Final answer:
Fluorine is the halogen that undergoes radical halogenation with ethane most rapidly.
Step-by-step explanation:
In halogenation reactions, the reactivity of the halogens decreases from top to bottom in Group 17. This means that fluorine is the most reactive halogen, while iodine is the least reactive. Therefore, the halogen that undergoes radical halogenation with ethane most rapidly is fluorine.