Cubic equations of state

Cubic equations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and which can be rewritten as a cubic function of the molar volume.
Equations of state are generally applied in the fields of physical chemistry and chemical engineering, particularly in the modeling of vapor–liquid equilibrium and chemical engineering process design.

Van der Waals equation of state

The van der Waals equation of state may be written as
where is the absolute temperature, is the pressure, is the molar volume and is the universal gas constant. Note that, where is the volume, and, where is the number of moles, is the number of particles, and is the Avogadro constant. These definitions apply to all equations of state below as well.
Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this equation, usually is called the attraction parameter and the repulsion parameter. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data for vapor-liquid equilibria is limited. The van der Waals equation is commonly referenced in textbooks and papers for historical and other reasons, but since its development other equations of only slightly greater complexity have been since developed, many of which are far more accurate.
The van der Waals equation may be considered as an ideal gas law which has been "improved" by the inclusion of two non-ideal contributions to the equation. Consider the van der Waals equation in the form
as compared to the ideal gas equation
The form of the van der Waals equation can be motivated as follows:

Molecules are thought of as particles which occupy a finite volume. Thus the physical volume is not accessible to all molecules at any given moment, raising the pressure slightly compared to what would be expected for point particles. Thus, an "effective" molar volume, is used instead of in the first term.
While ideal gas molecules do not interact, real molecules will exhibit attractive van der Waals forces if they are sufficiently close together. The attractive forces, which are proportional to the density, tend to retard the collisions that molecules have with the container walls and lower the pressure. The number of collisions that are so affected is also proportional to the density. Thus, the pressure is lowered by an amount proportional to, or inversely proportional to the squared molar volume.

The substance-specific constants and can be calculated from the critical properties and as:
Expressions for written as functions of may also be obtained and are often used to parameterize the equation because the critical temperature and pressure are readily accessible to experiment. They are
With the reduced state variables, i.e., and, the reduced form of the van der Waals equation can be formulated:
The benefit of this form is that for given and, the reduced volume of the liquid and gas can be calculated directly using Cardano's method for the reduced cubic form:
For and, the system is in a state of vapor–liquid equilibrium. In that situation, the reduced cubic equation of state yields 3 solutions. The largest and the lowest solution are the gas and liquid reduced volume. In this situation, the Maxwell construction is sometimes used to model the pressure as a function of molar volume.
The compressibility factor is often used to characterize non-ideal behavior. For the van der Waals equation in reduced form, this becomes
At the critical point,.

Redlich–Kwong equation of state

Introduced in 1949, the Redlich–Kwong equation of state was considered to be a notable improvement to the van der Waals equation. It is still of interest primarily due to its relatively simple form.
While superior to the van der Waals equation in some respects, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating vapor–liquid equilibria. However, it can be used in conjunction with separate liquid-phase correlations for this purpose. The equation is given below, as are relationships between its parameters and the critical constants:
Another, equivalent form of the Redlich–Kwong equation is the expression of the model's compressibility factor:
The Redlich–Kwong equation is adequate for calculation of gas phase properties when the reduced pressure is less than about one-half of the reduced temperature,
The Redlich–Kwong equation is consistent with the theorem of corresponding states. When the equation expressed in reduced form, an identical equation is obtained for all gases:
where is:
In addition, the compressibility factor at the critical point is the same for every substance:
This is an improvement over the van der Waals equation prediction of the critical compressibility factor, which is . Typical experimental values are , , and .

Soave modification of Redlich–Kwong

A modified form of the Redlich–Kwong equation was proposed by Soave. It takes the form
where ω is the acentric factor for the species.
The formulation for above is actually due to Graboski and Daubert. The original formulation from Soave is:
for hydrogen:
By substituting the variables in the reduced form and the compressibility factor at critical point
we obtain
thus leading to
Thus, the Soave–Redlich–Kwong equation in reduced form only depends on ω and of the substance, contrary to both the VdW and RK equation which are consistent with the theorem of corresponding states and the reduced form is one for all substances:
We can also write it in the polynomial form, with:
In terms of the compressibility factor, we have:
This equation may have up to three roots. The maximal root of the cubic equation generally corresponds to a vapor state, while the minimal root is for a liquid state. This should be kept in mind when using cubic equations in calculations, e.g., of vapor-liquid equilibrium.
In 1972 G. Soave replaced the term of the Redlich–Kwong equation with a function α involving the temperature and the acentric factor. The α function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.
Note especially that this replacement changes the definition of a slightly, as the is now to the second power.

Volume translation of Peneloux et al. (1982)

The SRK EOS may be written as
where
where and other parts of the SRK EOS is defined in the SRK EOS section.
A downside of the SRK EOS, and other cubic EOS, is that the liquid molar volume is significantly less accurate than the gas molar volume. Peneloux et alios proposed a simple correction for this by introducing a volume translation
where is an additional fluid component parameter that translates the molar volume slightly. On the liquid branch of the EOS, a small change in molar volume corresponds to a large change in pressure. On the gas branch of the EOS, a small change in molar volume corresponds to a much smaller change in pressure than for the liquid branch. Thus, the perturbation of the molar gas volume is small. Unfortunately, there are two versions that occur in science and industry.
In the first version only is translated, and the EOS becomes
In the second version both and are translated, or the translation of is followed by a renaming of the composite parameter. This gives
The c-parameter of a fluid mixture is calculated by
The c-parameter of the individual fluid components in a petroleum gas and oil can be estimated by the correlation
where the Rackett compressibility factor can be estimated by
A nice feature with the volume translation method of Peneloux et al. is that it does not affect the vapor–liquid equilibrium calculations. This method of volume translation can also be applied to other cubic EOSs if the c-parameter correlation is adjusted to match the selected EOS.

Peng–Robinson equation of state

The Peng–Robinson equation of state was developed in 1976 at The University of Alberta by Ding-Yu Peng and Donald Robinson in order to satisfy the following goals:

The parameters should be expressible in terms of the critical properties and the acentric factor.
The model should provide reasonable accuracy near the critical point, particularly for calculations of the compressibility factor and liquid density.
The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature, pressure, and composition.
The equation should be applicable to all calculations of all fluid properties in natural gas processes.

The equation is given as follows:
In polynomial form:
For the most part the Peng–Robinson equation exhibits performance similar to the Soave equation, although it is generally superior in predicting the liquid densities of many materials, especially nonpolar ones. Detailed performance of the original Peng-Robinson equation has been reported for density, thermal properties, and phase equilibria. Briefly, the original form exhibits deviations in vapor pressure and phase equilibria that are roughly three times as large as the updated implementations. The departure functions of the Peng–Robinson equation are given on a separate article.
The analytic values of its characteristic constants are:

Peng–Robinson–Stryjek–Vera equations of state

PRSV1

A modification to the attraction term in the Peng–Robinson equation of state published by Stryjek and Vera in 1986 significantly improved the model's accuracy by introducing an adjustable pure component parameter and by modifying the polynomial fit of the acentric factor.
The modification is:
where is an adjustable pure component parameter. Stryjek and Vera published pure component parameters for many compounds of industrial interest in their original journal article. At reduced temperatures above 0.7, they recommend to set and simply use. For alcohols and water the value of may be used up to the critical temperature and set to zero at higher temperatures.