One mole mixture of `CH_4` and air (containing 80% `N_2` 20 % `O_2` by volume ) of a composition such that when underwent combustion gave maximum heat (assume combustion of only `CH_4`).Then which of the statements are correct , regarding composition of initial mixture ? (X presents mole fraction )

Options

A. `X_(CH_4)=1/11, X_(O_2)=2/11,X_(N_2)=8/11`
B. `X_(CH_4)=3/8, X_(O_2)=1/8,X_(N_2)=1/2`
C. `X_(CH_4)=1/6, X_(O_2)=1/6,X_(N_2)=2/3`
D. Data insufficient

Answer

To solve the problem, we need to determine the composition of a mixture of methane (CH₄) and air that will yield maximum heat upon combustion. The air is composed of 80% nitrogen (N₂) and 20% oxygen (O₂) by volume.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( X \) be the number of moles of CH₄ in the mixture.
- The total mixture is 1 mole, so the moles of air will be \( 1 - X \).

2. **Determine the Composition of Air**:
- Since air consists of 20% O₂ and 80% N₂ by volume, the moles of O₂ in the air can be expressed as:
\[ \text{Moles of O₂} = 0.2 \times (1 - X) = 0.2 - 0.2X \]
- The moles of N₂ in the air can be expressed as:
\[ \text{Moles of N₂} = 0.8 \times (1 - X) = 0.8 - 0.8X \]

3. **Combustion Reaction**:
- The combustion of methane can be represented as:
\[ \text{CH₄} + 2 \text{O₂} \rightarrow \text{CO₂} + 2 \text{H₂O} \]
- For \( X \) moles of CH₄, the required moles of O₂ for complete combustion will be:
\[ \text{Required O₂} = 2X \]

4. **Set Up the Equation**:
- To find the maximum heat, we need to ensure that the available O₂ from the air is equal to the required O₂ for combustion:
\[ 0.2 - 0.2X = 2X \]

5. **Solve for X**:
- Rearranging the equation gives:
\[ 0.2 = 2X + 0.2X \]
\[ 0.2 = 2.2X \]
\[ X = \frac{0.2}{2.2} = \frac{1}{11} \]

6. **Determine the Moles of O₂ and N₂**:
- Now, substituting \( X \) back to find the moles of O₂:
\[ \text{Moles of O₂} = 2X = 2 \times \frac{1}{11} = \frac{2}{11} \]
- And for N₂:
\[ \text{Moles of N₂} = 0.8 - 0.8X = 0.8 - 0.8 \times \frac{1}{11} = 0.8 - \frac{0.8}{11} = \frac{8}{11} \]

7. **Calculate Mole Fractions**:
- The mole fractions are:
- Mole fraction of CH₄:
\[ X_{CH₄} = \frac{1}{11} \]
- Mole fraction of O₂:
\[ X_{O₂} = \frac{2}{11} \]
- Mole fraction of N₂:
\[ X_{N₂} = \frac{8}{11} \]

### Conclusion:
The mole fractions of the components in the mixture are:
- Mole fraction of CH₄ = \( \frac{1}{11} \)
- Mole fraction of O₂ = \( \frac{2}{11} \)
- Mole fraction of N₂ = \( \frac{8}{11} \)