$(30$ Points$)$ In the following table, cases are shown and based on the checked conditions for these cases

give the initial assumptions for the conditions imposed. No nonexpansion work is allowed in these

scenarios. Now derive expressions that you would use to evaluate $q,w,U,H,$ and $S$ for the

following cases seen in the photo $($sets of conditions$)$:

Start from the basic assumptions for Case $1$ given below and show the reasons for each change. In some

cases you may not be able to calculate all quantities, indicate when there is not enough information to go to a

form with measurable quantities.

IMPORTANT: INFORMATION FOR CASE $1$ IS GIVEN IN THE SECOND PHOTO AND PLEASE SHOW ALL WORK IN DERIVING THE EQUATIONS FOR ALL OTHER CASES!!! PLEASE DERIVE ALL CASES $2-7$ AND NOT JUST CASE $1!$

Here we have no material or conditions to apply and only the most general results are possible.

$w=-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{P}_{\text{e}\text{x}\text{t}}dV,U=q+w,H=U+(PV),dS=\frac{d{q}_{\text{r}\text{e}\text{v}}}{T}$

note :$C_P=C_V+TV(\frac{{}^2}{{}_T}),=\frac{1}{V}(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P,{}_T=-\frac{1}{V}(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{P}}{)}_T$

$U(T_f,V_f)-U(T_i,V_i)=U={\displaystyle \underset{T_1}{\overset{T_2}{}}}{C}_{V}dT,U=U(T)$ for all substances

$H(T_f,P_f)-H(T_i,P_i)={\displaystyle \underset{T_1}{\overset{T_2}{}}}{C}_{P}dT,$ and $H=H(T),$ liquids and solids