Free Entropy Research Articles

Co‐Pyrolytic Recycling of Bakelite and Polymethyl Methacrylate: Kinetic, Synergistic, and Thermodynamic Analysis

AbstractBakelite, a thermosetting plastic, has limited feasibility for thermal recycling processes such as pyrolysis, unlike thermoplastics, due to its inherent tendency to get charred upon heating and difficulty in being converted into valuable fuel and chemical compounds. In this study, the co‐pyrolysis of bakelite and poly methyl methacrylate is explored to improve the thermal degradation and thus recycling feasibility of bakelite by pyrolysis. The kinetics and thermodynamics of the co‐pyrolysis of blended samples are analyzed using different kinetics models. The thermal degradation experiments of the sample are conducted from ambient to 1000 °C at five various heating rates of 5, 10, 20, 30, and 50 °C/min in an inert N2 gas environment. The results showed that adding poly methyl methacrylate to bakelite changes the pyrolytic degradation temperature and weight loss trend. The kinetic analysis reveals that the thermal degradation at 220–345 °C follows an F2.5 order‐based model with an average activation energy of 181 kJ/mol, while the degradation at 345–475 °C follows an F2 order‐based model with an average activation energy of 318 kJ/mol. The enthalpy changes and free energy change associated with thermal degradation exhibit a positive trend while that of entropy change shows a negative trend. With blending the free energy change and entropy change decreased while enthalpy change increased. During batch pyrolysis at 450 °C, the PMMA‐Bakelite blend generates oil (49.17 %), wax (12.36 %), residue (15.36 %), and gas (23.11 %). According to FTIR and GC‐MS analyses, the pyrolytic oil produced by the co‐pyrolysis of the blended sample at 450 °C comprises 32.23 % saturated hydrocarbons (C9–40), 20.97 % unsaturated hydrocarbons (C6–24), and 46.80 % oxygenated material (C5–19). These findings can assist in the optimization of pyrolysis reactor design for the recycling of thermosetting plastics.

Deciphering the Entropy-Driven Host-Guest Interactions within Single Covalent Organic Frameworks for Trapping of 131I.

By combining dark-field optical microscopy with an in situ thermochemical aqueous solution system, we report a single-particle imaging strategy to investigate the host-guest interactions between covalent organic framework-300 (COF-300) as a representative COF host and a series of linear-chain fatty amines. The thermodynamic parameters, such as dissociation constant, Gibbs free energy changes, enthalpy changes, and entropy changes for the binding events within COF-300 are quantified. Correlation between the hydrophobicity of various amines and other data suggests that the mechanism of the host-guest bindings arises from the entropy-driven noncovalent interactions such as hydrogen bonds and van der Waals forces. These mechanistic insights allow for the rational design and preparation of COF-300-encapsulated n-octylamine with enhanced trapping performance of radioactive 131I-. This study not only provides thermodynamic insights into the host-guest interactions within the COF framework but also establishes a structure-property relationship between fatty amines and energetic magnitude information.

Entropy in catalyst dynamics under confinement.

Entropy during the dynamic structural evolution of catalysts has a non-trivial influence on chemical reactions. Confinement significantly affects the catalyst dynamics and thus impacts the reactivity. However, a full understanding has not been clearly established. To investigate catalyst dynamics under confinement, we utilize the active learning scheme to effectively train machine learning potentials for computing free energies of catalytic reactions. The scheme enables us to compute the reaction free energies and entropies of O2 dissociation on Pt clusters with different sizes confined inside a carbon nanotube (CNT) at the timescale of tens of nanoseconds while keeping ab initio accuracy. We observe an entropic effect owing to liquid-to-solid phase transitions of clusters at finite temperatures. More importantly, the confinement effect enhances the structural dynamics of the cluster and leads to a lower melting temperature than those of the bare cluster and cluster outside the CNT, consequently facilitating the reaction to occur at lower temperatures and preventing the catalyst from forming unfavorable oxides. Our work reveals the important influence of confinement on structural dynamics, providing useful insight into entropy in dynamic catalysis.

Novel tempo oxidized polyvinyl alcohol/ cellulose nanocrystal-based nanocomposite membrane for malachite green dye removal.

In this study, in-situ modification by TEMPO oxidation was performed after nanocomposite synthesis to improve its properties toward dye molecule removal. The unoxidized and oxidized polymeric-based nanocomposite was denoted as PNC6 and PNC6O respectively. The nanocomposites were characterized using FESEM, FTIR, contact angle, XRD and BET analysis. Measurements of swelling ratio and chemical stability were also performed to provide insight into the durability of the nanocomposites. The effects of changing variables included contact duration, pH of aqueous solution, initial pollutant concentration, and temperature were observed. The kinetic study showed that the experimental data is best fitted with pseudo-second-order kinetics (R2 = 0.988 and 0.997 respectively), whereas on observing isotherm data, in both unoxidized and oxidized nanocomposite it fits well with Langmuir isotherm (R2 = 0.951 and 0.993 respectively). In addition, the effects on Gibb's free energy, Enthalpy, and Entropy were measured in terms of thermodynamic characteristics, it was established that dye molecules adsorption mechanism is endothermic and spontaneous in behaviour. To check regeneration tendency of the nanocomposite seven consecutive adsorption desorption cycles were run and about 90% and 80%, regeneration ability could be seen in an unoxidized state (PNC6) and an oxidized state (PNC6O) respectively upto 5th cycle after that the tendency get reduced. This study suggests that this novel polymeric nanocomposite can be employed as an efficient and relatively inexpensive adsorbent for dye removal from aqueous solutions.

Two-state free-volume model for anomalous dynamics in supercooled water macromolecules undergoing liquid–liquid phase transition

Modelling anomalous dynamics of complex liquid water is a huge challenge due to its various condensed molecular structures and the associated liquid–liquid phase transitions (LLPTs). In this study, we considered, for the first time, the influences of free volume and entropy on LLPTs for both low-density liquid (LDL) and high-density liquid (HDL) waters. Firstly, we proposed a two-state free-volume model and combined with the Adam-Gibbs models to investigate the dynamic equilibria with free diffusion coefficient, viscosity, glass transition temperature and the anomalous dynamics in the two-state water. Free-energy equation was then developed to explore the constitutive relationships of free volume, entropy and anomalously dynamic behaviors. Finally, analytical results of the proposed two-state free-volume model were compared with the experimental data of two-state water reported in literature, and good agreements between them were demonstrated. This study offers a new physical insight into free volume for the anomalous dynamics and LLPTs in the two-state water.

Thermo-Magnetic properties of non-relativistic particles under the effect of energy-dependent Hellmann potential

This manuscript presents an analysis of the thermal and magnetic properties of particles moving in two dimensions under the influence of an external magnetic field and Aharanov-Bohm flux, considering an energy-dependent Hellmann potential. The energy eigenvalue is obtained using the asymptotic iteration method. Next, we model the system as a canonical ensemble and derive the partition function to acquire analytical expressions for the Helmholtz free energy, entropy, mean energy, and specific heat. We observe that an increase in the external magnetic field and AB flux generally increases the energy eigenvalues, with a stronger effect at lower flux values, while energy eigenvalues decrease with an increase in the screening parameter. Additionally, Helmholtz free energy and entropy functions are sensitive to temperature, particularly in the low-temperature regime, while specific heat exhibits a Schottky anomaly. We also compute the magnetisation and susceptibility functions, finding that both magnetisation and susceptibility increase with the external magnetic field strength and the AB flux.

Optimization study of methylene blue dye adsorption on Chamaerops humilis fibers biosorption using a central composite design

This research explores the effectiveness of Chamaerops humilis (CH) fibers as a biosorbent for the adsorption of Methylene Blue (MB) dye from aqueous solutions. The fibers were characterized using X-ray diffraction (XRD), Fourier-transform infrared spectroscopy (FTIR). To optimize the dye removal process, a Design of Experiments methodology, specifically Central Composite Design (CCD), was applied. Factors such as solution pH, adsorbent dosage, contact time, and temperature were varied. The CCD analysis identified the optimal conditions for maximum removal efficiency (99.87 %) of a 10 mg/L MB solution, which were: 1.06 g/L of CH fiber dosage, pH 9.6, and a contact time of 60 min. The adsorption process aligned well with a pseudo-second-order kinetic model, and the maximum adsorption capacity of CH fibers was determined to be 9.422 mg/g. Thermodynamic parameters, including Gibbs free energy (ΔG°), enthalpy (ΔH°), and entropy (ΔS°) changes, were calculated, confirming that the adsorption is spontaneous and endothermic. The study highlights the potential of Chamaerops humilis fibers as an efficient and eco-friendly adsorbent for MB dye removal, offering a sustainable and cost-effective solution for wastewater treatment, particularly in industrial effluent management.

Sctudies on adsorption of Brilliant Green from aqueous solution onto nutraceutical industrial pepper seed spent

The study proposed the removal of Brilliant Green, a cationic dye, by adsorption process from wastewater solution utilizing a low-cost adsorbent such as Nutraceutical Industrial Pepper Seed Spent (NIPSS). The study comprises an investigation of the parametric influence on the adsorption process. The parameters identified are pH, dye concentration, process temperature, quantity of the adsorbent, and particle size. The study of statistics found from experiments was carried out by incorporating Freundlich, Brouers-Sotolongo, Langmuir, Toth, Sips, Jovanovic, and Redlich-Peterson isotherm models. The adsorption kinetics were determined by implementing pseudo-first-order and second-order models, diffusion film models, and Dumwald-Wagner and Weber-Morris models. The experimental adsorption capacity qe was found to be about 130 mg/g. This value was closest to the maximum adsorption of 144.6mg/g predicted by the Brouers-Sotolongo isotherm which had a correlation coefficient (R2) of 0.998. The adsorption kinetics data was confirmed to be a pseudo-second-order model. The change in free energy, enthalpy change, and entropy change were vital thermodynamic factors in concluding that adsorption is almost spontaneous and endothermic process. Change in enthalpy (ΔH°) reduced value indicates the physical nature of the process. The adsorption of BG dye on the adsorbent surface was authenticated by FTIR spectroscopy and SEM imaging. A Central Composite Design (CCD) Quadratic model under Response Surface Methodology (RSM) was implemented for statistical optimization of adsorption capacity for the five parameters studied, namely, time, temperature, concentration of the dye, weight of the adsorbent, and pH. Software Design Expert 7.0 was used to evaluate 3D contour plots. The process of optimization yielded a value of 350 mg/g. Thus, incrementing the adsorption process by 84.2 %. The study provides insights on various dye and adsorbent interaction possibilities and derives that NIPSS is an efficient adsorbent to extract BG dye from wastewater solutions.

Estimation of Rheological Coefficients of Acacia nilotica Gum: A Versatile Biopolymer for Biomedical and Food Industries

Introduction: Polysaccharides are widely used in the biomedical and food industries as thickening, gelling, emulsifying, hydrating, and suspending agents. Polysaccharides have adequate viscoelastic properties and flow characteristics. The purpose of this study was to determine various rheological parameters of Acacia nilotica (Babool) gum. Methods: Understanding the influence of temperature on rheological properties is quite important for polymeric materials to be considered as pharmaceutical excipients. Thus, a polymeric solution of purified Babool gum was prepared, and the influence of temperature on its rheological behaviour (viscosity and surface tension) was investigated to develop a better understanding of the structural organization of the gum. Furthermore, viscosity, surface tension, temperature coefficient, activation energy, Gibbs free energy, Reynolds number, and entropy of fusion were calculated using the Arrhenius, Gibbs–Helmholtz, Frenkel–Eyring, and Eotvos equations, respectively. Results: The activation energy of the gum was 3.81 ± 0.18 kJ/mol. Changes in entropy and enthalpy were 0.56 ± 0.23 and 4.27 ± 0.81 kJ/mol, respectively. The calculated amount of entropy of fusion was found to be 0.014 ± 0.01 kJ mol−1 K−1. Conclusion: The study outcomes showed that the viscosity and surface tension increased as the temperature decreased. The good rheological properties of Babool gum make it a suitable excipient for its applications in the food and pharmaceutical industries.

Uptake of selected heavy metals from contaminated waters utilizing cost-effective and environmentally friendly biosorbents prepared from the residues of a traditionally fermented Ethiopian alcoholic beverage (Tella)

In the current study, the adsorption capacity of Tella residues (residues of fermented alcoholic beverage) for quantitative uptake of Cu(II), Cd(II), Zn(II) and Pb(II) was evaluated. Chemical treatment of the local beer residue (LBR) has improved the removal efficiency of the adsorbent, which was achieved at pH = 5, 1.0 g adsorbent, 50 mg/L initial concentration, 180 min contact time and agitation speed of 100 rpm. The adsorption was found to fit the Langmuir adsorption isotherm model, and the theoretical equilibrium capacities were well fitted with the experimental equilibrium capacities, resulting in chemical adsorption (chemisorptions) on the adsorbent surface while the equilibrium kinetics follows the pseudo-second-order. The adsorption capacity (Qo) of LBR decreases in the following order: Zn(II) > Cu(II) > Pb(II) > Cd(II) as metal concentration ranged from 20-200 mg/L. Thermodynamic parameters, including standard free energy (ΔG°), enthalpy (ΔH°) and entropy (ΔS°) were calculated to predict the nature of adsorption. The negative values of ΔG° and the positive value of ΔH° indicate that the adsorption process was spontaneous and endothermic. Adsorption capacities were found to increase when the temperature ranged from 25-60 °C. Thus, the findings suggest a promising application of LBR as an alternative low-cost novel adsorbent for the removal of toxic heavy metals from wastewater.

Experimental measurement, thermodynamic analysis, and mathematical modeling for budesonide solubility in 1-propanol + water mixtures at T = (293.2 to 313.2) K.

Budesonide (BDS) a steroid-based anti-inflammatory drug widely prescribed for various diseases, has a low aqueous solubility. In this study, we investigated cosolvency approach to study the thermodynamic specifications related to the solubility of BDS at the temperature range of 293.2-313.2K in (1-propanol + water) mixtures applying the shaking flask method. The predictive power of different mathematical models for experimental data in the cosolvency systems was evaluated. For this purpose, the linear and nonlinear mathematical equations such as van't Hoff model (as a linear model), Buchowski-Ksiazczak equation (as a non-linear), CNIBS/R-K and MRS models (as a linear model for solvent composition at an isothermal condition), modified Wilson model (as a non-linear model for isothermal condition), the Jouyban-Acree model (as a model that considers temperature and solvent composition), and Jouyban-Acree-van't Hoff model (as a model with no further input data) were studied. Also, the Williams-Amidon excess Gibbs energy model was investigated. In addition, the related apparent thermodynamics of the BDS dissolution process in the desired temperature such as Gibbs free energy, enthalpy, and entropy, were computed by the corresponding equations. Moreover, based on the inverse Kirkwood-Buff integrals, it is demonstrated that BDS is preferentially solvated by water in water-rich mixtures. The accuracy of the fitness was evaluated with mean relative deviations (MRDs%) for back-calculated molar BDS solubility data. The result showed that the maximum solubility of BDS was obtained at 0.7 mass fraction of 1-propanol at all temperatures. Thermodynamic studies demonstrated that BDS dissolution procedures were obtained as endothermic and entropy-driven in almost all cases. The overall MRDs% values for the back-computed BDS solubility in the aqueous mixture of 1-propanol based on van't Hoff model, Buchowski-Ksiazczak equation, CNIBS/R-K model, modified Wilson model, Jouyban-Acree model, Jouyban-Acree-van't Hoff model, MRS model, and Williams-Amidon excess Gibbs energy model were found 1.93%, 1.80%, 11.68%, 33.32%, 12.30%, 9.24%, 10.70%, and 6.57%, respectively.

An elementary proof of the inequality $\chi\leq\chi^{*}$ for conditional free entropy

Through the study of large deviations theory for matrix Brownian motion, Biane, Capitaine, and Guionnet proved the inequality \chi(X)\leq\chi^{*}(X) that relates two analogs of entropy in free probability defined by Voiculescu. We give a new proof of \chi\leq\chi^{*} that is elementary in the sense that it does not rely on stochastic differential equations and large deviations theory. Moreover, we generalize the result to conditional microstates and non-microstates free entropy.

Recent advances in applications of animal biowaste-based activated carbon as biosorbents of water pollutants: a mini-review.

Advances in green engineering and technology have revealed a number of environmentally acceptable alternatives for water purification. In line with this, recent advances in biosorption of pollutants from aqueous solutions using animal biowaste-based activated carbon (AC) are reported herein. Apart from the fish scale-derived AC which is extensively documented, animal bones, among the rest others, have been studied most widely, followed by hair and feathers. Out of the various target water pollutants, removal of heavy metals has been mostly studied. Majority of the reports showed the Freundlich isotherm and pseudo second order as the best fit. Few investigations on the thermodynamics of the adsorption studies and reports on the Gibbs free energy change (ΔG°), enthalpy change (ΔH°), and entropy change (ΔS°) have also been discussed in this report. It has been concluded that while plant-based AC has gained wide interest, the same is not true for the animal-based counterpart albeit the latter's potential for high sorption efficiency as seen in the present report.

Exploring the thermodynamics and dynamics of lithium ions inserted into Li1-xMn2O4 electrode at different temperatures

Efficient lithium extraction via electrochemical adsorption holds great promise for resource utilization and energy storage. However, the mechanism of how temperature influences the electrochemical adsorption process, particularly the thermodynamics and dynamics in the liquid, liquid–solid interface, and solid remains unclear. To address this challenge, we explored the impact of temperature (273.15 to 328.15 K) using Li1-xMn2O4 as an adsorbent. For the liquid phase, the conductivity increased from 0.89 to 0.95 S/m following lnΛ = -0.11(1000/T) + 0.27 in thermodynamics; the solution viscosity decreased from 1.71 × 10-3 to 5.15 × 10-4 Pa·s conforming lnη = 1.96(1000/T) − 13.54 in dynamics. For the liquid–solid interface, the charge transfer resistance (Rct) decreased from 42.03 to 25.17 ohm obeying ln(1/Rct) = -0.82(1000/T) − 0.69 in thermodynamics; the exchange current (i0) showed a positive correlation with temperature in fitting Logi0 = -0.50(1000/T) − 1.40 in dynamics. For the solid phase, the increase in reduction peak voltage (ERP) and the decrease in Gibbs free energy (G), entropy (S), and enthalpy (H) all indicate that the increase of temperature promotes lithium ions to inserted into Li1-xMn2O4 in thermodynamics; the increased diffusion coefficient (DLi+(S)) and higher reduction peak current (IRP) suggest that higher temperature strengthens lithium ions diffusion and insertion in dynamics. In the temperature range, the adsorption capacity gradually increased from 1.15 to 7.37 mg/g, the initial current enhanced from −3.40 × 10-4 to −1.85 × 10-3 A, the terminal current raised from −1.59 × 10-4 to −7.40 × 10-4 A, and the current efficiency improved from 31.28 % to 66.11 % in the temperature range. This work provides a deeper understanding of the impact of temperature on the electrochemical lithium adsorption process from thermodynamics and dynamics. This work also offers crucial guidance for the upgrading of electrochemical lithium adsorption technology and can be applied to other fields such as energy storage, catalysis, and synthesis, providing a valuable reference for optimizing reaction parameter design in thermodynamics and dynamics.

Ab-Initio calculations on physical properties of Dirac semimetal AMgBi (A=K, Rb, Cs)

This study presents a comprehensive first-principles investigation of the structural, mechanical, vibrational, thermodynamic and electronic properties of AMgBi (A = K, Rb, Cs) compounds using Density Functional Theory. Also, the effect of substituting alkali atoms on the physical properties has been discussed. The tetragonal PbClF-type structure has been confirmed by the analysis and the results revealed good agreement between calculated and experimental lattice parameters for KMgBi. The mechanical properties have been investigated for the first time for RbMgBi and CsMgBi. The mechanical stability of the materials in the ground state has been confirmed through the use of obtained elastic constants. Furthermore, derived parameters from elastic constants such as bulk modulus, shear modulus, and Poisson's ratio indicated that the materials are brittle and exhibited anisotropic mechanical behavior due to their layered structure. This study conducts a detailed analysis of phonon modes, explores their connections to thermal and elastic properties, visualizes the movements of phonon modes at the gamma point, determines Born effective charges, and discusses LO/TO splitting, which is for the first time for RbMgBi and CsMgBi. Phonon dispersion calculations confirmed the dynamical stability of the compounds and revealed the presence of phonon band gaps, supporting their quasi-two-dimensional nature. Investigation of thermodynamic properties using the quasi-harmonic approximation has shown the temperature dependence of internal energy, Helmholtz free energy, specific heat, and entropy for the first time for all compounds. The materials exhibited relatively low thermal conductivity, following the order KMgBi > RbMgBi > CsMgBi. The calculated Grüneisen parameter values were found to be 1.42, 1.44, and 1.53 for KMgBi, RbMgBi, and CsMgBi, respectively, suggesting relatively weak anharmonicity within the materials.

Determination of Kinetic Parameters and Thermal Characteristics of Epoxy Casuarina Bio Composites

By reinforcing casuarina equisetifolia leaf, a flexible bio fibre, in different volume proportions of 4%, 8%, 12%, 16%, and 20% into the epoxy resin, a bio-based composite was produced utilizing the hand layup method. FT IR spectrum was used to verify the production of epoxy casuarina composite. The produced composites have undergone an extensive thermal investigation using Differential Thermal Analysis (DTA), Thermogravimetric Analysis (TGA), Derivative Thermogravimetric (DTG) study and Differential Scanning Calorimetry (DSC) analysis. It has been observed that increasing the percentage of casuarina fibre content enhances the thermal stability of epoxy resin. Horowitz- Metzger (HM), Broido and Coats-Redfern (CR) models were worked and the result revealed that the decomposition of casuarina epoxy composite followed one Dimensional Diffusion Model with coefficient of correlation value close to 1. The thermodynamic parameters including Activation energy (Ea), frequency factor (A), Gibbs free energy (ΔG), entropy (ΔS) and enthalpy (ΔH) of epoxy casuarina composites were found to increase with increase in fibre contents. This makes the composite more suitable for making high performance engineering material for various applications.

Investigation on the structural, elastic, electronic, thermodynamic, and vibrational properties of the full heusler Sc2XAl (X =Cd and Zn): An ab initio study

The structural, mechanical, electronic, thermodynamic, and phonon characteristics of Sc2CdAl and Sc2ZnAl in L21 phase were the main focuses on this investigation. The density of states (DOS) and electronic band spectra signify the metallic behavior of the L21 phase of both materials. The impact of atomic arrangement with respect to the Wyckoff sites on the mechanical stability were additionally studied. The elastic constants of Sc2CdAl and Sc2ZnAl alloys indicated that these alloys convened Born mechanical stability criteria. Using the density functional perturbation theory's first-principle linear response method, complete phonon spectra have been acquired. Moreover, the quasi-harmonic approximation, specific heat capacity at constant volume, the internal free energy, entropy, and vibrational free energy variations of Sc2CdAl and Sc2ZnAl as full Heusler alloys have been utilized in examination the change over the temperature between 0 and 800 K. Both alloys might find application in real industrial applications.

Thermodynamic characteristics of multi-component crystals: Experimental evaluation, prediction and relationship with crystal parameters

In this study, we have analyzed the thermodynamic formation functions for four multi-component crystals based on the solubility data in the range of 293.15–313.15 K. The investigation was performed for the cocrystals of fluconazole (FLZ) with 4-hydroxybenzoic (4OHBA) and vanillic (VA) acids, and salts of fenbendazole (FNB) with maleic (Mal) and oxalic (Ox) acids. A database on the thermodynamic parameters of the multi-component molecular crystals (Gibbs free energy, enthalpy, and entropy) obtained in this study and borrowed from the literature was created. These parameters were determined using the experimental data on the solubility of the individual components and the product of the solubilities of the multi-component crystals at 298.15 K. A regression model was proposed to estimate the solubility product of the two-component crystals using the intrinsic solubility of the individual components as an independent variable. A detailed analysis of the database disclosed the relationship between the entropy term and the molecular volume of the multi-component crystal formation.

A novel adsorbent for the removal of chromium form aqueous solution by Acacia Fumosa seed shell activated carbon

In this experimental study, the activated carbon was synthesized by acid impregnation and digestion methods using Acacia Fumosa seed shell. The chromium removal was examined by altering the process conditions like carbon dose, chromium concentration, pH, and temperature and interaction time. The viability of the sorption process was examined using Freundlich and Langmuir equations. Evaluating the isotherm equations, the Langmuir equation shows more concordance compared with Freundlich equation. The feasibility of the adsorption was studied using Lagergren's kinetics equations of pseudo order equations of first and second order equations. Experimental investigation implies that the chromium removal follows pseudo-second order equation and it confines the investigation was subjected to time and concentration of chromium. The temperature effect was checked by chromium with Acacia Fumosa seed shell carbon. The Acacia Fumosa seed shell carbon alone was firmed using Gibbs free energy, enthalpy and entropy equations. The FTIR study of the synthesized activated carbon of before and after spectrum confirms the chromium adsorption. The SEM surface of the superficial layer change of carbon of initial and after chromium loaded state show viable change.

Shannon entropy and thermodynamic properties of an inversely quadratic Yukawa potential model

An inversely quadratic Yukawa potential is an exponential-type potential that has received little attention only in the bound state to the best of our understanding. This study obtained the solution of the Schrödinger equation for an inversely quadratic Yukawa potential via parametric Nikiforov-Uvarov method and supersymmetric approach. Some thermodynamic properties (vibrational enthalpy, vibrational Gibbs free energy and vibrational entropy) for the Inversely quadratic Yukawa potential are examined. The study is also applied to shannon entropy as the theoretic quantity. The results obtained showed that the energy eigenvalue for the potential goes in the opposite direction with the quantum state and the screening parameter. The Shannon entropy obtained as a function of the potential strength only obeyed the Heinsenberg principle at the ground state and excited states. In both cases, the also satisfied Bialynick-Birula, Mycielski inequality. It is noted that the thermal properties studied as a function of the temperature, even though the studied potential is not a molecular potential, the results followed the trend of the molecular potential.