Cubic equations of state

The van der Waals equation of state may be written as

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

where ${\displaystyle T}$ is the absolute temperature, ${\displaystyle p}$ is the pressure, ${\displaystyle V_{\text{m}}}$ is the molar volume and ${\displaystyle R}$ is the universal gas constant. Note that ${\displaystyle V_{\text{m}}=V/n}$, where ${\displaystyle V}$ is the volume, and ${\displaystyle n=N/N_{\text{A}}}$, where ${\displaystyle n}$ is the number of moles, ${\displaystyle N}$ is the number of particles, and ${\displaystyle N_{\text{A}}}$ is the Avogadro constant. These definitions apply to all equations of state below as well.

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

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

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

Expressions for ${\displaystyle (a,b)}$ written as functions of ${\displaystyle (T_{\text{c}},p_{\text{c}})}$ may also be obtained and are often used to parameterize the equation because the critical temperature and pressure are readily accessible to experiment.[1] They are

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

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

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 textbooks 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 including two non-ideal contributions to the equation. Consider the van der Waals equation in the form

${\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a}{V_{\text{m}}^{2}}}}$

as compared to the ideal gas equation

${\displaystyle p={\frac {RT}{V_{\text{m}}}}}$

The form of the van der Waals equation can be motivated as follows:

1. 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 (${\displaystyle V_{\text{m}}-b}$), an "effective" molar volume, is used instead of ${\displaystyle V_{\text{m}}}$ in the first term.
2. 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 ${\displaystyle \rho }$, 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 ${\displaystyle \rho ^{2}}$, or inversely proportional to the squared molar volume.

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

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

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

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

For ${\displaystyle P_{\text{r}}<1}$ and ${\displaystyle T_{\text{r}}<1}$, 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 ${\displaystyle Z=PV_{\text{m}}/RT}$ is often used to characterize non-ideal behavior. For the van der Waals equation in reduced form, this becomes

${\displaystyle Z={\frac {V_{\text{r}}}{V_{\text{r}}-{\frac {1}{3}}}}-{\frac {9}{8V_{\text{r}}T_{\text{r}}}}}$

At the critical point, ${\displaystyle Z_{\text{c}}=3/8=0.375}$.

Introduced in 1949,[2] 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:

${\displaystyle {\begin{aligned}p&={\frac {R\,T}{V_{\text{m}}-b}}-{\frac {a}{{\sqrt {T}}\,V_{\text{m}}\left(V_{\text{m}}+b\right)}}\\[3pt]a&={\frac {\Omega _{a}\,R^{2}T_{\text{c}}^{\frac {5}{2}}}{p_{\text{c}}}}\approx 0.42748{\frac {R^{2}\,T_{\text{c}}^{\frac {5}{2}}}{P_{\text{c}}}}\\[3pt]b&={\frac {\Omega _{b}\,RT_{\text{c}}}{P_{\text{c}}}}\approx 0.08664{\frac {R\,T_{\text{c}}}{p_{\text{c}}}}\\[3pt]\Omega _{a}&=\left[9\left(2^{1/3}-1\right)\right]^{-1}\approx 0.42748\\[3pt]\Omega _{b}&={\frac {2^{1/3}-1}{3}}\approx 0.08664\end{aligned}}}$

Another, equivalent form of the Redlich–Kwong equation is the expression of the model's compressibility factor:

${\displaystyle Z={\frac {pV_{\text{m}}}{RT}}={\frac {V_{\text{m}}}{V_{\text{m}}-b}}-{\frac {a}{RT^{3/2}\left(V_{\text{m}}+b\right)}}}$

The Redlich–Kwong equation is adequate for calculation of gas phase properties when the reduced pressure (defined in the previous section) is less than about one-half of the ratio of the temperature to the reduced temperature,

${\displaystyle P_{\text{r}}<{\frac {T}{2T_{\text{c}}}}.}$

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:

${\displaystyle P_{\text{r}}={\frac {3T_{\text{r}}}{V_{\text{r}}-b'}}-{\frac {1}{b'{\sqrt {T_{\text{r}}}}V_{\text{r}}\left(V_{\text{r}}+b'\right)}}}$

where ${\displaystyle b'}$ is:

${\displaystyle b'=2^{1/3}-1\approx 0.25992}$

In addition, the compressibility factor at the critical point is the same for every substance:

${\displaystyle Z_{\text{c}}={\frac {p_{\text{c}}V_{\text{c}}}{RT_{\text{c}}}}=1/3\approx 0.33333}$

This is an improvement over the van der Waals equation prediction of the critical compressibility factor, which is ${\displaystyle Z_{\text{c}}=3/8=0.375}$ . Typical experimental values are ${\displaystyle Z_{\text{c}}=0.274}$ (carbon dioxide), ${\displaystyle Z_{\text{c}}=0.235}$ (water), and ${\displaystyle Z_{\text{c}}=0.29}$ (nitrogen).

A modified form of the Redlich–Kwong equation was proposed by Soave.[3] It takes the form

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

${\displaystyle a={\frac {\Omega _{a}\,R^{2}T_{\text{c}}^{2}}{P_{\text{c}}}}={\frac {0.42748\,R^{2}T_{\text{c}}^{2}}{P_{\text{c}}}}}$

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

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

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

${\displaystyle \Omega _{a}=\left[9\left(2^{1/3}-1\right)\right]^{-1}\approx 0.42748}$

${\displaystyle \Omega _{b}={\frac {2^{1/3}-1}{3}}\approx 0.08664}$

where ω is the acentric factor for the species.

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

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

for hydrogen:

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

By substituting the variables in the reduced form and the compressibility factor at critical point

${\displaystyle \{p_{\text{r}}=p/P_{\text{c}},T_{\text{r}}=T/T_{\text{c}},V_{\text{r}}=V_{\text{m}}/V_{\text{c}},Z_{\text{c}}={\frac {P_{\text{c}}V_{\text{c}}}{RT_{\text{c}}}}\}}$

we obtain

${\displaystyle p_{\text{r}}P_{\text{c}}={\frac {R\,T_{\text{r}}T_{\text{c}}}{V_{\text{r}}V_{\text{c}}-b}}-{\frac {a\alpha \left(\omega ,T_{\text{r}}\right)}{V_{\text{r}}V_{\text{c}}\left(V_{\text{r}}V_{\text{c+}}b\right)}}={\frac {R\,T_{\text{r}}T_{\text{c}}}{V_{\text{r}}V_{\text{c}}-{\frac {\Omega _{b}\,RT_{\text{c}}}{P_{\text{c}}}}}}-{\frac {{\frac {\Omega _{a}\,R^{2}T_{\text{c}}^{2}}{P_{\text{c}}}}\alpha \left(\omega ,T_{\text{r}}\right)}{V_{\text{r}}V_{\text{c}}\left(V_{\text{r}}V_{\text{c}}+{\frac {\Omega _{b}\,RT_{\text{c}}}{P_{\text{c}}}}\right)}}=}$

${\displaystyle ={\frac {R\,T_{\text{r}}T_{\text{c}}}{V_{\text{c}}\left(V_{\text{r}}-{\frac {\Omega _{b}\,RT_{\text{c}}}{P_{\text{c}}V_{\text{c}}}}\right)}}-{\frac {{\frac {\Omega _{a}\,R^{2}T_{\text{c}}^{2}}{P_{\text{c}}}}\alpha \left(\omega ,T_{\text{r}}\right)}{V_{\text{r}}V_{\text{c}}^{2}\left(V_{\text{r}}+{\frac {\Omega _{b}\,RT_{\text{c}}}{P_{\text{c}}V_{\text{c}}}}\right)}}={\frac {R\,T_{\text{r}}T_{\text{c}}}{V_{\text{c}}\left(V_{\text{r}}-{\frac {\Omega _{b}}{Z_{\text{c}}}}\right)}}-{\frac {{\frac {\Omega _{a}\,R^{2}T_{\text{c}}^{2}}{P_{\text{c}}}}\alpha \left(\omega ,T_{\text{r}}\right)}{V_{\text{r}}V_{\text{c}}^{2}\left(V_{\text{r}}+{\frac {\Omega _{b}}{Z_{\text{c}}}}\right)}}}$

thus leading to

${\displaystyle p_{\text{r}}={\frac {R\,T_{\text{r}}T_{\text{c}}}{P_{\text{c}}V_{\text{c}}\left(V_{\text{r}}-{\frac {\Omega _{b}}{Z_{\text{c}}}}\right)}}-{\frac {{\frac {\Omega _{a}\,R^{2}T_{\text{c}}^{2}}{P_{\text{c}}^{2}}}\alpha \left(\omega ,T_{\text{r}}\right)}{V_{\text{r}}V_{\text{c}}^{2}\left(V_{\text{r}}+{\frac {\Omega _{b}}{Z_{\text{c}}}}\right)}}={\frac {T_{\text{r}}}{Z_{\text{c}}\left(V_{\text{r}}-{\frac {\Omega _{b}}{Z_{\text{c}}}}\right)}}-{\frac {{\frac {\Omega _{a}}{Z_{\text{c}}^{2}}}\alpha \left(\omega ,T_{\text{r}}\right)}{V_{\text{r}}\left(V_{\text{r}}+{\frac {\Omega _{b}}{Z_{\text{c}}}}\right)}}}$

Thus, the Soave–Redlich–Kwong equation in reduced form only depends on ω and ${\displaystyle Z_{\text{c}}}$ 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:

${\displaystyle p_{\text{r}}={\frac {T_{\text{r}}}{Z_{\text{c}}\left(V_{\text{r}}-{\frac {\Omega _{b}}{Z_{\text{c}}}}\right)}}-{\frac {{\frac {\Omega _{a}}{Z_{\text{c}}^{2}}}\alpha \left(\omega ,T_{\text{r}}\right)}{V_{\text{r}}\left(V_{\text{r}}+{\frac {\Omega _{b}}{Z_{\text{c}}}}\right)}}}$

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

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

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

In terms of the compressibility factor, we have:

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

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[4] replaced the ${\textstyle {\frac {1}{\sqrt {T}}}}$ 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 ${\displaystyle T_{\text{c}}}$ 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

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

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)[5] proposed a simple correction for this by introducing a volume translation

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

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 ${\displaystyle V_{{\text{m}},{\text{SRK}}}}$ is translated,[6] [7] and the EOS becomes

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

In the second version both ${\displaystyle V_{{\text{m}},{\text{SRK}}}}$ and ${\displaystyle b_{\text{SRK}}}$ are translated, or the translation of ${\displaystyle V_{{\text{m}},{\text{SRK}}}}$ is followed by a renaming of the composite parameter b − c.[8] This gives

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

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.[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.

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:[10]

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.

The equation is given as follows:

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

In polynomial form:

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

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

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

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

The analytic values of its characteristic constants are:

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

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

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

Babalola [14] 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’ was considered to be a constant with respect to pressure in the Peng–Robinson equation of state. 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 the Peng–Robinson equation of state for heavier fluids particularly at high pressure ranges (>30MPa) and eliminates the need for tuning the original Peng–Robinson equation of state.

The Elliott–Suresh–Donohue (ESD) equation of state was proposed in 1990.[15] 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 any molecule and can be directly extended to polymers with molecular parameters characterized in terms of solubility parameter and liquid volume instead of using critical properties (as shown here). The EOS itself was developed through comparisons with computer simulations and should capture the essential physics of size, shape, and hydrogen bonding.

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

where:

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

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

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

For non-spherical molecules, the following relation between the shape factor and the acentric factor is suggested:

${\displaystyle c=1+3.535\omega +0.533\omega ^{2}}$.

The reduced number density ${\displaystyle \eta }$ is defined as ${\displaystyle \eta =b\rho }$, where

${\displaystyle b}$ is the characteristic size parameter [cm³/mol], and

${\displaystyle \rho ={\frac {1}{V_{\text{m}}}}=N/(N_{\text{A}}V)}$ is the molar density [mol/cm³].

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

${\displaystyle b={\frac {RT_{\text{c}}}{P_{\text{c}}}}\Phi }$

where

${\displaystyle \Phi ={\frac {Z_{\text{c}}^{2}}{2A_{q}}}{[-B_{q}+{\sqrt {B_{q}^{2}+4A_{q}C_{q}}}]}}$

${\displaystyle 3Z_{\text{c}}=([(-0.173/{\sqrt {c}}+0.217)/{\sqrt {c}}-0.186]/{\sqrt {c}}+0.115)/{\sqrt {c}}+1}$

${\displaystyle A_{q}=[1.9(9.5q-k_{1})+4ck_{1}](4c-1.9)}$

${\displaystyle B_{q}=1.9k_{1}Z_{\text{c}}+3A_{q}/(4c-1.9)}$

${\displaystyle C_{q}=(9.5q-k_{1})/Z_{\text{c}}}$

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

${\displaystyle Y_{\text{c}}=({\frac {RT_{\text{c}}}{bP_{\text{c}}}})^{2}{\frac {Z_{\text{c}}^{3}}{A_{q}}}}$

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

${\displaystyle z_{\text{m}}=9.5}$ 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 with non-associating components. Details are in the paper by J.R. Elliott, Jr. et al. (1990).[15]

Noting that ${\displaystyle 4(k_{3}-1)/k_{3}}$ = 1.900, ${\displaystyle Z^{\text{rep}}}$ can be rewritten in the SAFT[16][17] form as:

${\displaystyle Z^{\rm {rep}}=4q\eta g-(q-1){\frac {\eta }{g}}{\frac {dg}{d\eta }}={\frac {4q\eta }{1-1.9\eta }}-{\frac {(q-1)1.9\eta }{1-1.9\eta }};g={\frac {1}{1-1.9\eta }}}$

If preferred, the "q" can be replaced by "m" in SAFT notation and the ESD EOS can be written:

${\displaystyle Z=1+m({\frac {4\eta }{1-1.9\eta }}-{\frac {9.5Y\eta }{1+k_{1}Y\eta }})-{\frac {(m-1)1.9\eta }{1-1.9\eta }}}$

In this form, SAFT's segmental perspective is evident and all the results of Michael Wertheim[16][17][18] are directly applicable and relatively succinct. In SAFT's segmental perspective, each molecule is conceived as comprising m spherical segments floating in space with their own spherical interactions, but then corrected for bonding into a tangent sphere chain by the (m − 1) term. When m is not an integer, it is simply considered as an "effective" number of tangent sphere segments.

Solving the equations in Wertheim's theory can be complicated, but simplifications can make their implementation less daunting. Briefly, a few extra steps are needed to compute ${\displaystyle Z^{\rm {assoc}}}$given density and temperature. For example, when the number of hydrogen bonding donors is equal to the number of acceptors, the ESD equation becomes:

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

where:

${\displaystyle Z^{\rm {assoc}}=-gN^{\text{AD}}(1-X^{\text{AD}});X^{\text{AD}}=2/[1+{\sqrt {1+4N^{\text{AD}}\alpha ^{\text{AD}}}}];\alpha ^{\text{AD}}=\rho N_{\text{A}}K^{\text{AD}}[\exp {(\epsilon ^{\text{AD}}/kT)-1]}}$

${\displaystyle N_{\text{A}}}$ is the Avogadro constant, ${\displaystyle K^{\text{AD}}}$ and ${\displaystyle \epsilon ^{\text{AD}}}$ are stored input parameters representing the volume and energy of hydrogen bonding. Typically, ${\displaystyle K^{\text{AD}}=\mathrm {0.001\ nm^{3}} }$ and ${\displaystyle \epsilon ^{\text{AD}}/k_{\text{B}}=\mathrm {2000\ K} }$ are stored. ${\displaystyle N^{\text{AD}}}$ is the number of acceptors (equal to number of donors for this example). For example, ${\displaystyle N^{\text{AD}}}$ = 1 for alcohols like methanol and ethanol. ${\displaystyle N^{\text{AD}}}$ = 2 for water. ${\displaystyle N^{\text{AD}}}$ = degree of polymerization for polyvinylphenol. So you use the density and temperature to calculate ${\displaystyle \alpha ^{\text{AD}}}$ then use ${\displaystyle \alpha ^{\text{AD}}}$ to calculate the other quantities. 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 cubic-plus-association (CPA) equation of state combines the Soave–Redlich–Kwong equation with the association term from SAFT[16][17] based on Chapman's extensions and simplifications of a theory of associating molecules due to Michael Wertheim.[18] The development of the equation began in 1995 as a research project that was funded by Shell, and published in 1996.[19][20]

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

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

The cubic-plus-chain (CPC)[21][22][23] equation of state hybridizes the classical cubic equation of state with the SAFT chain term.[16][17] 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 CPC monomer term is not restricted to one classical cubic EOS form, instead many forms can be used within the same framework. The cubic-plus-chain (CPC) equation of state is written in terms of the reduced residual Helmholtz energy (F^CPC) as:

${\displaystyle F^{CPC}={\frac {A^{R}(T,V,{\text{n}})}{RT}}=m(a^{rep}+a^{att})+a^{chain}}$

where A^R is the residual Helmholtz energy, m 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. Using Redlich−Kwong (RK) for the monomer term, CPC can be written as:

${\displaystyle P={\frac {nRT}{V}}{\Biggl (}1+{\frac {{\bar {m}}^{2}B}{V-{\bar {m}}B}}{\Biggr )}-{\frac {{\bar {m}}^{2}A}{V(V+{\bar {m}}B)}}-{\frac {nRT}{V}}\left[\sum _{i}n_{i}(m_{i}-1)\beta {\frac {g'(\beta )}{g(\beta )}}\right]}$

where A is the molecular interaction energy parameter, B is the co-volume parameter, ${\displaystyle {\bar {m}}}$ is the mole-average chain length, g(β) is the radial distribution function (RDF) evaluated at contact, and β is the reduced volume.

The CPC model combines the simplicity and speed compared to other complex models used to model polymers. Sisco et al.[21][23] applied the CPC equation of state to model different well-defined and polymer mixtures. They analyzed different factors including elevated pressure, temperature, solvent types, polydispersity, etc. The CPC model proved to be capable of modeling different systems by testing the results with experimental data.

Alajmi et al.[24] 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.