Reconsidering Problem $3$ in Homework #$1.$ An monatomic ideal gas at $300K$ has a volume of $15$ liters at a pressure of $15$atm. The total amount of gas in the system is $9\mathrm{.}14$ mole.

Calculate the a$-$c when the gas undergoes following three separate paths:

i$.$ A reversible isothermal expansion to a pressure of $10$atm $($to state $2$: $P_2=10$atm, $V_2=22\mathrm{.}5$ liters, $T_2=300K).$

ii$.$ A reversible adiabatic expansion to a pressure of $10$ atm $($to state $3$: $P_3=10$ atm, $V_3=19\mathrm{.}13$ liters, $T_3=255K).$

iii. An irreversible adiabatic free expansion to final volume of $22\mathrm{.}5$ liters during which it performs $0J$ of work. $($to state $2$: $P_2=10$atm, $V_2=22\mathrm{.}5$ liters, $T_2=300K).$

a$.$ The entropy change of the system $S_{sys}.$

b$.$ The entropy change of its surroundings $S_{\text{s}\text{u}\text{r}\text{r}\text{o}\text{u}\text{n}\text{d}\text{i}\text{n}\text{g}}.$

c$.$ The total entropy change of the universe $S_{\text{t}\text{o}\text{t}\text{a}\text{l}}.$

$R$ is gas constant: $R=0\mathrm{.}082057\frac{\text{l}\text{i}\text{t}\text{e}\text{r}*\text{a}\text{t}\text{m}}{K*mol}=8\mathrm{.}3144\frac{J}{K*mol}.$ Integration: ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{x}dx=lnx+C$;