Methane gas at 550K, 5 bar is expanded adiabatically and reversibly to 1 bar. Find the final temperature of the gas.

I have made little progress with this...

I've calculated the initial volume, and that is all so far. Obviously to find the work usually I'd integrate pressure with respect to volume, after substituting PVγ=const but I can't do this because the final volume isn't available to me...

Help would be appreciated!