Enthalpy Change

To solve this problem, we need to understand how the properties of alkaline earth metals change as we move down the group (increasing atomic number).

1. Analyzing the Trend of Alkaline Earth Metals:
The alkaline earth metals (Group 2) include elements like beryllium (Be), magnesium (Mg), calcium (Ca), strontium (Sr), barium (Ba), and radium (Ra). As we move down the group, certain properties change due to the increasing atomic number.

2. Ionization Enthalpy:
Ionization enthalpy (or ionization energy) decreases as we move down the group. This is because the outermost electrons are farther from the nucleus and are more shielded by inner electrons, making it easier to remove them. Hence, ionization enthalpy does not increase with atomic number.

3. Electronegativity:
Electronegativity decreases as we move down the group. This is because the atomic size increases, and the outer electrons are less tightly held, making the atom less electronegative. Thus, electronegativity does not increase with atomic number.

4. Solubility of Hydroxide in Water:
The solubility of the alkaline earth metal hydroxides in water increases as we move down the group. For example, beryllium hydroxide (Be(OH)₂) is insoluble in water, but calcium hydroxide (Ca(OH)₂) is more soluble, and the solubility increases further down the group.

5. Solubility of Sulphate in Water:
The solubility of the alkaline earth metal sulphates in water generally decreases as we move down the group. For instance, barium sulfate (BaSO₄) is insoluble, while magnesium sulfate (MgSO₄) is highly soluble. Therefore, solubility of sulphates does not increase with atomic number.

Final Answer:
The property of alkaline earth metals that increases with their atomic number is (C) "Solubility of their hydroxide in water."