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
N2O with 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.

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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. Here we
only use the Lennard-Jones (6-12) pair potential for vibrational quantities and the quadrupole
-quadrupole potential for rotational quantities. We calculated some 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 above mentioned crystals at different temperatures and
zero pressure. Some obtained values are in relatively good agreement with the experimental data.
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Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son
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