A solution reacts with crushed egg – shells to give a gas that turns lime water milky?  The solution contains

Crushed Egg shells are made up of$"{\text{CaCO}}_{\text{3}}"$ . Any carbonate react with acid to form salt, ${\mathrm{CO}}_{\mathrm{2}}\&{\mathrm{H}}_{\mathrm{2}}\mathrm{O}$${\text{CO}}_2$ .  turns lime water to milky white. 
$\begin{array}{l}{\mathrm{CaCO}}_3+2\mathrm{HCl}\to {\mathrm{CaCl}}_2+{\mathrm{CO}}_2\uparrow +{\mathrm{H}}_2\mathrm{O}\\ \mathrm{Ca}(\mathrm{OH}{)}_2+{\mathrm{CO}}_2\to {\mathrm{CaCO}}_3\downarrow +{\mathrm{H}}_2\mathrm{O}\end{array}$