Which one is the most stable carbonate?

The correct option is BaCO₃.

To understand why barium carbonate (${\mathrm{BaCO}}_3$) is the most stable carbonate among the given alkaline earth metal carbonates, we need to analyze the periodic trends of Group 2 elements (Be, Mg, Ca, Sr, Ba) and their relation to thermal stability.

1. Periodic Trend of Ionic Size:
As we move down Group 2 in the periodic table, the atomic and ionic radii of the metal cations increase due to the addition of extra electron shells. The order of ionic size for the given Group 2 cations is:
${\mathrm{Be}}^{2+}<{\mathrm{Mg}}^{2+}<{\mathrm{Ca}}^{2+}<{\mathrm{Ba}}^{2+}$

2. Polarizing Power and Fajan's Rules:
According to Fajan's rules, smaller cations have a higher charge density and thus a greater polarizing power. The highly polarizing small cations (like ${\mathrm{Be}}^{2+}$ or ${\mathrm{Mg}}^{2+}$) strongly distort the electron cloud of the large, polarizable carbonate anion (${\mathrm{CO}}_3^{2-}$). This polarization weakens the carbon-oxygen bonds within the carbonate ion, making the compound share covalent character and decompose easily upon heating to form the metal oxide and carbon dioxide (${\mathrm{CO}}_2$).

3. Thermal Stability of Carbonates:
Barium (${\mathrm{Ba}}^{2+}$) is the largest cation among the choices. Because of its large size, it has the lowest charge density and the weakest polarizing power. Consequently, it causes the least distortion/polarization of the carbonate anion. This preserves the stability of the carbonate group, resulting in high thermal stability.

Therefore, the thermal stability of Group 2 carbonates increases down the group in the following order:
${\mathrm{BeCO}}_3<{\mathrm{MgCO}}_3<{\mathrm{CaCO}}_3<{\mathrm{BaCO}}_3$
This makes BaCO₃ the most stable carbonate.