Here is a review of Equations of States (EOS) that was discussed in class:

And in equation form:

**Statistical Associating Fluid Theory (SAFT) EOS development**

These four empirical equations are mentioned because they show the major contributions to EOS development and more specifically, they focus on attractive term advancements.

Developments to the repulsive term by Carnahan and Starling (CS) [1972] coupled with attractive term improvement (R-K) are shown as the following:

(1)
\begin{align} \frac{P\nu} {RT} =\frac{1+y+y^2-y^3} {(1-y)^3} - \frac{a} {RT(\nu+b)} \end{align}

**Chain molecules**

Perturbed Hard Chain Theory (PHCT) [1957]

PHCT is derived from a molecular vibrational and rotational partition function using free-volume concepts:

(2)
\begin{align} Z = Z(hard chain) - \frac{a} {RTV} \end{align}

Perturbed Anisotropic Chain Theory (PACT) [1985-86]

PACT includes anisotropic multipolar forces in the PHCT equation and accounts for size, shape, and intermolecular effects:

(3)
\begin{equation} Z = 1+Z^{rep}+Z^{iso}+Z^{ani} \end{equation}

**Hard-Sphere Chain Concept**

- The backbone of many systematic improvements to EOS
- Thermodynamic perturbation theory (TPT), proposed by Wertheim [1987], which accommodates hard-chain molecules
- Chapman generalized it, Compressibility factor of hard-chain of m segments,

(4)
\begin{align} Z^{hc} = mZ^{hs} - (m-1) (1+\eta \frac{\partial ln g^{hs} \sigma} {\partial\eta}) \end{align}

**Associating Fluids**

The associated perturbed anisotropic chain theory (APACT) [1986] is derived from the infinite equilibrium model and monomer-dimer model into the PACT and accounts for isotropic repulsive and attractive interactions, anisotropic molecular interactions, and is capable of predicting thermodynamic properties of pure associating components as well as mixtures,

(5)
\begin{equation} Z = 1+Z^{rep}+Z^{att}+Z^{assoc} \end{equation}

The SAFT EOS is developed from Wertheim's theory of Helmholtz energy expansion and is expressed as residual Helmholtz energy and it describes hard-sphere repulsive forces, chain formation (for nonspherical molecules) and association,

(6)
\begin{equation} a^{res} = a^{seg}+a^{chain}+Z^{assoc} \end{equation}

The following diagram shows the interrelationship of EOS

Wei and Sadus 2000

For a pdf version of most diagrams click here: EOS Review

The key to the SAFT EOS is the molecular model and solution parameters. The following diagram shows the steps in the fluid model and molecular parameters:

Example of prototype spheres:

- AA-bonding when distance and orientation are favorable
- association bond strength is quantified with a square-well potential with center on the A site characterized by,
- $\epsilon^AA$, association energy depth
- $\kappa^AA$, interaction volume, which corresponds with r
^{AA}, well width

**Approximations**

- 1st order theory: allows for chainlike or treelike associated clusters (at least two sites are needed to form clusters of three segments or more) But no ringlike clusters are allowed
- fluid properties are independent of site angle and the site activity is independent of bonding at other sites on the same molecule
- some steric hindrance effects are ignored

Steric Hindrance Approximations:

- if molecules i and j are close enough that site A on molecule i bonds to site B of molecule j, then repulsive cores i, j, and k prevent any site on molecule k from coming close enough to bond to either site A or B
- no site on molecule i can bond simultaneously to 2 sites on molecule j
- no double bonding between molecules is allowed, But this can be relaxed for further development

**Chain Formation**

Multi-segmented chain molecules are formed from imposing strong covalent bonds

- fluid is made up of LJ spheres where 1 can only bond to 2 and 2 can only bond to 1 and 3, so m bonds to m-1
- stoichiometric sphere ratio is required
- spheres are forced to bond and create a chain

The figures are taken from Chapman et al. 1990 and a pdf is available: SAFT Parameters

**Lennard-Jones (LJ) Segments**

LJ segments are characterized by diameter, $\sigma$ and number of segments, m

$\sigma$ is temperature independent, but can be related to a (temperature dependent) hard sphere diameter, d (Barker-Henderson Theory, 1967)

(7)
\begin{align} d = \sigma f (\frac{\kappa T} {\epsilon},m) \end{align}

where $f \frac{\kappa T} {\epsilon}$ is a generic function of the reduced temperature and $\epsilon$ is an LJ interaction energy

$f \frac{\kappa T} {\epsilon}$ is found by fitting d, m and $\epsilon$ to saturated liquid densities and vapor pressure data

Cotterman et al. 1986 fit the data to 10K above the triple point and 10K below the critical point for alkanes up to n-octane,

(8)
\begin{align} f \frac{\kappa T} {\epsilon} = \frac {1+0.2977\frac{\kappa T} {\epsilon}} {1+0.33163 \frac {\kappa T} {\epsilon} + f(m)\frac {\kappa T} {\epsilon}^2} \end{align}

and

(9)
\begin{align} f(m) = 0.0010477+0.025337\frac{m-1} {m} \end{align}

**Nonassociation Mixtures**

To evaluate nonassociating mixtures, the Helmholtz energy of the fluid mixture of spherical LJ segments that exist before bonding and association takes place must be calculated.

Mole fraction of LJ sphere of species i,

(10)
\begin{align} y_i = \frac{x_i m} {\sum_j x_j m_j} \end{align}

where x_{i} = the mole fraction of chain molecules of species i (formed after bonding takes place) having m_{i} segments

For a mixture of LJ segments, van der Waals one-fluid theory can be used and has good agreement for LJ spheres of similar size. It may be derived from the perturbation theory that defines parameters $\sigma_x$ and $\epsilon_x$ of a hypothetical pure fluid, X, having the same residual properties as the mixture of interest.

2 mixing rules for $\sigma_x$ and $\epsilon_x$ to be equivalent to $\sigma$ and $\epsilon$ for the mixture of interest,

(11)
\begin{align} \sigma_x^3 = \frac{\sum_i \sum_j x_i x_j m_i m_j \sigma_{ij}^3} {(\sum_i x_i m_i)^2} \end{align}

and

(12)
\begin{align} \epsilon_x \sigma_x^3 = \frac{\sum_i \sum_j x_i x_j m_i m_j \sigma_ij^3 \epsilon_{ij}} {(\sum_i x_i m_i)^2} \end{align}

where x_{i} = the mole fraction of component i and the unlike-interaction energy parameter $\epsilon_ij$ is determined from a modified geometric average,

(13)
\begin{align} \epsilon_{ij} = \xi_{ij} (\epsilon_{ii} \epsilon_{jj})^\frac{1} {2} \end{align}

and the unlike-interation size parameter $\sigma_{ij}$ is determined from an arthmetic average

(14)
\begin{align} \sigma_{ij} = \frac{\sigma_{ii}+\sigma_{jj}} {2} \end{align}

when $\xi_{ij}$ = 1 these two equations reduce to the Lorentz-Berthelot rules.

The effective hard diameter of the hypothetical pure fluid,

(15)
\begin{align} d_x = \sigma_x f (\frac{\kappa T} {\epsilon_x},m_x) \end{align}

where m_{x} it the effective chain length of the conformal fluid, approximated by,

(16)
\begin{align} m_x = \sum_i x_i m_i \end{align}

(17)
\begin{equation} a^{res} = a^{seg}+a^{chain}+Z^{assoc} \end{equation}

a^{seg} = segment-segment interactions such as LJ interactions

a^{chain} - is from the presence of covalent chain-forming bonds among the segments, i.e. LJ segments

a^{assoc} - accounts for the increment of a^{res} due to the presence of site-site specfic interactions among segments, i.e. hydrogen bonding interactions

and more specifically the SAFT EOS is,

(18)
\begin{align} a^{res} = a^{seg}(m \rho, T; \sigma, \epsilon)+a^{chain}(\rho; d, m)+Z^{assoc}(\rho, T; \epsilon^{AB}, \kappa^{AB}) \end{align}

**Association Term**

**Pure Components**

(19)
\begin{align} \frac {a^{assoc}} {R T} = \sum_A [ln X^A - \frac {X^A} {2}] + \frac {1} {2} \end{align}

where,

- M = number of sites on each molecule
- X
^{A} = mole fraction of molecules NOT bonded at site A
- $\sum_A$ = a sum over over all associating sites on the molecule
- Examples:
- One site, $\frac {a^{assoc}} {R T} = ln X^A - \frac {X^A} {2} + \frac {1} {2}$
- Two sites, $\frac {a^{assoc}} {R T} = ln X^A - \frac {X^A} {2} + ln X^A - \frac {X^A} {2} + 1$

Mole fraction of terms NOT bonded at site A,

(20)
\begin{align} X^A = [1 + N_{AV} \sum_B \rho X^B \Delta^{AB}]^{-1} \end{align}

(summation over all sites, A, B, C,…)

- N
_{AV} = Avogadro's number
- $\Delta^{AB}$ = association strength

Association Strength

(21)
\begin{align} \Delta^{AB} = 4 \pi F^{AB} \int_0^r_c r^2 g(r)^{seg} dr \end{align}

(22)
\begin{align} F^{AB} = exp(\frac {\epsilon^{AB}} {T}) - 1 \end{align}

also,

(23)
\begin{align} \Delta^{AB} = 4 \pi F^{AB} \int_0^r_c r^2 \omega(r) g(r)^{seg} dr \end{align}

where $4 \pi r^2 \omega(r) dr$ is the bonding site overlap volume

- bonding is assumed to occur between hard sphere contact and r
_{c}

Chapman 1988 and Chapman et al. 1988 approximated,

(24)
\begin{align} \Delta^{AB} = d^3 g(d)^{seg} \kappa^{AB} [exp(\frac {\epsilon^{AB}} {T}) - 1] \end{align}

- $(\frac {\epsilon^{AB}} {T})$ only explicit temperature dependence
- depends on segment diameter and segment radial distribution
- also dependent on association energy and volume discussed in the sphere example

(25)
\begin{align} g(d)^{seg} = g(d)^{hs} \frac {1-\frac {1} {2} \eta}{(1-\eta)^3} \end{align}

- $\eta$ = reduced density $\eta = \frac {\pi N_{AV}} {6} \rho d^3 m$
- $\rho$ = molar density of molecules, only density dependent term

**Mixtures**

(26)
\begin{align} \frac {a^{assoc}} {R T} = \sum_i X_i [\sum_A_i [ln X^Az-i - \frac {X^A_i} {2}] + \frac {1} {2} M_i] \end{align}

where X_i = the mole fraction of molecules i that are NOT bonded at site A,

(27)
\begin{align} X^A_i = [1 + N_{AV} \sum_j \sum_B \rho_j X^B_j \Delta^{A_iB_j}]^{-1} \end{align}

($\sum_{Bj}$ over ALL sites on molecule j, A_{j}, B_{j}, C_{j},…; $\sum_j$ over all components)

Association Strength

(28)
\begin{align} \Delta^{A_iB_j} = d_{ij}^3 g_{ij} (d_{ij})^{seg} \kappa^{A_iB_j}[exp(\frac {\epsilon ^{A_i B_j}} {\kappa T})-1] \end{align}

where $d_{ij} = (d_{ii} + d_{jj})/2$

Much like in equations 24 and 25 for pure components, a hard spheres expression is derived for mixtures (Reed and Gubbins, 1973),

(29)
\begin{align} g_{ij}(d_{ij})^{seg} = g_{ij}(d_{ij})^{hs} = \frac {1} {1-\xi_3}+ \frac {3 d_{ii} d_{jj}} {d_{ii} + d_{jj}} \frac {\xi_2} {(1-\xi_3)^2} + 2 [\frac {d_{ii} d_{jj}} {d_{ii} + d_{jj}}]^2 \frac {\xi_2^2} {(1-\xi_3)^3} \end{align}

for like segments,

(30)
\begin{align} g_{ii}(d_{jj})^{seg} = g_{ii}(d_{ii})^{hs} = \frac {1} {1-\xi_3}+ \frac {3 d_{ii}} {2} \frac {\xi_2} {(1-\xi_3)^2} + 2 [\frac {d_{ii}} {2}]^2 \frac {\xi_2^2} {(1-\xi_3)^3} \end{align}

These hard sphere distribution functions depend on the effective sphere diameter and on a function of density $\xi_{\kappa = 0,1,2,3,}$ defined as,

(31)
\begin{align} \xi_{\kappa} = \frac {\pi N_{Av}} {6} \rho \sum_i X_i m_i d_{ii}^{\kappa} \end{align}

$\xi_3$ (not $\xi_0, \xi_1, \xi_2$) is equivalent to the packing fraction

**Chain Term**

(32)
\begin{align} \frac {a^{chain}} {R T} = \sum_i X_i (1-m_i) ln (g_{ii}(d_{ii}^{hs}) \end{align}

where g_{ii} is the hard sphere pair correlation function for the interaction of two spheres i in the mixture of spheres evaluated at contact

**LJ Segment Term**

(33)
\begin{align} a^{seg} = a_0^{seg} \sum_i X_i m_i \end{align}

where,

- a
_{0}^{seg} = residual Helmholtz energy of nonassociated spherical segments
- $\sum_i X_i m_i$ = a ratio of the number of segments to the number of molecules in the fluid

a_{0}^{seg} may be further broken down. If the segments are LJ spheres a_{0}^{seg} may be described with two parts, reference + perturbation

(34)
\begin{equation} a_0^{seg} = a_0^{hs}+a_0^{disp} \end{equation}

The hard term, a_{0}^{hs} may be calculated as proposed by Carnahan and Starling (1969) for mixtures and pure components,

(35)
\begin{align} \frac {a_0^{seg}} {RT} \frac {4 \eta - 3 \eta^2} {(1 - \eta)^2} \end{align}

$\eta$ is the segment packing fraction (reduced density), $\eta = \frac {\pi N_{Av}} {6} \rho d^3 m$ for pure components and $\eta = \frac {\pi N_{Av}} {6} \rho d^3 \sum_i X_i m_i$ for mixtures

SAFT derivation from Chapman et al. 1990

The vapor pressure and density data were used to verify SAFT EOS predicted values. The following plots show propane and n-butane data compared well with the SAFT EOS predicted values.

**EOS improvements**

- Simplified SAFT (SSAFT): simplifies the dispersion part of aseg with an attraction term for a square-well fluid
- Hard-sphere SAFT (HS-SAFT): molecules are viewed as chains of hard spheres with van der Waals interactions
- Lennard-Jones SAFT (LJ-SAFT): includes effects from dipole interactions and modifications to the hard sphere term
- Square-well SAFT (SW-SAFT): extended thermodynamic perturbation to include square-well chains which quantifies monomer-monomer potential
- SAFT with variable range (SAFT-VR): addresses chain contribution with segments of variable range potentials
- SAFT with crossover technique (SAFT-VRX): SAFT-VR combined with a crossover function to improve critical data prediction

Blas and Vega 1998 use a LJ fluid accounting for dispersive and repulsive forces and the EOS is extended to predict fluid behavior of heteronuclear chains,

methane = circles, propane = diamonds, n-octane = triangles

Gross and Sadowski 2001 employ a hard-chain description of molecules involving thermodynamic perturbation theory, using a simplified molecular model, which involves chains composed of spherical segment pair potential derived from modified square well potentials,

Here are plots by Kruska and Gubbins 1996 employing the LJ-SAFT EOS for methane and ethane

McCabe and Kiselev 2003, in addition to incorporating variable range potential, a crossover function was included, which is derived from Helmholtz free energy, renormalizing ΔT and Δν to give correct non-analytical asymptotic behavior of real fluids in the critical region,

The dotted line represents a crossover SAFT-HR (Kiselev and Ely), the dashed-dotted line is SAFT-VR with rescaled parameters, the dashed line is SAFT-VR, and the solid line is the SAFT-VRX for methane.

**General EOS**

Ya Song Wei and Richard J. Sadus. Equations of Sates for the Calculation of Fluid-Phase Equilibria. Thermodynamics Journal Review AIChE Journal. 46(1):169-196 (2000).

E.A.Guggenheim. The Principle of Corresponding States. The Journal of Chemical Physics. 13(7):253-261 (1945).

Dr. Schaefer’s Equations of State handout. ME3007 Fall 2007.

Schaum Outline of Theory and Problems of Thermodynamics with Chemical Applications by Abbott and Van Ness. Second Edition, 1989.

**SAFT EOS**

Chapman, W.G., Gubbins, K.E., Jackson, G., Radosz, M. SAFT: Equation-of-State Solution Model for Associating Fluids. Fluid Phase Equilibria. 52:31-38 (1989). Paper available from Handouts and Links: SAFT: Equation-of-State Solution Model for Associating Fluids

Chapman, W.G., Gubbins, K.E., Jackson, G., Radosz, M. New Reference Equation of State for Associating Liquids. Industrial and Engineering Chemistry Research. 29:1709-1721 (1990). Paper available from Handouts and Links: Another SAFT paper

**SAFT EOS improvements**

This group of papers discuss more implementation of the EOS, how it compares with others, and improvements to the EOS.

Kraska, T., Gubbins, K.E. Phase Equilibria Calculations with a Modified SAFT Equation of State. 1. Pure Alkanes, Alkanols, and Water. Industrial and Engineering Chemistry Research. 35:4727-4737 (1996).

Blas, F.J., Vega, L.F. Prediction of Binary and Ternary Diagrams Using the Statistical Associating Fluid Theory (SAFT) Equation of State. Industrial and Engineering Chemistry Research. 37:660-674 (1998)

Kiselev, S.B., Ely, J.F. Simplified crossover SAFT equation of state for pure fluids and fluid mixtures. Fluid Phase Equilibria. 147:93-113 (2000).

Gross, J., Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Industrial and Engineering Chemistry Research. 40:1244-1260 (2001).

McCabe, C., Kiselev, S.B. A crossover SAFT-VR equation of state for pure fluids: preliminary results for light hydrocarbons. Fluid Phase Equilibria. 219:3-9 (2004).

Ries, R.A., Paredes, M.L.L., Castier, M., Tavares, F.W. Evaluation of mixing and of combining rules for asymmetric Lennard-Jones chain mixtures: Effect of segment diameter, energy interaction, and chain length. Fluid Phase Equilibria. 259:123-134 (2007).