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Does it somehow depend upon the lattice energy of the compound? My textbook says that Lithium carbonate is not so stable to heat and forms more stable $\ce{Li2O}$ and $\ce{CO2}$. Could it depend on the electropositive character? Because my textbook further sates that the stability of carbonates increases down the group.

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As we move down the alkali metal group, the electropositive character increases. This causes an increase in the stability of alkali carbonates. However, lithium carbonate is not so stable to heat. This is because lithium carbonate is covalent. Lithium ion, being very small in size, polarizes a large carbonate ion, leading to the formation of more stable lithium oxide.

Therefore, lithium carbonate decomposes at a low temperature while a stable sodium carbonate decomposes at a high temperature.

The smaller the size of the ion, the higher the lattice energy and the greater the extent of the polarization. Li+ has a smaller size compared to $\ce{Na+}$, hence polarizes the $\ce{CO3-}$ ion at a greater extent compared to $\ce{Na+}$. $\ce{Li2CO3}$ therefore requires less energy to break while $\ce{Na2CO3}$ needs a higher energy making it more stable than $\ce{Li2CO3}$.

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