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Which of the following statements is correct?
A. Lithium carbonate is soluble in water.
B. Carbonates of Ca and Ba are soluble in water.
C. Carbonates of \[N{{a}^{+}}\], \[{{K}^{+}}\], and \[N{{H}^{4+}}\] are soluble in water.
D. All the above

Answer
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Hint: Solubility (mostly in water) of any compound depends on the lattice energy (energy needed to change ionic solid compound into a gaseous state or can say energy needed to break bond). As greater the lattice energy so more energy we need to supply to break the bond between compounds and thus solubility decreases. But if hydration energy is high as compared to lattice energy (the energy released while hydrogen molecules form bonds with compound) then it can easily compensate for lattice energy.

Complete step by step solution:
Carbonates of metal are formed when carbonic acid reacts with metal ion but not with metal and carbonic acid is formed when carbon dioxide is made to react with water such as
\[C{{O}_{2}}+{{H}_{2}}O\to {{H}_{2}}C{{O}_{3}}(Carbonic)\]
\[{{H}_{2}}C{{O}_{3}}+C{{a}^{2+}}\to CaC{{O}_{3}}+{{H}_{2}}\\ \]
\[\begin{matrix}
   (Calcium & Carbonate) \\
\end{matrix}\]

Carbonate is the weak base or can say somewhere slightly acidic (nucleophile). Lithium metal is small in size and thus it does not tend to give a free electron to any electron deficient (less electromotive or electrophile) or can say it has a large tendency to attract an electron. Due to this lithium carbonate (carbonate give and lithium accept) bond becomes strong. Because of the strong interaction, we need to supply high energy to break the bond to make it dissolve in water (lattice energy is high).

Lithium has very high lattice energy as compared to all the elements which are of the same group (1st group) from which lithium belongs. Due to this sodium ion (\[N{{a}^{+}}\]) and potassium ion (\[{{K}^{+}}\]) has less lattice energy so they both can easily dissolve or be soluble in water. Also, \[N{{H}^{4+}}\]acts like an acid, and also carbonate is a good acid thus, the bond formed between the two will be weak and so the lattice will be less. Due to this, it is also soluble in water.

Mostly all carbonates are insoluble in water. Calcium and barium both are of group 2nd due to which the electronegativity of the can and ca is greater than lithium, sodium, and potassium. So, it tends to attract more electrons as compared to the group first element, and this forms a stable carbonate whose lattice energy will be high. Because of this calcium and barium carbonate are insoluble in water.
Thus, the correct option is C whose all ions are soluble in water.

Note: It is important to note that lattice energy increases with the increase of electronegativity (an increase from left to right and from bottom to the top of the group) or can say increase with a decrease in size of the element. Also, lattice energy increases with an increase in charge of the element.