Increasing order of stability of-
I. `K_(2)CO_(3)` II. `MgCO_(3)` III `Na_(2)CO_(3)`

Options

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

Explanation

To determine the increasing order of stability for the carbonates \( K_2CO_3 \), \( MgCO_3 \), and \( Na_2CO_3 \), we need to analyze the thermal stability of these compounds based on the degree of polarization of the cations involved.

### Step-by-Step Solution:

1. **Understanding Thermal Stability**:
- Thermal stability refers to the ability of a compound to withstand heat without decomposing. For carbonates, this often involves the breakdown into oxides and carbon dioxide upon heating.

2. **Degree of Polarization**:
- The thermal stability of carbonates is inversely related to the degree of polarization of the cation. A higher degree of polarization indicates a more covalent character, which leads to lower thermal stability.

3. **Identifying the Cations**:
- The cations in question are \( K^+ \) (potassium), \( Na^+ \) (sodium), and \( Mg^{2+} \) (magnesium).
- Potassium and sodium are alkali metals, while magnesium is an alkaline earth metal.

4. **Comparing Cation Sizes**:
- As we move down the group in the periodic table, the size of the cations increases. Larger cations have a lower charge density, which results in less polarization.
- Therefore, \( K^+ \) is larger than \( Na^+ \), and both are larger than \( Mg^{2+} \).

5. **Covalent vs Ionic Character**:
- According to Fajans' rules, larger cations (like \( K^+ \)) will have more ionic character and less covalent character compared to smaller cations (like \( Mg^{2+} \)).
- Thus, \( K^+ \) has the least covalent character, followed by \( Na^+ \), and \( Mg^{2+} \) has the most covalent character.

6. **Determining Stability Order**:
- Since \( K^+ \) has the least covalent character, \( K_2CO_3 \) will be the most thermally stable.
- Next, \( Na_2CO_3 \) will have moderate stability, and finally, \( MgCO_3 \) will be the least stable due to its higher covalent character.

7. **Final Order of Stability**:
- The increasing order of stability is:
\[ MgCO_3 < Na_2CO_3 < K_2CO_3 \]

### Answer:
The increasing order of stability is:
\[ MgCO_3 < Na_2CO_3 < K_2CO_3 \]