$(35$ pts$)2)$ Gaseous propane is compressed in a piston$-$cylinder apparatus from an initial state of $P_1=25lb\frac{f}{i}{n}^2,V_1=3\mathrm{.}3f{t}^3,$ to a state of $P_2=70lb\frac{f}{i}{n}^2,V_2=1\mathrm{.}5f{t}^3.$ The propane behaves in a polytropic manner such that $P{V}^n=$ constant.

Determine:

$(3600\frac{16f}{f{t}^2})(3\mathrm{.}3f{t}^3{)}^n=(10080\frac{16f}{f{t}^2})(1\mathrm{.}5f{t}^3{)}^n$

$($a$)$ the polytropic constant $n$

$ln(\frac{25}{70})=ln(\frac{1\mathrm{.}5}{3\mathrm{.}3}{)}^x*n\frac{ln(\frac{25}{70})}{ln(\frac{1\mathrm{.}5}{3\mathrm{.}3})}=n$

$P_1{V}_1^n=P_2{V}_2^n$

$(25\frac{1bf}{i{n}^2}\frac{144in}{1ft})(3\mathrm{.}3f{t}^3{)}^n=(70\frac{16f}{i{n}^2}\frac{144ii{n}^2}{1f{t}^2})(1\mathrm{.}5f{t}^3{)}^n$

$n=1\mathrm{.}306$

$($b$)$ the work done by the propane during the process $($ft$-$lbf$)$

$P_2{V}_2-P_1{V}_1,(10080\frac{16t}{f{t}^2})(1\mathrm{.}5f{t}^3)-(3600\frac{16f}{f{t}^2})(3\mathrm{.}3f{t}^3)$