To avoid confusion, let's clarify the concept of half-reactions. In a redox (reduction-oxidation) reaction, the overall reaction can be split into two half-reactions:
1. The oxidation half-reaction (where loss of electrons occurs).
2. The reduction half-reaction (where gain of electrons occurs).
First, we need to split the given overall reaction [tex]\( C + O_2 \rightarrow CO_2 \)[/tex] into its respective oxidation and reduction half-reactions.
### Oxidation Half-Reaction:
The carbon (C) is being oxidized because it is losing electrons in the process of forming carbon dioxide ([tex]\(CO_2\)[/tex]):
[tex]\[ C \rightarrow CO_2 + 4e^- \][/tex]
### Reduction Half-Reaction:
The oxygen ([tex]\(O_2\)[/tex]) is being reduced because it is gaining electrons to form carbon dioxide ([tex]\(CO_2\)[/tex]):
[tex]\[ O_2 + 4e^- \rightarrow 2O^{2-} \][/tex]
However, since we are forming [tex]\(CO_2\)[/tex], we should represent the reduction half-reaction directly for this compound:
[tex]\[ O_2 + 4e^- \rightarrow 2CO_2 \][/tex]
But actually, the reaction to carbon monoxide could be simplified directly to the CO2 in our initial reaction. Let’s rephrase our half-reactions from the given options.
### Evaluating given options:
#### Option 1:
[tex]\[
\begin{array}{l}
C \rightarrow CO_2 + 4e^- \\
O_2 + 4e^- \rightarrow CO_2
\end{array}
\][/tex]
This pair correctly represents the oxidation half-reaction as [tex]\( C \rightarrow CO_2 + 4e^- \)[/tex], where carbon loses 4 electrons. The reduction half-reaction is correctly represented as [tex]\( O_2 + 4e^- \rightarrow CO_2 \)[/tex], where oxygen gains 4 electrons to form [tex]\( CO_2 \)[/tex].
#### Option 2:
[tex]\[
\begin{array}{l}
C + 4e^- \rightarrow CO_2 \\
O_2 \rightarrow CO_2 + 4e^-
\end{array}
\][/tex]
This cannot be correct because in the first equation, it incorrectly states that carbon is gaining electrons (which is not true for oxidation). In the second equation, it incorrectly states that oxygen is losing electrons (which is not true for reduction).
#### Option 3:
[tex]\[
\begin{array}{l}
C \rightarrow CO_2 + 2e^- \\
O_2 + 2e^- \rightarrow CO_2 \\
C + 2e^- \rightarrow CO_2 \\
O_2 \rightarrow CO_2 + 2e^-
\end{array}
\][/tex]
This option contains four incorrect half-reactions since the electron transfer must add up to the correct overall charge balance, and for [tex]\( CO_2 \)[/tex], the number of transferred electrons should be 4, not 2.
So, the correct pair of half-reactions, representing the oxidation and reduction processes for the reaction [tex]\( C + O_2 \rightarrow CO_2 \)[/tex], is:
[tex]\[
\begin{array}{l}
C \rightarrow CO_2 + 4e^- \\
O_2 + 4e^- \rightarrow CO_2
\end{array}
\][/tex]
