Equal weights of methane and oxygen are mixed in an empty container at `25^(@)C`. The fraction of the total pressure exerted by oxygen is

Options

A. `1/3`
B. `1/2`
C. `2/3`
D. `1/3 xx 273/298`.

Explanation

To solve the problem of finding the fraction of the total pressure exerted by oxygen when equal weights of methane (CH₄) and oxygen (O₂) are mixed in an empty container at 25°C, we can follow these steps:

### Step 1: Define the masses of the gases
Let the mass of methane (CH₄) be \( m_{CH4} = X \) grams. Since equal weights are mixed, the mass of oxygen (O₂) will also be \( m_{O2} = X \) grams.

### Step 2: Calculate the number of moles of each gas
To find the number of moles, we use the formula:
\[ \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \]

- For methane (CH₄):
- Molar mass of CH₄ = 16 g/mol
- Moles of CH₄ = \( \frac{X}{16} \)

- For oxygen (O₂):
- Molar mass of O₂ = 32 g/mol
- Moles of O₂ = \( \frac{X}{32} \)

### Step 3: Calculate the total number of moles
The total number of moles in the mixture is the sum of the moles of methane and oxygen:
\[ \text{Total moles} = \text{Moles of CH₄} + \text{Moles of O₂} = \frac{X}{16} + \frac{X}{32} \]

To add these fractions, we need a common denominator:
- The least common multiple of 16 and 32 is 32.
- Convert \( \frac{X}{16} \) to \( \frac{2X}{32} \).

Now we can add:
\[ \text{Total moles} = \frac{2X}{32} + \frac{X}{32} = \frac{3X}{32} \]

### Step 4: Calculate the mole fraction of oxygen
The mole fraction of oxygen (χ_O₂) is given by:
\[ \chi_{O₂} = \frac{\text{Moles of O₂}}{\text{Total moles}} = \frac{\frac{X}{32}}{\frac{3X}{32}} = \frac{1}{3} \]

### Step 5: Calculate the fraction of total pressure exerted by oxygen
According to Dalton's law of partial pressures, the partial pressure of a gas is equal to its mole fraction multiplied by the total pressure:
\[ P_{O₂} = \chi_{O₂} \times P_{total} \]

The fraction of the total pressure exerted by oxygen is simply its mole fraction:
\[ \text{Fraction of total pressure by O₂} = \chi_{O₂} = \frac{1}{3} \]

### Conclusion
The fraction of the total pressure exerted by oxygen is \( \frac{1}{3} \).

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