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. 20 ml
B. 10 ml
C. 25 ml
D. 15 ml

Answer

To solve the problem, we need to find the volume of oxygen (O₂) in the original mixture of 30 ml of oxygen and ozone (O₃). We will follow these steps:

### Step 1: Define Variables
Let the volume of O₂ be \( x \) ml. Therefore, the volume of O₃ will be \( 30 - x \) ml (since the total volume of the mixture is 30 ml).

### Step 2: Write the Decomposition Reaction
The decomposition of ozone (O₃) can be represented as:
\[ 2 \, \text{O}_3 \rightarrow 3 \, \text{O}_2 \]
From this reaction, we can see that 2 moles of O₃ produce 3 moles of O₂.

### Step 3: Relate the Volumes of O₃ and O₂
From the stoichiometry of the reaction, we can derive the relationship between the volumes:
- 1 mole of O₃ produces \( \frac{3}{2} \) moles of O₂.
- Therefore, \( 30 - x \) ml of O₃ will produce:
\[ \frac{3}{2} \times (30 - x) \, \text{ml of O}_2 \]

### Step 4: Total Volume After Decomposition
After the decomposition of O₃, the total volume of the gas mixture will be the original volume of O₂ plus the volume of O₂ produced from the decomposition of O₃:
\[ \text{Total Volume} = x + \frac{3}{2} \times (30 - x) \]
According to the problem, this total volume expands to 40 ml after cooling:
\[ x + \frac{3}{2} \times (30 - x) = 40 \]

### Step 5: Solve the Equation
Now we will solve the equation:
1. Distribute \( \frac{3}{2} \):
\[ x + \frac{3}{2} \times 30 - \frac{3}{2} x = 40 \]
\[ x + 45 - \frac{3}{2} x = 40 \]

2. Combine like terms:
\[ \left(1 - \frac{3}{2}\right)x + 45 = 40 \]
\[ -\frac{1}{2} x + 45 = 40 \]

3. Isolate \( x \):
\[ -\frac{1}{2} x = 40 - 45 \]
\[ -\frac{1}{2} x = -5 \]
\[ x = 10 \, \text{ml} \]

### Conclusion
The volume of O₂ in the original mixture is **10 ml**.