grizclk320

Hi, I have a 2004 CLK320. I put a new battery in it last fall and then it was parked for the winter. I now can jump the car to get it running but the battery is completely dead after running the car for a half hour. I am assuming the alternator is OK since it ran for the half hour but it is not charging the battery. Can a battery die if left in the car over the winter? I assumed a jump and it would be fine after a little run time???

Also, I jumped and running can I drive this car ~ 20 miles to get it home? Will it run off the alternator once running?