Ten moles of `(O_2)` gas are kept at temperature `T`. At some higher temperature `2T`, fourty percent of molecular oxygen breaks into atomic oxygen. Find change in internal energy of the gas.

Explanation

To find the change in internal energy of the gas when 40% of molecular oxygen breaks into atomic oxygen upon heating from temperature \( T \) to \( 2T \), we can follow these steps:

### Step 1: Determine the initial conditions
We have 10 moles of \( O_2 \) gas at temperature \( T \). Since \( O_2 \) is a diatomic gas, its degrees of freedom \( F \) is 5.

### Step 2: Calculate the initial internal energy \( U_1 \)
The formula for internal energy \( U \) for a gas is given by:
\[ U = n \cdot \frac{F}{2} \cdot R \cdot T \]
For the initial state (case 1):
- \( n = 10 \) moles
- \( F = 5 \) (for diatomic gas)
- \( R \) is the universal gas constant
- \( T \) is the initial temperature

Thus, the initial internal energy \( U_1 \) is:
\[ U_1 = 10 \cdot \frac{5}{2} \cdot R \cdot T = 25RT \]

### Step 3: Determine the final conditions
At temperature \( 2T \), 40% of the \( O_2 \) gas dissociates into atomic oxygen. Therefore:
- Moles of \( O_2 \) remaining = \( 60\% \) of \( 10 \) moles = \( 6 \) moles
- Moles of atomic oxygen produced = \( 40\% \) of \( 10 \) moles = \( 4 \) moles

Each mole of \( O_2 \) produces 2 moles of \( O \) upon dissociation, so \( 4 \) moles of \( O_2 \) produce \( 8 \) moles of atomic oxygen.

### Step 4: Calculate the final internal energy \( U_2 \)
For the remaining \( O_2 \) (6 moles):
\[ U_{O_2} = 6 \cdot \frac{5}{2} \cdot R \cdot (2T) = 6 \cdot \frac{5}{2} \cdot R \cdot 2T = 30RT \]

For the \( O \) atoms (8 moles):
Since \( O \) is a monoatomic gas, its degrees of freedom \( F \) is 3:
\[ U_{O} = 8 \cdot \frac{3}{2} \cdot R \cdot (2T) = 8 \cdot \frac{3}{2} \cdot R \cdot 2T = 24RT \]

Now, the total internal energy \( U_2 \) is:
\[ U_2 = U_{O_2} + U_{O} = 30RT + 24RT = 54RT \]

### Step 5: Calculate the change in internal energy \( \Delta U \)
The change in internal energy \( \Delta U \) is given by:
\[ \Delta U = U_2 - U_1 = 54RT - 25RT = 29RT \]

### Final Answer
The change in internal energy of the gas is:
\[ \Delta U = 29RT \]

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