a$)$ Write the total differential for entropy, $S$ as a function of temperature,T and

pressure, P$.$ Given that,

$(\frac{\text{d}\text{e}\text{l}\text{S}}{\text{d}\text{e}\text{l}\text{T}}{)}_P=\frac{C_P}{T}=\frac{5R}{2T}$ and $(\frac{\text{d}\text{e}\text{l}\text{S}}{\text{d}\text{e}\text{l}\text{P}}{)}_T=-(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P=-\frac{R}{P}$ for $1$ mole of ideal gas.

b$)$ Calculate work done and heat absorbed by the ideal gas if $P_1=1$atm,$T_1=300K,$

$P_2=1\mathrm{.}5$atm,$T_2=320K.$

c$)$ Calculate $U$ for the change from state $A$ to $B.$

Note that work, w$=-$PdV$,$ heat, $q_{\text{r}\text{e}\text{v}}=TdS,dU=w+q$