To determine which equation describes the reduction half-reaction, it is important to understand that "reduction" in a redox reaction involves the gain of electrons by a species.
Given the overall redox reaction:
[tex]\[ Mg (s) + Fe^{2+} (aq) \rightarrow Mg^{2+} (aq) + Fe (s) \][/tex]
We can split this reaction into two half-reactions: oxidation and reduction.
1. Oxidation half-reaction: This will involve the loss of electrons.
2. Reduction half-reaction: This will involve the gain of electrons.
First, let's identify the oxidation half-reaction. Magnesium (Mg) is going from a solid state (0 oxidation state) to a [tex]\( Mg^{2+} \)[/tex] state:
[tex]\[ Mg (s) \rightarrow Mg^{2+} (aq) + 2e^{-} \][/tex]
This is the oxidation half-reaction because magnesium is losing two electrons (being oxidized).
Now, for the reduction half-reaction, we need to see which species is gaining electrons. Here, [tex]\( Fe^{2+} \)[/tex] is gaining two electrons to form solid iron (Fe):
[tex]\[ Fe^{2+} (aq) + 2e^{-} \longrightarrow Fe (s) \][/tex]
This is our reduction half-reaction because iron ions [tex]\( Fe^{2+} \)[/tex] are gaining electrons (being reduced).
Comparing this equation to the given options:
Option 1: [tex]\[ Fe^{2+}(aq) + 2e^{-} \longrightarrow Fe (s) \][/tex]
Thus, the correct equation for the reduction half-reaction in the given redox reaction is:
[tex]\[ Fe^{2+}(aq) + 2e^{-} \longrightarrow Fe (s) \][/tex]
This states that [tex]\( Fe^{2+} \)[/tex] gains two electrons to become Fe, which fits the definition of reduction. Therefore, the first option is the correct answer.
