Step 1: Understanding the Concept:
Brine is a saturated solution of NaCl containing impurities like CaCl\(_2\), MgCl\(_2\), etc. The process involves precipitating pure NaCl from this solution.
Step 2: Detailed Explanation:
When HCl gas is passed through the brine solution, it dissociates to give H\(^+\) and Cl\(^-\) ions. The high concentration of Cl\(^-\) ions (common ion) from HCl suppresses the dissociation of NaCl (\(NaCl (s) \rightleftharpoons Na^+ (aq) + Cl^- (aq)\)) by the common ion effect. This shifts the equilibrium backwards, leading to the precipitation of pure NaCl. The impurities, being more soluble, remain in the solution.
Step 3: Final Answer:
Hence, the purification is based on the common ion effect, option (A).