To determine the final balanced equation for the redox reaction, we need to follow these steps:
### Given Half-Reactions:
1. Oxidation half-reaction:
[tex]\[
2 \text{Cl}^- \rightarrow \text{Cl}_2 + 2 e^-
\][/tex]
2. Reduction half-reaction:
[tex]\[
\text{Cr}^{3+} + 3 e^- \rightarrow \text{Cr}
\][/tex]
### Step-by-Step Solution:
#### Step 1: Balance the electrons in both half-reactions
- The first half-reaction (oxidation) involves [tex]\(2 e^-\)[/tex].
- The second half-reaction (reduction) involves [tex]\(3 e^-\)[/tex].
To combine these half-reactions, the number of electrons lost in the oxidation half-reaction must equal the number of electrons gained in the reduction half-reaction. Find the least common multiple (LCM) of 2 and 3 to achieve this:
- The LCM of 2 and 3 is 6.
Therefore, we need to balance the half-reactions by multiplying them by appropriate factors to get 6 electrons in each half-reaction.
#### Step 2: Multiply each half-reaction by appropriate factors
- Multiply the oxidation half-reaction by 3:
[tex]\[
3 \times (2 \text{Cl}^- \rightarrow \text{Cl}_2 + 2 e^-) = 6 \text{Cl}^- \rightarrow 3 \text{Cl}_2 + 6 e^-
\][/tex]
- Multiply the reduction half-reaction by 2:
[tex]\[
2 \times (\text{Cr}^{3+} + 3 e^- \rightarrow \text{Cr}) = 2 \text{Cr}^{3+} + 6 e^- \rightarrow 2 \text{Cr}
\][/tex]
#### Step 3: Add the balanced half-reactions together
Now add the two half-reactions ensuring the electrons cancel each other out:
[tex]\[
6 \text{Cl}^- \rightarrow 3 \text{Cl}_2 + 6 e^-
\][/tex]
[tex]\[
2 \text{Cr}^{3+} + 6 e^- \rightarrow 2 \text{Cr}
\][/tex]
Combine the equations:
[tex]\[
2 \text{Cr}^{3+} + 6 \text{Cl}^- \rightarrow 2 \text{Cr} + 3 \text{Cl}_2
\][/tex]
### Step 4: Write the final balanced equation
[tex]\[
2 \text{Cr}^{3+}(aq) + 6 \text{Cl}^-(aq) \rightarrow 2 \text{Cr}(s) + 3 \text{Cl}_2(g)
\][/tex]
In conclusion, the final balanced equation for the redox reaction is:
[tex]\[
2 \text{Cr}^{3+}(aq) + 6 \text{Cl}^-(aq) \rightarrow 2 \text{Cr}(s) + 3 \text{Cl}_2(g)
\][/tex]
Thus, the correct choice is:
[tex]\[
2 \text{Cr}^{3+}(aq) + 6 \text{Cl}^-(aq) \rightarrow 2 \text{Cr}(s) + 3 \text{Cl}_2(g)
\][/tex]
