Calculation of Entropy for a Two-Dimensional (2D) Ideal Gas with a Statistical Thermodynamic Method

Abstract

Predicting adsorbate entropy is an essential prerequisite for surface science, heterogeneous catalysis, and microkinetic modeling. The linear correlation between adsorbate and gas-phase entropy has been widely found in various experiments, and it is considered as a simple and effective method to predict adsorbate entropy. In this work, the two-dimensional (2D) ideal gas model was used to calculate adsorbate entropy with the statistical thermodynamic method. The rotational entropy was obtained by solving the Schrodinger equation for a two-dimensional rotator. Combining the translational and vibrational entropy, an extended reversible adiabatic equation for 2D ideal gas can be derived, which is identical in form to the three-dimensional (3D) equation. Then the linear correlation between the 2D and 3D entropy can be proved theoretically, and the slope of this linear correlation is the heat capacity ratio. According to the characteristics of vibrational contribution to the heat capacity, the temperature is divided into three ranges, each of which has its own linear correlation. From the statistical thermodynamic computation data of benzene, perfect linear correlations can be fitted, and each slope is much close to the predicted value from our derived equation. The widely used Campbell–Sellers correlation matches our correlation in the low temperature range. We predict that there may be another linear correlation for different molecules in the medium temperature range, which is more appropriate in most heterogeneous catalytic reactions.

Derivation of an Adsorption Isotherm, Chemical Potential, and Entropy for 2D Gas Adsorbates with Packing Exclusions and Attractive Interactions

“…While making this separation introduces a significant approximation, it enables the derivation to be more easily applied to cases with coverage dependent interactions. The results of several studies that have considered unhindered translations versus hindered translations suggest that effects of the approximation will only moderately affect equilibrium concentrations, − which is likely a smaller effect relative to other approximations. A common practice today for computational calculations of adsorption energetics is to treat all of the degrees of freedom of the adsorbate (including vibrational, rotational, and translational) as harmonic vibrations, , and that reduction of all modes to harmonic is similarly severe to the approximation made in eq .…”

The Entropy of van der Waals Fluid

“…The suggested molar area is kept at A m °=7.18×10 6 m 2 mol −1 . Accordingly, the linear correlation between the entropy of 2D and 3D ideal gas can be proved [33] . The slope can be C A ,m / C p ,m or C A ,m / C V ,m , which depends on whether the 3D ideal gas is at constant pressure or constant volume.…”

“…These parts are the same as those of ideal gas, and the calculation equations can be found in references or textbooks. [33,38] Equations 27, 33 and 34 for diatomic or polyatomic molecules are the same as monatomic molecules. Ethane is selected as the specimen to calculate the entropy of polyatomic molecules due to its industrial importance.…”

“…Under Campbell standard, [15] the slope value 0.68 can also be obtained in low temperature range. [33] However, the translational entropy term 320 320 320 320 320 320 320 420 420 420 420 420 420 420 is required to divide equally between x, y and z components. This requirement is unnecessary in our work.…”

“…where γ = C p ,m / C V ,m . It can be extended to the following equation, which can be applied to any process [32,33] $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr TV_{\rm{m}}^{\gamma - 1} = L\exp \left( {{{S_{\rm{m}} } \over {C_{V,{\rm{m}}} }}} \right)\hfill\cr}}$$$ …”