The following compounds have been arranged in order of their increasing stabilities. Identify the correct order.
(I) `K_(2)CO_(3)` (II) `MgCO_(3)`
(III) `CaCO_(3)` (IV) `BeCO_(3)`

Options

A. `I lt II lt III lt IV`
B. `IV lt II lt III lt I`
C. `IV lt II lt I lt III`
D. `II lt IV lt III lt I`

Explanation

To determine the correct order of stability for the compounds \( K_2CO_3 \), \( MgCO_3 \), \( CaCO_3 \), and \( BeCO_3 \), we need to analyze the stability of these compounds based on their group classifications and the properties of the cations involved.

### Step-by-Step Solution:

1. **Identify the Groups**:
- \( K_2CO_3 \) is a compound of Group 1 (alkali metal).
- \( MgCO_3 \), \( CaCO_3 \), and \( BeCO_3 \) are compounds of Group 2 (alkaline earth metals).

2. **Understand Stability Trends**:
- Generally, the thermal stability of carbonates increases as you move down the group in the periodic table. This is due to the increase in ionic size and decrease in polarizing power of the cations.
- Group 1 carbonates (like \( K_2CO_3 \)) are typically more thermally stable than Group 2 carbonates.

3. **Analyze the Group 2 Carbonates**:
- Among the Group 2 carbonates, the stability order is influenced by the size of the cation:
- \( BeCO_3 \) is the least stable because beryllium has a small ionic radius and high charge density, leading to a high polarizing power, which destabilizes the carbonate ion.
- \( MgCO_3 \) is more stable than \( BeCO_3 \) but less stable than \( CaCO_3 \) due to the increasing ionic size and decreasing polarizing power.
- \( CaCO_3 \) is more stable than both \( BeCO_3 \) and \( MgCO_3 \) due to its larger ionic size.

4. **Combine the Information**:
- Since \( K_2CO_3 \) is a Group 1 compound, it will be more stable than all Group 2 carbonates.
- The order of stability among the Group 2 carbonates is \( CaCO_3 > MgCO_3 > BeCO_3 \).

5. **Final Order of Stability**:
- Therefore, the overall order of increasing stability is:
\[ BeCO_3 < MgCO_3 < CaCO_3 < K_2CO_3 \]

### Conclusion:
The correct order of increasing stability for the compounds is:
\[ BeCO_3 < MgCO_3 < CaCO_3 < K_2CO_3 \]