1 Introduction

Several industries use dyes to manufacture their products, especially the textile, tannery, paper and food sectors [1]. Synthetic dyes are widely used due to their advantages over natural dyes [2]. The yearly production of synthetic dyes reaches 700,000 tons/year, and about 10 to 15% of the dyes used by industries reach the effluents [3]. It is expected that the textile industry alone releases approximately 100 tons/year of dyes in aqueous effluents [4]. Depending on concentration and exposure time, dyes may cause allergies, dermatitis and even cancer [5]. Upon release into effluents, they produce an unwanted color in water bodies, reducing photosynthesis processes and harming aquatic life [6].

Effluent treatment containing acid dyes has been widely studied as they represent a growing market and are efficient for coloring cotton. Nevertheless, more than 30% of the dye applied in the dyeing process is lost by hydrolysis in alkaline medium used, and conventional effluent treatment plants are not effective for anionic dye removal [3, 7]. There exists a number of procedures that may be employed to wastewater treatment, as photocatalysis [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23], electro-coagulation [24, 25] and membrane technology [17, 26].

The adsorption technique has become popular among effluent purification processes owing to their efficiency in removing pollutants resistant to other methods such as dyes. With increasing studies on low-cost adsorbents, this technique has become more feasible [2, 5, 7]. Low-cost adsorbents are considered interesting because they are abundant in nature, available in large quantities and low-priced. Among low-cost adsorbents, biossorbents are those produced from wastes or natural by-products [27]. They were obtained from a variety of materials, since fungi and algae yield by-products or fruit wastes [28].

Lignocellulosic materials appear to behave as good adsorbents, and different physical and/or chemical treatments have been applied to these materials to improve their adsorption capacity [7, 29]. Among these materials, fruit peels produced by processing in the food industry may represent an economical and renewable source of adsorbents for effluents treatment [30]. Various fruit peels have been tested as a low-cost adsorbent, such as orange peel, banana peel and passion fruit peel [3, 29]. Other non-lignocellulosic low-cost adsorbents include bagasse fly ash [31], waste rubber tires [32,33,34], de-oiled soya [35], Fe3O4 nanoparticles [36], guar gum-nano zinc oxide [37], activated carbon [38] among others.

Brazil is the world´s largest passion fruit producer with production outstripping 695,000 tons in 2016 [39,40,41]. The predominant species is Passiflora edulis f. flavicarpa, the popular name of which is yellow passion fruit [42]. Passion fruit juice is the third most consumed beverage in Brazil and keeps the second rank in export [43]. The peel representing 40 to 50% of the fruit weight is considered an industrial waste [43]. This material is rich in essential oils and pectin, and the residue from their extraction may be used as adsorbent.

Pavan et al. [44,45,46] tested yellow passion fruit peel for methylene blue adsorption and found an adsorption capacity of 44.70 mg g−1. Jacques et al. [47] studied the adsorption of Cr(III) and Pb(II) on passion fruit peel. The maximum adsorption capacity was 85.1 mg g−1 and 151.6 mg g−1 for Cr(III) and Pb(II), respectively. Gerola et al. [48] studied the adsorption of Pb(II) on passion fruit peel modified with sodium hydroxide and citric acid, verifying an increase in the adsorption capacity for the sample modified with citric acid. The adsorption of acid dyes in fruit peels is scarcely researched in the literature, and there are no reports of this dye adsorption class using passion fruit peel. Eriochrome black T (CI N° 14645, mordant black 11) is used as a dye for wool, silk and nylon fiber and as a complexometric indicator [49].

This article investigates the effect of solution pH parameters, contact time and initial dye concentration in the Eriochrome black adsorption on passion fruit peel treated with nitric acid and sodium hydroxide. The equilibrium data were studied according to the equations of Langmuir, Freundlich, Temkin, Sips and Dubinin–Radushkevich. The process kinetics was studied per the models of pseudo-first order, pseudo-second order, intraparticle diffusion, Elovich and Avrami. The thermodynamic parameters were investigated in order to determine the adsorption process spontaneity.

2 Materials and methods

The Eriochrome black T (molar mass 461.38 g mol−1) was provided by Vetec Química Ltda. Aqueous solutions of concentration ranging from 100 to 1000 mg L−1 were prepared with deionized water. Figure 1 shows the dye structure.

Fig. 1
figure 1

Source: Ref. [49]

Structure of the Eriochrome black T. dye.

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2.1 Adsorbent treatments

The passion fruit peel samples were ground, washed with deionized water and dried in an oven with air circulation at 150 °C for 24 h to remove more volatile essential oils. A portion of this material was treated with nitric acid 0.1 mol L−1 for 90 min at 100 °C under stirring, and another portion was treated with sodium hydroxide 0.1 mol L−1 under the same conditions of time and temperature (adapted from Kulkarni et al. [50]). The resulting material was filtered, washed with deionized water and dried at 105 °C for 24 h in ovens with controlled air circulation. The samples were named MF02 (without chemical treatment), MF03 (acid treated) and MF04 (base treated). Each sample underwent grain size separation for further evaluation. The samples were sieved on a vibrating table using 40, 60, 100 and 200 mesh sieves, corresponding to 0.420; 0.250; 0.149 and 0.074 mm particle sizes, respectively.

2.2 pH of zero charge point

The zero charge pH of the samples was determined by the solid addition method as described by Vieira et al. [51]. Twelve vials containing solutions in the pH range of 2 to 13 (pH0) and 0.1 g of adsorbent were stirred for 24 h at room temperature and the final pH was measured with a 10-MB MARTE potentiometer and a glass electrode. The difference between the initial pH and the final pH (DpH) was plotted versus the initial pH and the point where DpH = 0 was taken as pH of zero charge point (pHPZC).

2.3 Effect of mass and granulometry of adsorbent

In order to set the mass to be used in kinetic and thermodynamic studies, a preliminary test was made using Eriochrome black solutions with a concentration of 100 mg L−1, at room temperature, in the four grain sizes obtained for this work. The adsorbent mass varied from 50 to 1500 mg, and the samples were shaken in Erlenmeyers of 100 mL containing 25 mL of the dye solution during 24 h. The sample used for this test was crushed passion fruit peel, washed and dried at 150 °C (MF02), and was used a 6.76 pH, which is equivalent to that of the solution prepared with deionized water.

2.4 Effect of solution pH on adsorption

With the purpose of studying the effect of pH on the adsorption, the pH of the dye solutions was adjusted in the range from 2 to 12 by addition of 0.1 M NaOH or 0.1 M HCl. The solutions, with an initial concentration of 100 mg/L, were then stirred with 0.1 g of adsorbent at room temperature for 24 h. The final concentrations were measured with a spectrophotometer UV/VIS FEMTO 700 Plus and the pH was measured using a 10-MB MARTE potentiometer with a glass electrode. This test used the MF02 sample. No color change of Eriochrome black solution in the pH range used could be observed.

2.5 Kinetic, thermodynamic and adsorption isotherms experiments

In order to evaluate the kinetic models and adsorption isotherms, batch studies were performed using 100-mL Erlenmeyers containing 25 mL of the dye test solution, with initial concentration ranging from 100 to 1000 mg L−1. The adsorbent mass was defined in other experiments as approximately 0.1 g, and the pH of the experiments was always that of the initial solution, ranging from 6.7 to 6.8. The mixture was stirred at room temperature at different contact times. After stirring, the solutions were filtered with paper filter and the final dye concentration was determined using a FEMTO 700 Plus UV/VIS spectrophotometer. Contact times ranged from 15 to 240 min, and experiments with 24 h of agitation were performed, taken as equilibrium time. The adsorption isotherms were built with contact time of 24 h. The effect of temperature on adsorption was investigated in the range of 298–348 K (25–75 °C) as previously described for the batch experiments.

3 Results and discussion

3.1 Effect of mass and granulometry of the adsorbent

Figure 2 shows the effect of adsorbent mass variation on dye removal percentage in the grain sizes obtained for this research. It was observed that percentage of removal increases with decreasing particle size, which was already expected, as smaller particles minimize diffusion limitations and provide greater contact of the adsorbent with the dye, increasing the removal rate. There is an increase in the percentage removal by increasing the adsorbent mass to about 0.1 g. For larger masses, there is barely any increase in dye removal. In accordance with the results of these preliminary tests, the kinetic and thermodynamic experiments used a mass of about 0.1 g of adsorbent in the 0.074 mm particle size for 25 mL of dye solution at concentrations ranging from 100 to 1000 mg/L−1.

Fig. 2
figure 2

Removal percentage of Eriochrome black with adsorbent mass for different particle sizes. Ci = 100 mg L−1, pH = 6.70; T = 298 K, sample MF02

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3.2 Effect of solution pH on adsorption and determination pH of zero charge (PZC) of adsorbent

Adsorption of dye molecules is extremely dependent on pH since the functional groups of the adsorbent surface that are responsible for interaction with these molecules can be protonated or deprotonated to produce different surface charges in solutions with different pH values [52,53,54]. The pH of the solution may also affect the adsorbate molecule as most dyes go through protonation/deprotonation depending on the medium pH [55].

The pH of the point of zero charge (pHPZC) indicates the adsorbent surface ionization degree and its possible interaction with the adsorbate [55]. Usually, the adsorbent surface becomes positive if it accepts protons from the solution or negative if it loses protons to the solution. The pH where the adsorbent surface is neutral is represented as pHPZC [55]. Figure 3 shows the results of determination of the pH of zero charge point for the passion fruit peel samples used in this study. Table 1 shows the values found for each sample.

Fig. 3
figure 3

pH of zero charge point for passion fruit peel samples

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Table 1 pH of point of zero charge of yellow passion fruit peel samples
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The pHPZC of samples MF02 and MF03 is lower than 7, indicating predominantly acidic surfaces. Sample MF04 showed pHPZC higher than 7, indicating that NaOH treatment turned the surface more basic, whereas pHPZC from sample MF03 indicates that the surface became more acidic after HNO3 treatment. These results suggest that samples MF02 and MF03 would be good adsorbents for basic pollutants, whereas sample MF04 would be a good adsorbent for acid pollutants.

Figure 4 shows the effect of pH variation on Eriochrome black sorption on passion fruit samples.

Fig. 4
figure 4

Effect of pH on Eriochrome black adsorption. T = 298 K, Ci = 100 mg L−1

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As already expected from the results obtained for pHPZC (Table 1), adsorption is favored for pH ≤ 4.0 except for sample MF04, which presented pHPZC = 9.84 ± 0.01. Srivastava et al. [56] state that cation adsorption is favored at pH > pHPZC because the surface becomes negatively charged, whereas anion adsorption is favored at pH < pHPZC because the surface becomes positively charged. The results obtained in this work confirm this statement and suggest the influence of polarization effects on the adsorption of the colored anion studied. Similar results were found by Munagapati et al. [57] for the adsorption of Congo red over orange peel, Mokhtari et al. [58] for the adsorption of methyl orange on activated carbon impregnated with CuS, Gupta et al. [59] for the adsorption of crystal violet and brilliant green on kaolinite.

Figure 4 shows that the MF02 samples (without treatment) and MF03 (acid treated) exhibit a lower percentage removal across the pH range studied. For these samples, the pH does not influence the adsorption in the pH range 5–7, and a decrease occurs after dye removal with increasing pH. The MF04 sample presented the highest removal rates, with the percentage of removal practically constant in the pH range from 2 to 7, decreasing only from pH = 8.00. Taking into account that between pH = 5.0 and pH = 8.0 the adsorption appears to undergo significant influence of pH, even experiencing a decrease in the percentage removal for MF02 and MF03 samples, kinetic and thermodynamic experiments were accomplished in the original pH of the aqueous solution, which was always close to the pH of the deionized water ranging from 6.5 to 6.8. Such choice simplifies the adsorption system and reduces the cost of adding reagents to adjust the solution pH.

3.3 Study of adsorption isotherms

The Eriochrome black adsorption isotherms on the passion fruit peel samples are shown in Fig. 5, for a contact time of 24 h, pH = 6.8 and room temperature of (298 K). The isotherm model that best fits the adsorption process was studied by adjusting the data obtained from the Langmuir, Freundlich, Dubinin–Radushkevich and Temkin models. The models where the linear regression correlation coefficient (R2) was closest to the unit were considered adequate to describe the Eriochrome black adsorption system on passion fruit peel.

Fig. 5
figure 5

Adsorption isotherms for Eriochrome black. T = 298 K; pH = 6.8; t = 24 h

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The Langmuir isotherm and its linearized form may be described by Eqs. 1 and 2, respectively. The Langmuir isotherm presumes monolayer adsorption on sites with equivalent energy [60, 61].

$$Q_{e} = \frac{{Q_{\hbox{max} } K_{L} C_{e} }}{{1 + K_{L} C_{e} }}$$
(1)
$$\frac{{C_{e} }}{{Q_{e} }} = \frac{{C_{e} }}{{Q_{\hbox{max} } }} + \frac{1}{{K_{l} Q_{\hbox{max} } }}$$
(2)

The model of Langmuir allows the use of a variable called separation factor (RL), given by Eq. 3:

$$R_{L} = \frac{1}{{1 + K_{L} C_{0} }}$$
(3)

If RL is between zero and 1, that is, 0 < RL < 1, adsorption is said to be favorable. When RL > 1, there is an indication that the solute prefers the liquid phase and adsorption is unfavorable. If RL = 1, we have a linear isotherm, where the amount adsorbed is directly proportional to the equilibrium concentration of adsorbate in the liquid phase [61].

The model of Freundlich considers heterogeneous solid and multilayer adsorption, with adsorption sites of different energies [62, 63]. The isotherm of Freundlich and its linearized equation are represented by Eqs. 4 and 5, respectively.

$$Q_{e} = K_{F} C_{e}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 n}}\right.\kern-0pt} \!\lower0.7ex\hbox{$n$}}}}$$
(4)
$$\ln Q_{e} = \frac{1}{{K_{F} }} + \frac{1}{n}\ln C_{e}$$
(5)

Usually, a favorable adsorption tends to have an n value (Freundlich constant) between 1 and 10. The higher the n value, the stronger the interaction between adsorbate and adsorbent [64].

The Dubinin–Radushkevich isotherm was developed for adsorption of gases and solids and does not consider aspects inherent in the adsorption solution, such as pH and interactions between solute and solvent [64, 65]. The isotherm is given by Eq. 6.

$$Q_{e} = Q_{s} \exp \left( { - B_{DR} \varepsilon^{2} } \right)$$
(6)

And in the linearized form:

$$\ln Q_{e} = \ln Q_{s} - B_{DR} \varepsilon^{2}$$
(7)

where

$$\varepsilon = RT\ln \left( {1 + \frac{1}{{C_{e} }}} \right)$$
(8)

The value of the constant k may be associated with average adsorption energy through the equation:

$$E = \frac{1}{{\sqrt {2.B_{DR} } }}$$
(9)

And it may be used to differentiate adsorption as being chemical or physical [66].

The Temkin isotherm takes into account adsorbent–adsorbate interactions and considers that the adsorption heat decreases linearly with increasing surface coverage [67,68,69,70]. The expression of the Temkin isotherm is given by Eq. 10 and its linearized form is shown in Eq. 11.

$$Q_{e} = \frac{RT}{b}\ln \left( {a_{T} C_{e} } \right)$$
(10)
$$Q_{e} = B\ln a_{T} + B\ln C_{e}$$
(11)
$$B = \frac{RT}{b}$$
(12)

The constant B is related to the heat of adsorption and aT is an equilibrium constant.

Figure 6 shows the linearized isotherms according to the studied models. The parameters achieved for each model are shown in Table 2. From the linear correlation coefficients obtained after the isotherms linearization, we can assume that the Eriochrome black adsorption follows the Langmuir model. The values found for the maximum adsorption capacities (Qmax) calculated from the Langmuir isotherm (Table 2) show that treating the samples with acid or with base improves the adsorption capacity of the sample for Eriochrome black.

Fig. 6
figure 6

Linearization of isotherms according to Langmuir (a), Freundlich (b), Dubinin–Radushkevich (c) and Temkin (d). T = 298 K, pH = 6.8, t = 24 h

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Table 2 Parameters obtained from adsorption isotherms for Eriochrome black on passion fruit samples
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The RL values achieved at from the Langmuir isotherm are in the range of 0 < RL < 1 indicating that the Eriochrome black adsorption on passion fruit peel is favorable for all samples. This is confirmed by the n values of the Freundlich equation, which are higher than the unit. The values achieved for the paramenter E of the Dubinin-Radushkevich equation indicate that for all samples the adsorption process is physical. This is confirmed by the factor B values of the Temkin equation.

The adsorbed amount results obtained for samples MF03 and MF04 are significantly higher than for the sample without chemical treatment. The hot treatment of fruit peel in acid medium is one of the methods for pectin extraction found in the literature [71]. As a result, the increased adsorption capacity for the MF03 sample may be attributed to surface modification by pectin extraction, which may have increased the access of molecules to the adsorption sites. The increased adsorption capacity for sample MF04 may be attributed to the surface basicity increase, which favors acid dyes adsorption.

The pH used in the adsorption experiments may also have influenced the Qmax results, since the samples MF02 and MF03 have negatively charged surface at pH above 4.50 and 2.45, respectively (Table 1). As the adsorption experiments were performed at pH = 6.8, the surfaces of these materials would be negatively charged, decreasing the adsorption efficiency for anionic dyes. As the sample MF04 has pHPZC = 9.84, its surface would be positively charged at the pH used in the adsorption experiments, thus increasing the amount of Eriochrome black adsorbed by this sample. A pH adjustment of the solution, however, would introduce more process costs due to the need for more reagents, thereby failing to meet the requirements of low-cost adsorption processes.

Table 3 shows a comparison of the maximum adsorption capacity of Langmuir to Eriochrome black over various adsorbents. Adsorption capacities are shown in increasing order. As can be seen, although there are adsorbents with higher adsorption capacity than passion fruit peel, the NaOH-treated material has one of the highest adsorption capacities of the materials presented here. Comparison was made for adsorption of Eriochrome black in similar conditions. Although our materials are not the best adsorbent, it has better adsorption capacity than numerous materials, including graphene and lamellar double hydroxides. Among lignocellulosic materials found to adsorb Eriochrome black, our material is the best one.

Table 3 Maximum adsorption capacities (Langmuir) of Eriochrome black on various adsorbents
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3.3.1 Kinetic studies

The Eriochrome black adsorption was studied as a function of contact time to determine the kinetic model that best describes the adsorption process. The results of variation in amount adsorbed over time and with initial concentrations are shown in Fig. 7. The results in Fig. 7 show that Eriochrome black adsorption reaches equilibrium in about 60 min for all samples passion fruit peel. The adsorbed amount increases with increasing initial concentration for all samples.

Fig. 7
figure 7

Changes in the amount of Eriochrome black adsorbed with the contact time and with the initial concentrations for the passion fruit peel samples. T = 298 K, pH = 6.8, Ci = 100 to 1000 mg L−1

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The kinetics of this process of adsorption was studied by the models of pseudo-first order, pseudo-second order, Elovich, Boyd and intraparticle diffusion. The adsorption kinetic modeling is shown here for the initial concentration of Eriochrome black 100 mg/L−1.

Equation 13 shows the pseudo-first-order model proposed by Lagergren in its linear form [80].

$$\log \left( {Q_{e} - Q_{t} } \right) = \log Q_{e} - \frac{{k_{ad} }}{2,303}t$$
(13)

where Qe and Qt are the adsorbed amounts at equilibrium and time t (mg/g); kad is the adsorption velocity constant (min−1); and t is the time (min).

Ho et al. [80,81,82,83] proposed that the pseudo-second-order model better describes the adsorption of dyes on low-cost materials. Equation 14 shows one form of the linear model proposed by Ho [83].

$$\frac{t}{{Q_{t} }} = \frac{1}{{k_{2} .Q_{e}^{2} }} + \frac{1}{{Q_{e} }}.t$$
(14)

where Qe and Qt are the adsorbed amounts (mg g−1) at equilibrium and at time t (min) and k is the pseudo-second-order rate constant (g mg−1 min−1).The linear form of the Elovich equation is given by:

$$Q = \frac{1}{\beta }\ln \left( {\alpha .\beta } \right) + \frac{1}{\beta }\ln t$$
(15)

where Q is the amount adsorbed per gram of adsorbent (mg/g), α is the initial rate of adsorption (mg/g.min), β is a constant related to desorption of the surface coverage and the activation energy adsorption (g/mg) [84]. The Elovich equation neglects desorption and seems to better describe adsorption on heterogeneous surfaces [85].

The following figures show the graphs obtained for the kinetic modeling of the adsorption process studied in this research. Table 4 shows the parameters obtained for this modeling. Due to room unavailability, graphs and data are shown only for the initial concentration of 100 mg L−1.

Table 4 Parameters obtained for the kinetic modeling of Eriochrome black adsorption on passion fruit bark
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Figure 8 shows the graph obtained for pseudo-first-order modeling at room temperature and initial concentration of Eriochrome black 100 mg/L.

Fig. 8
figure 8

Plot of pseudo-first order for Eriochrome black adsorption on passion fruit peel. Ci = 100 mg/L, T = 298 K, pH = 6.8

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The linear plots of log10(Qe− Qt) versus t showed very low linear correlation coefficients, indicating that this model does not describe the Eriochrome black adsorption kinetics on different samples passion fruit peel. Even in the samples in which the correlation coefficient approached unity (between 0.95 and 0.97), the calculation of the adsorbed amount at equilibrium (Qe) showed results very far from those obtained experimentally, confirming that the pseudo-first-order model is not proper to describe the results achieved.

Figure 9 shows the graph obtained for pseudo-second-order modeling for Eriochrome black adsorption on passion fruit peel samples at room temperature and initial dye concentration of 100 mg L−1.

Fig. 9
figure 9

Pseudo-second-order plot for adsorption of Eriochrome black on passion fruit bark. Ci = 100 mg/L, T = 298 K, pH = 6.8

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The results obtained for the correlation coefficients were higher than 0.999 for all samples tested, indicating that the pseudo-second-order model describes well the Eriochrome black adsorption on passion fruit peel. The pseudo-second-order model is considered more experimental than theoretical, being applicable to systems where the experimental simulation is close to real conditions. The pseudo-second-order model also assumes that there may be ion exchange between adsorbate and adsorbent. However, the thermodynamic data shown in the next section does not indicate which chemical processes are occurring in this system. Accordingly, the pseudo-second-order model is a good mathematical description of the process, but obscures the adsorption mechanism taking place. This assumption is supported by several works in the literature, as the adsorption of Eriochrome black [79, 86].

Figure 10 shows the graph obtained for the study of Eriochrome black adsorption data on passion fruit peel according to the linearized Elovich equation.

Fig. 10
figure 10

Elovich plot for adsorption of Eriochrome black on passion fruit samples. Ci = 100 mg/L; T = 298 K, pH = 6.8

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The R2 values for Elovich modeling were very low compared to the pseudo-second-order model, indicating that this equation is not proper to describe the Eriochrome black adsorption kinetics on passion fruit peel. The α values were extremely high, which would correspond to a very fast initial adsorption that is confirmed by the low values of t0.

The results obtained for the other initial concentrations showed that the expected increase in the constants α and β with the increase in the initial concentration [84] did not occur, and the obtained values varied randomly. Yet, the values of the Elovich constants found in this paper do not agree with values obtained in similar conditions [87, 88]. Based on these results, it is considered that the Elovich model does not describe the adsorption process kinetics studied in this work.

The adsorption mechanism is not always clear from the models used in this study. Furthermore, liquid-phase adsorption is always subject to rate-limiting steps such as film diffusion and intraparticle diffusion. In order to study the rate-limiting step of this process, models of Boyd and Weber and Morris were used.

Boyd diffusion model [89] determines how much resistance to mass transfer is in the film surrounding the adsorbent particle or the diffusion inside the pores. This model can be expressed by equation:

$$f = 1 - \frac{6}{{\pi^{2} }}\mathop \sum \limits_{n = 1}^{\infty } \frac{1}{{n^{2} }}\exp \left( { - n^{2} Bt} \right)$$
(16)
$$f = \frac{{Q_{t} }}{{Q_{e} }}$$
(17)

where f is the equilibrium fraction reached at different times (t) and Bt is a function of f. Qe and Qt are the quantities adsorbed at time t and at equilibrium, respectively (mg/g).

Reichenberg (1953) [90] obtained the following approximations to allow the calculation of Bt:

$${\text{For}}\; \, f > 0.85\;\;\;\;Bt = - 0.4977 - \ln \left( {1 - f} \right)$$
(18)
$${\text{For}}\; \, f < 0.85\;\;\;\;Bt = \left( {\sqrt \pi - \sqrt {\pi - \frac{{\pi^{2} f}}{3}} } \right)^{2}$$
(19)

A plot of Bt versus time can be used to obtain information about the adsorption process mechanism. If the graph is linear and passes through the origin, the pore diffusion controls the mass transfer rate. If the graph is nonlinear or linear, but does not pass through the origin, film diffusion or chemical reactions control the adsorption rate [91].

Figure 11 shows the graph obtained for applying the Boyd equation [89] to the experimental data for initial concentration of 100 mg L−1. Table 5 summarizes the values found for B and for the diffusion coefficient. (Di), calculated by the expression:

$$B = \frac{{\pi^{2} D_{i} }}{{r^{2} }}$$
(20)

where r is the radius of the molecule (m) assuming it has a spherical shape.

Fig. 11
figure 11

Boyd plots for adsorption of Eriochrome black on passion fruit bark, Ci = 100 mg L−1, T = 298 °C, pH = 6.8

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Table 5 Parameters obtained for mathematical modeling by equations of Boyd and Weber–Morris
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The plot of Bt versus t for Eriochrome black adsorption on passion fruit peel showed low linearity and does not go through the origin. This behavior indicates that the adsorption is controlled by external mass transfer and not by particle pore diffusion, which was already expected, since the passion fruit peel particles do not have a defined pore structure. This consideration is reinforced by the low values of B found, which resulted in values for the diffusion coefficient, Di, also very low, indicating that the diffusion of the molecule in the solution is slow.

Diffusion processes can also be investigated using the Weber–Morris model [92], described by Eq. 21:

$$Q_{t} = k_{i} \sqrt t + c$$
(21)

where ki is the intraparticle diffusion velocity constant (mol g−1 min−1/2). A graph of Qt versus t1/2 must be a straight line, the slope of which gives the value of the diffusion constant. If c ≠ 0 intraparticle diffusion is not the only process controlling the velocity [80].

Figure 12 shows Weber–Morris plots for the Eriochrome black adsorption process on the samples tested at an initial concentration of 100 mg/L, pH = 6.8 and room temperature (298 K).

Fig. 12
figure 12

Weber–Morris diagrams for adsorption of Eriochrome black on passion fruit bark. Ci = 100 mg/L, T = 298 K, pH = 6.8

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It can be observed that the graphs are linear, but do not go through the origin, indicating that intraparticle diffusion is not the main factor determining the adsorption velocity. The graphs show two diffusion steps, which can be attributed to solution diffusion and diffusion in the liquid film surrounding the particle. The result set shows that the limiting step in the process was probably diffusion in the solution.

As a result, the kinetic studies indicate that the process kinetics can be mathematically described by the pseudo-second-order model, but that the diffusional limitations in the solution, which may be due to the agitation speed used in the experiments, are influencing the process.

3.4 Thermodynamic studies

Figures 13, 14 and 15 show the Eriochrome black adsorption isotherms, respectively, for samples MF02 to MF04, at the temperatures studied in this work.

Fig. 13
figure 13

Adsorption isotherms for the sample MF02 at the temperatures used in this work

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Fig. 14
figure 14

Adsorption isotherms for the sample MF03 at the temperatures used in this work

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Fig. 15
figure 15

Adsorption isotherms for the sample MF04 at the temperatures used in this work

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The amount adsorbed decreases with increasing temperature as expected for an exothermic process. Thermodynamic parameter values, enthalpy, entropy and Gibbs free energy, as well as the equilibrium constants, were determined using Eqs. 22 to 24 [53, 93] and are summarized in Table 6.

Table 6 Thermodynamic parameters obtained for adsorption of Eriochrome black on passion fruit peels
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$$\log K_{C} = - \frac{{\Delta H^{0} }}{2.303RT} + \frac{{\Delta S^{0} }}{2.303R}$$
(22)
$$\Delta G^{0} = - RT\ln K_{C}$$
(23)
$$K_{C} = \frac{{C_{\text{ads}} }}{{C_{e} }}$$
(24)

Thermodynamic studies’ results indicate that the adsorption process is exothermic for all samples tested, supporting the results of decreased adsorption capacity with increasing temperature shown in Figs. 13, 14 and 15. The process occurs with decreased entropy, which was expected as the molecules are more organized when adsorbed on the surface than in solution.

The results obtained for Gibbs free energy indicate that the Eriochrome black adsorption on the various samples is spontaneous at low temperatures, becoming less and less spontaneous as the temperature increases. This result confirms the assumption that the process is primarily physical adsorption. Other works reported the same behavior, considering that interaction of Eriochrome black with adsorbents is from electrostatically nature [73].

4 Conclusions

Passion fruit peel proved to be a promising adsorbent for Eriochrome black dye, with adsorption capacities ranging from 196 to 303 mg dye per gram of adsorbent. Generally, the treatment of the sample with acid or base improves the adsorption capacity, but the base treatment was more effective in increasing the adsorption capacity. The process kinetics may be described mathematically by the pseudo-second-order equation, and diffusion studies have shown that diffusion in the solution is influencing the results, being the process velocity a limiting step. Eriochrome black adsorption on passion fruit peel is an exothermic process, which occurs with decreased entropy, and the obtained Gibbs free energies indicate that the process is spontaneous at temperatures close to room temperatures (298 K).

Comparison with other Eriochrome black adsorption processes reported in the literature shows that passion fruit peel can be employed as adsorbent for this dye and that the base treatment produces the best material for the adsorption process at room temperature. The results of this study show that the passion fruit peel may be a good adsorbent for other acid dyes beyond the Eriochrome black.