`CaCO_(3)(s) hArrCaO(s)+CO_(2)(g)`
At equilibrium in the above case, `'a'` moles of `CaCO_(3),'b'` moles of `CaO` and `'c'` moles of `CO_(2)` are found. Then, identify the wrong statement `:`

Options

A. Moles of `CaCO_(3)` will decrease with the addition of inert gas at constant pressure
B. Moles of `CaCO_(3)` will remain constant with the increases in volume.
C. If volume of the vessel in halved, then moles of `CaCO_(3)` will increases
D. Moles of `CaO` will decreases with the increases in pressure.

Answer

To solve the question, we need to analyze the equilibrium reaction and the statements provided. The reaction is:

\[ \text{CaCO}_3(s) \rightleftharpoons \text{CaO}(s) + \text{CO}_2(g) \]

At equilibrium, we have 'a' moles of CaCO3, 'b' moles of CaO, and 'c' moles of CO2. We need to identify the wrong statement among the options given.

### Step-by-Step Solution:

**Step 1: Understand the Reaction and Equilibrium**
- The reaction involves solid calcium carbonate decomposing into solid calcium oxide and gaseous carbon dioxide.
- At equilibrium, the amounts of reactants and products remain constant.

**Step 2: Analyze the Change in Moles (ΔNG)**
- The change in the number of moles of gas (ΔNG) is calculated as:

\[ \Delta N_G = \text{moles of gaseous products} - \text{moles of gaseous reactants} \]

- In this reaction, we have:
- Gaseous products: CO2 (c moles)
- Gaseous reactants: None (0 moles)

Therefore,

\[ \Delta N_G = c - 0 = c \]

**Step 3: Apply Le Chatelier's Principle**
- Since ΔNG is positive (c > 0), according to Le Chatelier's principle, if we increase the volume (or decrease the pressure), the equilibrium will shift to the right to produce more CO2, which means more CaCO3 will decompose.

**Step 4: Evaluate Each Statement**
1. **Statement 1**: "Moles of CaCO3 will decrease with the addition of inert gas at constant pressure."
- **Evaluation**: This statement is correct. Adding an inert gas at constant pressure increases the volume, shifting the equilibrium to produce more CO2, thus decreasing CaCO3.

2. **Statement 2**: "Moles of CaCO3 will remain constant with increase in volume."
- **Evaluation**: This statement is incorrect. Increasing the volume shifts the equilibrium to the right, leading to a decrease in CaCO3.

3. **Statement 3**: "If volume of the vessel is halved, then moles of CaCO3 will increase."
- **Evaluation**: This statement is correct. Halving the volume increases pressure, shifting the equilibrium to the left, thus increasing CaCO3.

4. **Statement 4**: "Moles of calcium oxide will decrease with increase in pressure."
- **Evaluation**: This statement is correct. Increasing pressure shifts the equilibrium to the side with fewer gas moles, which is the left side (CaCO3), leading to a decrease in CaO.

**Conclusion**: The wrong statement is **Statement 2**.