To determine which of the statements provided is correct, we will analyze each statement one by one.
### Step 1: Analyze Statement 1
**Statement 1:** "The hydration energy of the alkali metals increases down the group."
- **Explanation:** Hydration energy refers to the energy released when gaseous ions are surrounded by water molecules to form hydrated ions. For alkali metals (Li, Na, K, Rb, Cs), as we move down the group, the size of the ions increases. Larger ions have a lower charge density, which results in weaker interactions with water molecules. Therefore, hydration energy actually **decreases** down the group, making this statement **incorrect**.
### Step 2: Analyze Statement 2
**Statement 2:** "Except Li2CO3, all the carbonates of the alkali metals decompose on heating."
- **Explanation:** Lithium carbonate (Li2CO3) is known to be stable upon heating compared to other alkali metal carbonates, which typically decompose into their respective oxides and carbon dioxide upon heating. However, Li2CO3 does decompose upon heating to form Li2O and CO2. Thus, this statement is also **incorrect**.
### Step 3: Analyze Statement 3
**Statement 3:** "Li does not form nitride on burning in air."
- **Explanation:** Lithium does react with nitrogen when burned in air to form lithium nitride (Li3N). The reaction can be represented as:
\[
6 \text{Li} + \text{N}_2 \rightarrow 2 \text{Li}_3\text{N}
\]
Therefore, this statement is **incorrect**.
### Step 4: Analyze Statement 4
**Statement 4:** "The solubility of the sulfates of group 2 elements decreases down the group."
- **Explanation:** For alkaline earth metals (group 2), the solubility of their sulfates (like BeSO4, MgSO4, CaSO4, SrSO4, BaSO4) generally decreases down the group. This is due to the decreasing hydration enthalpy as the size of the cations increases, which outweighs the lattice enthalpy. Therefore, this statement is **correct**.
### Conclusion
Based on the analysis of all statements, the correct statement is **Statement 4**: "The solubility of the sulfates of group 2 elements decreases down the group."
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