### Abstract

An explicit expression for dispersion in activity coefficient models can be derived from cubic equations of state (cEoS). Here we show that all the two-parameter cEoS deliver a van Laar type of equation. The difference between these equations can be characterized by a single parameter K, which can be computed directly from the cEoS characteristic parameters. The theoretical values for K are always higher than experimental activity coefficient data of alkane mixtures indicate. We show that mixtures of linear and branched alkanes require K=4.13 and K=3.04, respectively, while the lowest theoretical value, K=9, is given by the van der Waals equation. This mismatch in results is caused by the assumptions, which are made in the derivation of the van der Waals equation of state and which remain present in later developed cEoS. One of these is that all molecules are spherical, which leads to the inconsistency that the ratio of the covolume and the van der Waals volume is always 4, while this ratio for linear alkanes decreases rapidly to nearly 2 with increasing chain length. Another assumption is that all molecules experience the same number of external interactions, which neglects the fact that polyatomic molecules have less intermolecular interactions per spherical segment due to presence of covalent bonds and the occurrence of intramolecular interaction. Therefore, the van Laar type of activity coefficient equations are limited in their use as predictive model for dispersion. Perturbed hard-sphere chain equation of state will be discussed in part 2.

Original language | English |
---|---|

Article number | 112275 |

Number of pages | 13 |

Journal | Fluid Phase Equilibria |

Volume | 501 |

DOIs | |

Publication status | Published - 1 Dec 2019 |

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### Keywords

- Activity model
- Cubic equation of state
- Dispersion
- Van Laar

### Cite this

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**Dispersion activity coefficient models. Part 1 : cubic equations of state.** / Krooshof, Gerard J.P. (Corresponding author); Tuinier, Remco; de With, Gijsbertus.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Dispersion activity coefficient models. Part 1

T2 - cubic equations of state

AU - Krooshof, Gerard J.P.

AU - Tuinier, Remco

AU - de With, Gijsbertus

PY - 2019/12/1

Y1 - 2019/12/1

N2 - An explicit expression for dispersion in activity coefficient models can be derived from cubic equations of state (cEoS). Here we show that all the two-parameter cEoS deliver a van Laar type of equation. The difference between these equations can be characterized by a single parameter K, which can be computed directly from the cEoS characteristic parameters. The theoretical values for K are always higher than experimental activity coefficient data of alkane mixtures indicate. We show that mixtures of linear and branched alkanes require K=4.13 and K=3.04, respectively, while the lowest theoretical value, K=9, is given by the van der Waals equation. This mismatch in results is caused by the assumptions, which are made in the derivation of the van der Waals equation of state and which remain present in later developed cEoS. One of these is that all molecules are spherical, which leads to the inconsistency that the ratio of the covolume and the van der Waals volume is always 4, while this ratio for linear alkanes decreases rapidly to nearly 2 with increasing chain length. Another assumption is that all molecules experience the same number of external interactions, which neglects the fact that polyatomic molecules have less intermolecular interactions per spherical segment due to presence of covalent bonds and the occurrence of intramolecular interaction. Therefore, the van Laar type of activity coefficient equations are limited in their use as predictive model for dispersion. Perturbed hard-sphere chain equation of state will be discussed in part 2.

AB - An explicit expression for dispersion in activity coefficient models can be derived from cubic equations of state (cEoS). Here we show that all the two-parameter cEoS deliver a van Laar type of equation. The difference between these equations can be characterized by a single parameter K, which can be computed directly from the cEoS characteristic parameters. The theoretical values for K are always higher than experimental activity coefficient data of alkane mixtures indicate. We show that mixtures of linear and branched alkanes require K=4.13 and K=3.04, respectively, while the lowest theoretical value, K=9, is given by the van der Waals equation. This mismatch in results is caused by the assumptions, which are made in the derivation of the van der Waals equation of state and which remain present in later developed cEoS. One of these is that all molecules are spherical, which leads to the inconsistency that the ratio of the covolume and the van der Waals volume is always 4, while this ratio for linear alkanes decreases rapidly to nearly 2 with increasing chain length. Another assumption is that all molecules experience the same number of external interactions, which neglects the fact that polyatomic molecules have less intermolecular interactions per spherical segment due to presence of covalent bonds and the occurrence of intramolecular interaction. Therefore, the van Laar type of activity coefficient equations are limited in their use as predictive model for dispersion. Perturbed hard-sphere chain equation of state will be discussed in part 2.

KW - Activity model

KW - Cubic equation of state

KW - Dispersion

KW - Van Laar

UR - http://www.scopus.com/inward/record.url?scp=85070880802&partnerID=8YFLogxK

U2 - 10.1016/j.fluid.2019.112275

DO - 10.1016/j.fluid.2019.112275

M3 - Article

AN - SCOPUS:85070880802

VL - 501

JO - Fluid Phase Equilibria

JF - Fluid Phase Equilibria

SN - 0378-3812

M1 - 112275

ER -