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基于SO分子振转能级计算其宏观气体摩尔热容

文琳 樊群超 蹇君 范志祥 李会东 付佳 马杰 谢锋

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基于SO分子振转能级计算其宏观气体摩尔热容

文琳, 樊群超, 蹇君, 范志祥, 李会东, 付佳, 马杰, 谢锋

Calculating macroscopic gas molar heat capacity of SO molecule based on rovibrational energy level

Wen Lin, Fan Qun-Chao, Jian Jun, Fan Zhi-Xiang, Li Hui-Dong, Fu Jia, Ma Jie, Xie Feng
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  • 本文在研究SO宏观气体摩尔热容的工作中, 进一步考虑了分子内部的转动贡献, 通过联立能获得分子某电子态完全振动能级的变分代数法 (variational algebraic method, VAM) 和RKR (Rydberg-Klein-Rees) 方法构建了SO电子基态的势能函数, 解析求解获得了该体系的振转能级, 进而采用量子统计系综理论计算得到了300—6000 K温度范围内SO宏观气体的摩尔热容. 将本文的计算结果与其他几种理论模型的计算结果进行比较分析, 结果表明: 当采用基于全程势能曲线求解的完全振转能级来计算热力学性质时, 得到的摩尔热容与实验结果更为吻合. 本文利用分子完全振转能级计算摩尔热容的思路, 弥补了前一阶段工作中仅采用近似模型表征分子转动行为来计算热容的不足, 为基于微观统计过程求解宏观热力学量提供了新的研究范式.
    Sulfur oxide (SO) is a kind of well-known diatomic molecule which becomes one of the major pollutants in the atmosphere. Control of the heat capacity of SO molecule is of great significance for elucidating its macroscopic evolution process. In the research of macroscopic systems composed of many particles as well as several matters, it is an important approach to obtain macroscopic thermodynamic quantities of the system by constructing a partition function from the microscopic information of molecule. For diatomic molecules in a certain electronic state, the partition function can directly be obtained by calculating the rovibrational energy of the system to acquire the macroscopic molar heat capacities.In this work, the contribution of rotational behavior to molar heat capacity is further considered. The potential energy function for the ground electronic state of SO is constructed by the variational algebraic method (VAM) and RKR (Rydberg-Klein-Rees) method, in which the former one can determine the complete vibrational energy levels of an electronic state of a molecule. The rovibrational energy level of the system is obtained by analytical solution, and then the molar heat capacity of SO macroscopic gas in the temperature range of 300–6000 K is calculated by quantum statistical ensemble theory The above calculation depends only on the experimental vibrational energy, experimental rotational spectral constant and the dissociation energy of SO molecule. Fortunately, through comparison between theoretical calculation results and experimental data, we find that the molar heat capacity of gaseous SO molecule can be well predicted by employing the full set of rovibrational energy to describe the internal vibration and rotation of SO molecule. The idea of calculating the molar heat capacity by using the full set of rovibrational energy makes up for the shortcomings of previous work where molar heat capacity is calculated by using the approximate model characterizing the molecular rotational behavior, and also provides a new research paradigm for solving macro thermodynamic quantities based on micro statistical processes .
      通信作者: 樊群超, fanqunchao@mail.xhu.edu.cn ; 范志祥, fanzhixiang235@126.com
    • 基金项目: 中央引导地方科技发展面上项目 (批准号: 2021ZYD0050)、国家自然科学基金 (批准号: 61722507, 11904295) 和极端光学省部共建协同创新中心开放课题 (批准号: KF2020003) 资助的课题.
      Corresponding author: Fan Qun-Chao, fanqunchao@mail.xhu.edu.cn ; Fan Zhi-Xiang, fanzhixiang235@126.com
    • Funds: Project supported by the Fund for the Program of Science and Technology of Sichuan Province of China (Grant No. 2021ZYD0050), the National Natural Science Foundation of China (Grant Nos. 61722507, 11904295), and the Open Research Fund Program of the Collaborative Innovation Center of Extreme Optics, China (Grant No. KF2020003).
    [1]

    McBride B J, Zehe M J, Gordon S 2002 NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species (Cleveland: National Aeronautics and Space Administration) p1

    [2]

    汪志诚 2013 热力学·统计物理 (第五版) (北京: 高等教育出版社) 第1页

    Wang Z C 2013 Thermodynamic·statistical physics (Vol. 5) (Beijing: Higher Education Press) p1 (in Chinese)

    [3]

    Gabriel V O, Luis A A H 2018 Int. J. Quantum Chem. 118 1Google Scholar

    [4]

    Irwin A W 1988 Astron. Astrophys. Suppl. Ser. 74 145

    [5]

    Fischer J, Gamache R R, Goldman A, Rothman L S, Perrin A 2003 J. Quant. Spectrosc. Radiat. Transfer 82 401Google Scholar

    [6]

    伍冬兰, 万慧军, 谢安东, 程新路, 杨向东 2009 物理学报 58 7410Google Scholar

    Wu D L, Wan H J, Xie A D, Cheng X L, Yang X D 2009 Acta Phys. Sin. 58 7410Google Scholar

    [7]

    Jia C S, Zhang L H, Wang C W 2017 Chem. Phys. Lett. 667 211Google Scholar

    [8]

    Maltsev M A, Kulikov A N, Morozov I V 2016 J. Phys. Conf. Ser. 774 012023Google Scholar

    [9]

    Maltsev M A, Morozov I V, Osina E L 2019 High Temp. 57 335Google Scholar

    [10]

    Ikot A N, Chukwuocha E O, Onyeaju M C, Onate C A, Ita B I, Udoh M E 2018 Pramana-J. Phys. 90 22Google Scholar

    [11]

    王小霞, 刘鑫, 张琼, 陈宏善 2017 物理学报 66 103601Google Scholar

    Wang X X, Liu X, Zhang Q, Chen H S 2017 Acta Phys. Sin. 66 103601Google Scholar

    [12]

    Maltsev M A, Morozov I V, Osina E L 2019 High Temp. 57 42Google Scholar

    [13]

    Horchani R, Jelassi H 2020 Chem. Phys. 532 1Google Scholar

    [14]

    Zúñiga J, Bastida A, Requena A, Cerezo J 2021 J. Phys. Chem. A 125 9226Google Scholar

    [15]

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

    [16]

    Fu J, Fan Q C, Liu G Y, Li H D, Xu Y G, Fan Z X, Zhang Y 2017 Comput. Theor. Chem. 1115 136Google Scholar

    [17]

    McDowell R S 1988 J. Chem. Phys. 88 356Google Scholar

    [18]

    蹇君, 雷娇, 樊群超, 范志祥, 马杰, 付佳, 李会东, 徐勇根 2020 物理学报 69 053301Google Scholar

    Jian J, Lei J, Fan Q C, Fan Z X, Ma J, Fu J, Li H D, Xu Y G 2020 Acta Phys. Sin. 69 053301Google Scholar

    [19]

    Rydberg R 1932 Z. Phys. 73 376Google Scholar

    [20]

    Klein V O 1932 Z. Phys. 76 226Google Scholar

    [21]

    Rees A L G 1947 Proc. Phys. Soc. 59 998Google Scholar

    [22]

    Pathria R K 1977 Statistical Mechanics (London: Pcrgamon Press) p100

    [23]

    Le Roy R J 2017 J. Quant. Spectrosc. Radiat. Transfer 186 167Google Scholar

    [24]

    Li C L, Li Y C, Ji Z H, Qiu X B, Lai Y Z, Wei J L, Zhao Y T, Deng L H, Chen Y Q, Liu J J 2018 Phys. Rev. A 97 062501Google Scholar

    [25]

    Laurendeau N M 2005 Statistical Thermodynamics: Fundamentals and Applications (England: Cambridge University Press) p1

    [26]

    Huang K 1965 Phys. Today 18 92

    [27]

    Babou Y, Rivière P, Perrin M Y, Soufiani A 2009 Int. J. Thermophys. 30 416Google Scholar

    [28]

    Jaffe R L 1987 AIAA 22nd Thermophysics Conference (New York: American Institute of Aeronautics and Astronautics) p1633

    [29]

    Capitelli M, Colonna G, Giordano D, Maraffa L, Casavola A, Minelli P, Pagano D, Pietanza L D, Taccogna F 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) p3

    [30]

    Huber K P, Herzberg G 1950 Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules (New York: Van Nostrand Reinhold Company) p9

    [31]

    朱华, 谢代前, 鄢国森 1999 高等学校化学学报 20 1910Google Scholar

    Zhu H, Xie D Q, Yan G S 1999 Chem. J. Chin. Univ. 20 1910Google Scholar

    [32]

    James B B, Edward R L, Philip D H, Carleton J H 1987 J. Mol. Spectrosc. 124 379Google Scholar

    [33]

    Gottlieb C A, Gottlieb E W, Litvak M M, Ball J A, Penfield H 1978 Astrophys. J. 219 77Google Scholar

    [34]

    Clyne M A A, Mcdermid I S 1979 J. Chem. Soc. , Faraday Trans. 2 75 905Google Scholar

    [35]

    Peterson K A, Woods R C 1990 J. Chem. Phys. 93 1876Google Scholar

    [36]

    Martin-Drumel M A, Hindle F, Mouret G, Cuisset A, Cernicharo J 2015 Astrophys. J. 799 115Google Scholar

    [37]

    Chase M W 1998 Journal of Physical and Chemical Reference DataMonograph (Vol. 9) (New York: National Institute of Standards and Technology Gaithersburg) p1726

  • 图 1  基于VAM和RKR方法构建的势能曲线与实验势能曲线的对比

    Fig. 1.  Comparisons of the potential energy curves constructed based on the VAM and RKR method with those experimentally.

    图 2  不同摩尔热容误差的比较

    Fig. 2.  Comparisons of the errors of different molar heat capacities.

    表 1  不同方法所得SO分子电子基态的振动光谱常数 (单位: cm–1)

    Table 1.  Vibrational spectral constants of SO in the ground electronic state obtained by different methods (in cm–1)

    $ {\omega _0} $$ {\omega _{\text{e}}} $$ {\omega _{\text{e}}}{x_{\text{e}}} $$ {\omega _{\text{e}}}{y_{\text{e}}} $$ {\omega _{\text{e}}}{z_{\text{e}}} $$ {\omega _{\text{e}}}{t_{\text{e}}} $
    实验[34]1148.196.12
    CASS-
    CF[35]
    1161.806.50
    CI-SD[35]1263.505.35
    MP4
    SDQ[35]
    1173.105.09
    VAM0.751147.715.99–1.55×10–27.59×10–4–1.30×10–5
    下载: 导出CSV

    表 2  不同摩尔热容与实验值的误差 (单位: J·mol–1·K–1)

    Table 2.  Errors between different molar heat capacities and observed experimentally (in J·mol–1·K–1)

    T/K${ C_\upsilon ^{ {\text{expt} } } }^{\rm\; a}$${ \Delta C_{\upsilon {\text{, RKR} } }^{ {\text{cal} } } } ^{\rm\; b}$${ \Delta C_{\upsilon {\text{, VAM} } }^{ {\text{cal} } } }^{\rm\; c}$${ \Delta C_{\upsilon \_r{\text{, RKR} } }^{ {\text{cal} } } } ^{\rm\; d}$${ \Delta C_{ {\text{This work} } }^{ {\text{cal} } } }^{\rm\; e}$
    30030.197–0.025–0.0240.0710.006
    40031.560–0.041–0.0400.0920.009
    50032.826–0.057–0.0560.0800.011
    60033.838–0.073–0.0720.0540.011
    70034.612–0.088–0.0870.0270.011
    80035.206–0.108–0.108–0.0010.005
    90035.672–0.135–0.136–0.032–0.008
    100036.053–0.177–0.177–0.071–0.035
    110036.379–0.235–0.235–0.123–0.079
    120036.672–0.313–0.314–0.192–0.143
    130036.946–0.412–0.413–0.280–0.228
    140037.210–0.532–0.533–0.386–0.333
    150037.469–0.670–0.670–0.508–0.455
    160037.725–0.822–0.823–0.645–0.593
    170037.980–0.989–0.989–0.795–0.743
    180038.232–1.163–1.164–0.952–0.902
    190038.482–1.345–1.346–1.116–1.068
    200038.727–1.530–1.530–1.282–1.236
    210038.967–1.715–1.716–1.449–1.404
    220039.200–1.899–1.900–1.614–1.571
    230039.425–2.079–2.079–1.775–1.733
    240039.641–2.254–2.254–1.929–1.889
    250039.847–2.422–2.421–2.077–2.038
    260040.043–2.582–2.581–2.217–2.180
    270040.229–2.735–2.733–2.349–2.313
    280040.404–2.879–2.876–2.472–2.436
    290040.568–3.014–3.009–2.585–2.549
    300040.721–3.140–3.133–2.689–2.652
    310040.864–3.258–3.248–2.784–2.746
    320040.996–3.367–3.352–2.869–2.830
    330041.119–3.469–3.449–2.946–2.904
    340041.232–3.562–3.536–3.015–2.969
    350041.336–3.649–3.615–3.076–3.025
    360041.432–3.729–3.686–3.130–3.072
    370041.520–3.804–3.749–3.178–3.111
    380041.601–3.874–3.806–3.220–3.143
    390041.676–3.940–3.857–3.259–3.168
    400041.745–4.002–3.902–3.293–3.187
    410041.810–4.063–3.943–3.326–3.202
    420041.871–4.122–3.980–3.356–3.211
    430041.929–4.180–4.014–3.387–3.217
    440041.986–4.241–4.047–3.419–3.221
    450042.042–4.303–4.079–3.453–3.224
    460042.098–4.367–4.111–3.490–3.226
    470042.156–4.437–4.144–3.533–3.229
    480042.217–4.512–4.181–3.581–3.235
    490042.282–4.593–4.221–3.637–3.244
    500042.352–4.682–4.266–3.702–3.258
    510042.429–4.781–4.318–3.778–3.279
    520042.514–4.891–4.377–3.865–3.307
    530042.608–5.012–4.445–3.966–3.344
    540042.712–5.146–4.523–4.080–3.392
    550042.829–5.295–4.614–4.211–3.453
    560042.959–5.459–4.718–4.359–3.527
    570043.104–5.641–4.837–4.526–3.617
    580043.265–5.841–4.971–4.714–3.724
    590043.444–6.061–5.123–4.923–3.849
    600043.620–6.280–5.273–5.133–3.973
    $ \Delta {C_{{\text{aver}}}} $f2.8962.7212.3632.084
    注: a. 实验热容; b. 基于实验振动能级运用近似模型所得热容值与实验值的误差; c. 基于VAM完全振动能级运用近似模型所得热容值与实验值的误差; d. 基于实验振转能级所得热容值与实验值的误差; e. 基于完全振转能级所得热容值与实验值的误差; f. $\Delta {C_{ {\text{aver} } } } = \dfrac{ {\text{1} } }{w} \displaystyle\sum {\left| { {C_{\text{m} } } - {C_{ {\text{expt} } } } } \right|}$, w为参与计算的热容值个数.
    下载: 导出CSV
  • [1]

    McBride B J, Zehe M J, Gordon S 2002 NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species (Cleveland: National Aeronautics and Space Administration) p1

    [2]

    汪志诚 2013 热力学·统计物理 (第五版) (北京: 高等教育出版社) 第1页

    Wang Z C 2013 Thermodynamic·statistical physics (Vol. 5) (Beijing: Higher Education Press) p1 (in Chinese)

    [3]

    Gabriel V O, Luis A A H 2018 Int. J. Quantum Chem. 118 1Google Scholar

    [4]

    Irwin A W 1988 Astron. Astrophys. Suppl. Ser. 74 145

    [5]

    Fischer J, Gamache R R, Goldman A, Rothman L S, Perrin A 2003 J. Quant. Spectrosc. Radiat. Transfer 82 401Google Scholar

    [6]

    伍冬兰, 万慧军, 谢安东, 程新路, 杨向东 2009 物理学报 58 7410Google Scholar

    Wu D L, Wan H J, Xie A D, Cheng X L, Yang X D 2009 Acta Phys. Sin. 58 7410Google Scholar

    [7]

    Jia C S, Zhang L H, Wang C W 2017 Chem. Phys. Lett. 667 211Google Scholar

    [8]

    Maltsev M A, Kulikov A N, Morozov I V 2016 J. Phys. Conf. Ser. 774 012023Google Scholar

    [9]

    Maltsev M A, Morozov I V, Osina E L 2019 High Temp. 57 335Google Scholar

    [10]

    Ikot A N, Chukwuocha E O, Onyeaju M C, Onate C A, Ita B I, Udoh M E 2018 Pramana-J. Phys. 90 22Google Scholar

    [11]

    王小霞, 刘鑫, 张琼, 陈宏善 2017 物理学报 66 103601Google Scholar

    Wang X X, Liu X, Zhang Q, Chen H S 2017 Acta Phys. Sin. 66 103601Google Scholar

    [12]

    Maltsev M A, Morozov I V, Osina E L 2019 High Temp. 57 42Google Scholar

    [13]

    Horchani R, Jelassi H 2020 Chem. Phys. 532 1Google Scholar

    [14]

    Zúñiga J, Bastida A, Requena A, Cerezo J 2021 J. Phys. Chem. A 125 9226Google Scholar

    [15]

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

    [16]

    Fu J, Fan Q C, Liu G Y, Li H D, Xu Y G, Fan Z X, Zhang Y 2017 Comput. Theor. Chem. 1115 136Google Scholar

    [17]

    McDowell R S 1988 J. Chem. Phys. 88 356Google Scholar

    [18]

    蹇君, 雷娇, 樊群超, 范志祥, 马杰, 付佳, 李会东, 徐勇根 2020 物理学报 69 053301Google Scholar

    Jian J, Lei J, Fan Q C, Fan Z X, Ma J, Fu J, Li H D, Xu Y G 2020 Acta Phys. Sin. 69 053301Google Scholar

    [19]

    Rydberg R 1932 Z. Phys. 73 376Google Scholar

    [20]

    Klein V O 1932 Z. Phys. 76 226Google Scholar

    [21]

    Rees A L G 1947 Proc. Phys. Soc. 59 998Google Scholar

    [22]

    Pathria R K 1977 Statistical Mechanics (London: Pcrgamon Press) p100

    [23]

    Le Roy R J 2017 J. Quant. Spectrosc. Radiat. Transfer 186 167Google Scholar

    [24]

    Li C L, Li Y C, Ji Z H, Qiu X B, Lai Y Z, Wei J L, Zhao Y T, Deng L H, Chen Y Q, Liu J J 2018 Phys. Rev. A 97 062501Google Scholar

    [25]

    Laurendeau N M 2005 Statistical Thermodynamics: Fundamentals and Applications (England: Cambridge University Press) p1

    [26]

    Huang K 1965 Phys. Today 18 92

    [27]

    Babou Y, Rivière P, Perrin M Y, Soufiani A 2009 Int. J. Thermophys. 30 416Google Scholar

    [28]

    Jaffe R L 1987 AIAA 22nd Thermophysics Conference (New York: American Institute of Aeronautics and Astronautics) p1633

    [29]

    Capitelli M, Colonna G, Giordano D, Maraffa L, Casavola A, Minelli P, Pagano D, Pietanza L D, Taccogna F 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) p3

    [30]

    Huber K P, Herzberg G 1950 Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules (New York: Van Nostrand Reinhold Company) p9

    [31]

    朱华, 谢代前, 鄢国森 1999 高等学校化学学报 20 1910Google Scholar

    Zhu H, Xie D Q, Yan G S 1999 Chem. J. Chin. Univ. 20 1910Google Scholar

    [32]

    James B B, Edward R L, Philip D H, Carleton J H 1987 J. Mol. Spectrosc. 124 379Google Scholar

    [33]

    Gottlieb C A, Gottlieb E W, Litvak M M, Ball J A, Penfield H 1978 Astrophys. J. 219 77Google Scholar

    [34]

    Clyne M A A, Mcdermid I S 1979 J. Chem. Soc. , Faraday Trans. 2 75 905Google Scholar

    [35]

    Peterson K A, Woods R C 1990 J. Chem. Phys. 93 1876Google Scholar

    [36]

    Martin-Drumel M A, Hindle F, Mouret G, Cuisset A, Cernicharo J 2015 Astrophys. J. 799 115Google Scholar

    [37]

    Chase M W 1998 Journal of Physical and Chemical Reference DataMonograph (Vol. 9) (New York: National Institute of Standards and Technology Gaithersburg) p1726

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出版历程
  • 收稿日期:  2021-12-08
  • 修回日期:  2022-04-21
  • 上网日期:  2022-08-13
  • 刊出日期:  2022-09-05

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