$($Show work and circle$/$box final answers$)$Problem $1$

From $Tds$ equations show that for a Calorically perfect Gas $($CPG$),$

a$)s_2-s_1=c_{v}ln(\frac{T_2}{T_1})+Rln(\frac{v_2}{v_1})$ and

b$)s_2-s_1=c_{p}ln(\frac{T_2}{T_1})-Rln(\frac{p_2}{p_1})$

c$)$ Moreover, for an isentropic process, show that $(\frac{p_2}{p_1})=(\frac{T_2}{T_1}{)}^{}.$ Express $""$ in term of $$ only

d$)$ Deduce that for an isentropic flow, $(\frac{{}_2}{{}_1})=(\frac{T_2}{T_1}{)}^{\frac{1}{-1}}.$ Combine this equation with your

result in Part $($c$)$ above to show that $(\frac{p_2}{p_1})=(\frac{{}_2}{{}_1}{)}^{}$ for the isentropic flow of a perfect gas.