The average speed of nitrogen molecules in a gas is `v`. If the temperature is doubled and the `N_(2)` molecule dissociate into nitrogen atoms, then the average speed will be

Options

A. `v`
B. `vsqrt(2)`
C. `2v`
D. `4v`

Answer

To solve the problem, we need to analyze how the average speed of nitrogen molecules changes when the temperature is doubled and the nitrogen molecules dissociate into nitrogen atoms.

### Step-by-Step Solution:

1. **Initial Conditions**:
- Let the average speed of nitrogen molecules (N₂) be \( v \).
- The initial temperature is \( T \).

2. **Molar Mass of Nitrogen Molecule**:
- The molar mass of nitrogen (N₂) is \( 28 \, \text{g/mol} \) (since each nitrogen atom has a mass of \( 14 \, \text{g/mol} \)).

3. **Average Speed Formula**:
- The root mean square speed \( v_{\text{rms}} \) of gas molecules is given by the formula:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature, and \( M \) is the molar mass.

4. **Average Speed of Nitrogen Molecules (N₂)**:
- For nitrogen molecules:
\[ v = \sqrt{\frac{3RT}{28}} \]

5. **New Conditions After Temperature is Doubled**:
- When the temperature is doubled, the new temperature \( T' = 2T \).

6. **Dissociation of Nitrogen Molecules**:
- When N₂ dissociates into nitrogen atoms (N), the molar mass of nitrogen atoms is \( 14 \, \text{g/mol} \).

7. **Average Speed of Nitrogen Atoms (N)**:
- The average speed of nitrogen atoms at the new temperature can be calculated as:
\[ v' = \sqrt{\frac{3R(2T)}{14}} \]

8. **Simplifying the Expression**:
- Substitute \( T' = 2T \) into the equation:
\[ v' = \sqrt{\frac{3R(2T)}{14}} = \sqrt{\frac{6RT}{14}} = \sqrt{\frac{3RT}{7}} \]

9. **Relating the Speeds**:
- We can relate \( v' \) to \( v \):
\[ v = \sqrt{\frac{3RT}{28}} \]
- To find the ratio:
\[ \frac{v'}{v} = \frac{\sqrt{\frac{3RT}{7}}}{\sqrt{\frac{3RT}{28}}} = \sqrt{\frac{28}{7}} = \sqrt{4} = 2 \]

10. **Conclusion**:
- Therefore, the average speed of nitrogen atoms after the temperature is doubled and the molecules dissociate is:
\[ v' = 2v \]

### Final Answer:
The average speed will be \( 2v \).