Do I have to buy the game again?

Probably a really stupid question, but:


I bought a new laptop with a touchscreen display (really want to try it with the fall patch), but like most modern laptops, it doesn't have a cd/dvd drive. I only have a physical copy of Civ V. Do I have to buy a new digital copy of the game, or is it possible to download the game with Steam for free, and then enter my old product code or something like that?