A mathematical approach to chemical equilibrium theory for gaseous systems IV: a mathematical clarification of Le Chatelier’s principle

A long-standing scientific problem about G and ∂G/∂ξ, involving the Gibbs-Duhem equation, demonstrates that wrong concepts are sometimes dominant in the literature even though corrections are readily available. This fact shows that the Gibbs-Duhem equation and especially its applications are among the most difficult topics in thermodynamics. Moreover, it demonstrates that there are often weak points in existing methods which necessitate new perspectives. There exist several different derivations of the Gibbs-Duhem equation. A novel mathematical derivation of the equation is developed here and is demonstrated to be more natural, direct, and intuitive than its thermodynamics counterpart. Different perspectives of mathematical and thermodynamic derivations are illustrated along with discussions concerning methods for solving specific scientific problems. The introduction of different self-consistent methods to solve a specific problem is vital in providing researchers with a variety of perspectives, thus, Legendre transformations are used here to introduce thermodynamic functions since the method is unfamiliar to many researchers in this field, also because it is a good method

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“…7. [20,21] The proof can be extended to real systems where xk in Eq. 34 is replaced by the activity or fugacity ak…”

Section: A General Expression System For Theoretical Studymentioning

confidence: 99%

Natural Mathematical Derivation of the Gibbs–Duhem Equation Related to ΔG and ∂G/∂ξ

Liu

Drew

2022

Int J Thermophys

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“…The environmental equivalence from translation operations has been confused with asymmetric unit associated with symmetry operations other than translations and the classical definition of Miller indices has been modified incorrectly, and these wrong practices have a big influence [34,35]. The problems with Le Chaterlier's principle were identified long ago by many researches from different perspectives [36] but the principle is still prevalent in modern classrooms.…”

Section: Introductionmentioning

confidence: 99%

A theoretical investigation of the quarter-wavelength model-part 2: verification and extension

2022

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The quarter-wavelength model states that the reflection of microwaves is minimized if the thickness of the film is mλ/4 where m is an odd integer and λ is the wavelength within the film . In part 1 of this work, published previously, it was shown that the theory behind this model is flawed and that it needs to be replaced using a wave superposition approach involving principles not beyond physics at college level. Through the verification of the results from formulae familiar to material scientists, it is shown here that many other concepts in current mainstream microwave absorption theory are used inappropriately including that reflection loss RL has been used to characterize microwave absorption from material instead of film and also that the results have been rationalized incorrectly by impedance matching theory. Impedance matching theory is developed from transmission-line theory for scattering parameter s 11 but cannot be applied to RL which is an innate property only for metal-backed film.

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“…Particularly, its specific surface area is as high as 1658 m 2 ·g –1 , which is even higher than that of reported MOF-199 prepared at a similar temperature but for long time (Table ). By transferring the byproducts out of the reaction system, the reaction equilibrium will further move to the positive reaction direction and trigger the increase of the conversion of reactants according to Le Chatelier’s principle . In chemical kinetics, the rate constant “ k ” is an important parameter, which directly determines the reaction rate .…”

Section: Results and Discussionmentioning

confidence: 99%

“…By transferring the byproducts out of the reaction system, the reaction equilibrium will further move to the positive reaction direction and trigger the increase of the conversion of reactants according to Le Chatelier's principle. 42 In chemical kinetics, the rate constant "k" is an important parameter, which directly determines the reaction rate. 43 It is herein calculated at 303.7 mol −1 •h −1 for the sample MOF-199- 2S using eq 4, which is obviously higher than reported ones (Table 2).…”

Section: ■ Experimental Sectionmentioning

confidence: 99%

Alkali-Induced Self-Transferring Byproduct Strategy for Strengthening Sustainable Synthesis of MOF-199 Without Waste Discharge

Xue

Zhang

Huang

et al. 2020

ACS Sustainable Chem. Eng.

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To eliminate the effect of the byproduct on the yield and mother liquor in synthesizing metal–organic framework (MOF), an alkali-induced self-transferring byproduct (AISB) method is proposed by introducing alkali NaOH as a quasi-reactant which indirectly makes contact with the reactants. The quasi-reactant well captures and immobilizes the byproducts volatilized from mother liquor, simultaneously reducing reaction resistance and improving concentration gradient of reactants and intensifying the crystallization. It is found that ethanol not only acts as a solvent but also is partially oxidized by the primary byproduct HNO3. Thus, the reaction environment with a strong acid is weakened, which is convenient for the formation and existence of MOF-199. It is first confirmed by the appearance of Na2CO3 and NaNO2 in the quasi-byproduct. An improved reaction mechanism of the MOF-199 material is also proposed. Also, the resultant mother liquor can be directly used for next preparation. The reaction rate constant of the AISB method reaches up to 1201.2 mol–1·h–1. A high dry base yield up to 97.3% can be calculated based on the added Cu(NO3)2. Especially, the obtained sample MOF-199-2S shows a specific surface area of 1734 m2·g–1, which is obviously better than the samples obtained from traditional hydrothermal methods. It exhibits a CO2 static adsorption capacity of 9.03 mmol·g–1 at 1.0 bar and 273 K. Its cyclic CO2 adsorption capacity in simulated flue gas flowing through the fixed bed is stable at around 1.25 mmol·g–1 with a small pressure drop of 1.6 bar, which benefits from its large size (0.7 mm). Notably, by replacing the quasi-reactant NaOH with Cu(OH)2, this method provides a sustainable and ingenious strategy to synthesize MOF with zero discharge.

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A mathematical approach to chemical equilibrium theory for gaseous systems IV: a mathematical clarification of Le Chatelier’s principle

Cited by 11 publications

References 24 publications

Natural Mathematical Derivation of the Gibbs–Duhem Equation Related to ΔG and ∂G/∂ξ

Natural Mathematical Derivation of the Gibbs–Duhem Equation Related to ΔG and ∂G/∂ξ

A theoretical investigation of the quarter-wavelength model-part 2: verification and extension

Alkali-Induced Self-Transferring Byproduct Strategy for Strengthening Sustainable Synthesis of MOF-199 Without Waste Discharge

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