Why did Thorin go down the tunnel in chapter 13 of The Hobbit?
Thorin wanted to go into the tunnel in order to get fresh air.
The dwarves hide in the tunnel because they are afraid that the dragon is going to find them. Bilbo is frightened and keeps reminding the dwarves that they are in danger, but Thorin is in charge and is not pleased with being cooped up.
Thorin spoke: "Let us try the door!" he said. "I must feel the wind on my face soon or die. I think
I would rather be smashed by Smaug in the open than suffocate in here!" (ch 13)
They find that they are trapped, and the only way out is through the tunnel. Thorin is brave and the leader of the bunch, so he is the first one to go down the tunnel as they all go together.
This incident demonstrates that while Thorin is brave and a good leader, he can get frustrated and act hastily. However, Thorin does not give up easily. He gets upset when he learns that they are trapped, but then he follows Bilbo. He is one of the first ones through the tunnel because he can't wait to get out, but also because he feels it is his duty as leader.