[Thermo] Derivation of compressibility factor vs reduced pressure

[-Nita]

Homework Statement

derivation of compressibility factor vs. reduced pressure
I am supposed to derive the graph by solving equations

Homework Equations

Van der Waals equation of state
compressibility factor, Z = (Pv)/(RT)
reduced pressure = P/critical pressure
Z = f(T_r, P_r)

The Attempt at a Solution

I sat for 12 hours attempting to find a solution but just spent time trying to understand what I was doing instead.
Is there a way to get the graph mathematically without using any values for critical pressure or temperature?

Thank you!

 

[RTW69]

Compressibility charts are derived from experimental data from 10 gases such as propane, Nitrogen, Carbon Dioxide etc. Select a gas you have properties for such as propane. Tc propane is 370 K. Pc propane is 42.7 bar. Assume a constant temperature of let's say 555 K to generate the Tr=1.5 line. Vary pressure from 42.7 to 300 bars (Pr 1 to 6). Use propane tables to find v (specific volume) solve for Z. Plot Tr(Z,Pr)

 

[-Nita]

After solving the Van der Waals equation for the compressibility factor and
deriving the critical temperature, pressure, and volume

knowing that a horizontal inflection point occurs on the isotherm at critical point

am I supposed to solve PV³-(Pb+RT)V²+aV-ab=0
for the cubic root?