Thermophysical Properties of a Krypton Gas

Transcription

1 CHINESE JOURNAL OF PHYSICS VOL. 52, NO. 3 June 214 Thermophysical Properties of a Krypton Gas C. Benseddik, 1 M. T. Bouazza, 1 and M. Bouledroua 2 1 Laboratoire LAMA, Badji Mokhtar University, B. P. 12, Annaba 23, Algeria 2 Laboratoire de Physique des Rayonnements, Badji Mokhtar University, B. P. 12, Annaba 23, Algeria (Received October 21, 213; Revised January 2, 214) In this work, we use the Chapman-Enskog model to evaluate the transport parameters of a dilute gas formed of monatomic krypton Kr(4p). We calculate the coefficients of selfdiffusion, viscosity, and thermal conductivity when the krypton atoms are in their fundamental electronic configurations. The calculations are carried out quantum mechanically, making use of the potential energy curve, relative to the 1 Σ + molecular state. The data points upon which the construction is made are smoothly connected to the long- and short-range forms. They are supposed to behave analytically like 1/R n and α exp( βr), respectively. The spectroscopic data, r e and D e are in accordance with what is available in the literature. The second virial coefficients are also calculated at lower temperatures when the quantum effects are very important and for moderate and high temperatures. Our computation yields a value of the Boyle temperature T B 555 K. Generally, the results of the transport parameters with temperature show an excellent agreement with the available published data. DOI: /CJP PACS numbers: d, 34.2.Cf I. INTRODUCTION In the last few years, the determination of the thermophysical properties of a dilute gas has been the object of several experimental and theoretical works [1 7]. Such properties are mainly required in the qualitative and quantitative descriptions of several transport phenomena occurring, for instance, in the upper atmosphere, discharge lamps, and natural and artificial cold plasmas [5, 8, 9]. For such a field of investigation, several researchers calculated the transport properties at higher and lower temperatures of atomic and molecular gases by adopting very recent interatomic potentials. This work will focus on the determination of the thermophysical properties of krypton gas, namely, the second virial coefficients, self-diffusion coefficients, viscosity coefficients, and thermal conductivity. In particular, by adopting the Chapman-Enskog model [1], we will determine the collision integrals (integral of diffusion and integral of viscosity), deduce the transport coefficients, and examine their variation law with temperature. More attention will especially be given to the quantum-mechanical determination of the second virial coefficients of the Kr gas at low temperatures when the quantum effects is important. We also calculated the Boyle temperature and compare it with published works. All the physical quantities are given hereafter in atomic units (a.u.); in particular, h = 1, energies are in terms of the Hartree Energy (E h ), and distances in terms of the Bohr radius (a ) c 214 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

2 VOL. 52 C. BENSEDDIK, M. T. BOUAZZA, AND M. BOULEDROUA 13 II. THEORY The temperature-dependent thermophysical coefficients, namely, diffusion D, viscosity η, and thermal conductivity λ, for a dilute gas made of like monatoms are generally calculated by using the Chapman-Enskog model [1]. For one temperature T and atomic density n, these coefficients are given by the formulas D(T ) = 3 k B T 16nµ Ω (1,1) (T ), (1) η(t ) = 5 k B T 8 Ω (2,2) (T ), (2) λ(t ) = 5 C v η(t ), 4 µ (3) where k B is Boltzmann s constant, µ is the reduced mass of two colliding atoms, and C v is the specific heat per atom, generally taken as C v 3k B T for monatomic gases. All these expressions depend on the collision integrals [1], Ω (p,q) k B T = x 2q+3 Q p (x) exp ( x 2) dx, (4) 2πµ with the variable x 2 = E/k B T being related to the relative energy E. In this expression, Q p (x) are the transport cross sections defined, in atomic units, as and Q p=1 = 4π k 2 Q p=2 = 4π k 2 (l + 1) sin 2 (δ l+1 δ l ) (5) l= l= (l + 1) (l + 2) (2l + 3) sin 2 (δ l+2 δ l ), (6) where k = 2µE is the wave number of the relative motion and δ l = δ l (E) are the elastic phase shifts for energy E and angular momentum l. The phase shifts δ l are computed quantum mechanically by solving the radial wave equation [ ] k 2 2µV (r) Ψ (r) = (7) d 2 Ψ(r) dr 2 + l(l + 1) r 2 and imposing on the radial wave function Ψ(r) the asymptotic form ( Ψ(r) sin kr l ) 2 π + δ l (8) at large internuclear distances r. In the above Equation (7), V (r) is the interaction energy between the colliding atoms.

3 14 THERMOPHYSICAL PROPERTIES OF A KRYPTON GAS VOL. 52 III. POTENTIAL The resolution of Equation (7) requires a knowledge of the interatomic potential V (r). In our case, we consider the interaction between two ground krypton atoms Kr(4p) leading to the singlet 1 Σ + molecular state only. We have constructed the potential-energy curve relative to this state from recent ab initio data which are available in literature. In fact, we used the energy values from Ref. [11] for various distances r in the range a. The authors have obtained the Kr-Kr potential from coupled-cluster calculations, using the CCSD(T) level of theory and two successive correlation consistent basis sets, aug-cc-pvtz and -pqtz. For the short-range region, the potential is given by the mathematical form V (r) = A exp( αr), (9) where A and α are two constants. The long-range region is characterized by ( C6 V (r) = r 6 + C 8 r 8 + C ) 1 r 1, (1) where C 6 = , C 8 = , and C 1 = are the dispersion coefficients, in a.u., taken from Mitroy and Zhang [12]. The borrowed data points are smoothly connected to the forms characterizing the long- and short-range regions. The connection with the short-range region yields A = and α = The singlet potential-energy curve built up as described above is drawn in Figure Σ +.4 Kr(4p)+Kr(4p) potential (a.u.) Distance r (a.u.) FIG. 1: Singlet potential-energy curve of krypton dimer.

4 VOL. 52 C. BENSEDDIK, M. T. BOUAZZA, AND M. BOULEDROUA 15 TABLE I: Spectroscopic constants of ground state Kr 2 dimer of the ground molecular state compared with values from literature. Kr 2 ( 1 Σ + ) D e (µe h ) r e (Å) Refs This work [11] [13] [14] [15] [16] [17] The well depth D e as well as the equilibrium distance r e have been determined for this curve. The present values are displayed in Table I together with values from the literature. The agreement is satisfying. IV. SECOND VIRIAL COEFFICIENTS Having constructed the unique potential-energy curve relative to the Kr-Kr molecular state, it is judicious to analyze its accuracy and determine some of the physical parameters it can output at higher and lower temperatures. We have particularly examined the second virial coefficients B 2 (T ) for T ranging from 1 to 3 K. They are defined as [1] ( ) ( ) 4π 2 4π 2 2 B 2 (T ) B cl (T ) + B I (T ) + B II(T ) +... (11) m m with B cl (T ) being the classical second virial coefficient given by B cl (T ) = {1 exp [ V (r)/k B T ]} r 2 dr. (12) The remaining terms B I and B II represent the first and second quantum corrections, which are mainly significant at low temperatures [1]. They are given by the expressions B I (T ) = 1 24π (k B T ) 3 [ V (r) ] 2 exp [ V (r)/kb T ] r 2 dr (13)

5 16 THERMOPHYSICAL PROPERTIES OF A KRYPTON GAS VOL. 52 and B II (T ) = 1 96π 3 (k B T ) 4 { [V (r) ] 2 2 [ + V r 2 (r) ] k B T } 5 [ V 36 (k B T ) 2 (r) ] 4 1 [ V (r) ] 3 r exp [ V (r)/k B T ] r 2 dr, (14) where V (r) and V (r) are the first and second derivatives of the potential V (r) with respect to r. We list in Table II our results of the second virial coefficients, with and without the quantum effects, computed from Equation (11). It is easy to notice that the quantum effects seem to be very important at low temperatures, more exactly from 1 to 2 K. We compare in Table III our results of the second virial coefficients with those given by Kesten et al. [18] at a few temperatures higher than 5 K. A good agreement is noticed. The calculations yield, in particular, the Boyle temperature T B 555 K, at which B 2 (T B ) =, which is comparable to T B = 57 K of Ref. [19] and T B = 575 K of Ref. [11]. V. THERMOPHYSICAL PROPERTIES The knowledge of the potential V (r) enables us to calculate the phase shifts δ l and, therefore, the diffusion and viscosity cross sections (5) and (6), as well as the collision integrals (4) requested in the evaluation of the thermophysical properties, such as the selfdiffusion, viscosity, and thermal conductivity coefficients. Practically, to carry out the calculations of the transport cross sections, the phase shifts δ l = δ l (E), for each energy E and angular momentum l, are determined quantum mechanically by using the Numerov procedure to solve numerically the radial wave equation (7). Beyond a given large value of l, the phase shifts are rather calculated with the semi-classical expression [2] δ l 3π µc 6 k 4 16 l 5. (15) V-1. Self-diffusion For a monatomic dilute gas of density n and at temperature T, the self-diffusion coefficients are given in the Chapman-Enskog model by Equation (1). They have a direct relationship with the collision integrals for diffusion Ω (1,1) (T ), given in Equation (4) for p = 1 and q = 1, and thus with the cross section of diffusion Q 1. The calculations of D(T ) for the Kr-Kr system are performed for moderate and higher temperatures lying in the interval 2 3 K. We use Equation (5) to represent in Figure 2 the variations with energy of the total and diffusion cross sections. The shape resonances are particularly due to the contribution of the l = 4 and l = 5 partial waves. The self-diffusion coefficient is calculated from Equation (1) at the standard conditions of density and pressure, n m 3 and p = 76 torr. It is important to note that the pressure dependence of D(T ) is obtained from Equation (1), using the relationship of a perfect gas p = nk B T.

6 VOL. 52 C. BENSEDDIK, M. T. BOUAZZA, AND M. BOULEDROUA 17 TABLE II: Variation with temperature of the second virial coefficients, B 2 (T ), for the system Kr(4p) + Kr(4p). (a) with quantum effects, (b) without quantum effects. Temperature Kr 2 (cm 3 mol 1 ) T (K) (a) B cl (T ) (b) B 2 (T ) Table IV displays the collision integrals and the self-diffusion coefficient of krypton gas for several temperatures under standard conditions of density and pressure. Our results are in good agreement with those obtained by Sevast yanov and Zykov [21]. These authors have calculated the self-diffusion coefficient of krypton monatomic vapor at a pressure of 1 atm. In order to examine the variation law of the coefficient of diffusion D(T ) with temperature T, we accomplished a linear fit of the diffusion data relative to the pressure p in the temperature range 2 3 K. The variation law is taken as D αt β exp ( ξ T ). (16)

7 18 THERMOPHYSICAL PROPERTIES OF A KRYPTON GAS VOL. 52 TABLE III: Second virial coefficient, B 2 (T ), for the system Kr(4p) + Kr(4p) compared with experimental values of Kestin et al. [18] for several temperatures. Temperature B 2 (T ) (cm 3 mol 1 ) T (K) This work Ref. [18] l=4 Diffusion Elastic 4 Total cross sections (a.u.) 3 2 l= Energy E (a.u.) FIG. 2: Variation with energy of the elastic cross section.

8 VOL. 52 C. BENSEDDIK, M. T. BOUAZZA, AND M. BOULEDROUA 19 TABLE IV: Collision integrals and self-diffusion coefficients for the system Kr(4p) + Kr(4p), at some specific temperatures. Temperature D ( cm 2 s 1) T (K) Ω ( ) (1,1) a 2 n = n p = p The numerical fit leads to the constant parameters α = (7.84 ±.26) 1 1, β = ±.4, and ξ = 3.92 ± Figure 3 shows the fitting results for the Kr dimer. V-2. Viscosity and thermal conductivity The model of Chapman-Enskog defines also the viscosity coefficients of a monatomic gas which are given as a function of the collision integrals Ω (2,2) (T ) for p = 2 and q = 2 by the expression (2). These integrals are expressed in terms of the viscosity cross section (6). Figure 4 shows the variation with energy of the viscosity cross section. This viscosity cross section leads to the calculation of the collision integrals Ω (2,2) (T ) for each temperature. Our numerical values of Ω (2,2) (T ) are listed in Table VI. Moreover, the calculations of the collision integrals effective in viscosity allow us to estimate the viscosity coefficients η(t ) expressed by Equation (2), and consequently the coefficients of thermal conductivity λ(t ) given in Eq. (3). We report in Tables VII and VIII our computed values of viscosity and thermal conductivity respectively for several temperatures. In the same tables, we make a comparison with those calculated by Sevast yanov and Zykov [21] and Bich et al. [19] and with the experimental values given by Hanley [22]. A good agreement is observed with our results and the difference with the results of these authors does not exceed 2.5%. Furthermore, it is interesting to find out the variation law through which the viscosity coefficient of the Kr system varies with temperature. Using the same analytical formula used above for the self-diffusion coefficients (16), it is easy to see that this law applies very

9 11 THERMOPHYSICAL PROPERTIES OF A KRYPTON GAS VOL. 52 TABLE V: Self-diffusion coefficients for the system Kr(4p) + Kr(4p) compared with results from Sevast yanov and Zykov [21]. Temperature D ( cm 2 s 1) T (K) This work Ref. [21] well for temperatures ranging approximately from 2 to 3 K. If η(t ) is in µpa s and T in K, the fitting algorithm we have used generates the values α = (9.933 ±.26) 1 7, β = (6.298 ±.4) 1 1, and ξ = ±.271. VI. CONCLUSION In the present work, we have been interested in the quantal collisions between two krypton atoms in their fundamental electronic configurations. To fulfil this task, we solved numerically the radial wave equation to determine the phase shifts δ l (E). These phase shifts allowed us to calculate various cross sections necessary in the determination of the

10 VOL. 52 C. BENSEDDIK, M. T. BOUAZZA, AND M. BOULEDROUA Self diffusion Coefficient D (m 2 s 1 ) n = n p = p numerical fit Temperature T (K) FIG. 3: Numerical fit of the self-diffusion coefficient D(T ) for fixed density n = n and pressure p = p. The circles and triangles correspond to our theoretical values. The dashed lines represent the numerical fit obtained with Eq. (16). 1 8 Viscosity cross section (a.u.) Energy E (a.u.) FIG. 4: Variation with energy of the viscosity cross section of the Kr(4p)-Kr(4p) system. transport parameters. We have particularly constructed the potential energy curve relative to the singlet 1 Σ + molecular state. The accuracy of the constructed potential is evaluated by comparing between its data and those already published and by the calculation of the second virial coefficients, with quantum correction included. The obtained results

11 112 THERMOPHYSICAL PROPERTIES OF A KRYPTON GAS VOL. 52 TABLE VI: Integrals of viscosity for the Kr 2 system for some temperatures. T (K) Ω ( ) (2,2) a TABLE VII: Coefficient of viscosity of the Kr 2 system for some temperatures compared with results from Refs. [19, 21, 22]. Temperature η(t ) (µpas) T (K) This work Ref. [21] Ref. [22] Ref. [19] generally agree quite well with those already published. Finally, within the framework of the Chapman-Enskog model for dilute gases, we have performed the calculations of the self-diffusion coefficients at moderate and higher temperatures. The viscosity and thermal conductivity calculations have been also performed in the temperature range 2 3 K. The results demonstrated a good agreement with the available theoretical and experimental data. Finally, the variation laws of D and η with temperature have been examined with the analytical form αt β e ξ/t.

12 VOL. 52 C. BENSEDDIK, M. T. BOUAZZA, AND M. BOULEDROUA 113 TABLE VIII: Coefficients of thermal conductivity of the Kr 2 system compared with results from Refs. [19, 21, 22]. Temperature λ (mwk 1 m 1 ) T (K) This work Ref. [21] Ref. [22] Ref. [19] References [1] M. T. Bouazza and M. Bouledroua, Mol. Phys. 15, 51 (27). doi: 1.18/ [2] M. Bouledroua and T. H. Zerguini, Phys. Plas. 9, 4348 (22). doi: 1.163/ [3] K. E. Grew and W. A. Wakeham, J. Phys. B 11, 245 (1978). [4] W. L. Taylor and D. Cain, J. Chem. Phys. 78, 622 (1983). doi: 1.163/ [5] P. M. Holland and L. Biolsi, J. Chem. Phys. 89, 323 (1988). doi: 1.163/ [6] J. J. Hurly and M. R. Moldover, J. Res. Natl. Inst. Stand. Technol. 15, 667 (2). [7] M. S. A. El Kader, J. Phys. B 35, 421 (22). doi: 1.188/ /35/19/36 [8] R. K. Janev, in Atomic and Molecular Processus in Fusion Edge Plasmas (Plenum Press, New York, 1995). [9] L. Troudi et al., J. Phys. D 37, 61 (24). doi: 1.188/ /37/4/12 [1] J. O. Hirschfelder, C. F. Curtis, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley and Sons, Inc., New York, 1964). [11] A. E. Nasrabad and U. K. Deiters, J. Chem. Phys. 119, 947 (23). doi: 1.163/ [12] J. Mitroy and J.-Y. Zhang, Phy. Rev. A 76, 3276 (27). doi: 1.113/PhysRevA [13] R. A. Aziz and M. J. Slaman, Mol. Phys. 58, 679 (1986). doi: 1.18/ [14] F. Tao, J. Chem. Phys. 111, 247 (1999). doi: 1.163/ [15] A. A. Madej and B. P. Stoicheff, Phys. Rev. A 38, 3456 (1988). doi: 1.113/PhysRevA [16] A. K. Dham, A. R. Allnatt, W. J. Meath, and R. A. Azziz, Mol. Phys. 67, 1291 (1989). doi: 1.18/ [17] T. P. Haley and S. M. Cybulski, J. Chem. Phys. 119, 5487 (23). doi: 1.163/ [18] J. Kestin et al., J. Phys. Chem. Ref. Data 13, 229 (1984). doi: 1.163/ [19] E. Bich, J. Millat, and E. Vogel, J. Phys. Chem. Ref. Data 19, 1289 (199).

13 114 THERMOPHYSICAL PROPERTIES OF A KRYPTON GAS VOL. 52 doi: 1.163/ [2] A. Dalgarno, Proc. Phys. Soc. A 262, 132 (1961). doi: 1.198/rspa [21] R. M. Sevast yanov and N. A. Zykov, Inzhenrno-Fizicheskii Zhurnal 34, 79 (1978). [22] H. J. M. Hanley, J. Phys. Chem. Ref. Data 2, 619 (1973). doi: 1.163/