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采用量子统计系综理论, 研究了基态NO分子宏观气体摩尔熵、摩尔内能、摩尔热容等热力学性质. 首先应用课题组前期建立的变分代数法(variational algebraic method, VAM)计算获得了基态NO分子的完全振动能级, 得到的VAM振动能级作为振动部分, 结合欧拉-麦克劳林渐进展开公式的转动贡献, 应用于经典的热力学与统计物理公式中, 从而计算得到了1000—5000 K温度范围内NO宏观气体的摩尔内能、摩尔熵和摩尔热容. 将不同方法计算得到的摩尔热容结果分别与实验值进行比较, 结果表明基于VAM完全振动能级获得的结果优于其他方法获得的理论结果. 振动部分采用谐振子模型对无限能级求和计算热力学性质的方法有一定的局限性, 应当使用有限的完全振动能级进行统计求和.
Nitric oxide (NO) is one of atmospheric molecules of interest and has attracted considerable attention due to its important role in the chemical process taking place in a flow field of hypersonic vehicle, in which the thermodynamic properties are required in the calculation of the aerothermodynamic flow field. Moreover, the total internal partition function is the key to calculating the thermodynamic properties of high-temperature gases. For diatomic molecules, according to the product approximation, the total internal partition function is split into three parts: electronic, vibration and rotation partition function. In this paper, by using the quantum statistical ensemble theory based on some classical thermodynamic and statistical formulae, the thermodynamic properties of NO are analyzed and discussed. Firstly, in order to obtain an accurate energy of molecule, the variational algebraic method (VAM) is employed to calculate the full vibrational energy, the resultis in good agreement with the experimental result and thus yielding the realistic predictions of the unobserved higher vibrational energy that converges to the dissociation limit. Secondly, an attempt is to use the full VAM vibrational energy, the Rydberg-Klein-Rees (RKR) vibrational energy, the simple Harmonic oscillator (SHO) model and the quantum-mechanical vibrational energy obtained by the multiconfiguration self-consistent-field (MCSCF) to calculate the vibrational partition function. Then, with the rotational contributions from the Müller-McDowell formula, the internal partition function can be determined by combining the product of electronic, vibration and rotation partition functions. Thirdly, according to the thermodynamic and statistical formulae, it is easy to calculate the internal energy, entropy and heat capacity for the NO molecule in a range of 1000－5000 K. Comparison of different calculated heat capacities with the experimental ones reveals the heat capacity, of which vibrational contributions determined by the full VAM vibrational energy accord better with the experimental ones, with the maximum relative error being no more than 2.4%, whereas it can be seen that those thermodynamic results evaluated from the SHO model attest to a failure for the summation of infinite vibrational energy. The thermodynamic results of NO may have proper applications in areas that can be of great importance in theoretical and (or) experimental aspects. -
Keywords:
- NO molecule /
- variational algebraic method /
- vibrational energy /
- thermodynamic properties
[1] Brown W A, King D A 2000 J. Phys. Chem. B 104 2578
[2] Palmer R M J, Ferrige A G, Moncada S 1987 Nature 327 524
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[13] 邓伦华, 李传亮, 朱圆月, 何文艳, 陈扬骎 2012 物理学报 61 194208
Google Scholar
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图 1 VAM完全振动能级和实验能级的对比图
Fig. 1. Comparison between the VAM full vibrational energies and experimental ones
图 2 不同热容相对误差的比较
Fig. 2. Comparison of the relative errors of different heat capacities
表 1 基态NO分子不同振动能级间的比较 (单位: cm–1)
Table 1. Comparison of different vibrational levels of NO in the ground state (in cm–1).
$\upsilon $ $E_{\rm{\upsilon }}^{{\rm{exp}}}$[17] $E_{\rm{\upsilon }}^{{\rm{MCSCF}}}$[16] $E_{\rm{\upsilon }}^{{\rm{VAM}}}$ $E_{\rm{\upsilon }}^{{\rm{exp}}} - E_{\rm{\upsilon }}^{{\rm{MCSCF}}}$ $E_{\rm{\upsilon }}^{{\rm{exp}}} - E_{\rm{\upsilon }}^{{\rm{VAM}}}$ υ $E_{\rm{\upsilon }}^{{\rm{VAM}}}$ 0 948.50 948.60 948.50 –0.10 0 26 40240.70 1 2824.50 2824.60 2824.50 –0.10 0 27 41310.84 2 4627.30 4672.40 4672.27 –45.10 –44.97 28 42340.94 3 6491.90 6492.10 6491.90 –0.20 0 29 43329.70 4 8283.50 8283.90 8283.43 –0.40 0.07 30 44275.74 5 10046.90 10047.80 10046.90 –0.90 0 31 45177.59 6 11782.30 11783.70 11782.30 –1.40 0 32 46033.69 7 13489.40 13491.70 13489.60 –2.30 –0.20 33 46842.41 8 15171.80 15168.73 34 47602.00 9 16823.90 16819.60 35 48310.62 10 18448.00 18442.06 36 48966.35 11 20044.00 20035.93 37 49567.15 12 21611.80 21600.99 38 50110.90 13 23151.30 23136.97 39 50595.35 14 24662.40 24643.56 40 51018.15 15 26144.80 26120.38 41 51376.86 16 27598.50 27567.04 42 51668.92 17 29023.20 28983.04 43 51891.64 18 30367.88 44 52042.23 19 31720.95 20 33041.62 21 34329.18 22 35582.84 23 36801.78 24 37985.07 25 39131.73 $D_{\rm{e}}^{{\rm{exp}}}$ 52155.68 $D_{\rm{e}}^{{\rm{cal}}}$ 52155.68 注: $E_{\rm{\upsilon }}^{{\rm{VAM}}}$中表示VAM计算所需的已知实验振动能级用黑体标出. 
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表 2 使用RKR, MCSCF, SHO和VAM振动能级计算的摩尔内能 (单位: J·K–1·mol–1)
Table 2. Calculated molar internal energy using RKR, MCSCF, SHO, and VAM vibrational energies as the vibrational contributions, respectively (in J·K–1·mol–1).
T/K $U_{{\rm{RKR}}}^{{\rm{cal}}}$ $U_{{\rm{MCSCF}}}^{{\rm{cal}}}$ $U_{{\rm{SHO}}}^{{\rm{cal}}}$ $U_{{\rm{VAM}}}^{{\rm{cal}}}$ 1000 21.28 21.28 21.27 21.28 1100 22.61 22.61 22.59 22.61 1200 23.98 23.99 23.95 23.98 1300 25.39 25.40 25.34 25.39 1400 26.84 26.84 26.77 26.84 1500 28.31 28.31 28.23 28.31 1600 29.80 29.80 29.70 29.80 1700 31.31 31.31 31.20 31.31 1800 32.84 32.84 32.71 32.84 1900 34.38 34.39 34.24 34.38 2000 35.94 35.94 35.77 35.94 2100 37.51 37.51 37.32 37.51 2200 39.09 39.09 38.88 39.09 2300 40.68 40.68 40.45 40.68 2400 42.28 42.28 42.03 42.28 2500 43.89 43.89 43.61 43.89 2600 45.50 45.50 45.19 45.50 2700 47.12 47.12 46.79 47.12 2800 48.75 48.75 48.38 48.75 2900 50.38 50.38 49.99 50.38 3000 52.01 52.01 51.59 52.01 3100 53.66 53.65 53.20 53.66 3200 55.30 55.30 54.81 55.30 3300 56.95 56.95 56.43 56.95 3400 58.60 58.60 58.05 58.60 3500 60.26 60.26 59.67 60.26 3600 61.92 61.92 61.29 61.92 3700 63.58 63.58 62.91 63.59 3800 65.25 65.24 64.54 65.25 3900 66.91 66.91 66.17 66.92 4000 68.58 68.58 67.80 68.60 4100 70.26 70.25 69.43 70.27 4200 71.93 71.93 71.07 71.95 4300 73.61 73.61 72.70 73.63 4400 75.29 75.28 74.34 75.32 4500 76.97 76.96 75.97 77.01 4600 78.65 78.64 77.61 78.69 4700 80.33 80.32 79.25 80.39 4800 82.01 82.01 80.89 82.08 4900 83.69 83.69 82.53 83.78 5000 85.38 85.37 84.17 85.48
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表 3 使用RKR, MCSCF、SHO和VAM振动能级计算的摩尔熵(单位: J·K–1·mol–1)
Table 3. Calculated molar entropy using RKR, MCSCF, SHO, and VAM vibrational energies as the vibrational contributions, respectively (in J·K–1·mol–1).
T/K $S_{{\rm{RKR}}}^{{\rm{cal}}}$ $S_{{\rm{MCSCF}}}^{{\rm{cal}}}$ $S_{{\rm{SHO}}}^{{\rm{cal}}}$ $S_{{\rm{VAM}}}^{{\rm{cal}}}$ 1000 60.50 60.50 60.42 60.50 1100 61.76 61.76 61.67 61.76 1200 62.96 62.96 62.86 62.96 1300 64.09 64.09 63.97 64.09 1400 65.16 65.16 65.03 65.16 1500 66.17 66.17 66.03 66.17 1600 67.13 67.13 66.99 67.13 1700 68.05 68.05 67.89 68.05 1800 68.92 68.92 68.76 68.92 1900 69.76 69.76 69.58 69.76 2000 70.56 70.56 70.37 70.56 2100 71.32 71.32 71.13 71.32 2200 72.06 72.06 71.85 72.06 2300 72.77 72.77 72.55 72.77 2400 73.45 73.45 73.22 73.45 2500 74.10 74.10 73.87 74.10 2600 74.73 74.73 74.49 74.73 2700 75.35 75.35 75.09 75.35 2800 75.94 75.94 75.67 75.94 2900 76.51 76.51 76.23 76.51 3000 77.06 77.06 76.78 77.06 3100 77.60 77.60 77.30 77.60 3200 78.12 78.12 77.82 78.12 3300 78.63 78.63 78.31 78.63 3400 79.12 79.12 78.80 79.13 3500 79.60 79.60 79.27 79.61 3600 80.07 80.07 79.72 80.07 3700 80.53 80.53 80.17 80.53 3800 80.97 80.97 80.60 80.97 3900 81.41 81.40 81.03 81.41 4000 81.83 81.83 81.44 81.83 4100 82.24 82.24 81.84 82.25 4200 82.65 82.64 82.24 82.65 4300 83.04 83.04 82.62 83.05 4400 83.43 83.42 83.00 83.43 4500 83.80 83.80 83.36 83.81 4600 84.17 84.17 83.72 84.18 4700 84.53 84.53 84.08 84.55 4800 84.89 84.89 84.42 84.90 4900 85.24 85.23 84.76 85.25 5000 85.58 85.57 85.09 85.60
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表 4 不同摩尔热容及其与实验值的相对误差 (单位: J·K–1·mol–1)
Table 4. Comparisons of different molar capacities with observed experimental
${C_{{\rm{exp}}}}$ (in J·K–1·mol–1).T/K ${C_{{\rm{exp}}}}$ $C_{{\rm{RKR}}}^{{\rm{cal}}}$ $C_{{\rm{MCSCF}}}^{{\rm{cal}}}$ $C_{{\rm{SHO}}}^{{\rm{cal}}}$ $C_{{\rm{VAM}}}^{{\rm{cal}}}$ ${\delta _{{\rm{RKR}}}}$a ${\delta _{{\rm{MCSCF}}}}$b ${\delta _{{\rm{SHO}}}}$c ${\delta _{{\rm{VAM}}}}$d 1000 13.20 13.04 13.04 10.72 13.04 1.22% 1.22% 18.82% 1.22% 1100 13.68 13.52 13.52 11.24 13.52 1.18% 1.18% 17.85% 1.18% 1200 14.09 13.93 13.93 11.73 13.93 1.16% 1.16% 16.77% 1.16% 1300 14.44 14.27 14.27 12.17 14.27 1.17% 1.17% 15.69% 1.17% 1400 14.74 14.56 14.56 12.58 14.56 1.19% 1.19% 14.66% 1.19% 1500 14.99 14.81 14.81 12.94 14.81 1.21% 1.21% 13.70% 1.21% 1600 15.22 15.03 15.03 13.26 15.03 1.23% 1.23% 12.83% 1.23% 1700 15.41 15.22 15.21 13.55 15.22 1.26% 1.26% 12.04% 1.26% 1800 15.58 15.38 15.38 13.81 15.38 1.29% 1.29% 11.35% 1.29% 1900 15.73 15.52 15.52 14.04 15.52 1.33% 1.33% 10.73% 1.33% 2000 15.86 15.65 15.64 14.25 15.65 1.36% 1.37% 10.18% 1.36% 2100 15.98 15.76 15.76 14.43 15.76 1.41% 1.41% 9.71% 1.41% 2200 16.09 15.86 15.86 14.59 15.86 1.45% 1.45% 9.29% 1.45% 2300 16.18 15.95 15.94 14.74 15.95 1.48% 1.49% 8.92% 1.48% 2400 16.27 16.03 16.03 14.88 16.03 1.52% 1.53% 8.59% 1.52% 2500 16.35 16.10 16.10 15.00 16.10 1.56% 1.57% 8.31% 1.56% 2600 16.43 16.17 16.17 15.10 16.17 1.61% 1.61% 8.07% 1.61% 2700 16.50 16.23 16.23 15.20 16.23 1.65% 1.65% 7.85% 1.65% 2800 16.56 16.28 16.28 15.29 16.28 1.69% 1.70% 7.67% 1.69% 2900 16.62 16.34 16.33 15.38 16.34 1.74% 1.74% 7.51% 1.73% 3000 16.68 16.38 16.38 15.45 16.39 1.77% 1.78% 7.37% 1.77% 3100 16.73 16.43 16.43 15.52 16.43 1.82% 1.83% 7.25% 1.80% 3200 16.78 16.47 16.47 15.58 16.47 1.87% 1.88% 7.16% 1.85% 3300 16.83 16.51 16.51 15.64 16.51 1.91% 1.92% 7.08% 1.88% 3400 16.88 16.55 16.54 15.69 16.55 1.96% 1.97% 7.01% 1.92% 3500 16.92 16.58 16.58 15.74 16.59 2.01% 2.02% 6.96% 1.95% 3600 16.96 16.61 16.61 15.79 16.62 2.06% 2.07% 6.91% 1.99% 3700 17.00 16.64 16.64 15.83 16.66 2.12% 2.13% 6.89% 2.03% 3800 17.04 16.67 16.67 15.87 16.69 2.18% 2.19% 6.86% 2.06% 3900 17.08 16.69 16.69 15.91 16.72 2.24% 2.25% 6.85% 2.09% 4000 17.11 16.72 16.71 15.94 16.75 2.31% 2.32% 6.85% 2.13% 4100 17.15 16.74 16.74 15.97 16.78 2.38% 2.40% 6.86% 2.16% 4200 17.18 16.76 16.75 16.00 16.81 2.46% 2.48% 6.86% 2.18% 4300 17.21 16.77 16.77 16.03 16.83 2.55% 2.57% 6.88% 2.21% 4400 17.24 16.79 16.79 16.05 16.86 2.64% 2.66% 6.90% 2.24% 4500 17.28 16.80 16.80 16.08 16.89 2.74% 2.76% 6.93% 2.26% 4600 17.31 16.81 16.81 16.10 16.91 2.85% 2.86% 6.96% 2.29% 4700 17.34 16.82 16.82 16.12 16.94 2.96% 2.99% 7.00% 2.31% 4800 17.36 16.83 16.83 16.14 16.96 3.09% 3.11% 7.04% 2.32% 4900 17.39 16.83 16.83 16.16 16.99 3.22% 3.25% 7.08% 2.35% 5000 17.42 16.84 16.83 16.18 17.01 3.37% 3.38% 7.13% 2.36% 注: a, ${\delta _{{\rm{RKR}}}} = \left| {C_{{\rm{exp}}}^{} - C_{{\rm{RKR}}}^{{\rm{cal}}}} \right|/C_{{\rm{exp}}}^{} \times 100\% $; b, ${\delta _{{\rm{MCSCF}}}} = \left| {C_{{\rm{exp}}}^{} - C_{{\rm{MCSCF}}}^{{\rm{cal}}}} \right|/C_{{\rm{exp}}}^{} \times 100\% $; c, $ {\delta _{{\rm{SHO}}}} = \left| {C_{{\rm{exp}}} - C_{{\rm{SHO}}}^{{\rm{cal}}}} \right|/C_{{\rm{exp}}} \times 100\% $; d,$ {\delta _{{\rm{VAM}}}} = \left| {C_{{\rm{exp}}} - C_{{\rm{VAM}}}^{{\rm{cal}}}} \right|/C_{{\rm{exp}}} \times 100\% .$
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[1] Brown W A, King D A 2000 J. Phys. Chem. B 104 2578
[2] Palmer R M J, Ferrige A G, Moncada S 1987 Nature 327 524
[3] Wang Z Z, Zhao Y Y, Sun R, et al. 2019 Fuel 253 1424
Google Scholar
[4] Goldstein I, Lue T F, Padma-Nathan H, et al. 1998 N. Engl. J. Med. 338 1397
Google Scholar
[5] Meng Q T, Liu X G, Zhang Q G, Han K L 2005 Chem. Phys. 316 93
Google Scholar
[6] 王德华, 王雅静, 薛艳丽, 李洪云, 林圣路 2007 物理学报 56 6209
Google Scholar
Wang D L, Wang Y J, XueY L, Li H Y, Lin S L 2007 Acta Phys. Sin. 56 6209
Google Scholar
[7] Shrestha K P, Seidel L, Zeuch T, Mauss F 2019 Combust. Sci. Technol. 191 1628
Google Scholar
[8] Chen L, Wang D, Wang J D, Weng D, Cao L 2019 J. Rare Earths 37 829
Google Scholar
[9] Matzkin A, Raoult M, Gauyacq D 2003 Phys. Rev. A 68 061401
Google Scholar
[10] Jaffe R L 1987 AIAA 22nd Thermophysics Conference New York, USA, June 8–10, 1999 p1633
[11] Capitelli M, Colonna G, Giordano D, et al. 2005 Tables of Internal Partition Functions and Thermodynamic Properties of High-temperature Mars-atmosphere Species from 50 K to 50000 K (Netherlands: European Space Agency Publications Division) pp3–19
[12] Babou Y, Rivière P, Perrin M Y, Soufiani A 2009 Int. J. Thermophys. 30 416
Google Scholar
[13] 邓伦华, 李传亮, 朱圆月, 何文艳, 陈扬骎 2012 物理学报 61 194208
Google Scholar
Deng L H, Li C L, Zhu Y Y, He W Y, Chen Y Q 2012 Acta Phys. Sin. 61 194208
Google Scholar
[14] Pathria R K1977 Statistical Mechanics (London: Pergamon press) pp100–107
[15] Huber K P, Herzberg G 1950 Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules (New York: Van Nostrand Reinhold Company) pp9–11
[16] Billingsley F P 1975 J. Chem. Phys. 62 864
Google Scholar
[17] Reddy R R, Ahammed Y N, Basha D B, et al. 2006 J. Quant. Spectrosc. Radiat. Transfer 97 344
Google Scholar
[18] Qin Z, Zhao J M, Liu L H 2018 J. Quant. Spectrosc.Radiat. Transfer 210 1
Google Scholar
[19] Zhang Y, Sun W G, Fu J, Fan Q C, et al. 2014 Spectrochim. Acta Part A 117 442
Google Scholar
[20] 周金伟, 李吉成, 石志广, 陈小天, 卢晓卫 2014 光学学报 34 130001
Google Scholar
Zhou J W, Li J C, Shi Z G, Chen X T, Lu X W 2014 Acta Optic. Sin. 34 130001
Google Scholar
[21] Gamachea R R, Kennedya S, Hawkinsb R, Rothmanb L S 2000 J. Mol. Struct. 517 407
[22] McDowell R S 1988 J. Chem. Phys. 88 356
Google Scholar
[23] Amiot C 1982 J. Mol. Spectrosc. 94 150
Google Scholar
[24] Sun W G, Hou S L, Feng H, Ren W Y 2002 J. Mol. Spectrosc. 215 93
Google Scholar
[25] Wang J K, Yang Z Y, Wu Z S 2010 Acta Photonica Sin. 39 1312
Google Scholar
[26] Chase M W 1998 Journal of Physical and Chemical Reference DataMonograph No.9 (New York: National Institute of Standards and Technology Gaithersburg) pp641–643
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