processes
Article Cyclic Sequential Removal of Alizarin Red S Dye and Cr(VI) Ions Using Wool as a Low-Cost Adsorbent
Mustafa I. Khamis 1, Taleb H. Ibrahim 2,*, Fawwaz H. Jumean 1, Ziad A. Sara 1 and Baraa A. Atallah 1
1 Department of Biology, Chemistry and Environmental Sciences, American University of Sharjah, Sharjah 26666, UAE; [email protected] (M.I.K.); [email protected] (F.H.J.); [email protected] (Z.A.S.); [email protected] (B.A.A.) 2 Department of Chemical Engineering, American University of Sharjah, Sharjah 26666, UAE * Correspondence: [email protected]; Tel.: +971-507769239
Received: 22 April 2020; Accepted: 5 May 2020; Published: 9 May 2020
Abstract: Alizarin red S (ARS) removal from wastewater using sheep wool as adsorbent was investigated. The influence of contact time, pH, adsorbent dosage, initial ARS concentration and temperature was studied. Optimum values were: pH = 2.0, contact time = 90 min, adsorbent dosage = 8.0 g/L. Removal of ARS under these conditions was 93.2%. Adsorption data at 25.0 ◦C and 90 min contact time were fitted to the Freundlich and Langmuir isotherms. R2 values were 0.9943 and 0.9662, respectively. Raising the temperature to 50.0 ◦C had no effect on ARS removal. Free wool and wool loaded with ARS were characterized by Fourier Transform Infrared Spectroscopy (FTIR). ARS loaded wool was used as adsorbent for removal of Cr(VI) from industrial wastewater. ARS adsorbed on wool underwent oxidation, accompanied by a simultaneous reduction of Cr(VI) to Cr(III). The results hold promise for wool as adsorbent of organic pollutants from wastewater, in addition to substantial self-regeneration through reduction of toxic Cr(VI) to Cr(III). Sequential batch reactor studies involving three cycles showed no significant decline in removal efficiencies of both chromium and ARS.
Keywords: Alizarin Red S; wool; adsorption isotherms; dyes; chromium; cyclic process
1. Introduction Large amounts of dye-contaminated wastewaters are released yearly from leather, cosmetics, as well as the pharmaceutical, plastics and textile industries. Most of these dyes are skin irritants, mutagenic or carcinogenic [1,2]. Dye-polluted water decreases photosynthesis since light penetration is inhibited [3]. These dyes have complex aromatic structures, which give them thermal, optical and physicochemical stability, and thus, they could not be easily biodegraded by natural substances [4–6]. Alizarin red S (ARS) (Figure1), or 1,2-dihydroxy-9,10-anthra-quinonesulfonic acid sodium salt, is a water soluble anthraquinone dye originally derived from the root of the madder plant [7]. It has been extensively employed since ancient times in dyeing textiles [8]. It is a strong oxidizing agent, and hence, must be stored away from moisture and heat [9].
Processes 2020, 8, 556; doi:10.3390/pr8050556 www.mdpi.com/journal/processes Processes 2019, 7, x FOR PEER REVIEW 2 of 14
solid phase, thereby decreasing effluent volume. Native mustard husk has been used in ARS removal from aqueous solutions, with thermodynamic studies indicating that the process is spontaneous and endothermic [14]. Pentaerythritol modified multi-walled carbon nanotubes (ox-MWCNT-PER) has been shown to be highly efficient in ARS removal [15]. Removal of ARS from wastewater using alumina as adsorbent has been investigated at optimum conditions [16]. Moreover, activated carbon Processesis a good2020 adsorbent, 8, 556 for removal of ARS from wastewater, as indicated by favorable thermodynamic2 of 15 parameters [17].
Figure 1. AlizarinAlizarin red S (ARS).(ARS).
AHeavy wide metals, range ofincluding physical chromium, and chemical are widely treatment distributed technologies in the haveenvironment been investigated as a result forof removingnumerous dyes industrial from wastewater. applications. They These include include coagulation, electroplating, precipitation, chromate adsorption, manufacture, membrane wood filtration,preservation, electrochemical galvanization, techniques steel industry, and ozonation paint, textile [10]. production, Among these, ox adsorptionidative dyeing, is the cooling most widely water employed,towers, leather due tanning, to its high corrosion efficiency, inhibitors non-toxicity, and batteries readily [18]. available As a result, adsorbents, heavy metals low cost,are foun eased ofin recoverymany industrial and environmental wastewaters sustainability[19–21]. Chromium [11–13 is]. a Applicability metallic element largely listed depends by the on Environmental the cost and eProtectionfficiency of Agency adsorbent. (EPA) Currently, as one of various 129 priority potential pollutants adsorbents [22,23 are]. It used is found for removal in the air, of specificwater and organic soil, compoundsand occurs in from several wastewater. oxidation The states adsorption ranging of from dyes Cr(II) transfers to Cr(VI), them fromwith waterthe trivalent effluent and to solidhexaval phase,ent therebystates being decreasing the most effl uentstable volume. and common Native mustard[1-3]. In husknatural has waters, been used chro inmium ARS removal is present from in aqueous several solutions,forms, the with most thermodynamic common of which studies are indicatingCr(0), Cr(III), that and the processCr(VI). Cr(III) is spontaneous has very andlow endothermic solubility and [14 is]. Pentaerythritolrelatively stable, modified whereas multi-walled Cr(VI) is environmentally carbon nanotubes mobile (ox-MWCNT-PER) and highly toxic. has Exposure been shown to Cr(VI) to be highlycauses e variousfficient in health ARS problemremovals [,15 including]. Removal skin of ARS and fromstomach wastewater irritation, using dermatitis, alumina liver as adsorbent damage, haskidney been circulation investigated and at nerve optimum and conditionstissue damage [16]. [24 Moreover,–26]. Numerous activated technologies carbon is a goodhave adsorbentalready been for removalapplied in of the ARS removal from wastewater, of Cr(VI) from as indicated aqueous bysolutions. favorable These thermodynamic include adsorption, parameters biosorpti [17].on, ion- exchange,Heavy foam metals, flotation, including electrolysis, chromium, surface are widely adsorption distributed precipitation, in the environment reverse osmosis, as a result sand offiltration, numerous chemical industrial reduction/oxidation, applications. These electrochemical include electroplating, precipitation, chromate membrane manufacture, filtration wood and preservation,solvent extraction galvanization, [27–30]. steel industry, paint, textile production, oxidative dyeing, cooling water towers,In previous leather tanning, studies corrosion by our group, inhibitors sheep and wool batteries has been [18 ].found As a to result, be efficient heavy metals in Cr(VI) are removal, found in manywith subsequent industrial wastewatersreduction to Cr(III) [19–21 ].[31,32 Chromium]. This study is a metallic reports on element the results listed of by using the Environmentalsheep wool for Protectionthe removal Agency of ARS (EPA) from aswastewater one of 129 and priority the possibility pollutants of [22 regenerating,23]. It is found the incontamin the air,ated water wool. and soil,The andoutcome occurs of in the several research oxidation could states be utilized ranging in from a sequential Cr(II) to Cr(VI), batch with reactor the trivalentfor removal and ofhexavalent the two statespollutants being from the wastewaters. most stable and common [1–3]. In natural waters, chromium is present in several forms, the most common of which are Cr(0), Cr(III), and Cr(VI). Cr(III) has very low solubility and is2. relativelyMaterials stable,and Methods whereas Cr(VI) is environmentally mobile and highly toxic. Exposure to Cr(VI) causes various health problems, including skin and stomach irritation, dermatitis, liver damage, kidney circulation2.1. Materials and nerve and tissue damage [24–26]. Numerous technologies have already been applied in the removal of Cr(VI) from aqueous solutions. These include adsorption, biosorption, ion-exchange, All chemicals were of analytical grade and used without further purification. Solutions were foam flotation, electrolysis, surface adsorption precipitation, reverse osmosis, sand filtration, chemical prepared using distilled deionized water (DDW) and their concentrations determined reduction/oxidation, electrochemical precipitation, membrane filtration and solvent extraction [27–30]. spectrophotometrically. ARS was from BDH (Turkey). Potassium dichromate was from Riedel De- In previous studies by our group, sheep wool has been found to be efficient in Cr(VI) removal, Haen (Germany). Acetone and 5-di-phenyl carbazide were from Sigma Aldrich (USA). Sheep wool with subsequent reduction to Cr(III) [31,32]. This study reports on the results of using sheep wool (Sharjah animal market) was trimmed and riffled, then washed with water and detergent for two for the removal of ARS from wastewater and the possibility of regenerating the contaminated wool. days. The outcome of the research could be utilized in a sequential batch reactor for removal of the two pollutants from wastewaters. Processes 2020, 8, 556 3 of 15
2. Materials and Methods
2.1. Materials All chemicals were of analytical grade and used without further purification. Solutions were prepared using distilled deionized water (DDW) and their concentrations determined spectrophotometrically. ARS was from BDH (Turkey). Potassium dichromate was from Riedel De-Haen (Germany). Acetone and 5-di-phenyl carbazide were from Sigma Aldrich (USA). Sheep wool (Sharjah animal market) was trimmed and riffled, then washed with water and detergent for two days.
2.2. Instrumentation ARS and Cr(VI) concentrations were determined spectrophotometrically using Cary 50 (Varian, Australia). Total chromium was determined using Spectra AA220FS (Varian, Australia). Samples were shaken using Edmund Buhler shaker (KS-15/TH-15, Bodelshausen, Germany) at 25.0 0.1 C. pH was ± ◦ measured on a Thermo-Orion 210A + pH meter (USA) equipped with a combined glass electrode. IR spectra were obtained on a Spectrum One FTIR (Perkin Elmer, Waltham, MA, USA). ARS oxidation byproducts were detected using HPLC (Shimadzu, LC-2040C, Kyoto, Japan).
2.3. Methods
2.3.1. Reagent Preparation A 1000 mg/L stock ARS solution was prepared at pH 2.0 and used to prepare standard solutions in the range 1–20 mg/L. Stock solutions containing 1000 mg/L Cr(VI) were prepared by dissolving potassium dichromate in DDW. Stock ligand solutions were prepared by mixing 0.20 mL of 0.05% of 1,5 diphenyl carbazide (in acetone) with 2 drops of 6.0 M sulfuric acid and 0.10 mL of sample. Standard Cr(VI) solutions were in the range 1–10 mg/L.
2.3.2. Determination of Cr ions Concentration Total Cr concentration was determined by atomic absorption spectroscopy (AAS). The 1,5-diphenyl carbazide method could not be used in the determination of Cr(VI) alone, because Cr(VI) removal by wool-ARS complex was accompanied by formation of ARS oxidation by-products. ARS degradation by oxidation involves the cleavage of dye-chromophore components [33,34]. FTIR analysis of residual 1 products from dye oxidation gave IR bands at 1717, 1623, 1387, 1105 and 1045 cm− , attributed to 2 >C=O (carbonyl), >C=C< (alkenes),-C-C-C (alkanes), SO4 − and –C-O-C- groups, respectively. These by-products interfere with analysis of Cr(VI) by 1,5-diphenyl carbazide. Hence, only total Cr could be analyzed by AAS. The highest removal of Cr(VI) was 93%, as detected by AAS. However, previous studies by our group revealed that, at this short term contact, slight amounts of Cr(III) were released into solution [31]. This process causes incomplete removal of Cr(VI).
2.3.3. Adsorption Studies Batch adsorption studies were carried out in flasks containing 50 mL of test solutions at the desired initial ARS concentrations. λmax for dye solutions was 261 nm. Batch adsorption studies were then carried out. Wool loaded with ARS was placed in flasks containing the desired initial Cr(VI) concentration and the contents shaken. Shaking conditions were: 1.5 h, 25.0 ◦C and 175 rpm. AAS was used to measure total Cr concentrations. Oxidation by-products of ARS by Cr(VI) were detected using HPLC under the following conditions: mobile phase 40:60% methanol:phosphate buffer, Pinnacle D8 C8 5 µm (RESTEK, Bellefonte, PA, USA), analytical wavelength 540 nm and 1.5 mL/min flow rate.
2.3.4. Regeneration In-Place Studies Wool loaded with ARS was prepared by shaking wool with 100 mg/L ARS solution at optimum conditions. The resulting wool-ARS was washed with DDW and dried. The remaining ARS solution Processes 2020, 8, 556 4 of 15 was analyzed spectrophotometrically. ARS removal was 88%. Wool-ARS was shaken with 50 mg/L Cr(VI) solution at optimum conditions for 4 days. Subsequently, the wool-ARS-Cr was collected, washed with DDW and dried. The remaining Cr(VI) solution was analyzed by AAS, giving a removal efficiency of 86.8%. The cycle was repeated three times.
3. Results and Discussion
3.1. Adsorption Studies
ProcessesRemoval 2019, 7, x of FOR ARS PEER was REVIEW calculated using 4 of 14
% Removal = ((Co Ce)/Co) 100 (1) % Removal = ((Co−-Ce)/Co)*100∗ (1) where Co and Ce areare initial initial and equilibrium equilibrium ARS concentrations (mg/L), (mg/L), respectively.
3.1.1. Optimization Studies on ARS Removal by Wool Effect of Contact Time Effect of Contact Time Figure2 shows the e ffect of contact time on ARS removal. At all three pH values used, removal Figure 2 shows the effect of contact time on ARS removal. At all three pH values used, removal increases with increasing contact time until a maximum is reached at ca. 90 min, selected as the increases with increasing contact time until a maximum is reached at ca. 90 min, selected as the optimum time. optimum time.
100 90 80 70 60 50 40 30 % Removal % ARS of Removal 20 10 0 0 50 100 150 200 Time (min)
pH 2.0 pH 3.2 pH 5.7
Figure 2. Effect of contact time on ARS removal. Adsorbent dosage = 8.0 g/L, T = 25.0 °C, shaking Figure 2. Effect of contact time on ARS removal. Adsorbent dosage = 8.0 g/L, T = 25.0 ◦C, shaking speed = 175175 rpm. rpm.
EffectEffect of pH pH ARS is employed as an indicator in acid-baseacid-base titration titration,, and changes color in the pH range 4.0 (yellow)-6.0(yellow)-6.0 (red) [[3535].]. At pH < 4.6,4.6, the acidic form predominates, whereas at pH >> 6.0, it is mostly in the the the basic basic form. form. Figure Figure3 shows 3 shows the etheffect effect of pH of on pH ARS on removal. ARS removal. As pH increases,As pH increases, removal decreasesremoval decreasesfrom 90% from at pH 90% 2.0 reachingat pH 2.0 a reaching minimum a minimum of <20% at of pH <20%> 5.0. at HencepH > 5.0. the Hence acidic the form acidic ARS form is the ARS one ismore the favorablyone more adsorbed favorably by adsorbed wool. pH by 2.0 wool. was thuspH 2.0 selected was thus as optimum selected in as subsequent optimum measurements.in subsequent measurements.
100 80 60 40 20 % Removal % ARS of Removal 0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 pH Figure 3. Effect of pH on ARS removal by wool. Adsorbent dosage = 8.0 g/L, initial [ALS] = 100 mg/L, T = 25.0 °C, contact time = 90 min, shaking speed = 175 rpm.
Processes 2019, 7, x FOR PEER REVIEW 4 of 14
% Removal = ((Co-Ce)/Co)*100 (1) where Co and Ce are initial and equilibrium ARS concentrations (mg/L), respectively.
3.1.1. Optimization Studies on ARS Removal by Wool
Effect of Contact Time Figure 2 shows the effect of contact time on ARS removal. At all three pH values used, removal increases with increasing contact time until a maximum is reached at ca. 90 min, selected as the optimum time.
100 90 80 70 60 50 40 30 % Removal % ARS of Removal 20 10 0 0 50 100 150 200 Time (min)
pH 2.0 pH 3.2 pH 5.7
Figure 2. Effect of contact time on ARS removal. Adsorbent dosage = 8.0 g/L, T = 25.0 °C, shaking speed = 175 rpm.
Effect of pH ARS is employed as an indicator in acid-base titration, and changes color in the pH range 4.0 (yellow)-6.0 (red) [35]. At pH < 4.6, the acidic form predominates, whereas at pH > 6.0, it is mostly in the the basic form. Figure 3 shows the effect of pH on ARS removal. As pH increases, removal decreases from 90% at pH 2.0 reaching a minimum of <20% at pH > 5.0. Hence the acidic form ARS is theProcesses one2020 more, 8, 556 favorably adsorbed by wool. pH 2.0 was thus selected as optimum in subsequent5 of 15 measurements.
100 80 60 40 20 % Removal % ARS of Removal 0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 pH
FigureFigure 3. 3.EffectEffect of ofpH pH on on ARS ARS removal removal by by wool. wool. Adsorbent Adsorbent dosagedosage= = 8.08.0 gg/L,/L, initial initial [ALS] [ALS]= =100 100 mg mg/L,/L,
ProcessesT =T 2019 25.0= 25.0, 7°,C, x◦ C,FORcontact contact PEER time timeREVIEW = =9090 min, min, shaking shaking speed speed == 175175 rpm. rpm. 5 of 14
EffectEffect of ofWool Wool Dosage Dosage FigureFigure 44 shows shows the the effect e ffect of of wool wool dosage dosage on ARS removalremoval at at optimum optimum conditions. conditions. ARS ARS removal removal increasesincreases with with increasing increasing wool wool dosage dosage until until a a maximum 8.08.0 gg/L./L. This This dosage dosage was was selected selected as as optimal optimal and used in further experiments. and used in further experiments.
120
100
80
60
40 % Removal % ARS of Removal
20
0 0.0 2.0 4.0 6.0 8.0 Wool dosage (g/L)
FigureFigure 4. 4.EffectEffect of of adsorbent adsorbent dosage dosage o onn removal removal of ARSARS byby woolwool at at di differentfferent wool wool dosages. dosages. Initial Initial
[ARS][ARS] = 100= 100 mg/L, mg/L, pH pH = =2.0,2.0, contact contact time time == 9090 min, min, T T == 25.025.0 °◦C,C, shaking shaking speed = 175 rpm.
EffectEffect of ofTemperature Temperature ForFor initial initial ARS ARS concentrations concentrations ofof 60 60 mg mg/L/L, 80, 80 mg mg/L/L and and 100 mg100/L, mg/L, increasing increasing the temperature the temperature from from25.0 25.0 to 50.0 to ◦ 50.C0 has °C little has eff littleect on effect ARS on removal ARS removal (Figure5). (Figure The temperature 5). The temperature of 25.0 ◦C was of selected25.0 °C as was selectedoptimal as in optimal adsorption in adsorption isotherms. isotherms.
100
90
80
70
% Removal % ARS of Removal 60
50 25 30 35 40 45 50 T (o C)
100 ppm 80 ppm 60 ppm
Figure 5. Effect of temperature and initial ARS concentration on removal of ARS by wool. Adsorbent dosage = 8.0 g/L, contact time = 90 min, pH = 2.0, shaking speed = 175 rpm.
3.1.2. Adsorption Isotherms
The adsorption capacity (qe) for ARS removal was evaluated using
qe = (Co−Ce) × V/m (2)
Processes 2019, 7, x FOR PEER REVIEW 5 of 14
Effect of Wool Dosage Figure 4 shows the effect of wool dosage on ARS removal at optimum conditions. ARS removal increases with increasing wool dosage until a maximum 8.0 g/L. This dosage was selected as optimal and used in further experiments.
120
100
80
60
40 % Removal % ARS of Removal
20
0 0.0 2.0 4.0 6.0 8.0 Wool dosage (g/L)
Figure 4. Effect of adsorbent dosage on removal of ARS by wool at different wool dosages. Initial [ARS] = 100 mg/L, pH = 2.0, contact time = 90 min, T = 25.0 °C, shaking speed = 175 rpm.
Effect of Temperature For initial ARS concentrations of 60 mg/L, 80 mg/L and 100 mg/L, increasing the temperature fromProcesses 25.0 to2020 50., 8,0 556 °C has little effect on ARS removal (Figure 5). The temperature of 25.06 of°C 15 was selected as optimal in adsorption isotherms.
100
90
80
70
% Removal % ARS of Removal 60
50 25 30 35 40 45 50 T (o C)
100 ppm 80 ppm 60 ppm
FigureFigure 5. Effect 5. Eff ofect temperature of temperature and and initial initial ARS ARS concentration concentration on on removal removal of of ARS ARS by by wool. wool. Adsorbent Adsorbent dosagedosage = 8.0= g/L,8.0 gcontact/L, contact time time = 90= 90min, min, pH pH = =2.0,2.0, shaking shaking speed speed == 175 rpm.rpm.
3.1.2.3.1.2. Adsorption Adsorption Isotherms Isotherms
The adsorption capacity (qe) for ARS removal was evaluated using The adsorption capacity (qe) for ARS removal was evaluated using
qe = (Co Ce) V/m (2) qe = (Co−Ce) ×× V/m (2)
where qe is the equilibrium adsorption capacity (mg adsorbate/g adsorbent), V is the volume of solution (L) and m is the mass of adsorbent (g). At optimum adsorption parameters, qe was evaluated at several initial concentrations and the results fitted to the linearized forms of the Langmuir (Equation (3)) and Freundlich (Equation (4)) isotherms [36]. These are
Ce/qe = 1/Qb + Ce/Q (3)
1/n qe = Kf Ce (4)
where Ce is the equilibrium concentration (mg/L), qe is the amount adsorbed at equilibrium, in mg/g adsorbent, Q (mg/g) and b (L/mg) are the Langmuir constants, representing adsorption capacity and energy, respectively. Kf and n are the Freundlich constants. The poor fit to the Langmuir isotherm in Figure6a indicates that adsorption does not follow this isotherm. This may be attributed to the nonequivalence adsorption sites on wool. Figure6b shows the data fitted to the Freundlich isotherm. The good linearity of this plot clearly reveal that adsorption best fits the Freundlich isotherm, indicating that adsorption sites are not equivalent. The Freundlich constants, Kf and n, were 3.38 and 1.70, respectively. Processes 2019, 7, x FOR PEER REVIEW 6 of 14 where qe is the equilibrium adsorption capacity (mg adsorbate/g adsorbent), V is the volume of solution (L) and m is the mass of adsorbent (g). At optimum adsorption parameters, qe was evaluated at several initial concentrations and the results fitted to the linearized forms of the Langmuir (Equation (3)) and Freundlich (Equation (4)) isotherms [36]. These are
Ce/qe = 1/Qb + Ce/Q (3)
qe = Kf Ce1/n (4) where Ce is the equilibrium concentration (mg/L), qe is the amount adsorbed at equilibrium, in mg/g adsorbent, Q (mg/g) and b (L/mg) are the Langmuir constants, representing adsorption capacity and energy, respectively. Kf and n are the Freundlich constants. The poor fit to the Langmuir isotherm in Figure 6a indicates that adsorption does not follow this isotherm. This may be attributed to the nonequivalence adsorption sites on wool. Figure 6b shows the data fitted to the Freundlich isotherm. The good linearity of this plot clearly reveal that adsorption best fits the Freundlich isotherm, indicatingProcesses 2020 that, 8, 556 adsorption sites are not equivalent. The Freundlich constants, Kf and n, were 3.387 and of 15 1.70, respectively.
a 3.5
3.0
2.5
e 2.0 /q e
C 1.5
1.0
0.5
0.0 0 50 100 150 200 250 300
Ce, mg/L
b 2.5 y = 0.5333x + 0.6431 2.0 R² = 0.9943
1.5 e
log q 1.0
0.5
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log Ce
Figure 6. (a) Langmuir and (b) Freundlich plots for adsorption of ARS by wool. Adsorbent dosage = Figure 6. (a) Langmuir and (b) Freundlich plots for adsorption of ARS by wool. Adsorbent dosage = 8.0 g/L, T = 25.0 °C, contact time = 90 min, pH = 2.0, shaking speed = 175 rpm. 8.0 g/L, T = 25.0 ◦C, contact time = 90 min, pH = 2.0, shaking speed = 175 rpm.
3.1.3. Adsorption Kinetics The kinetics of adsorption of ARS by wool was studies at pH 2 and were fitted to both Lagergren pseudo first order model (Equation (5)) and pseudo second order model (Equation (6)) [37,38].
ln (qe qt) = ln qe k t (5) − − t 1 t = + (6) 2 qt k2qe qe Our data was found to best fit the pseudo second order model with R2 = 1 yielding a value for the 1 1 second order rate constant (k2) of 0.134 g mg− min− at 25.0 ◦C. Processes 2019, 7, x FOR PEER REVIEW 7 of 14
3.1.3. Adsorption Kinetics The kinetics of adsorption of ARS by wool was studies at pH 2 and were fitted to both Lagergren pseudo first order model (Equation (5)) and pseudo second order model (Equation (6)) [37,38].
ln ( e− t ) = ln e− k t (5)