Two vessels having equal volumes conain `H_(2)` and He at 1 and 2 atm respectively at same temperature. Then which one correct :
(i) `U_(rms), H_(2) = U_(rms) He`, (ii) `r_(H_(2)) = r_(He)/sqrt(2)`, (iii) `U_(rms) H_(2) = sqrt(2)U_(rms) He`, (iv) `r_(H_(2)) = sqrt(8) xx r_(He)`

Options

A. I,ii,iii
B. ii,iii
C. I,iv
D. iii,iv

Answer

To solve the problem, we need to analyze the given conditions and apply the relevant equations for root mean square (RMS) speed and diffusion rates of gases.

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- We have two vessels of equal volume containing Hydrogen (H₂) at 1 atm and Helium (He) at 2 atm, both at the same temperature.

2. **Formula for RMS Speed**:
- The RMS speed (U_rms) of a gas is given by the formula:
\[ U_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

3. **Calculating U_rms for H₂ and He**:
- For Hydrogen (H₂), the molar mass \( M_{H2} = 2 \, \text{g/mol} \).
- For Helium (He), the molar mass \( M_{He} = 4 \, \text{g/mol} \).
- Since both gases are at the same temperature, we can compare their RMS speeds:
\[ \frac{U_{rms, H2}}{U_{rms, He}} = \sqrt{\frac{M_{He}}{M_{H2}}} = \sqrt{\frac{4}{2}} = \sqrt{2} \]
- Therefore, we can express this as:
\[ U_{rms, H2} = \sqrt{2} \cdot U_{rms, He} \]

4. **Analyzing the Options**:
- (i) \( U_{rms, H2} = U_{rms, He} \) → **False** (we found \( U_{rms, H2} = \sqrt{2} U_{rms, He} \)).
- (ii) \( r_{H2} = \frac{r_{He}}{\sqrt{2}} \) → This is a relation derived from the diffusion rates. Since \( r \) is inversely proportional to the square root of the molar mass, this is **True**.
- (iii) \( U_{rms, H2} = \sqrt{2} U_{rms, He} \) → **True** (as derived above).
- (iv) \( r_{H2} = \sqrt{8} \cdot r_{He} \) → **False** (this does not match our derived relations).

5. **Conclusion**:
- The correct options are (ii) and (iii).

### Final Answer:
The correct statements are:
- (ii) \( r_{H2} = \frac{r_{He}}{\sqrt{2}} \)
- (iii) \( U_{rms, H2} = \sqrt{2} U_{rms, He} \)