$ $\Delta\text{S} > 0$ $\text{(entropy increases during relaxation).}$
$\text{Therefore, stretching a rubber band decreases entropy.}$
$\text{4. Dissolution of sugar in water:}$
$\text{When sugar dissolves in water, the ordered crystalline structure of solid sugar breaks down, and the sugar molecules become dispersed throughout the water. This increases the randomness or disorder of the system. Although there is some ordering of water molecules around the sugar molecules (hydration shells), the overall entropy change is positive due to the increased translational freedom of the sugar molecules. Therefore, dissolution of sugar in water results in an increase in entropy, not a decrease.}$
$\text{After analyzing all options, we can conclude that only the stretching of a rubber band involves a decrease in entropy due to the increased molecular alignment and order in the stretched state.}$
$\text{The correct answer is option 3: Stretching of rubber band.}$