PREDICTION OF THE DENSITIES AND PRESSURES OF ETHANE ON THE COEXISTENCE

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1 PREDICTION OF THE DENSITIES AND PRESSURES OF ETHANE ON THE COEXISTENCE A. Abbai *, Faulté des Sienes, Département de Chimie, Université Badji-Mokhtar, B. P.12, El-Hadjar, Annaba (23200), Algeria, azzedineabbai@hotmail.om, Tél : , Fax: Abstrat: The oexisting liquid and vapor densities and the pressure of ethane are predited from a global equation of state in the viinity of the ritial point. Speifially, a rossover thermodynami potential is used to represent the thermodynami data of ethane aurately. The hook of the retilinear diameter very lose to the ritial point is reprodued using this model and the saling laws. Keywords: Coexisting densities, ritial point, rossover model, ethane, equation of state, pressure. INTRODUCTION: The ritial thermodynami behavior of fluid systems has been the subjet of several investigators [1], onduted in analogy with the 3-dimensional Ising-like systems. The thermodynami surfae of fluids exhibits a singularity at the ritial point. This behavior an be haraterized in terms of saling laws with universal ritial exponents and universal saling funtions [1-3]. Unlike 3-dimensional Ising-like systems, fluids exhibit a lak of vaporliquid symmetry in its oexistene-urve. In the present work, we give the appliation of the rossover model to the oexistene-urve diameter to ethane, and show that we an reprodue the liquid-vapor oexistene urve diameter only by using a newly formulated equation of state for ethane [4]. FUNDAMENTAL EQUATION: Starting from earlier work Nioll et al. [5-7], we have developed a rossover model to represent the thermodynami properties of fluids in the ritial region [5]. This rossover model is based on the renormalization group theory of ritial phenomena to inlude the ooperative effets assoiated with the long-range ritial flutuations up to a maximum mirosopi wavenumber. Let ρ be the density, T the temperature, P the pressure, µ the hemial potential and A/V the of Helmholtz free energy per unit volume. We make these properties dimensionless with the aid of the ritial parameters [4, 8]: T ρ = ρ ρ, T = T T, P = PT P T, µ = µρ P T, Α= AT P VT (1)

2 Α = Α ρ µ ( T ) Α 0 0 ( T ). (3) In addition we define ρ = ρ 1, T = T + 1, µ = µ - µ 0 ( T) and (2) Α=Α ρ µ 0( T) Α0( T). Here µ 0 ( T) and A 0 ( T) are analyti bakground funtions of T subjet to the onditions that at the ritial temperature µ ( T = T ) = 0 and A 0 ( T = T) = 1. Classial equations of state for the Helmholtz free energy density A imply that the lassial part A l has an asymptoti expansion of the form: (3) Α l 1 u a a a a = tm + M + M + M + tm + t M 2 4! 5! 6! 4! 2!2!. (4) W here t and M are temperature-like and density-like variables related to T and ρ in a manner to be speified below. The oeffiient u 0 of the M 4 term in (4) is u0 =uλ, where Λ a dimensionless utoff wave number [9] is. In order to obtain a fundamental equation that an be applied in a large range of densities and temperatures around the ritial point we renormalize t, M and u 0 to obtain: Αr 1 2 u a /2 a /2 = tm TD + M D U + M D VU + M D U 2 4! 5! 6! a /2 a / tm TD U + t M T DU - t K 4! 2!2! 2 where the funtions T, D, U, V and K are defined by T = Y (2-1/ ν)/, D = Y η /, U = Y1/ (2a 1)/2 V = Y, K = ν αuλ Y α / ν 1 The rossover funtion Y in Eq. (6) is to be determined from: 1/2 1 (1 u) Y = u 1+ Λ2 κ 2 Y1/ with (5) (6) (7) κ 2 = tt uλm 2 DU, (8)

3 The onstants ν, η, and u * a are universal ritial exponents, is another universal onstant presented in Table 1. The variable κ is related to the inverse orrelation length and is a measure of distan e from the ritial point. The effet of the lak of symmetry in fluids an be inorporated by the mixing of the field variables t and M through the following d efinition Α = Αr Αr M t Αr t M (9) where is the mixing parameter. The variables t and M are related to T and ρ as: t = t T + Αr M, t M = ρ( ρ d1 T) + ( Αr t) M (10) (11) where d 1 is a onstant representing global symmetry. The expression of (9) for A is to be substituted into (3). To speify the Helmholtz free energy density ompletely the analyti bakground funtions µ 0 ( T) and A 0 ( T) are represented by trunated Taylor expansions through the relations: j= 4 j= 4 µ 0 ( T ) = µ ( ) j and Α 0 ( T ) = 1+ Α j( T ) j j T j= 1 j= 1 (12) APPLICATION TO ETHANE: Ethane is an important substane due to Its presene In most natural gases and petroleum. The rossover paramete rs u and Λ, the saling-field parameters, t, ρ and d 1, the lassial parameters a 05, a 06, a 14 a 22 and the bakground parameters à j whih an be determined by fitting the rossover model to the P-ρ-T data of Funke et al. [10] assoiated with those of Claus et al. [11]. The system dependent parameters are presented into Table 1. We denote the liquid density by ρ liq. and the vapor density by ρ vap., ρ as the ritial density, and T the redued temperature. Then lose to the ritial point, the renormalization-group theory predits [12] the following equation: ρd = ( ρ + ρvap) liq 2ρ (1-α ) (1-α + ) = 1+ d1 T + ds1 T + ds2 T +..., (13) where α is the ritial exponent that haraterizes the divergene of the speifi heat at onstant volume,, and a = a ν are other ritial exponents given in Table 2. A omparison of the obtained equation of state [4] with experimental data of Pestak et al. [13] and Douslin

4 and Harrison [14] and those of Funke et al. [10] retilinear diameter is presented in Figure 1. In this omparison the offset of Pestak s Finally, we present in Figs. 1-3, the omparison of the rossover model with the experimental data of the retilinear diameter, liquid and vapor densities and their differenes, as reported by Funke et al. [10], Pestak et al. [13], Douslin and Harrison [14], and those reported by Khazanova and Sominskaya [15]. The data sets reported by Funke et al. [10] show satisfatory onsisteny, if one keeps in mind that none of the saturated density-data for ethane were used to determine any of the system-dependent parameters in the rossover equation of state. The offset of Pestak s data is probably due to the fat that these density-data were obtained from measurements of the refrative index, then the data of this latter were onverted to densities using values from Lorentz-Lorentz formula. However, it is well known that the Lorentz-Lorentz formula is only valid at low densities, and even at moderate densities the deviations of the Lorentz- Lorentz formula from the true values ould amount to as muh as a few perent. Table 1: Critial exponents ν= 0.630, η= 0.033,α=2-3ν=0.110, =0.51, a=2.1, u * =0.47 Table 2: System-dependent onstants for C 2 H 6 Critial parameters: T = , P = MPa, ρ = 6.86 mol L -1 Crossover parameters: u = , Λ = Saling-field parameters: t = , ρ = , = P-ρ-T bakground parameters: Ã 0 = -1, Ã 1 = , Ã 2 = 3.988, Ã 3 = , Ã 4 = 7.541, d 1 = Classial Parameters: a 05 = , a 06 = 1.453, a 14 = 0.299, a 22 = Range of validity of the equation of state: T and ρ/ ρ 1.68

5 ρd This work Pestak et al. [13] Funke et al. [10] Douslin and Harrison [14] T /T 1 Figure 1: Coexistene diameter liq vap ρ d = ( ρ.+ ρ.) for ethane as a funtion of redued 2ρ temperature. The squares indiate the experimental data obtained by Pestak et al., Funke et al. and Douslin and Harrison. The Solid urve represents the values from the rossover model ρ This work Pestak et al. [13] Funke et al. [10] Douslin and Harrison [14] Khazanova and Sominskaya [15] ρ This work 0.7 Pestak et al. [13] Funke et al. [10] Douslin and Harrison [14] T/T T/T Figure 2: Liquid and vapor densities of ethane as a funtion of redued temperature. The squares indiate the experimental data obtained by Pestak et al., Funke et al. and Douslin and Harrison. The Solid urve represents the values from the rossover model. Figure 3: Density differenes liq vap ρ d = ( ρ. ρ.) of ethane as a funtion of 2ρ redued temperature. The squares indiate the experimental data obtained by Pestak et al., Funke et al. and Douslin and Harrison. The Solid urve represents the values from the rossover model.

6 REFERENCES: [1] SENGERS J. V., and SENGERS LEVELT, J. M. H., Ann. Rev. Phys. Chem., Vol. 37, 1986, p [2] SENGERS J. V., and SENGERS LEVELT, J. M. H., Int. J. Thermophys., Vol. 5, 1984, p. 195 [3] ABBACI, A., J. So. Alger. Chim, Vol. 4(1), 1994, p. 97. [4] ABBACI, A., J. Mol. Liq., in press. [5] NICOLL, J. F., Phys. Rev. A, Vol. 24, 1981, p [6] NICOLL, J. F., and BHATTACHARJEE, J. K., Phys. Rev. B, Vol. 23, 1981, p [7] NICOLL, J. F., and ALBRIGHT, P.C. Phys. Rev. B, Vol. 31, 1985, p [8] ABBACI, A., Ph.D. Thesis, University of Maryland at College Park, [9] CHEN, Z. Y., ABBACI, A., TANG, S. and SENGERS, J. V., Phys. Rev. A, Vol. 42, 1990, p [10] FUNKE, M., KLEINRAHM, R., and WAGNER, W., J. Chem. Thermodynamis, Vol 34, 2002, p [11] CLAUS, P., KLEINRAHM, R., and WAGNER, W., J. Chem. Thermodynamis, Vol. 35, 2003, p [12] LEY-KOO, M., and GREEN, M. S., Phys. Rev. A., Vol. 23, 1981, p [13] PESTAK, M. W., GOLDSTEIN, R. E., CHAN, M. H. W., de BRUYN, J. R., D. A. BALZARINI, and ASHCROFT, N. W., Phys. Rev. B, Vol. 36, 1987, p [14] DOUSLIN, D. R, and HARRISON, R. H., J. Chem. Thermodynamis, Vol. 5, 1973, p [15] KHAZANOVA, N. E., and SOMINSKAYA, E. E., Russ. J. Phys. Chem., Vol. 45, 1971, p. 88.