Step 1: Understand the composition of egg-shell.
Egg-shells are primarily composed of calcium carbonate (CaCO$_3$).
Step 2: Recall the reaction of carbonates with acids.
Carbonates react with acids to produce carbon dioxide (CO$_2$) gas, water (H$_2$O), and a salt. This is a characteristic test for the presence of carbonates.
Step 3: Analyze the given options.
(1) MgCl$_2$ (Magnesium Chloride): This is a salt and typically does not react with calcium carbonate to produce CO$_2$ gas.
(2) NaCl (Sodium Chloride): This is a salt and typically does not react with calcium carbonate to produce CO$_2$ gas.
(3) KCl (Potassium Chloride): This is a salt and typically does not react with calcium carbonate to produce CO$_2$ gas.
(4) HCl (Hydrochloric Acid): This is a strong acid. When HCl reacts with calcium carbonate, it produces calcium chloride (a salt), water, and carbon dioxide gas.
Step 4: Write the chemical reaction.
CaCO$_3$(s) + 2HCl(aq) $\rightarrow$ CaCl$_2$(aq) + H$_2$O(l) + CO$_2$(g)
Step 5: Conclude the correct solution.
Since the reaction with egg-shell (calcium carbonate) produces CO$_2$ gas, the solution must contain an acid. Among the given options, HCl is an acid.
$$(4) HCl$$