Question $5(43$ marks$)$

The figure below shows a diatomic ideal gas sample in a closed system undergoes a cycle process ABCA such that $AB$ is adiabatic and $BC$ is isochoric.

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$($a$)$ Find the followings:

$($i$)$ number of mole n of the gas sample,

$(3$ marks$)$

$($ii$)$ temperature $T_A.$

$(3$ marks$)$

$($b$)$ Find the works by the system $W_{BC}$ and $W_{CA}$ along paths BC and CA $.$

$(6$ marks$)$

$($c$)$ Find the gas pressure at state B$.$

$(6$ marks$)$

$($d$)$ Find the internal energy when the system is at A and B respectively.

$(6$ marks$)$

$($e$)$ Find the work $W_{AB}$ and the change of entropy $S_{AB}$ along the path $AB$

$(9$ marks$)$

$($f$)$ Find the total work done by the system $W_{\text{A}\text{B}\text{C}\text{A}},$ the total change of internal energy $\frac{?}{{?}_{\frac{?}{?}}{E}_{\text{i}\text{n}\text{t}\text{,}\text{A}\text{B}\text{C}\text{A}}},$ the total heat absorbed by the system $Q_{\text{A}\text{B}\text{C}\text{A}},$ and the total change of entropy $\frac{?}{{?}_{\frac{?}{?}}{S}_{\text{A}\text{B}\text{C}\text{A}}}$ along the path ABCA $.$

$(10$ marks$)$