To determine which of the given reactions results in decreased entropy, we need to analyze the change in the number of gas molecules (or states of matter) from reactants to products. Entropy (a measure of disorder or randomness in a system) generally increases when a substance goes from a more ordered state (e.g., solid or fewer gas molecules) to a less ordered state (e.g., liquid, aqueous ions, or more gas molecules).
Let's analyze each option:
Option A: [tex]\( N_2(g) + 3 H_2(g) \rightarrow 2 NH_3(g) \)[/tex]
- Reactants: [tex]\( N_2(g) \)[/tex] (1 mole of gas) + [tex]\( H_2(g) \)[/tex] (3 moles of gas) → Total = 4 moles of gas
- Products: [tex]\( NH_3(g) \)[/tex] (2 moles of gas)
In this reaction, we go from 4 moles of gas (reactants) to 2 moles of gas (products). Since the number of gas molecules decreases, the entropy of the system decreases (entropy decreases when the disorder decreases).
Option B: [tex]\( MgCl_2(s) \rightarrow Mg^{2+}(aq) + 2 Cl^-(aq) \)[/tex]
- Reactants: [tex]\( MgCl_2(s) \)[/tex] (solid; more ordered state)
- Products: [tex]\( Mg^{2+}(aq) \)[/tex] + [tex]\( 2 Cl^-(aq) \)[/tex] (aqueous ions; less ordered state)
In this reaction, a solid breaks down into aqueous ions, which increases the disorder. Thus, the entropy increases.
Option C: [tex]\( 2 CH_4(g) + 3 O_2(g) \rightarrow 2 CO(g) + 4 H_2O(g) \)[/tex]
- Reactants: [tex]\( CH_4(g) \)[/tex] (2 moles of gas) + [tex]\( O_2(g) \)[/tex] (3 moles of gas) → Total = 5 moles of gas
- Products: [tex]\( CO(g) \)[/tex] (2 moles of gas) + [tex]\( H_2O(g) \)[/tex] (4 moles of gas) → Total = 6 moles of gas
In this reaction, we go from 5 moles of gas (reactants) to 6 moles of gas (products). Since the number of gas molecules increases, the entropy of the system increases.
Option D: [tex]\( PCl_5(g) \rightarrow PCl_3(g) + Cl_2(g) \)[/tex]
- Reactants: [tex]\( PCl_5(g) \)[/tex] (1 mole of gas)
- Products: [tex]\( PCl_3(g) \)[/tex] (1 mole of gas) + [tex]\( Cl_2(g) \)[/tex] (1 mole of gas) → Total = 2 moles of gas
In this reaction, we go from 1 mole of gas (reactants) to 2 moles of gas (products). Since the number of gas molecules increases, the entropy of the system increases.
Based on the analysis, the reaction that results in decreased entropy is:
Option A: [tex]\( N_2(g) + 3 H_2(g) \rightarrow 2 NH_3(g) \)[/tex]
As we transition from 4 moles of gas (reactants) to 2 moles of gas (products), there is a decrease in the number of gas molecules, which corresponds to decreased entropy.
