The rate of diffusion of methane is twice that of X. The molecular mass of X is

Options

A. 16
B. 32
C. 80
D. 64

Explanation

To find the molecular mass of gas X given that the rate of diffusion of methane is twice that of gas X, we can use Graham's law of diffusion. Here’s a step-by-step solution:

### Step 1: Understand Graham's Law of Diffusion
Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it can be expressed as:
\[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \]
where \( r_1 \) and \( r_2 \) are the rates of diffusion of gases 1 and 2, and \( M_1 \) and \( M_2 \) are their molar masses.

### Step 2: Set Up the Equation
Let’s denote:
- \( r_{CH_4} \) as the rate of diffusion of methane (gas 1),
- \( r_X \) as the rate of diffusion of gas X (gas 2).

According to the problem, the rate of diffusion of methane is twice that of gas X:
\[ r_{CH_4} = 2 \cdot r_X \]

### Step 3: Substitute into Graham's Law
Using Graham's law, we can write:
\[ \frac{r_X}{r_{CH_4}} = \sqrt{\frac{M_{CH_4}}{M_X}} \]
Substituting \( r_{CH_4} = 2 \cdot r_X \) into the equation gives:
\[ \frac{r_X}{2 \cdot r_X} = \sqrt{\frac{M_{CH_4}}{M_X}} \]
This simplifies to:
\[ \frac{1}{2} = \sqrt{\frac{M_{CH_4}}{M_X}} \]

### Step 4: Square Both Sides
Now, square both sides to eliminate the square root:
\[ \left(\frac{1}{2}\right)^2 = \frac{M_{CH_4}}{M_X} \]
This results in:
\[ \frac{1}{4} = \frac{M_{CH_4}}{M_X} \]

### Step 5: Rearrange to Find Molar Mass of X
Rearranging the equation gives:
\[ M_X = 4 \cdot M_{CH_4} \]

### Step 6: Calculate Molar Mass of Methane
The molar mass of methane (\( CH_4 \)) can be calculated as follows:
- Carbon (C) has a molar mass of 12 g/mol.
- Hydrogen (H) has a molar mass of 1 g/mol, and there are 4 hydrogen atoms.

Thus, the molar mass of methane is:
\[ M_{CH_4} = 12 + (4 \times 1) = 16 \text{ g/mol} \]

### Step 7: Substitute to Find Molar Mass of X
Now substitute \( M_{CH_4} \) into the equation for \( M_X \):
\[ M_X = 4 \cdot 16 = 64 \text{ g/mol} \]

### Conclusion
The molecular mass of gas X is 64 g/mol.