Consider a real gas that is known to follow the following equation of state:

$v=\frac{RT}{P}+a{P}^{1\mathrm{.}5}$

where, $a=0\mathrm{.}02\frac{L}{m}ol-$ bar ${?}^{1\mathrm{.}5},P$ is in bars, $T$ is in Kelvin, and $v$ is in $\frac{L}{m}ol.R$ is the

universal gas constant. The ideal gas heat capacity of the gas is $40\frac{J}{m}ol.$

One mole of the gas is compressed isothermally at $400K$ from initial pressure of $1$ bar

to a final pressure of $5$ bars

$($a$)$ Determine the enthalpy change for the gas from the initial to the final state.

$($b$)$ Determine the internal energy change for the gas from the initial to the final state.

$($c$)$ Determine the work done for compression.

$($d$)$ Determine the heat transferred for the compression process.

$($e$)$ Refer to part $($d$),$ indicate if heat is added to or removed from the system.

Please solve it clearly with steps.