When do halogens displace each other in solutions of their salts?

More reactive halogens displace less reactive halogens. The more reactive halogens are those at the top of the group, as their outer electron shells (negatively charged) are closer to the nucleus (positively charged), meaning that there is a greater attraction between them and the halogen is more likely to gain an extra election = an ionic bond. For example, chlorine will displace iodine: Cl2(aq) + 2KI(aq) → I2 (aq) + 2KCl(aq)