Implicit function theorem and Jacobians in solvation and adsorption

Shimizu, Seishi; Matubayasi, Nobuyuki

doi:10.1016/j.physa.2021.125801

Cited by 8 publications

(22 citation statements)

References 54 publications

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“…The basic equation of the preferential solvation theory (Eq. (6.2)) [9,11,12,15,16] was revealed to be the statistical variable transformation on a mixed deviation vector. This picture is much more straightforward than the approaches based on thermodynamic variable transformation from the Kirkwood-Buff tradition [55].…”

Section: Statistical Variable Transformation In Inhomogeneous Solutionsmentioning

confidence: 98%

“…However, the discrepancy between the isochoric "observatory" subsystem and the ensemble of experimental convenience (i.e., isobaric) necessitated thermodynamic variable transformation between the two ensembles, leading to significant calculus and algebra [6][7][8][9][10]. For the fluctuation solution theory to remain a useful tool for even more complex solutions [5,13,14], facilitating calculation [15,16] is indispensable.…”

Section: Introductionmentioning

confidence: 99%

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Ensemble transformation in the fluctuation theory

Shimizu

Matubayasi

2022

Physica A: Statistical Mechanics and its Applications

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Section: Statistical Variable Transformation In Inhomogeneous Solutionsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Ensemble transformation in the fluctuation theory

Shimizu

Matubayasi

2022

Physica A: Statistical Mechanics and its Applications

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“…To do so, as a first step, let us note that varying T and a 2 under constant ϵ does not change the isotherm because of eq . Consequently Applying the implicit function theorem , on eq yields Noting the equivalence between constant ϵ and constant ⟨ n 2 ⟩/ n 1 and using again the definition (eq ), we obtain This left-hand side of eq is reminiscent of the procedure for obtaining the energy from the free energy (ϵ in this case). This derivative is also ϵ; hence, eq signifies the purely energetic nature of the adsorption potential, ϵ, ,− ,, which is in agreement with the original assumption by Polanyi. − ,− …”

Section: Theorymentioning

confidence: 96%

Temperature Dependence of Sorption

Shimizu

Matubayasi

2021

Langmuir

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Understanding how sorption depends on temperature on a molecular basis has been made difficult by the coexistence of isotherm models, each assuming a different sorption mechanism and the routine application of planar, multilayer sorption models (such as Brunauer–Emmett–Teller (BET) and Guggenheim–Anderson–de Boer (GAB)) beyond their premises. Furthermore, a common observation that adsorption isotherms measured at different temperatures fall onto a single “characteristic curve” when plotted against the adsorption potential has not been given a clear explanation, due to its ambiguous foundation. Extending our recent statistical thermodynamic fluctuation theory of sorption, we have generalized the classical isosteric theory of sorption into a statistical thermodynamic fluctuation theory and clarified how sorption depends on temperature. We have shown that a characteristic curve exists when sorbate number increment contributes purely energetically to the interface, whereas the correlation between sorbate number and entropy drives the temperature dependence of an isotherm. This theory rationalizes the opposite temperature dependence of water vapor sorption on activated carbons with uniform versus broad pore size distributions and can be applied to moisture sorption on starch gels. The adsorption potential is a convenient variable for sorption in its ability to unify sorbate–sorbate fluctuation and the isosteric thermodynamics of sorption.

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“…Using the implicit function theorem, which can be observed in [13,14], (1d) can be differentiated with respect to t:…”

Section: Fourth-order Hessenberg Daes Problem Formulationmentioning

confidence: 99%

Functional Approach for Solving Reduced Order of Index‐Four Hessenberg Differential‐Algebraic Control System

Ghazwa

2022

Journal of Mathematics

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This research investigates differential-algebraic equations with higher index (index four). Specifically, a functional analytic approach is proposed to find the solution of (index four) Hessenberg differential-algebraic equations (DAEs). The approximate solution of the proposed functional approach for a suitable separable Hilbert space is obtained with the help of the direct optimisation technique and Ritz basis functions method. Illustrations have been ranked from an 8 × 8 test system of index-four Hessenberg linear DAEs to a 4 × 4 DAE of rotating masses as well as an 8 × 8 differential-algebraic generator model, where it reformulated index-four linear Hessenberg DAEs. Their approximate solutions were obtained using the present approach with comparisons. The numerical results demonstrate the simplicity of the proposed approach and express suitable accuracy and efficiency.

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Implicit function theorem and Jacobians in solvation and adsorption

Cited by 8 publications

References 54 publications

Ensemble transformation in the fluctuation theory

Ensemble transformation in the fluctuation theory

Temperature Dependence of Sorption

Functional Approach for Solving Reduced Order of Index‐Four Hessenberg Differential‐Algebraic Control System

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