Balance the following reaction
`aZn+NO_(3)+bH_(2)Ooverset("OH")tocZn^(2+)+NH_(4)^(+)+dOH^(-)`

Options

A. a = 4 , b = 7, c= 4, d = 10
B. a = 2, b = 5, c = 6 , d = 8
C. a = 3, b = 3 , c = 1 , d = 9
D. a = 1 , b = 3 , c = 7 , d = 8

Answer

To balance the given redox reaction, we will follow a systematic approach by identifying the oxidation and reduction half-reactions, balancing them, and then combining them to get the final balanced equation.

### Step 1: Identify Oxidation and Reduction
- **Oxidation**: Zinc (Zn) is oxidized to Zinc ion (Zn²⁺).
- **Reduction**: Nitrate ion (NO₃⁻) is reduced to Ammonium ion (NH₄⁺).

### Step 2: Write Half-Reactions
1. **Oxidation Half-Reaction**:
\[ \text{Zn} \rightarrow \text{Zn}^{2+} + 2e^- \]

2. **Reduction Half-Reaction**:
\[ \text{NO}_3^- + 8e^- + 10\text{H}_2\text{O} \rightarrow \text{NH}_4^+ + 10\text{OH}^- \]

### Step 3: Balance the Oxidation Half-Reaction
- The oxidation half-reaction is already balanced in terms of mass and charge:
\[ \text{Zn} \rightarrow \text{Zn}^{2+} + 2e^- \]

### Step 4: Balance the Reduction Half-Reaction
- Start with the reduction half-reaction:
\[ \text{NO}_3^- + 8e^- + 10\text{H}_2\text{O} \rightarrow \text{NH}_4^+ + 10\text{OH}^- \]
- Here, we have balanced the oxygen by adding water molecules and then balancing the hydrogen by adding hydroxide ions.

### Step 5: Equalize the Number of Electrons
- The oxidation half-reaction produces 2 electrons, while the reduction half-reaction consumes 8 electrons. To equalize, we multiply the oxidation half-reaction by 4:
\[ 4\text{Zn} \rightarrow 4\text{Zn}^{2+} + 8e^- \]

### Step 6: Combine the Half-Reactions
- Now we can add the two half-reactions:
\[ 4\text{Zn} + \text{NO}_3^- + 10\text{H}_2\text{O} \rightarrow 4\text{Zn}^{2+} + \text{NH}_4^+ + 10\text{OH}^- \]

### Step 7: Identify Coefficients
From the balanced equation, we can identify the coefficients:
- \( a = 4 \) (for Zn)
- \( b = 10 \) (for H₂O)
- \( c = 4 \) (for Zn²⁺)
- \( d = 10 \) (for OH⁻)

### Final Balanced Equation
The final balanced equation is:
\[ 4\text{Zn} + \text{NO}_3^- + 10\text{H}_2\text{O} \rightarrow 4\text{Zn}^{2+} + \text{NH}_4^+ + 10\text{OH}^- \]

### Summary of Coefficients
- \( a = 4 \)
- \( b = 10 \)
- \( c = 4 \)
- \( d = 10 \)