When the equation
______`MnO_(4)^(-)+`___`SO_(3)^(2-)+`_______`H^(+)to`______`Mn^(2+)+`_______`SO_(4)^(2-)+`________`H_(2)O`
is balanced correctly with the smallest whole number coefficients, what is the coefficient for `H_(2)O`?

Options

A. 3
B. 5
C. 8
D. 10

Answer

To balance the redox reaction given by the equation:

\[ \text{MnO}_4^- + \text{SO}_3^{2-} + \text{H}^+ \rightarrow \text{Mn}^{2+} + \text{SO}_4^{2-} + \text{H}_2\text{O} \]

we will follow these steps:

### Step 1: Identify Oxidation and Reduction
- **Oxidation half-reaction**: The sulfite ion \( \text{SO}_3^{2-} \) is oxidized to sulfate \( \text{SO}_4^{2-} \).
- **Reduction half-reaction**: The permanganate ion \( \text{MnO}_4^- \) is reduced to manganese ion \( \text{Mn}^{2+} \).

### Step 2: Write the Half-Reactions
1. **Oxidation half-reaction**:
\[ \text{SO}_3^{2-} \rightarrow \text{SO}_4^{2-} + 2 \text{H}^+ + 2 \text{e}^- \]

2. **Reduction half-reaction**:
\[ \text{MnO}_4^- + 8 \text{H}^+ + 5 \text{e}^- \rightarrow \text{Mn}^{2+} + 4 \text{H}_2\text{O} \]

### Step 3: Equalize the Number of Electrons
To balance the electrons transferred, we need to multiply the oxidation half-reaction by 5:

\[ 5 \text{SO}_3^{2-} \rightarrow 5 \text{SO}_4^{2-} + 10 \text{H}^+ + 10 \text{e}^- \]

### Step 4: Combine the Half-Reactions
Now, we can combine the two half-reactions:

\[ \text{MnO}_4^- + 8 \text{H}^+ + 5 \text{e}^- + 5 \text{SO}_3^{2-} \rightarrow \text{Mn}^{2+} + 4 \text{H}_2\text{O} + 5 \text{SO}_4^{2-} + 10 \text{H}^+ + 10 \text{e}^- \]

### Step 5: Cancel Out Electrons and Combine
After canceling out the electrons and combining the terms, we have:

\[ \text{MnO}_4^- + 5 \text{SO}_3^{2-} + 6 \text{H}^+ \rightarrow \text{Mn}^{2+} + 5 \text{SO}_4^{2-} + 3 \text{H}_2\text{O} \]

### Final Balanced Equation
The final balanced equation is:

\[ 2 \text{MnO}_4^- + 5 \text{SO}_3^{2-} + 16 \text{H}^+ \rightarrow 2 \text{Mn}^{2+} + 5 \text{SO}_4^{2-} + 8 \text{H}_2\text{O} \]

### Step 6: Identify the Coefficient for \( \text{H}_2\text{O} \)
From the balanced equation, the coefficient for \( \text{H}_2\text{O} \) is **8**.