The root mean spuare (rms) speed of hydrogen molecules at a certain temperature is 300m/s. If the temperature is doubled and hydrogen gas dissociates into atomic hydrogen the rms speed will become

Options

A. `424.26m//s`
B. `300m//s`
C. `600m//s`
D. `150m//s`

Answer

To solve the problem, we need to find the new root mean square (rms) speed of hydrogen atoms after the temperature is doubled and the hydrogen gas dissociates into atomic hydrogen.

### Step-by-step Solution:

1. **Understand the rms speed formula**: The rms speed \( V_{rms} \) 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.

2. **Initial conditions**: We know that the initial rms speed of hydrogen molecules (H₂) at a certain temperature is \( 300 \, \text{m/s} \). The molar mass of hydrogen molecule (H₂) is \( 2 \, \text{g/mol} \) or \( 0.002 \, \text{kg/mol} \).

3. **Doubling the temperature**: If the temperature is doubled, we denote the initial temperature as \( T \) and the final temperature as \( 2T \).

4. **Dissociation of hydrogen**: When hydrogen gas (H₂) dissociates into atomic hydrogen (H), the molar mass changes. The molar mass of atomic hydrogen (H) is \( 1 \, \text{g/mol} \) or \( 0.001 \, \text{kg/mol} \).

5. **New rms speed calculation**: We need to find the new rms speed \( V'_{rms} \) for atomic hydrogen at the new temperature:
\[ V'_{rms} = \sqrt{\frac{3R(2T)}{M'}} \]
where \( M' \) is the molar mass of atomic hydrogen.

6. **Relating initial and final speeds**: We can relate the initial and final rms speeds using the formula:
\[ \frac{V_{initial}}{V_{final}} = \frac{T_{initial}}{T_{final}} \cdot \frac{M_{final}}{M_{initial}} \]
Plugging in the values:
\[ \frac{300}{V'_{rms}} = \frac{T}{2T} \cdot \frac{1}{2} \]
Simplifying gives:
\[ \frac{300}{V'_{rms}} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]

7. **Solving for \( V'_{rms} \)**:
\[ 300 = \frac{1}{4} V'_{rms} \]
Therefore:
\[ V'_{rms} = 300 \times 4 = 1200 \, \text{m/s} \]

8. **Final answer**: The new rms speed of atomic hydrogen after doubling the temperature is \( 1200 \, \text{m/s} \).