Problem $1.(2$ points$)$ Recall the Peng$-$Robinson equation of state

$(P+\frac{a(T)}{v(v+b)+b(v-b)})(v-b)=RT.$

Given parameters $a$ and $b$$,$ derive the critical pressure, critical molar volume, critical temperature, and critical compressibility factor for the Peng$-$Robinson fluid. Discuss whether the critical compressibility factor depends on the parameters $a$ or $b$$.$ Derive the Peng$-$Robinson equation in terms of reduced pressure, reduced molar volume, and reduced temperature.

I want you to write the derivation in details and find Vc$,$ Tc$,$ Pc$,$ and Zc for PengRobinson fluids