Partition function of improved Tietz oscillators
TL;DR: In this paper, a closed-form expression of the vibrational partition function for the improved Tietz potential energy model is presented, which is applicable to many issues in chemical physics and engineering.
About: This article is published in Chemical Physics Letters.The article was published on 2017-05-16. It has received 134 citations till now. The article focuses on the topics: Vibrational partition function & Partition function (mathematics).
Citations
More filters
TL;DR: In this article, an analytical representation for the prediction of the molar Gibbs free energies of gaseous diatomic molecule substances was established. But the present model is related to three molecular constants and away from the need of lots of experimental spectroscopy data.
108 citations
TL;DR: In this article, the molar entropy and Gibbs free energy of pure gaseous substances were predicted in the temperature range of 100 to 6,000 K using a closed-form representation.
106 citations
TL;DR: In this paper, an explicit representation of molar entropy for gaseous substances was established based on the improved Rosen-Morse oscillator for describing the internal vibration of a molecule, using the dissociation energy, equilibrium internuclear distance and harmonic vibrational frequency.
101 citations
05 Nov 2019
TL;DR: Three reliable analytical expressions of the entropy, enthalpy and Gibbs free energy of carbon dioxide (CO2) are reported and 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 are performed.
Abstract: 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.
87 citations
TL;DR: In this article, an explicit representation of molar entropy for gaseous substances is presented, and the molar entropies can be successfully predicted by applying experimental values of only four molecular constants for the carbon monoxide, hydrogen chloride, hydrogen fluoride, deuterium fluoride and nitrogen monoxide.
83 citations
References
More filters
TL;DR: A new discussion of the complex branches of W, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W are presented.
Abstract: The LambertW function is defined to be the multivalued inverse of the functionw →we
w
. It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.
5,591 citations
TL;DR: The results demonstrate the ability to change the intrinsic catalytic properties of Cu for this notoriously difficult reaction by growing interconnected nanocrystallites from the constrained environment of an oxide lattice, demonstrating the feasibility of a two-step conversion of CO2 to liquid fuel that could be powered by renewable electricity.
Abstract: The electrochemical conversion of CO and H2O into liquid fuel is made feasible at modest potentials with the use of oxide-derived nanocystalline Cu as the catalyst. Renewable electricity is often produced when it is not needed. If the surplus could be harnessed to drive the conversion of CO2 and water into liquid fuel, the energy would not go to waste and a use would be found for CO2 produced by carbon capture. All this requires efficient electrocatalysts that reduce CO2 not only to CO, but also further into fuel chemicals. Copper does this but with low efficiency and selectivity. Christina Li et al. now show that the intrinsic catalytic properties of copper can be improved by producing it from its oxide as interconnected nanocrystallites. Their enhanced catalyst generates primarily ethanol, demonstrating that a two-step conversion of CO2 to liquid fuel powered by renewable electricity might be possible. The electrochemical conversion of CO2 and H2O into liquid fuel is ideal for high-density renewable energy storage and could provide an incentive for CO2 capture. However, efficient electrocatalysts for reducing CO2 and its derivatives into a desirable fuel1,2,3 are not available at present. Although many catalysts4,5,6,7,8,9,10,11 can reduce CO2 to carbon monoxide (CO), liquid fuel synthesis requires that CO is reduced further, using H2O as a H+ source. Copper (Cu) is the only known material with an appreciable CO electroreduction activity, but in bulk form its efficiency and selectivity for liquid fuel are far too low for practical use. In particular, H2O reduction to H2 outcompetes CO reduction on Cu electrodes unless extreme overpotentials are applied, at which point gaseous hydrocarbons are the major CO reduction products12,13. Here we show that nanocrystalline Cu prepared from Cu2O (‘oxide-derived Cu’) produces multi-carbon oxygenates (ethanol, acetate and n-propanol) with up to 57% Faraday efficiency at modest potentials (–0.25 volts to –0.5 volts versus the reversible hydrogen electrode) in CO-saturated alkaline H2O. By comparison, when prepared by traditional vapour condensation, Cu nanoparticles with an average crystallite size similar to that of oxide-derived copper produce nearly exclusive H2 (96% Faraday efficiency) under identical conditions. Our results demonstrate the ability to change the intrinsic catalytic properties of Cu for this notoriously difficult reaction by growing interconnected nanocrystallites from the constrained environment of an oxide lattice. The selectivity for oxygenates, with ethanol as the major product, demonstrates the feasibility of a two-step conversion of CO2 to liquid fuel that could be powered by renewable electricity.
1,256 citations
TL;DR: In this paper, an extension of sinc interpolation to algebraically decaying functions is presented, where the algebraic order of decay of a function's decay can be estimated everywhere in the horizontal strip of complex plane around the complex plane.
Abstract: An extension of sinc interpolation on $\mathbb{R}$ to the class of
algebraically decaying functions is developed in the paper. Similarly to the
classical sinc interpolation we establish two types of error estimates. First
covers a wider class of functions with the algebraic order of decay on
$\mathbb{R}$. The second type of error estimates governs the case when the
order of function's decay can be estimated everywhere in the horizontal strip
of complex plane around $\mathbb{R}$. The numerical examples are provided.
1,000 citations
TL;DR: In this article, a solution of the wave equation for the nuclear motion of a diatomic molecule with a Morse potential function and the rotational term included is given, and the wave functions are found to have the same form as the functions obtained when the rotation term is neglected.
Abstract: A solution of the wave equation for the nuclear motion of a diatomic molecule with a Morse potential function and the rotational term included is given. The wave functions are found to have the same form as the functions obtained when the rotational term is neglected. The constants ${D}_{e}$ and ${\ensuremath{\alpha}}_{e}$ in the equations ${B}_{v}={B}_{e}\ensuremath{-}{\ensuremath{\alpha}}_{e}(v+\frac{1}{2}),$ ${D}_{v}={D}_{e}+{\ensuremath{\beta}}_{e}(v+\frac{1}{2}),$ are found to be given by the relations ${D}_{e}=\ensuremath{-}\frac{4{{B}_{e}}^{3}}{{{\ensuremath{\omega}}_{e}}^{2}}$ ${\ensuremath{\alpha}}_{e}=2{x}_{e}{B}_{e}(3{[\frac{{B}_{e}}{{x}_{e}{\ensuremath{\omega}}_{e}}]}^{\frac{1}{2}}\ensuremath{-}\frac{3{B}_{e}}{{x}_{e}{\ensuremath{\omega}}_{e}}),$ a result which can also be derived from Dunham's formulas. Eq. (2) differs from the corresponding relation in Kratzer's formula by the term in parenthesis. This term is fairly constant for a number of molecules and has an average value of 0.7\ifmmode\pm\else\textpm\fi{}0.1 as was found empirically by Birge. The values of ${\ensuremath{\alpha}}_{e}$ given by (2) show satisfactory agreement with experimental values for many molecules.
595 citations
TL;DR: A cobalt protoporphyrin immobilized on a pyrolytic graphite electrode that reduces carbon dioxide in an aqueous acidic solution at relatively low overpotential, with an efficiency and selectivity comparable to the best porphyrIn-based electrocatalyst in the literature.
Abstract: The electrochemical conversion of carbon dioxide and water into useful products is a major challenge in facilitating a closed carbon cycle. Here we report a cobalt protoporphyrin immobilized on a pyrolytic graphite electrode that reduces carbon dioxide in an aqueous acidic solution at relatively low overpotential (0.5 V), with an efficiency and selectivity comparable to the best porphyrin-based electrocatalyst in the literature. While carbon monoxide is the main reduction product, we also observe methane as by-product. The results of our detailed pH-dependent studies are explained consistently by a mechanism in which carbon dioxide is activated by the cobalt protoporphyrin through the stabilization of a radical intermediate, which acts as Bronsted base. The basic character of this intermediate explains how the carbon dioxide reduction circumvents a concerted proton–electron transfer mechanism, in contrast to hydrogen evolution. Our results and their mechanistic interpretations suggest strategies for designing improved catalysts.
428 citations