10970
J. Phys. Chem. B 2007, 111, 10970-10974
Equations of State for Fluids: Empirical Temperature Dependence of the Second Virial Coefficients Jianxiang Tian*,†,‡ and Yuanxing Gui‡ Department of Physics, Qufu Normal UniVersity, Qufu 273165, People’s Republic of China, and Department of Physics, Department of Applied Mathematics, Dalian UniVersity of Technology, Dalian 116024, People’s Republic of China ReceiVed: March 14, 2007; In Final Form: July 12, 2007
In this paper, an empirical dependence of the second virial coefficients is derived from equations of state. The second virial coefficient B2 is found to be a linear function of 1/T1+β, where T is the temperature and β is a constant and has different value for different substances. Excellent experimental supports to this relationship are reported for nonpolar fluids, polar fluids, heavy globular molecule fluids, and quantum fluid He-4.
Introduction Equations of state play an important role in the predication of the thermodynamic properties for pure components and fluid mixtures. Historically, the van der Waals equation of state was the first equation to predict vapor-liquid coexistence.1 Later, the Redlich-Kwong equation of state2 improved the accuracy of the van der Waals equation by introducing temperaturedependence for the attractive term. Soave3 and Peng-Robinson4 proposed additional modification to more accurately predict the vapor pressure, liquid density, and equilibria ratios. Carnahan and Starling,5 Guggenheim,6 Malijevsky and Veverka,7 and Yelash et al.8 modified the repulsive term of the van der Waals equation of state to obtain accurate expression for hard sphere repulsion. All of the above equations follow the framework of the van der Waals type equations: z ) zrep + zattr. The alternative framework of equations of state is the Dieterici type equations with z ) zrep*zattr such as the Dieterici equation9 and the Carnahan-Starling-Dieterici10-13 equation. Recently, Roman and his co-workers14 introduced a variable exponent β in the power-law temperature dependence of the attractive term to both the van der Waals-type equations and Dieterici-type equations, and obtained accurate results for the vapor-liquid equilibrium properties. An important application of equations of state is to predict the virial coefficient of pure substances and fluid mixtures. The equation of state of a gas can be described by the virial expansion. In this expansion the compressibility factor is given in powers of the density, and the coefficients of the expansion are known as the virial coefficients. These virial coefficients are always temperature-dependent. Virial coefficients of real systems can be measured experimentally by a number of different techniques and methods.15-18 In the last century it was shown that the virial coefficients can be determined if the intermolecular forces between the molecules are known.19-20 Second, third, and fourth virial coefficients can be computed by evaluating certain integrals involving two, three, and four molecules, respectively. The expression for the second virial * To whom correspondence should be addressed.
[email protected]. Phone: 086-411-84706203. † Qufu Normal University. ‡ Dalian University of Technology.
E-mail:
coefficient, B2, is especially simple since it is given by minus one-half of the integral of the angle-averaged Mayer function over all possible values of the separation between the center of mass (i.e., reference points) of the two molecules. The way of determining B2 is by numerically evaluating the integrals. But the second virial coefficient can be obtained quite easily for molecules interacting through a pairwise potential of spherical symmetry. For hard spheres, square well potentials, and convex bodies, B2 can be determined analytically by using the equation of state. In this paper, we study the second virial coefficients of pure substances predicted by equations of state for hard spheres. The linearity of B2 vs 1/T1+β is derived under the work in ref 14. Excellent experimental supports of 26 pure fluids including nonpolar gases, polar fluids, quantum fluid He-4, and heavy globular molecular fluids are reported. Experimental data show β has different values for different fluids. Equations of State and the Second Virial Coefficients In this section we present the relation between the second virial coefficients and the temperature predicted by equations of state. For the sake of simplicity, we use the dimensionless variable y ) b/(4V), the packing fraction, instead of the molar volume V, with b being the so-called covolume. And the equation of state is described by the compressibility factor z instead of the pressure p, with z ) pV/(RT), where T is the temperature and R is the gas constant. Within two types of equations, the commonly used repulsive terms are the van der Waals repulison1 (vdWR), the CarnahanStarling repulsion5 (CSR), and the recently pubished MalijevskyVeverka repulsion7 (MVR), which is developed and found to be more accurate than the CSR at fluid packing fractions from y ) 0.30 to 0.49. The commonly used attractive term are the van der Waals attraction1 (vdWA), the Redlich-Kwong attraction2 (RKA), and the Dieterici attraction9 (DA). These repulsions and attractions are summarized in Table 1. Combinations of the attractions vs the repulsions can yield widely used equations of state such as the van der Waals equation, the CarnahanStarling-van der Waals equation of state,5 the RK equation of state,2 the Dieterici equation of state,9 the Carnahan-StarlingDieterici equation of state,10-13 etc. In our following calculation,
10.1021/jp072049y CCC: $37.00 © 2007 American Chemical Society Published on Web 08/28/2007
Equations of State for Fluids
J. Phys. Chem. B, Vol. 111, No. 37, 2007 10971
TABLE 1: Commonly Used Attractive and Repulsive Termsa repulsive terms
a
attractive terms
1/(1 - 4y) [1 + y + y2 - y3]/(1 - y)3 (1 - y)-3[(1 + 1.0560y + 1.6539y2 + 0.3262y3)/ (1 + 0.0560y + 0.5979y2 + 0.3076y3)]
vdWR CSR MVR
4ay/bRT1+β 4ay/[bRT1+β(1 + 4y)] exp(- 4ay/bRT1+β)
vdWA-β RKA-β DA-β
vdWA, RKA, and DA can be obtained by setting β special values as 0 or 0.5.
TABLE 2: Results of (br,ar) for Equations of State equations of state
br
-ar
critical value of B2r
vdWR + vdWA - β CSR + vdWA - β MVR + vdWA - β vdWR + RKA - β CSR + RKA - β MVR + RKA - β vdWR*DA - β CSR*DA - β MVR*DA - β
1/3 0.5217755369 0.5204252461 0.2599210499 0.3325774769 0.3321626935 1/2 1.528527531 1.535435537
9/8 1.382865235 1.381015473 1.282440701 1.462978022 1.461824317 2 3.199601253 3.215859768
-0.791666667 -0.861089698 -0.860590227 -1.022519651 -1.130400545 -1.129661623 -1.5 -1.671073722 -1.680424231
the T-1-β temperature dependence of the attractions is used. This means that the β-family of equations of state14 are used to calculate the second virial coefficient. In general, the nth virial coefficient (Bn) can be obtained from any equation of state via the following relationship:
Bn )
( )
1 ∂n-1z b 4 (n - 1)! ∂yn-1
(1)
y)0
For all van der Waals-type and Dieterici-type equations of state which are the combinations of the attractions and repulsions illustrated in Table 1, application of eq 1 yields the following general relationship for the second virial coefficient:
The parameters a and b for each equation can be obtained by solving the critical conditions of a pure fluid. Then the reduced second virial coefficient is derived as
B2r ) br +
ar
(3)
T1+β r
or
B2r′ ) br′ +
ar ′
(4)
T1+β r
with B2r ) B2/Vc, B2r′ ) B2Pc/(RTc), Tr ) T/Tc, etc. Results of (br,ar) for each equation are summarized in Table 2 for comparison. Historically, the form of eq 4 with β ) 1 was proposed to describe the second virial coefficient correlated with the Berthelot equation21 and recently used to analyze the second virial coefficient predicted by a modified RK equation of state.22 The prediction12 of the second virial coefficient by the Carnahan-Starling-Dieterici equation of state has been stated by using eq 3 with β ) 0. Equation 2 implies that the second virial coefficient B2 is a linear function of1/T1+β. In the following section, we will give the experimental supports for this conclusion. Results and Discussions
a B2 ) b - 1+β RT
(2)
We have collected the experimental data of B2 of 26 pure fluids including nonpolar fluids, polar fluids, quantum fluid He-
TABLE 3: The Slope ar, the Intercept br, and the Power β for Substancesa substance
br
ar
β
Tc
∆T (K)
Nop
AAD
ref
propene ethyne ethylene ethylene methane methane oxygen nitrogen argon CO2 He-4 BF3 SiCl4 water ethane benzene ammonia bromomethane diethyl ether dimethyl ether methanol fluoromethane phenol sulfur dioxide 1, 4-dimethylbenzene 1, 3-dimethylbenzene 1-butanol 1-propanol
-0.0001 -0.1942 0.2185 0.1677 0.3652 0.4045 0.2861 0.3420 0.3193 0.2243 0.0207 -2.9065 -3.0204 0.0295 0.2200 -0.2985 -0.3226 -1.1663 -0.8592 -1.0424 -0.5081 -0.0691 -0.5619 -0.0705 -0.6752 -0.5477 -0.7230 -0.8015
-1.1651 -0.9915 -1.4166 -1.3884 -1.5342 -1.5447 -1.4440 -1.4911 -1.4675 -1.4707 -1.3157 0.5131 1.1203 -1.5127 -1.4249 -0.9844 -1.1641 -0.3519 -0.5641 -0.5634 -0.7840 -1.5334 -1.0598 -1.2473 -0.7269 -0.5656 -0.5240 -0.4600
1.1632 2.7844 0.7889 0.8679 0.4588 0.3645 0.6470 0.5305 0.4963 0.8352 0.0717 -1.8740 -1.3840 1.8071 0.8421 1.7067 2.3893 3.3843 3.0316 2.8096 4.2393 1.6118 2.4268 1.7217 2.7317 3.4781 3.9653 4.7569
365.57029 308.330 282.3530 282.65031 190.56432 190.56432 154.58130 126.19230 150.68736 304.128232 5.195336 260.9038 507.0038 647.1030 305.3230 562.0530 405.430 464.030 466.630 400.030 512.530 315.030 694.330 430.830 616.230 617.130 563.030 536.830
90-575 200-310 200-470 273.15-625 100-640 273.15-623.15 70-495 98-340 120-340 283.14-363.15 2-40 200-900 200-900 500-1000 195-620 300-625 240-595 250-330 285-430 275-370 315-625 275-420 430-620 270-470 350-560 380-435 350-573.15 295-570
72 12 13 33 48 16 9 29 26 7 13 71 71 17 14 12 13 9 9 8 13 8 8 10 10 5 12 9
0.0053 0.0074 0.0063 0.0018 0.0599 0.0017 0.0863 0.0315 0.0040 0.0070 0.0328 0.0050 0.0016 0.0073 0.0294 0.0092 0.0221 0.0008 0.0055 0.0013 0.0353 0.0060 0.0026 0.0015 0.0065 0.0003 0.0519 0.0182
29 16 16 31 33 34 16 35 36 37 16 39 39 40 16 16 16 16 16 16 16 16 16 16 16 16 16 16
a
Superscripts in the Tc column denote the source literatures of the critical parameters of substances; Nop ) number of data points.
10972 J. Phys. Chem. B, Vol. 111, No. 37, 2007
Tian and Gui
Figure 1. The linearity of B2r vs 1/Tr1+β for nonpolar fluid argon in the temperature range [120, 340] K. Dotted line: Experimental data from ref 34. Solid line: This work with β ) 0.4963.
Figure 2. The linearity of B2r vs 1/Tr1+β for polar fluid water in the temperature range [500, 1000] K. Dotted line: Experimental data from ref 40. Solid line: This work with β ) 1.8071.
4, and heavy globular molecular fluids to test the linearity of B2 vs 1/T1+β. The results are stated in the figures and the tables. Figure 1 is the linearity of B2 vs 1/T1+β with β ) 0.4963 reported for the standard testing fluid argon. One can conclude that the linearity holds well, with the average absolute deviation (AAD) of 0.0040. As is established in Table 3, for nonpolar gases, β experimentally has a positive value less than 1. Figure 2 shows the linearity for polar fluid water with AAD of 0.0073. Table 3 tells us that β experimentally has a value of more than 1 for polar fluids. Figure 3 is the linearity of B2 vs 1/T1+β with β ) -1.3840 for heavy globular molecular fluid SiCl4 with AAD of 0.0016. A negative value of β is also suggested for heavy globular molecular fluid BF3 in Table 3. All that can be said about quantum fluid He-4 is that the experimental second virial coefficient given in ref 16 is crude, although the AAD of 0.0328 is obtained. Although the signs of the slopes for the heavy globular molecular fluids are opposite to the ones for other fluids, we have dBr/dTr > 0 for all the fluids considered here. The reason is that the signs of the power (-1-β) for the
heavy globular molecular fluids are opposite to those for other fluids, as is clearly shown in Table 3. In general, the second virial coefficient is expressed as the equations of the square well form,23 or a polynomial24-26 of the reduced inverse temperature 1/Tr, or a polynomial27 of 1/xTr, and other forms.18,28 Comparing with previous forms, eq 3 seems to behave with both simplicity and accuracy. Table 3 tells us that the relationship of the second virial coefficient vs the temperature experimentally obeys the linearity of B2 vs 1/T1+β with high accuracy, not only for nonpolar fluids but also for polar fluids, even for heavy globular molecular fluids. The intercept and slope of eq 3 for each substance are different from the ones from each equation of state, as established in Tables 2 and 3. Additionally, the p - T isochores from these equations with β apart from zero are clearly inconsistent with the known near linearity of p vs T at constant density,41 over the entire range from the perfect gas to the compressed liquid, although they lead to accurate thermodynamic properties at liquid-vapor equilibrium.14 Thus new
Equations of State for Fluids
J. Phys. Chem. B, Vol. 111, No. 37, 2007 10973 where Ci are the constant coefficients. This result coincides with the current popularly used expressions for B2 as polynomial functions of 1/T for real fluids.16 Our work denotes that eqs 2 and 8 have identical effects in describing the temperaturedependent second virial coefficients. We think it is a mathematical problem. Conclusions
Figure 3. The linearity of B2r vs 1/Tr1+β for heavy globular molecular fluid SiCl4 in the temperature range [200, 900] K. Dotted line: Experimental data from ref 39. Solid line: This work with β ) -1.3840.
equations of state should be developed to accurately predict the second virial coefficient of pure substances. But both the van der Waals-type and the Dieterici-type equations of state give out the qualitatively correct relationship of the second virial coefficient vs the temperature as eqs 2 and 3. Calculations of B2 have commonly been performed by treating the molecules as rigid bodies. In this case, only intermolecular degrees of freedom are relevant, which can be considered in the quasiclassical approach:42,43
B2 )
1 16π2
∫ (1 - e-U(R,Ω)/kT) dR dΩ
(5)
where U(R,Ω) is the interaction energy of the pair, Ω represents the Euler angles, R is the separation, and k is Boltzman’s constant. For molecules with a spherically symmetric intermolecular potential energy function, B2 is given by16,44
B2 ) -2πNA
∫ (e-U(R)/kT - 1)R2 dR
(6)
where NA is the Avogadro number. When the interaction energy of the pairs is available, B2 can be obtained by integrating eq 5 or eq 6 directly. The interaction energy originates from longrange dispersion forces, overlap repulsive forces, and valence forces, and can be determined by solving the electronic Schrodinger equation for all relevant values of the molecule coordinates.44 Until now, the interaction energy between real molecules is known to fall off fairly rapidly with separation and tends to be zero at large molecule separations.44 Put in another way, instead of the direct integration to eq 5, one can first expand the integrated function at U(R,Ω) ) 0 and obtain
B2 ) 1 16π2
∫
(
)
U(R,Ω) U2(R,Ω) U3(R,Ω) + + ... dR dΩ (7) kT 2k2T2 6k3T3
By leaving out the expressions of U(R,Ω), the integration above behaves such forms as
B 2 ) C0 +
C1 C 2 C3 + 2 + 3 + ... T T T
(8)
In this paper, the linearity of of B2 vs 1/T1+β is derived from equations of state. The validity of this linearity is tested only on pure substances. It is found that this linearity holds not only for nonpolar fluids, but also for polar fluids, even for heavy globular molecular fluids and quantum fluid He-4. β has different values for difference fluids and is very sensitive to the accuracy of the experimental data. Experimental data of the second virial coefficient denote that this linearity is valid in a large temperature range mostly across the critical point. For some fluids such as propene, the linearity holds in the whole temperature range from the subcritical state to the supercritical state where the experimental data of the second virial coefficient are available. Because the second virial coefficient is related to the pairwise intermolecular forces,45 this linearity offers us a way to better understand the pairwise interactions in fluids, which is an incompletely known problem. At the same time, this linearity will be helpful for the development of certain equations of state such as the Ihm-Song-Mason equation of state,46-47 in which the second virial coefficient is involved as a parameter. Also some thermodynamic properties of fluids related to the second virial coefficient will be correlated.16 For example, following this work, the zero-pressure isothermal Joule-Thomsom coefficient may be temperature dependent in such a form as φ ) b - (2 + β)a/(T1+β). In the future, we also would like to consider these problems and the applications of this linearity to the excess and cross second virial coefficient of mixtures. These are our works in progress. Acknowledgment. The National Natural Science Foundation of China under Grant No. 10573004, China Postdoctoral Foundation, the Natural Science Foundation of Shandong Province under Grant No. Y2006A06, and the Science Research Starting-up Foundation from QFNU and DUT support this work. J.T. thanks Prof. Yunjie Xia at QFNU, Dr. Hua Jiang at USTC, and Dr. Shenming Wang at the University of Laval for their kind help and comments on this paper. References and Notes (1) van der Waals, J. D. On the Continuity of the Gaseous and Liquid States; Rowlinson, J. S., Ed.; North Holland: Amsterdam, The Netherlands, 1988. (2) Redlich, O.; Kwong, J. N. S. Chem. ReV. 1949, 44, 233. (3) Soave, G. Chem. Eng. Sci. 1972, 27, 1197. (4) Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. (5) (a) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (b) Carnahan, N. F.; Starling, K. E. AIChE J. 1972, 18, 1184. (6) Guggenheim, E. A. Mol. Phys. 1965, 9, 199. (7) Malijevsky, A.; Veverka, J. Phys. Chem. Chem. Phys. 1999, 1, 4267. (8) Yelash, L. V.; Kraska, T.; Deiters, U. K. J. Chem. Phys. 1999, 110, 3079. (9) Dieterici, C. Ann. Phys. Chem. Wiedemanns Ann. 1899, 69, 685. (10) Sadus, R. J. J. Chem. Phys. 2002, 116, 5913. (11) Sadus, R. J. J. Chem. Phys. 2001, 115, 1460. (12) Sadus, R. J. Phys. Chem. Chem. Phys. 2002, 4, 919. (13) Sadus, R. J. Fluid Phase Equilib. 2003, 212, 31. (14) Roman, F. L.; Mulero, A.; Cuadros, F. Phys. Chem. Chem. Phys. 2004, 6, 5402. (15) Fontalba, F.; Marsh, K. N.; Holste, J. C.; Hall, K. R. Fluid Phase Equilib. 1988, 41, 141.
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