- Email: [email protected]
Prediction of Gibbs free energy for the gases Cl2, Br2, and HCl
Chemical Physics Letters 726 (2019) 83–86
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
Prediction of Gibbs free energy for the gases Cl2, Br2, and HCl ⁎
⁎
T
Rui Jiang, Chun-Sheng Jia , Yong-Qing Wang , Xiao-Long Peng, Lie-Hui Zhang State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, People's Republic of China
H I GH L IG H T S
present an efficient closed-form representation of molar Gibbs free energy for gaseous substances. • We molar Gibbs free energy calculation model only involves three molecular constants. • Present • We excellently predict molar Gibbs free energy values for the gases Cl , Br , and HCl. 2
2
A R T I C LE I N FO
A B S T R A C T
Keywords: Gibbs free energy Analytical representation Chlorine gas Bromine gas Hydrogen chloride
We develop a new closed-form representation for the molar Gibbs free energies of gaseous substances. The predicted molar Gibbs free energies are in excellent agreement with the experimental data in a wide temperature range for three gases under consideration, and the average absolute deviations of the predicted results from the experimental data are 0.06155%, 0.0708% and 0.16696% for the gases Cl2, Br2, and HCl, respectively. The proposed model is away from the need of a large amount of experimental spectroscopy data, and only related to the dissociation energy, equilibrium bond length, and equilibrium vibration frequency.
1. Introduction An available closed-form representation of the molar Gibbs free energy of a system is central to the development of a simple and efficient way to deal with minimization of the total Gibbs free energy of the system. However, obtaining a universal closed-form representation of the Gibbs free energy of the system has been the challenging subject in chemistry and engineering. So far, one has not reported a universal closed-form representation governing the molar Gibbs free energy for the gases Cl2, Br2, and HCl. These gases often exist in combustion and gasification of solid fuels, such as coal, biomass, and municipal solid waste [1–5]. The content of chlorine or bromine is over 1% in some types of coal and biomass [1]. Hydrochloric acid (HCl) has been the most commonly used as an effective stimulation treatment to increase hydrocarbon production in carbonate reservoirs [6–10]. The Gibbs free energy plays an important role in addressing many issues, including the chemical reaction equilibrium constant [11–15], phase equilibrium constant [16–18], and adsorption equilibrium distribution coefficient [19–25]. The equilibrium constants are directly related to the Gibbs free energies of systems. The equilibrium state of a macroscopic system refers to a condition of minimization of the total Gibbs free energy subject to a closed system at system temperature and pressure. To achieve phase equilibrium in a multi-phase multi-component system,
⁎
one may formulate and solve a global minimization of the total Gibbs free energy of the system, at given temperature and pressure, with respect to the number of moles of each component in each phase. The knowledge of adsorption free energy is important to understand adsorbate-adsorbent interactions. The change in the Gibbs free energy shows the degree of spontaneity of an adsorption process, and a higher negative value indicates a more energetically favorable adsorption. The Gibbs free energy change due to adsorption process can provide useful information for design and fabrication of suitable adsorbents. Key to the success of constructing explicit representations of thermochemical quantities is the choice of suitable molecular oscillator model. With the help of introducing the dissociation energy and equilibrium bond length as explicit parameters, improved versions have been devised for some well-known oscillators, including the RosenMorse, Tietz, Frost-Musulin, Pöschl-Teller, and Manning-Rosen oscillators [26–30]. The improved oscillators have received a considerable attention in dealing with the thermodynamic properties of some diatomic substances afterwards [30–44]. The principal objective of the present work is the development of an explicit representation of the molar Gibbs free energy for gaseous diatomic molecule substances that is of reasonable accuracy, and that is based on the improved ManningRosen oscillator in describing the internal vibration of a molecule. Predictions of molar Gibbs free energy values of the gases Cl2, Br2 and
Corresponding authors. E-mail addresses: [email protected] (C.-S. Jia), [email protected] (Y.-Q. Wang).
https://doi.org/10.1016/j.cplett.2019.04.040 Received 19 March 2019; Received in revised form 12 April 2019; Accepted 13 April 2019 Available online 15 April 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.
Chemical Physics Letters 726 (2019) 83–86
R. Jiang, et al.
HCl are used to test the suitability of the proposed model. The results are compared with the experimental data from the National Institute of Standards and Technology (NIST) gas phase thermochemistry database [45]. The average absolute deviations of the predicted values from the NIST data are 0.06155%, 0.0708% and 0.16696% for the gases Cl2, Br2, and HCl, respectively. This prediction accuracy demonstrates that the molar Gibbs free energy for some gaseous diatomic molecule substances can be satisfactorily predicted on a molecular level.
characteristic temperature, Θr =
∂ ln Q ⎞ G = H − TS = kTV ⎛ − kT ln Q, ⎝ ∂V ⎠T
πkT ⎛ ⎛ λ ⎞ λ ⎞ c1⎟ − erfi ⎜⎛ c2⎟ ⎜erfi ⎜ λ ⎝ ⎝ kT ⎠ kT ⎝ ⎠
2λa 2λa λ λ ⎤ −e− kT erfi ⎛⎜ (2a + c1) ⎞⎟ + e− kT erfi ⎛⎜ (2a + c2) ⎞⎟ ⎞⎟ ⎥, kT kT ⎝ ⎠ ⎝ ⎠⎠⎦
3
λ=
2μ
,
(1)
(3)
a b c1 = − , b 2
(4)
⎜
2μ 1 + W ⎜⎛−πcωe re De re ⎝
1⎛ ⎜1 + 2⎝
1+
2λa 2λa λ λ ⎞⎞ −e− kT erfi ⎜⎛ (2a + c1) ⎟⎞ + e− kT erfi ⎛⎜ (2a + c2) ⎞⎟ ⎟⎞ ⎟ , ⎟ kT kT ⎝ ⎠ ⎝ ⎠⎠⎠⎠
where R represents the universal gas constant, and P represents gas pressure. Following the above three equations, the total molar Gibbs free energy is obtained as
(5)
2μ −πcωe re e De
⎠
8μDe (e αre − 1)2 ⎞ ⎟, ħ2α 2 ⎠
Knowing the values of three molecular constants De , re , and ωe , and using the above expressions, we can easily determine the temperature dependence of the total molar Gibbs free energy for the gaseous diatomic molecule substances. 3. Applications
(6) To check the above molar Gibbs free energy prediction model, we predict the variation of the reduced molar Gibbs free energy, Gr = −(G − H298.15)/ T , with respect to temperature T for the gases Cl2, Br2, and HCl. The experimental values of three molecular constants De , re , and ωe are given in Table 1 by referring to literature [47–49]. Values of the constants are h = 6.6260755 × 10−34 J·s, c = 2.99792 × 108 m·s−1, k = 1.380658 × 10−23 J·K−1, and R = 8.314511 J·mol−1·K−1. From the NIST database [45], we know that the reduced molar Gibbs free energy of the gas Cl2, −(G − H298.15)/ T , is 223.1 J·mol−1·K−1 at 298 K. The molar Gibbs free energy calculated from expression (15) is −53457.57696 J·mol−1 at 298 K. From these two values, we obtain the molar enthalpy value of 13026.2230 J·mol−1 at the temperature of 298.15 K. At the pressure of 1 bar and in the temperature range of 298 K to 6000 K, we calculate the reduced molar Gibbs free energy, and show the predicted results in Fig. 1(A). To facilitate the comparison with the experimental data from the NIST database [45], we also show those in Fig. 1(A). The NIST data are taken from the NIST-JANAF
(7)
(8)
where c represents the speed of light, ωe refers to the harmonic vibration frequency, vmax refers to the maximum vibration quantum number, and [n] means the maximum integer referring to n . The Lambert function W and imaginary error function erfi are defined as z = W (z ) eW (z ) , z 2 2 anderfi(z ) = π ∫0 et dt , respectively. We now proceed to address translational and rotational partition functions for a diatomic molecule. For simplicity, we assume there is no chemical or physical interaction between two molecules, and treat diatomic molecules as rigid rotors. Following these assumptions, the translational and rotational partition functions can be determined as follows, respectively [46],
Table 1 Experimental values of molecular constants for the molecules Cl2, Br2, and HCl.
3 2
2πmkT ⎞ V, Qt = ⎛ ⎝ h2 ⎠
(9) 2
Molecule
De (cm−1)
re (Å)
ωe (cm−1)
Ref.
Cl2 Br2 HCl
20276.48 15894.546 37,243
1.937200 2.2810213 1.27456303
559.7507 325.314194 2990.875
[47] [48] [49]
3
T ⎛ 1 Θr 1 Θr 4 Θr ⎞ ⎛ ⎞ + ⎛ ⎞ , 1+ + σ Θr ⎝ 3T 15 ⎝ T ⎠ 315 ⎝ T ⎠ ⎠ ⎜
(15)
G = Gt + G r + G v.
2μ De ⎞ , ⎟
8μDe (e αre − 1)2 2μDe (e 2αre − 1) 1 1 1+ vmax = ⎡ − − ⎤ , 2 2 2α 2 ⎢ 2⎥ ħ α 2 ħ ⎣ ⎦
Qr =
(13)
2 λc22 πkT ⎛ ⎛ λ ⎞ λ ⎞ ⎛ 1 De λc1 G v = −RT ln ⎜ e− kT ⎜⎛e kT − e kT + c1⎟ − erfi ⎜⎛ c2⎟ ⎜erfi ⎜ 2 λ ⎝ ⎝ kT ⎠ kT ⎠ ⎝ ⎝ ⎝
where ħ = h/2π , h represents the Planck constant, μ represents the reduced mass of a diatomic molecule, re refers to the equilibrium bond length, and parameters α , b and vmax are determined by the following three expressions,
b=
⎟
(14)
a v +1+b − max , vmax + 1 + b 2
α = πcωe
(12)
T ⎛ 1 Θr 1 Θr 2 4 Θr 3⎞ ⎞ ⎛ ⎞ + ⎛ ⎞ ⎟, G r = −RT ln ⎜⎛ 1+ + 3T 15 ⎝ T ⎠ 315 ⎝ T ⎠ ⎠ ⎠ ⎝ σ Θr ⎝
(2)
μ a = 2 2 De (e 2αre − 1), ħα
c2 =
5
(2πm) 2 (kT ) 2 ⎞ Gt = −RT ln ⎜⎛ ⎟, h3P ⎝ ⎠
in which De means the dissociation energy, k denotes the Boltzmann’s constant, and T is the temperature. Here parameters λ , a , c1, and c2 are defined by the following expressions,
ħ2α 2
(11)
in which H and S are the enthalpy and entropy, respectively. When we apply the improved Manning-Rosen oscillator to represent the internal vibration of the molecule, the molar translational, rotational, and vibrational Gibbs free energies are obtained as, respectively,
For the improved Manning-Rosen oscillator the vibrational partition function can be written in the form [30], λc22 1 − De ⎡ λc12 e kT e kT − e kT + ⎢ 2 ⎣
, and the values of σ are one
and two when diatomic molecules are heteronuclear and homonuclear molecules, respectively. In order to derive an analytical representation of the total molar Gibbs free energy, we start from deriving the translational, rotational, and vibrational Gibbs free energy through the thermodynamic relationship [46],
2. Analytical representation of Gibbs free energy
Qv =
h2 8π 2μ re2 k
⎟
(10)
where m is molecule mass, V is the volume, Θr is the rotational 84
Chemical Physics Letters 726 (2019) 83–86
R. Jiang, et al.
is increased by 1% and values of De and ωe remain unchanged, the average absolute deviation varies from 0.06155% to 0.06883%. When we increase the value of ωe by 1% and keep the original values of De and ωe , the average absolute deviation turns to 0.06653% from 0.06155%. It is obvious that the sensitivity of predicted results most depends on the value of the equilibrium bond length. The vibrational partition function (1) obtained in Ref. [30] by employing the Poisson summation formula only contains the lowest-order approximation contributions. Due to this reason, the excellent agreement with the experiment will deteriorate with decreasing temperature. For example, the relative deviations of the predicted reduced molar Gibbs free energies from the experimental values for HCl are 0.6137% and 0.7023% at the temperatures of 900 K and 700 K, respectively. In the practical applications, one can stipulate the limits of applicability of the present model according to the required precision. 4. Conclusions A straightforward method for the prediction of the molar Gibbs free energies of gaseous diatomic molecule substances has been presented on the basis of representing the internal vibration of the molecule with the improved Manning-Rosen oscillator. The proposed model has been applied to predict the molar Gibbs free energies of the gases Cl2, Br2, and HCl. Predicted results are compared with the experimental measurements and the excellent agreement is satisfying and therefore the availability of the proposed model is verified. The Gibbs free energy for some gaseous diatomic molecule substances can be predicted through the use of the dissociation energy, equilibrium bond length, and harmonic vibration frequency. One of advantages of the proposed model is that without a large amount of experimental spectroscopy data are used. Declaration of interest statement
Fig. 1. Temperature variation of molar reduced Gibbs free energy for (A) Cl2, (B) Br2, and (C) HCl. The green solid line refers to the theoretically predicted results, and the blue solid circles represent the experimental data. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The authors declare no conflict of interest. Acknowledgments We would like to thank the kind referees for positive and invaluable suggestions which have greatly improved the manuscript. This work was supported by the Key Program of National Natural Science Foundation of China under Grant No. 51534006 and the Sichuan Province Foundation of China for Fundamental Research Projects under Grant No. 2018JY0468.
Thermochemical Tables given by Chase [50]. The values in the NISTJANAF Tables are based on theoretical or experimental spectroscopic constants. The excellent agreement indicates that the prediction model is quite successful, and proves that, in a wide range of temperature, the molar Gibbs free energy can be simply predicted by means of only three molecular constants. The NIST data have contributions of the ground state and an excited state (1Σg+, 3Πo+) for Cl2. The NIST data are taken from the NIST-JANAF Thermochemical Tables given by Chase [50]. The developed model from microscope is based on the three molecular constants, away from the need of a great number of experimental spectroscopy data, and provides a rapid and reliable method for estimating the Gibbs free energy of the system. This conclusion will be further supported with simulations of the reduced molar Gibbs free energy for Br2 and HCl. The corresponding predicted results are presented in Fig. 1(B) and (C), respectively. The NIST data include contributions of the ground state and four excited states (1Σg+, 3Π2u, 3Π1u, 3Π - , 3Π+ ) for Br , and the NIST data have the contribution of the 1Σ 2 ou ou state for HCl [50]. In order to confirm the effect of the proposed model, we calculate the average absolute deviation of the predicted values from the NIST data. The corresponding average absolute deviations are 0.06155%, 0.0708% and 0.16696% for the gases Cl2, Br2, and HCl, respectively. These deviations strongly demonstrate that the present model can provide accurate predictions for the molar Gibbs free energies of Cl2, Br2 and HCl in a simple and straightforward way. When the value of De is increased by 1% and values of re and ωe are not changed, the average absolute deviation becomes 0.06279% from 0.06155%. If the value of re
References [1] S.V. Vassilev, G.M. Eskenazy, C.G. Vassileva, Contents, modes of occurrence and origin of chlorine and bromine in coal, Fuel 79 (2000) 903–921. [2] G. Dixon-Lewis, P. Marshall, B. Ruscic, A. Burcat, E. Goos, A. Cuoci, A. Frassoldati, T. Faravelli, P. Glarborg, Inhibition of hydrogen oxidation by HBr and Br2, Combust. Flame 159 (2012) 528–540. [3] M. Pelucchi, A. Frassoldati, T. Faravelli, B. Ruscic, P. Glarborg, High-temperature chemistry of HCl and Cl2, Combust. Flame 162 (2015) 2693–2704. [4] X. Wei, Y. Wang, D. Liu, H. Sheng, Influence of HCl on CO and NO emissions in combustion, Fuel 88 (2009) 1998–2003. [5] Y. Wang, J. Zhao, X. Wei, S. Li, Effect of HCl on NO formation during CO/NH3 combustion in an entrained flow reactor at 1023–1223 K, Energy Fuels 31 (2017) 3281–3287. [6] A.S. Zakaria, H.A. Nasr-El-Din, A novel polymer-assisted emulsified-acid system improves the efficiency of carbonate matrix acidizing, SPE J. 21 (2016) 1–14. [7] N. Liu, M. Liu, Simulation and analysis of wormhole propagation by VES acid in carbonate acidizing, J. Petrol. Sci. Eng. 138 (2016) 57–65. [8] R. Kumar, J. He, M. Bataweel, H. Nasr-El-Din, New insights on the effect of oil saturation on the optimal acid-injection rate in carbonate acidizing, SPE J. 23 (2018) 969–984. [9] A.A. Garrouch, A.R. Jennings Jr., A contemporary approach to carbonate matrix acidizing, J. Petrol. Sci. Eng. 158 (2017) 129–143. [10] M. Babaei, M. Sedighi, Impact of phase saturation on wormhole formation in rock matrix acidizing, Chem. Eng. Sci. 177 (2018) 39–52. [11] J. Solsvik, T. Haug-Warberg, H.A. Jakobsen, Implementation of chemical reaction
85
Chemical Physics Letters 726 (2019) 83–86
R. Jiang, et al.
[12] [13]
[14] [15] [16]
[17] [18]
[19]
[20] [21]
[22] [23] [24]
[25]
[26]
[27] [28]
[29] [30]
Chem. Phys. Lett. 667 (2017) 211–215. [31] X.Q. Song, C.W. Wang, C.S. Jia, Thermodynamic properties for the sodium dimer, Chem. Phys. Lett. 673 (2017) 50–55. [32] C.S. Jia, C.W. Wang, L.H. Zhang, X.L. Peng, R. Zeng, X.T. You, Partition function of improved Tietz oscillators, Chem. Phys. Lett. 676 (2017) 150–153. [33] J.F. Wang, X.L. Peng, L.H. Zhang, C.W. Wang, C.S. Jia, Entropy of gaseous boron monobromide, Chem. Phys. Lett. 686 (2017) 131–133. [34] C.S. Jia, C.W. Wang, L.H. Zhang, X.L. Peng, H.M. Tang, J.Y. Liu, Y. Xiong, R. Zeng, Predictions of entropy for diatomic molecules and gaseous substances, Chem. Phys. Lett. 692 (2018) 57–60. [35] C.S. Jia, C.W. Wang, L.H. Zhang, X.L. Peng, H.M. Tang, R. Zeng, Enthalpy of gaseous phosphorus dimer, Chem. Eng. Sci. 183 (2018) 26–29. [36] C.S. Jia, R. Zeng, X.L. Peng, L.H. Zhang, Y.L. Zhao, Entropy of gaseous phosphorus dimer, Chem. Eng. Sci. 190 (2018) 1–4. [37] X.L. Peng, R. Jiang, C.S. Jia, L.H. Zhang, Y.L. Zhao, Gibbs free energies of gaseous phosphorus dimer, Chem. Eng. Sci. 190 (2018) 122–125. [38] R. Jiang, C.S. Jia, Y.Q. Wang, X.L. Peng, L.H. Zhang, Prediction of enthalpy for the gases CO, HCl, and BF, Chem. Phys. Lett. 715 (2019) 186–189. [39] C.S. Jia, X.T. You, J.Y. Liu, L.H. Zhang, X.L. Peng, Y.T. Wang, L.S. Wei, Prediction of enthalpy for the gases Cl2, Br2, and gaseous BBr, Chem. Phys. Lett. 717 (2019) 16–20. [40] C.S. Jia, L.H. Zhang, X.L. Peng, J.X. Luo, Y.L. Zhao, J.Y. Liu, J.J. Guo, L.D. Tang, Prediction of entropy and Gibbs free energy for nitrogen, Chem. Eng. Sci. 202 (2019) 70–74. [41] Z. Ocak, H. Yanar, M. Salti, O. Aydogdu, Relativistic spinless energies and thermodynamic properties of sodium dimer molecule, Chem. Phys. 513 (2018) 252–257. [42] R. Khordad, A. Avazpour, A. Ghanbari, Exact analytical calculations of thermodynamic functions of gaseous substances, Chem. Phys. 517 (2019) 30–35. [43] M. Buchowiecki, Partition functions of thermally dissociating diatomic molecules and related momentum problem, Chem. Phys. Lett. 687 (2017) 227–232. [44] M. Buchowiecki, Ro-vibrational coupling in high temperature thermochemistry of the BBr molecule, Chem. Phys. Lett. 692 (2018) 236–241. [45] National Institute of Standards and Technology (NIST), NIST Chemistry WebBook, NIST Standard Reference Database Number 69, 2017.
equilibrium by Gibbs and Helmholtz energies in tubular reactor models: application to the steam–methane reforming process, Chem. Eng. Sci. 140 (2016) 261–278. V. Van, V. Daněk, Formation Gibbs energy of K2NbF7, K2NbOF5, and K2NbO2F3 from electrochemical measurements, Thermochim. Acta 373 (2001) 181–186. H.S. Roh, Thermodynamic chemical energy transfer mechanisms of non-equilibrium, quasi-equilibrium, and equilibrium chemical reactions, Energy 89 (2015) 1029–1049. A. Basu, G.S. Kumar, Thermodynamic characterization of proflavine–DNA binding through microcalorimetric studies, J. Chem. Thermodyn. 87 (2015) 1–7. J. van Baten, R. Szczepanski, A thermodynamic equilibrium reactor model as a CAPE-OPEN unit operation, Comput. Chem. Eng. 35 (2011) 1251–1256. H. Tang, K. Kitagawa, Supercritical water gasification of biomass: thermodynamic analysis with direct Gibbs free energy minimization, Chem. Eng. J. 106 (2005) 261–267. S. Sharma, P.G. Debenedetti, Evaporation rate of water in hydrophobic confinement, Proc. Natl. Acad. Sci. USA 109 (2012) 4365–4370. J.D. Olson, Thermodynamic consistency testing of PTx-data via the GibbsHelmholtz equation 2: methodology and limitations, Fluid Phase Equilib. 418 (2016) 50–56. V.R.N. Telis, A.L. Gabas, F.C. Menegalli, J. Telis-Romero, Water sorption thermodynamic properties applied to persimmon skin and pulp, Thermochim. Acta 343 (2000) 49–56. B. Guo, L. Chang, K. Xie, Adsorption of carbon dioxide on activated carbon, J. Nat. Gas Chem. 15 (2006) 223–229. S. Karaca, A. Gürses, M. Açıkyıldız, M. Ejder (Korucu), Adsorption of cationic dye from aqueous solutions by activated carbon, Micropor. Mesopor. Mater. 115 (2008) 376–382. J. Liu, C.J. Peach, H. Zhou, C.J. Spiers, Thermodynamic models for swelling of unconfined coal due to adsorption of mixed gases, Fuel 157 (2015) 151–161. V.K. Singh, E.A. Kumar, Measurement and analysis of adsorption isotherms of CO2 on activated carbon, Appl. Therm. Eng. 97 (2016) 77–86. B.J. Sherman, A.F. Ciftja, G.T. Rochelle, Thermodynamic and mass transfer modeling of carbon dioxide absorption into aqueous 2-piperidineethanol, Chem. Eng. Sci. 153 (2016) 295–307. F. Raganati, M. Alfe, V. Gargiulo, R. Chirone, P. Ammendola, Isotherms and thermodynamics of CO2 adsorption on a novel carbon-magnetite composite sorbent, Chem. Eng. Res. Des. 134 (2018) 540–552. C.S. Jia, Y.F. Diao, X.J. Liu, P.Q. Wang, J.Y. Liu, G.D. Zhang, Equivalence of the Wei potential model and Tietz potential model for diatomic molecules, J. Chem. Phys. 137 (2012) 014101. G.D. Zhang, J.Y. Liu, L.H. Zhang, W. Zhou, C.S. Jia, Modified Rosen-Morse potential-energy model for diatomic molecules, Phys. Rev. A 86 (2012) 062510. C.S. Jia, T. Chen, L.Z. Yi, S.R. Lin, Equivalence of the deformed Rosen-Morse potential energy model and Tietz potential energy model, J. Math. Chem. 51 (2013) 2165–2172. C.S. Jia, L.H. Zhang, X.L. Peng, Improved Pöschl-Teller potential energy model for diatomic molecules, Int. J. Quantum. Chem. 117 (2017) e25383. C.S. Jia, L.H. Zhang, C.W. Wang, Thermodynamic properties for the lithium dimer,
86
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
Prediction of Gibbs free energy for the gases Cl2, Br2, and HCl ⁎
⁎
T
Rui Jiang, Chun-Sheng Jia , Yong-Qing Wang , Xiao-Long Peng, Lie-Hui Zhang State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, People's Republic of China
H I GH L IG H T S
present an efficient closed-form representation of molar Gibbs free energy for gaseous substances. • We molar Gibbs free energy calculation model only involves three molecular constants. • Present • We excellently predict molar Gibbs free energy values for the gases Cl , Br , and HCl. 2
2
A R T I C LE I N FO
A B S T R A C T
Keywords: Gibbs free energy Analytical representation Chlorine gas Bromine gas Hydrogen chloride
We develop a new closed-form representation for the molar Gibbs free energies of gaseous substances. The predicted molar Gibbs free energies are in excellent agreement with the experimental data in a wide temperature range for three gases under consideration, and the average absolute deviations of the predicted results from the experimental data are 0.06155%, 0.0708% and 0.16696% for the gases Cl2, Br2, and HCl, respectively. The proposed model is away from the need of a large amount of experimental spectroscopy data, and only related to the dissociation energy, equilibrium bond length, and equilibrium vibration frequency.
1. Introduction An available closed-form representation of the molar Gibbs free energy of a system is central to the development of a simple and efficient way to deal with minimization of the total Gibbs free energy of the system. However, obtaining a universal closed-form representation of the Gibbs free energy of the system has been the challenging subject in chemistry and engineering. So far, one has not reported a universal closed-form representation governing the molar Gibbs free energy for the gases Cl2, Br2, and HCl. These gases often exist in combustion and gasification of solid fuels, such as coal, biomass, and municipal solid waste [1–5]. The content of chlorine or bromine is over 1% in some types of coal and biomass [1]. Hydrochloric acid (HCl) has been the most commonly used as an effective stimulation treatment to increase hydrocarbon production in carbonate reservoirs [6–10]. The Gibbs free energy plays an important role in addressing many issues, including the chemical reaction equilibrium constant [11–15], phase equilibrium constant [16–18], and adsorption equilibrium distribution coefficient [19–25]. The equilibrium constants are directly related to the Gibbs free energies of systems. The equilibrium state of a macroscopic system refers to a condition of minimization of the total Gibbs free energy subject to a closed system at system temperature and pressure. To achieve phase equilibrium in a multi-phase multi-component system,
⁎
one may formulate and solve a global minimization of the total Gibbs free energy of the system, at given temperature and pressure, with respect to the number of moles of each component in each phase. The knowledge of adsorption free energy is important to understand adsorbate-adsorbent interactions. The change in the Gibbs free energy shows the degree of spontaneity of an adsorption process, and a higher negative value indicates a more energetically favorable adsorption. The Gibbs free energy change due to adsorption process can provide useful information for design and fabrication of suitable adsorbents. Key to the success of constructing explicit representations of thermochemical quantities is the choice of suitable molecular oscillator model. With the help of introducing the dissociation energy and equilibrium bond length as explicit parameters, improved versions have been devised for some well-known oscillators, including the RosenMorse, Tietz, Frost-Musulin, Pöschl-Teller, and Manning-Rosen oscillators [26–30]. The improved oscillators have received a considerable attention in dealing with the thermodynamic properties of some diatomic substances afterwards [30–44]. The principal objective of the present work is the development of an explicit representation of the molar Gibbs free energy for gaseous diatomic molecule substances that is of reasonable accuracy, and that is based on the improved ManningRosen oscillator in describing the internal vibration of a molecule. Predictions of molar Gibbs free energy values of the gases Cl2, Br2 and
Corresponding authors. E-mail addresses: [email protected] (C.-S. Jia), [email protected] (Y.-Q. Wang).
https://doi.org/10.1016/j.cplett.2019.04.040 Received 19 March 2019; Received in revised form 12 April 2019; Accepted 13 April 2019 Available online 15 April 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.
Chemical Physics Letters 726 (2019) 83–86
R. Jiang, et al.
HCl are used to test the suitability of the proposed model. The results are compared with the experimental data from the National Institute of Standards and Technology (NIST) gas phase thermochemistry database [45]. The average absolute deviations of the predicted values from the NIST data are 0.06155%, 0.0708% and 0.16696% for the gases Cl2, Br2, and HCl, respectively. This prediction accuracy demonstrates that the molar Gibbs free energy for some gaseous diatomic molecule substances can be satisfactorily predicted on a molecular level.
characteristic temperature, Θr =
∂ ln Q ⎞ G = H − TS = kTV ⎛ − kT ln Q, ⎝ ∂V ⎠T
πkT ⎛ ⎛ λ ⎞ λ ⎞ c1⎟ − erfi ⎜⎛ c2⎟ ⎜erfi ⎜ λ ⎝ ⎝ kT ⎠ kT ⎝ ⎠
2λa 2λa λ λ ⎤ −e− kT erfi ⎛⎜ (2a + c1) ⎞⎟ + e− kT erfi ⎛⎜ (2a + c2) ⎞⎟ ⎞⎟ ⎥, kT kT ⎝ ⎠ ⎝ ⎠⎠⎦
3
λ=
2μ
,
(1)
(3)
a b c1 = − , b 2
(4)
⎜
2μ 1 + W ⎜⎛−πcωe re De re ⎝
1⎛ ⎜1 + 2⎝
1+
2λa 2λa λ λ ⎞⎞ −e− kT erfi ⎜⎛ (2a + c1) ⎟⎞ + e− kT erfi ⎛⎜ (2a + c2) ⎞⎟ ⎟⎞ ⎟ , ⎟ kT kT ⎝ ⎠ ⎝ ⎠⎠⎠⎠
where R represents the universal gas constant, and P represents gas pressure. Following the above three equations, the total molar Gibbs free energy is obtained as
(5)
2μ −πcωe re e De
⎠
8μDe (e αre − 1)2 ⎞ ⎟, ħ2α 2 ⎠
Knowing the values of three molecular constants De , re , and ωe , and using the above expressions, we can easily determine the temperature dependence of the total molar Gibbs free energy for the gaseous diatomic molecule substances. 3. Applications
(6) To check the above molar Gibbs free energy prediction model, we predict the variation of the reduced molar Gibbs free energy, Gr = −(G − H298.15)/ T , with respect to temperature T for the gases Cl2, Br2, and HCl. The experimental values of three molecular constants De , re , and ωe are given in Table 1 by referring to literature [47–49]. Values of the constants are h = 6.6260755 × 10−34 J·s, c = 2.99792 × 108 m·s−1, k = 1.380658 × 10−23 J·K−1, and R = 8.314511 J·mol−1·K−1. From the NIST database [45], we know that the reduced molar Gibbs free energy of the gas Cl2, −(G − H298.15)/ T , is 223.1 J·mol−1·K−1 at 298 K. The molar Gibbs free energy calculated from expression (15) is −53457.57696 J·mol−1 at 298 K. From these two values, we obtain the molar enthalpy value of 13026.2230 J·mol−1 at the temperature of 298.15 K. At the pressure of 1 bar and in the temperature range of 298 K to 6000 K, we calculate the reduced molar Gibbs free energy, and show the predicted results in Fig. 1(A). To facilitate the comparison with the experimental data from the NIST database [45], we also show those in Fig. 1(A). The NIST data are taken from the NIST-JANAF
(7)
(8)
where c represents the speed of light, ωe refers to the harmonic vibration frequency, vmax refers to the maximum vibration quantum number, and [n] means the maximum integer referring to n . The Lambert function W and imaginary error function erfi are defined as z = W (z ) eW (z ) , z 2 2 anderfi(z ) = π ∫0 et dt , respectively. We now proceed to address translational and rotational partition functions for a diatomic molecule. For simplicity, we assume there is no chemical or physical interaction between two molecules, and treat diatomic molecules as rigid rotors. Following these assumptions, the translational and rotational partition functions can be determined as follows, respectively [46],
Table 1 Experimental values of molecular constants for the molecules Cl2, Br2, and HCl.
3 2
2πmkT ⎞ V, Qt = ⎛ ⎝ h2 ⎠
(9) 2
Molecule
De (cm−1)
re (Å)
ωe (cm−1)
Ref.
Cl2 Br2 HCl
20276.48 15894.546 37,243
1.937200 2.2810213 1.27456303
559.7507 325.314194 2990.875
[47] [48] [49]
3
T ⎛ 1 Θr 1 Θr 4 Θr ⎞ ⎛ ⎞ + ⎛ ⎞ , 1+ + σ Θr ⎝ 3T 15 ⎝ T ⎠ 315 ⎝ T ⎠ ⎠ ⎜
(15)
G = Gt + G r + G v.
2μ De ⎞ , ⎟
8μDe (e αre − 1)2 2μDe (e 2αre − 1) 1 1 1+ vmax = ⎡ − − ⎤ , 2 2 2α 2 ⎢ 2⎥ ħ α 2 ħ ⎣ ⎦
Qr =
(13)
2 λc22 πkT ⎛ ⎛ λ ⎞ λ ⎞ ⎛ 1 De λc1 G v = −RT ln ⎜ e− kT ⎜⎛e kT − e kT + c1⎟ − erfi ⎜⎛ c2⎟ ⎜erfi ⎜ 2 λ ⎝ ⎝ kT ⎠ kT ⎠ ⎝ ⎝ ⎝
where ħ = h/2π , h represents the Planck constant, μ represents the reduced mass of a diatomic molecule, re refers to the equilibrium bond length, and parameters α , b and vmax are determined by the following three expressions,
b=
⎟
(14)
a v +1+b − max , vmax + 1 + b 2
α = πcωe
(12)
T ⎛ 1 Θr 1 Θr 2 4 Θr 3⎞ ⎞ ⎛ ⎞ + ⎛ ⎞ ⎟, G r = −RT ln ⎜⎛ 1+ + 3T 15 ⎝ T ⎠ 315 ⎝ T ⎠ ⎠ ⎠ ⎝ σ Θr ⎝
(2)
μ a = 2 2 De (e 2αre − 1), ħα
c2 =
5
(2πm) 2 (kT ) 2 ⎞ Gt = −RT ln ⎜⎛ ⎟, h3P ⎝ ⎠
in which De means the dissociation energy, k denotes the Boltzmann’s constant, and T is the temperature. Here parameters λ , a , c1, and c2 are defined by the following expressions,
ħ2α 2
(11)
in which H and S are the enthalpy and entropy, respectively. When we apply the improved Manning-Rosen oscillator to represent the internal vibration of the molecule, the molar translational, rotational, and vibrational Gibbs free energies are obtained as, respectively,
For the improved Manning-Rosen oscillator the vibrational partition function can be written in the form [30], λc22 1 − De ⎡ λc12 e kT e kT − e kT + ⎢ 2 ⎣
, and the values of σ are one
and two when diatomic molecules are heteronuclear and homonuclear molecules, respectively. In order to derive an analytical representation of the total molar Gibbs free energy, we start from deriving the translational, rotational, and vibrational Gibbs free energy through the thermodynamic relationship [46],
2. Analytical representation of Gibbs free energy
Qv =
h2 8π 2μ re2 k
⎟
(10)
where m is molecule mass, V is the volume, Θr is the rotational 84
Chemical Physics Letters 726 (2019) 83–86
R. Jiang, et al.
is increased by 1% and values of De and ωe remain unchanged, the average absolute deviation varies from 0.06155% to 0.06883%. When we increase the value of ωe by 1% and keep the original values of De and ωe , the average absolute deviation turns to 0.06653% from 0.06155%. It is obvious that the sensitivity of predicted results most depends on the value of the equilibrium bond length. The vibrational partition function (1) obtained in Ref. [30] by employing the Poisson summation formula only contains the lowest-order approximation contributions. Due to this reason, the excellent agreement with the experiment will deteriorate with decreasing temperature. For example, the relative deviations of the predicted reduced molar Gibbs free energies from the experimental values for HCl are 0.6137% and 0.7023% at the temperatures of 900 K and 700 K, respectively. In the practical applications, one can stipulate the limits of applicability of the present model according to the required precision. 4. Conclusions A straightforward method for the prediction of the molar Gibbs free energies of gaseous diatomic molecule substances has been presented on the basis of representing the internal vibration of the molecule with the improved Manning-Rosen oscillator. The proposed model has been applied to predict the molar Gibbs free energies of the gases Cl2, Br2, and HCl. Predicted results are compared with the experimental measurements and the excellent agreement is satisfying and therefore the availability of the proposed model is verified. The Gibbs free energy for some gaseous diatomic molecule substances can be predicted through the use of the dissociation energy, equilibrium bond length, and harmonic vibration frequency. One of advantages of the proposed model is that without a large amount of experimental spectroscopy data are used. Declaration of interest statement
Fig. 1. Temperature variation of molar reduced Gibbs free energy for (A) Cl2, (B) Br2, and (C) HCl. The green solid line refers to the theoretically predicted results, and the blue solid circles represent the experimental data. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The authors declare no conflict of interest. Acknowledgments We would like to thank the kind referees for positive and invaluable suggestions which have greatly improved the manuscript. This work was supported by the Key Program of National Natural Science Foundation of China under Grant No. 51534006 and the Sichuan Province Foundation of China for Fundamental Research Projects under Grant No. 2018JY0468.
Thermochemical Tables given by Chase [50]. The values in the NISTJANAF Tables are based on theoretical or experimental spectroscopic constants. The excellent agreement indicates that the prediction model is quite successful, and proves that, in a wide range of temperature, the molar Gibbs free energy can be simply predicted by means of only three molecular constants. The NIST data have contributions of the ground state and an excited state (1Σg+, 3Πo+) for Cl2. The NIST data are taken from the NIST-JANAF Thermochemical Tables given by Chase [50]. The developed model from microscope is based on the three molecular constants, away from the need of a great number of experimental spectroscopy data, and provides a rapid and reliable method for estimating the Gibbs free energy of the system. This conclusion will be further supported with simulations of the reduced molar Gibbs free energy for Br2 and HCl. The corresponding predicted results are presented in Fig. 1(B) and (C), respectively. The NIST data include contributions of the ground state and four excited states (1Σg+, 3Π2u, 3Π1u, 3Π - , 3Π+ ) for Br , and the NIST data have the contribution of the 1Σ 2 ou ou state for HCl [50]. In order to confirm the effect of the proposed model, we calculate the average absolute deviation of the predicted values from the NIST data. The corresponding average absolute deviations are 0.06155%, 0.0708% and 0.16696% for the gases Cl2, Br2, and HCl, respectively. These deviations strongly demonstrate that the present model can provide accurate predictions for the molar Gibbs free energies of Cl2, Br2 and HCl in a simple and straightforward way. When the value of De is increased by 1% and values of re and ωe are not changed, the average absolute deviation becomes 0.06279% from 0.06155%. If the value of re
References [1] S.V. Vassilev, G.M. Eskenazy, C.G. Vassileva, Contents, modes of occurrence and origin of chlorine and bromine in coal, Fuel 79 (2000) 903–921. [2] G. Dixon-Lewis, P. Marshall, B. Ruscic, A. Burcat, E. Goos, A. Cuoci, A. Frassoldati, T. Faravelli, P. Glarborg, Inhibition of hydrogen oxidation by HBr and Br2, Combust. Flame 159 (2012) 528–540. [3] M. Pelucchi, A. Frassoldati, T. Faravelli, B. Ruscic, P. Glarborg, High-temperature chemistry of HCl and Cl2, Combust. Flame 162 (2015) 2693–2704. [4] X. Wei, Y. Wang, D. Liu, H. Sheng, Influence of HCl on CO and NO emissions in combustion, Fuel 88 (2009) 1998–2003. [5] Y. Wang, J. Zhao, X. Wei, S. Li, Effect of HCl on NO formation during CO/NH3 combustion in an entrained flow reactor at 1023–1223 K, Energy Fuels 31 (2017) 3281–3287. [6] A.S. Zakaria, H.A. Nasr-El-Din, A novel polymer-assisted emulsified-acid system improves the efficiency of carbonate matrix acidizing, SPE J. 21 (2016) 1–14. [7] N. Liu, M. Liu, Simulation and analysis of wormhole propagation by VES acid in carbonate acidizing, J. Petrol. Sci. Eng. 138 (2016) 57–65. [8] R. Kumar, J. He, M. Bataweel, H. Nasr-El-Din, New insights on the effect of oil saturation on the optimal acid-injection rate in carbonate acidizing, SPE J. 23 (2018) 969–984. [9] A.A. Garrouch, A.R. Jennings Jr., A contemporary approach to carbonate matrix acidizing, J. Petrol. Sci. Eng. 158 (2017) 129–143. [10] M. Babaei, M. Sedighi, Impact of phase saturation on wormhole formation in rock matrix acidizing, Chem. Eng. Sci. 177 (2018) 39–52. [11] J. Solsvik, T. Haug-Warberg, H.A. Jakobsen, Implementation of chemical reaction
85
Chemical Physics Letters 726 (2019) 83–86
R. Jiang, et al.
[12] [13]
[14] [15] [16]
[17] [18]
[19]
[20] [21]
[22] [23] [24]
[25]
[26]
[27] [28]
[29] [30]
Chem. Phys. Lett. 667 (2017) 211–215. [31] X.Q. Song, C.W. Wang, C.S. Jia, Thermodynamic properties for the sodium dimer, Chem. Phys. Lett. 673 (2017) 50–55. [32] C.S. Jia, C.W. Wang, L.H. Zhang, X.L. Peng, R. Zeng, X.T. You, Partition function of improved Tietz oscillators, Chem. Phys. Lett. 676 (2017) 150–153. [33] J.F. Wang, X.L. Peng, L.H. Zhang, C.W. Wang, C.S. Jia, Entropy of gaseous boron monobromide, Chem. Phys. Lett. 686 (2017) 131–133. [34] C.S. Jia, C.W. Wang, L.H. Zhang, X.L. Peng, H.M. Tang, J.Y. Liu, Y. Xiong, R. Zeng, Predictions of entropy for diatomic molecules and gaseous substances, Chem. Phys. Lett. 692 (2018) 57–60. [35] C.S. Jia, C.W. Wang, L.H. Zhang, X.L. Peng, H.M. Tang, R. Zeng, Enthalpy of gaseous phosphorus dimer, Chem. Eng. Sci. 183 (2018) 26–29. [36] C.S. Jia, R. Zeng, X.L. Peng, L.H. Zhang, Y.L. Zhao, Entropy of gaseous phosphorus dimer, Chem. Eng. Sci. 190 (2018) 1–4. [37] X.L. Peng, R. Jiang, C.S. Jia, L.H. Zhang, Y.L. Zhao, Gibbs free energies of gaseous phosphorus dimer, Chem. Eng. Sci. 190 (2018) 122–125. [38] R. Jiang, C.S. Jia, Y.Q. Wang, X.L. Peng, L.H. Zhang, Prediction of enthalpy for the gases CO, HCl, and BF, Chem. Phys. Lett. 715 (2019) 186–189. [39] C.S. Jia, X.T. You, J.Y. Liu, L.H. Zhang, X.L. Peng, Y.T. Wang, L.S. Wei, Prediction of enthalpy for the gases Cl2, Br2, and gaseous BBr, Chem. Phys. Lett. 717 (2019) 16–20. [40] C.S. Jia, L.H. Zhang, X.L. Peng, J.X. Luo, Y.L. Zhao, J.Y. Liu, J.J. Guo, L.D. Tang, Prediction of entropy and Gibbs free energy for nitrogen, Chem. Eng. Sci. 202 (2019) 70–74. [41] Z. Ocak, H. Yanar, M. Salti, O. Aydogdu, Relativistic spinless energies and thermodynamic properties of sodium dimer molecule, Chem. Phys. 513 (2018) 252–257. [42] R. Khordad, A. Avazpour, A. Ghanbari, Exact analytical calculations of thermodynamic functions of gaseous substances, Chem. Phys. 517 (2019) 30–35. [43] M. Buchowiecki, Partition functions of thermally dissociating diatomic molecules and related momentum problem, Chem. Phys. Lett. 687 (2017) 227–232. [44] M. Buchowiecki, Ro-vibrational coupling in high temperature thermochemistry of the BBr molecule, Chem. Phys. Lett. 692 (2018) 236–241. [45] National Institute of Standards and Technology (NIST), NIST Chemistry WebBook, NIST Standard Reference Database Number 69, 2017.
equilibrium by Gibbs and Helmholtz energies in tubular reactor models: application to the steam–methane reforming process, Chem. Eng. Sci. 140 (2016) 261–278. V. Van, V. Daněk, Formation Gibbs energy of K2NbF7, K2NbOF5, and K2NbO2F3 from electrochemical measurements, Thermochim. Acta 373 (2001) 181–186. H.S. Roh, Thermodynamic chemical energy transfer mechanisms of non-equilibrium, quasi-equilibrium, and equilibrium chemical reactions, Energy 89 (2015) 1029–1049. A. Basu, G.S. Kumar, Thermodynamic characterization of proflavine–DNA binding through microcalorimetric studies, J. Chem. Thermodyn. 87 (2015) 1–7. J. van Baten, R. Szczepanski, A thermodynamic equilibrium reactor model as a CAPE-OPEN unit operation, Comput. Chem. Eng. 35 (2011) 1251–1256. H. Tang, K. Kitagawa, Supercritical water gasification of biomass: thermodynamic analysis with direct Gibbs free energy minimization, Chem. Eng. J. 106 (2005) 261–267. S. Sharma, P.G. Debenedetti, Evaporation rate of water in hydrophobic confinement, Proc. Natl. Acad. Sci. USA 109 (2012) 4365–4370. J.D. Olson, Thermodynamic consistency testing of PTx-data via the GibbsHelmholtz equation 2: methodology and limitations, Fluid Phase Equilib. 418 (2016) 50–56. V.R.N. Telis, A.L. Gabas, F.C. Menegalli, J. Telis-Romero, Water sorption thermodynamic properties applied to persimmon skin and pulp, Thermochim. Acta 343 (2000) 49–56. B. Guo, L. Chang, K. Xie, Adsorption of carbon dioxide on activated carbon, J. Nat. Gas Chem. 15 (2006) 223–229. S. Karaca, A. Gürses, M. Açıkyıldız, M. Ejder (Korucu), Adsorption of cationic dye from aqueous solutions by activated carbon, Micropor. Mesopor. Mater. 115 (2008) 376–382. J. Liu, C.J. Peach, H. Zhou, C.J. Spiers, Thermodynamic models for swelling of unconfined coal due to adsorption of mixed gases, Fuel 157 (2015) 151–161. V.K. Singh, E.A. Kumar, Measurement and analysis of adsorption isotherms of CO2 on activated carbon, Appl. Therm. Eng. 97 (2016) 77–86. B.J. Sherman, A.F. Ciftja, G.T. Rochelle, Thermodynamic and mass transfer modeling of carbon dioxide absorption into aqueous 2-piperidineethanol, Chem. Eng. Sci. 153 (2016) 295–307. F. Raganati, M. Alfe, V. Gargiulo, R. Chirone, P. Ammendola, Isotherms and thermodynamics of CO2 adsorption on a novel carbon-magnetite composite sorbent, Chem. Eng. Res. Des. 134 (2018) 540–552. C.S. Jia, Y.F. Diao, X.J. Liu, P.Q. Wang, J.Y. Liu, G.D. Zhang, Equivalence of the Wei potential model and Tietz potential model for diatomic molecules, J. Chem. Phys. 137 (2012) 014101. G.D. Zhang, J.Y. Liu, L.H. Zhang, W. Zhou, C.S. Jia, Modified Rosen-Morse potential-energy model for diatomic molecules, Phys. Rev. A 86 (2012) 062510. C.S. Jia, T. Chen, L.Z. Yi, S.R. Lin, Equivalence of the deformed Rosen-Morse potential energy model and Tietz potential energy model, J. Math. Chem. 51 (2013) 2165–2172. C.S. Jia, L.H. Zhang, X.L. Peng, Improved Pöschl-Teller potential energy model for diatomic molecules, Int. J. Quantum. Chem. 117 (2017) e25383. C.S. Jia, L.H. Zhang, C.W. Wang, Thermodynamic properties for the lithium dimer,
86