(a) When dilute hydrochloric acid reacts with sodium carbonate, then sodium chloride, carbon dioxide and water are formed.     $$Na_2CO_3 (s) + 2HCl(aq) \rightarrow 2NaCl (aq) + CO_2 (g) + H_2O (l)$$ (b) $$CO_2$$ gas is liberated during the reaction. When carbon dioxide gas formed in the form of brisk effervescence is passed through lime water, it turns the lime water milky. If excess carbon dioxide gas is passed through the milky lime water, the solution becomes clear again. This confirms the presence of carbon dioxide gas.