1896
J. Phys. Chem. 1996, 100, 1896-1899
A New Property of the ISM Equation of State† Mohammad Hadi Ghatee* and Ali Boushehri Department of Chemistry, Shiraz UniVersity, Shiraz 71454, Iran ReceiVed: June 26, 1995; In Final Form: October 3, 1995X
With a new development of corresponding states principle involving quasi-second virial coefficients of alkali metals, the free parameter of the ISM analytical equation of state, Γ, is shown to incorporate quantum effects. This observation is justified by the assumption that Γ adjusts itself with the value of a second virial coefficient so much that both describe the compressibility behavior of the system. Calculations along with the present data show that Li, and to some extent Na, deviates from the corresponding states behavior, succeeded by K, Rb, and Cs. By developing a corresponding states principle involving ∆Hv, which the scaling factor used in the calculation of the quasi-second virial coefficients of alkali metals, the quantum effect in Γ is well justified. The same behaviors are observed among noble gases, where only He deviates. The statistical mechanical view of the open-shell electronic structure of alkalis must be considered as a special difference in physical behavior with noble gases.
Introduction In a recent paper we demonstrated that the analytical equation of state derived by Ihm, Song, and Mason (ISM) can be applied to alkali metals.1 The equation is based on the statistical mechanical perturbation theory of liquid state in the perturbation scheme of Week-Chandler-Anderson.2,3 In the final form it reads as
P/ρkT ) 1 - [(R - B2)ρ/(1 + 0.22Γbρ)] + RρG(bρ) (1) where
G(bρ) ≈ 1/(1 - Γbρ)
(2)
and P is the pressure, F is the density, and kT is the thermal energy of one molecule. B2 is the second virial coefficient, R is a parameter that takes care of the softness of the repulsive side of the potential function, and b is the van der Waals covolume. In eq 1 the characteristics constants B2, R, and b are all temperature dependent. R and b vary slightly with temperature and do not depend on the details of the potential function used for their calculations. B2 depends highly on temperature and has the major role to represent an interaction coefficient of the system in the equation. G(bF) is the pair distribution function for hard convex bodies at contact.4 G(bF)-1 is a linear function of bF only, with the slope of Γ. Γ is a free parameter that is known to compensate for any uncertainties in B2(T). The value of R and b can be calculated from the potential function by integration. If an accurate potential function is not known, both can be calculated from the experimental second virial coefficient by means of the two-constant scaling rule.2 This can be done because R and b depend on the intermolecular repulsive forces and are relatively insensitive to the details of the intermolecular potential. Thus the second virial coefficient serves to predict the entire equation of state for fluids in terms of two scaling constants, over the whole range of temperatures and pressures, excluding the nonanalytical critical point and the two-phase region. † This paper is dedicated to the memory of Professor E. A. Mason of Brown University. * Corresponding author. X Abstract published in AdVance ACS Abstracts, December 1, 1995.
0022-3654/96/20100-1896$12.00/0
Equation 1 is well applied to alkali metals. It reproduces the experimental liquid densities within 5%, from freezing temperature to several hundred degrees above boiling temperature.1 The values of B2(T), in applying eq 1 to alkalis, were calculated from the corresponding states correlation given for normal fluids that follow Lennard-Jones (12-6) potential.5 For this purpose, ∆Hv and Ff, the latent heat of vaporization and liquid density at the freezing temperature, respectively, are the only input (scaling) data. This seems to be a good solution where the experimental values of B2(T) are not available, and its calculation is complicated due to polyatom formation in the vapor phase of alkalis.6 Here, two points are noticeable. First, the alkalis in the vapor state interact by two possible singletand triplet-type potentials, and second, alkalis do not follow the Lennard-Jones potential. In other words, alkali metals have the characteristics of interacting through singlet and triplet potentials so that the treatment by a single potential here is fortuitous. Alkali metals, therefore, would not follow a corresponding states for normal fluids. By this procedure then, only quasi-second virial coefficients are obtained. This was the rationale for the results we obtained. The procedure, however, successfully produces the three temperature-dependent parameters to be used in eq 1. A Physical Picture for Γ Γ is known as the free parameter of the equation of state that primarily compensate for the uncertainty of the B2(T). Now we develop the idea that Γ varies slightly with temperature to behave in a structural interaction conformity with the quasisecond virial coefficient in such a way that both describe the compressibility behavior of the system. However, at low temperatures, we would expect that Γ does not keep an appreciable conformity with R(T) and b(T), because they vary slightly with temperature, while B2(T) shows a sharp variation. On the other hand, at high temperatures, B2(T), R(T), and b(T) vary slightly with temperature, and since their values are in the same order of magnitude, one expects Γ to vary in accord with the net effect of all three constants. Equation 1 has a feature that can be written as
P/ρkT ) 1 - [(R* - B2*)/(1 + 0.22Γb*)] + R*G(b*) (3) where asterisks indicates that the parameters are reduced with © 1996 American Chemical Society
ISM Equation of State
J. Phys. Chem., Vol. 100, No. 5, 1996 1897
TABLE 1: Values of Γ for Alkali Metals and Noble Gases Γ
substance
d (nm)
/k (K)
Li Na K Rb
0.314a 0.384a 0.476a 0.504a
2350a 1970a 1760a 1600a
0.512c 0.485c 0.472c 0.466c
Cs He Ne Ar Kr Xe
0.540a 0.2610b 0.2755b 0.3350b 0.3571b 0.3885b
1550a 10.4b 42.0b 141.5b 197.8b 274.0b
0.462c 0.366d,e 0.384d,e 0.381d 0.385d,e 0.381d
0.454f 0.454f 0.454f 0.454f
0.396g 0.395g 0.396g 0.395g
a Reference 12. b Reference 8. c Reference 1. d Reference 5. e The same calculations were carried out as in ref 5 (this work). f The calculations involve the use of Aziz potential function for the second virial coefficients.3,10 g The calculations involve the use of correlation for the second virial coefficients in which surface tension and liquid density at freezing temperature are used as scaling constants.11
respect to F at the particular temperature, T. It is not crucial to treat Γ as a constant or an imaginary reduced quantity, but taking Γ as a reduced quantity implies that a corresponding states principle, including a functional form of Γ, can exist. In other words, referring to the argument above, at low temperatures Γ(T) is adjusted with B2*(T), and at high temperatures it is adjusted with the net effects of B2*(T), R*(T), and b*(T). By this argument, Γ(T*) is related to B*(T*), where T* ) T/(/k) and (/k) is the potential well depth in units of kelvin, leading to a corresponding states principle for different alkalis. G(bF)-1 is virtually the strong corresponding state in the equation of state. For noble gases (Ne, Ar, Kr, Xe) a plot of G(bF)-1 vs bF is linear with the unique slope of Γ. The results of the calculation for Γ of noble gases by theoretical methods and by different correlations are listed in Table 1. Provisionally, as much as thermophysical properties are concerned, one may expect to observe the same behavior, if eq 1 is applied to molten alkalis, that is, all the metals of the group follow the same relation with a nearly unique Γ. However, from Table 1, it can be seen that although values of Γ for K, Rb, and Cs are nearly the same, Li and to some extent Na show deviations and therefore do not follow the principle of corresponding states perfectly. In view that Γ is related to B2(T), and therefore to the nature of the metals, we propose that quantum effects should be incorporated with Γ. The objective of this paper is to verify a quantal approach to Γ. In particular we use the quantum mechanical law of corresponding states to consider such a property in Γ. The establishment of the law of corresponding states for the second virial coefficient resolves the conformity of Γ with B2(T). Finally, by establishment of a law of corresponding states involving ∆Hv, it is shown that the results with the quasi-second virial coefficient is in a real sense. Calculations, Supporting Materials, and Discussion The quantum mechanical law of corresponding states permits one to relate a reduced parameter, parametrically with the reduced de Boer parameter Λ* ) h/(md2)1/2 where h, m, d, and are the Plank’s constant, mass, collision diameter, and potential well depth, respectively. Generally this parameter is a measure of the effect of quantum mechanics. If Λ* ) 0, one can apply classical statistics to give a reduced equation of state in accord with the law of corresponding states.7 We use Λ* to estimate the effects of quantum mechanics on Γ. The variation of Γ(T*)0.4) with Λ* is shown in Figure 1. The quantum effects in Γ are evident by the fact that Li and to some extent Na do not follow the trend that is succeeded by K, Rb, and Cs.
Figure 1. Variation of Γ with Λ* for alkali metals at T* ) 0.4. 9, Li; [, Na; 4, K; O, Rb; 0, Cs. The dashed line is a linear fit to K, Rb, and Cs.
Figure 2. Variation of Β2Ff with ln T* for alkali metals. 9, Li; [, Na; 4, K; O, Rb; 0, Cs. Li and Na, for T* e 0.4, do not follow the corresponding states principle.
We should mention that the deviations for Li and Na started at T* e 0.4. Since B2(T) is used to predict the entire equation of state and because B2(T) has the major role to represent the intermolecular interaction in eq 1, we may pursue the quantum property of Γ in B2(T). Figure 2 demonstrates the variation of B2(T)Ff, with ln T*. The second virial coefficients used here are quasi-ones, that is, they are calculated, whatever they are, from the corresponding states correlation developed from experimental second virial coefficients of normal fluids that follow Lennard-Jones potential function.1,5 In this case also we see that for T* e 0.4, Li and to some extent Na do not follow the corresponding states succeeded by K, Rb, and Cs. We shall prove this conclusion that the physical properties due to the intermolecular forces, particularly the quantum effects, are contained in Γ. In other words, we shall show that Γ is more than an ad hoc constant. The behavior of noble gases here is considered by comparing their Γ values listed in Table 1. They are from different correlations, using scaling constants such a ∆Hv - Ff, TB VB, and γf - Ff where TB, VB, and γf are Boyle’s temperature, Boyle’s volume, and surface tension at the freezing temperature, respectively (see the footnotes of Table 1 for specific references). It is interesting to see that, for a given correlation, Ne, Ar, Kr, and Xe have almost the same Γ values. For He, we calculated the values of Γ from eq 1 using experimental B2(T) values8 and
1898 J. Phys. Chem., Vol. 100, No. 5, 1996
Figure 3. Variation of Β2* ()B2/0.666πΝd3) with ln T* for noble gases. 9, He; [, Ne; 4, Ar; O, Kr; 0, Xe. Only He does not follow the corresponding states principle.
the Lennard-Jones potential function along with ∆Hv and Ff as scaling constants to calculate values of R(T) and b(T).5 For neon, experimental data on B2(T) are available only down to T* ) 1,8 and so we used the correlation given by Najafi et al.9 to calculate B2(T) at the freezing temperature. We see that the value of Γ for He markedly deviates from other elements of the group. The corresponding states behavior of the noble gases, demonstrated in Figure 3, show that the elements of the group are strongly correlated, except He. From the values of Γ for noble gases, it can be said that only He shows quantum effects. This is consistent with the observation that its second virial coefficient does not even moderately correlate with the second virial coefficient of the elements of its group (see Figure 3). These observations may be used to estimate that only small quantum effects associated with the second virial coefficient of Ne at its freezing temperature, T* ) 0.59. Thus the available data on Γ and the principle of corresponding states for noble gases and alkalis (that we already developed) convince us that both Γ and the second virial coefficients describe the compressibility behaviors including the quantum effects. However, this justification on alkalis is questionable in that we come up with this conclusion by using the quasisecond virial coefficients calculated from the correlation that uses ∆Hv and Ff as scaling constants, while we used the experimental second virial coefficient for noble gases. This leads us to examine the scaling constants from which the correlation for B2(T) of normal fluids originate. This should finalize the thoroughness of the conclusion that we seek. The quantum effects on the scaling constants of alkalis can be verified from the corresponding states principle involving (experimental) ∆Hv demonstrated in Figure 4. This in particular justifies the conclusion that Γ incorporates quantum effects. In other words, the quantum effects that are contained in Γ, the free parameter of the eq 1, quantitatively originate from the fact that Li and Na deviate from the corresponding state principle involving ∆Hv of alkalis. Since the quantum effects adhere to the nature of the systems, this conclusion about the alkalis finds generality regardless of the correlation from which B2(T) is calculated. In Figure 5 we see that noble gases (except He) follow a promising correlation based on the corresponding states involving ∆Hv in all temperature ranges. Since noble gases (except He) have the same Γ values, this supports the idea that Γ incorporates the quantum effects among alkalis and noble gases.
Ghatee and Boushehri
Figure 4. Variation of ∆HV/RT with ln T* for alkali metals. 9, Li; [, Na; 4, K; O, Rb; 0, Cs. Li and Na do not follow the corresponding states principle.
Figure 5. Variation of ∆HV/RT with ln T* for noble gases. 9, He; [, Ne; 4, Ar; O, Kr; 0, Xe. Only He does not follow the corresponding states principle.
Using the de Boer parameter as a criteria for the extents of quantum effects, one expect that Ne incorporates larger quantum effects than Na while in this work the reverse is observed (inspect Figures 2-5). This may be attributed to the fact that (/k) values for alkali metals are much larger than those of noble gases. The singlet and triplet type potentials of the alkalis reflect the nature of interactions; however, they do not apparently enter our results. The open-shell electronic structure of alkali metals is responsible for the extensive quantum effects. The de Boer constant takes the mass, size, and energy effects as criteria for the estimation of the degree of quantum statistical properties of a species. This should work quite well for noble gases, while for alkalis the degree of contribution of the free electron to the such properties distinctively cannot be stated. Such a contribution presumably leads to the reason that the values of Γ for K and Rb are almost the same as can be seen in Figure 1. Conclusions For both alkali metals and noble gases, Γ adjusts itself with the values of B2(T) so much so that both describe the compressibility behavior of the system through eq 1. The corresponding states principle involving experimental ∆Hv, in which Li and Na in the alkalis and He in noble gases show deviations, demonstrates the quantum effects, as they do in the corresponding states principle involving B2(T). This justifies that Γ incorporates quantum effects in a real sense even if the
ISM Equation of State quasi-second virial coefficients are used in the equation of state. This property of the ISM equation of state as yet was not perceived. Until now accurate second virial coefficients, for alkali metals especially at low temperatures, had been not reported. This work also provides a means of establishing a law of corresponding states involving the quasi second virial coefficient for alkali metals in a real sense. Acknowledgment. We are indebted to the Research Committee of the Shiraz University for supporting this work. References and Notes (1) Ghatee, M. H.; Boushehri, A. Int. J. Thermophys., in press. (2) Song, Y.; Mason, E. A. J. Chem. Phys. 1989, 91, 7840.
J. Phys. Chem., Vol. 100, No. 5, 1996 1899 (3) Ihm, G.; Song, Y.; Mason, E. A. J. Chem. Phys. 1991, 93, 3839. (4) Equations 1 and 2 appear to work as well for arbitrary dimensionality as they do for three dimensions; that is, they work for one-dimensional hard rods and hard spheres in three dimensions as well: Song, Y., Mason, E. A. J. Chem. Phys. 1990, 93, 686. (5) Boushehri, A.; Mason, E. A. Int. J. Thermophys. 1993, 14, 685. (6) Nieto de Castro, C. A.; Fareleira, J. M. N. A.; Matias, P. M.; Ramires, M. L. V.; Canelas, A. A. C.; Varandas, A. J. C. Ber Bunsen-Ges. Phys. Chem. 1990, 74, 53. (7) de Boer, J. Physica 1948, 14, 139. (8) Dymond, J. H.; Smith, E. B. The Virial Coefficient of Pure Gases and Mixtures; Clarendon Press: Oxford, 1980. (9) Najafi, B.; Mason, E. A.; Kestin, J. Physica 1983, 119A, 387. (10) Ihm, G.; Song, Y.; Mason, E. A. Fluid Phase Equilib. 1992, 75, 117. (11) Ghatee, M. H.; Boushehri, A. Int. J. Thermophys., to be published. (12) Pasternak, A. D. Mater. Sci. Eng. 1968/69, 3, 65.
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