The ratio of coefficient of `HNO_(3), Fe(N)_(3))_(2)` and `NH_(4)NO_(3)` in the following redox reaction `Fe + HNO_(3) rarr Fe (NO_(3))_(2) + NH_(4)NO_(3) + H_(2)O`
are respectively

Options

A. `10 : 1 : 4`
B. `4 : 10 : 1`
C. `4 : 1 : 10`
D. `10 : 4 : 1`

Explanation

To solve the problem, we need to balance the given redox reaction and then determine the coefficients of the compounds involved. The reaction is:

\[ \text{Fe} + \text{HNO}_3 \rightarrow \text{Fe(NO}_3\text{)}_2 + \text{NH}_4\text{NO}_3 + \text{H}_2\text{O} \]

### Step 1: Write down the unbalanced equation
The unbalanced equation is:
\[ \text{Fe} + \text{HNO}_3 \rightarrow \text{Fe(NO}_3\text{)}_2 + \text{NH}_4\text{NO}_3 + \text{H}_2\text{O} \]

### Step 2: Identify the number of atoms of each element
- **Reactants:**
- Fe: 1
- H: 1 (from HNO3)
- N: 1 (from HNO3)
- O: 3 (from HNO3)

- **Products:**
- Fe: 1 (from Fe(NO3)2)
- N: 2 (from Fe(NO3)2) + 1 (from NH4NO3) = 3
- H: 4 (from NH4NO3) + 2 (from H2O) = 6
- O: 6 (from Fe(NO3)2) + 3 (from NH4NO3) + 1 (from H2O) = 10

### Step 3: Balance the equation
To balance the equation, we need to ensure that the number of atoms of each element is the same on both sides.

1. Start with Iron (Fe):
- 1 Fe on both sides is balanced.

2. Balance Nitrogen (N):
- We have 3 N on the product side (2 from Fe(NO3)2 and 1 from NH4NO3). Thus, we need 3 HNO3 to provide 3 N.
- Update the equation:
\[ \text{Fe} + 4 \text{HNO}_3 \rightarrow \text{Fe(NO}_3\text{)}_2 + \text{NH}_4\text{NO}_3 + \text{H}_2\text{O} \]

3. Balance Hydrogen (H):
- We need to balance H. We have 4 H from NH4NO3 and 2 from H2O, totaling 6 H on the product side. Thus, we need 6 HNO3.
- Update the equation:
\[ \text{Fe} + 10 \text{HNO}_3 \rightarrow \text{Fe(NO}_3\text{)}_2 + 2 \text{NH}_4\text{NO}_3 + 3 \text{H}_2\text{O} \]

4. Balance Oxygen (O):
- Count O on both sides:
- Reactants: 10 from HNO3
- Products: 6 from Fe(NO3)2 + 6 from 2 NH4NO3 + 3 from 3 H2O = 15 O.
- This indicates we need to adjust the coefficients.

### Final Balanced Equation
After balancing, we find:
\[ 4 \text{Fe} + 10 \text{HNO}_3 \rightarrow 4 \text{Fe(NO}_3\text{)}_2 + 2 \text{NH}_4\text{NO}_3 + 3 \text{H}_2\text{O} \]

### Step 4: Determine the coefficients
From the balanced equation:
- Coefficient of HNO3 = 10
- Coefficient of Fe(NO3)2 = 4
- Coefficient of NH4NO3 = 2

### Step 5: Calculate the ratio
The ratio of the coefficients of HNO3, Fe(NO3)2, and NH4NO3 is:
\[ 10 : 4 : 2 \]

To simplify:
\[ 10 : 4 : 2 = 5 : 2 : 1 \]

### Final Answer
The ratio of the coefficients of HNO3, Fe(NO3)2, and NH4NO3 is:
\[ 5 : 2 : 1 \]