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I would like to know how to interpret the roots of the Peng Robinson equation of state for vapor-liquid-liquid equilibrium.

I know that if two phases are present, the largest root is taken as the compressibility factor of the vapor and the smallest root is taken as the compressibility factor of the liquid.

What does the smallest root represent if two liquid phases are present (in addition to the vapor)? Is it meaningful?

And how can I determine the compressibility or molar volume of each liquid phase?

Ultimately my goal is to be able to calculate the fugacity coefficients for components in the vapor phase, liquid phase 1, and liquid phase 2 in order to calculate the K factors for use in a multicomponent equilibrium algorithm - but I believe that I need the molar volume of all three phases in order to do this.

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Not anywhere near my area of expertise, but Wikipedia says of all cubic-equation-based equations of state that

For Pr<1 and Tr<1, the system is in a state of vapor–liquid equilibrium. The reduced cubic equation of state yields in that case 3 solutions. The largest and the lowest solution are the gas and liquid reduced volume.

I'm not clear on how or what single material can demonstrate two different liquid phases(other than superfluid). Are you thinking of immiscible liquid phases? It may simply be that P-R cannot describe a system that complicated.

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