where Z=Pv/(RT) is the compressibility factor, P is the pressure, v is the molar volume, R denotes the gas constant, T is the absolute temperature, A is the Helmholtz free energy, N is the total number of molecules, k is the Boltzmann constant, and superscripts hc, and pert denote the hard-sphere chain reference equation of state, and the perturbation contribution, respectively. In this terminology the reference equation of state reduces to ideal-gas behavior at the zero-density limit.
Based on Wertheim’s [2-5] thermodynamic perturbation theory of first order Chapman et al. [6] developed an equation of state, which for hard-sphere chains comprising m segments is given by
(1)
(2)
(3)
where xi is the mole fraction of chains of component i, mi is the number of segments in a chain of component i, r is the total number density of molecules, giihs is the radial pair distribution function for segments of component i in the hard sphere system, and superscript 'hs' indicates quantities of the hard-sphere system. Expressions of Boublik [29] and Mansoori et al. [30] are used for mixtures of the hard-sphere reference system in Eq. (1) and (2), given by
where
m = {0,1,2,3}
with di being a temperature dependent segment diameter of component i , according to
.
In the above equations mi, sii, and eii are the pure component parameter (segment number, segment diameter, and interaction-energy parameter).
The packing fraction h is defined by
.
The second-order perturbation theory of Barker and Henderson was extended to chain molecules. The perturbation contribution is the sum of the first- and second-order term, according to
.
Van der Waals one fluid mixing rules are adopted here to extend the perturbation terms to mixtures.
.
Conventional combining rules are employed to determine the parameters between a pair of unlike segments
We also apply the one-fluid mixing concept to the compressibility term of the second order perturbation term, i.e.
In these equations, we have replaced the integrals over the radial pair distribution function of chain-molecules by power series in density of sixth order
and
where ai(m) and bi(m) are coefficients of the power series in density, each depending upon segment number. We found, that the dependence of each of the power series coefficients on segment number can accurately be described with a relation proposed by Liu and Hu
.
These model-constants a0i, a1i, and a2i as well as b0i, b1i, and b2i were fitted to thermophysical properties of pure n-Alkanes. They are given in the following two tables:
| i | a0i | a1i | a2i |
| 0 | 0.91056314452 | -0.30840169183 | -0.09061483510 |
| 1 | 0.63612814495 | 0.18605311592 | 0.45278428064 |
| 2 | 2.68613478914 | -2.50300472587 | 0.59627007280 |
| 3 | -26.5473624915 | 21.4197936297 | -1.72418291312 |
| 4 | 97.7592087835 | -65.2558853304 | -4.13021125312 |
| 5 | -159.591540866 | 83.3186804809 | 13.7766318697 |
| 6 | 91.2977740839 | -33.7469229297 | -8.67284703680 |
| i | b0i | b1i | b2i |
| 0 | 0.72409469413 | -0.57554980753 | 0.09768831158 |
| 1 | 2.23827918609 | 0.69950955214 | -0.25575749816 |
| 2 | -4.00258494846 | 3.89256733895 | -9.15585615297 |
| 3 | -21.0035768149 | -17.2154716478 | 20.6420759744 |
| 4 | 26.8556413627 | 192.672264465 | -38.8044300521 |
| 5 | 206.551338407 | -161.826461649 | 93.6267740770 |
| 6 | -355.602356122 | -165.207693456 | -29.6669055852 |
The compressibility factor is given by
and the perturbation terms of first- and second-order are given by
with
and
where
and where C1 and C2 are abbreviations defined as
