Thermodynamic Properties for Carbon Dioxide.

We first report three reliable analytical expressions of the entropy, enthalpy and Gibbs free energy of carbon dioxide (CO2) and perform predictions of these three thermodynamic quantities on the basis of the proposed analytical expressions and in terms of experimental values of five molecular constants for CO2. The average relative deviations of the calculated values from the National Institute of Standards and Technology database over the temperature range from 300 to 6000 K are merely 0.053, 0.95, and 0.070%, respectively, for the entropy, enthalpy, and Gibbs free energy. The present predictive expressions are away from the utilization of plenty of experimental spectroscopy data and are applicable to treat CO2 capture and storage processes.

Introduction

Carbon dioxide (CO2) as a greenhouse gas has been regarded a main contributor of global warming. Davis et al.[1] suggested that the people would continue to confront increasing global mean temperatures and sea level rise if current greenhouse gas concentrations remain constants in the atmosphere. As CO2 has a key influence to global climate change, CO2 capture and storage (CCS) technologies are becoming more and more important topics for reducing CO2 emission. CO2 capture technologies can be classified as three types: post-combustion, pre-combustion, and oxy-fuel combustion. Adsorbing CO2 is one of post-combustion capture technologies.[2] Many efforts have been made to develop efficient approaches of adsorbing CO2.[3−35] To optimally design such processes, we require accurate and reliable thermodynamic property data on CO2. For some diatomic substances, the improved versions of some well-known oscillators such as the improved Tietz oscillator have been applied to represent the internal vibration of a diatomic molecule for constructions of universal analytical expressions of thermodynamic properties.[36−50] A CCS process simulation needs a combination of an equation of state (EoS) with the Helmholtz free energy or Gibbs free energy.[51−55] The accurate three-parameter EoS proposed by Jaubert et al.[56] and the EoS developed by Span and Wagner[57] on the basis of the Helmholtz free energy are available in CCS process simulators. An analytical expression of the thermodynamic property can combine the simplicity of the analytical expression with the solid theoretical background of the statistical thermodynamic theory. However, it is difficult for chemists and engineers to obtain reliable analytical expressions of the thermodynamic quantities for gaseous substances. Although Span and Wagner[57] proposed an empirical polynomial analytical representation for the ideal-gas part of the Helmholtz free energy of CO2, it includes 13 adjustable parameters, which were determined by fitting plenty of experimental data points. To date, one has not reported any analytical expressions governing the entropy, enthalpy, and Gibbs free energy for CO2 from the view of the first principle. Therefore, it is of considerable interest to explore suitable analytical expressions of the thermodynamic properties for CO2.

The aim of this work is to establish suitable analytical expressions of the thermodynamic properties for the triatomic substance CO2. Compared with a diatomic substance, the generation of analytical expressions of the thermodynamic properties shows considerably greater challenges. To verify the effectiveness of the present predictive models, we apply three analytical expressions to predict the variations of the entropy, enthalpy, and Gibbs free energy with respect to temperature for CO2 and compare the predictive results with the available data from the National Institute of Standards and Technology (NIST) database.[58] The NIST data are compiled by the utilization of NIST-JANAF Thermochemical Tables, which is constructed by Chase[59] in terms of plenty of theoretical or experimental spectroscopic constants.

Results and Discussion

To show the accuracy and reliability of the proposed predictive models, we calculate the entropy, enthalpy, and Gibbs free energy values at the pressure of 1 bar and different temperatures. This performance needs experimental data on the O–CO bond energy, O–C equilibrium bond length, and stretching and bending vibrational frequencies in the CO2 molecule. We consult the experimental data given by Itikawa,[60] and list the required molecular constant values for the ground electronic state of CO2: De = 5.451 eV, reCO = 1.160 Å, ωes = 1388 cm–1, ωea = 2349 cm–1, and ωeb = 667.4 cm–1. Taking q = – 10 and employing the experimental data of the five molecular constants De, reCO, ωes, ωea, and ωeb, the predictive values of molar entropy, enthalpy, and reduced Gibbs free energy for CO2 are produced in terms of eqs –15 from 298 up to 6000 K. The predictive results are depicted in Figure , in which the green solid lines represent the theoretically predictive results and the blue solid circles refer to the NIST data.[58] From Figure , we observe that the predictive values show excellent agreement with the NIST data in the entire temperature range. The average relative deviations of the predictive data from the NIST data in a wide range of temperatures from 300 up to 6000 K for the entropy, enthalpy, and reduced Gibbs free energy are 0.053, 0.95, and 0.070%, respectively. These results indicate that our three predictive models are of very high accuracy to predict the entropy, enthalpy, and Gibbs free energy for CO2. In our calculations, the reduced molar Gibbs free energy, Gr = – (G – H298.15)/T, of CO2 from the NIST database[58] is 213.8 J·mol–1·K–1 at 298 K. The molar Gibbs free energy calculated by the aid of eq is −22,901.65 J·mol–1 at 298 K. On the basis of these two values, we obtain the value of 40,810.75 J·mol–1 at 298.15 K for the molar enthalpy. When T = 6000 K and ωe = ωeb, the value of e–Θ reaches at the maximum, e–Θ = 0.852 < 1; thus, eq is always valid in our calculations.

Figure 1

Temperature variation of the thermodynamic properties of CO2 for (A) molar entropy, (B) molar enthalpy, and (C) reduced molar Gibbs free energy.

Conclusions

In summary, we first present suitable analytical expressions of the molar entropy, enthalpy, and Gibbs free energy for CO2. The average relative deviations between the theoretical predictive values and the NIST data ranging from the temperatures of 300 to 6000 K are 0.053, 0.95, and 0.070%, respectively, for the entropy, enthalpy, and reduced Gibbs free energy of CO2. The present predictive expressions exhibit satisfactorily accuracies and simple and straightforward characteristics and are away from the requirements of plenty of experimental spectroscopy data. The methodology developed in our work opens up a new pathway serving as constructions of accurate and efficient analytical expressions of thermodynamic properties for other triatomic substances such as water and hydrogen sulfide. It is expected that the present predictive models could aid the design and optimization of CCS processes.

Computational Methods

Figure shows that CO2 is a centrosymmetric linear molecule and possesses symmetric stretching, antisymmetric stretching, and bending vibration modes. However, as a diatomic molecule is consisted of two atoms, it merely possesses a kind of stretching vibration. Hence, addressing vibrations of the CO2 molecule is much more difficult than treating the vibration of the diatomic molecule such as the nitrogen molecule. Here, we treat the internal symmetric stretching vibration of the CO2 molecule in terms of the improved Tietz oscillator[61] and deal with the antisymmetric stretching and bending vibrations with the help of the harmonic oscillator. According to the vibrational partition function given in eq 14 of ref (50) for the improved Tietz oscillator, the partition function corresponding to the symmetric stretching vibration for CO2 is expressed asin which De denotes the energy serving as dissociating CO2 into O–CO, k refers to the Boltzmann constant, T is the temperature, symbol erfi represents the imaginary error function, , , , , , and . Here, h is the Planck constant, μOO refers to the reduced mass of two oxygen atoms, reOO = 2reCO (reCO represents the C–O equilibrium length), q denotes a dimensionless variable, and the symbol ± stands for the plus and minus in the cases of a positive value and negative value of q, respectively. The parameter α reads as , where , c is the speed of light, and ωes represents the symmetric stretching vibration frequency. The contributions of the excited states to the partition function are represented by the factor . The highest vibrational quantum number, vmax, reads as where [x] refers to the greatest integer, which is less than x.

Figure 2

Schematic diagram of the vibration modes of the carbon dioxide molecule.

The vibrational partition function for a harmonic oscillator is given by[36]where Θv refers to the vibrational characteristic temperature and is expressed as Θv = hcωe/k. Here, ωe denotes the vibrational frequency.

We consider the canonical ensemble. From the following basic thermodynamic relationshipswe obtain the molar vibrational entropy, enthalpy, and Gibbs free energy referring to the symmetric stretching vibration for CO2, respectivelywhere R is the universal gas constant.

With the help of thermodynamic formulas above (eqs –5), we obtain the molar vibrational entropy, enthalpy, and Gibbs free energy serving as contributions of the antisymmetric stretching and bending vibrations for CO2, respectivelyin which Θva and Θvb are the respective vibrational characteristic temperatures for the antisymmetric stretching mode and bending mode. They are defined as Θva = hcωea/k and Θvb = hcωeb/k, where ωea and ωeb are the vibrational frequencies corresponding to the antisymmetric stretching vibration and bending vibration, respectively.

To simplify the mathematical calculation, it is reasonable to assume that the interaction between two molecules is not considered and the CO2 molecule is regarded as a rigid rotor. The molar translational and rotational entropies are given respectively by[37]where mCO stands for the mass of the CO2 molecule, P represents the gas pressure, Θr refers to the rotational characteristic temperature, . Here, μCO represents the reduced mass of carbon and oxygen atoms. The molar translational and rotational enthalpies can be written as follows,[36] respectivelyThe molar translational and rotational Gibbs free energies are expressed in terms of the following two representations,[38] respectivelyThe total molar entropy, enthalpy, and Gibbs free energy for CO2 can be determined from the following formulasBy inputting the experimental values of De, reCO, ωes, ωea, and ωeb in our predictive models, we can predict the molar entropy, enthalpy, and Gibbs free energy for carbon dioxide. Our prediction approaches rely only on requirements of experimental data of five molecular constants for CO2. It is superior to the conventional computation models, which need a large amount of experimental spectroscopy data.