A cylinder with thermally insulated walls and a frictionless piston has methane at an initial temperature of $25C$ and an initial specific volume of $0\mathrm{.}02\frac{m^3}{m}ol.$ The gas in the cylinder is slowly compressed until the volume has decreased by $80\%.$ The value of $c_{p}ca{n}^c$ be assumed to be independent of temperature and equal to the first term. The PVT behavior of methane is best described by the van der Waals equation where $a=0\mathrm{.}2303Pa*m^6*mo{l}^{-2}$ and $b=4\mathrm{.}306{10}^{-5}\frac{m^3}{m}ol.$

a$.(20$ points$)$ What is the final temperature of the gas after compression?

b$.(15$ points$)$ How much work per mole is done on the gas?