At 300K, the vapor pressure of propylene is higher than the system pressure of 10 bar. The ideal gas volume can be used as starting value for the root function:
Similar to the Vieta's formula for cubic equations, Cardano'f formula allows to solve cubic equations analytically (see Appendix C, A10).
In the first step the equation should be braucht into the normal form:
The polyroots-Function allows to calculate all roots of a polynomial via the LaGuerre or the companion matrix method:
All three methods naturally lead to the same results. The highest calculated volume is the vapor volume. As propylene is below its vapor prealculated asssure, the liquid solution is not stable. The medium value solution is always unstable. The molar density can be calculated as