Adsorption Equilibrium Constant Research Articles

Adsorption behavior of methyl palmitate onto silica particle surface

Adsorption behavior of methyl palmitate (MP), as a model substance of biodiesel fuel, onto silica surface in the presence of methanol and toluene was investigated, assuming the dry washing purification of biodiesel production process. Methanol and toluene affected amount absorbed of MP, especially methanol greatly affected amount absorbed of MP. The adsorption model was proposed based on the Langmuir type adsorption and two assumptions for competitive adsorption. The equilibrium adsorption constant, K, and the maximum number of the adsorption sites, XS, were determined by fitting the data to the proposed model. The proposed model agreed well experimental data. The magnitude of K correlated with the dielectric constant of each substance. The occupied surface area of silanol group estimated from XS was mostly the same as the reported values. These adsorption parameters were verified experimentally with other conditions.

Enantioseparation of cloprostenol on the polysaccharide chiral stationary phase: Influence of the mobile phase on enantioselective adsorption

Cloprostenol (CLO) is an important chiral drug widely used in veterinary practice. In high-performance liquid chromatography (HPLC), the acetonitrile/water system is more effective in the chiral resolution of CLO. The change in the ratio of acetonitrile and water has a great influence on the chiral selectivity of CLO. The enantioselective adsorption and nonselective adsorption of CLO on the polysaccharide stationary phase with three groups of mobile phases (Acetonitrile / 0.1 TFA; 95 Acetonitrile/5 H2O/0.1 TFA; 45 Acetonitrile/55 H2O/0.1 TFA) were determined and fitted with the subtraction fitting method (SFM). Adding a small amount of water to the mobile phase mainly reduced the enantioselective equilibrium adsorption constants Ks of (-)-CLO and thus changed the selectivity. Among them, the true separation factor (αtrue) of the 95 Acetonitrile/5 H2O mobile phase was as high as 92.86. Chiral preparation was performed on the basis of this mobile phase. The preparation performance in kkd (kilograms per kilogram of stationary phase purified compound per day) was 0.25 kg racemate/kg CSP/day.

Aza-18-crown-6 modifiye karbon nanotüplerin sentezi, karakterizasyonu ve adsorpsiyon kapasitesinin belirlenmesi

Karbon nanotüpler (CNT) olağanüstü eşsiz özellikleri sebebiyle keşfinden günümüze birçok uygulama alanı bulmuştur. Yüzeylerinde gerçekleştirilen kovalent modifikasyonlarla kimyasal ve fiziksel özellikleri geliştirilebilmektedir. Bu çalışma ile 1-aza-18-crown-6 modifiye karbon nanotüpler Steglich esterleşmesi ile sentezlenmiştir. Sentezlenen bileşik FTIR, 1H-NMR, 13C-NMR, TGA ve TEM yöntemleriyle karakterize edilmiştir. Elde edilen bileşiğin ve modifiye edilmemiş CNT bileşiğinin Fe3+, Mn2+, Co2+, Cr3+, Cu2+, Zn2+ metal iyonlarını adsobsiyonu incelenmiştir. Adsorpsiyon yüzdesi (% A.Y.), adsorpsiyon kapasitesi (qe), maksimum adsorpsiyon kapasitesi (qmax), adsorpsiyon denge sabiti (Kd), Serbest Gibbs Enerjisi (ΔGo) ve izoterm türü belirlenmiştir. Adsorpsiyon çalışmalarına göre, monoaza-18-crown-6 modifiye edilmiş çok duvarlı karbon nanotüp (MWCNT-CO-(MA-18-Crown-6)) bileşiği, tüm metal iyonlarını ham çok duvarlı karbon nanotüpten (MWCNT) çok daha iyi yüzdelerde adsorbe ettiği gözlenmiştir. MWCNT-CO-(MA-18-Crown-6) bileşiği, çözelti içindeki Fe3+ iyonlarının yaklaşık olarak tamamını (% 96.15) adsorbe etmiştir. Bu sonuçlar ışığında elde edilen MWCNT-CO-(MA-18-Crown-6) bileşiği metal iyonlarının sulu ortamlardan adsorpsiyonunda başarılı bir şekilde uygulanabileceği söylenebilir.

Thermodynamic and Adsorption Study of the Corrosion Inhibition of Mild Steel by Aframomum chrysanthum Extract in 0.1 M Hydrochloric Acid Solution

The study reports the corrosion inhibition activity of methanol extract of Aframomum chrysanthum on mild steel in 0.1 M HCl, using gravimetry analysis. The weight loss of the mild steels was observed to increase with increasing immersion time. The inhibition efficiency (%IE) was also observed to have increased with increasing concentrations of the inhibitor but decreases with increasing immersion time. The effect of temperature change on the inhibition efficiency was also studied and it was observed that for every increase in temperature there was a corresponding increase in weight loss and decreased in the %IE. The highest values of %IE; 46.66, 56.66, 60.0, 80.0 & 93.33 was observed at temperature 303 K for 0.2, 0.4, 0.6, 0.8 & 1.0 g/L respectively. Activation energy (Ea) values and the enthalpy values reviews that the adsorption process followed a physisorption’s mechanism. Change in enthalpy (ΔH) and entropy change (ΔS) of the reaction was positive indicating the endothermic nature and the spontaneity of the reaction. Three adsorption isotherms were tried on the inhibition process and only the Temkin isotherm gave the best fit with R2 value of 0.903, describing the best adsorption mechanism. The adsorption equilibrium constants K­ads were positive, indicating the feasibility of the adsorption of the inhibitor to the metal surface. Gibb’s Free Energy change of adsorption, ΔGads are negative indicating that the adsorption of the extract of Aframomum chrysanthum on mild steel surface is spontaneous. The values of ΔGads shows physisorption mechanism. All confirming that Aframomumm chrysanthum extract is a good corrosion inhibitor on mild steel in 0.1 M HCl.

Adsorption of Cetylpyridinium Chloride at Silica Nanoparticle/Water Interfaces (I): Dependence of Adsorption Equilibrium on Particle Size.

In the current work, a size-effect model was developed to describe the particle size-dependence of adsorption at solid/liquid interfaces. A parameter, ΔQad, was introduced, defined as the change of the product of the solid/liquid interfacial tension and the molar volume of solid surface components caused by adsorption. The model predicts that with a decrease in particle radius (r), the saturation adsorption amount per unit area (Γm, mol/m2) decreases, while the change of the adsorption equilibrium constant (Kad) is determined by the ΔQad, namely, it decreases if ΔQad > 0 but increases if ΔQad < 0. There exists a critical r at which the saturation adsorption amount per unit mass (Γmg, mol/g) attains a maximum. In addition, the adsorption of cetylpyridinium chloride (CPyCl), a cationic surfactant, on silica nanoparticles with different r (ca. 6-61 nm) values was determined at 298 K and pH 9, showing an obvious size-dependence. With a decrease in r, Kad and Γm decrease, indicating a decrease in the affinity of silica particles toward CPyCl. The size-dependent adsorption data can be well described using our model. Adsorption can affect the molar volume of the solid surface phase, which plays an important role in the size-dependence of adsorption. This work provides a better understanding of the size-dependent adsorption phenomenon at solid/liquid interfaces.

Insight into Organic Pollutant Adsorption Characteristics on a g-C3N4 Surface by Attenuated Total Reflection Spectroscopy and Molecular Dynamics Simulation

Herein the adsorption characteristics of zwitterionic dye pollutant Rhodamine B (Rh+B-) on a g-C3N4 surface were investigated by both an attenuated total reflection spectroscopy (ATRS) experiment and a molecular dynamics simulation (MDS). For experimental investigation, g-C3N4 was coated on a silica optical fiber (SOF) surface to fabricate an adsorption film. According to the ATRS response, adsorption thermodynamics and thermodynamics results were in situ obtained and evaluated. The isothermal Langmuir model was used to calculate the adsorption equilibrium constants (Kads) and adsorption energies (ΔGads) for Rh+B- as 27.25 × 104 M-1 and -31.01 kJ mol-1, respectively, which indicated the spontaneous adsorption behavior of Rh+B- at the g-C3N4 surface. Using dynamic Elovich modeling, the rate constants of Rh+B- were found to be k1 = 0.0063 min-1 and k2 = 0.0004 min-1, which indicated two-stage adsorption at the g-C3N4 surface. For theoretical simulation, adsorption configurations and adsorption energies were systematically calculated by a molecular dynamics simulation (MDS) . Rh+B- molecules were inclined to orient in a parallel position at the g-C3N4 surface during low concentration but a perpendicular position at the g-C3N4 surface during high concentration. Combined with experimental and calculation results, this work revealed the microscopic adsorption performance and elucidated the intermolecular interaction between localized interfaces of g-C3N4 and hazardous dye pollutant. We propose an adsorption model to explain the process of surface interaction, which is based on molecular orientation and a force-driven mechanism. Electrostatic attraction and π-π interaction dominated the adsorption interaction with an adsorption energy of ΔGlow(ads) = -38.96 kJ mol-1 for low Rh+B- concentration, and electrostatic attraction dominated the adsorption interaction with an adsorption energy of ΔGhigh(ads) = -25.76 kJ mol-1 for high Rh+B- concentration. This work can provide a fundamental basis for a dye-pollutants removal application by g-C3N4 in both adsorption and photocatalyzation.

Mechanism of coke formation and corresponding gas fraction characteristics in biochar-catalyzed tar reforming during Corn Straw Pyrolysis

Formation of coke deposition is the main problem that restricts the development of tar reforming and even biomass thermal conversion. Biochar, prepared from gasification at 800 °C, was used for the catalytic reforming of biomass pyrolysis tar at 650 °C, monitoring the syngas formation during coke deposition. The principal consequences and conclusions are as follows: the coke yield increases from 10.9% to 18.9%, and the tar removal efficiency decreases from 89.0% to 56.7% when the feeding time of biomass is extended (10–50 min). The number of O-containing groups on the biochar surface decreases, and the proportion of minor aromatic rings increases. Biochar is more efficient in the catalytic removal of aliphatic components from tar than aromatic components. Catalytic reforming leads to the development of tar in the direction of increased aromatization. The addition of biochar has essentially no effect on CO yield and CO2 yield, while the CH4 yield decreases and the H2 yield increases. In addition to tar reforming, CH4 cracking is one of the pathways for coke generation on the biochar surface. The number of physical adsorption sites (nN2) and adsorption equilibrium constant (b) of biochar are obtained based on the N2 adsorption test.

Electrostatic properties of C–S–H and C-A-S-H for predicting calcium and chloride adsorption

The adsorption capacity of cement hydrates considerably affects the ionic ingress into cementitious materials. In this study, the surface electrostatic properties of calcium silicate hydrate (C–S–H) and calcium aluminosilicate hydrate (C-A-S-H) were determined to understand the effects of the properties on calcium and chloride adsorption. The density of the surface functional groups was determined by analysing the structure of C–S–H and C-A-S-H through 27Al and 29Si MAS NMR. The surface sites of ≡SiOH and ≡AlOH are available in C-A-S-H whereas C–S–H has ≡SiOH groups for ionic adsorption. We found that the incorporation of aluminium decreases the number of total adsorption sites in C-A-S-H. Furthermore, the site density increased with Ca/(Si + Al). To understand the C-A-S-H/solution interface, a triple-layer surface complexation model was developed and the associated equilibrium constants for deprotonation, calcium, and chloride adsorption were determined by fitting the experimental data of potentiometric titration and zeta potential measurement results. The estimated surface complexation modelling parameters were verified by predicting the experimental data of calcium and chloride adsorption on C–S–H and C-A-S-H.

Bio-oils/FCC co-processing: Insights into the adsorption of guaiacol on Y zeolites with distinct acidity and textural properties

The guaiacol adsorption capacity of several Y zeolites with different physicochemical properties was tested by performing breakthrough adsorption experiments, in order to investigate the guaiacol adsorption on Fluid Catalytic Cracking catalysts during bio-oils/FCC feedstocks co-processing. X-Ray diffraction, nitrogen sorption measurements and pyridine adsorption followed by Infrared Spectroscopy were used to determine the framework Si/Al ratio, the textural parameters, and finally the nature and amount of acid sites. Klinkenberg model was used to fit the experimental data and to obtain the guaiacol adsorption capacity and ka (overall coefficient of mass transfer) and Ke (adsorption equilibrium constant) parameters, which are directly related to, respectively, kinetic and thermodynamic aspects. Ke values, and so guaiacol adsorption, were observed to increase with the total number of Brønsted and Lewis acid sites on the zeolites, as well as with the amount of Na exchange. Conversely, mesoporosity increases the diffusion rate of guaiacol inside the zeolite structure, leading to higher ka values and decreasing guaiacol retention. Overall, data show that guaiacol adsorption on the FCC catalysts, and so its impact on the activity, might decrease with the catalyst age, owing to the changes on the FCC catalyst properties taking place during operation.

Numerical Investigation of Process Enhancement Using a Bifunctional Catalyst in a Dual Fluidized-Bed Reactor

The paper outlines the concept of process intensification and integration, with a particular focus on sorption-enhanced, solid-catalyzed chemical processes. An alternative and attractive solution to a system of parallel fixed-bed apparatuses is evaluated, which utilizes the solids’ circulation in a dual fluidized-bed reactor–regenerator system. This allows for continuous mode operation and greatly simplifies the control procedures. To illustrate some aspects related to the steady-state operation of such a dual system, a simplified mathematical model of two interconnected fluidized beds operating in the bubbling regime was developed. A generic reversible chemical reaction of the overall second-order, catalyzed by bifunctional pellets, integrating catalytic active sites and adsorption sites, was considered as a test case. The model was used to study the effects of the bed hydrodynamics, as well as of the chemical reaction and physical adsorption equilibrium constants. It was shown how the superposition of various chemical, physical and hydrodynamical phenomena affects the performance of the system.

Atomically Dispersed Pt on TiO2 Nanosheets for Catalytic Gaseous Acetaldehyde Abatement

Increasing volatile organic compounds (VOCs) in the atmosphere have triggered intensive research on photocatalytic oxidation technology to safely photodegrade these volatile organic pollutants. Downsizing metal catalysts up to an atomic level on oxide supports can provide more active sites on the surface for different reactions, i.e., CO oxidation, CO2 reduction, removal of organic pollutants as well as decrease the use of precious metals. Here, we designed atomically dispersed metal catalysts (ADMCs), i.e., Pt on TiO2 nanosheets, which produced a robust and stable photocatalytic degradation of the gas-phase acetaldehyde. The loading of Pt was carried out using different deposition times, such as 2, 4, and 6 h, designated as Pt/T xh at 70 °C. This was followed by vacuum drying (60 °C) and subsequent calcination at 125 °C in an inert atmosphere. The high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) analysis confirmed the presence of atomically dispersed Pt on TiO2 nanosheets. The sample Pt/T 4h displayed a high photodegradation efficiency (100%) than pure TiO2 (38%) for 500 ppm of acetaldehyde with a flow rate of 10 sccm. Also, 700 ppm of CO2 was produced by Pt/T 4h in contrast to 340 ppm by pure TiO2. The experimental data for the sample Pt/T 4h was further characterized using Langmuir–Hinshelwood kinetic models to determine the reaction rates, adsorption equilibrium constants, and water adsorption constant. This study provides a unique opportunity to develop metal-decorated TiO2-based photocatalysts for the robust and efficient removal and mineralization of acetaldehyde from the indoor atmosphere.

Enzyme- and acid-extracted sugar beet pectin as green corrosion inhibitors for mild steel in hydrochloric acid solution

Pectin extracted from sugar beet pulp using enzymatic or acid hydrolysis was evaluated as corrosion inhibitor for mild steel under acidic conditions. Effect of pectin concentration on corrosion rate and efficiency was studied via weight loss and electrochemical measurements. The effect of temperature on corrosion inhibition was studied in temperature range from 298 to 333 K. The thermodynamic parameters of corrosion process such as activation energy, equilibrium constant of adsorption (Kads), and standard free energy of adsorption (ΔGads) were also calculated. Scanning electron microscopy (SEM) coupled with Energy Dispersive X-Ray analysis (EDX) was used to examine the surface of mild steel after corrosion test. The results showed dependence of corrosion efficiency on method of pectin extraction. Enzymatic-extracted pectin showed moderately higher inhibition efficiency than the acid-extracted pectin. Polarization curves indicated that both types of pectin acted as mixed-type inhibitors with predominant inhibition at the cathodic site.

Kinetics simulation of propylene epoxidation over different Ti species in TS‐1

AbstractA thorough experimental investigation on the kinetic behavior of liquid‐phase propylene epoxidation over TS‐1 and tetrapropylammonium hydroxide (TPAOH) treated TS‐1 catalysts was conducted in a fixed‐bed reactor. The amounts of different coordinated Ti species in the catalysts were quantified by spectroscopies, and their catalytic performances of the epoxidation and alcoholysis of propylene oxide were measured by kinetic modeling. The study shows that the TPAOH treatment converted some of the tetrahedrally coordinated Ti to octahedrally coordinated Ti, and both species were active for the epoxidation and alcoholysis. The superior catalytic performance observed over the TPAOH treated TS‐1 is due to two factors, the increased percentage of active sites, and reduced energy barrier for epoxidation on the octahedrally coordinated Ti. In addition, as the H2O2 conversion increases, the adsorption equilibrium constant of propylene oxide plays a more decisive role than the activation energy for the selectivity of propylene glycol monomethyl ethers.

Mechanistic investigation in ultrasound‐assisted interesterification using non‐edible oil blends and heterogeneous catalyst

AbstractThis study explores the mechanistic issues of ultrasound‐assisted interesterification through identification of links between physics and chemistry of the process. Notably, the interesterification reactions have been carried out using blended feedstock of non‐edible oils—keeping in view requirements for large‐scale processes. The reaction system consisted of blended non‐edible oils, methyl acetate and a solid (Cu2O) catalyst. Initial optimization of the process parameters was done using statistical design. To get insight into physical mechanism of interesterification process, reactants/products profiles in control (mechanical agitation) and test (sonication) experiments were analysed using the Langmuir–Hinshelwood–Hougen–Watson type kinetic model. The relative adsorption equilibrium constants required (to fit the kinetic model) were obtained using binary composite isotherm coupled with the non‐random two‐liquid (NRTL) model for activity coefficients. Kinetic constants of interesterification showed two times rise with sonication, while the adsorption equilibrium constants reduced slightly. Activation energy of reaction reduced from 89.04 to 66.97 kJ/mol, which is attributed to reduction in mass transfer barriers due to intense microconvection generated by sonication.

Intensification the Features of Interaction Between Components of Pollutants of Industrial Waste Waters with Modified Zeolites Based on the Results of Integrated Physical and Chemical Researches

The results of a study on the adsorption of oil products and heavy metal ions by modified zeolites from industrial wastewater are presented. As a modifier of zeolites of the Holinsky deposit, a sulfur polymer is proposed that is obtained from waste during the production of epichlorohydrin, the main components of which is 1,2,3-trichloropropane. The mechanisms of adsorption and the results of investigations by the methods of infra-red pectroscopy and energy differential X-ray spectroscopy of the initial samples of materials are described in detail. On the Irspectres new absorption bands are considered. These bands reflect the nature of fixation of modifier on the surface of sorbent. Also the results of original investigations of the thermodynamic laws of sorption extraction of Ni2+, Zn2+, Cu2+ from wastewater are presented. The limiting parameters of the adsorption process are described by the Langmuir equation. The graphical method the limiting value of adsorption, the adsorption equilibrium constant, the differential heat of adsorption, the adsorption rate constant, the activation energy, the Gibbs energy was founded. It is proved that the investigated processes obey the laws of physical adsorption and are spontaneous.

Adsorptive removal of methylene blue (MB) and malachite green (MG) dyes from aqueous solutions using graphene oxide (GO)

Abstract Graphene oxide (GO) synthesized via modified Hummers method was studied for adsorption of methylene blue (MB) and malachite green (MG) dyes from aqueous solutions. It is hypothesized that electrostatic interactions between dye molecules and surface of GO will facilitate charge carrier movements and degrade the dye in an efficient way. The as synthesized GO was characterized using various characterization techniques such as XRD, Raman, FTIR, UV–Vis, SEM and EDAX. The experimental results suggest that dye removal percentage will increase with increase in adsorbent dosage, time as well as solution pH and the process was exothermic in nature. The adsorption data at 293 K could be fitted by Langmuir equation with a maximum adsorption amount of 119.04, 102.4 mg/g and Langmuir adsorption equilibrium constant of 1.58, 0.867 L/mg for MB and MG dyes, respectively. The outcomes of present article will help not only to understand the adsorption characteristics of GO on MB and MG dyes but also paves path towards development of highly oxidized GO surface for degradation of complex dyes.

Possibilities of the Molecular Modeling of Kinetic Processes under Supercritical Conditions

Current possibilities of modeling the kinetics of supercritical processes are considered, based on the theory of an absolute rate of the reaction for non-ideal reaction systems, which considers intermolecular interactions that change the effective energy of activation of elementary stages. This allows the theory to describe the rates of elementary stages for arbitrary temperatures and densities of the reagent in different phases. Application of this theory in a wide range of state parameters (pressure and temperature) is examined while calculating elementary bimolecular reactions and dissipative coefficients under supercritical conditions. Generalized dependences within the law of corresponding states are calculated for the compressibility, viscosity, and thermal conductivity coefficients of pure substances and those of the compressibility, self- and mutual diffusion, and viscosity of binary mixtures. The effect density and temperature have on the rates of elementary stages under supercritical conditions is demonstrated for a reaction’s effective energies of activation, diffusion and share coefficients, and equilibrium constants of adsorption. Differences between models with effective parameters and the prospects for developing them by allowing for differences in size and contributions from the vibrational motions of components are described, along with ways of improving the accuracy of describing correlation effects.

Kinetics of H2 Adsorption at the Metal-Support Interface of Au/TiO2 Catalysts Probed by Broad Background IR Absorbance.

H2 adsorption on Au catalysts is weak and reversible, making it difficult to quantitatively study. We demonstrate H2 adsorption on Au/TiO2 catalysts results in electron transfer to the support, inducing shifts in the FTIR background. This broad background absorbance (BBA) signal is used to quantify H2 adsorption; adsorption equilibrium constants are comparable to volumetric adsorption measurements. H2 adsorption kinetics measured with the BBA show a lower Eapp value (23 kJ mol-1 ) for H2 adsorption than previously reported from proxy H/D exchange (33 kJ mol-1 ). We also identify a previously unreported H-O-H bending vibration associated with proton adsorption on electronically distinct Ti-OH metal-support interface sites, providing new insight into the nature and dynamics of H2 adsorption at the Au/TiO2 interface.

Kinetics of H2 Adsorption at the Metal–Support Interface of Au/TiO2 Catalysts Probed by Broad Background IR Absorbance

AbstractH2 adsorption on Au catalysts is weak and reversible, making it difficult to quantitatively study. We demonstrate H2 adsorption on Au/TiO2 catalysts results in electron transfer to the support, inducing shifts in the FTIR background. This broad background absorbance (BBA) signal is used to quantify H2 adsorption; adsorption equilibrium constants are comparable to volumetric adsorption measurements. H2 adsorption kinetics measured with the BBA show a lower Eapp value (23 kJ mol−1) for H2 adsorption than previously reported from proxy H/D exchange (33 kJ mol−1). We also identify a previously unreported H‐O‐H bending vibration associated with proton adsorption on electronically distinct Ti‐OH metal‐support interface sites, providing new insight into the nature and dynamics of H2 adsorption at the Au/TiO2 interface.

Effects of Molecular Properties on Adsorption of Six-Carbon VOCs by Activated Carbon in a Fixed Adsorber.

Gravimetric adsorption equipment with a microbalance was used to measure the adsorption of volatile organic compounds (VOCs) by activated carbon from 288 to 313 K. VOCs [n-hexane, cyclohexane, 1-hexene, 2-methylpentane, 3-methylpentane, 2,2-dimethylbutane, acetone, butanone, and 2-pentanone (Pentan-2-one)] were used as adsorbates in the adsorption system. Considering the geometric barrier, the critical diameter, and the boiling point, the adsorption capacities for six-carbon (C6) alkane isomers decrease in the order of n-hexane, 3-methylpentane, and 2-methylpentane. The adsorbates, including nonpolar or weakly polar substances, and substances with smaller geometric obstacles and smaller molecular weights, were more easily adsorbed by the activated carbon. However, the dipole–dipole interactive force at higher pressures resulted in a higher adsorption capacity for 1-hexene than for n-hexane. Both polarity and molecular size should be considered in the analysis of the adsorption of ketones by activated carbon. The adsorption equilibrium constants decreased with increases in temperature because a higher temperature was unfavorable for adsorption. The results for the Toth adsorption isotherm model fitted by the adsorption data showed that the experimental data and the Toth adsorption isotherm model were consistent with each other, as evidenced by the low deviation between the experimental data and those from the fitted model.