The root mean square speed of `N_(2)` molecules in sample at temperature T is 'x'. If the temperature is doubled, then nitrogen molecules dissociate into atoms, the root mean square speedof nitrogen atoms becomes n times of 'x' find the value of n here?

Answer

To solve the problem, we will follow these steps:

### Step 1: Understanding the root mean square speed formula
The root mean square speed (\( V_{rms} \)) of gas molecules is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 2: Initial conditions
We are given that the root mean square speed of \( N_2 \) molecules at temperature \( T \) is \( x \). Therefore, we can write:
\[ x = \sqrt{\frac{3RT}{M}} \]

### Step 3: Doubling the temperature
If the temperature is doubled, the new temperature \( T' \) becomes:
\[ T' = 2T \]

### Step 4: Change in molecular mass due to dissociation
When the nitrogen molecules dissociate into atoms, the molecular mass changes. The molar mass of \( N_2 \) is \( M \), and when it dissociates into nitrogen atoms, the molar mass of each nitrogen atom becomes \( \frac{M}{2} \).

### Step 5: New root mean square speed calculation
Now, we can calculate the new root mean square speed (\( V'_{rms} \)) after the temperature is doubled and the nitrogen molecules dissociate:
\[ V'_{rms} = \sqrt{\frac{3R(2T)}{\frac{M}{2}}} \]
This simplifies to:
\[ V'_{rms} = \sqrt{\frac{3R \cdot 2T \cdot 2}{M}} = \sqrt{\frac{6RT}{M}} = \sqrt{4} \cdot \sqrt{\frac{3RT}{M}} = 2 \cdot x \]

### Step 6: Relating the new speed to \( n \)
According to the problem, the new root mean square speed of nitrogen atoms becomes \( n \) times \( x \):
\[ V'_{rms} = n \cdot x \]
From our calculation, we found that:
\[ V'_{rms} = 2x \]
Thus, we can equate:
\[ n \cdot x = 2x \]

### Step 7: Solving for \( n \)
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ n = 2 \]

### Conclusion
The value of \( n \) is:
\[ \boxed{2} \]