A Method To Determine Virial Coefficients from Experimental (p,Ï,T

Article pubs.acs.org/jced

A Method To Determine Virial Coefficients from Experimental (p,ρ,T) Measurements Diego E. Cristancho,† Pedro L. Acosta-Perez, Ivan D. Mantilla,‡ James C. Holste,* and Kenneth R. Hall§ Artie McFerrin Department of Chemical Engineering, Texas A&M University College Station, College Station, Texas 77843-312, United States

Gustavo A. Iglesias-Silva Departamento de Ingeniería Química, Instituto Tecnológico de Celaya, 38010 Celaya, Guanajuato, Mexico ABSTRACT: This paper proposes a method to determine the virial coefficients of a substance from experimental density measurements. The method consists of a technique to establish the proper number of terms in the truncated series for a given set of data or to establish the proper data range for a given truncated series. The new methodology when applied to experimental (Pρ-T) data for argon provides results comparable to those reported previously in the literature using curve-fitting techniques or advanced graphical analyses. The proposed method uses additional information from independent measurements of molar mass to provide an objective criterion for selecting the number of terms in the virial expansion and/or the required number of experimental data to determine second and third virial coefficients accurately. The method uses mass densities (as produced by magnetic suspension densimeters, for example) and adjusts the fits of the data until the proper molecular mass results. The paper provides analyses of data from the literature and comparison to virial coefficients provided in the literature.

■

INTRODUCTION Because of its precise basis in statistical mechanics, the virial equation of state (VEOS) has become immensely important in the study of thermophysical properties of fluids. The VEOS first appeared as an empirical expression proposed by Thiesen1 that Kamerlingh Ohnes2,3 expanded upon and used. Later, Ursell4 provided the fundamental statistical-mechanical basis. The principal advantage of the VEOS is that the coefficients can be calculated from the intermolecular potential function. However, with some exceptions, the intermolecular potential function is unknown, although Garberoglio and Harvey5 and Moldover and McLinden6 have made remarkable progress in the calculation of the coefficients for helium. It is possible to obtain the virial coefficients from experimental P−ρ−T data, but, because the VEOS is an infinite series, it must be truncated after a finite number of terms. In the conventional analysis of experimental data, this step always is somewhat subjective. Kleinrahm et al.7 and Gilgen et al.8 use a truncated series expansion to correlate the experimental data (compression factor) over a defined range of temperature and pressure using the evolutionary optimization method (EOM) developed by Ewers and Wagner9 to determine the functional form of the series. After achieving a good correlation of the experimental data, the estimated terms of the truncated series provide the second and third virial coefficients. Cencek et al.10 use computational statistical mechanics methods to calculate the second and third virial coefficients. Although this methodology has a basis in fundamental principles, it does require a considerable amount of computational time, and it becomes © XXXX American Chemical Society

prohibitively complex for polyatomic molecules. This paper suggests a more practical objective method to determine second and third virial coefficients from volumetric data.

■

THEORETICAL CONSIDERATIONS The most common expression for the VEOS provides pressure as a function of temperature and molar density ∞

Z = P /RTρ = 1 +

∞

∑ Bk ρk− 1 = ∑ Bk ρk− 1 k=2

k=1

(1)

in which Z is the ratio of the ideal gas density to the real fluid density, P is the pressure, T is the temperature, ρ is the molar density, R is the gas constant and the Bk(T) are the virial coefficients (B1 = 1). When written in terms of the mass density, ρm, this expression is Z = MP /RTρm ∞

=1+

∑ (Bk /M k− 1)ρmk− 1 k=2

∞

=

∑ (Bk /M k− 1)ρmk− 1 k=1

(2)

Special Issue: Memorial Issue in Honor of Anthony R. H. Goodwin Received: July 21, 2015 Accepted: October 20, 2015

A

DOI: 10.1021/acs.jced.5b00629 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

in which M is the molar mass. One rearrangement of eq 2 is ∞

P /RT = (ρm /M ) +

The number of terms required depends upon the maximum experimental density, ρmax, used in the curve fit. In this case, ρmax does not imply the highest measured density, but rather the highest density selected for the curve fit. In the limit as the number of terms becomes insufficient, the deviations of the equation from the data exceed the experimental errors, and in the limit as the number of terms becomes excessive, one or more of the fit parameters will not be statistically significant, indicated when the uncertainty of the coefficient exceeds the value of the coefficient. For a range near the optimum, valid fits result that satisfy both criteria. Figure 1 illustrates the general behavior. The challenge in determining the appropriate number of terms arises because both N and the experimental ρmax are discrete quantities; however, it appears that using the molecular mass allows an accurate determination of N or ρmax. This work indicates that a good way to establish the proper number of coefficients for a given ρmax is to fit eq 4 while varying N and the experimental range for the selected ρmax until B1 provides a value of M as close as possible to the accepted value. Then, obtaining final values for the virial coefficients results from refitting the data using the accepted value of M and the value of N determined previously. Of course, when seeking virial coefficients, it is possible to use forms other than eq 4. Two other applicable forms are

∞

∑ (Bk /M k)ρmk = ∑ (Bk /M k)ρmk k=2

k=1

(3)

Equation 3 is a polynomial in density with zero intercept. In this equation, P, T, and ρm are experimental observations and R and M are accepted experimental values. The adjustable parameters for a fit of eq 3 to experimental data are the coefficients (Bk/Mk). The first coefficient is 1/M, so the molar mass becomes an adjustable parameter. In effect, this allows finding M from the data. It is now necessary to truncate the series to N terms N

P /RT =

∑ (Bk /M k)ρmk

(4)

k=1

N

P /RT − ρm /M =

∑ (Bk /M k)ρmk

(5)

k=2

which has an intercept at (0,0), thus no extrapolation, and P /RT − ρm /M ρm2

N

=

∑ (Bk /M k)ρmk− 2

(6)

k=2

for which the extrapolated intercept at ρm = 0 is B2/M and the limiting slope as ρm → 0 is B3/M3. Clearly, eqs 4, 5, and 6 all provide the virial coefficients, but they do not necessarily result in 2

Figure 1. General behavior of statistical fits for various maximum experimental densities and number of terms included in the virial equation of state.

Table 1. Molecular Masses Resulting from Fits with Various Combinations of Number of Virial Coefficients (Nv)a and Maximum Density at T = 157 K. The Combination Chosen as the Best Fit Is Shown in Bold-Faced Italics ρmax kg·m−3 891 863 825 761 679 600 561 519 481 442 400 361 320 281 240 200 177 145 93 a

Nv = 2

Nv = 3

Nv = 4

39.7575

39.8921

39.8061

39.9576

39.8496

39.9619

Nv = 5

Nv = 6

40.1575 40.0539

40.1165 40.0263 39.9438 39.9288

39.9833

39.9371

Nv = 7 39.8718 39.8654 39.8653 39.8842 39.9232

39.9551 39.9141 39.9546 39.8982 39.9142

39.9504 39.9485 39.9494

39.9319 39.9431

NV = N − 1 for N as defined in eq 4. B



■

RESULTS AND DISCUSSION Table 1 demonstrates the procedure for the VEOS applied to the 157 K isotherm of Gilgen et al.7 The molar masses obtained by the least-squares method appear in the table. The fit with the best value for molar mass has eight observations and four virial coefficients. In fact, this is the only combination that achieved the correct molar mass. All combinations not shown had statistically insignificant parameters or deviations from the experimental data that exceed the experimental error. Table 2 contains the selected combinations of maximum density and number of terms for each isotherm measured by

the same values. The following sections examine use of all three equations. We also can represent any polynomial as a Padè approximant, so that eq 2 or eq 3 can be a ratio of polynomial functions ∞

P /RT =

∑k = 1 nk (ρm /M )k ∞

∑k = 1 dk(ρm /M )k − 1

∞

=

∑k = 1 (nk /M k)ρmk ∞

∑k = 1 (dk /M k)ρmk − 1

(7)

in which n1 = 1= d1. Again, the question is how many terms are necessary in the rational function. Usually, it is desirable to use a square Padè, that is, the same power in the numerator and denominator (but this is not necessary). N

P /RT =

∑k = 1 (nk /M k)ρmk N ∑k = 1 (dk /M k)ρmk − 1

Table 2. Selected Combinations of Maximum Density and Number of Virial Coefficients for the Argon Measurements Reported by Gilgen et al.8 and McLinden.11

(8)

The second and third virial coefficients are calculated from the rational equation of state (REOS) using

B2 = n 2 − d 2 B3 = n3 − d3 − n2d 2 + d 22

virial EOS

(9)

rational EOS

T

ρmax

MEOS

parameters

MREOS

K

kg m−3

NV

g mol−1

Pade Form

g mol−1

31.6 53.4 NPa NPa 146.7 180.5 133.1 239.9 192.9b 239.6 216.3 210.2 221.6 340.3 338.7 102.6 148.4 108.5c 188.5 182.1 129.3

1 2 NPa NPa 3 3 3 4 3 4 3 3 3 4 3 2 2 2 3 3 3

39.9477 39.9479 NPa NPa 39.9487 39.9491 39.9473 39.9486 39.9478 39.9485 39.9481 39.9485 39.9487 39.9480 39.9486 39.9484 39.948 39.9485 39.9477 39.9478 39.9476

virial n2, d2 NPa NPa n2, n3, d3 n2, n3, d3 n2, n3, d3 n2, n3, d3 n2, n3, d3 n2, n3, d3 n2, n3, d2 n2, n3, d2 n2, n3, d2 n2, n3, d2, d3 n2, n3, d2 virial virial n2, d2, d3 n2, n3, d2 n2, n3, d2 n2, n3, d2

30.9476 NPa NPa 39.9484 39.9479 39.9477 39.9479 39.9472 39.9476 39.9479 39.9484 39.9486 39.9483 39.9485 virial virial 39.9474c 39.9477 39.9476 39.9475

136.0 130.4 101.8 101.8 53.6 66.6 66.6 56.4

2 2 2 2 2 2 2 2

39.9483 39.9480 39.9479 39.9477 39.9480 39.9482 39.9481 39.9480

virial virial virial virial n2, d3 virial virial virial

39.9483 39.9480 39.9479 39.9477 39.9480 39.9482 39.9481 39.9480

Gilgen et al:8 110 120 135 140 143 146 148 150.7 153 157 165 175 180 190 200 220 250 280 310 325 340 McLinden:11 232.32 273.16 293.15-I 293.15-II 360 429.75-I 429.75-II 505.32

(10)

The challenges for determining the appropriate number of terms and the method used are the same as described above for the VEOS. The procedure to determine the virial coefficients for each isotherm is as follows: (1) For a given N and ρmax, fit both the VEOS (eq 4) and the REOS (eq 8) to all experimental observations with ρm ≤ ρmax. Eliminate any obvious outliers and refit if necessary. The number of independent experimental observations always should be at least 2N. In this step, the molar mass is a fit parameter (step 4 corrects the value to the accepted molar mass). (2) Repeat step 1 for various combinations of N and ρmax. Retain only the results for fits in which all fit parameters are statistically significant and the deviations from the data are within the estimated experimental error (the shaded region in Figure 1. (3) Choose the N, ρmax combination that yields a molar mass closest to the accepted value. The values should be essentially the same for both the VEOS and the REOS. (4) Finally, fix the molar mass at the accepted experimental value and repeat the fit using the N, ρmax combination chosen in step 3 to determine the virial coefficients using the VEOS or the REOS. As an example of the method for determining virial coefficients, this paper uses measurements for argon (molar mass = 39.948 ± 0.001, critical temperature Tc = 150.687 K, critical pressure Pc = 4.863 MPa, and mass density ρc = 535.6 kg·m−3). Gilgen et al.8 provide both experimental densities and derived second and third virial coefficients, and McLinden11 reports experimental densities. Cencek et al.10 report second and third virial coefficients derived from the McLinden measurements, and Tegeler et al.12 provide revised values for second and third virial coefficients derived from the Gilgen et al.8 measurements. The least-squares fitting procedure used in this work to find the virial coefficients is a Marquardt method contained in SAS13 used with equal weighting of observations. The measure of goodness of fit is the standard deviation in pressure, defined as ⎡ ∑ (P − P )2 ⎤1/2 exp cal ⎥ σ=⎢ ⎢⎣ ⎥⎦ n−m

Article

No combination of NV and ρmax yielded an appropriate value for the molar mass. bThe REOS fit contains one additional measurement (ρmax = 242.2 kg m−3). cThe REOS fit contains six additional measurements (ρmax = 184.1 kg m−3). a

Gilgen et al.8 and McLinden.11 Table 2 also includes the parameters needed in a REOS. In most cases the same number of data points are necessary as in the VEOS to correlate the REOS. At T = (135 and 140) K it is not possible to obtain the correct value for the molar mass with either equation. The REOS always uses a lower power in density than the VEOS. In some cases, as in isotherms (120, 220, and 250) K for Gilgen et al.8 and

(11)

in which Pexp is the measured pressure, Pcal is the calculated pressure, n is the number of observations, and m is the number of parameters. C



Article

Table 3. Second Virial Coefficients Derived from Gilgen et al.8 Measurements this work virial EOS Gilgen et al. T

B2

u(B2)

8

Tegeler et al. B2

u(B2)

12

eq 4 and 5 B2

eq 6

eq 8

u(B2)

B2

u(B2)

0.03 0.15 0.07 0.06 0.04 0.03 0.05 0.05 0.02 0.02 0.08 0.02 0.02 0.02 0.02 0.02 0.03 0.02 0.03 0.04 0.02 0.04

−152.37 −130.32 −105.28 −98.37 −94.39 −90.75 −88.32 −85.37 −82.98 −78.97 −76.07 −71.93 −63.81 −60.11 −53.61 −47.93 −38.51 −27.72 −19.66 −13.30 −10.57 −8.23

0.04 0.32 0.14 0.09 0.04 0.06 0.03 0.07 0.02 0.01 0.10 0.16 0.05 0.02 0.01 0.06 0.07 0.02 0.05 0.03 0.04 0.03

B2

u(B2)

−130.49

0.19

−94.36 −90.69 −88.31 −85.42 −82.96 −78.97

0.04 0.04 0.05 0.02 0.02 0.01

−71.67 −63.70 −60.10 −53.62 −47.99

0.02 0.04 0.02 0.02 0.02

−19.60 −13.28 −10.62 −8.29

0.04 0.04 0.02 0.03

−1

cm ·mol 3

K 110 120 135 140 143 146 148 150.7 153 157 160 165 175 180 190 200 220 250 280 310 325 340

rational EOS

−152.72 −130.70 −105.40 −98.49 −94.65 −91.00 −88.68 −85.67 −83.21 −79.15 −76.28 −71.78 −63.75 −60.16 −53.68 −48.00 −38.54 −27.71 −19.62 −13.34 −10.69 −8.31

1.3 0.6 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

−130.28 −105.28 −98.29 −94.46 −90.83 −88.51 −85.60 −83.04 −79.04 −76.27 −71.68 −63.71 −60.13 −53.62 −48.00 −38.52 −27.74 −19.59 −13.27 −10.66 −8.30

1.30 0.30 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

−152.38 −130.46 −105.19 −98.48 −94.38 −90.72 −88.32 −85.42 −82.99 −78.96 −76.12 −71.67 −63.70 −60.10 −53.62 −47.99 −38.53 −27.70 −19.59 −13.28 −10.62 −8.28

Table 4. Third Virial Coefficients Derived from Gilgen et al.8 Measurements this work virial EOS Gilgen et al.8 T

B3

Tegeler et al.12 u(B3)

B3

u(B3)

B3

a

eq 6 u(B3)

eq 8

B3

u(B3)

B3

u(B3)

1615 2392a 2393a 2228 2201 2090 2082 2130 2093 1996a 2200 1980 1844 1698 1600 1455 1288 1187 1055 943 930

420 148 39 35 45 30 76 15 16 109 130 41 15 9 34 30 10 26 25 43 23

1807

125

2194 2152 2077 2126 2106 2085

26 23 43 7 8 3

2026 1902 1840 1705 1625

79 73 60 12 23

1171 1041 987 972

19 64 18 27

cm3·mol−1

K 120 135 140 143 146 148 150.7 153 157 160 165 175 180 190 200 220 250 280 310 325 340

rational EOS

eq 4 and 5

2671 2620 2575 2522 2485 2432 2386 2306 2247 2152 1977 1899 1761 1645 1466 1287 1169 1085 1051 1021

60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 65 70 75 80 80

2389 2344 2326 2314 2289 2294 2216 2182 2171 2053 1913 1864 1718 1635 1456 1293 1157 1052 1034 1005

60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 65 70 75 80 80

1757 2344a 2430a 2223 2184 2096 2121 2134 2077 2043a 2023 1901 1840 1707 1625 1460 1279 1157 1039 982 968

121 35 27 25 18 39 29 12 13 48 12 11 10 13 7 8 6 11 26 11 22

Calculated with number of points giving the molar mass closest to 39.948. D



Article

all except 360 K for McLinden,11 the d coefficients are zero, so the REOS reduces to the VEOS. Tables 3 and 4 present the second and third virial coefficients, respectively, together with their uncertainties, derived using the VEOS by fitting eqs 4 to 6, respectively, to the experimental values of Gilgen et al.8 These tables also contain derived values reported by Gilgen et al.8 and Tegeler et al.12 for comparison. The current second virial coefficients agree with the reported values within their uncertainties. The virial coefficients in this procedure should be statistically valid within a 95 % confidence interval. The tables contain the value of half the interval to be consistent with Gilgen et al.8 Tables 3 and 4 also contain the second and third virial coefficients derived from the measurements of Gilgen et al.8 using the REOS (eq 8). The standard deviations for the fits of eqs 4, 5, 6, and 8, which are shown in Table 5, indicate no significant differences in goodness of fit for the VEOS and REOS. Table 6 shows the results for the data of

Figure 2. Differences between second virial coefficients derived using different methods and the reference EOS of Tegeler et al.12 This work: virial EOS (+), rational EOS (×). Previous work: Gilgen et al.8 measurements: Gilgen et al.8 (●), Tegeler et al.12 (○). McLinden11 measurements: Cencek et al.10 (△). Values for the 110 K isotherm are not included.

Table 5. Standard Deviations for Fits of Different Forms of the Virial EOS (eq 4, 5, and 6) and the Rational EOS (eq 8) to the Measurements Reported by Gilgen et al.8

a

T

σP (eq 4 and 5)

σP (eq 6)

σP (eq 8)

K

Pa

Pa

Pa

110 120 135 140 143 146 148 150.7 153 157 160 165 175 180 190 200 220 250 280 310 325 340

4.1 15 19 157 21 19 10 25 15 6.7 65 32 30 36 66 98 58 61 55 93 47 67

7.5 52 103 377 24 57 10 132 21 23 147 828a 273a 41 97 537 92 117 167 122 219 172

15

19 31 10 22 25 7.5 32 31 36 66 98

Figure 3. Differences between third virial coefficients derived using different methods and the reference EOS of Tegeler et al.12 This work: virial EOS (+), rational EOS (×). Previous work: Gilgen et al.8 measurements: Gilgen et al.8 (●), Tegeler et al.12 (○). McLinden11 measurements: Cencek et al.10 (△).

82 92 45 67

McLinden11 using eq 4, and, for comparison, the derived values reported by Cencek et al.10 Figures 2 and 3 compare the second and third virial coefficients obtained in this work using the VEOS (eq 4) and the REOS (eq 8) to those reported by Gilgen et al.,8 Tegeler et al.,12 and Cencek et al.10 The basis for the comparison is the reference

A statistically invalid virial coefficient found during the fit.

Table 6. Second and Third Virial Coefficients Derived from McLinden11 Measurements Cencek et al.10

this work T

M

K

NP

g mol−1

234.32 273.16 293.15-I 293.15-II 360.00 429.75-I 429.75-II 505.32

11 36 26 26 19 12 29 18

39.9483 39.9480 39.9479 39.9477 39.9480 39.9482 39.9481 39.9480

B2 (eq 4)

u(B2)

B3 (eq 4)

cm3·mol−1 −32.98 −21.27 −16.64 −16.63 −5.45 1.83 1.85 7.13

u(B3)

cm6·mol−2

0.0132 0.0066 0.0069 0.0067 0.0082 0.0128 0.0039 0.0194

1373 1187 1112 1110 944 932 924 853 E

B2

u(B2) cm3·mol−1

4.2 2.5 3.3 3.2 7.3 9.4 2.5 14.1

−32.89 −21.23 −16.64 −16.64 −5.50 1.82 1.82 7.03

B3

u(B3) cm6·mol−2

0.16 0.13 0.20 0.20 0.27 0.38 0.40 0.59

1330 1153 1092 1091 949 907 911 893

40 32 67 68 113 177 190 115



■

equation of Tegeler et al.,12 which is available for convenient calculations within REFPROP.14 The values from this work are consistently slightly lower than the other values near and below the critical temperature, but all results agree within the combined uncertainties of the measurements, with the largest deviations occurring for the Gilgen et al.8 analysis at the lowest temperatures. It is reasonable to expect unusual behavior in the critical region and larger deviations at low temperatures. However, because the results from all fits are within the combined uncertainties of the measurements, it is not possible to say with certainty which technique is better in these regions. The current technique appears to have an advantage in implementation. The values of the second virial coefficients have less correlation to the value of the molar mass than the third virial coefficients. It appears that using the molar mass as a control parameter provides a sensitive test when seeking third virial coefficients. The Padè form of the REOS (square rational) can correlate the pressure with the same degree of accuracy as the normal VEOS but with a lower power in the density. For example, the correlation of pressure for the 190 K isotherm uses 20 data points. Using the VEOS requires five virial coefficients (5th power in density) to obtain a molar mass close to the true value. With the same degrees of freedom, the Padè approximant, or REOS is cubic in density in the numerator and quadratic in the denominator. The residuals of pressure for both VEOS and REOS fits are nearly identical, with only insignificant differences (∼5 Pa) between fits. The molar mass from the fit is 39.9476 for the VEOS and 39.9478 for the rational EOS.

REFERENCES

(1) Thiesen, M. Untersuchungen über die Zustandsgleichung. Ann. Phys. 1885, 24, 467−492. (2) Kamerlingh Onnes, H. Expression of the Equation of State of Gases and Liquids by Means of Series. Commun. Phys. Lab. Univ. Leiden 1901, 71, 3−25. (3) Kamerlingh Onnes, H. Expression of the Equation of State of Gases and Liquids by Means of Series. P. K. Ned. Akad. Wetensc. 1902, 4, 125− 147. (4) Ursell, H. D. The Evaluation of Gibbs’ Phase-integral for Imperfect Gases. Math. Proc. Cambridge Philos. Soc. 1927, 23, 685−697. (5) Garberoglio, G.; Harvey, A. H. First Principles Calculation of the Third Virial Coefficient of Helium. J. Res. Natl. Inst. Stand. Technol. 2009, 114, 249−262. (6) Moldover, M. R.; McLinden, M. O. Using Ab Initio ‘‘Data” to Accurately Determine the Fourth Density Virial Coefficient of Helium. J. Chem. Thermodyn. 2010, 42, 1193−1203. (7) Kleinrahm, R.; Duschek, W.; Wagner, W. Measurement and Correlation of the (Pressure, Density, Temperature) Relation of Methane in the Temperature Range from 273.15 to 323.15 K at Pressures up to 8 MPA. J. Chem. Thermodyn. 1988, 20, 621−631. (8) Gilgen, R.; Kleinrahm, R.; Wagner, W. Measurement and Correlation of the (Pressure, Density, Temperature) Relation of Argon. The Homogeneous Gas and Liquid Regions in the Temperature Range from 90 to 340 K at Pressures up to 12 MPa. J. Chem. Thermodyn. 1994, 26, 383−398. (9) Ewers, J.; Wagner, W. A Method for Optimizing the Structure of Equations of State and its Application to an Equation of State for Oxygen. VDI-Forschungsheft 1982, 609, 27−34. (10) Cencek, W.; Garberoglio, G.; Harvey, A. H.; McLinden, M. O.; Szalewicz, K. Three-Body Nonadditive Potential for Argon with Estimated Uncertainties and Third Virial Coefficient. J. Phys. Chem. A 2013, 117, 7542−7552. (11) McLinden, M. O. Densimetry for Primary Temperature Metrology and a Method for the in situ Determination of Densimeter Sinker Volumes. Meas. Sci. Technol. 2006, 17, 2597−2612. (12) Tegeler, Ch.; Span, R.; Wagner, W. A New Equation of State for Argon Covering the Fluid Region for Temperatures from the Melting Line to 700 K at Pressures up to 1000 MPa. J. Phys. Chem. Ref. Data 1999, 28, 779−850. (13) SAS Institute Inc., SAS OnlineDoc, version 9.1.3; SAS Institute Inc.: Cary, NC, 2002−2005. (14) Lemmon, E. W., Huber, M. L., McLinden, M. O. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, version 9.0; Standard Reference Data Program; National Institute of Standards and Technology: Gaithersburg, 2010.

■

CONCLUSIONS This paper presents a new procedure to calculate virial coefficients using the molar mass as an indicator. The indicator selects a data set given a number of virial coefficients or vice versa. In addition, a rational EOS can determine virial coefficients; however, with higher asymptotic standard errors for the parameters. The advantages of using the proposed approach compared to previous approaches are that it utilizes an independently measured property to limit objectively the number of experimental data points and/or series terms required for correlation of the experimental data. The new method does not require a significant amount of computing time to obtain fundamentally consistent results based upon experimental data.

■

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +1.979.845.3384. Fax: +1.979.845.6446. Present Addresses †

D.E.C.: DOW Chemical, Freeport TX USA. I.D.M.: Bryan Research & Engineering, Bryan TX USA. § K.R.H.: Currently on assignment at Texas A&M University at Qatar, Doha QATAR. ‡

Funding

The authors gratefully acknowledge financial support from the Jack E. & Frances Brown Chair Endowment and the Texas A&M Engineering Experiment Station. Notes

The authors declare no competing financial interest. F


A Method To Determine Virial Coefficients from Experimental (p,Ï,T

Recommend Documents

A Method To Determine Virial Coefficients from Experimental (p,Ï,T