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The Sublimation of Argon, Krypton, and Xenon

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The Sublimation of Argon, Krypton, and Xenon J. Chem. Thermodynamics 40 (2008) 1621–1626 Contents lists available at ScienceDirect J. Chem. Thermodynamics journal homepage: www.elsevier.com/locate/jct The sublimation of argon, krypton, and xenon A.G.M. Ferreira, L.Q. Lobo * Departamento de Engenharia Química, Universidade de Coimbra, Coimbra 3030-290, Portugal article info abstract Article history: In this paper, the sublimation pressure of Ar, Kr, and Xe are obtained as functions of temperature from an Received 18 June 2008 exactly integrated form of the Clapeyron equation. No fitting to experimental data of the equilibrium Received in revised form 30 July 2008 pressure has been necessary. The deviation plots of the sublimation pressure show that the results are Accepted 30 July 2008 satisfactory. The derived enthalpy of sublimation of the three rare gases from T = 0 K up to their respec- Available online 8 August 2008 g s tive triple point temperatures are asymmetric, distorted parabolas showing maxima for Cp;m ¼ Cp;m. The g D Hm ðT ¼ 0KÞ, which is a measure of the cohesion energy of the solid crystals, is easily calculated. A gen- Keywords: s eral equation has been obtained for DgH as a function of temperature which also gives the enthalpies of Rare gases: Ar, Kr, Xe, (Ne, Rn) s m sublimation of neon and radon. The (s + ‘ + g) triple point coordinates of Rn are reassessed. Sublimation Pressure Ó 2008 Elsevier Ltd. All rights reserved. Enthalpy: Clapeyron equation 1. Introduction In this paper, we examine some of the sublimation properties of the rare gases from a more fundamental approach, since the ancil- Not long ago, we reported on the physico-mathematical lary data necessary for the calculations are available in the litera- description of the phase equilibria of pure substances from an ex- ture for Ar, Kr, and Xe. Expressions for the sublimation curves of actly integrated form of the Clapeyron equation [1]. A number of the three substances are obtained from the knowledge of those substances from rare gases to metals, and from polar compounds data with no need of fitting to any experimental values of the equi- to hydrocarbons (either linear, or cyclic, or aromatic) were put to librium pressure. The derived enthalpies of sublimation are then test with satisfactory results. Almost all first order transitions viz. easily calculated, namely at zero kelvin whose values render direct sublimation, vaporization, fusion, and solid to solid, were exam- assessment of the cohesion energy of the solid crystals of each of ined in that light. Of course, such ab initio treatment of the phase the three elements. equilibrium curves requires the knowledge of certain primary ther- modynamic properties of the pure substances involved and of their 2. Theory phases: heat capacities, molar volumes, enthalpies of transition, and the coordinates of reference points (usually taken from those From the exact integration of the Clapeyron equation of the triple points of the substances). In the absence of measured g g values of these properties, one can try and describe the equilibrium dp ðDs Hm=Ds V mÞðdT=TÞ¼0; ð1Þ curves through empirical equations. Using this latter procedure, where p and T are, respectively, the equilibrium pressure and tem- we recently reported on the vapour pressure of radon [2] and on g g perature, Ds Hm is the molar enthalpy of sublimation, and Ds V m is the fusion curves of Xe, Kr, and Ar [3]. In the years of 1960s, Ziegler the molar volume change along the (solid + gas) equilibrium curve, and co-workers [4–6] analysed the experimental sublimation data an analytical representation of the sublimation curve of pure sub- for argon, krypton, and xenon, and devised a method to conjugate stances is obtained in the form [1]: the information available to describe the (solid + vapour) equilib- X4 ria. Not many experimental results on the same subject have been iÀ1 ln p ¼ A À B=T þ C ln T þ DiT þ EðTÞp=T: ð2Þ published subsequently. Leming and Pollack [7] interpreted their i¼2 own measurements on the sublimation pressure of the three hea- vier classical rare gases over large temperature intervals by using In this equation A, B, C, Di, and E(T) are parameters that can be cal- the (microscopic) principle of corresponding states. culated ab initio from measured properties of the phases at equilib- rium: the heat capacities and molar volumes of the two phases, the (p,T) coordinates of a reference point on the sublimation curve (usually those of the s + ‘ + g triple point), and the enthalpy of phase * Corresponding author. Tel.: +351 239 798 733; fax: +351 239 798 703. transition at that point. All these quantities contribute to the values E-mail address: [email protected] (L.Q. Lobo). of A and B while the calculation of parameter C only involves the 0021-9614/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2008.07.023 1622 A.G.M. Ferreira, L.Q. Lobo / J. Chem. Thermodynamics 40 (2008) 1621–1626 molar heat capacities of the two phases. The Di depends on the mo- adopted. The coordinates of the triple point of krypton were both lar volume of the solid and the difference between the molar heat taken from reference [8] after converting the temperature into capacities of gas and solid. The E(T), which is a function of temper- ITS-90 by recommended methods [9,10] giving Tt = 115.772 K, ature, is calculated from the molar volumes of both phases. The ex- and pt = (72.999 ± 0.012) kPa. The same procedure was adopted act expressions relating all these parameters and properties are for Xe yielding Tt = 161.404 K, and pt = (81.675 ± 0.011) kPa, also given and discussed in reference [1] as well as the relative magni- from reference [8]. For the present purposes, this value of Tt (Xe) tude of the terms in equation (2). The term A, which is a positive is virtually the same as Tt = (161.40596 ± 0.032) K arrived at by Hill constant, is smaller than terms involving B and C, while the summa- and Steele [11] in a recent most careful study intended to investi- tion over the Di terms is negative in every case. Although consider- gate the possible establishment of the exact triple point tempera- able partial cancellation between such terms always occurs, all their ture of Xe as a fixed point in the ITS-90. contributions are relatively important in magnitude. The term The molar enthalpy of sublimation at the triple point g involving E(T) is only relevant at temperatures not far below the tri- Ds Hm ðT ¼ TtÞ can be obtained from the general relationship ple point, since in the low temperature region the values of E(T) are small and also because they are to be multiplied by the relatively g ‘ g Ds Hm ¼ DsHm þ D‘ Hm ðat T ¼ TtÞ; ð5Þ small values of the sublimation pressure p and divided by the equi- g librium temperature T. since no direct measurements of Ds Hm ðT ¼ TtÞ seem to have been g reported, as far as we are aware. The enthalpy of vaporization of ar- The molar enthalpy of sublimation Ds Hm is obtained as a func- tion of temperature from equation (1) by using equation (2) to cal- gon was determined experimentally by Flubacher et al. [12] at the culate (dp/dT). The following expression is obtained [1]: temperature of T = 85.67 K, slightly above Tt. Since the difference ()between the heat capacities of the vapour and the liquid phases X4 g g i À2 D‘ Cp;m is known as function of temperature, a short extrapolation Ds Hm ¼ RBþ CT þ ði À 1ÞDiT þfdEðTÞ=dTgpT : ð3Þ g À1 gives D Hm ðT ¼ TtÞ¼6595 J Á mol . Flubacher et al. also mea- i¼2 ‘ sured the enthalpy of fusion of argon at its triple point obtaining Figure 2 in reference [1] illustrates the results for several sub- ‘ 3 À1 DsHm ðT ¼ TtÞ¼1:190 Á 10 J Á mol [12], not far from the value stances. A direct conclusion can be drawn from equation (3): that of 1175 J Á molÀ1 at the same temperature given by Clusius [13] the enthalpy of sublimation at T = 0 K is given simply by from his own experiments. Combining the average of these two fig- ‘ g ures for DsHm ðT ¼ TtÞ with the above mentioned value of the en- Ds Hm ðT ¼ 0KÞ¼RB: ð4Þ thalpy of vaporization at Tt, the enthalpy of sublimation at the This quantity, which is easily assessed in this way, is a measure of g triple point is obtained from equation (5) as Ds Hm ðT ¼ TtÞ¼ the energy necessary to transform the solid crystal at zero kelvin 7:778 Á 103 J Á molÀ1. The enthalpy of sublimation of krypton at into the (perfect) gas at the same temperature, and it is, of course, its triple point was determined by Beaumont et al. [14] as closely related to the so-called lattice cohesion energy of the solid, g 3 À1 Ds Hm ðT ¼ TtÞ¼10:791 Á 10 J Á mol from their measurements a quantity of interest in the study of intermolecular forces and g ‘ of D‘ Hm ðT ¼ 116:85 KÞ and DsHm ðT ¼ TtÞ. The situation for xenon otherwise. is similar to that of argon; Clusius and Riccoboni [15] measured Equation (3) enables the calculation of DgH from zero kelvin to g 3 À1 s m D‘ Hm ðT ¼ 165:13 KÞ¼12:636 Á 10 J Á mol , a few Kelvin above the triple point once sufficiently accurate values of the quantities the triple point, which after a short extrapolation using the differ- necessary to the ab initio evaluation of the sublimation pressure g ence between the molar heat capacities of both phases D‘ Cp;m gives through equation (2) are available, as happens for Ar, Kr, and Xe.
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