Keywords

1.1 Introduction

The widespread contamination of water from different pollutants has stimulated the development of effective technologies able to remove them or, at least, to significantly reduce their impact. Nowadays, water resources are threatened by many xenobiotic substances, commonly referred as micropollutants, including both organic (chlorinated, aromatic, etc.) and inorganic (mainly heavy metals) compounds. These substances are usually characterized by a high degree of toxicity and recalcitrance; for this reason, they are typically resistant to classical biological treatments and are not effectively removed by physicochemical methods, as those commonly adopted in municipal wastewater plants. In this context, adsorption has attracted the attention of several researchers due to its high efficiency, low cost, high selectivity at the molecular level, besides presenting a low energy consumption. Adsorption phenomena are spontaneous and occur when a solid surface is exposed to a gas or liquid stream in which the target molecule (adsorbate) is present, resulting in an increase of molecule density in the proximity of the material surface (adsorbent). Being a mass transfer phenomenon that involves the outer surface of solid materials, porous solids are commonly employed such as activated carbons, silica gel, some synthetic polymers, alumina, natural materials (e.g., zeolite , clays, etc.), or waste materials (e.g., lignocellulosic wastes, and red mud) (Babel and Kurniawan 2003; Bertocchi et al. 2006; Iovino et al. 2015; Sellaoui et al. 2018; Toumi et al. 2018; Li et al. 2019). The optimal choice of the adsorbent derives from a thorough evaluation of the nature of the pollutant and of the physical and chemical properties of the polluted stream.

From a practical point of view, the process configuration commonly adopted for the treatment of polluted water provides for the passage of the liquid to be purified through a column in which is present a fixed bed of porous material, capable of retaining pollutants on its surface. During the operation, the concentration of pollutants on the outer surface of the adsorbent solid increases, until a complete saturation, that is, exhaustion of the adsorption capacity. In order to be reused, these materials must be replaced or treated, aiming at restoring, albeit in part, the ability to bind pollutants on their surface. Regeneration consists in the desorption of pollutants, which is its removal from the solid matrix that can be achieved in different ways: by desorption with a (hot) inert gas passing through the bed of adsorbent; by thermal treatment, using a stream of water vapor or microwaves; or by using an acid/basic/saline solution for the extraction of the pollutant (generally for heavy metals) (Di Natale et al. 2013; Marques et al. 2017; Oladejo et al. 2020). From an economic point of view, it is worth using a nondestructive desorption agent, with high pollutant removal efficiency, not toxic to humans and to environment, and that does not alter the adsorption characteristics of the material. At the end of the regeneration treatment, the adsorbing capacity is not fully restored, so that after a series of cycles of adsorption and regeneration the material must be disposed of. In order to ensure a continuous operation of the system, the polluted stream is treated in two or more alternate treatment units, so that the exhausted one can be replaced, possibly reusing the adsorbent after a proper regeneration step.

Meantime, adsorption process presents some problems regarding sustainability. The main costs associated with this treatment rely on the adsorbent acquisition, and on the efficiency of regeneration techniques, while the operational costs are less intensive. For this reason, adsorbents deriving from low-cost sources capable to efficiently remove specific hazardous substances and to assure a cost-effective water treatment represent one of the most important challenges in adsorption intensification. Moreover, the optimization of the operational parameters for both thermodynamic and kinetic aspects of the adsorption process is the starting point for an effective application of this important technology for water treatment.

In this context, this chapter aims to address the fundamental aspects of adsorption in liquid phase in order to guide the reader to a better understanding of subsequent chapters of this book.

1.2 Adsorption Operation

For a thorough adsorption design and optimization process, a complete and accurate understanding of the effects of the main operational parameters must be accomplished. In general, the performance of an adsorption treatment depends on the thermodynamic aspects of solute–solvent–sorbent interactions and on the diffusive–convective transport phenomena involving the adsorbent (Ruthven, 1984; Vocciante et al. 2014). Adsorption equilibrium defines the limits of the applicability of the process and of a particular adsorbent, while the kinetic and transport phenomena allow defining the real efficiency and the overall extent of the intervention. To this aim, a thorough knowledge of the adsorption mechanism occurring at liquid–solid interface, which comprises the interactions between the adsorbent, the solvent (water), and adsorbate, is indispensable. The mass transfer phenomena can occur chemically, in which electrons are shared or exchanged between solute (adsorbate) and adsorbent; it is commonly referred as chemisorption. Alternatively, adsorption can involve weak physical forces with very low binding energy, such as Van der Waals forces or dispersion forces, and it is called physisorption.

In the case of adsorption in liquids, and in particular in electrolytic solutions, the description of adsorption phenomena is more difficult. In fact, while in gas systems the chemical form in which the molecule is present and, consequently, also adsorbed is known apriori, in aqueous solutions this is not true, because the chemical species can vary (e.g., for acid/base dissociation, hydrolysis reactions, or oxidation/reduction phenomena on the adsorbent surface) (Benjamin 2002). The phenomenon is dependent on the composition of the solution and on typical parameters such as pH, ionic strength, and temperature, which define the relative concentrations of ionic species in solution. Furthermore, the specific interactions between the solid surface and each of the ionic species present can determine significant modifications of the overall adsorption performances (Stumm and Morgan 1996).

The adsorption of organic compounds from aqueous solutions presents peculiar characteristics, which differentiate from inorganic compounds because, especially for nonionic species, some parameters have less influence (e.g., ionic strength and pH).

In general, adsorption from water streams is closely related to the following factors (Ruthven 1984; Do 1998):

  • Physical characteristics of the adsorbent: specific surface area, total pore volume, structure, size and distribution of pores, etc.

  • Chemical characteristics of the adsorbent: composition, presence of functional groups superficial, pH at point of zero charge (pHPZC), hydrophilicity/hydrophobicity, etc.

  • Chemical characteristics of the pollutant : nature, concentration, water solubility, molecular weight, etc.

  • Characteristics of the aqueous matrix: temperature, pH, salinity, presence of natural organic matter (NOM), presence of other micropollutants, etc.

As adsorption is a surface phenomenon, the importance of a well-developed porous structure of the adsorbents is of utmost importance and the distribution and size of the porous matrix influence the adsorption mechanism occurring. Simultaneously, chemical interactions can occur on the surface of the solid due to the presence of surface functional groups variously reactive. In liquid adsorption, the chemistry of the adsorbent can be also characterized by the pH at point of zero charge (pHPZC). It is defined as the pH value of a water/solid suspension in correspondence of which an equal adsorption of H+ and OH− ions is observed. Consequently, for solution pH below the pHPZC value, the outer surface of the adsorbent will be positively charged; thus, it will tend to attract anions. On the contrary, for solution pH higher than pHPZC the outer surface will be negatively charged, then attracting cations (Noh and Schwarz 1990).

The composition of the liquid phase also plays a very important role in the adsorption of a stated compound or for its adsorption at the maximum possible extent, which can be evaluated from the specific solute–adsorbent interactions (Nascimento et al. 2014).

From a thermodynamic point of view, adsorption is a spontaneous process (ΔG < 0) and is characterized by a decrease in the entropy of the adsorbed molecule/ion, which is included into the adsorbent matrix (ΔS < 0). Since, at constant temperature and pressure (Ruthven 1984):

$$ \Delta G=\Delta H-T\Delta S $$
(1.1)

It results that adsorption is an exothermic process, that is, characterized by decrease of enthalpy, also referred as heat of adsorption, (ΔH < 0) and, therefore, it is favored by low temperatures. However, in the presence of water, the overall phenomenon can be coupled by a water desorption step, for to allow the target-compound to be adsorbed. As desorption is an endothermic process, in these cases the overall process can result as endothermic.

The main thermodynamic parameter, expressing the maximum adsorption amount of a pollutant that an adsorbent can bind on its surface at equilibrium, is the adsorption capacity. The evaluation of the thermodynamic characteristics of the phenomenon (i.e., adsorption capacity) and the influence that the characteristic parameters have on the phenomenon itself are independent of the particular plant configuration chosen for tests; therefore, for the sake of simplicity, the experiments aimed at investigating these properties are conducted in batch mode.

The adsorption capacity (q) is expressed as moles or mass of pollutant per mass of adsorbent (mol/g or mg/g) and can be calculated by using a material balance on the pollutant , as in Eq. (1.2):

$$ q=\frac{\left({C}_0-{C}_{\mathrm{eq}}\right)}{m}\ast V $$
(1.2)

where C0 is the initial adsorbate concentration (mg/L); Ceq is the final (equilibrium) adsorbate concentration (mg/L); m is the mass of adsorbent (mg); and V is the volume of batch adsorbate solution (L).

From a kinetic point of view, the mass transfer from the solution to the surface of the adsorbent solid can be schematized with a series of successive steps, each of which representing one of the contributions to the global resistance. In particular, adsorption mechanism (Fig. 1.1) is rate-controlled by one of these steps or by their combination (Sanghi and Bhattacharya 2002; Plazinski et al. 2009; Vocciante et al. 2014):

  1. 1.

    Solute transportation in the bulk of the solution

  2. 2.

    Solute diffusion through the liquid film surrounding the solid particles

  3. 3.

    Solute diffusion inside the pores (intraparticle diffusion)

  4. 4.

    Adsorption/desorption equilibrium (reaction)

Fig. 1.1
figure 1

Adsorption mechanism in terms of mass transfer phenomena

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Similarly, in a batch adsorption test, the evolution of the adsorbed amount on the adsorbent surface (qt) as a function of contact time with the adsorbent can be expressed by Eq. (1.3):

$$ {q}_t=\frac{\left({C}_0-{C}_t\right)}{m}\ast V $$
(1.3)

where C0 is the initial adsorbate concentration (mg/L); Ct is the adsorbate concentration at time t (mg/L); m is the mass of adsorbent (mg); and V is the volume of adsorbate solution (L).

1.3 Adsorption Kinetics in Liquid Phase: Theoretical Aspects

In a heterogeneous system such as liquid–solid mixtures, the knowledge of adsorption–desorption kinetic is necessary to evaluate the rate of the process, the limiting steps of the mass transfer, and the influence of the textural properties of adsorbent material (Azizian 2004). Besides, this study provides useful information about the specific adsorption mechanism between adsorbent and adsorbate, considering that the adsorption process follows three main steps, as previously observed:

  1. 1.

    Transport of analyte from the bulk to the surface of the materials and diffusion in the liquid film surrounding the solid particle

  2. 2.

    Diffusion of analyte in the pores of the solid particle(pore diffusion/surface diffusion)

  3. 3.

    Formation of physical or chemical interaction between surface atoms of the adsorbent and the adsorbate

In this context, in order to understand the solid–liquid interactions as well as adsorbate–adsorbent complex dynamics, different mathematical models have been developed, which consider the dependence of adsorption pseudo-reaction rates with the concentration of pollutant either on the adsorbent or in the liquid contacted. In turn, the best fitting to one of the existing models provides information about the most probable reaction mechanism (Largitte and Pasquier 2016). The diverse kinetic mathematical models that have been formulated in the literature are summarized in the following.

1.3.1 Crank Model

This model assumes that the internal diffusion is the rate-limiting step and the adsorbent is considered as a sphere, assuming a constant surface diffusivity for all the adsorbent particles. The General Crank equation (Eq. 1.4) is described as (Largitte and Pasquier 2016; Westwater and Drickamer 1957):

$$ \left({q}_t/{q}_e\right)=1-\left(6/{\pi}^2\right){\sum}_{n-1}^{\infty}\left(1/{n}^2\right)\exp \left({\mathrm{D}}_{\mathrm{S}}{n}^2{\pi}^2t/{R}^2\right) $$
(1.4)

where qt is the amount of adsorbate adsorbed at time t (mg/g), qe is the maximum amount of analyte adsorbed at equilibrium (mg/g), n is a fitting parameter (−), DS is a surface diffusion coefficient (cm2/s), t is time (s), and R (cm) is the radius of the particle of adsorbent, assumed as spherical.

1.3.2 Weber and Morris Model

In this model, the limiting step of the mass transfer is the intraparticle diffusion, and the plot of the amount of analyte (adsorbate) adsorbed at any time (mg/g) vs the square root of time (t1/2) provides information about the rate of the process. This model is given by Eq. (1.5) (Yousef et al. 2011):

$$ {q}_{\mathrm{t}}={k}_{id}{t}^{1/2}+{C}_i $$
(1.5)

where kid is the intraparticle diffusion rate constant (mg/g h0.5), t is time (h), and Ci corresponds to the boundary layer thickness (Zhou et al. 2017).

However, the plot of this model shows three curves, the first stage associated with faster or instantaneous process, followed by intra-particle diffusion stage, and finally equilibrium stage (Naseem et al. 2019). In this sense, if in the plot of qt vs t1/2 is a line passing through the origin, the more important and limiting process is the intraparticle diffusion but if the line does not pass through the origin, this stage is not the determining step (Yousef et al. 2011).

1.3.3 Pseudo First-Order Model (PFO)

In this model, also known as Lagergren model, the adsorption pseudo-reaction is the rate-limiting step or the slowest ones. This model is described by a nonreversible equation (Largitte and Pasquier 2016), assuming that the relationship between the adsorbate adsorbed onto the solid and the active site is 1:1 (Boparai et al. 2011). Moreover, adsorption does not involve the interaction between adsorbate–adsorbent and the process occurs on local sites. Adsorption bond takes place by means of the formation of monolayer on solid surface, where the energy of this process is independent on surface coverage (Largitte and Pasquier 2016).

The mathematical representation of the PFO model is shown in Eq. (1.6) (Azizian 2004; Sen Gupta and Bhattacharyya 2011; Yousef et al. 2011):

$$ \frac{d_{q_t}}{d_t}={k}_1\left({q}_e-{q}_t\right) $$
(1.6)

Integrating this equation with the boundary conditions (qt = 0 at t = 0; and qt = qeq at t = teq), it is obtained:

$$ {q}_t={q}_e\left(1-{e}^{-{\mathrm{k}}_1\mathrm{t}}\right) $$
(1.7)

which can be rewritten in linear form as:

$$ \ln \left({q}_e-{q}_t\right)=\ln\ {q}_e-{k}_1t $$
(1.8)

where qt and qe (mg/g) are the amounts of analyte adsorbed at any times t (min) and at equilibrium, respectively, and k1 is the rate constant of PFO reaction (1/min), which depends on the initial concentration of adsorbate. It is opportune to mention that Lagergren equation correlates better with the kinetic data for the adsorption systems that are not far from equilibrium.

1.3.4 Pseudo Second-Order (PSO) Model

In this case, the model assumes that adsorbate adsorption takes place by a second-order pseudo-reaction, that is, two active sites bind one adsorbate molecule. Moreover, the pollutant concentration is constant, the total number of active site depends on the ions adsorbed at equilibrium and considered that the mechanism follows a chemisorption kinetic , expressed as (Ho and McKay 1998, 1999):

$$ \frac{d_{q_t}}{d_t}={k}_2{\left({q}_e-{q}_t\right)}^2 $$
(1.9)

Considering the boundary conditions (qt = 0 at t = 0 and qt = qeq at t = teq), the equation has the form:

$$ \frac{t}{q_t}=\left(\frac{t}{q_e}\right)+\frac{1}{\left({k}_2{q}_e^2\right)} $$
(1.10)

or the linear formulation:

$$ {q}_{t=}\frac{\left({k}_2{q}_e^2\right)\ t}{\left(1+{k}_2{q}_et\right)} $$
(1.11)

The initial adsorption rate is obtained by defining h = k2qe2 directly from the intercept of the curve. Furthermore, qt and qe (mg/g) are the amount of analyte adsorbed at any time t (min) and at equilibrium, respectively, and k2 is the overall rate constant for PSO reaction and depends on the operating conditions, such as analyte concentration, pH, and temperature.

1.3.5 Elovich Model

This mathematical model considers that in the adsorption process the adsorbent surface has a heterogeneous energy (Sen Gupta and Bhattacharyya 2011). The adsorption mechanism involves interactions between ions and adsorbent and there is a linear relationship between the energy of adsorption and the surface coverage (Largitte and Pasquier 2016). The equation has the form:

$$ \frac{d_{q_t}}{d_t}=\alpha \exp \left(-\upbeta {\mathrm{q}}_{\mathrm{e}}\right) $$
(1.12)

Assuming qt = 0 at t = 0 and t = t and qt = qeq and αβt >> 1, the linear expression is:

$$ {q}_t=\beta \ln \left(\alpha \beta \right)+\beta \ln t $$
(1.13)

And the nonlinear form is (Largitte and Pasquier 2016):

$$ {q}_t=\left(\frac{1}{\beta}\right)\ln \left(\alpha \beta t\right) $$
(1.14)

where α and β are the Elovich coefficients that correspond to the initial adsorption rate (g/mg/min2) and desorption coefficient (mg/g/min), respectively, and qt is the amount of analyte adsorbed at any time t (min).

1.3.6 Experimental Procedure

The effect of contact time (kinetic study) can be performed by batch adsorption technique, using different glass flasks, capped polyethylene bottles, or Falcon tubes. Each flask consists in a volume containing a defined concentration of the pollutant in aqueous solution and a specific mass of adsorbent. They are put in contact and agitated in a thermostat shaker at constant temperature and, periodically, the supernatant is sampled at different time intervals, to quantify the analyte present in the solution and the adsorbed amount according to Eq. (1.3) (AL-Othman and Naushad 2012).

After the experimentation, the amount of adsorbate adsorbed per unit mass of adsorbent (qt) can be plotted vs the adsorption time (t). In order to assess the best fitting model of the experimental data, a fitting procedure can be performed and different statistical parameters can be calculated for a fair evaluation of the fitting quality. For example, R2, chi-square test (χ2), root mean square error (RMSE), and standard deviation Δq (%) are excellent statistical parameters to individuate and validate the best fitting model. These parameters are calculated as (Harja and Ciobanu 2018; Wang et al. 2010):

$$ {\chi}^2=\sum \limits_{i=1}^N\frac{{\left({q}_{\mathrm{e},\kern0.5em \exp }-{q}_{\mathrm{e},\mathrm{cal}}\right)}^2}{q_{\mathrm{e},\mathrm{cal}}} $$
(1.15)
$$ {R}^2=1-\frac{\sum \limits_{\mathrm{i}=1}^N{\left({q}_{e,\exp }-{q}_{e,\mathrm{cal}}\right)}^2\ }{\sum \limits_{\mathrm{i}=1}^N{\left({q}_{e,\exp }-\overline{q_{e,\exp }}\right)}^2} $$
(1.16)
$$ \mathrm{RMSE}=\sqrt{\frac{1}{N-2}\sum \limits_{1=1}^N{\left({q}_{e,\exp }-{q}_{e,\mathrm{cal}}\right)}^2} $$
(1.17)
$$ \Delta q\left(\%\right)=100\sqrt{\frac{\sum {\left(\left({q}_{e,\exp }-{q}_{e,\mathrm{cal}}\right)/{q}_{e,\exp}\right)}^2}{N-1}} $$
(1.18)

where qe,exp and qe,cal correspond to the experimental and calculate amount of analyte adsorbed on the adsorbent (mg/g) and N is the number of measurements. A low value of Δq suggests that the model has a good mathematical fitting and, therefore, that the parameters deriving from the model adequately describe the experimental evidences.

In Table 1.1, different kinetic experiments and the related best-fitting models are resumed as retrieved from the literature and taken as example.

Table 1.1 Kinetic models and parameters determined for different analyte adsorption
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1.4 Adsorption Equilibrium: Theoretical Aspects of Adsorption Isotherms

The study of adsorption equilibrium defines the potentiality of the specific application and it also helps to characterize the kinetic aspects of the process, as reported in the previous section.

The main instrument for to assess the thermodynamic properties of an adsorption system is the experimental retrieving of adsorption isotherms, which represent a correlation between the mass of the analyte (adsorbate) bond per unit mass of the adsorbent and its concentration in the solution, at constant temperature (Do 1998).

A modeling analysis of the experimental adsorption isotherms can be carried out in order to define a theoretical interpretation of the observed evidences.

In the case of solid–liquid systems, the models deriving from Langmuir theory have been extensively used to study the adsorption phenomenon for their wide applicability and undoubtable success (Chung et al. 2015). The use has been related to the interpretation of the adsorption of both organic and inorganic compounds. As an example, Hu et al. (2014) observed that by the study of the isotherms it is possible determining the distribution of ions of a given analyte between liquid and solid phase, at equilibrium. In order to characterize this distribution, different mathematical models have been proposed, which are summarized below.

1.4.1 Langmuir Model

This model assumes that the surface of adsorbent is homogeneous, all the active sites present in the substrate have similar energy or affinity for the investigated analyte and, therefore, the adsorption is homogeneous and occurs without any interaction among adsorbed analytes on the adsorbent surface (Ghosal and Gupta 2017). Besides, the Langmuir isotherm considers the formation of a monolayer and adsorption is assumed as reversible (Naseem et al. 2019). The mathematical expression is (Uslu et al. 2016):

$$ {q}_e={q}_{\mathrm{max}}\frac{K_{\mathrm{L}}\ {C}_e}{1+{K}_{\mathrm{L}}\ {C}_e} $$
(1.19)

which can be also linearized as:

$$ \frac{C_e}{q_e}=\frac{1}{q_{\mathrm{max}}{K}_{\mathrm{L}}}+\frac{C_e}{q_{\mathrm{max}}} $$
(1.20)

where Ce is the concentration of adsorbate at equilibrium (mol/L), qe is the analyte mass adsorbed per unit of adsorbent mass at equilibrium (adsorption capacity) (mg/g), qmax is the maximum adsorption capacity (mg/g), and KL is the Langmuir constant, which is related to the affinity between adsorbate and adsorption sites or the binding energy between adsorbate and adsorbent (L/mol).

Besides, to evaluate the favorability of an adsorption process, a separation factor (RL) parameter can be calculated, as given below (Atta et al. 2015; Naseem et al. 2019):

$$ {R}_{\mathrm{L}}=\frac{1}{\left(1+b{C}_e\right)} $$
(1.21)

Depending on RL value, it is possible determining the feasibility of the adsorption process. If RL > 1, the process is unfavorable, for RL = 0, adsorption is irreversible, and if 0 < RL < 1 adsorption process is favorable (Chaudhry et al. 2017).

1.4.2 Freundlich Model

This model is used when the adsorption sites cannot be hypothesized as uniform and an energetic distribution exists; usually, increasing their energy level results in a decrease in the number of active sites that can be associated with that level (i.e., a decreasing exponential distribution of energies of the active sites can be assumed) (Chaudhry et al. 2017). In fact, the Freundlich isotherm is applied when heterogeneous adsorbents are considered, and it is expressed by the following equation (Dada et al. 2012):

$$ {q}_e={K}_f{C}_e^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.} $$
(1.22)

which can be also linearized:

$$ \mathrm{Log}{q}_e=\mathrm{Log}{K}_f+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.\mathrm{Log}{C}_e $$
(1.23)

where qe is the adsorption capacity at equilibrium (mg/g), Kf is the Freundlich constant and corresponds to the maximum adsorption capacity, approximately, Ce corresponds to the equilibrium concentration of pollutant (mg/L), and n is a Freundlich constant related to the binding energy (Mohan and Karthikeyan 1997). In particular, when n > 1 or 1/n < 1 the adsorption condition is favorable (Aslan et al. 2016; Dada et al. 2012; El-Mallah and Hassouba 2014).

1.4.3 Temkin Model

This isotherm model assumes that in heterogeneous systems the adsorbent–adsorbate interaction generates a linearly and not logarithmic decreasing trend of the heat of adsorption of molecules on the surface with the coverage. Moreover, the process occurs by uniform distribution of binding energy up to some maximum binding energy (Varmazyar et al. 2017). The model is expressed as (Dada et al. 2012; Goel et al. 2015):

$$ {q}_{\mathrm{e}}=B\ \ln \left({K}_T{C}_e\right) $$
(1.24)

while the linearized expression is written as

$$ {q}_e=B\ln {K}_T+B\ln {C}_e $$
(1.25)

where \( B=\frac{\mathrm{RT}}{b} \) is a constant associated with the heat of adsorption, in which b (kJ/mol), R is the universal gas constant (8.314 J/K/mol), T is absolute temperature expressed in kelvin (K), KT (L/atm or L/g) is the Temkin constant related to equilibrium binding constant, and Ce is the equilibrium concentration of adsorbate (mg/L).

1.4.4 Dubinin–Radushkevich Model

This model was developed for microporous solids and it hypothesizes the existence of a force field in the immediate proximity of the surface of the adsorbent, assuming an adsorption potential defined as the work done by the adsorption forces in binding a certain molecule and equals to the variation of free energy of a substance between its state in liquid bulk and the condition of adsorbed molecule. It can be applied when it is possible assuming a Gaussian energy distribution onto a heterogeneous surface, where the principal characteristic is its independence on the temperature (Sengupta et al. 2016). The mathematical equation is (Dada et al. 2012):

$$ {q}_{\mathrm{e}}=\left({q}_{\mathrm{s}}\right)\exp \left(-\beta {\varepsilon}^2\right) $$
(1.26)

The linearized form is:

$$ \ln {q}_e=\ln {q}_s-\beta {\varepsilon}^2 $$
(1.27)

where qe is the adsorption capacity at equilibrium (mg/g), qs corresponds to the theoretical isotherm saturation capacity (mg/g), β is the Dubinin–Radushkevich isotherm constant (mol2/kJ2) related to the mean free energy of adsorption, and ɛ is the Polanyi potential associated with the equilibrium concentration (Ce) (Chaudhry et al. 2017):

$$ \varepsilon =\mathrm{RTln}\left(1+\frac{1}{C_e}\right) $$
(1.28)

where R (8.314 J/K/mol) is the gas universal constant and T is the absolute temperature (K). Finally, the energy (E) per molecule of adsorbate (adsorption energy) can be obtained by (Albadarin et al. 2012):

$$ E=\frac{1}{\sqrt{-2}\beta } $$
(1.29)

1.4.5 Sips Model

The Sips isotherm is an extension of the Langmuir isotherm equation, which includes some of the properties and the mathematical form of the Freundlich isotherm . It overcomes the limits of Freundlich equation, by assuming a limit to the maximum value of the adsorption capacity for a stated adsorbent–adsorbate couple (Hamdaoui and Naffrechoux 2007; Febrianto et al. 2009). The model proposed by Sips (1948) can be expressed by the following equation:

$$ {q}_e=\frac{q_S{\left({K}_S{C}_e\right)}^{{\mathrm{m}}_{\mathrm{S}}}}{1+{\left({K}_S{C}_e\right)}^{{\mathrm{m}}_{\mathrm{S}}}} $$
(1.30)

where qe is the adsorption capacity at equilibrium (mg/g), qs is the maximum adsorption capacity of the Sips model (mg/g), Ce is the liquid concentration of the adsorbate at equilibrium (mg/L), KS is the equilibrium constant of the Sips isotherm model (L/mg), and ms is the exponent of the model of the Sips isotherm .

1.4.6 Redlich–Peterson Model

Redlich–Paterson is an empirical equation with three parameters that can accurately represent adsorption equilibria for a wide concentration range. This model incorporates the characteristics of the Langmuir and Freundlich isotherms into a single equation and hypothesizes a hybrid adsorption mechanism, not following the ideal monolayer adsorption as for Langmuir model. The isotherm of Redlich–Peterson is expressed by the following equation:

$$ {q}_e=\frac{C\mathrm{e}{K}_{\mathrm{rp}}}{\left(1+{a}_{\mathrm{rp}}{C\mathrm{e}}^{{\mathrm{b}}_{\mathrm{rp}}}\right)} $$
(1.31)

where krp and arp are the R-P constant (L/g and Lβ/mgβ, respectively) and β is the exponent, which can vary between 1 and 0.

As previously reported in kinetics models, the best fitting among the isotherm models is assessed by the coefficient of determination (R2), chi-square test (χ2), root mean square error (RMSE), and standard deviation Δq (%) (see Eqs. 1.15, 1.16, 1.17, and 1.18).

1.4.7 Batch Experimental Procedure

As for the kinetic study, also the assessment of the equilibrium condition is based on experimental studies, usually carried out in batch mode (for the sake of simplicity).

The batch experiments can be carried out in glass flasks, capped polyethylene bottles, or Falcon tubes containing an appropriated concentration of adsorbent material and a constant volume of solution. In the study, the concentration of adsorbate should vary so to cover the range of interest, which depends on the specific analyte. The suspension or reaction mixture obtained is shaken for a constant time and kept at constant temperature. Finally, the samples are centrifuged and filtered to quantify the residual concentration of the pollutant in the supernatant and the adsorption capacity, using Eq. (1.2).

It is necessary to mention that this analysis should be carried out with optimized but constant physicochemical parameters for each data set, such as pH, temperature, ionic strength, and background electrolyte. In addition, the isotherm study can be made at different temperatures in order to determine important thermodynamic parameters.

As examples retrieved in the literature, different researchers have used isotherm studies to determine the removal mechanism of different pollutants present in water. Harja and Ciobanu (2018) reported an isotherm analysis for the study of adsorption from aqueous solution using 10–300 mg/L of oxytetracycline, assuming a set of constant adsorption parameters (pH = 8, adsorbent concentration 2 g/L, contact time 1 h and temperature = 20 °C). Similarly, Ding et al. (2016) performed an isotherm study of Pb2+ and Cd2+ adsorption from polluted water, conducted at different concentrations of analytes (50–900 mg/L), with 0.1 g of adsorbent (biochar ), pH 5.0, using 50 mL of solution, at 30 °C and shaking the samples for 24 h. Zhou et al. (2017) worked on a Fe3O4 magnetic polypyrrole–graphene oxide nanocomposite as adsorbent, at 0.05 g/L concentration, with an adsorption solution at pH 7.0 having a Hg concentration of 20–100 mg/L, in the temperature range 300–320 K.

In Table 1.2, some isotherm studies and the corresponding best-fitting models are reported together with the indications of the experimental conditions.

Table 1.2 Isotherm studies, best fitting isotherm model, and equilibrium constant for different adsorption studies
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1.5 Thermodynamics of Adsorption

From thermodynamic (equilibrium) tests carried out at different temperatures, it is possible to retrieve indications about important thermodynamic parameters, such as Gibbs free energy (ΔG°), enthalpy (ΔH°), and entropy (ΔS°), which characterize the adsorption of the investigated system.

From the values assumed by these parameters, it is possible to determine the adsorption characteristics of the investigated system, that is, if adsorption is endothermic or exothermic, favorable, spontaneous, if there is an increase or a reduction in system disorder, the process nature (i.e., physisorption or chemisorption), as well as if enthalpy or entropy control the operation (Do 1998; Piccin et al. 2017).

A thermodynamic analysis of the adsorption phenomena can include the measurement of the magnitude of the heat of adsorption and its variation with surface loading, which can provide useful indications about the nature of the adsorbent surface and the adsorbed molecules/ions interactions. The heat of adsorption determined at constant amounts of sorbate adsorbed is commonly known as the “isosteric heat of adsorption” or “differential heat of adsorption” (ΔH, kJ/mol). It is defined as the ratio of the infinitesimal change in the adsorbate enthalpy and the infinitesimal change in the amount adsorbed (Do 1998).

The isosteric heat of adsorption can be calculated from the following thermodynamic relationship, derived from the Van’t Hoff equation (Do 1998):

$$ \frac{d\ln (C)}{dT}=-\frac{\Delta H}{RT^2}\kern0.5em \mathrm{or}\kern0.5em \frac{d\ln (C)}{d\left(1/T\right)}=\frac{\Delta H}{R} $$
(1.32)

The adsorbate equilibrium concentrations (C) at constant adsorption capacity (q) are taken from the adsorption isotherm data at different temperatures. Following this path, the ΔH is calculated from the slope of the plot of ln(C) versus (1/T) for different amounts of q (see Eq. 1.32).

In many applications, in order to evaluate the thermodynamic parameters of the process (ΔG°, ΔH°, and ΔS°) and to investigate the adsorption mechanism, a basic assumption can be made, that is, the independence of the thermodynamic parameters on temperature. This simplified approach allows building a relationship between ln(Ke) and 1/T, in which Ke is the adsorption constant determined starting from one of the models presented in the previous section (e.g., Langmuir constant):

$$ \Delta {G}^0=- RT\ln\ {K}_e $$
(1.33)
$$ \Delta {G}^0=\Delta {H}^0-T\Delta {S}^0 $$
(1.34)

Starting from this relationship, it is possible calculating ΔH° and ΔS° as the slope and the intercept, respectively (Dotto et al. 2013; Milonjić 2007):

$$ \ln \left({k}_e\right)=\frac{-\Delta {H}^0}{RT}+\frac{\Delta {S}^0}{R} $$
(1.35)

where R (8.314 J/mol/K) is the universal constant of gases, T (K) is the absolute temperature, and Ke is the equilibrium constant obtained by the isotherm model that best fitted the equilibrium data. Ke must be dimensionless, while it is expressed by L/mol, L/g, or L/mg, then it should be multiplied by a correction factor. When the value is expressed in L/mol, it can be multiplied by 55.55 mol/L, equivalent to the number of moles of water per L of solution. However, when it is expressed in L/g, it must be multiplied by water density (Milonjić 2007).

As examples, in Table 1.3 some studies reported in the literature are listed, with the indication of the retrieved thermodynamic parameters.

Table 1.3 Thermodynamic parameters in different adsorption studies
Full size table

1.6 Conclusions

Adsorption is a fundamental technique that can be adopted for the removal of different compounds, both organic and inorganic, from polluted water. It is commonly known as an efficient, versatile, and cost-effective treatment that has a very wide applicability, due to a general unselectively and adaption to different polluting scenarios. Large applications can be recognized worldwide, mostly based on fixed-bed reactors in which many different adsorbents, depending on the adsorbate characteristics and on polluted stream properties, can be proficiently adopted.

Adsorption phenomena depend on many different parameters, among which are the physical and chemical characteristics of the adsorbent, the chemical characteristics of the pollutant , and the characteristics of the aqueous matrix. In general, adsorption is a surface phenomenon, and hence the adsorbents are porous solids with defined physical (e.g., porosity) and chemical properties (e.g., composition). The influence of all these properties is of fundamental importance in the practical application of this technique. The performance of an adsorption process mainly depends on the thermodynamic aspects of solute–solvent–sorbent interactions and on the diffusive–convective transport phenomena within the porous adsorbent. The characterization of an adsorption process can be done either from a thermodynamic point of view, in order to have information about the equilibrium (maximum) adsorption capacity and its dependence on the main process parameters, as those previously indicated, or by a kinetic standpoint, in order to determine the fluid-dynamic properties and the mass transfer from the fluid to adsorbent and, more in general, the rate of the overall process.

A meaningful application of this technique requires the knowledge of both the thermodynamic and kinetic properties of the investigated adsorbent/adsorbate couple, in the specific conditions of the polluted water. To this aim, dedicated experimental campaigns might be made in order to investigate the properties of the adsorption system and to retrieve important information about the adsorption mechanisms. The interpretation of the retrieved experimental data can be made by the fitting of a wide number of possible adsorption models, separately for the thermodynamic (equilibrium) and kinetic analysis. The availability of reliable theoretical models for the interpretation of adsorption data is an invaluable tool for process optimization (i.e., determination of the best operating parameters set) and for design purposes.