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 need to analyze the situation step by step.

### Step 1: Understand the Initial Conditions
We start with 3 moles of a diatomic gas at temperature T. When one mole of this diatomic gas dissociates, it breaks into two moles of monoatomic gas. Therefore, after dissociation, we have:
- 2 moles of diatomic gas
- 2 moles of monoatomic gas (from the dissociation of 1 mole of diatomic gas)

### Step 2: Calculate the Change in Internal Energy
For a gas, the change in internal energy (ΔU) can be calculated using the formula:
\[ \Delta U = \frac{f}{2} n R \Delta T \]
where:
- \( f \) is the degrees of freedom,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant,
- \( \Delta T \) is the change in temperature.

**For Monoatomic Gas:**
- Degrees of freedom \( f = 3 \)
- Number of moles \( n = 2 \)
- Initial temperature \( T \) and final temperature \( 2T \) gives \( \Delta T = 2T - T = T \)

So, the change in internal energy for the monoatomic gas is:
\[ \Delta U_{mono} = \frac{3}{2} \times 2 \times R \times T = 3RT \]

**For Diatomic Gas:**
- Degrees of freedom \( f = 5 \)
- Number of moles \( n = 2 \)
- Change in temperature \( \Delta T = T \)

So, the change in internal energy for the diatomic gas is:
\[ \Delta U_{diatomic} = \frac{5}{2} \times 2 \times R \times T = 5RT \]

### Step 3: Total Change in Internal Energy
Now, we can find the total change in internal energy:
\[ \Delta U_{total} = \Delta U_{mono} + \Delta U_{diatomic} = 3RT + 5RT = 8RT \]

### Step 4: Relate Heat Supplied to Change in Internal Energy
According to the first law of thermodynamics, since the container is rigid, there is no work done (W = 0). Therefore, the heat supplied (Q) is equal to the change in internal energy:
\[ Q = \Delta U_{total} = 8RT \]

### Step 5: Find the Value of x
The problem states that the heat supplied is \( x(RT) \). Therefore, we can equate:
\[ x(RT) = 8RT \]
Dividing both sides by \( RT \):
\[ x = 8 \]

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

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