What motivates Bilbo to give the Arkenstone to Bard and the Elvenking? use text evidence.

Bilbo wishes the elves and dwarves to make peace..... the Arkenstone is something to barter with.

"This is the Arkenstone of Thrain," said Bilbo, "the Heart of the Mountain; and it is also the heart of Thorin. He values it above a river of gold. I give it to you. It will aid you in your bargaining." Then Bilbo, not without a shudder, not without a glance of longing, handed the marvellous stone to Bard, and he held it in his hand, as though dazed.