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Compare the solubility and thermal stability of alkali metals carbonates with those of the alkaline earth metals nitrates.

Answer
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Hint: We need to know what are the alkali metals and alkaline earth metals and study the solubility and thermal stability of their carbonates and nitrates accordingly. Alkali metals are the six elements which belong to the group 1 of the periodic table and are found in the leftmost column. The alkaline earth metals are the six elements which belong to group 2 of the periodic table.

Complete answer:
Alkali Metals: We have to remember that the alkali metals consist of the following elements: Lithium (Li), Sodium (Na), Potassium (K), Rubidium (Rb), Cesium (Cs) and Francium (Fr).
Carbonates of alkali metals: We need to know that the carbonates of the alkali metals are \[L{i_2}C{O_3}\] , \[N{a_2}C{O_3}\] , \[{K_2}C{O_3}\] , \[R{b_2}C{O_3}\] , \[C{s_2}C{O_3}\] and \[F{r_2}C{O_3}\].
Solubility of the carbonates of alkali metals: We must have to note that all the carbonates of alkali metals are soluble in water except for \[L{i_2}C{O_3}\]. However, their solubilities increases as we move down the group, i.e. the order of their solubilities are \[N{a_2}C{O_3}\]<\[{K_2}C{O_3}\]<\[R{b_2}C{O_3}\]<\[C{s_2}C{O_3}\]<\[F{r_2}C{O_3}\].
Thermal stability of carbonates of alkali metals: We need to know that the carbonates of alkali metals are exceptionally stable towards heat, that is they do not decompose on application of heat. This however does not stand true for \[L{i_2}C{O_3}\] as it decomposes to its oxide on application of heat. The reaction is as follows:
\[L{i_2}C{O_3}\xrightarrow{\Delta }L{i_2}O + C{O_2}\]
Nitrates of alkali metals: We need to remember that the nitrates of the alkali metals are \[LiN{O_3}\] , \[NaN{O_3}\] ,\[KN{O_3}\] ,\[RbN{O_3}\] , \[CsN{O_3}\] and \[FrN{O_3}\].
Solubility of the nitrates of alkali metals: We have to know that all the nitrates of alkali metals are soluble in water and the solubility increases as we move down the group. The order of their solubilities are: \[LiN{O_3}\]<\[NaN{O_3}\]<\[KN{O_3}\]<\[RbN{O_3}\]<\[CsN{O_3}\]<\[FrN{O_3}\].
Thermal stability of nitrates of alkali metals: Nitrates of alkali metals decompose on application of heat to give nitrites.
For example: \[2KN{O_3}\xrightarrow{\Delta }2KN{O_2} + {O_2}\]
There is an exception for \[LiN{O_3}\] as it decomposes to give oxides instead of nitrites: \[2LiN{O_3}\xrightarrow{\Delta }L{i_2}O + 2N{O_2} + {O_2}\]
Alkaline earth metals: The alkaline earth metals consist of the following elements: Beryllium (Be), Magnesium (Mg), Calcium (Ca), Strontium (Sr), Barium (Ba), and Radium (Ra).
Carbonates of alkaline earth metals: The carbonates of the alkaline earth metals are\[BeC{O_3}\] , \[MgC{O_3}\] , \[CaC{O_3}\] , \[SrC{O_3}\] , \[BaC{O_3}\] , and \[RaC{O_3}\].
Solubility of the carbonates of alkaline earth metals: All the carbonates of alkaline earth metals are insoluble in water.
Thermal stability of carbonates of alkaline earth metals: All the carbonates of alkaline earth metals decompose on heating to form oxide and carbon dioxide.
For example: \[MgC{O_3}\xrightarrow{\Delta }MgO + C{O_2}\].
Nitrates of alkaline earth metals: The nitrates of the alkaline earth metals are \[BeN{O_3}\] , \[MgN{O_3}\] , \[CaN{O_3}\] , \[SrN{O_3}\] , \[BaN{O_3}\] , and \[RaN{O_3}\].
Solubility of the nitrates of alkaline earth metals: Nitrates of alkaline earth metals are soluble in water and the solubility increases as we move down the group. Therefore, the order of their solubilities are: \[BeN{O_3}\]<\[MgN{O_3}\]<\[CaN{O_3}\]<\[SrN{O_3}\]<\[BaN{O_3}\]<\[RaN{O_3}\].
Thermal stability of nitrates of alkaline earth metals: The nitrates of alkaline earth metals decompose to give oxides on application of heat. 
\[2Ca(N{O_3})\xrightarrow{\Delta }2CaO + 4N{O_2} + {O_2}\]
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Note: We must be noted that the strange behavior of Li as an alkali metal is due to the small ionic size of lithium hence the stronger electrostatic attraction between the cation and ion makes it insoluble in water. Alkaline earth metals that have larger cations will be more soluble because of the decreasing strength of the electrostatic force of attraction.