To determine which reaction occurs most rapidly at standard conditions, we must consider the nature of the reactions, reactants, and the states of matter involved.
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A. \( \text{H}_2(g) + \text{I}_2(s) \rightarrow 2\text{HI}(g) \)
- This is a gas-phase reaction, and the reactants are quite reactive. The formation of hydrogen iodide (\(HI\)) is fairly rapid under standard conditions but depends on the activation energy.
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B. \( \text{Cu}(s) + \text{S}(s) \rightarrow \text{CuS}(s) \)
- This is a solid-state reaction, and generally, solid-solid reactions can be quite slow because they are often limited by diffusion. This reaction tends to occur at higher temperatures.
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C. \( \text{C}6\text{H}{12}\text{O}_6(s) + 6\text{O}_2(g) \rightarrow 6\text{CO}_2(g) + 6\text{H}_2\text{O}(g) \)
- This reaction involves the combustion of glucose, which is exothermic and usually occurs quickly in biological systems (for example, in cellular respiration) but may not be rapid under all conditions, particularly in the absence of a catalyst (e.g., enzymes).
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D. \( 5\text{C}_2\text{O}_4^{2-}(aq) + 2\text{MnO}_4^{-} + 16\text{H}^{+}(aq) \rightarrow 10\text{CO}_2(g) + 2\text{Mn}^{2+}(aq) + 2\text{H}_2\text{O}(l) \)
- This is a redox reaction that is often rapid in acidic solutions due to the involvement of permanganate ions, which are strong oxidizing agents.
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E. \( \text{Ag}^+(aq) + \text{I}^-(aq) \rightarrow \text{AgI}(s) \)
- This is a precipitation reaction that occurs quickly due to the low solubility of silver iodide (\(AgI\)).
Considering the characteristics of these reactions, Option E (the precipitation of silver iodide) is typically very rapid because the formation of a solid from ions in solution occurs almost immediately due to the driving force of the formation of an insoluble compound.
Answer: E. \( \text{Ag}^+(aq) + \text{I}^-(aq) \rightarrow \text{AgI}(s) \)
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