Two vessels have equal volums. One of them contains hydrogen at one atmosphere and the other helium at two atmosphere. If both the samples are at the same temperature, the `rms` velocity of the hydrogen molecules is

Options

A. Equal to that of helium
B. Twice that of helium
C. Half that of helium
D. `sqrt(2)` times that of helium

Answer

To find the root mean square (RMS) velocity of hydrogen molecules in comparison to helium molecules, we can follow these steps:

### Step 1: Understand the relationship of RMS velocity
The RMS velocity (v_rms) of a gas can be expressed using the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass (molecular weight) of the gas.

### Step 2: Identify the gases and their molecular weights
- For hydrogen (H₂), the molecular weight \( M_H \) is approximately 2 g/mol.
- For helium (He), the molecular weight \( M_{He} \) is approximately 4 g/mol.

### Step 3: Set up the ratio of RMS velocities
Since both gases are at the same temperature, we can compare their RMS velocities:
\[ \frac{v_{rms, H}}{v_{rms, He}} = \sqrt{\frac{M_{He}}{M_H}} \]

### Step 4: Substitute the molecular weights into the equation
Substituting the molecular weights into the equation gives:
\[ \frac{v_{rms, H}}{v_{rms, He}} = \sqrt{\frac{4}{2}} = \sqrt{2} \]

### Step 5: Express the RMS velocity of hydrogen
From the ratio, we can express the RMS velocity of hydrogen in terms of helium:
\[ v_{rms, H} = v_{rms, He} \cdot \sqrt{2} \]

### Step 6: Conclusion
Thus, the RMS velocity of hydrogen is \(\sqrt{2}\) times that of helium.

### Final Answer
The RMS velocity of the hydrogen molecules is \(\sqrt{2}\) times the RMS velocity of the helium molecules.

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