A container is filled with 20 moles of an ideal diatomic gas at absolute temperature T. When heat is supplied to gas temperature remains constant but 8 moles dissociate into atoms. Heat energy given to gas is

Options

A. 4 RT
B. 6 RT
C. 3 RT
D. 5 RT

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the initial energy of the diatomic gas
The initial energy \( U_i \) of the diatomic gas can be calculated using the formula:
\[ U_i = n \cdot \frac{f}{2} \cdot R \cdot T \]
where:
- \( n = 20 \) moles (initial number of moles),
- \( f = 5 \) (degrees of freedom for diatomic gas: 3 translational + 2 rotational),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature.

Substituting the values:
\[ U_i = 20 \cdot \frac{5}{2} \cdot R \cdot T = 50RT \]

### Step 2: Determine the final state after dissociation
When 8 moles of the diatomic gas dissociate, they form 16 moles of monoatomic gas. Thus, the final composition of the gas is:
- 12 moles of diatomic gas,
- 16 moles of monoatomic gas.

### Step 3: Calculate the final energy of the system
The final energy \( U_f \) can be calculated as the sum of the energy of the remaining diatomic gas and the energy of the newly formed monoatomic gas.

1. **Energy of 12 moles of diatomic gas**:
\[ U_{diatomic} = 12 \cdot \frac{5}{2} \cdot R \cdot T = 30RT \]

2. **Energy of 16 moles of monoatomic gas**:
\[ U_{monoatomic} = 16 \cdot \frac{3}{2} \cdot R \cdot T = 24RT \]

Adding these energies together gives:
\[ U_f = U_{diatomic} + U_{monoatomic} = 30RT + 24RT = 54RT \]

### Step 4: Calculate the heat energy supplied to the gas
The heat energy \( Q \) given to the gas is the difference between the final energy and the initial energy:
\[ Q = U_f - U_i = 54RT - 50RT = 4RT \]

### Final Answer
The heat energy given to the gas is:
\[ \boxed{4RT} \]

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