A sample containing $2$ moles of of a mixture of ideal gases $x$ and $Y,$ with a total mass of $21$ grams, initially has a volume of $0\mathrm{.}6m^3$ and a pressure of $10$kPa. This sample undergoes a cyclic process consisting of four steps:

Step $1$: Isothermal reversible expansion from State A to State B$,$ resulting in a volume increase to $1\mathrm{.}0m^3.$

Step $2$: Adiabatic expansion from State B to State C$,$ reaching a final temperature of $298K.$

Step $3$: Isothermal reversible compression from State C to State D$,$ leading to a final pressure of $6\mathrm{.}2$kPa.

Step $4$: Adiabatic compression from State D to State A$,$ resulting in a final volume of $0\mathrm{.}6m^3.$

$($a$)$ Complete the following table by filling in the values, and clearly show your calculations.

$\backslash$table$[[,$State A$,$State B$,$State C$,$State D$],[P(kPa),10,,,6\mathrm{.}2],[V(m^3),0\mathrm{.}6,1\mathrm{.}0,,],[T(K),,,298,],[n(mol),2,2,2,2]]$

$(16$ marks$)$

$($b$)$ Draw the P$-$V graph for this thermodynamic cycle.

$(6$ marks$)$

$($c$)$ Calculate the values for $q,w,U,$ and $H$ for steps $1$ and $2,$ expressed in kilojoules $(kJ).$

$(14$ marks$)$

$($d$)$ Determine the change in internal energy $(U)$ for the entire cyclic process. Provide an explanation for your answer.

$(2$ marks$)$

$($e$)$ Determine the partial pressure of gas $x$ at state $C$ in kPa, considering the molar masses of $x$ and $Y$ as $4\frac{g}{m}ol$ and $40\frac{g}{m}ol,$ respectively.

$(4$ marks$)$