At what temperature the effective speed of gaseous `H_2` molecules is equal to that of oxygen molecules at 320 K?

Options

A. `20 K`
B. `50 K`
C. `40 K`
D. `30 K`

Explanation

To solve the problem of finding the temperature at which the effective speed of gaseous \( H_2 \) molecules is equal to that of oxygen molecules at 320 K, we will use the formula for the root mean square (RMS) speed of gas molecules.

### Step-by-Step Solution:

1. **Understand the RMS Speed Formula**:
The RMS speed (\( v_{rms} \)) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

2. **Calculate RMS Speed for Oxygen (\( O_2 \))**:
For oxygen, the molar mass \( M \) is 32 g/mol, which is equivalent to \( 32 \times 10^{-3} \) kg/mol. The temperature \( T \) is given as 320 K. Thus, the RMS speed for oxygen is:
\[ v_{rms, O_2} = \sqrt{\frac{3R \cdot 320}{32 \times 10^{-3}}} \]

3. **Calculate RMS Speed for Hydrogen (\( H_2 \))**:
For hydrogen, the molar mass \( M \) is 2 g/mol, which is equivalent to \( 2 \times 10^{-3} \) kg/mol. Let the temperature of hydrogen be \( T \). Thus, the RMS speed for hydrogen is:
\[ v_{rms, H_2} = \sqrt{\frac{3R \cdot T}{2 \times 10^{-3}}} \]

4. **Set the Two RMS Speeds Equal**:
According to the problem, the effective speed of \( H_2 \) molecules is equal to that of \( O_2 \) molecules:
\[ v_{rms, O_2} = v_{rms, H_2} \]
This gives us the equation:
\[ \sqrt{\frac{3R \cdot 320}{32 \times 10^{-3}}} = \sqrt{\frac{3R \cdot T}{2 \times 10^{-3}}} \]

5. **Square Both Sides**:
Squaring both sides to eliminate the square roots, we get:
\[ \frac{3R \cdot 320}{32 \times 10^{-3}} = \frac{3R \cdot T}{2 \times 10^{-3}} \]

6. **Cancel Common Terms**:
The \( 3R \) terms cancel out from both sides:
\[ \frac{320}{32 \times 10^{-3}} = \frac{T}{2 \times 10^{-3}} \]

7. **Cross Multiply**:
Cross multiplying gives:
\[ 320 \cdot 2 \times 10^{-3} = T \cdot 32 \times 10^{-3} \]

8. **Simplify the Equation**:
Simplifying this, we have:
\[ 640 \times 10^{-3} = T \cdot 32 \times 10^{-3} \]
Dividing both sides by \( 10^{-3} \):
\[ 640 = T \cdot 32 \]

9. **Solve for \( T \)**:
Now, divide both sides by 32:
\[ T = \frac{640}{32} = 20 \text{ K} \]

### Final Answer:
The temperature at which the effective speed of gaseous \( H_2 \) molecules is equal to that of oxygen molecules at 320 K is **20 K**.