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NO分子宏观气体热力学性质的理论研究

蹇君 雷娇 樊群超 范志祥 马杰 付佳 李会东 徐勇根

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NO分子宏观气体热力学性质的理论研究

蹇君, 雷娇, 樊群超, 范志祥, 马杰, 付佳, 李会东, 徐勇根

Theoretical study on thermodynamic properties of NO gas

Jian Jun, Lei Jiao, Fan Qun-Chao, Fan Zhi-Xiang, Ma Jie, Fu Jia, Li Hui-Dong, Xu Yong-Gen
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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.
      通信作者: 樊群超, fanqunchao@mail.xhu.edu.cn ; 范志祥, fanzhixiang235@126.com
    • 基金项目: 省部级-四川省杰出青年学术与技术带头人计划(2019JDJQ0050和2019JDJQ0051)
      Corresponding author: Fan Qun-Chao, fanqunchao@mail.xhu.edu.cn ; Fan Zhi-Xiang, fanzhixiang235@126.com
    [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 1424Google Scholar

    [4]

    Goldstein I, Lue T F, Padma-Nathan H, et al. 1998 N. Engl. J. Med. 338 1397Google Scholar

    [5]

    Meng Q T, Liu X G, Zhang Q G, Han K L 2005 Chem. Phys. 316 93Google Scholar

    [6]

    王德华, 王雅静, 薛艳丽, 李洪云, 林圣路 2007 物理学报 56 6209Google Scholar

    Wang D L, Wang Y J, XueY L, Li H Y, Lin S L 2007 Acta Phys. Sin. 56 6209Google Scholar

    [7]

    Shrestha K P, Seidel L, Zeuch T, Mauss F 2019 Combust. Sci. Technol. 191 1628Google Scholar

    [8]

    Chen L, Wang D, Wang J D, Weng D, Cao L 2019 J. Rare Earths 37 829Google Scholar

    [9]

    Matzkin A, Raoult M, Gauyacq D 2003 Phys. Rev. A 68 061401Google 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 416Google Scholar

    [13]

    邓伦华, 李传亮, 朱圆月, 何文艳, 陈扬骎 2012 物理学报 61 194208Google Scholar

    Deng L H, Li C L, Zhu Y Y, He W Y, Chen Y Q 2012 Acta Phys. Sin. 61 194208Google 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 864Google Scholar

    [17]

    Reddy R R, Ahammed Y N, Basha D B, et al. 2006 J. Quant. Spectrosc. Radiat. Transfer 97 344Google Scholar

    [18]

    Qin Z, Zhao J M, Liu L H 2018 J. Quant. Spectrosc.Radiat. Transfer 210 1Google Scholar

    [19]

    Zhang Y, Sun W G, Fu J, Fan Q C, et al. 2014 Spectrochim. Acta Part A 117 442Google Scholar

    [20]

    周金伟, 李吉成, 石志广, 陈小天, 卢晓卫 2014 光学学报 34 130001Google Scholar

    Zhou J W, Li J C, Shi Z G, Chen X T, Lu X W 2014 Acta Optic. Sin. 34 130001Google 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 356Google Scholar

    [23]

    Amiot C 1982 J. Mol. Spectrosc. 94 150Google Scholar

    [24]

    Sun W G, Hou S L, Feng H, Ren W Y 2002 J. Mol. Spectrosc. 215 93Google Scholar

    [25]

    Wang J K, Yang Z Y, Wu Z S 2010 Acta Photonica Sin. 39 1312Google 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

  • 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}}}$
    0 948.50 948.60948.50–0.1002640240.70
    1 2824.50 2824.602824.50–0.1002741310.84
    24627.304672.404672.27–45.10–44.972842340.94
    3 6491.90 6492.106491.90–0.2002943329.70
    48283.508283.908283.43–0.400.073044275.74
    5 10046.90 10047.8010046.90–0.9003145177.59
    6 11782.30 11783.7011782.30–1.4003246033.69
    7 13489.40 13491.7013489.60–2.30–0.203346842.41
    815171.8015168.733447602.00
    916823.9016819.603548310.62
    1018448.0018442.063648966.35
    1120044.0020035.933749567.15
    1221611.8021600.993850110.90
    1323151.3023136.973950595.35
    1424662.4024643.564051018.15
    1526144.8026120.384151376.86
    1627598.5027567.044251668.92
    1729023.2028983.044351891.64
    1830367.884452042.23
    1931720.95
    2033041.62
    2134329.18
    2235582.84
    2336801.78
    2437985.07
    2539131.73
    $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.
    下载: 导出CSV

    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}}}$
    100021.2821.2821.2721.28
    110022.6122.6122.5922.61
    120023.9823.9923.9523.98
    130025.3925.4025.3425.39
    140026.8426.8426.7726.84
    150028.3128.3128.2328.31
    160029.8029.8029.7029.80
    170031.3131.3131.2031.31
    180032.8432.8432.7132.84
    190034.3834.3934.2434.38
    200035.9435.9435.7735.94
    210037.5137.5137.3237.51
    220039.0939.0938.8839.09
    230040.6840.6840.4540.68
    240042.2842.2842.0342.28
    250043.8943.8943.6143.89
    260045.5045.5045.1945.50
    270047.1247.1246.7947.12
    280048.7548.7548.3848.75
    290050.3850.3849.9950.38
    300052.0152.0151.5952.01
    310053.6653.6553.2053.66
    320055.3055.3054.8155.30
    330056.9556.9556.4356.95
    340058.6058.6058.0558.60
    350060.2660.2659.6760.26
    360061.9261.9261.2961.92
    370063.5863.5862.9163.59
    380065.2565.2464.5465.25
    390066.9166.9166.1766.92
    400068.5868.5867.8068.60
    410070.2670.2569.4370.27
    420071.9371.9371.0771.95
    430073.6173.6172.7073.63
    440075.2975.2874.3475.32
    450076.9776.9675.9777.01
    460078.6578.6477.6178.69
    470080.3380.3279.2580.39
    480082.0182.0180.8982.08
    490083.6983.6982.5383.78
    500085.3885.3784.1785.48
    下载: 导出CSV

    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}}}$
    100060.5060.5060.4260.50
    110061.7661.7661.6761.76
    120062.9662.9662.8662.96
    130064.0964.0963.9764.09
    140065.1665.1665.0365.16
    150066.1766.1766.0366.17
    160067.1367.1366.9967.13
    170068.0568.0567.8968.05
    180068.9268.9268.7668.92
    190069.7669.7669.5869.76
    200070.5670.5670.3770.56
    210071.3271.3271.1371.32
    220072.0672.0671.8572.06
    230072.7772.7772.5572.77
    240073.4573.4573.2273.45
    250074.1074.1073.8774.10
    260074.7374.7374.4974.73
    270075.3575.3575.0975.35
    280075.9475.9475.6775.94
    290076.5176.5176.2376.51
    300077.0677.0676.7877.06
    310077.6077.6077.3077.60
    320078.1278.1277.8278.12
    330078.6378.6378.3178.63
    340079.1279.1278.8079.13
    350079.6079.6079.2779.61
    360080.0780.0779.7280.07
    370080.5380.5380.1780.53
    380080.9780.9780.6080.97
    390081.4181.4081.0381.41
    400081.8381.8381.4481.83
    410082.2482.2481.8482.25
    420082.6582.6482.2482.65
    430083.0483.0482.6283.05
    440083.4383.4283.0083.43
    450083.8083.8083.3683.81
    460084.1784.1783.7284.18
    470084.5384.5384.0884.55
    480084.8984.8984.4284.90
    490085.2485.2384.7685.25
    500085.5885.5785.0985.60
    下载: 导出CSV

    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
    100013.2013.0413.0410.7213.041.22%1.22%18.82%1.22%
    110013.6813.5213.5211.2413.521.18%1.18%17.85%1.18%
    120014.0913.9313.9311.7313.931.16%1.16%16.77%1.16%
    130014.4414.2714.2712.1714.271.17%1.17%15.69%1.17%
    140014.7414.5614.5612.5814.561.19%1.19%14.66%1.19%
    150014.9914.8114.8112.9414.811.21%1.21%13.70%1.21%
    160015.2215.0315.0313.2615.031.23%1.23%12.83%1.23%
    170015.4115.2215.2113.5515.221.26%1.26%12.04%1.26%
    180015.5815.3815.3813.8115.381.29%1.29%11.35%1.29%
    190015.7315.5215.5214.0415.521.33%1.33%10.73%1.33%
    200015.8615.6515.6414.2515.651.36%1.37%10.18%1.36%
    210015.9815.7615.7614.4315.761.41%1.41%9.71%1.41%
    220016.0915.8615.8614.5915.861.45%1.45%9.29%1.45%
    230016.1815.9515.9414.7415.951.48%1.49%8.92%1.48%
    240016.2716.0316.0314.8816.031.52%1.53%8.59%1.52%
    250016.3516.1016.1015.0016.101.56%1.57%8.31%1.56%
    260016.4316.1716.1715.1016.171.61%1.61%8.07%1.61%
    270016.5016.2316.2315.2016.231.65%1.65%7.85%1.65%
    280016.5616.2816.2815.2916.281.69%1.70%7.67%1.69%
    290016.6216.3416.3315.3816.341.74%1.74%7.51%1.73%
    300016.6816.3816.3815.4516.391.77%1.78%7.37%1.77%
    310016.7316.4316.4315.5216.431.82%1.83%7.25%1.80%
    320016.7816.4716.4715.5816.471.87%1.88%7.16%1.85%
    330016.8316.5116.5115.6416.511.91%1.92%7.08%1.88%
    340016.8816.5516.5415.6916.551.96%1.97%7.01%1.92%
    350016.9216.5816.5815.7416.592.01%2.02%6.96%1.95%
    360016.9616.6116.6115.7916.622.06%2.07%6.91%1.99%
    370017.0016.6416.6415.8316.662.12%2.13%6.89%2.03%
    380017.0416.6716.6715.8716.692.18%2.19%6.86%2.06%
    390017.0816.6916.6915.9116.722.24%2.25%6.85%2.09%
    400017.1116.7216.7115.9416.752.31%2.32%6.85%2.13%
    410017.1516.7416.7415.9716.782.38%2.40%6.86%2.16%
    420017.1816.7616.7516.0016.812.46%2.48%6.86%2.18%
    430017.2116.7716.7716.0316.832.55%2.57%6.88%2.21%
    440017.2416.7916.7916.0516.862.64%2.66%6.90%2.24%
    450017.2816.8016.8016.0816.892.74%2.76%6.93%2.26%
    460017.3116.8116.8116.1016.912.85%2.86%6.96%2.29%
    470017.3416.8216.8216.1216.942.96%2.99%7.00%2.31%
    480017.3616.8316.8316.1416.963.09%3.11%7.04%2.32%
    490017.3916.8316.8316.1616.993.22%3.25%7.08%2.35%
    500017.4216.8416.8316.1817.013.37%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\% .$
    下载: 导出CSV
  • [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 1424Google Scholar

    [4]

    Goldstein I, Lue T F, Padma-Nathan H, et al. 1998 N. Engl. J. Med. 338 1397Google Scholar

    [5]

    Meng Q T, Liu X G, Zhang Q G, Han K L 2005 Chem. Phys. 316 93Google Scholar

    [6]

    王德华, 王雅静, 薛艳丽, 李洪云, 林圣路 2007 物理学报 56 6209Google Scholar

    Wang D L, Wang Y J, XueY L, Li H Y, Lin S L 2007 Acta Phys. Sin. 56 6209Google Scholar

    [7]

    Shrestha K P, Seidel L, Zeuch T, Mauss F 2019 Combust. Sci. Technol. 191 1628Google Scholar

    [8]

    Chen L, Wang D, Wang J D, Weng D, Cao L 2019 J. Rare Earths 37 829Google Scholar

    [9]

    Matzkin A, Raoult M, Gauyacq D 2003 Phys. Rev. A 68 061401Google 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 416Google Scholar

    [13]

    邓伦华, 李传亮, 朱圆月, 何文艳, 陈扬骎 2012 物理学报 61 194208Google Scholar

    Deng L H, Li C L, Zhu Y Y, He W Y, Chen Y Q 2012 Acta Phys. Sin. 61 194208Google 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 864Google Scholar

    [17]

    Reddy R R, Ahammed Y N, Basha D B, et al. 2006 J. Quant. Spectrosc. Radiat. Transfer 97 344Google Scholar

    [18]

    Qin Z, Zhao J M, Liu L H 2018 J. Quant. Spectrosc.Radiat. Transfer 210 1Google Scholar

    [19]

    Zhang Y, Sun W G, Fu J, Fan Q C, et al. 2014 Spectrochim. Acta Part A 117 442Google Scholar

    [20]

    周金伟, 李吉成, 石志广, 陈小天, 卢晓卫 2014 光学学报 34 130001Google Scholar

    Zhou J W, Li J C, Shi Z G, Chen X T, Lu X W 2014 Acta Optic. Sin. 34 130001Google 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 356Google Scholar

    [23]

    Amiot C 1982 J. Mol. Spectrosc. 94 150Google Scholar

    [24]

    Sun W G, Hou S L, Feng H, Ren W Y 2002 J. Mol. Spectrosc. 215 93Google Scholar

    [25]

    Wang J K, Yang Z Y, Wu Z S 2010 Acta Photonica Sin. 39 1312Google 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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出版历程
  • 收稿日期:  2019-11-11
  • 修回日期:  2019-12-26
  • 刊出日期:  2020-03-05

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