1 mole of `CaC_(2), Mg_(2)C_(3) and Al_(4)C_(3` react with water in separate flask. Work done by the system in the three cases respectively are `w_(1), w_(2) and w_(3)`. Then which of the following is/are incorrect.

Options

A. `w_(3) gt w_(2)`
B. `w_(1) gt w_(3)`
C. `w_(3) gt w_(1)`
D. `w_(1) = w_(2)`

Answer

To solve the problem, we need to analyze the reactions of calcium carbide (CaC₂), magnesium carbide (Mg₂C₃), and aluminum carbide (Al₄C₃) with water. We will determine the work done by the system in each case based on the number of moles of gas produced.

### Step-by-Step Solution:

1. **Identify the Reactions**:
- **Calcium Carbide (CaC₂)**:
\[ \text{CaC}_2 + 2 \text{H}_2\text{O} \rightarrow \text{Ca(OH)}_2 + \text{C}_2\text{H}_2 \text{ (1 mole of gas)} \]
- **Magnesium Carbide (Mg₂C₃)**:
\[ \text{Mg}_2\text{C}_3 + 4 \text{H}_2\text{O} \rightarrow 2 \text{Mg(OH)}_2 + \text{C}_3\text{H}_4 \text{ (1 mole of gas)} \]
- **Aluminum Carbide (Al₄C₃)**:
\[ \text{Al}_4\text{C}_3 + 12 \text{H}_2\text{O} \rightarrow 4 \text{Al(OH)}_3 + 3 \text{CH}_4 \text{ (3 moles of gas)} \]

2. **Count the Moles of Gas Produced**:
- For **CaC₂**, 1 mole of gas (C₂H₂).
- For **Mg₂C₃**, 1 mole of gas (C₃H₄).
- For **Al₄C₃**, 3 moles of gas (3 moles of CH₄).

3. **Determine Work Done (W)**:
The work done (W) by the system is related to the number of moles of gas produced. The formula for work done is:
\[ W = -P \Delta V \]
where \( \Delta V \) is the change in volume. Since \( \Delta V \) is proportional to the number of moles of gas produced, we can say:
- \( W_1 \) (for CaC₂) is proportional to 1 mole of gas.
- \( W_2 \) (for Mg₂C₃) is proportional to 1 mole of gas.
- \( W_3 \) (for Al₄C₃) is proportional to 3 moles of gas.

4. **Compare Work Done**:
- Since \( W_1 \) and \( W_2 \) both correspond to 1 mole of gas, we have:
\[ W_1 = W_2 \]
- Since \( W_3 \) corresponds to 3 moles of gas, we have:
\[ W_3 > W_1 \quad \text{and} \quad W_3 > W_2 \]

5. **Identify Incorrect Statements**:
The statements provided are:
- \( W_3 > W_2 \) (True)
- \( W_1 > W_3 \) (False)
- \( W_3 > W_1 \) (True)
- \( W_1 = W_2 \) (True)

Therefore, the incorrect statement is:
\[ W_1 > W_3 \]

### Final Answer:
The incorrect statement is \( W_1 > W_3 \).