Question Statement:
A gaseous fuel mixture comprising 3 moles of methane and 2 moles of ammonia
is combusted in X moles of pure oxygen in stoichiometric amount. Assuming
complete combustion, with only CO2, H2O and N2 in the product gases, the value
of X is ___.
3 CH4 + 2 NH3 + X O^2 → Products (CO2, H2O, N^2)
Options:
-
(A) 7.5
-
(B) 5.5
-
(C) 8.5
-
(D) 9.5
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.33)**
**1. Intuitive Physical Setup & Given Parameters**:
Combustion of a gaseous fuel mixture in pure oxygen:
- 3 moles of methane ($\text{CH}_4$)
- 2 moles of ammonia ($\text{NH}_3$)
- $X$ moles of oxygen ($\text{O}_2$)
- Complete stoichiometric combustion producing only $\text{CO}_2$, $\text{H}_2\text{O}$, and $\text{N}_2$:
3\,\text{CH}_4 + 2\,\text{NH}_3 + X\,\text{O}_2 \to a\,\text{CO}_2 + b\,\text{H}_2\text{O} + c\,\text{N}_2
We need to determine the stoichiometric moles of oxygen $X$.
**2. Step-by-Step Chemical Stoichiometry Derivation**:
1. **Carbon (C) Balance**:
- Reactants: $3 \times 1 = 3$ moles of C.
- Products: $a$ moles of $\text{CO}_2$.
a = 3
2. **Nitrogen (N) Balance**:
- Reactants: $2 \times 1 = 2$ moles of N.
- Products: $2c$ moles of N in $c\,\text{N}_2$.
2c = 2 \implies c = 1
3. **Hydrogen (H) Balance**:
- Reactants:
- From $3\,\text{CH}_4$: $3 \times 4 = 12$ atoms of H.
- From $2\,\text{NH}_3$: $2 \times 3 = 6$ atoms of H.
- Total H atoms $= 12 + 6 = 18$ atoms.
- Products: $2b$ atoms of H in $b\,\text{H}_2\text{O}$.
2b = 18 \implies b = 9
4. **Oxygen (O) Balance**:
- Reactants: $2X$ atoms of O in $X\,\text{O}_2$.
- Products:
- In $3\,\text{CO}_2$: $3 \times 2 = 6$ atoms of O.
- In $9\,\text{H}_2\text{O}$: $9 \times 1 = 9$ atoms of O.
- Total O atoms in products $= 6 + 9 = 15$ atoms.
- Equating Oxygen atoms:
2X = 15 \implies X = \frac{15}{2} = 7.5\text{ moles}
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 7.5**: ✅ **CORRECT**. $X = 3(2) + 2(0.75) = 6 + 1.5 = 7.5\text{ moles}$.
- **(B) 5.5**: ❌ **INCORRECT**. Misses water from ammonia.
- **(C) 8.5**: ❌ **INCORRECT**. Overcounts oxygen atoms.
- **(D) 9.5**: ❌ **INCORRECT**. Arithmetic error in hydrogen summation.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Independent mole balance: $\text{CH}_4 + 2\text{O}_2 \to \text{CO}_2 + 2\text{H}_2\text{O}$ (needs 2 $\text{O}_2$/mol), $\text{NH}_3 + 0.75\text{O}_2 \to 0.5\text{N}_2 + 1.5\text{H}_2\text{O}$ (needs 0.75 $\text{O}_2$/mol). Total $\text{O}_2 = 3(2) + 2(0.75) = 7.5$.