If the ratio of molar masses of two gases A and B is 1 : 4. What is the ratio of the average speeds ?

Options

A. 2
B. 4
C. 1
D. 4

Answer

To solve the problem of finding the ratio of the average speeds of two gases A and B, given that the ratio of their molar masses is 1:4, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between average speed and molar mass**:
The average speed (U) of a gas is inversely proportional to the square root of its molar mass (M). This relationship can be expressed mathematically as:
\[ U \propto \frac{1}{\sqrt{M}} \]

2. **Set up the ratio of average speeds**:
Let \( U_A \) be the average speed of gas A and \( U_B \) be the average speed of gas B. The ratio of the average speeds can be expressed as:
\[ \frac{U_A}{U_B} = \sqrt{\frac{M_B}{M_A}} \]

3. **Substitute the given molar mass ratio**:
We are given that the ratio of the molar masses \( M_A : M_B = 1 : 4 \). This means:
\[ M_A = 1 \quad \text{and} \quad M_B = 4 \]

4. **Calculate the ratio of average speeds**:
Substitute the values of \( M_A \) and \( M_B \) into the equation:
\[ \frac{U_A}{U_B} = \sqrt{\frac{M_B}{M_A}} = \sqrt{\frac{4}{1}} = \sqrt{4} = 2 \]

5. **Conclusion**:
Therefore, the ratio of the average speeds of gases A and B is:
\[ \frac{U_A}{U_B} = 2 : 1 \]

### Final Answer:
The ratio of the average speeds of gases A and B is \( 2 : 1 \).

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