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Comment on “An equation of state for gaseous argon determined from the speed of sound”
Chemical Physics 411 (2013) 43–44
Contents lists available at SciVerse ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Comment
Comment on ‘‘An equation of state for gaseous argon determined from the speed of sound’’ J. Wiebke Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey University Albany, Private Bag 102904, North Shore City, Auckland 0745, New Zealand
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 15 December 2011 In final form 25 January 2012 Available online 7 February 2012
Ó 2012 Elsevier B.V. All rights reserved.
Keywords: Argon Virial equation of state Mayer cluster expansion Many-body interaction potentials
In a contribution to Chem. Phys., Monago reported [1] a new semi-empirical parametrization of a fourth-order virial Equation of State (EOS) for Argon. This parametrization was obtained by fitting Argon 2- and 3-body potentials – and, in effect, thermal virial coefficients B, C, D computed from these – to experimental acoustic data by Estrada-Alexander and Trusler [2]. Agreement with reference EOS data by Gilgen et al. [3,4] has been claimed to be within ±5 104 in terms of PV=RT ¼ 1 þ B=V þ C=V 2 þ up to sub- and supercritical molar volumes V of ca. 300 and 200 cm3 mol1, respectively. We show here that the analysis of Ref. [1] is flawed and consequently leads to an erroneous parametrization. The approach chosen in Ref. [1] is approximate in the sense that, first, the total potential’s many-body expansion is truncated after the third-order term and, second, 3-body contributions to D are considered in an ‘‘approximate’’ way only. (A choice of fixed functional forms for 2- and 3-body potentials Uij and Uijk is a limitation that cannot be avoided.) Whereas it is sensible to omit 4-body contributions D4 to D ¼ D2 þ D3 þ D4 [5], Ref. [1]’s approximation to the 3-body part
obtained by discarding all terms with more than one 3-body f-function fijk :¼ ebUijk 1 (in the usual diagrammatic notation [1,6,7] symbolized by a shaded triangle), is unnecessary and inconsistent. E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2012.01.023
For b = 1/kBT > 0, fijk ? 0 if and only if Uijk ? 0; but for configurations where 0 > Uijk 1/b, fijk 0. Such configurations exist [8,9] also for weak triple-dipole potentials as employed in Ref. [1] and, thus, contribute to the full-space integral D. We know of no rigorous P P argument to infer exp b i
44
J. Wiebke / Chemical Physics 411 (2013) 43–44
Fig. 1. Temperature-dependence of D2 þ D3 for our set of ab initio potentials [16,17], Ref. [1]’s fitted Maitland–Smith and Axilrod–Teller potentials, and Ref. [1]’s Lennard–Jones and Axilrod–Teller potentials (in black, red, and blue, respectively). Empty symbols are D2 þ D3 values from Ref. [1]’s Table 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
model for Argon. However, we could not reproduce the D2 þ D3 values given in Ref. [1]’s Table 4 with neither expression for D3 , using any of Ref. [1]’s sets of potentials; these values are also plotted in Fig. 1 for comparison. We are certain that our calculations are correct as, first, our code reproduces both experimental Ar EOS [3,4] and ab initio Ar EOS data [14] computed with an independently implemented Gauss–Legendre integration scheme [14] and, second, experimental EOS data for Neon from only ab initio potentials [15]. We also carefully re-computed all integrals given in Tables 1 and 2 of Ref. [1] and compared with reference data [6,7] cited therein. For Ref. [1]’s LJ potential (/kB = 143.3163 K, r = 6.3343a0), at T⁄ = 1 we compute values of 4.11 ± 0.01 and 2.684 ± 0.004 for the reduced first and second integrals of Eq. (2), respectively, as compared to reference values of 4.1438 and 2.71 [6,7]. (Note that the column captions of Tables 1 and 2 are misleading and occasionally wrong. All entries correspond to the
unweighted diagrams; but similar to Casanova et al. [6], the first diagram of Eq. (2) is tabulated incuding a factor of 32.) We do not want to speculate about how the D2 þ D3 values of Ref. [1]’s Table 4 must have been derived from the apparently correct diagrams listed in Tables 1 and 2. Instead, we conclude that the analysis of Ref. [1] is inconsistent. The temperature-dependence of D2 þ D3 D given in Ref. [1]’s Table 4 could not be reproduced and, moreover, is incorrect. We strongly discourage from its use, particularly as correct parametrizations for fourth- [14] and higher-order [18] virial EOS for Argon have been reported recently. We also discourage from employing ad hoc ‘‘approximations’’ [7,10–13] to the 3-body or, generally, the non-additive contributions to D, and have shown that such ‘‘approximations’’ may, in fact, overestimate these contributions significantly. We consequently disagree with Ref. [1]’s conclusion that the ‘‘non-additive three-body forces in Argon are vanishingly weak, so that the approximation employed here led to a good estimate of [the ordinary fourth virial coefficient]’’ [1]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
K.O. Monago, Chem. Phys. 316 (2005) 9. A.F. Estrada-Alexander, J.P.M. Trusler, J. Chem. Thermodyn. 27 (1995) 1075. R. Gilgen, R. Kleinrahm, W. Wagner, J. Chem. Thermodyn. 26 (1994) 383. R. Gilgen, R. Kleinrahm, W. Wagner, J. Chem. Thermodyn. 26 (1994) 399. J. Wiebke, P. Schwerdtfeger, E. Pahl, in preparation. G. Casanova, R.J. Dulla, D.A. Jonah, J.S. Rowlinson, G. Saville, Mol. Phys. 18 (1970) 589. R.J. Dulla, J.S. Rowlinson, W.R. Smith, Mol. Phys. 21 (1971) 29900315. V.F. Lotrich, K. Szalewicz, J. Chem. Phys. 106 (1997) 9688. M.J. Elrod, R.J. Saykally, Chem. Rev. 94 (1994) 1975. K.O. Monago, Chem. Phys. 337 (2007) 125. K.O. Monago, Can. J. Chem. Eng. 88 (2010) 55. K.O. Monago, Korean. J. Chem. Eng. 27 (2010) 590. J.P.M. Trusler, Int. J. Thermophys. 18 (1997) 635. J. Wiebke, P. Schwerdtfeger, G.E. Moyano, E. Pahl, Chem. Phys. Lett. 514 (2011) 164. J. Wiebke, in preparation. B. Jäger, R. Hellmann, E. Bich, E. Vogel, Mol. Phys. 107 (2009) 2181. P. Schwerdtfeger, B. Assadollahzadeh, A. Hermann, Phys. Rev. B 82 (2010) 205111. B. Jäger, R. Hellmann, E. Bich, E. Vogel, J. Chem. Phys. 135 (2011) 084308-1.
Contents lists available at SciVerse ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Comment
Comment on ‘‘An equation of state for gaseous argon determined from the speed of sound’’ J. Wiebke Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey University Albany, Private Bag 102904, North Shore City, Auckland 0745, New Zealand
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 15 December 2011 In final form 25 January 2012 Available online 7 February 2012
Ó 2012 Elsevier B.V. All rights reserved.
Keywords: Argon Virial equation of state Mayer cluster expansion Many-body interaction potentials
In a contribution to Chem. Phys., Monago reported [1] a new semi-empirical parametrization of a fourth-order virial Equation of State (EOS) for Argon. This parametrization was obtained by fitting Argon 2- and 3-body potentials – and, in effect, thermal virial coefficients B, C, D computed from these – to experimental acoustic data by Estrada-Alexander and Trusler [2]. Agreement with reference EOS data by Gilgen et al. [3,4] has been claimed to be within ±5 104 in terms of PV=RT ¼ 1 þ B=V þ C=V 2 þ up to sub- and supercritical molar volumes V of ca. 300 and 200 cm3 mol1, respectively. We show here that the analysis of Ref. [1] is flawed and consequently leads to an erroneous parametrization. The approach chosen in Ref. [1] is approximate in the sense that, first, the total potential’s many-body expansion is truncated after the third-order term and, second, 3-body contributions to D are considered in an ‘‘approximate’’ way only. (A choice of fixed functional forms for 2- and 3-body potentials Uij and Uijk is a limitation that cannot be avoided.) Whereas it is sensible to omit 4-body contributions D4 to D ¼ D2 þ D3 þ D4 [5], Ref. [1]’s approximation to the 3-body part
obtained by discarding all terms with more than one 3-body f-function fijk :¼ ebUijk 1 (in the usual diagrammatic notation [1,6,7] symbolized by a shaded triangle), is unnecessary and inconsistent. E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2012.01.023
For b = 1/kBT > 0, fijk ? 0 if and only if Uijk ? 0; but for configurations where 0 > Uijk 1/b, fijk 0. Such configurations exist [8,9] also for weak triple-dipole potentials as employed in Ref. [1] and, thus, contribute to the full-space integral D. We know of no rigorous P P argument to infer exp b i
44
J. Wiebke / Chemical Physics 411 (2013) 43–44
Fig. 1. Temperature-dependence of D2 þ D3 for our set of ab initio potentials [16,17], Ref. [1]’s fitted Maitland–Smith and Axilrod–Teller potentials, and Ref. [1]’s Lennard–Jones and Axilrod–Teller potentials (in black, red, and blue, respectively). Empty symbols are D2 þ D3 values from Ref. [1]’s Table 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
model for Argon. However, we could not reproduce the D2 þ D3 values given in Ref. [1]’s Table 4 with neither expression for D3 , using any of Ref. [1]’s sets of potentials; these values are also plotted in Fig. 1 for comparison. We are certain that our calculations are correct as, first, our code reproduces both experimental Ar EOS [3,4] and ab initio Ar EOS data [14] computed with an independently implemented Gauss–Legendre integration scheme [14] and, second, experimental EOS data for Neon from only ab initio potentials [15]. We also carefully re-computed all integrals given in Tables 1 and 2 of Ref. [1] and compared with reference data [6,7] cited therein. For Ref. [1]’s LJ potential (/kB = 143.3163 K, r = 6.3343a0), at T⁄ = 1 we compute values of 4.11 ± 0.01 and 2.684 ± 0.004 for the reduced first and second integrals of Eq. (2), respectively, as compared to reference values of 4.1438 and 2.71 [6,7]. (Note that the column captions of Tables 1 and 2 are misleading and occasionally wrong. All entries correspond to the
unweighted diagrams; but similar to Casanova et al. [6], the first diagram of Eq. (2) is tabulated incuding a factor of 32.) We do not want to speculate about how the D2 þ D3 values of Ref. [1]’s Table 4 must have been derived from the apparently correct diagrams listed in Tables 1 and 2. Instead, we conclude that the analysis of Ref. [1] is inconsistent. The temperature-dependence of D2 þ D3 D given in Ref. [1]’s Table 4 could not be reproduced and, moreover, is incorrect. We strongly discourage from its use, particularly as correct parametrizations for fourth- [14] and higher-order [18] virial EOS for Argon have been reported recently. We also discourage from employing ad hoc ‘‘approximations’’ [7,10–13] to the 3-body or, generally, the non-additive contributions to D, and have shown that such ‘‘approximations’’ may, in fact, overestimate these contributions significantly. We consequently disagree with Ref. [1]’s conclusion that the ‘‘non-additive three-body forces in Argon are vanishingly weak, so that the approximation employed here led to a good estimate of [the ordinary fourth virial coefficient]’’ [1]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
K.O. Monago, Chem. Phys. 316 (2005) 9. A.F. Estrada-Alexander, J.P.M. Trusler, J. Chem. Thermodyn. 27 (1995) 1075. R. Gilgen, R. Kleinrahm, W. Wagner, J. Chem. Thermodyn. 26 (1994) 383. R. Gilgen, R. Kleinrahm, W. Wagner, J. Chem. Thermodyn. 26 (1994) 399. J. Wiebke, P. Schwerdtfeger, E. Pahl, in preparation. G. Casanova, R.J. Dulla, D.A. Jonah, J.S. Rowlinson, G. Saville, Mol. Phys. 18 (1970) 589. R.J. Dulla, J.S. Rowlinson, W.R. Smith, Mol. Phys. 21 (1971) 29900315. V.F. Lotrich, K. Szalewicz, J. Chem. Phys. 106 (1997) 9688. M.J. Elrod, R.J. Saykally, Chem. Rev. 94 (1994) 1975. K.O. Monago, Chem. Phys. 337 (2007) 125. K.O. Monago, Can. J. Chem. Eng. 88 (2010) 55. K.O. Monago, Korean. J. Chem. Eng. 27 (2010) 590. J.P.M. Trusler, Int. J. Thermophys. 18 (1997) 635. J. Wiebke, P. Schwerdtfeger, G.E. Moyano, E. Pahl, Chem. Phys. Lett. 514 (2011) 164. J. Wiebke, in preparation. B. Jäger, R. Hellmann, E. Bich, E. Vogel, Mol. Phys. 107 (2009) 2181. P. Schwerdtfeger, B. Assadollahzadeh, A. Hermann, Phys. Rev. B 82 (2010) 205111. B. Jäger, R. Hellmann, E. Bich, E. Vogel, J. Chem. Phys. 135 (2011) 084308-1.