30 ml of a mixture of oxygen `(O_(2))` and ozone `(O_(3))` was heated till ozone was completely decomposed. The mixture on cooling was found to expand 40 ml. The volume of `O_(2)` in original mixture was :

Options

A. 10
B. 8
C. 6
D. 4

Answer

To solve the problem step by step, we will follow the decomposition reaction of ozone and apply the concept of volume relationships in gases.

### Step 1: Write the decomposition reaction of ozone
The decomposition of ozone can be represented as:
\[ 2 \, O_3 \rightarrow 3 \, O_2 \]
This means that 2 volumes of ozone produce 3 volumes of oxygen.

### Step 2: Define variables for the volumes
Let:
- \( x \) = volume of ozone (\( O_3 \)) in the original mixture (in ml)
- \( 30 - x \) = volume of oxygen (\( O_2 \)) in the original mixture (in ml)

### Step 3: Calculate the volume of oxygen produced from the decomposition of ozone
From the reaction, we know that:
- 2 ml of \( O_3 \) produces 3 ml of \( O_2 \)
- Therefore, \( x \) ml of \( O_3 \) will produce \( \frac{3}{2}x \) ml of \( O_2 \).

### Step 4: Write the total volume after decomposition
After the decomposition of ozone, the total volume of the gas mixture will be the sum of the remaining oxygen and the oxygen produced from the decomposition:
\[ \text{Total Volume} = \text{Remaining } O_2 + \text{Produced } O_2 \]
\[ \text{Total Volume} = (30 - x) + \frac{3}{2}x \]

### Step 5: Set up the equation based on the final volume
According to the problem, after cooling, the total volume of the gas mixture expands to 40 ml:
\[ (30 - x) + \frac{3}{2}x = 40 \]

### Step 6: Simplify the equation
Now, simplify the equation:
\[ 30 - x + \frac{3}{2}x = 40 \]
Combine like terms:
\[ 30 + \frac{1}{2}x = 40 \]

### Step 7: Solve for \( x \)
Subtract 30 from both sides:
\[ \frac{1}{2}x = 10 \]
Multiply both sides by 2:
\[ x = 20 \, \text{ml} \]

### Step 8: Find the volume of oxygen in the original mixture
Now, substitute \( x \) back to find the volume of \( O_2 \):
\[ \text{Volume of } O_2 = 30 - x = 30 - 20 = 10 \, \text{ml} \]

### Final Answer
The volume of \( O_2 \) in the original mixture is **10 ml**.

---