Ind. Eng. Chem. Res. 2002, 41, 1107-1112
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Accurate Correlation of the Second Virial Coefficients of Some 40 Chemicals Based on an Anisotropic Square-Well Potential Jennifer Howard McFall, David Scott Wilson, and Lloyd L. Lee* School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019
A statistical mechanics inspired formulation for the second virial coefficient (B2) correlation is presented. We revisit the formula proposed some 40 years ago based on the simple square-well potential (Hirschfelder, J. O.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley: New York, 1964). To account for polar and anisotropic effects, we reformulate in terms of an anisotropic square-well potential that also depends on the Euler angles of orientation of the moleulces. A new mean-field second virial formula is derived. Only four parameters are needed in the correlation: i.e., the repulsive volume, Br, the attractive (range) volume, Va, the interaction energy (well depth), , and a temperature correction factor, Dt, due to anisotropy. For some 40 chemicals, such as methane, ethane, propane (hydrocarbons), CO, CO2, oxygen, nitrogen, hydrogen, and water vapor, high accuracy in the correlation is achieved: most deviations are within 1-5% for the gases studied from low temperatures to temperatures above the Boyle temperature. The parameters Br, Va, , and Dt can be expressed in terms of the critical volume, critical temperature, and acentric factor. Comparison with similar industrial correlations (e.g., Hayden, J. G.; O’Connell, J. P. Ind. Eng. Chem. Process Des. Dev. 1975, 14, 209; Tsonopoulos, C. AIChE J. 1974, 20, 263) shows that the present approach is generally more accurate. I. Introduction Second virial coefficients are important in gas property estimation, e.g., in equations of state, P-V-T properties, phase equilibrium (chemical potentials), free energy, and enthalpy calculations. The compressibility factor, Z ) PV/nRT, can be approximated up to about 1/ of the critical density with the help of the second 3 virial coefficients alone. Their fundamental relation to thermodynamics and intermolecular forces cannot be overemphasized. A number of useful correlations for industrially important gases have been proposed in the last 30 years.1-6 They were based on different principles: some were of molecular origin; others were based on chemical association and/or on empirical curve fitting. It is desirable to have formulas that are based on the molecular-level theories and have a sound physical basis in the sense that, although the database used for the correlation is limited, the correlation parameters can be related to the molecular characteristics (such as the molecular size, attractive energy depth and range, and some measures of anisotropyspolarity or nonsphericity), so the properties of new gases with similar characteristics can also be estimated with confidence. We first review some of the existing approaches below. Hirschfelder et al.1 proposed a second virial formula for gases based on the square-well potential. This formula captures the essential features of the temperature dependence of most second virial coefficients. For example, the negative region at low temperatures is given by an inverted exponential relation Va[1 - exp(β)], the rise of values up to and beyond the Boyle temperature, and then the positive tail at very high temperatures giving the residual hard-core volume Br. In a mean-field way, it embodies the entire second virial behavior. This behavior was captured outstandingly in Sherwood and Prausnitz2 for some 10 gases in 1964.
Nothnagel et al.3 applied chemical theory to vapor imperfections. This theory attributes deviations from the ideal behavior to dimerization, or chemical association. Dimerization equilibrium constants for 178 pure fluids were obtained from experimental data. The method is particularly useful for highly polar and hydrogen-bonding fluids. Tsonopoulos4 modified the Pitzer-Curl5 correlation to predict additional polar compounds. The second virial is a function of the reduced temperature, the acentric factor, the dipole moment, and the critical pressure. For polar substances, an extra term is added to the nonpolar expression. These equations are capable of high accuracy. Another generalized expression was developed by Hayden and O’Connell.6 The correlation is based on the collision theory and requires as inputs critical constants and dipole moments. A number of the parameters are derived from the mean radius of gyration, which is not always available for a given compound. Otherwise, the method gives accurate correlations for many polar and nonpolar gases. In section II, we shall present a formulation based fundamentally on statistical mechanical principles. The interaction potential, starting with the spherical case, is extended to a nonspherical (anisotropic) potential form. This form is useful for treating the polar forces. In addition, we consider a multistep form of the squarewell potential based on the core-collapse models introduced sometime ago by Stell et al.7 in the 1970s. For example, the two-step form is capable of modeling the maxima in the second virial coefficients. In section III, the simplified formula is applied to some 40 common industrially important gases. Accuracy of the second virial calculation is determined by comparison with experimental data. Section IV presents a generalization of the parameters. Conclusions are given in section V.
10.1021/ie0104666 CCC: $22.00 © 2002 American Chemical Society Published on Web 02/02/2002
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
(for nitrogen, as an example). Br corresponds to the remnant volume exhibited by the second virial coefficient at infinite temperature (equivalent to 4 times the hard-sphere volume). Angle-Dependent Potentials. For angle-dependent pair potentials (such as for polar fluids), the nonspherical effects can be approximated by an angle-dependent square-well potential, i.e.
u(r,ω1,ω2) ) +∞
for r < σ(ω1ω2)
u(r,ω1,ω2) ) -(r,ω1,ω2)
for σ(ω1ω2) < r < λ σ(ω1ω2)
u(r,ω1,ω2) ) 0
for r > λσ(ω1ω2)
(6)
Both the size σ and range λ parameters are now dependent on the angles ω1 and ω2. This potential can approximate the polar interactions in a mean-field sense (just like the approximation of the Lennard-Jones-like potentials by the square wells9). The second virial is then given by angular averaging Figure 1. Behavior of the second virial coefficients (nitrogen). Data from Dymond and Smith.10 Note that B2 is essentially negative at low temperatures. The functional form is similar to an inverted exponential decay. This feature is captured by the square-well formula of Hirschfelder et al.1 At infinite temperature, T f ∞ or 1/T f 0 (by extrapolation), one can define a hard-core excluded volume, Br, as indicated.
II. Derivation of Formulas The second virial coefficients we shall treat here are based on the expansion of pressure P in terms of density F
P ) F + B2(T) F2 + B3(T) F3 + ... RT
(1)
where B2 is the second virial coefficient, B3 the third virial coefficient, etc. The statistical mechanical expression for B2 is well-known,8 for example, in terms of a spherical pair interaction potential, u(r). Spherically Symmetric Potentials.
∫0∞dr r2[1 - exp(-βu(r)]
B2 ) 2π
(2)
where β ) 1/kT (k ) Boltzmann constant, and T is the absolute temperature). For a square-well potential u(r) with hard-sphere diameter σ, well depth , and “attractive” range λσ,
u(r) ) +∞
for r < σ
u(r) ) -
for σ < r < λσ
u(r) ) 0
for r > λσ
(3)
the second virial becomes (after integration according to eq 2)
B2 )
2πσ3 {1 - (λ3 - 1)[exp(β) - 1]} 3
(4)
This can be transcribed to
B2 ) Br - Va[exp(β) - 1]
(5)
where Br ) 2πσ3/3 is called the “repulsive volume” and Va ) 2πσ3(λ3 - 1)/3 is the “attractive-range volume”. The behavior of such an equation is exemplified in Figure 1
B2 )
∫0∞dr dω1 dω2 r2[1 - exp(-βu(12)]
2π (8π2)2
(7)
where ωi’s are the Euler angles (θ, φ, χ) for molecule i and dωi is to be integrated over the three Euler angles. “(12)” stands for the relative distance r and the two sets of Euler angles between molecules 1 and 2. Substituting (6) into (7) gives after application of the mean-value theorem to
B2 ) Br - Va[exp(β(T)) - 1]
(8)
where (T) is the mean-value energy. The angular averaging in (7) produces a temperature dependence in . Rearranging (8) gives
[
ln 1 +
]
Br - B2 (T) ) Va kT
(9)
This working equation can be used to fit (T) as a function of T by plotting the left-hand side vs 1/T using experimental data. The slope will be the energy parameter /k. For simple gases, is a constant. For polar fluids, is a function of temperature. In Figure 2, we plotted the data for the noble gas argon and the highly polar methanol. We can see that B2 of argon is well represented by a constant , while B2 of methanol shows a deviation from the linear relation, especially at lower temperatures (or high 1/T). However, the dependence on 1/T is quite benign and can be accounted for by a simple correction. After a few trials, we found that a coefficient Dt, added to the temperature, is sufficient to account for this temperature dependence of (T). Namely,
B2 ) Br - Va{exp[/[k(T + Dt)]] - 1}
(10)
Figure 2 shows that, after this addition, the relation (9) becomes linear for methanol. Multistep Square-Well Potentials. In 1976 and the following years, Stell et al.7 introduced a core-collapse potential that exhibits phase transitions for the fluid of concern. Lee8 also formulated a two-step square-well potential for correlation of the equation of state of light hydrocarbons. Both potentials introduce “steps” (either mounds or wells) into the hard-core interaction. For the square-well potential with two steps (one in the repul-
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1109
Figure 2. Testing of the linear relation for (9). The slope is the energy parameter, /k. For argon (circles, experimental data from ref 10), a constant slope is obtained. For methanol (triangles), a highly polar gas, there is curvature at low temperatures (or at high 1/T). Thus, /k for methanol is not constant and should be a function of temperature. After using a temperature correction, Dt (eq 10), the plot for B2 of methanol is visibly linearized (open triangles).
sive region and the other in the attractive region), the interaction potential can be defined as
u(r) ) +∞
for r < σ
u(r) ) +1
for σ < r < λ1σ
u(r) ) -2
for λ1σ < r < λ2σ
u(r) ) 0
for r > λ2σ
(11)
Substitution of (11) into (2) gives the second virial for this two-step potential. It can written in the form8
B2 ) Br - Vrexp(-β1(T)) - 1 Vaexp(β2(T)) - 1 (12) It can be easily shown that this two-step B2 is able to exhibit a maximum with respect to temperature, when such behavior also occurs in data. Therefore, (10) (the one-step square-well form which cannot produce any extrema in B2) can be considered a special case of (12). For the gases we are studying, the high-temperature data are either not precise enough or lacking in existence, for the sake of pinpointing this maximum. Therefore, for simplicity, we revert back to the one-step case, eq 10, in our correlations below. (Situation warranting, we can always return to (12) for treatment of these extrema.) III. Correlation of Second Virial Coefficients We proceed by three steps: (1) correlation of the second virial cofficients for a small number of simple molecules with (10) and generalization of the parameters Br and ; (2) least-squares fit of the other two parameters, Va and Dt, and examination of the accuracy of fit; (3) testing the generalization on new substances. The data source we have used is based on the compilation of Dymond and Smith.10 Before the data contained therein are used, they have been evaluated (i) according to the recommendations by Dymond and Smith themselves (i.e., expert opinions) and (ii) by agreement among different sets of experiments (i.e., mutual consistency among different experimentalists when using different
Figure 3. Generalization of the repulsive volume, Br0, in terms of the critical volume, Vc. A least-squares fit for a straight line is obtained; see (13). Table 1. Repulsive Volume, Br0, Obtained from Extrapolating the Experimental Data, “The Raw Values” compound
Br0 (cm3/gmol)
compound
Br0 (cm3/gmol)
methane hydrogen nitrogen carbon dioxide oxygen krypton argon carbon monoxide water
46.0 22.0 41.0 44.0 36.0 44.0 35.0 45.0 25.0
ammonia hydrogen sulfide nitrous oxide ethane propane sulfur dioxide ethylene propylene methanol
35.0 45.0 45.0 75.0 90.0 60.0 60.0 88.0 51.0
methods). Data with discrepancies higher than, say, 10 cm3/gmol from the norm are excluded from the correlation. First, we use the evaluated data to obtain Br. Eighteen small-molecule substances are examined (e.g., methane, nitrogen, argon, etc.) The plot for nitrogen (a small molecule) is shown in Figure 1; namely, the B2 of N2 is plotted against 1/T. As T f ∞ or as 1/T f 0, the asymptotic value is the residual hard-core contribution to B2. We call it here Br0 (the raw value). For nitrogen, we obtained Br0 ≈ 41 cm3/gmol. Larger molecules incur more uncertainties, because first the high-temperature data are lacking and one has to extrapolate over broad regions of values of 1/T. This was, therefore, not done for large molecules. This procedure is carried out for the other small molecules. These Br0’s constitute the “raw data” for the parameter Br. They are listed in Table 1. To generalize Br0 in terms of, say, the critical volume (Vc), we plot Br0 with Vc in Figure 3. We obtain a linear least-squares fit:
Br (cm3/gmol) ) 479.534Vc (cm3/gmol) - 1.4995 (13) This generalized hard-core Br will be used in later correlations. After substituting these Br of (13) in (10), we obtained best fits for the other three parameters, , Va, and Dt for the 40 gases. A least-squares regression program based on Marquardt’s modified steepest descent method is used.11 We obtained thusly the “raw data” of , Va, and Dt. (These raw data are not shown
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 Table 2. Second Virial Coefficient Molecular Parameters for Use in (10) compound
Figure 4. Generalization of the energy parameter, /k, in terms of the critical temperature, Tc. A 45° line is drawn, and there is some scatter around the diagonal. However, the majority falls near or on the diagonal. Thus, we set /k ) Tc.
here but are contained in ref 12.) The raw values of /k are then plotted against the critical temperatures (Figure 4). Although there is some scatter, a linear relation could be constructed via a least-squares fit. Because the line is very close to a 45° angle, we set /k ) Tc (i.e., /k is forced to be equal to the critical temperature of the substance).
/k ) Tc
(14)
A new regression is carried out to obtain Va and Dt based on Br and /k as calculated from (13) and (14). The final results are presented in Table 2. Also shown are the λ values for these 40 gases. They range from 1.26 to 1.67, clustering around the value λ ) 1.5, a value often quoted for real substances. We note that, in Table 2, Br and /k are generalized values from (13) and (14), while Va and Dt values were obtained from regression with data and are specific for each gas species. At this juncture, we remark that nongeneralized parameters are constantly being used in chemical engineering (for example, the Antoine equation parameters for the vapor pressure and the ideal heat capacities for different substances). This does not constitute a “handicap”. The quality of the fit is examined in Table 3. We list also the temperature ranges where this formula applies. Two types of error statistics are provided: the rootmean-squared deviations (rmsd) and the average absolute deviations (AAD %).
∑ | i)1 n
AAD )
calci - expti expti
|
/n
(15)
n
rmsd ) [
(calci - expti)2/(n - 1)]1/2 ∑ i)1
(16)
where calci is the calculated value (eq 10) of the ith datum and expti the experimental value. The reason for providing rmsd is that many B2 values are very small (especially near the Boyle temperature, where B2 ) 0) and are thus subject to large experimental error. Small
acetone acetylene ammonia argon benzene 1-butene CCl4 CO CO2 CS2 cyclohexane cis-2-butene ethane ethanol ethylene hydrogen H2S isobutane isopentane krypton methane methanol n-butane n-heptane n-hexane n-octane n-pentane neon neopentane nitrogen N2O oxygen propadiene propane propylene SO2 toluene trans-2-butene vinyl chloride water
Bra Va (cm3/gmol) (cm3/gmol) 98.72 52.69 33.27 34.42 122.70 113.59 130.85 43.15 43.58 80.02 146.20 110.71 69.47 78.58 60.36 29.67 45.74 124.62 145.24 42.23 45.97 55.09 120.78 205.66 175.93 234.43 144.28 18.50 143.80 41.42 45.21 33.70 76.19 95.85 85.30 57.00 150.03 112.63 79.54 25.35
171.89 82.89 65.91 80.50 252.79 213.55 368.54 87.74 76.93 184.60 306.93 192.36 145.46 80.86 133.12 93.58 86.86 227.19 306.43 98.93 104.55 73.23 236.71 387.12 344.68 432.12 293.18 44.99 293.13 88.70 93.24 72.27 177.90 200.03 178.06 107.20 285.73 230.15 295.88 44.50
/kb (K)
Dt (K)
λc
508.1 308.3 405.6 150.8 562.1 419.6 556.4 132.9 304.2 552.0 553.4 435.6 305.4 516.2 282.4 33.2 373.2 408.1 460.4 209.4 190.6 512.6 425.2 540.2 507.4 568.8 469.6 44.4 433.8 126.2 309.6 154.6 393.0 369.8 365.0 430.8 591.7 428.6 429.7 647.3
-102.51 -60.05 -65.13 14.48 -15.00 -22.70 27.27 -9.74 -34.42 16.24 -5.55 -38.98 4.25 -156.83 9.60 12.08 -24.44 -34.26 -14.14 17.87 14.66 -142.99 -22.19 -51.91 -36.23 -64.95 -22.49 6.36 -12.17 4.22 -8.59 5.23 10.83 1.37 -1.92 -43.58 -50.29 -19.64 146.62 -110.55
1.39 1.37 1.43 1.49 1.45 1.42 1.56 1.44 1.40 1.48 1.45 1.39 1.45 1.26 1.47 1.60 1.42 1.41 1.45 1.49 1.48 1.32 1.43 1.42 1.43 1.41 1.44 1.50 1.44 1.46 1.45 1.46 1.49 1.45 1.45 1.42 1.42 1.44 1.67 1.40
a B is generalized according to (13). b /k is generalized accordr ing to (14). c Attractive range parameter (in units of σ, i.e., Va ) 2πσ3(λ3 - 1)/3).
errors will give unusually large AAD. Thus, AAD is not a good measure under these circumstances. We then use the rmsd, which is a measure of the absolute errors (rather than the relative errors). We see that, for most gases, AAD % is from 0.5% (isopentane, etc.) to 6% (CO and CO2), with many of the calculated values lying within 1-2% (n-butane, acetone, n-octane, toluene, etc.). The largest AAD % is for hydrogen (23%), a quantum gas (from 14 to 473 K), but here we managed without explicit quantum corrections. This large error is also largely due to distortion from the small B2 values mentioned (e.g., 0.41 cm3/gmol expt vs 0.15 cm3/gmol calc at 109.01 K) and the fact that we used diverse sources of data (149 data points from 19 different researchers). Its rmsd is, on the other hand, only 11 cm3/ gmol, less than those of acetone (rmsd ) 35 cm3/gmol) and isopentane (rmsd ) 16 cm3/gmol). For methane, AAD % is 5.84% and rmsd ) 1.91 cm3/gmol. Here we used 136 data points from 17 different experimental sources. A detailed comparison for methane can be found in ref 12. In Table 4, we compare the present correlation with other commonly used correlations (Tsonopoulos4 and Hayden-O’Connell6). Both AAD % and rmsd are listed. We see that, for most cases (about 38 out 40 gases), the
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1111 Table 3. Error Analyses of the Second Virial Equation, Eq 10, Using Parameters from Table 2a compound
no. of points
temp range (K)
rmsd (cm3/gmol)
AAD (%)
acetone acetylene ammonia argon benzene 1-butene CCl4 CO CO2 CS2 cyclohexane cis-2-butene ethane ethanol ethylene hydrogen H2S isobutane isopentane krypton methane methanol n-butane n-heptane n-hexane n-octane n-pentane neon neopentane nitrogen N2O oxygen propadiene propane propylene SO2 toluene trans-2-butene vinyl chloride water
36 26 31 197 129 17 15 29 174 18 29 6 67 5 97 149 6 13 18 122 136 19 61 37 43 27 40 36 37 122 22 72 11 67 56 25 31 9 5 57
295-473 199-313 273-573 77-1223 280-628 243-420 315-418 213-573 203-1100 281-432 283-573 250-343 199-623 313-393 199-448 14-473 277-444 273-510 273-473 107-873 110-623 313-573 244-560 300-700 325-700 300-700 298-573 44-973 303-498 70-748 258-423 85-373 222-353 211-548 223-510 265-473 413-583 243-333 309-416 293-1248
35.9 22.9 4.23 1.6 22.9 23.8 34.6 1.1 6.9 9.0 13.4 25.7 4.5 60.4 0.7 11.1 1.3 5.8 16.4 2.2 1.9 25.8 11.0 16.2 18.3 35.2 11.5 0.9 10.7 5.8 1.9 5.3 6.1 4.2 3.3 6.3 14.7 15.9 10.6 37.7
2.2 5.7 2.6 4.0 1.8 1.9 2.3 6.1 5.9 0.8 1.4 2.0 1.9 3.2 0.4 23.4 0.6 0.5 1.1 4.4 5.8 1.8 1.5 1.2 1.1 1.0 1.4 5.2 1.3 4.7 1.0 6.9 1.2 1.2 1.0 1.2 1.1 1.7 2.4 5.4
a Deviations are from experimental data.10 rmsd ) root-meansquared deviation. AAD ) average absolute deviation.
square-well-based formula gives better agreement on the same data set. For methane, a major component of the natural gas, (10) gives AAD % ) 5.8% and rmsd ) 1.91 cm3/gmol, while Hayden-O’Connell’s work gives AAD % ) 10.6% and rmsd ) 1.95 cm3/gmol. Similarly, for Tsonopoulos, AAD % ) 6.3% and rmsd ) 1.46 cm3/ gmol. This shows that different correlations emphasize different regions. While (10) is the best in AAD %, Tsonopoulos’ work is a little better in rmsd. This would happen if the Boyle temperature region is forced to give a good fit (where B2 f 0). Note for comparison that Tsonopoulos’ work used correlations depending on three functions, f 0, f 1, and f 2, with generalized parameters depending on the critical pressure, critical temperature, acentric factor, and parameters a and b, which are related to the dipole moment. Hayden-O’Connell’s work uses critical pressure, temperature, mean radius of gyration, dipole moment, and chemical association parameters b0, /k, and η. For the major components in natural gas (methane through octane, plus nitrogen, carbon dioxide, and hydrogen sulfide), the average deviation from (10) is about 2.33%, while it is 3.70% from Tsonopoulos’ work and 18.21% from Hayden-O’Connell’s work. The present correlation is more accurate for highly polar compounds (e.g., acetone, water, and ethanol). It only requires four
Table 4. Comparison of Different Correlations: Work by Tsonopoulos4 Hayden-O’Connell6 and This Work (Eq 10)a Hayden6 compound
AAD %
acetone acetylene ammonia argon benzene 1-butene CCl4 CO CO2 CS2 cyclohexane cis-2-butene ethane ethanol ethylene hydrogen H2S isobutane isopentane krypton methane methanol n-butane n-heptane n-hexane n-octane n-pentane neon neopentane nitrogen N2O oxygen propadiene propane propylene SO2 toluene trans-2-butene vinyl chloride water
14.9 17.8 16.1 20.7 2.3 2.1 3.4 65.6 93.0 16.6 6.3 6.1 2.1 8.4 1.7 56.9 8.1 39.1 3.5 11.2 10.6 8.6 4.2 2.7 2.4 3.0 2.2 39.8 1.7 71.2 4.6 12.2 4.1 1.8 1.3 19.5 6.9 12.4 7.5 32.4
correlation4,b
rmsd
AAD %
236.3 64.9 20.6 2.9 32.9 42.9 52.0 4.0 12.9 112.7 156.6 100.0 5.2 216.9 2.3 144.2 23.1 228.7 27.8 5.8 2.0 123.4 32.7 30.2 22.6 43.6 13.5 2.3 12.6 5.6 6.9 3.6 16.8 5.3 5.9 95.1 82.2 120.9 35.3 134.4
2.4 26.4 12.2 11.3 5.2 1.8 5.3 7.8 5.9 5.0 5.2 2.7 1.9 4.1 1.0 40.7 4.7 5.4 3.7 9.6 6.3 2.5 2.5 1.5 2.4 1.2 1.9 33.7 1.7 10.7 1.1 3.72 11.2 1.7 1.2 7.2 2.7 3.3 8.6 16.5
this work
rmsd
AAD %
rmsd
44.9 125.8 27.8 2.4 94.3 22.9 70.3 0.8 4.1 30.6 153.5 49.5 4.0 67.9 1.7 58.7 14.9 51.9 32.8 2.8 1.5 15.8 20.7 37.1 34.9 83.8 18.0 1.6 12.6 5.0 2.5 2.5 102.8 11.3 5.6 34.0 36.7 30.7 42.5 398.6
2.2 5.7 2.6 4.1 1.8 1.9 2.3 6.1 5.9 0.8 1.4 2.1 1.9 3.2 0.4 23.4 0.6 0.5 1.1 4.5 5.8 1.8 1.5 1.2 1.1 1.0 1.4 5.2 1.3 4.7 1.0 7.0 1.2 1.2 1.0 1.2 1.0 1.7 2.4 5.4
35.9 22.9 4.2 1.6 22.9 23.8 34.6 1.1 6.9 9.0 13.4 25.7 4.5 60.4 0.7 11.1 1.3 5.8 16.4 2.2 2.0 25.9 11.0 16.1 18.3 35.2 11.5 0.9 10.7 5.8 1.9 5.3 6.2 4.2 3.3 6.3 14.7 16.0 10.5 37.7
a rmsd ) root-mean-squared deviation. AAD ) average absolute deviation. b Correlation ) from Tsonopoulos.4
parameters, two of which, Br and /k, are generalized. The parameters Va and Dt can be further generalized if so desired with some deterioration of accuracy. IV. Generalization of Parameters In Figure 5, we plot Va vs Br. Except for a few species, the following equation holds:
Va (cm3/gmol) ) 1.925Br (cm3/gmol) + 10.035 (17) Parameter Dt can also be generalized with respect to the acentric factor ω:
Dt ) -237.717ω + 22.782
(18)
However, this generalization suffers a lot of scatter. We test the new generalized correlations (17) and (18) on five substances not included in the previous correlation database. This is, thus, a test of the generalization. The substances are polar compounds and hydrocarbons: methyl chloride, fluorobenzene, butylethylene, 2-propanol, cyclopropane, and 1-pentene. These parameters are presented in Table 5 when Va and Dt are obtained
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
square-well potential. It has been tested on 40 common gases and found to be more accurate than conventional formulas (e.g., Hayden-O’Connell and Tsonopoulos). The formula (10) is exceptionally compact and simple to use, with only four parameters: Br, , Va, and Dt. These parameters possess physical meanings because they are derived from the molecular formulas of the square-well potential. Br is the repulsive volume at high temperatures where the molecular exclusion core volume is exposed. is the mean-value energy of interaction between a pair of molecules. Va is the attractiverange volume, and Dt accounts for the polar interactions and/or the anisotropy of the molecules. Table 2 is recommended for use for calculation of highly accurate B2 values for the gases studied. Literature Cited
Figure 5. Generalization of the attractive-range volume, Va, in terms of Br. A least-squares fit line is constructed; see (17). Table 5. Predictions of Second Virial Coefficients for New Compounds (Not Included in the regression) Based on Generalized Parameters: (17) and (18)
compound
no. of points
methyl chloride fluorobenzene butylethylene isopropanol cyclopropane 1-pentene
30 38 11 13 14 15
generalized Va and Dt AAD % rmsd 8.4 2.0 9.4 66.1 28.0 12.2
19.1 15.2 112.3 540.8 85.8 105.2
Tsonopoulos4 AAD % rmsd 10.7 2.8 1.1 6.8 3.8 11.6
29.1 26.6 18.7 133.7 14.0 104.5
from the above equations. Table 5 shows the quality of the fits and also comparisons with other correlations. For methyl chloride for example, Tsonopoulos4 gives AAD % ) 10% (rmsd ) 29.1 cm3/gmol), while generalization here gives AAD % ) 8.4% (rmsd ) 19.1 cm3/ gmol). In some other cases, correlation4 is better. V. Conclusions The correlation for the second virial coefficients has been improved by considering the anisotropic effects in the statistical mechanical treatment of the anisotropic
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Received for review May 25, 2001 Revised manuscript received November 26, 2001 Accepted December 10, 2001 IE0104666
