A mixture of `NH_(4)NO_(3)` and `(NH_(4))_(2)HP_(4)` coitain 30.40% mass per cent of nitrogen. What is the mass ratio of the two components in the mixture ?

Options

A. `2:1`
B. `1:2`
C. `3:4`
D. `4:1`

Answer

To solve the problem, we need to determine the mass ratio of the two components in the mixture, which are ammonium nitrate (NH₄NO₃) and ammonium hydrogen phosphate ((NH₄)₂HPO₄). We know that the mixture contains 30.40% mass percent of nitrogen.

### Step-by-Step Solution:

1. **Define Variables**:
Let:
- \( X \) = mass of ammonium nitrate (NH₄NO₃)
- \( Y \) = mass of ammonium hydrogen phosphate ((NH₄)₂HPO₄)

2. **Determine Molar Masses**:
- Molar mass of NH₄NO₃ = 14 (N) + 4 (H) + 14 (N) + 3 (O) = 80 g/mol
- Molar mass of (NH₄)₂HPO₄ = 2(14 (N) + 4 (H)) + 31 (P) + 4 (O) = 132 g/mol

3. **Calculate Moles of Each Component**:
- Moles of NH₄NO₃ = \( \frac{X}{80} \)
- Moles of (NH₄)₂HPO₄ = \( \frac{Y}{132} \)

4. **Calculate Moles of Nitrogen**:
- Each mole of NH₄NO₃ contains 1 mole of nitrogen (N), so moles of nitrogen from NH₄NO₃ = \( \frac{X}{80} \)
- Each mole of (NH₄)₂HPO₄ contains 2 moles of nitrogen, so moles of nitrogen from (NH₄)₂HPO₄ = \( 2 \times \frac{Y}{132} \)

5. **Total Moles of Nitrogen**:
- Total moles of nitrogen = \( \frac{X}{80} + 2 \times \frac{Y}{132} \)

6. **Total Mass of the Mixture**:
- Total mass = \( X + Y \)

7. **Mass Percent of Nitrogen**:
- The mass of nitrogen = (moles of nitrogen) × (molar mass of nitrogen)
- Molar mass of nitrogen = 14 g/mol
- Mass of nitrogen = \( 14 \left( \frac{X}{80} + 2 \times \frac{Y}{132} \right) \)

8. **Set Up the Equation**:
- According to the problem, the mass percent of nitrogen is 30.40%. Therefore:
\[ \frac{14 \left( \frac{X}{80} + 2 \times \frac{Y}{132} \right)}{X + Y} = 0.304 \]

9. **Multiply Through by Total Mass**:
\[ 14 \left( \frac{X}{80} + 2 \times \frac{Y}{132} \right) = 0.304 (X + Y) \]

10. **Simplify the Equation**:
- Multiply both sides by 80 × 132 to eliminate the denominators and simplify further.

11. **Solve for the Ratio \( \frac{X}{Y} \)**:
- Rearrange the equation to isolate \( \frac{X}{Y} \).

12. **Final Calculation**:
- After simplifying, you will arrive at the mass ratio of the two components.