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

Options

A. `1//12`
B. `8//9`
C. `1//6`
D. `16//17`

Explanation

To solve the problem of finding the fraction of the total pressure exerted by hydrogen when equal weights of methane (CH₄) and hydrogen (H₂) are mixed in an empty container at 25°C, we can follow these steps:

### Step 1: Define the Mass of Each Gas
Assume we have equal masses of methane and hydrogen. Let the mass of each gas be \( x \) grams.

### Step 2: Calculate the Molar Masses
- Molar mass of hydrogen (H₂) = 2 g/mol
- Molar mass of methane (CH₄) = 16 g/mol (calculated as 12 for carbon + 4 for four hydrogens)

### Step 3: Calculate the Number of Moles
Using the formula for the number of moles:
\[ \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} \]
- For hydrogen:
\[ n_{H_2} = \frac{x}{2} \]
- For methane:
\[ n_{CH_4} = \frac{x}{16} \]

### Step 4: Calculate the Total Number of Moles
The total number of moles in the mixture is:
\[ n_{total} = n_{H_2} + n_{CH_4} = \frac{x}{2} + \frac{x}{16} \]
To add these fractions, find a common denominator (16):
\[ n_{total} = \frac{8x}{16} + \frac{x}{16} = \frac{9x}{16} \]

### Step 5: Calculate the Mole Fraction of Hydrogen
The mole fraction of hydrogen (\( X_{H_2} \)) is given by:
\[ X_{H_2} = \frac{n_{H_2}}{n_{total}} = \frac{\frac{x}{2}}{\frac{9x}{16}} \]
This simplifies to:
\[ X_{H_2} = \frac{x}{2} \cdot \frac{16}{9x} = \frac{16}{18} = \frac{8}{9} \]

### Step 6: Determine the Fraction of Total Pressure Exerted by Hydrogen
According to Dalton's Law of Partial Pressures, the fraction of the total pressure exerted by hydrogen is equal to its mole fraction:
\[ \text{Fraction of total pressure by } H_2 = X_{H_2} = \frac{8}{9} \]

### Final Answer
The fraction of the total pressure exerted by hydrogen is \( \frac{8}{9} \).

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