Cubic equations of state

The Van der Waals equation of state may be written:

${\displaystyle \left(p+{\frac {a}{V_{m}^{2}}}\right)\left(V_{m}-b\right)=RT}$

where ${\displaystyle V_{m}}$ is molar volume. The substance-specific constants ${\displaystyle a}$ and ${\displaystyle b}$ can be calculated from the critical properties ${\displaystyle p_{c}}$, ${\displaystyle T_{c}}$, and ${\displaystyle V_{c}}$ (noting that ${\displaystyle V_{c}}$ is the molar volume at the critical point) as:

${\displaystyle a=3p_{c}V_{c}^{2}}$

${\displaystyle b={\frac {V_{c}}{3}}.}$

Also written as

${\displaystyle a={\frac {27(RT_{c})^{2}}{64p_{c}}}}$

${\displaystyle b={\frac {RT_{c}}{8p_{c}}}.}$

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 landmark equation ${\displaystyle a}$ is called the attraction parameter and ${\displaystyle b}$ the repulsion parameter or the effective molecular volume. 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 is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.

The van der Waals equation may be considered as the ideal gas law, "improved" due to two independent reasons:

1. Molecules are thought as particles with volume, not material points. Thus ${\displaystyle V_{m}}$ cannot be too little, less than some constant. So we get (${\displaystyle V_{m}-b}$) instead of ${\displaystyle V_{m}}$.
2. While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal gas law), so we write (${\displaystyle p+}$ something) instead of ${\displaystyle p}$. To evaluate this ‘something’, let's examine an additional force acting on an element of gas surface. While the force acting on each surface molecule is ~${\displaystyle \rho }$, the force acting on the whole element is ~${\displaystyle \rho ^{2}}$~${\displaystyle {\frac {1}{V_{m}^{2}}}}$.

With the reduced state variables, i.e. ${\displaystyle V_{r}=V_{m}/V_{c}}$, ${\displaystyle P_{r}=P/P_{c}}$ and ${\displaystyle T_{r}=T/T_{c}}$, the reduced form of the Van der Waals equation can be formulated:

${\displaystyle \left(P_{r}+{\frac {3}{V_{r}^{2}}}\right)\left(3V_{r}-1\right)=8T_{r}}$

The benefit of this form is that for given ${\displaystyle T_{r}}$ and ${\displaystyle P_{r}}$, the reduced volume of the liquid and gas can be calculated directly using Cardano's method for the reduced cubic form:

${\displaystyle V_{r}^{3}-\left({\frac {1}{3}}+{\frac {8T_{r}}{3P_{r}}}\right)V_{r}^{2}+{\frac {3V_{r}}{P_{r}}}-{\frac {1}{P_{r}}}=0}$

For ${\displaystyle P_{r}<1}$ and ${\displaystyle T_{r}<1}$, 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.

${\displaystyle {\begin{aligned}p&={\frac {R\,T}{V_{m}-b}}-{\frac {a}{{\sqrt {T}}\,V_{m}\left(V_{m}+b\right)}}\\[3pt]a&\approx 0.42748{\frac {R^{2}\,T_{c}^{\frac {5}{2}}}{p_{c}}}\\[3pt]b&\approx 0.08664{\frac {R\,T_{c}}{p_{c}}}\end{aligned}}}$

Introduced in 1949, the Redlich-Kwong equation of state was a considerable improvement over other equations of the time. It is still of interest primarily due to its relatively simple form. While superior to the van der Waals equation of state, 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 Redlich-Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the critical pressure (reduced pressure) is less than about one-half of the ratio of the temperature to the critical temperature (reduced temperature):

${\displaystyle {\frac {p}{p_{c}}}<{\frac {T}{2T_{c}}}.}$

${\displaystyle p={\frac {R\,T}{V_{m}-b}}-{\frac {a\alpha }{V_{m}\left(V_{m}+b\right)}}}$

${\displaystyle a={\frac {0.42747\,R^{2}T_{c}^{2}}{P_{c}}}}$

${\displaystyle b={\frac {0.08664\,RT_{c}}{P_{c}}}}$

${\displaystyle \alpha =\left(1+\left(0.48508+1.55171\,\omega -0.15613\,\omega ^{2}\right)\left(1-T_{r}^{0.5}\right)\right)^{2}}$

${\displaystyle T_{r}={\frac {T}{T_{c}}}}$

Where ω is the acentric factor for the species.

This formulation for ${\displaystyle \alpha }$ is due to Graboski and Daubert. The original formulation from Soave is:

${\displaystyle \alpha =\left(1+\left(0.480+1.574\,\omega -0.176\,\omega ^{2}\right)\left(1-T_{r}^{0.5}\right)\right)^{2}}$

for hydrogen:

${\displaystyle \alpha =1.202\exp \left(-0.30288\,T_{r}\right).}$

We can also write it in the polynomial form, with:

${\displaystyle A={\frac {a\alpha P}{R^{2}T^{2}}}}$

${\displaystyle B={\frac {bP}{RT}}}$

then we have:

${\displaystyle 0=Z^{3}-Z^{2}+Z\left(A-B-B^{2}\right)-AB}$

where ${\displaystyle R}$ is the universal gas constant and Z=PV/(RT) is the compressibility factor.

In 1972 G. Soave[3] replaced the 1/√T term of the Redlich-Kwong equation with a function α(T,ω) involving the temperature and the acentric factor (the resulting equation is also known as the Soave-Redlich-Kwong equation of state; SRK EOS). 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 ${\displaystyle T_{c}}$ is now to the second power.

The SRK EOS may be written as

${\displaystyle p={\frac {R\,T}{V_{m,{\text{SRK}}}-b}}-{\frac {a}{V_{m,{\text{SRK}}}\left(V_{m,{\text{SRK}}}+b\right)}}}$

where

${\displaystyle {\begin{aligned}a&=a_{c}\,\alpha \\a_{c}&\approx 0.42747{\frac {R^{2}\,T_{c}^{2}}{P_{c}}}\\b&\approx 0.08664{\frac {R\,T_{c}}{P_{c}}}\end{aligned}}}$

where ${\displaystyle \alpha }$ 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 (1982)[4] proposed a simple correction for this by introducing a volume translation

${\displaystyle V_{m,{\text{SRK}}}=V_{m}+c}$

where ${\displaystyle c}$ 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 ${\displaystyle V_{m,{\text{SRK}}}}$ is translated,[5] [6] and the EOS becomes

${\displaystyle p={\frac {R\,T}{V_{m}+c-b}}-{\frac {a}{\left(V_{m}+c\right)\left(V_{m}+c+b\right)}}}$

In the second version both ${\displaystyle V_{m,{\text{SRK}}}}$ and ${\displaystyle b_{\text{SRK}}}$ are translated, or the translation of ${\displaystyle V_{m,{\text{SRK}}}}$ is followed by a renaming of the composite parameter b − c.[7] This gives

${\displaystyle {\begin{aligned}b_{\text{SRK}}&=b+c\quad {\text{or}}\quad b-c\curvearrowright b\\p&={\frac {R\,T}{V_{m}-b}}-{\frac {a}{\left(V_{m}+c\right)\left(V_{m}+2c+b\right)}}\end{aligned}}}$

The c-parameter of a fluid mixture is calculated by

${\displaystyle c=\sum _{i=1}^{n}z_{i}c_{i}}$

The c-parameter of the individual fluid components in a petroleum gas and oil can be estimated by the correlation

${\displaystyle c_{i}\approx 0.40768\ {\frac {RT_{ci}}{P_{ci}}}\left(0.29441-Z_{{\text{RA}},i}\right)}$

where the Rackett compressibility factor ${\displaystyle Z_{{\text{RA}},i}}$ can be estimated by

${\displaystyle Z_{{\text{RA}},i}\approx 0.29056-0.08775\ \omega _{i}}$

A nice feature with the volume translation method of Peneloux et al. (1982) is that it does not affect the vapor-liquid equilibrium calculations.[8] 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.

${\displaystyle {\begin{aligned}p&={\frac {R\,T}{V_{m}-b}}-{\frac {a\,\alpha }{V_{m}^{2}+2bV_{m}-b^{2}}}\\[3pt]a&\approx 0.45724{\frac {R^{2}\,T_{c}^{2}}{p_{c}}}\\[3pt]b&\approx 0.07780{\frac {R\,T_{c}}{p_{c}}}\\[3pt]\alpha &=\left(1+\kappa \left(1-T_{r}^{\frac {1}{2}}\right)\right)^{2}\\[3pt]\kappa &\approx 0.37464+1.54226\,\omega -0.26992\,\omega ^{2}\\[3pt]T_{r}&={\frac {T}{T_{c}}}\end{aligned}}}$

In polynomial form:

${\displaystyle A={\frac {\alpha ap}{R^{2}\,T^{2}}}}$

${\displaystyle B={\frac {bp}{RT}}}$

${\displaystyle Z^{3}-(1-B)Z^{2}+\left(A-2B-3B^{2}\right)Z-\left(AB-B^{2}-B^{3}\right)=0}$

where ${\displaystyle \omega }$ is the acentric factor of the species, ${\displaystyle R}$ is the universal gas constant and ${\displaystyle Z=PV/nRT}$ is compressibility factor.

The Peng–Robinson equation of state (PR EOS) was developed in 1976 at The University of Alberta by Ding-Yu Peng and Donald Robinson in order to satisfy the following goals:[9]

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

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.[10] The departure functions of the Peng–Robinson equation are given on a separate article.

The analytic values of its characteristic constants are:

${\displaystyle Z_{c}={\frac {1}{32}}\left(11-2{\sqrt {7}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\frac {13}{7{\sqrt {7}}}}\right)\right)\right)\approx 0.307401}$

${\displaystyle b'={\frac {b}{V_{m,c}}}={\frac {1}{3}}\left({\sqrt {8}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\sqrt {8}}\right)\right)-1\right)\approx 0.253077\approx {\frac {0.07780}{Z_{c}}}}$

${\displaystyle {\frac {P_{c}V_{m,c}^{2}}{a\,b'}}={\frac {3}{8}}\left(1+\cosh \left({\frac {1}{3}}\operatorname {arcosh} (3)\right)\right)\approx 0.816619\approx {\frac {Z_{c}^{2}}{0.45724\,b'}}}$

Babalola [13] modified the Peng–Robinson Equation of state as:

${\displaystyle P=\left({\frac {RT}{v-b}}\right)-\left[{\frac {(a_{1}P+a_{2})\alpha }{v(v+b)+b(v-b)}}\right]}$

The attractive force parameter ‘a’, which was considered to be a constant with respect to pressure in Peng–Robinson EOS. The modification, in which parameter ‘a’ was treated as a variable with respect to pressure for multicomponent multi-phase high density reservoir systems was to improve accuracy in the prediction of properties of complex reservoir fluids for PVT modeling. The variation was represented with a linear equation where a₁ and a₂ represent the slope and the intercept respectively of the straight line obtained when values of parameter ‘a’ are plotted against pressure.

This modification increases the accuracy of Peng–Robinson equation of state for heavier fluids particularly at pressure ranges (>30MPa) and eliminates the need for tuning the original Peng-Robinson equation of state. Values for a

The Elliott, Suresh, and Donohue (ESD) equation of state was proposed in 1990.[14] The equation seeks to correct a shortcoming in the Peng–Robinson EOS in that there was an inaccuracy in the van der Waals repulsive term. The EOS accounts for the effect of the shape of a non-polar molecule and can be extended to polymers with the addition of an extra term (not shown). The EOS itself was developed through modeling computer simulations and should capture the essential physics of the size, shape, and hydrogen bonding.

${\displaystyle {\frac {pV_{m}}{RT}}=Z=1+Z^{\rm {rep}}+Z^{\rm {att}}}$

where:

${\displaystyle Z^{\rm {rep}}={\frac {4c\eta }{1-1.9\eta }}}$

${\displaystyle Z^{\rm {att}}=-{\frac {z_{m}q\eta Y}{1+k_{1}\eta Y}}}$

and

${\displaystyle c}$ is a "shape factor", with ${\displaystyle c=1}$ for spherical molecules

For non-spherical molecules, the following relation is suggested

${\displaystyle c=1+3.535\omega +0.533\omega ^{2}}$ where ${\displaystyle \omega }$ is the acentric factor.

The reduced number density ${\displaystyle \eta }$ is defined as ${\displaystyle \eta ={\frac {v^{*}n}{V}}}$

where

${\displaystyle v^{*}}$ is the characteristic size parameter

${\displaystyle n}$ is the number of molecules

${\displaystyle V}$ is the volume of the container

The characteristic size parameter is related to the shape parameter ${\displaystyle c}$ through

${\displaystyle v^{*}={\frac {kT_{c}}{P_{c}}}\Phi }$

where

${\displaystyle \Phi ={\frac {0.0312+0.087(c-1)+0.008(c-1)^{2}}{1.000+2.455(c-1)+0.732(c-1)^{2}}}}$ and ${\displaystyle k}$ is Boltzmann's constant.

Noting the relationships between Boltzmann's constant and the Universal gas constant, and observing that the number of molecules can be expressed in terms of Avogadro's number and the molar mass, the reduced number density ${\displaystyle \eta }$ can be expressed in terms of the molar volume as

${\displaystyle \eta ={\frac {RT_{c}}{P_{c}}}\Phi {\frac {1}{V_{m}}}.}$

The shape parameter ${\displaystyle q}$ appearing in the Attraction term and the term ${\displaystyle Y}$ are given by

${\displaystyle q=1+k_{3}(c-1)}$ (and is hence also equal to 1 for spherical molecules).

${\displaystyle Y=\exp \left({\frac {\epsilon }{kT}}\right)-k_{2}}$

where ${\displaystyle \epsilon }$ is the depth of the square-well potential and is given by

${\displaystyle {\frac {\epsilon }{k}}={\frac {1.000+0.945(c-1)+0.134(c-1)^{2}}{1.023+2.225(c-1)+0.478(c-1)^{2}}}}$

${\displaystyle z_{m}}$, ${\displaystyle k_{1}}$, ${\displaystyle k_{2}}$ and ${\displaystyle k_{3}}$ are constants in the equation of state:

${\displaystyle z_{m}=9.49}$ for spherical molecules (c=1)

${\displaystyle k_{1}=1.7745}$ for spherical molecules (c=1)

${\displaystyle k_{2}=1.0617}$ for spherical molecules (c=1)

${\displaystyle k_{3}=1.90476.}$

The model can be extended to associating components and mixtures of nonassociating components. Details are in the paper by J.R. Elliott, Jr. et al. (1990).[14]

The Cubic-Plus-Association (CPA) equation of state combines the Soave-Redlich-Kwong equation with the association term from SAFT[15][16] based on Chapman's extensions and simplifications of a theory of associating molecules due to Michael Wertheim.[17] The development of the equation began in 1995 as a research project that was funded by Shell, and in 1996 an article was published which presented the CPA equation of state.[17][18]

${\displaystyle P={\frac {RT}{(V-b)}}-{\frac {a}{V(V+b)}}+{\frac {RT}{V}}\rho \sum _{A}\left[{\frac {1}{X^{A}}}-{\frac {1}{2}}\right]{\frac {\partial X^{A}}{\partial \rho }}}$

In the association term ${\displaystyle X^{A}}$ is the mole fraction of molecules not bonded at site A.