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# The Law of Corresponding States for Hydrocarbons: the Two-Phase Region

Consider the following six chemical species: ethane, propane, -butane, -butane, -pentane, and -hexane. For each compound, this Demonstration uses arc-length continuation to plot (1) the compressibility factor versus reduced temperature for user-set values of the reduced pressure , or (2) the compressibility factor versus reduced pressure for user-set values of the reduced temperature . You can choose between two equations of state for EOS; namely the Soave–Redlich–Kwong EOS or the Peng–Robinson EOS (see the Details section).
The magenta curve, obtained for versus , is the spinodal line or locus of the turning points where . These points are readily obtained using the Mathematica built-in command WhenEvent. The spinodal curve envelops the regions where supercooling and superheating of a liquid can be achieved (i.e. metastable and unstable regions). The checkbox lets you zoom and clearly observe the black binodal curve. Also shown (1) are one of the two turning points (green) and (2) two blue dots corresponding to the saturation pressure at the user-set value of the reduced temperature, or . The binodal curve represents the curve where vapor and liquid coexist and the equilibrium state is stable.
The versus item of the second dropdown menu gives the limits of coexistence of vapor and liquid for a given value of the reduced pressure . These limits are indicated by the green dots. The dotted blue curve shows the limits of metastability (superheating and supercooling of a liquid).

### DETAILS

In the Soave–Redlich–Kwong (SRK) equation of state, the compressibility factor occurs as a solution of the cubic equation
,
where and , with , , , and .
In the Peng–Robinson (PR) equation of state, the compressibility factor occurs as a solution of the cubic equation
,
where and , with , , , and .
In the equations given above, is the acentric factor, and are the critical temperature and pressure, and is the reduced pressure.
Reference
[1] W.-G. Dong and J. H. Lienhard, "Corresponding States Correlation of Saturated and Metastable Properties," The Canadian Journal of Chemical Engineering, 64(1), 1986 pp. 158–161. doi:10.1002/cjce.5450640123.

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