At what temperature is the root mean square velocity of gaseous hydrogen molecules is equal to that of oxygen molecules at `47^(@)C`?

Options

A. `20 K`
B. `80 K`
C. `-73 K`
D. `3 K`

Explanation

To find the temperature at which the root mean square (rms) velocity of gaseous hydrogen molecules is equal to that of oxygen molecules at 47°C, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for RMS Velocity**:
The root mean square velocity (v_rms) of a gas 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 molar mass of the gas in kg/mol.

2. **Write the RMS Velocity for Oxygen (O2)**:
For oxygen, the molar mass \( M_{O2} \) is 32 g/mol, which is 0.032 kg/mol. The temperature \( T_{O2} \) is given as 47°C. First, convert this temperature to Kelvin:
\[ T_{O2} = 47 + 273 = 320 \text{ K} \]
Now, the rms velocity for oxygen can be expressed as:
\[ v_{rms, O2} = \sqrt{\frac{3R \cdot 320}{0.032}} \]

3. **Write the RMS Velocity for Hydrogen (H2)**:
For hydrogen, the molar mass \( M_{H2} \) is 2 g/mol, which is 0.002 kg/mol. The rms velocity for hydrogen can be expressed as:
\[ v_{rms, H2} = \sqrt{\frac{3RT_{H2}}{0.002}} \]

4. **Set the RMS Velocities Equal**:
According to the problem, we need to find the temperature \( T_{H2} \) at which the rms velocities are equal:
\[ v_{rms, O2} = v_{rms, H2} \]
This leads to the equation:
\[ \sqrt{\frac{3R \cdot 320}{0.032}} = \sqrt{\frac{3RT_{H2}}{0.002}} \]

5. **Square Both Sides**:
Squaring both sides to eliminate the square root gives:
\[ \frac{3R \cdot 320}{0.032} = \frac{3RT_{H2}}{0.002} \]

6. **Cancel Common Terms**:
The \( 3R \) terms on both sides cancel out:
\[ \frac{320}{0.032} = \frac{T_{H2}}{0.002} \]

7. **Cross Multiply**:
Cross multiplying gives:
\[ 320 \cdot 0.002 = T_{H2} \cdot 0.032 \]

8. **Calculate T_H2**:
\[ T_{H2} = \frac{320 \cdot 0.002}{0.032} = \frac{0.64}{0.032} = 20 \text{ K} \]

### Final Answer:
The temperature at which the root mean square velocity of gaseous hydrogen molecules is equal to that of oxygen molecules at 47°C is **20 K**.