Step 1: Understanding the Concept:
The value of \( x \) represents the number of water molecules of crystallization. This is largely determined by the hydration enthalpy of the metal cation. Hydration enthalpy depends on the charge density (charge-to-size ratio) of the ion. Step 2: Detailed Explanation:
1. All the given metals (\( Mg, Ca, Ba \)) are Alkaline Earth Metals (Group 2) and carry a \( +2 \) charge.
2. As we go down the group (\( Mg \to Ca \to Ba \)), the size of the cation increases.
3. Smaller ions have a higher charge density and thus a stronger ability to attract and hold water molecules in their crystal lattice.
4. Magnesium (\( Mg^{2+} \)) is the smallest of the three and typically forms Epsom salt: \( MgSO_4 \cdot 7H_2O \) (\( x=7 \)).
5. Calcium forms Gypsum: \( CaSO_4 \cdot 2H_2O \) (\( x=2 \)).
6. Barium (\( Ba^{2+} \)) has very low hydration enthalpy and \( BaSO_4 \) is generally anhydrous (\( x=0 \)). Step 3: Final Answer
The value of \( x \) is maximum for \( MgSO_4 \cdot xH_2O \).