Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law.
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 α function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.
can be estimated by A nice feature with the volume translation method of Peneloux et al. (1982) is that it does not affect the vapor–liquid equilibrium calculations.
[9] 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.
[11] Detailed performance of the original Peng-Robinson equation has been reported for density, thermal properties, and phase equilibria.
[12] Briefly, the original form exhibits deviations in vapor pressure and phase equilibria that are roughly three times as large as the updated implementations.
The analytic values of its characteristic constants are: A modification to the attraction term in the Peng–Robinson equation of state published by Stryjek and Vera in 1986 (PRSV) significantly improved the model's accuracy by introducing an adjustable pure component parameter and by modifying the polynomial fit of the acentric factor.
Stryjek and Vera published pure component parameters for many compounds of industrial interest in their original journal article.
While PRSV1 does offer an advantage over the Peng–Robinson model for describing thermodynamic behavior, it is still not accurate enough, in general, for phase equilibrium calculations.
[13] The highly non-linear behavior of phase-equilibrium calculation methods tends to amplify what would otherwise be acceptably small errors.
[citation needed] An extensive treatment of over 1700 compounds using the Twu method has been reported by Jaubert and coworkers.
[15] Detailed performance of the updated Peng-Robinson equation by Jaubert and coworkers has been reported for density, thermal properties, and phase equilibria.
[12] Briefly, the updated form exhibits deviations in vapor pressure and phase equilibria that are roughly a third as large as the original implementation.
[17] The variation was represented with a linear equation where a1 and a2 were the slope and the intercept respectively of the straight line obtained when values of parameter ‘a’ are plotted against pressure.
[19] The equation corrects the inaccurate van der Waals repulsive term that is also applied in the Peng–Robinson EOS.
The attractive term includes a contribution that relates to the second virial coefficient of square-well spheres, and also shares some features of the Twu temperature dependence.
[20] The EOS itself was developed through comparisons with computer simulations and should capture the essential physics of size, shape, and hydrogen bonding as inferred from straight chain molecules (like n-alkanes).
Solving the equations in Wertheim's theory can be complicated, but simplifications can make their implementation less daunting.
Technically, the ESD equation is no longer cubic when the association term is included, but no artifacts are introduced so there are only three roots in density.
The extension to efficiently treat any number of electron acceptors (acids) and donors (bases), including mixtures of self-associating, cross-associating, and non-associating compounds, has been presented here.
[24] [25] Detailed performance of the ESD equation has been reported for density, thermal properties, and phase equilibria.
[21][22] The addition of the chain term allows the model to be capable of capturing the physics of both short-chain and long-chain non-associating components ranging from alkanes to polymers.
The cubic-plus-chain (CPC) equation of state is written in terms of the reduced residual Helmholtz energy (
is the chain length, "rep" and "att" are the monomer repulsive and attractive contributions of the cubic equation of state, respectively.
The "chain" term accounts for the monomer beads bonding contribution from SAFT equation of state.
Sisco et al.[28][30] applied the CPC equation of state to model different well-defined and polymer mixtures.
Alajmi et al.[31] incorporate the short-range soft repulsion to the CPC framework to enhance vapor pressure and liquid density predictions.
They provided a database for more than 50 components from different chemical families, including n-alkanes, alkenes, branched alkanes, cycloalkanes, benzene derivatives, gases, etc.
This CPC version uses a temperature-dependent co-volume parameter based on perturbation theory to describe short-range soft repulsion between molecules.