6376
Ind. Eng. Chem. Res. 2007, 46, 6376-6378
Response to “Comments on ‘The Second Virial Coefficient and the Redlich-Kwong Equation’ ” Paul M. Mathias Fluor Corporation, 3 Polaris Way, Aliso Viejo, California 92656 Sir: The issues raised by Tian and Gui are useful to rebut, because they allow me to uncover the important, but largely forgotten, contributions of Daniel Berthelot to applied thermodynamics and to comment on the unfortunate practices of many scientists and engineers (including Tian, Gui, and myself) who read and cite secondary rather than original sources. In the series of papers of interest (1899 to 1907), Berthelot’s goal was to calculate very accurate departures from the idealgas equation of state at near-ambient conditions so that molecular weights could be calculated from vapor-phase massdensity measurements. At that time, it was of scientific interest to compare and evaluate molecular weights determined by chemical methods with those from physicochemical methods (i.e., densities). For example, Jones1 showed that the molecular weight of hydrogen determined from physicochemical and chemical methods agree to a high degree of precision. Here, I present a historical review of Berthelot’s methodology and the adoption of his results by later researchers. The molecular weight of a substance can be calculated from its experimental mass density as follows:
RT0 P0
MW ) Fmass(T0, P0) Z(T0, P0)
(1)
where Fmass is the mass density, Z the compressibility factor, and R the universal gas constant. Equation 1 is, of course, an exact equation at any temperature/pressure pair, but is typically applied under “normal” conditions (T0 ) 273.15 K and P0 ) 1 atm). In the late 1800s and early 1900s, it was expected that the Fmass(T0, P0) value could be measured to an accuracy of 0.01%,2 and, hence, Z must be established to a better accuracy to enable estimation of the MW value to an equivalent accuracy of 0.01%. Berthelot’s goal was to estimate Z in eq 1 accurately, using experimental data or generalized correlations, and the approach of eq 1 was recognized as ‘D. Berthelot’s “method of limiting densities.”’1 In the first Berthelot paper of interest,3 he analyzed gas-density measurements and found that the product of pressure and molar volume at a fixed temperature is not constant, as expected from the Boyle equation, but can be accurately represented as a linear function of pressure, up to the low pressures of interest. In today’s language, we would say that the virial equation of state truncated after the second term is sufficiently accurate.
Z≡
PB2 P ≈1+ FRT RT
(2)
where F is the molar density and B2 is the second virial coefficient, which is a function of temperature but independent of pressure. Initially, Berthelot3 estimated the second virial coefficient from experimental data; however, he later developed a generalized correlation for B2,4,5 which Berthelot called the method of “indirect limiting density” and Jones1 referenced as ‘Berthelot’s method of “critical constants.”’
As was common at his time, Berthelot started with the van der Waals (vdW) equation:
P)
RT a V - b V2
(3)
where V is the molar volume and a and b are model parameters that are dependent on the critical constants. The vdW model has a second virial coefficient (B2), which is given by
B2 ) b -
a RT
(4)
Through comparison with the experimental data, Berthelot4 concluded that the temperature dependence of B2 from the vdW equation (eq 3) was too weak and, hence, adopted the model of Clausius, who postulated that parameter a varies inversely with temperature; Berthelot called this model the “Modified vdW” equation. Berthelot further concluded that the Modified vdW model gave a second virial coefficient that was biased high, because the critical compressibility from the vdW equation (Zc ) 3/8) is too high. He reviewed the experimental critical compressibilities of the molecules he had studied (see Figure 1) and concluded that a better value of the “universal” critical compressibility is Zc ) 9/32 ≈ 0.281; scientists of that era had a predilection for rational fractions rather than real numbers! The various models proposed and analyzed by Berthelot are summarized in Table 1; note that the value of Zc for the “New” model is an empirical value. The model that was called “New” by Berthelot, which was correctly written in my paper6 as eq 7, is as follows:
B2Pc 9 54/128 ) RTc 128 T 2
(5)
R
where Tc is the critical temperature, Pc the critical pressure, and TR the reduced temperature (TR ≡ T/Tc). Equation 5 is widely known in the literature and textbooks as the Berthelot model. Figure 1 is a reconstruction of Figure 3 from Berthelot’s 1907 paper,4 and it demonstrates the remarkably high accuracy achieved by the Berthelot model (eq 5). The experimental values of B2 used in Figure 1 are those reported by Berthelot;4 however, I note here that they are in good quantitative agreement with the best values available today. The Berthelot model achieved success and recognition rapidly. Figure 2 presents the errors of various estimations of the molecular weights using eq 1. These estimations use “accepted values” of the gas density under normal conditions, as reported by Guye,2 and three methods to calculate the compressibility factor: (1) the ideal-gas approximation; (2) the Berthelot model for the second virial coefficient; and (3) the second virial equation (eq 2), together with current established values of the second virial coefficient at T0 ) 273.15 K taken from the DIPPR 801 database.7 The reference values of the molecular weights were computed from IUPAC recommended values of atomic weights.8 The Berthelot model clearly provides
10.1021/ie070715b CCC: $37.00 © 2007 American Chemical Society Published on Web 08/21/2007
Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6377
Figure 1. Comparison between experimental data and generalized models for the second virial coefficient proposed by Berthelot.
Figure 2. Calculation of the molecular weight using the physicochemical method (eq 1). Vapor densities used are “accepted values” reported by Guye.2 The three lines represent Z values calculated using (9) the ideal gas equation, ([) the Berthelot model, and (4) experimental second virial coefficients.
approximately an order-of-magnitude improvement in the estimation of the molecular weight over the ideal-gas approximation, and the estimation is equivalent in many cases to the expected density accuracy of 0.01%. The use of a highly accurate value of the second virial coefficient provides only a relatively modest improvement over the Berthelot model. The accuracy diminishes for NH3 and SO2, most probably because of limitations in the accuracy of the density measurements. Incidentally, Jones1 concluded that the atomic weight of hydrogen from both chemical and limiting density methods was 1.00775. This agrees with the current IUPAC value of 1.00794 8 to ∼0.02%. Today, atomic weights are determined neither by the chemical methods nor by the physicochemical methods of the early 1900s, but rather are derived from the measured
isotopic composition of elements and the atomic masses of the isotopes, and the atomic-weight scale shifted from Ar(O) ) 16 to Ar(12C) ) 12 in 1957.9 Nevertheless, the contributions of Berthelot to the scientific advances of his time should be evident. However, Berthelot’s contributions are not appreciated by all observers, because de Laeter and Peiser9 have written that “the only other significant method for measuring atomic weights at the start of the [last] century was based upon the density of chemically pure gases under the assumption that the perfect gas law applied”. The Berthelot model served well as the basis for advances in molecular thermodynamics. Lambert et al.10 used the Berthelot equation, which “is generally regarded as providing a satisfactory empirical approximation to an equation of state for vapors at pressures below 1 atm”, to infer two distinct classes of behavior. The measured second virial coefficients of class I (or nonpolar) substances (e.g., ethane, n-hexane, chloroform) were in agreement with the Berthelot equation, whereas those of class II (or polar) substances (e.g., acetone, acetonitrile, methanol) were considerably lower (more negative) than the Berthelot predictions. Lambert at al.10 concluded that polar substances have additional energies of interaction, which are due to dipole and hydrogen bonding, and Rowlinson11 further developed a model of the intermolecular potential that included dipolar forces to interpret and fit the second-virial-coefficient data for strongly polar substances such as water, methanol, and acetone. Pitzer and Curl12 started with the Berthelot equation as a base two-parameter corresponding-states model and developed the first three-parameter corresponding-states model, with the acentric factor (ω) in addition to Tc and Pc, for the second virial coefficients of “normal” (nonpolar) fluids. Pitzer and Curl12 noted “that the Berthelot model yields excellent agreement with data for ω in the range 0.15 to 0.20”. The work of Berthelot3-5 is a fine example of the work processes of effective applied science. He made inspired empirical adjustments guided by experimental data to solve a targeted problem. Next, Berthelot’s work served as a segue for later scientists to solve additional problems, mostly outside the originally envisioned scope. However, there are also lessons to be learned. Today, the seminal contributions of Berthelot have been largely forgotten. When references are made to Berthelot’s work, they are sometimes incorrect. In my paper,6 I correctly wrote the Berthelot equation, but should have cited the original source. The secondary reference I supplied (Hirschfelder et al.13) did not provide a link to the original source. Berthelot had proposed his model 96 years before the publication year of 2003 and not “over 100 years ago,” as I loosely stated. The objections of Tian and Gui clearly have no basis, because they started with the Clausius equation rather
Table 1. Various Generalized Models for the Second Virial Coefficient Proposed by Berthelot4 model
( ) ( ) ( )
a RT
( )( ) ( )( ) ( )( )
Zc
B2Pc B2 ≡ ZcBRTc VcB
van der Waals (vdW) equation
1 RTc 8 Pc
27 RTc 1 64 Pc TR
3 8
4 1.5 9 TR
Modified vdW model
1 RTc 8 Pc
27 RTc 1 64 Pc T 2 R
3 8
4 1.5 9 T 2
9 RTc 128 Pc
27 RTc 1 64 Pc T 2 R
9 32
1 1.5 4 T 2
“New” or Berthelot model
a
b
a
ZcB ≡ 9/32.
R
R
6378
Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007
than Berthelot’s new model. There are other examples of vague or incorrect citations to Berthelot’s work. Johnston and Weimer14 and Black15 incorrectly cited the 1899 paper3 as the source of eq 5. Shah and Thodos16 incorrectly stated that the 1899 paper3 proposed the Clausius equation, which incorrectly was attributed to Berthelot rather than Clausius. Lambert et al.10 and Redlich and Kwong17 did not provide any reference for the Berthelot model; apparently, the Berthelot model was sufficiently wellknown in the 1940s, so no original citation was needed. Textbooks present the correct form of the Berthelot model, but not necessarily any literature citation. The thermodynamics textbook revised by Pitzer and Brewer18 presents both the correct model and reference, whereas Glasstone19 presents a correct model and a good explanation, but no reference. As noted previously, Hirschfelder et al.13 presented the Berthelot model but no reference. It is admittedly somewhat subjective whether an original reference must be cited for very well-known theories and models. I intentionally did not provide original sources for the models attributed to Boyle, van der Waals, and Clausius. However, my conclusion is that original sources should be provided for less well-known work, such as that of Berthelot, to give proper credit and to avoid errors. This rebuttal has discussed, at some length, the important (but largely forgotten) work of Daniel Berthelot. My rebuttal has also identified the need to cite original sources, particularly for work that is less well-known.
Literature Cited
(2) Guye, P. A. Researches on the Density of Gases. J. Am. Chem. Soc. 1908, 30, 143. (3) Berthelot, D. Sur Une Me´thode Purement Physique Pour La De´termination des Poids Mole´culaires des Gaz et des Poids Atomiques de Leurs EÄ le´ments. J. Phys. 1899, 8, 263. (4) Berthelot, D. Sur les Thermome´tres a Gaz et Sur la Reduction de Leurs Indications a L’echelle Absolue des Temperatures. TraV. Mem. Bur. Int. Poids Mes. 1907, (13), 13. (5) Berthelot, D. Sur le Calcul de la Compressibilite´ des Gaz au Voisinage de la Pression Atmosphe´rique au Moyen des Constantes Critiques. Compt. Rend. 1907, 144, 194. (6) Mathias, P. M. The Second Virial Coefficient and the RedlichKwong Equation. Ind. Eng. Chem. Res. 2003, 42, 7037. (7) DIADEM 2004, Ver. 2.7.0, Brigham Young University, Provo, UT, 2004. (8) Wieser, M. E. Atomic Weights of the Elements 2005 (IUPAC Technical Report). Pure Appl. Chem. 2006, 78, 2051. (9) de Laeter, J. R.; Peiser, H. S. A Century of Progress in the Sciences Due to Atomic Weight and Isotopic Composition Measurements. Anal. Bioanal. Chem. 2003, 375, 62. (10) Lambert, J. D.; Roberts, G. A. H.; Rowlinson, J. S.; Wilkinson, V. J. The Second Virial Coefficients of Organic Vapors. Proc. R. Soc. A 1949, 196, 113. (11) Rowlinson, J. S. The Second Virial Coefficients of Polar Gases. Trans. Faraday Soc. 1949, 45, 974. (12) Pitzer, K. S.; Curl, R. F., Jr. The Volumetric and Thermodynamic Properties of Fluids. III. Empirical Equation for the Second Virial Coefficient. J. Am. Chem. Soc. 1957, 79, 2369. (13) Hirschfelder, J. O.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1964; p 252. (14) Johnston, H. L.; Weimer, H. R. Low Pressure Data of State of Nitric Oxide and of Nitrous Oxide Between Their Boiling Points and Room Temperature. J. Am. Chem. Soc. 1934, 56, 625. (15) Black, C. Vapor Imperfections in Vapor-Liquid Equilibria. Ind. Eng. Chem. 1958, 50, 391. (16) Shah, K. K.; Thodos, G. A Comparison of Equations of State. Ind. Eng. Chem. 1965, 57, 30. (17) Redlich, O.; Kwong, J. N. S. On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions. Chem. ReV. 1949, 44, 233. (18) Pitzer, K. S.; Brewer, L. Thermodynamics; McGraw-Hill: New York, 1961; p 187. (19) Glasstone, S. Textbook of Physical Chemistry; MacMillan: London, 1951; p 296.
(1) Jones, G. The Atomic Weight of Hydrogen. J. Am. Chem. Soc. 1910, 32, 513.
IE070715B
Acknowledgment I am pleased to acknowledge the insight, comments, and French-English translations of Dr. Jean-Charles de Hemptinne, Professor John O’Connell, Professor Fred Stein, and most especially Dr. Johanna M. H. Levelt Sengers, which helped me uncover the very interesting and useful work of Professor Daniel Berthelot (now correctly cited as being presented a century ago).
