Ind. Eng. Chem. Res. 2004, 43, 2271-2279
2271
Investigation of the Phase Equilibria for Nonpolar Chainlike Fluids by the Yukawa Potential and Renormalization-Group Theory Dong Fu and Yi-Gui Li* Department of Chemical Engineering, Tsinghua University, Beijing 100084, People’s Rebublic of China
An analytical equation of state based on the Yukawa potential is extended to chainlike fluids by use of the renormalization-group (RG) theory. The vapor-liquid equilibria and critical properties for both the spherical and chainlike nonpolar fluids are investigated. The average relative deviations for Tc, pc, and Fc obtained by RG correction are 0.5%, 1.7%, and 0.9%, respectively. Results show that the RG theory used in this paper is applicable to the vaporliquid equilibria from the low temperature up to the critical region. 1. Introduction In recent years, interest in the study on critical properties increased rapidly. The fluid becomes inhomogeneous near the critical point because the longwavelength density fluctuations are important to the Helmholtz free energy. Although the statistical associating fluid theory (SAFT)1,2 is widely used in the calculation of thermodynamic properties for chainlike fluids,3-6 it fails to describe the critical properties appropriately because the long-wavelength density fluctuation is ignored. A renormalization-group (RG) theory proposed by Wilson7 and developed by White and Zhang8-11 may be used to calculate the critical properties. Lue and Prausnitz12,13 developed an equation of state (EOS) for fluids close to and far from the critical points by combining liquid-state theory with RG theory and applied this approach successfully to square-well fluids for CH4, CO2, and n-C4H10 and their binary mixtures. Jiang and Prausnitz14 presented a global EOS for chain fluids (EOSCF) + RG equation, combining RG theory with the original Hu’s EOSCF15 to obtain n-alkanes as squarewell chain fluids. Duan et al.16 used the RG theory to obtain pure water and methanol as Stockmayer fluids under supercritical and subcritical conditions. Mi et al.17 applied it to chainlike fluids by combining the SAFT EOS and the Lennard-Jones (LJ) potential. Kiselev18 developed a general method to transform a classical EOS into a crossover EOS in the critical region. Kiselev and co-workers proposed the crossover cubic EOS18 and the crossover SAFT EOS.19,20 Hu et al.21 used the crossover SAFT EOS to calculate the thermodynamic properties of some common supercritical fluids. Recently, interests in the study on the Yukawa potential increased rapidly because it can describe both the attractive dispersion and the charged double-layer repulsive interactions instead of the classical Derjaguin-Landau-Verwey-Overbeek colloid theory. Moreover, the Yukawa potential can be solved analytically from the Ornstein-Zernike (OZ) equation by mean spherical approximation (MSA)22-24 while the LJ potential does not give a soluble OZ equation. Lin et al.25,26 used the two Yukawa potentials to describe the osmotic pressure for aqueous bovine serum albumin-NaCl * To whom correspondence should be addressed. Tel.: 86106278-4540.Fax: 8610-6277-0304.E-mail:
[email protected].
solutions and some other protein systems. Fu et al.27 used one Yukawa potential with the RG theory to study the phase behavior of colloidal model systems. On the basis of the analytical EOS with the Yukawa potential, Liu et al.6 extended it to chainlike and associating fluids in the framework of SAFT and used it to describe the saturated pressures and coexistence curves for 42 pure real fluids (including nonpolar, polar, chainlike, and associating fluids). However, in the region very near to critical points, the EOS still overestimates the critical points. In the present paper, an extended Yukawa EOS proposed by Liu et al.6 is combined with the RG theory to calculate the vapor-liquid equilibria and critical properties for pure spherical and chainlike nonpolar fluids. The results obtained by the RG correction are satisfactory. 2. Theory 2.1. RG Theory. For a closed system at temperature T with particle number N and volume V, the canonical partition function is expressed as
Q)
1 3N
N!Λ
∫drbN e-βE ) ∑e-βA[F(rb)]
(1)
F(r b)
where dr bN ≡ dr b1 dr b2 ‚‚‚ dr bN, Λ is the thermal wavelength, E represents the total potential energy, A[F(r b)] is the Helmholtz free energy, and F(r b) is the “instantaneous” density function, which can be expressed in Fourier space:
F(r b) )
∫
V dk B Fˆ (k B) e-ikBbr (2π)3
(2)
where ∧ stands for the Fourier transform and B k is a wave vector with the magnitude k ) 2π/λ, in which λ is the wavelength. Fˆ (k B) has the same dimensionality as F(r b) and represents the amplitude of a density wave packet. The summation ∑F(rb)e-βA[F(rb)] is made over all physically acceptable densities including both long- and shortwavelength contributions. Far away from the critical point, the fluid density fluctuation manifests only in a short length scale and the thermodynamic properties are dominated by the local structure. However, near the critical point, the fluid density is characterized by long-
10.1021/ie0307232 CCC: $27.50 © 2004 American Chemical Society Published on Web 03/31/2004
2272 Ind. Eng. Chem. Res., Vol. 43, No. 9, 2004 Table 1. Regressed Parameters for Some Nonpolar Fluidsa Ar Xe N2 CO2 CF4 CCl4 cyclohexane methylbenzene m-xylene methane ethane propane butane pentane hexane heptane octane nonane decane a
m
k-1/K
σ/10-10 m
ψ
1 1 1 1.92 1.34 1.49 1.35 2.10 2.25 1 1.05 1.35 1.37 1.69 1.89 2.20 2.40 2.42 3.16
127.95 247.01 107.80 178.66 164.17 377.14 404.74 339.13 337.39 162.50 253.27 265.27 301.36 300.09 305.15 300.58 300.47 312.01 288.49
3.46 4.05 3.69 2.82 3.81 4.53 4.92 4.10 4.22 3.83 4.24 4.26 4.57 4.45 4.48 4.42 4.47 4.63 4.20
6.01 5.44 5.73 10.05 7.59 8.20 6.61 9.94 10.47 5.52 5.53 7.36 7.50 8.08 8.45 9.51 9.65 10.50 11.00
The experimental data are taken from the literature.29,30
range correlations and long-wavelength components of F(r b) make most of the important contributions to the thermodynamic properties. To describe the role played by long-range correlations, a wavelength λs ) 2π/ks that is comparable to the size of a molecule is used to distinguish the short and long wavelengths. Thus, the partition function is rewritten as
Q)
∑ e-βA [Fj (rb)]-βU s
s
Fs(r b)
s
)
∑ e-β∫drb f [Fj (rb)]-βU s
s
s
(3)
Fs(r b)
where Fjs(r b) is the density function with only shortwavelength fluctuations (0 < λ e λs) and Fs(r b) is the density function with long wavelengths (λs < λ < ∞). As[Fjs(r b)] and fs[Fjs(r b)] represent the Helmholtz free energy and the Helmholtz free energy density due to the densities with wavelength λ from 0 to λs, respectively. They can be expressed by mean field theory. Us accounts for the Helmholtz energy contributed by the densities with wavelength λ from λs to ∞. The essential idea of the RG theory is that the contributions from the long-wavelength density can be removed successively from Us and incorporated into As[Fjs(r b)]. Subsequently, the set of function summation in the partition function becomes smaller. At the end of this sequence, Us disappears and an expression for the partition function that includes all levels of fluctuations is obtained. The contribution to the partition function from density fluctuations with wavelength longer than λs comes primarily from intermolecular attractions. The potential energy due to intermolecular attractions can be written by using the mean field approximation:
Φ)
∫ ∫drb′ F(rb) F(rb′) u(|rb-rb′|)
1 dr b 2
(4)
Supposing that the density wave packets with the wavelength λs < λ < λl have been subtracted from Us and incorporated into As[Fjs(r b)], the partition function becomes
Q)
l
l
(5)
l
Fl(r b)
where fl[Fjl(r b)] is the free energy density after one RG iteration, Fjl(r b) is the density function with wavelength b) is the density function fluctuations 0 < λ e λl, and Fl(r with wavelengths λl < λ < ∞. Ul accounts for the Helmholtz energy due to the densities with wavelength from λl to ∞. When the densities with wavelength λs < b)] in eq 3 become Ul λ < λl are subtracted, Us and fs[Fjs(r and fl[Fjl(r b)], respectively. Equation 5 means that, after the first RG iteration, the value of the free energy contributed by the densities with wavelength from 0 to λl has been obtained and only the densities with wavelength from λl to ∞ should be treated in the next iteration. In this iteration, the difference in free energy densities is equal to
b)] ) fl[Fjl(r b)] - fs[Fjs(r b)] δfl[Fjl(r
(6)
Equation 6 only shows the result of δfl[Fjl(r b)] but does not show the procedure for this result. By using the principle that the Helmholtz free energy remains invariant in the first iteration, we may express the procedure in detail as follows:
-
∑
A[F(r b)] ) -
0λsUs; hence, ∑∞>λsUs decreases and becomes ∑∞>λ′Us′. At the same time, fs[Fjs(r b)] increases and becomes fs′[Fjs′(r b)]. Equation 7 can also be written as
-
∑
A[F(r b)]
0
