[american institute of aeronautics and astronautics 8th aiaa/asme joint thermophysics and heat...

AIAA-2002-3339

A STUDY OF ASSOCIATING LENNARD-JONES CHAINS FOR HYDROCARBONS

Jurij Avsec* University of Maribor, Faculty of Mechanical Engineering, Smetanova 17, 2000 Maribor, P.O. BOX 224, SLOVENIA

The paper features the mathematical model of computing equilibrium thermophysical properties of state in the fluid domain for pure hydrocarbons with the help of classical thermodynamics, classical statistical thermodynamics and statistical associating chain theories. In the presented paper we calculated thermodynamic properties for butane as the example of hydocarbon substances. To calculate the thermodynamic properties of real fluid the models on the basis of Lennard-Jones intermolecular potential were applied. To calculate the thermodynamic properties of real fluid with help of classical thermodynamics we used Tillner-Roth-Watanabe-Wagner (TRWW) equation on the base of Helmhotz type. We developed the mathematical model for the calculation of all equilibrium thermodynamic functions of state for pure hydrocarbons. The analytical results obtained by statistical thermodynamics are compared with the TRWW model and show relatively good agreement.

NOMENCLATURE. *†

A free energy A* reduced free energy AAD absolute average deviation BWR Benedict-Webb-Rubin c0 velocity of sound Cp molar heat capacity at constant pressure CV molar heat capacity at constant volume CYJ Chunxi-Yigui-Jiufang model D hard sphere diameter Exp experiment f number of degrees of freedom G free enthalpy g0 hard sphere radial distribution function H enthalpy k Boltzmann constant LJ Lennard-Jones LLL Liu-Li-Lu m number of segments M molecular mass N number of molecules in system, Lennard-

Jones with influence of rotational contribution p pressure, momentum Rm

universal gas constant S entropy SAFT statistical associating fluid theory SEXP simplified exponential approximation

* Assistant Professor, Mechanical Engineer, Deparment of Thermodynamics, E-Mail: [email protected], AIAA Member Copyright © 2002 by the American Institute of Aeronautics and Astronautics Inc. All rights reserved.

T temperature T* reduced temperature TL Tang-Lu TTL Tang-Tong-Lu model r intermolecular distance TRWW Tillner-Roth-Wagner-Watanabe U internal energy v specific volume V volume Z partition function β Boltzmann factor βt coefficient of thermal expansion δ inversed reduced volume ε Lennard-Jones parameter εs segment interaction parameter η packing factor θ characteristic temperature ρ density ρ* reduced density τ inversed reduced temperature µJ Joule-Thomson coefficient χ isothermal compressibility σ Lennard-Jones parameter σs segment diameter Superscripts and subscripts 0 ground state assoc association c critical condition chain chain conf configurational el influence of electron excitation

1

American Institute of Aeronautics and Astronautics

8th AIAA/ASME Joint Thermophysics and Heat Transfer Conference24-26 June 2002, St. Louis, Missouri

AIAA 2002-3339

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

mailto:[email protected]

AIAA-2002-3339

ir internal rotation nuc influence of nuclear excitation pot potential energy res residual rot rotation seg segment trans translation vib vibration

INTRODUCTION In the engineering practice processes occuring in

liquid-gas region are of vital importance. In order to design devices of this field of activity, it is necessary to know the thermodynamic properties of state in a one and two phase environment for pure hydrocarbons and their mixtures.

In most cases the thermodynamic tables or diagrams or different empirical functions obtained from measurement are used (classical thermodynamics). Today, there are numerous equations of state (EOS) reported in the literature for describing the behaviours of fluid: (Van der Waals EOS (VDW), Peng-Robinson (PR), Redlich-Kwong (RK) EOS, Soave EOS..)1. However, these equations have exhibited some noticeable defects, such as poor agreement with experimental data at moderate densities. On the other side, we can use the complex equations of state with many constants (Benedict-Webb-Rubin1 (BWR) EOS, Lee-Kessler1 EOS, Benedict-Webb-Rubin-Starling-Nishiumi1 EOS, modified BWR1,21 (MBWR), Jacobsen-Stewart23 EOS (JS), Tillner-Roth-Watanabe-Wagner22,24,52,53 (TRWW)…). These equations are more complicated but they have no insight into the microstructure of matter and poor agreement with experimental data outside interpolation limits.

Table1. Fundamental characteristics of various EOS

obtained on the base of classical thermodynamics. EOS Number

of used constants

Influence of polarity

Type

VDW 2 NO Pressure RK 2 NO Pressure PR 3 YES Pressure BWR 8 NO Pressure BWRSN 16 YES Pressure MBWR 32 YES Pressure JS 75-150 YES Helmholtz TRWW >200 YES Helmholtz

The calculation of thermodynamic functions of state with help of classical thermodynamics is well-known matter and is not described in these paper. Table 1 shows only the fundamental characteristics of various EOS obtained by classical thermodynamics.

Statistical thermodynamics on the other hand,

calculates the properties of the state on the basis of molecular motions in a space and intramolecular interaction. The calculation of the thermodynamic functions of state is possible by many statistical theories. One of the most successfully is the perturbation theory. Several equations of state have been published that are based on perturbation theory2,3. The evolution of perturbation theory is well described in literature by Barker and Henderson4, Münster5, Lucas6 and Gray & Gubbins7.

In this paper we compared new Tang-Tong-Lu (TTL) model,8-10 Chunxi-Yigui-Jiufang (CYJ) model,11 with models obtained on the basis of statistical associating fluid theory (Liu-Li-Lu model (LLL) and Tang-Lu model (TL)). All these models are based on Lennard-Jones intermolecular potential function. The Lennard-Jones intermolecular potential is an important model for studying of simple fluids in one and two-phase region. It is widely used as a reference potential in perturbation theories for more complex potentials2,6,8.

In these paper we developed the mathematical model of computing the equilibrium properties of state. We compared the deviation of the results between various models for thermodynamic functions of state and for their derivatives (enthalpy, pressure, entropy, isothermal compressibility, the coefficient of thermal expansion, the heat capacities, the velocity of sound) too.

The results of the analysis are compared with TRWW model obtained on the basis of classical thermodynamics and show a relatively good agreement, especially for real gases. Somewhat larger deviations can however be found in the region of real liquid due to the large influence of the attraction and repulsion forces, since the Lennard- Jones potential is an approximation of the actual real intermolecular potential.

COMPUTATION OF THERMODYNAMIC

PROPERTIES OF THE STATE

To calculate thermodynamic functions of state we applied the canonical partition6. Utilising the semi-classical formulation for the purpose of the canonical ensemble for the N indistinguishable molecules the partition function Z can be expressed as follows5:

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AIAA-2002-3339

N21N21Nf pd..pdpdrd..rdrdkTHexp..

h!N1Z

rrrrrr⋅

−= ∫ ∫ (1)

where f stands for the number of degrees of freedom of individual molecule, H designates the Hamiltonian molecule system, vectors N21 r..r,r

rrr.. describe the

positions of N molecules and Np21 ...p,prrr

momenta, k is Boltzmann`s constant and h is Planck`s constant. The canonical ensemble of partition function for the system of N molecules can be expressed like this:

confnucelirrotvibtrans0 ZZZZZZZZZ = (2) Thus the partition function Z is a product of terms

of the ground state (0), the translation (trans), the vibration (vib), the rotation (rot), the internal rotation (ir),the influence of electrons excitation (el), the influence of nuclei excitation (nuc) and the influence of the intermolecular potential energy (conf).

Utilising the canonical theory for computating the thermodynamic functions of the state can be put as follows5,6:

Pressure VT

lnZkTp

∂∂

= ,

Internal energy V

2

TlnZkTU

∂∂

= ,

Freeenergy lnZkTA ⋅−=

Entropy

∂∂

+=VT

lnZTlnZkS , (3)

Free enthalpy

∂∂

−−=VT

lnZVlnZkTG ,

Enthalpy

∂∂

+

∂∂

=TV V

lnZVT

lnZTkTH ,

where T is temperature and V is volume of

molecular system. The computation of the individual terms of the

partition function and their derivatives except of the configurational integral is dealt with in the works of Lucas 6; Gray and Gubbins7 and McClelland12.

The various derivatives and expressions of the fundamental equations (3) have an important physical significance. These paper presents expressions which are very important for planning the energetic processes18,19,20. The various derivatives also prove to be of physical interest:

The coefficient of thermal expansion:

pT

VV1

∂∂

=β . (4)

The isothermal compressibility:

Tp

VV1

∂∂

−=χ . (5)

The heat capacity at constant volume per mole:

V

v TUC

∂∂

= . (6)

The heat capacity at constant pressure per mole:

pp T

HC

∂∂

=χβ

+=2

vTVC . (7)

The velocity of sound:

s

20 V

pM1Vc

∂∂

−=

v

p

p

p

p

2

pT

TC

TV

M1

VT

TC

V

∂∂

−

∂∂

∂∂

−= ,

where M is molecular mass. The Joule-Thomson coefficient:

−

∂∂

=µ VTVT

C1

ppJ . (8)

CLASSICAL STATISTICAL THERMODYNAMICS

Revisited Cotterman model (CYJ)11

Revisited Cotterman EOS is based on the hard sphere perturbation theory. The average relative deviation for pressure and internal energy in comparison with Monte-Carlo simulations are 2.17% and 2.62% respectively for 368 data points9. The configurational free energy is given by: (9) perths

conf AAA +=

( )2

2

m

hs

134

TRA

η−

η−η= , (10)

2*

)2(

*

)1(pert

T

AT

AA += (11)

m4

1mm2

m

)2(m4

1mm1

m

)1(A

TRA,A

TRA ∑∑

==

τη

=

τη

= , (12)

7405.0=τ , 6D3πρ

=η ,

where η is packing factor, D is hard-sphere diameter. With help of configurational free energy we can calculate all configurational thermodynamic properties.

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AIAA-2002-3339

Expressions for calculation of configurational entropy and internal energy are shown in literature9. We carried out all other expressions for calculation of thermophysical properties. Tang-Tong-Lu model8-10 Tang-Tong-Lu model uses as the intermolecular potential a new two-Yukawa function. This function is found to mimic very closely the Lennard-Jones potential. Tang-Tong-Lu analytical model calculates thermodynamic functions of state on the base of salvation of the Ornstein-Zernike equation with help of perturbation theory. Comparisons with the computer simulation data indicate that the expressions developed yield better results (pressure, internal energy, free energy) than theory of Weeks at al.3, Barker and Henderson2. Results of pressure, internal energy and free energy obtained by TTL model are comparable with the most recent modified Benedict-Webb-Rubin EOS. The configurational free energy is given by:

210conf aaa

NkTA

++= , (13)

where a0 represents the free energy of the hard sphere fluid, a1 and a2 are perturbed first and second order parts.

( )2

2

0 134aη−

η−η= , (14)

( )( ) ( )

( )( ) ( )

σ

+

σ

−

σ

ηβε

−

σ

−

σ

ηβε+

+−

η−

−

+−

η−ηβε−=

3612

0

612

22

2

222

2

22

21

1

122

1

11

31

D92

D31

D91g8

D31

D9148

zz1

DzQ1zDzL

k

zz1

DzQ1zDzLk

D12a

(15)

( ) ( )

( ) ( ) ( )

( ) ( )

σ

+

σ

−

σ

⋅

−εηβ

−

+

−−εηβ

−=

3612

222

12122

22

12

21

21

24

2

22

14

1

21

3

22

2

D92

D31

D91

DzQ

D/k

DzQ

D/k24

DzQDzQzz

kk2DzQz2

k

DzQz2

k

D6a

(16)

( )[ ]( )[ ]

[ ]( )

,t1

texp)t(L12)t(S)t(Q

,Dzexpkk,Dzexpkk

33

202

101

η−−⋅η+

=

−σ⋅=−σ⋅=

(17)

( ) ( ) ( )η+η−η+η−η+η−= 2112t18t16t1)t(S 2232

( )

,

201

2/1gη−

η+= , η++

η

+= 21t2

1)t(L ,kT1

=β ,

σ=σ=σ= /0167.14z,/9637.2z,1714.2k 210 . In Eq. (15) and (16) is β Boltzmann factor. With help of configurational free energy we can calculate all configurational thermodynamic properties. Expressions for calculation of configurational entropy and internal energy are shown in literature8. We carried out all other expressions for calculation of thermophysical properties. STATISTICAL ASSOCIATING FLUID THEORY

(SAFT)25-51

Over the last fifthy years, quiet accurate models on the basis of statistical thermodynamics have been developed for predicting the thermodynamic properties for simple molecules. By simple we mean molecules for which the most important intermolecular forces are repulsion and dispersion with weak electrostatic forces due to dipoles, qudrupoles and higher multipole moments. Many hydrocarbons, natural constituents, simple organic and simple inorganic molecules fall within this cathegory. But a lot of other components such as electrolytes, polar solvents, hydrogen-bonded fluids, polymers, liquid crystals, plasmas and particularly mixtures do not belong to this group. The reason for this is that, for such fluids, important new intermolecular forces come important: Coloumbic forces, strong polar forces, complexing forces, the effects of association and chain formation... An important sort of these complex fluids consists of those that associate to form relatively long-lived dimmers or higher n-mers. This sort of fluids includes hydrogen bonding, charge transfer other other types can occur. The intermolecular forces involved are stronger than those due to dispersion or weak electrostatic interactions but steal weaker than forces due to chemical bonds. In recent years thermodynamic theories based on statistical thermodynamics have been rapidly developed. Fluids with chain bonding and association have received much attention in recent years. Interests for these fluids are prompted by the fact that they cover

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AIAA-2002-3339

much wider range of real fluids than spherical ones. A good theory for these fluids will be very beneficial to chemical engineering applications by reducing the number of parameters and making them more physically meaningful and more predictable. In technical practice energy processes are of vital importance. To calculate the thermodynamic properties of real Lennard-Jones (LJ) fluid Liu-Li-Lu (LLL) (revisited Cotterman) equation of state based on simple perturbation theory and SAFT-VR equation of state for LJ chain fluid, complex Tang-Lu (TL) analytical model with new radial distribution function based on SEXP approximation and s expansion, was applied. The developed RDF has been applied to the development of a new SAFT model. The presented model has been used to calculate several typical properties of LJ chains and associating LJ chains. This paper the first time discusses the accuracy of presented models in real engineering practice. The original derivation of SAFT models can be showed by Wertheim papers.40-43 They requires a comprehensive knowledge of graph theory to be fully understand.

assocchainsegres AAAA ++= (18) The residual Helmholtz energy is consisted from three terms representing contribution from different intermolecular forces. The first term Aseg represents segment-segment interactions. In the presented paper are segment-segment interactions represented through Lennard-Jones interaction potential. Each segment is characterized by its diameter σs, segment interaction parameter εs and each molecule is characterized with the number of segments, m.

The second therm Achain is the result of presence of covalent chain-forming bonds between the LJ segments.

The th

)

ird therm Aassoc is the result of site-site interactions between segments, for example we can mention hydrogen bonding. For the hydrocarbons the association therm is with no importance and will be neglected in our equations.

( ) (( )ABABassoc

chains

segres

,,d,T,A

m,d,A,,T,mAA

κερ

+ρ+εσρ= (19)

where ρ is the molar density, Liu-Li-Lu model26 The presented model is developed on the basis of SAFT and perturbation theory around hard sphere with new coefficients by fitting reduced pressure and internal energy data from molecular simulation.

perthsseg AAA += (20)

( )2

2

m

hs

1

34mTR

A

η−

η−η= ,

2*

)2(

*

)1(pert

T

AmT

AmA += (21)

m4

1mm2

m

)2(m4

1mm1

m

)1(A

TRA,A

TRA ∑∑

==

τη

=

τη

= , (22)

7405.0=τ , m6d 3

sπρ=η , (23)

The effective segment diameter ds is determined on the basis of Barker perurbation theory. We use a function developed in the work of Chapman at al.27

m1m025337.00010477.0T33163.01

T2977.01d*

*

s −+++

+= (24)

where :

ε

=kTT* (25)

According to Wertheim first order thermodynamic perturbation theory, the contribution to free energy due to chain formation of the LJ system is expressed as:

( ) (σ−= LJchain

glnm1NkT

A ) (26)

Johnson et al.29 gave a correlation result of the radial distribution function for LJ fluids dependent on reduced temperature and reduced density:

( ) ( ) ( )∑∑= =

−ρ+=σ

5

1i

5

1j

j1*i*ijs

LJ Ta1g (27)

Tang-Lu model

)aaa(mNkTA

210

seg++= , (28)

where a0 represents the free energy of the hard sphere fluid, a1 and a2 are perturbed first and second order parts (see Eq. (14) to (17)).

m6d 3

sπρ=η (29)

In the presented paper we used simplified exponential approximation (SEXP) of radial distribution function (RDF). These function is obtained analytically with help of first order solution of the mean spherical approximation of RDF and improved by SEXP:25

( )( )rgexp)r(g)r(g 10SEXP = (30)

The advantage of SEXP approximationm is attested by yielding much better RDF values around r=σ.

5


AIAA-2002-3339

The contribution to free energy due to chain formation is calculated with the same model as is presented in Eq. (26). In the case of Tang-Lu model we used as RDF pure analytical function (see Eq. (30)).

CLASSICAL THERMODYNAMICS- TRWW MODEL

The TRWW equation of state for pure hydrocarbons is given in terms of the dimensionless Helmholtz free energy function:

TR

ATR

ATR

A

m

r

m

ig

m+= (31)

In equation (31) Aig is related to the free energy ideal gas function, Ar presents the residual part, which corrects the ideal –gas part to real fluid behaviour. T represents the temperature and Rm universal gas constant. The general structure of ideal gas part and residual part is written as:

( )( ) )TT,

VV (,nexp1lna

alnaaalnTR

A

cc16

11iii

10

4i

ini

03

02

01

m

ig

=τ=δτ−−+

τ+τ+τ++δ=

∑

∑

=

= (32)

( ) ( )∑

∑∑

=

==

γ−τβ−ν−δα−δτ+

δ−δτ+δτ=

60

46i

2ii

2iiidit

i

45

16iiidit

i

15

1i

iditi

m

r

)exp(a

)cexp(aaTR

A

(33)

The equations of state for the different refrigerants result from the use of coefficients ai and different exponents ti, di, ei, αi and βi. In Eqs. (32) and (33) δ and τ represent inverse reduced volume and inverse reduced volume.

RESULTS AND COMPARISON WITH EXPERIMENTAL DATA

The constants necessary for the computation such as the characteristic rotation-, vibration-, electronic etc. temperatures are obtained from experimental data14,15,16. The vibration constants are obtained in NIST Chemistry Web Book page. The inertia moments are obtained analytically by applying the knowledge of the atomic structure of the molecule. Constants for Lennard-Jones potential are obtained from the literature of, Hirschfelder,Curtiss and Bird17. We carried out calculations for butane (C4H10). The comparison of our calculations with TRWW model, is presented in Figures 1-6. The most important data for calculation are presented in Table 2.

Figures 1-6 show the deviation of the results for butane in the saturated gas region between the analytical computation (TTL-Tang-Tong-Lu (1997) model, CYJ Chunxi-Yigui-Jiufang (1997) model, Liu-Li-Lu model (LLL) and Tang-Lu model (TL)), and TRWW model obtained by classical thermodynamics. The results for all models obtained by statistical thermodynamics show relatively good agreement. The computed vapour pressure, isothermal compressibility , isobaric molar heat and velocity of sound conform well for all models, obtained by statistical thermodynamics. Somewhat larger deviations can be found in the region of critical conditions due to the large influence of fluctuation theory (Pelt and Sengers18) and singular behaviour of some thermodynamic properties in the near-critical condition. The perturbation models on the basis of SAFT theory (TTL and LLL) yield surprisingly good results. The models on the basis os SAFT theory give in comparison with models on the basis of classical statistical thermodynamics better results, especially in high temperature region.

Table 2. The important constants for analytical calculation

SAFT Classical statistical ther. σ, σs (m) 3.87E-10 5.046E-10 ε, εs (J) 3.84E-21 3.219E-21 M (-) 2.134 -

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-5.00E-02

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

3.00E-01

140 160 180 200 220 240 260 280 300 320 340 360 380

Temperature (K)

Rel

ativ

e de

viat

ion

TTLCYJTLLLL

Fig. 1: Vapour pressure for butane

-0.080

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

0.000140 160 180 200 220 240 260 280 300 320 340 360 380

Temperature (k)

Rel

ativ

e de

viat

ion

TTLCYJTLLLL

Fig. 2: Isoscoric molar heat for butane

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-0.250

-0.200

-0.150

-0.100

-0.050

0.000140 160 180 200 220 240 260 280 300 320 340 360 380

Temperature (K)

Rel

ativ

e de

viat

ion

TTLCYJTLLLL

Fig. 3: Isobaric molar heat for butane

0.000

0.050

0.100

0.150

0.200

0.250

0.300

140 160 180 200 220 240 260 280 300 320 340 360 380Temperature (K)

Rel

ativ

e de

viat

ion

TTLCYJTLLLL

Fig. 4: Velocity of sound for butane

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-0.700

-0.600

-0.500

-0.400

-0.300

-0.200

-0.100

0.000

0.100

140 160 180 200 220 240 260 280 300 320 340 360 380

Temperature (k)

Rel

ativ

e de

viat

ion

TTLCYJTLLLL

Fig. 5: Volumetric expansion coefficient for butane

-0.500

-0.400

-0.300

-0.200

-0.100

0.000

0.100

140 160 180 200 220 240 260 280 300 320 340 360 380

Temperature (K)

Rel

ativ

e de

viat

ion

TTLCYJTLLLL

Fig. 6: Isothermal compressibility for butane

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REFERENCES 1Walas,S.M., “Phase Equilibria in Chemical Engineering”, 1984, Buttenworth Publishers, Boston. 2Barker, J.A., Hendersom, D., 1967, “Perturbation Theory and Equation of State for Fluids II, The Journal of Chemical Physics, Vol 47, No.11, pp: 4714-4721. 3Weeks.J.D., Chandler,D.,1971, “Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids”, The Journal of Chemical Physics, Vol.54, pp. 5237-5245 4Barker, J. A., D. Henderson 1976, ˝What is Liquid˝, Reviews of Modern Physics Vol.48,pp.587-671. 5Münster, A., 1974, ˝Statistical Thermodynamics˝, Springer - Verlag, New York. 6Lucas, K., 1992, ˝Applied Statistical Thermodynamics˝, Springer - Verlag, New York. 7Gray, C.G., Gubbins, K.E., 1984, ˝Theory of Molecular Fluids˝, Clarendon Press, Oxford. 8Tang,Y., Tong, Z., Lu, B.C.-Y, ”Analytical Equation of State Based on the Ornstein-Zernike Equation, Fluid Pfase Equilibria, 1997 Vol. 134, pp. 20-42. 9Tang, Y., Lu, B.C.-Y., Analytical Description of the Lennard-Jones Fluid and Its Application, AIChE Journal, 1997, Vol. 43, No. 9, pp. 2215-2226. 10Tang, Y., Lu, B.C.-Y., Analytical Equation of State for Lennard-Jones Mixtures, Fluid Phase Equilibria, 1998, Vol. 146, pp. 73-92. 11Chunxi, L., Yigui, L.,Jiufang, L. “Investigation and Improvement of Equations of State for Lennard-Jones Fluid, Fluid Phase Equilibria, Vol. 127 (1997), pp. 71-81. 12McClelland, B.J., 1980, ˝Statistical Thermodynamics˝, Chapman & Hall, London. 13Cengel, Y.A., Boles M.A., 1994, ˝Thermodynamics˝, McGraw Hill, New York 14Bellamy, L.J., 1980, ˝The Infrared Spectra of Complex Molecules˝, Chapman & Hall, London. 15Herzberg G., 1966, ˝Electronic Spectra of Polyatomic Molecules˝, Van Nostrand Reinhold Company, London. Toronto, Melbourne. 16Herzberg, G., 1984, ˝Infrared and Raman Spectra of Polyatomic Molecules˝, Van Nostrand Reinhold Company, New York. 17Hirschfelder,O.O., Curtiss, C.F., Bird, R.B., 1954. ˝Molecular Theory of Gases and Liquids˝, John Wiley & Sons, London 18Pelt, A,, Sengers,J.V., 1995, “Thermodynamic Properties of 1,¸1 Difluoroethane (R152a) in the Critical Region, The Journal of Supercritical Fluids (1995), Vol 8, pp 81-99. 19Delfs, U. , “Berechnung der Idealgaswarmekapazität der Fluor-Chlorderivate von Methan und Ethan unter Verwendung eines Modifizierten Urey-Bradley-Modells, Wärmetechnik“, Kältetechnik, VDI

FORTSCHRITT-BERICHT, Reihe Nr. 81, Reihe 19, 1995. 20Setzmann, U., Wagner, W., “A New Equation of State and Tables of Thermodynamic Properies for Methane Covering the Range from the Melting Line to 625 K at Pressures up to 1000 Mpa”, 1995J. Phys. Chem. Ref. Data, Vol. 20,.pp. 1061-1155. 21Younglove, B.A. and Ely, J.F., "Thermophysical Properties of Fluids. II. Methane, Ethane, Propane, Isobutane and Normal Butane," J. Phys. Chem. Ref. Data, 16:577-798, 1987. 22Tillner-Roth, R., Yokezeki, A., Sato, H., Watanabe, K., “Thermodynamic Properties of Pure and Blended Hydrofluorocarbon (HFC) Refrigerants”, 1997, Japan Society of Refrigerating and Air Conditioning Enineers, Tokyo. 23Jacobsen, R.T., Stewart, R.B., Jahangiri, M., S.G. Penoncello, “A New fundamental Equation for Thermodynamis Property Correlations”, Advances in Cryogenic Engineering, Vol.31, 1986, pp. 1161-1169. 24 Tillner-Roth, R., 1993. Die Thermodynamischen Eigenschaften von R152a, R134a und ihren Gemischen, PhD Thesis, Hannover. 25Tang, Y., Lu, B. C.-Y., A Study of Associating Lennard-Jones Chains by a New Reference Radial Distribution Function, Fluid Phase Equilibria, Vol. 171, 2000, pp. 27-44. 26Liu, Z.-P., Li, Y.-G., Lu, J.-F., Comparison of Equation of State for Pure Lennard-Jones Fluids and Mixtures with Molecular Simulation, Fluid Phase equilibria, Vol 173, 2000, pp. 189-209 27Chapman, W.G., Gubbins, K.E., Jackson, G., Radosz, M., New Reference Equation of State for Associating Liquids, Ind. Eng. Chem. Res., Vol. 29, No. 8, 1990, pp. 1709-1721. 28Condo, P.D., Radosz, M., Equations of State for Polymers, Fluid Phase Equilibria, Vol. 117, 1996, pp. 1-10. 29Johnson, J.K., Gubbins, K.E., Phase Equilibria for Associating Lennard-Jones Fluids from Theory and Simulation, Molecular Physics, Vol. 77, No. 6., 1992, pp. 1033-1053. 30Wei Y. S., Sadus, R.J., Equations of State for the Calculation of Fluid-Phase Equilibria, Equations of State for the Calculation of Fluid-Phase Equilibria, AIChE Journal, Vol. 46, No. 1, 2000, pp. 169-196. 31Banaszak, M., Chen, C.K., Radosz, M., Copolymer SAFT Equation of State. Thermodynamic Perturbation Theory Extended to Heterobonded Chains, Macromolecules, Vol. 29, No. 20, 1996, pp. 6481-6486. 32Huang, S.H., Radosz, M., Equation of State for Small, Large, Polydisperze and Associating Molecules, Ind.

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