Three moles of an diatomic gas are in a closed rigid container at temperature T (in K). 1 mole of diatomic gas gets dissociated into atoms without appreciable change in temperature. Now heat is supplied to the gas and temperature becomes 2T. If the heat supplied to the gas is x(RT), find the value of x.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Initial Conditions
We have 3 moles of a diatomic gas in a closed rigid container at temperature T. When 1 mole of the diatomic gas dissociates, it converts into atoms, resulting in 2 moles of diatomic gas and 2 moles of monoatomic gas.

### Step 2: Identify the Final State
After dissociation, we have:
- 2 moles of diatomic gas
- 2 moles of monoatomic gas

The final temperature after heat is supplied is 2T.

### Step 3: Calculate the Change in Internal Energy for Monoatomic Gas
The formula for the change in internal energy (ΔU) is given by:
\[ \Delta U = \frac{f}{2} n R \Delta T \]
Where:
- \( f \) = degrees of freedom
- \( n \) = number of moles
- \( R \) = universal gas constant
- \( \Delta T \) = change in temperature

For a monoatomic gas, \( f = 3 \):
- Number of moles = 2 (from dissociation)
- \( \Delta T = 2T - T = T \)

So, for the monoatomic gas:
\[ \Delta U_{\text{mono}} = \frac{3}{2} \times 2 \times R \times T = 3RT \]

### Step 4: Calculate the Change in Internal Energy for Diatomic Gas
For a diatomic gas, \( f = 5 \):
- Number of moles = 2
- \( \Delta T = T \)

So, for the diatomic gas:
\[ \Delta U_{\text{di}} = \frac{5}{2} \times 2 \times R \times T = 5RT \]

### Step 5: Total Change in Internal Energy
The total change in internal energy for the system is the sum of the changes for both gases:
\[ \Delta U_{\text{total}} = \Delta U_{\text{mono}} + \Delta U_{\text{di}} = 3RT + 5RT = 8RT \]

### Step 6: Relate Heat Supplied to Change in Internal Energy
According to the first law of thermodynamics, since the container is rigid (no work done):
\[ Q = \Delta U \]
Where \( Q \) is the heat supplied. Given that the heat supplied is \( x(RT) \):
\[ x(RT) = 8RT \]

### Step 7: Solve for x
Dividing both sides by \( RT \):
\[ x = 8 \]

### Final Answer
The value of \( x \) is 8.

---