Abstract
Attempts have been made to calculate the thermodynamical quantities of diatomic molecules such as O2, S2, Se2, and Te2 from spectroscopic data with the help of partition function theory. The results have been calculated in the temperature range 100–3000°C. In order to increase accuracy of the calculated quantities, we have incorporated nonrigidity, anharmonicity, and stretching effects of molecules. The variation of these quantities with temperature have been studied and explained in terms of various modes of molecular motions.
1. Introduction
The diatomic molecules have gained increased interest over the past several years in both experiment and theoretical studies because of their importance in astrophysical processes and many chemical reactions. Thermodynamical quantities such as enthalpy, entropy, heat capacity, and free energy have their potential application in various fields of science. For instance, enthalpy is one of the most important thermophysical properties required for calculating heat loads in process design. Entropy data are used for heart disease diagnosis [1]. These quantities are required for a large number of experimental methods and processes such as chemical transport reaction like Van Arkel-deBoer process and Mond process in the purification of nickel [2]. Recently, these quantities are also used in fabrication of thermal sensors and applied in medical field [3]. With the use of heat capacities and entropy values of the gaseous substances involved, heat of reaction can be determined and converted to different temperatures, which is necessary for process optimization of such reactions. Enthalpy and entropy have provided valuable insights into microscopic factors that complement those provided by conventional structural techniques.
Investigations in high temperature chemistry, astrophysics, and other disciplines require the knowledge of the thermodynamic properties of diatomic molecules. The plausibility of predictive models obtained in such investigations relies on the accuracy of these data. The scrutiny of the literature reveals that thermodynamic data are often absent or have scattered values in different research articles and handbooks. The main requirements to thermodynamic values are their reliability, mutual consistency, and so forth. In our theoretical study, thermodynamic values are estimated by using spectroscopic data which are microscopic in nature, whereas thermodynamical quantities are macroscopic in nature.
The elements of VI group of periodic table (O, S, Se, and Te) have potential applications. Oxygen is essential for respiration of all plants and animals and for practically all combustion. Sulphur is used extensively in making phosphatic fertilizers, vulcanization of natural rubber, sulfite paper, and so forth. Selenium is very important for human health, photovoltaic action, xerography, photographic toner, and so forth. Tellurium is used in corrosive action, ceramics, p-type semiconductor, and for making thermoelectric devices.
Over the decade there had been increasing interest in the estimation of thermodynamical quantities. Tolman [4] was the first to estimate thermodynamical quantities from spectroscopic data using statistical mechanics. Hicks and Mitchell [5] applied the suggested method for the estimation of thermodynamical quantities of HCl molecule. Using stretching and interaction terms for the diatomic molecules Giauque and Overstreet [6] modified the methods and calculated various thermodynamical quantities of HCl, Br2, and NO, and so forth. Later on, several workers [7–12] had contributed for the development of subject by estimating thermodynamical quantities of different molecular species. In continuation, authors applied partition function theory and spectroscopic data for the estimation of thermodynamical quantities of dimer of chalcogenide. In the present account, thermodynamical quantities of O2, S2, Se2, and Te2 have been calculated in the temperature range 100–3000 K. The choice of temperature range 100–3000 K is due to the fact that it covers its applications in biological sciences, industry, and high temperature chemistry.
2. Method of Calculation
The energy of a molecular system can be divided into four categories: translational energy, due to the motion of the molecule’s centre of mass through space, rotational energy, due to the rotation of the molecule about its centre of mass, vibrational energy due to the vibration of the molecule’s constituent atoms, and electronic energy, due to the interactions between the molecule’s electrons and nuclei. Since in the real case of motion of the diatomic molecule, during vibration, rotation also occurs, therefore, instead of taking individual rotational and vibrational energy, rovibrational energy has to be used. Then, the total energy of a diatomic molecule will be the sum of translational, rovibrational, and electronic energy
Statistical Mechanics show that partition function is a measure of extent to which energy is partitioned among the different states. Partition function is concerned with degeneracy as well as the energy of that level mainly vibrational and rotational and these energies are determined very accurately by spectroscopic methods. The molecular partition function gives an indication of the average number of states that are thermally accessible to a molecule at the temperature of the system. The larger the value of the partition function, the larger the number of thermally accessible states is. The partition function for a set of energy levels in a molecule is given by where is the degeneracy of the th energy level, is the total energy of th level, is Boltzmann constant, is absolute temperature, and ranges over all quantum states.
Corresponding to three types of energies, there are three partition functions, namely, translational partition function (), rotational-vibrational partition function (), and electronic partition function (). The total partition function of a molecular system can be expressed as
The electronic energy levels contribute to the thermodynamic properties only at high temperature or if unpaired electrons are present. Therefore, total partition function will be the multiplication of translational partition function () and rovibrational partition function (). The total partition function can be expressed as
Translational contribution to the thermodynamical quantities can be calculated by the following relations [13].(i)Free energy .(ii)Enthalpy .(iii)Entropy .(iv)Heat capacity at constant pressure .Where represents enthalpy at a temperature of 298.15 K and is the gas constant.
With the help of vibrational-rotational partition function, its contribution in free energy, enthalpy, entropy, and heat capacity can be estimated by the following formulae [13].(i)Free energy .(ii)Enthalpy .(iii)Entropy .(iv)Heat capacity at constant pressure .Where represents the multiplicity of the ground state and in which
Finally, the thermodynamic quantities like free energy, enthalpy, entropy, and heat capacity of chalcogenide dimers are calculated by adding the translational contribution and the vibrational-rotational contribution.
3. Results and Discussion
The thermodynamical quantities (free energy, enthalpy, entropy, and heat capacity at constant pressure) of dimer of chalcogenide have been calculated in the temperature range 100–3000 K. The calculated values are listed in Tables 1, 2, 3, and 4. The spectroscopic data used for the present calculation are collected in Table 5 [14–18], whereas Table 6 shows the comparison of thermodynamical quantities with reported values in the literature [18]. From Table 6, it is clear that heat capacity shows the deviation from 0.056% to 0.626% from reported values while entropy shows the deviation from 0.078% to 0.175%. Variation of these quantities with temperature reveals that free energy, enthalpy, and entropy (Figures 1, 2, and 3) increase with temperature while heat capacity (Figure 4) becomes constant. At low temperature only translational motion of molecules contributes while as temperature increases rotational motion also occurs. On further increasing the temperature vibrational motion of the molecules gives its significant contribution. This explains the increase in the thermodynamical quantities like entropy, free energy, and enthalpy with temperature. After a certain value of temperature, there is no additional motion in the molecules which account for constancy of specific heat at a very high temperature. In this calculation, has been used instead of which involves the impact of stretching of bonds between the both atoms of diatomic molecules. Incorporation of instead of taking individual contribution of and increases the accuracy of the computed data. Dimers of chalcogenides have 3Σ ground state; therefore, ground state multiplicity has been included in the calculation of thermodynamical quantities which gives authentic values of thermodynamical quantities.
