Two vessels having equal volume contain molecular hydrogen at one atmosphere and helium at two atmospheres respectively. What is the ratio of rms speeds of hydrogen molecule to that of helium molecule if both the samples are at same temperature.

Options

A. `(V_(rms))_(n)=(V_(rms))_(He)`
B. `(V_(rms))_(He)=sqrt(2)(V_(rms))_(H)`
C. `(V_(rms))_(H)=sqrt(2)(V_(rms))_(He)`
D. `(V_(rms))_(H)=2(V_(rms))_(He)`

Explanation

To find the ratio of the root mean square (rms) speeds of hydrogen molecules to that of helium molecules, we can use the formula for rms speed:

\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]

where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 1: Identify the molar masses of the gases
- The molar mass of hydrogen (\( H_2 \)) is approximately \( 2 \, \text{g/mol} \) (since each hydrogen atom has a mass of about \( 1 \, \text{g/mol} \)).
- The molar mass of helium (\( He \)) is approximately \( 4 \, \text{g/mol} \).

### Step 2: Write the rms speed equations for both gases
For hydrogen:
\[ v_{\text{rms, H2}} = \sqrt{\frac{3RT}{M_{H2}}} \]

For helium:
\[ v_{\text{rms, He}} = \sqrt{\frac{3RT}{M_{He}}} \]

### Step 3: Set up the ratio of the rms speeds
We want to find the ratio \( \frac{v_{\text{rms, H2}}}{v_{\text{rms, He}}} \):
\[ \frac{v_{\text{rms, H2}}}{v_{\text{rms, He}}} = \frac{\sqrt{\frac{3RT}{M_{H2}}}}{\sqrt{\frac{3RT}{M_{He}}}} \]

### Step 4: Simplify the ratio
Since \( 3RT \) is common in both the numerator and denominator, it cancels out:
\[ \frac{v_{\text{rms, H2}}}{v_{\text{rms, He}}} = \sqrt{\frac{M_{He}}{M_{H2}}} \]

### Step 5: Substitute the molar masses
Substituting the values of the molar masses:
\[ \frac{v_{\text{rms, H2}}}{v_{\text{rms, He}}} = \sqrt{\frac{4 \, \text{g/mol}}{2 \, \text{g/mol}}} = \sqrt{2} \]

### Conclusion
Thus, the ratio of the rms speeds of hydrogen molecules to that of helium molecules is:
\[ \frac{v_{\text{rms, H2}}}{v_{\text{rms, He}}} = \sqrt{2} \]