The following compounds have been arranged in the order of their increasing thermal stabilities.Identify the correct order.
`K_2CO_3(I),Na_2CO_3(II),Rb_2CO_3(III),Li_2CO_3(IV)`

Explanation

To determine the increasing thermal stabilities of the given carbonates \( K_2CO_3 \) (I), \( Na_2CO_3 \) (II), \( Rb_2CO_3 \) (III), and \( Li_2CO_3 \) (IV), we need to consider the relationship between the size of the cation and the stability of the carbonate ion.

### Step-by-Step Solution:

1. **Identify the Cations and Their Sizes**:
- The cations involved are lithium (Li\(^+\)), sodium (Na\(^+\)), potassium (K\(^+\)), and rubidium (Rb\(^+\)).
- The order of cation sizes (from smallest to largest) is:
- Li\(^+\) < Na\(^+\) < K\(^+\) < Rb\(^+\).

2. **Understand the Stability of Carbonates**:
- The thermal stability of carbonates is influenced by the size of the cation. Larger cations can better stabilize the larger carbonate ion (CO\(_3^{2-}\)).
- Therefore, as the size of the cation increases, the thermal stability of the carbonate also increases.

3. **Arrange the Carbonates Based on Cation Size**:
- Since Li\(^+\) is the smallest cation, \( Li_2CO_3 \) (IV) will have the lowest thermal stability.
- Next, \( Na_2CO_3 \) (II) will be more stable than \( Li_2CO_3 \).
- Following that, \( K_2CO_3 \) (I) will be more stable than \( Na_2CO_3 \).
- Finally, \( Rb_2CO_3 \) (III) will have the highest thermal stability due to the largest cation.

4. **Final Order of Increasing Thermal Stability**:
- The correct order of increasing thermal stability is:
- \( Li_2CO_3 \) (IV) < \( Na_2CO_3 \) (II) < \( K_2CO_3 \) (I) < \( Rb_2CO_3 \) (III).

### Summary of the Order:
- \( Li_2CO_3 < Na_2CO_3 < K_2CO_3 < Rb_2CO_3 \)