Estimate the temperature at which the oxygen molecules will have the same rms velocity as hydrogen molecules at `150^(@)C`. Molecular weight of oxygen is 32 and that of hydrogen is 2.

Explanation

To solve the problem, we need to find the temperature at which the root mean square (rms) velocity of oxygen molecules equals the rms velocity of hydrogen molecules at 150°C.

### Step-by-Step Solution:

1. **Convert the Temperature of Hydrogen to Kelvin:**
The temperature of hydrogen is given as 150°C. To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Thus,
\[ T_{H2} = 150 + 273 = 423 \, K \]

**Hint:** Remember that to convert Celsius to Kelvin, you always add 273.

2. **Write the Formula for rms Velocity:**
The root mean square velocity (Vrms) is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molecular weight in kg/mol.

3. **Set the rms Velocities Equal:**
Since we want the rms velocity of oxygen (\( V_{rms(O2)} \)) to equal that of hydrogen (\( V_{rms(H2)} \)), we can set up the equation:
\[ \sqrt{\frac{3R T_{O2}}{M_{O2}}} = \sqrt{\frac{3R T_{H2}}{M_{H2}}} \]
Here, \( M_{O2} = 32 \, g/mol = 0.032 \, kg/mol \) and \( M_{H2} = 2 \, g/mol = 0.002 \, kg/mol \).

4. **Square Both Sides:**
Squaring both sides to eliminate the square root gives:
\[ \frac{3R T_{O2}}{M_{O2}} = \frac{3R T_{H2}}{M_{H2}} \]
The \( 3R \) cancels out:
\[ \frac{T_{O2}}{M_{O2}} = \frac{T_{H2}}{M_{H2}} \]

5. **Rearrange to Solve for \( T_{O2} \):**
Rearranging the equation gives:
\[ T_{O2} = T_{H2} \cdot \frac{M_{O2}}{M_{H2}} \]

6. **Substitute the Known Values:**
Substitute \( T_{H2} = 423 \, K \), \( M_{O2} = 32 \, g/mol \), and \( M_{H2} = 2 \, g/mol \):
\[ T_{O2} = 423 \cdot \frac{32}{2} \]
\[ T_{O2} = 423 \cdot 16 = 6768 \, K \]

7. **Convert the Temperature Back to Celsius:**
To convert \( T_{O2} \) back to Celsius:
\[ T_{O2}(°C) = T_{O2}(K) - 273 \]
\[ T_{O2}(°C) = 6768 - 273 = 6495 \, °C \]

### Final Answer:
The estimated temperature at which the oxygen molecules will have the same rms velocity as hydrogen molecules at 150°C is **6495°C**.