Thermodynamic properties of molecular cryocrystals of nitrogen type with fcc and HCP structures in harmonic approximation

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0042 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 129-136 This paper is available online at THERMODYNAMIC PROPERTIES OF MOLECULAR CRYOCRYSTALS OF NITROGEN TYPEWITH FCC AND HCP STRUCTURES IN HARMONIC APPROXIMATION Nguyen Quang Hoc1, Mai Thi La1, Vo Minh Tien2 and Dao Kha Son2 1Faculty of Physics, Hanoi National University of Education 2Tay Nguyen University Abstract. On the basis of results obtained in the previous paper, by combining the statistical moment method and the self-consistent field method, the thermodynamic quantities for molecular cryocrystals of nitrogen type such as solid α-N2, α-CO, CO2 and N2Owith face-centered cubic (FCC) structure and solid β-N2 with hexagonal close packed (HCP) structure are calculated. The obtained results for the molar heat capacity at constant volume are compared with the experimental data. Keywords: Statistical moment method, self-consistent field method, molecular cryocrystal. 1. Introduction The thermodynamic properties of molecular cryocrystals of nitrogen type have been considered in many experimental works [1]. The theoretical results calculated on the basis of the one-particle distribution functions [2], the self-consistent phonon theory [3], the Monte-Carlo simulation [4], etc. are relatively far from the experimental data. In the present paper we shall apply the theoretical results obtained in the previous paper to the case of α-N2, α-CO, CO2 and N2O crystals with face-centered cubic (FCC) structure and the β-N2 crystal with hexagonal close-packed (HCP) structure. In order to investigate them, we use the Lennard-Jones (6-12) pair potential for vibrational quantities and the quadrupole potential for rotational quantities. We have only considered the thermodynamic quantities such as the nearest neighbor distance, the lattice constant, the free energy, the energy, the entropy and the heat capacity at constant volume for the above mentioned crystals at different temperatures and zero pressure. Some obtained values are in relatively good agreement with the experimental data for the heat capacity at constant volume. Received August 19, 2015. Accepted October 12, 2015. Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn 129 Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son 2. Content 2.1. Numerical results of thermodynamic quantities for molecular cryocrystals of nitrogen type in harmonic approximation In order to apply the theoretical results obtained previously to molecular cryocrystals of nitrogen type in harmonic approximation, we use the Lennard-Jones interaction potential φ(r) = 4ε1 [(σ r )12 − (σ r )6] , (2.1) where σ is the distance in which φ(r) = 0 and ε is the depth of the potential well. The values of the parameters ε, σ are determined from experiments (Table 1) Table 1. The parameters ε, σ for crystals of N2 type [1] Crystal εkB ,K σ, 10 −10m α-N2 95.145 3.708 α-CO 100.145 3.769 CO2 218.913 3.829 N2O 235.507 3.802 β-N2 95.1 3.708 Therefore, using the coordinattion sphere method and the results previously, we obtain the values of parameters for α-N2, α-CO, CO2 and N2O crystals with FCC structure as follows: [5] kfcc = 4ε afcc2 ( σ afcc )6 [ 265.298 ( σ afcc )6 − 64.01 ] , γfcc = 16ε afcc4 ( σ afcc )6 [ 4410.797 ( σ afcc )6 − 346.172 ] , γfcc1 = 4ε afcc4 ( σ afcc )6 [ 803.555 ( σ afcc )6 − 40.547 ] , γfcc2 = 4ε afcc4 ( σ afcc )6 [ 3607.242 ( σ afcc )6 − 305.625 ] , (2.2) where afcc is the nearest neighbour distance of FCC crystal at temperature T and the values of parameters for β-N2 crystal with HCP structure are as follows: [6] k hcp x = 4ε ahcp2 ( σ ahcp )6 [ 614.6022 ( σ ahcp )6 − 162.8533 ] , k hcp z = 4ε ahcp2 ( σ ahcp )6 [ 286.3722 ( σ ahcp )6 − 64.7487 ] , γ hcp = − 4ε ahcp3 ( σ ahcp )6 [ 161.952 ( σ ahcp )6 − 24.984 ] , τ hcp 1 = 4ε ahcp4 ( σ ahcp )6 [ 6288.912 ( σ ahcp )6 − 473.6748 ] , τ hcp 2 = 4ε ahcp4 ( σ ahcp )6 [ 11488.3776 ( σ ahcp )6 − 752.5176 ] , τ hcp 3 = 4ε ahcp4 ( σ ahcp )6 [ 8133.888 ( σ ahcp )6 − 737.352 ] , τ hcp 4 = 4ε ahcp4 ( σ ahcp )6 [ 43409.3184 ( σ ahcp )6 − 4550.04 ] , τ hcp 5 = 4ε ahcp4 ( σ ahcp )6 [ 11315.6064 ( σ ahcp )6 − 1006.0428 ] , τ hcp 6 = 4ε ahcp4 ( σ ahcp )6 [ 40782.6048 ( σ ahcp )6 − 4189.6536 ] , (2.3) where ahcp is the nearest neighbour distance of HCP crystal at temperature T. The values of B,U0, η and ξ at various temperatures are given in Tables 2 and 3. 130 Thermodynamic properties of molecular cryocrystals of nitrogen type with FCC and HCP structures... Table 2. Values of B and U0 for crystals of N2 type [1] Crystal N2 CO CO2 N2O B,K 2.8751 2.7787 0.56355 0.60592 U0,K 325.6 688.2 7293.8 5844.5 Table 3. Values of η and ξ for crystals of N2 type at different temperatures in harmonic approximation [1] T, K 10 20 28 36 44 52 η 0.8769 0.8688 0.8523 0.8289 0.7995 0.764 ξ/kB , J 70.18 69.86 69.19 68.23 67.01 65.51 T, K 60 64 68 72 76 78.4 η 0.7208 0.6951 0.6652 0.6283 0.5761 0.4989 ξ/kB , J 63.63 62.48 61.12 59.41 56.88 52.94 Our calculated results for the vibrational free energies ψfccvib , ψ hcp vib , the rotational free energies ψfccrot , ψ hcp rot , the total free energies ψ fcc, ψhcp, the vibrational energies Efccvib , E hcp vib , the rotational energies Efccrot , E hcp rot , the total energies E fcc, Ehcp, the vibrational entropies Sfccvib , S hcp vib , the rotational entropies Sfccrot , S hcp rot , the total entropies S fcc, Shcp, the vibrational heat capacities at constant volume CfccV vib, C hcp V vib, the rotational heat capacities at constant volume C fcc V rot, C hcp V rot and the total heat capacities at constant volume CfccV , C hcp V for α-N2, α-CO, CO2, N2O, β-N2 crystals at different temperatures and zero pressure are shown in Figures 1-20. For solid α-N2, the rotational free energy is positive while the vibrational free energy and the total free energy are negative. From 10 to 35.6 K, the rotational free energy accounts for 12 to 18.65% of the total free energy, the rotational energy accounts for 14 to 15.8% of the total energy and the rotational heat capacity at constant volume accounts for 10 to 16.09% of the total heat capacity at constant volume. From 20 to 35.6 K, the rotational entropy and the total entropy are negative. The values of the total heat capacity at constant volume are in qualitative agreement with the experimental data [1] in the 10 to 35.6 K temperature interval. Figure 1. Graphs of ψfcc,harvib (T ), ψfcc,harrot (T ), ψ fcc,har(T ) for α-N2 Figure 2. Graphs of Efcc,harvib (T ), Efcc,harrot (T ), E fcc,har(T ) for α-N2 131 Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son Fig.3. Graphs of Sfcc,harvib (T ), Sfcc,harrot (T ), S fcc,har(T ) for α-N2 Fig.4. Graphs of Cfcc,harV vib (T ), Cfcc,harV rot (T ), C fcc,har V (T ) for α-N2 from SMM, SCFM, SMM+SCFM and EXPT [1] For solid α-CO, the rotational free energy is positive while the vibrational free energy and the total free energy are negative. From 20 to 60 K, the rotational free energy accounts for 0.72 to 7.56% of the total energy, the rotational energy accounts for 0.47 to 1.5% of the total free energy and the rotational heat capacity at constant volume accounts for 10 to 16.09% of the total heat capacity at constant volume. The rotational entropy is negative and the vibrational entropy is positive. The total entropy changes from negative to positive at about 52 K. In comparison with the experimental data [7], the discrepancy in the total heat capacity at constant volume is from 12.8 to 38.3% in the 36 to 52 K temperature interval. The calculated total heat capacity at constant volume is better than the calculated results in [8]. Fig.5. Graphs of ψfcc,harvib (T ), ψfcc,harrot (T ), ψ fcc,har(T ) for α-CO Fig.6. Graphs of Efcc,harvib (T ), Efcc,harrot (T ), E fcc,har(T ) for α-CO For solid CO2 from 20 to 76 K, the rotational free energy accounts for 75.5 to 70.7% of the total energy, the rotational energy accounts for 75.97 to 74.24% of the total free energy and the rotational heat capacity at constant volume accounts for 99.66 to 70.68% of the total heat capacity at constant volume. The rotational entropy is negative while the vibrational entropy and the total entropy are positive. In comparison with the experimental data [7], the discrepancy in the total heat capacity at constant volume is from 17.68 to 45.54% in the 20 to 36 K temperature interval. 132 Thermodynamic properties of molecular cryocrystals of nitrogen type with FCC and HCP structures... Fig. 7. Graphs of Sfcc,harvib (T ), Sfcc,harrot (T ), S fcc,har(T ) for α-CO Fig. 8. Graphs of Cfcc,harV vib (T ), Cfcc,harV rot (T ), C fcc,har V (T ) for α-CO from SMM, SCFM, SMM+SCFM, EXPT [7] and CAL [8] Fig. 9. Graphs of ψfcc,harvib (T ), ψfcc,harrot (T ), ψ fcc,har(T ) for CO2 Fig. 10. Graphs of Efcc,harvib (T ), Efcc,harrot (T ), E fcc,har(T ) for CO2 Fig.11. Graphs of Sfcc,harvib (T ), Sfcc,harrot (T ), S fcc,har(T ) for CO2 Fig.12. Graphs of Cfcc,harV vib (T ), Cfcc,harV rot (T ), C fcc,har V (T ) for CO2 from SMM, SCFM, SMM + SCFM and EXPT [7] For solid N2O from 10 to 76 K, the rotational free energy contributes to the total free energy from 59.5 to 54.7%,the rotational energy accounts for 60 to 58.9% of the total energy and 133 Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son the rotational heat capacity at constant volume accounts for 15 - 50% of the total heat capacity at constant volume. From 10 to 35.6 K, the rotational entropy increases and then decreases. The rotational entropy is negative while the vibrational entropy and the total entropy are positive. The values of the total heat capacity at constant volume are in relatively good agreement with the experimental data [1] in the 10 to 76 K temperature interval. Fig.13. Graphs of ψfcc,harvib (T ), ψfcc,harrot (T ), ψ fcc,har(T ) for N2O Fig.14. Graphs of Efcc,harvib (T ), E fcc,har rot (T ), Efcc,har(T ) for N2O Fig.15. Graphs of Sfcc,harvib (T ), Sfcc,harrot (T ), S fcc,har(T ) for N2O Fig.16. Graphs of Cfcc,harV vib (T ), Cfcc,harV rot (T ), C fcc,har V (T )for N2O from SMM, SCFM, SMM+SCFM and EXPT [1] For solid β-N2 from 36 to 60 K, the rotational free energy accounts for 7.55 to 7.67% of the total free energy, the rotational entropy accounts for 24 to 52.8% of the total entropy and the rotational heat capacity at constant volume accounts for 46.64 to 55.68% of the total heat capacity at constant volume. At 36 K, the rotational energy accounts for 7.01% of the total energy and when temperature increases, this contribution decreases. In comparison with the experimental data [9], the discrepancy in the total heat capacity at constant volume is 21.77%. The calculated total heat capacity at constant volume is better than the calculated results in [1]. 134 Thermodynamic properties of molecular cryocrystals of nitrogen type with FCC and HCP structures... Fig.17. Graphs of ψhcp,harvib (T ), ψhcp,harrot (T ), ψ hcp,har(T ) for β-N2 Fig.18. Graphs of Ehcp,harvib (T ), Ehcp,harrot (T ), E hcp,har(T ) for β-N2 Fig.19. Graphs of Shcp,harvib (T ), S hcp,har rot (T ), Shcp,har(T ) for β-N2 Fig.20. Graphs of Chcp,harV vib (T ), Chcp,harV rot (T ), C hcp,har V (T ) for β-N2 from SMM, SCFM, SMM+SCFM,CAL [1] and EXPT [9] 3. Conclusion The theoretical results obtained in the previous paper are applied to the case of α-N2, α-CO, CO2, N2O crystals with FCC structure and β-N2 crystal with HCP structure. 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