A mixture of two gases is contained in a vessel. The Gas 1 is monoatomic and gas 2 is diatomic and the ratio of their molecular masses M 1 ​ /M 2 ​ =1/4. the ratio of root mean square speeds of the molecules of two gases is

Options

A. (a)2
B. (b)4
C. (c)8
D. (d)16

Answer

To find the ratio of the root mean square speeds of the molecules of two gases, we can follow these steps:

### Step 1: Understand the formula for root mean square speed
The root mean square speed (v_rms) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
where:
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature,
- \( m \) is the molecular mass of the gas.

### Step 2: Write the formula for both gases
For gas 1 (monoatomic), the root mean square speed is:
\[ v_{rms1} = \sqrt{\frac{3kT}{M_1}} \]
For gas 2 (diatomic), the root mean square speed is:
\[ v_{rms2} = \sqrt{\frac{3kT}{M_2}} \]

### Step 3: Find the ratio of the root mean square speeds
To find the ratio of the root mean square speeds of the two gases, we can write:
\[ \frac{v_{rms1}}{v_{rms2}} = \frac{\sqrt{\frac{3kT}{M_1}}}{\sqrt{\frac{3kT}{M_2}}} \]
This simplifies to:
\[ \frac{v_{rms1}}{v_{rms2}} = \sqrt{\frac{M_2}{M_1}} \]

### Step 4: Substitute the given molecular mass ratio
From the problem, we know that the ratio of the molecular masses is given as:
\[ \frac{M_1}{M_2} = \frac{1}{4} \]
This implies:
\[ \frac{M_2}{M_1} = 4 \]

### Step 5: Calculate the ratio of root mean square speeds
Now substituting this into our ratio of speeds:
\[ \frac{v_{rms1}}{v_{rms2}} = \sqrt{4} = 2 \]

### Conclusion
Thus, the ratio of the root mean square speeds of the molecules of the two gases is:
\[ \frac{v_{rms1}}{v_{rms2}} = 2 \]