Basic principles of energy balances involving non$-$reactive processes

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

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 $hat(H)($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 $widehat(H)=3500ca\frac{l}{m}ol.$

If a system expands in volume by an amount $V(m^3)$ against a constant restraining pressure $P(\frac{N}{m^2}),$ a quantity $PV(J)$ of energy is transferred as expansion work from the system to its surroundings. Suppose that the following four conditions are satisfied for a closed system:

$($a$)$ the system expands against a constant pressure $($so that $\frac{?}{{?}_{\frac{?}{?}}P}=0)$;

$($b$)E_k=0$;

$($c$)E_p=0$; and

$($d$)$ the only work done by or on the system is expansion work. Prove that under these conditions, the energy balance simplifies to $Q=H.$