In "The Hobbit", how did the band of adventurers become trapped in the mountain in chapter 14?
The dwarves are in Lonely Mountain because, first, Smaug is gone and they have reclaimed the mountain and its treasure for themselves. This was their mission at the start of the story. Second, the dwarves are in the mountain using it as a fortress against the armies who want them to share the bounty of Smaug's hoarded treasure. The men, the elves, the goblins, the wargs, etc. -all want a part of the treasure and the dwarves don't want to share. The dwarves believe that since Smaug stole the treasure from their ancestors, the treasure is theirs. Dwarves are by nature, somewhat greedy and covetous. They are also stubborn creatures. They prize the gold for its intrinsic value. Thorin Oakenshield is particularly stubborn and refuses to share the gold, so he and the dwarves are prepared to go to war over the gold and the mountain makes a perfect fortress for them.