Write correctly balanced half reactions and overall equations for the following skeletal equations:
`NO_(3)^(ө)+Bi(s)rarrBi^(3+)+NO_(2)` ( in acid solution)

Explanation

To balance the given redox reaction \( \text{NO}_3^- + \text{Bi}(s) \rightarrow \text{Bi}^{3+} + \text{NO}_2 \) in acidic solution, we will follow these steps:

### Step 1: Identify the oxidation and reduction half-reactions
- **Reduction Half-Reaction**: The nitrate ion (\( \text{NO}_3^- \)) is reduced to nitrogen dioxide (\( \text{NO}_2 \)).
- **Oxidation Half-Reaction**: Bismuth (\( \text{Bi} \)) is oxidized to bismuth ion (\( \text{Bi}^{3+} \)).

### Step 2: Write the half-reactions
1. **Reduction**:
\[ \text{NO}_3^- \rightarrow \text{NO}_2 \]
2. **Oxidation**:
\[ \text{Bi} \rightarrow \text{Bi}^{3+} \]

### Step 3: Balance the reduction half-reaction
- **Oxidation States**: Nitrogen in \( \text{NO}_3^- \) is +5 and in \( \text{NO}_2 \) is +4.
- **Electrons**: To balance the change in oxidation state, we add 1 electron to the product side:
\[ \text{NO}_3^- + 1 \, e^- \rightarrow \text{NO}_2 \]
- **Oxygen Balance**: There are 3 oxygen atoms in \( \text{NO}_3^- \) and 2 in \( \text{NO}_2 \). We need to add 1 water molecule to the product side:
\[ \text{NO}_3^- + 1 \, e^- + 1 \, \text{H}_2\text{O} \rightarrow \text{NO}_2 + 2 \, \text{H}^+ \]

### Step 4: Balance the oxidation half-reaction
- **Electrons**: Bismuth is oxidized from 0 to +3, so we need to add 3 electrons to the reactant side:
\[ \text{Bi} \rightarrow \text{Bi}^{3+} + 3 \, e^- \]

### Step 5: Equalize the number of electrons
To combine the half-reactions, we need to have the same number of electrons. We will multiply the reduction half-reaction by 3:
\[ 3 \, \text{NO}_3^- + 3 \, e^- + 3 \, \text{H}_2\text{O} \rightarrow 3 \, \text{NO}_2 + 6 \, \text{H}^+ \]

### Step 6: Combine the half-reactions
Now we can add the two half-reactions together:
\[ 3 \, \text{NO}_3^- + 3 \, \text{H}_2\text{O} + \text{Bi} \rightarrow 3 \, \text{NO}_2 + 6 \, \text{H}^+ + \text{Bi}^{3+} \]

### Step 7: Write the overall balanced equation
The overall balanced equation is:
\[ 3 \, \text{NO}_3^- + \text{Bi} + 6 \, \text{H}^+ \rightarrow 3 \, \text{NO}_2 + 3 \, \text{H}_2\text{O} + \text{Bi}^{3+} \]