The ratio of the rate of diffusion of a given element to that of helium is 1:4. The molecular weight of the element is

Options

A. 32
B. 64
C. 16
D. None of these

Explanation

To find the molecular weight of the given element based on the ratio of its rate of diffusion to that of helium, 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 molecular weight. Mathematically, this can be expressed as:
\[ \frac{R_1}{R_2} = \frac{\sqrt{M_2}}{\sqrt{M_1}} \]
where:
- \( R_1 \) = rate of diffusion of the first gas (the given element)
- \( R_2 \) = rate of diffusion of the second gas (helium)
- \( M_1 \) = molecular weight of the first gas (the given element)
- \( M_2 \) = molecular weight of the second gas (helium)

### Step 2: Set Up the Ratio
According to the problem, the ratio of the rate of diffusion of the given element to that of helium is 1:4. Therefore, we can write:
\[ \frac{R_{\text{element}}}{R_{\text{He}}} = \frac{1}{4} \]
This means \( R_1 = 1 \) and \( R_2 = 4 \).

### Step 3: Substitute into Graham's Law
Using Graham's Law, we substitute the known values:
\[ \frac{1}{4} = \frac{\sqrt{M_{\text{He}}}}{\sqrt{M_{\text{element}}}} \]
The molecular weight of helium (\( M_{\text{He}} \)) is 4 g/mol.

### Step 4: Rearrange the Equation
Rearranging the equation gives:
\[ \frac{1}{4} = \frac{\sqrt{4}}{\sqrt{M_{\text{element}}}} \]
This simplifies to:
\[ \frac{1}{4} = \frac{2}{\sqrt{M_{\text{element}}}} \]

### Step 5: Cross Multiply
Cross multiplying gives:
\[ 1 \cdot \sqrt{M_{\text{element}}} = 4 \cdot 2 \]
This simplifies to:
\[ \sqrt{M_{\text{element}}} = 8 \]

### Step 6: Square Both Sides
Squaring both sides results in:
\[ M_{\text{element}} = 8^2 = 64 \]

### Conclusion
The molecular weight of the given element is 64 g/mol.