Prove that for an ideal gas, widehat$(U)$ and widehat$(H)$ are related as widehat$(H)=U+RT,$ where $R$ is the gas constant.

I

Then: $($a$)$ Taking as given that the specific internal energy of an ideal gas is independent of the gas pressure, justify the claim that $widehat(H)$ for a process in which an ideal gas goes from $(T_1,P_1)$ to $(T_2,P_2)$ equals $widehat(H)$ for the same gas going from $T_1$ to $T_2$ at a constant pressure of $P_1.$

$($b$)$ Calculate $widehat(H)($cal$)$ for a process in which the temperature of $2\mathrm{.}5$mol of an ideal gas is raised by $50C,$ resulting in a specific internal energy change $widehat(H)=3500ca\frac{l}{m}ol.$