A sample contains N moles of a diatomic gas at temperature T. Molecules of gas get dissociated into atoms temperature remaining constant, find change in internal energy of gas

Options

A. `NRT`
B. `(5)/(2)NRT`
C. `(NRT)/(2)`
D. `(3)/(2)NRT`

Answer

To find the change in internal energy of a diatomic gas when it dissociates into atoms while maintaining a constant temperature, we can follow these steps:

### Step 1: Understand the initial conditions
We have a sample of a diatomic gas containing \( N \) moles at temperature \( T \). The internal energy of a diatomic gas can be expressed using the formula:
\[ U_1 = n C_v T \]
where \( C_v \) for a diatomic gas is \( \frac{5}{2} R \).

### Step 2: Calculate the initial internal energy
Substituting the values into the equation:
\[ U_1 = N \left(\frac{5}{2} R\right) T = \frac{5}{2} N R T \]

### Step 3: Understand the dissociation process
When the diatomic gas dissociates into atoms, the number of moles doubles. Therefore, after dissociation, the number of moles becomes \( 2N \).

### Step 4: Determine the specific heat capacity after dissociation
For a monoatomic gas (which the atoms are after dissociation), the specific heat capacity \( C_v \) is \( \frac{3}{2} R \).

### Step 5: Calculate the final internal energy
Now, we can calculate the internal energy after dissociation:
\[ U_2 = (2N) \left(\frac{3}{2} R\right) T = 3 N R T \]

### Step 6: Calculate the change in internal energy
The change in internal energy (\( \Delta U \)) can be calculated as:
\[ \Delta U = U_2 - U_1 \]
Substituting the values we found:
\[ \Delta U = (3 N R T) - \left(\frac{5}{2} N R T\right) \]
\[ \Delta U = 3 N R T - \frac{5}{2} N R T \]
To combine these, we convert \( 3 N R T \) into a fraction with a common denominator:
\[ \Delta U = \frac{6}{2} N R T - \frac{5}{2} N R T = \frac{1}{2} N R T \]

### Final Result
Thus, the change in internal energy of the gas is:
\[ \Delta U = \frac{1}{2} N R T \]

---