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 need to analyze the given information step by step.

### Step 1: Understand the root mean square speed formula
The root mean square speed (vrms) 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,
- \( M \) 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:
\[ x = \sqrt{\frac{3RT}{M_{N_2}}} \]

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

### Step 4: Dissociation of nitrogen molecules
When the nitrogen molecules dissociate into atoms, the molar mass changes. The molar mass of \( N_2 \) is \( M_{N_2} \), and when it dissociates into nitrogen atoms, the molar mass \( M' \) becomes:
\[ M' = \frac{M_{N_2}}{2} \]

### Step 5: Calculate the new root mean square speed
Now we can find the new root mean square speed \( v'_{rms} \) for nitrogen atoms at the new temperature:
\[ v'_{rms} = \sqrt{\frac{3RT'}{M'}} = \sqrt{\frac{3R(2T)}{\frac{M_{N_2}}{2}}} \]
This simplifies to:
\[ v'_{rms} = \sqrt{\frac{3R(2T) \cdot 2}{M_{N_2}}} = \sqrt{\frac{6RT}{M_{N_2}}} \]

### Step 6: Relate new speed to initial speed
We know that:
\[ x = \sqrt{\frac{3RT}{M_{N_2}}} \]
To express \( v'_{rms} \) in terms of \( x \):
\[ v'_{rms} = \sqrt{2} \cdot \sqrt{\frac{3RT}{M_{N_2}}} = \sqrt{2} \cdot x \]

### Step 7: Find the value of \( n \)
According to the problem, the new root mean square speed of nitrogen atoms becomes \( n \) times \( x \):
\[ v'_{rms} = n \cdot x \]
From our earlier calculation, we have:
\[ \sqrt{2} \cdot x = n \cdot x \]
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ n = \sqrt{2} \]

### Conclusion
Thus, the value of \( n \) is:
\[ \boxed{\sqrt{2}} \]